proc_isotpand09_main-unnpress[1]

1

Proceedings of the Second International Seminar on Theoretical Physics & National

Development, July 5 - 8, 2009, Abuja, Nigeria

REVIEW OF 50 YEARS OF PSEUDOPOTENTIAL THEORY†

A.O.E. Animalu1, B. Ezekoye

2 and K.E. Essien

3

1Emeritus Professor: Dept. of Physics & Astronomy, University of Nigeria,

Nsukka & Chairman/CEO, International Centre for Basic Research, Abuja

2Dept. of Physics & Astronomy, University of Nigeria, Nsukka

3Dept. of Physics, University of Uyo, Uyo, Akwa Ibom

e-mail: 1

[email protected] 3 [email protected]

Abstract

The theoretical and practical aspects of the pseudopotential and model potential methods

in metal physics are reviewed. We start by showing how a formal statement of the

quantum mechanical many-body problem of interacting ions and valence electrons in a

metal lead, in the self-consistent field approximation, to the one-electron problem of

setting up the potential energy function for the motion of a valence electron.We then

review firstly the various pseudopotential theories for simple (non-transition) metals that

began in 1959 with Phillips and Kleinman‘s introduction of the pseudopotential concept

by re-interpretating Herring‘s 1940 orthogonalized plane wave (OPW) method of energy

band structure calculation as a (non-unitary) transformation that orthogonalizes plane

waves to the ion core wave function. Although the OPW-pseudoptential transformation

preserves the energy eigenvalue, it generates a generally non-unique repulsive potential

that cancels most of the deep attractive potential of the bare ions resulting in a net weak

effective potential which could be represented accurately enough by a model potential of

Heine-Abarenkov(HA) type. Secondly, we review the corresponding re-interpretation of

the augmented plane wave (APW) method of energy band calculation for the transition

and rare-earth metals that began in 1965 with Ziman‘s d-band resonance model in terms

of generalized OPW-pseudopotential and transition-metal model potential (TMMP) of

HA type, and tabulate for the first time the TMMP form factors for 27 transition and rare-

earth metals. Finally we review experimental verification of the pseudopotential theories

and a critique of the problems arising from the non-unitary character of pseudopotential

transformation and outline new vistas provided by the discipline of ―hadronic mechanics‖

for handling non-unitarity theories among other problematic aspects leading to fruitful

generalization of the standard (BCS) model for high-temperature superconductivity.

PACS numbers: 71.10.-w, 71.15 Dx, 71.15.-m

† African Journal of Physics Vol.2, pp. 1-45, (2009)

ISSN: PRINT: 1948-0229 CD ROM:1948-0245 ONLINE: 1948-0237

mailto:[email protected]


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1. GENERAL INTRODUCTION AND STATEMENT OF PROBLEM

1.0 General Introduction

In the past fifty years the modelling of the effective potential felt by a

valence electron in metals and semiconductors has assumed a key position in the

determination of the various aspects of the electronic structure of condensed

matter. The potential has been required not only for the calculation of the energy

band structure of the regular solid but also the phonon spectra, the structure

around a defect, scattering probabilities and hence transport properties of the

solid, liquid or alloy phases of condensed matter.

In this paper, we wish to rehearse the statement of the problem before

proceeding with the review (in Sec. 2) of the various aspects of the

pseudopotential formalism developed by Philip and Kleinman[1] in 1959 from the

orthogonaized plane wave (OPW) method of electronic energy band structure

calculation introduced by Herring[2] in 1940. The popularity of the OPW method

in the first six years (1959-1965) due to its successful application to theoretical

and experimental understanding of the geometry of the Fermi surface of metals

(Gold[3], Harrison[4], Heine and others[5]) climaxed with the introduction of a

model potential by Heine and Abarenkov[6]) which simplified the computational

aspects and led Animalu and Heine[7] to the production of tables of model

potential form factors for 25 non-transition elements which were described by

Harrison in his 1966-published book[8] as ―the best values currently available‖.

In Sec. 3 we shall describe the second high point dealing with the re-

interpretation of the augmented plane wave method (APW) (Loucks[9]) as a

phase-shift transition-metal pseudopotential method by Ziman[10] and others.

This led to the construction of a generalized transition-metal pseudopotantial by

Harrison [11] in 1969 and the corresponding transition-metal model potential

(TMMP) by Animalu [12] in 1973. The form factors obtained from mainframe

computers in ref.[12] were largely unpublished at the time but have recently been

recomputed on laptop with Fortran77 programme developed by Essien[13] who

also successfully applied it to the computation of the phonon spectrum of cobalt.

The form factors for 27 transition elements will be presented for the first time in

this paper and the most recent application of the TMMP form factors to laptop

computation of the phonon spectrum of cobalt will also be given.

Finally in Sec. 4, we shall review experimental verification of the pseudopotential

theory and a critique of well-known formal problem of the OPW-pseudopotential

method presently ignored by condensed matter physicists, namely the fact that the

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pseudopotential transformation is non-unitary and therefore does not conserve

probability. We shall highlight a new vista for resolving this problem permited by the

discipline of ―hadronic mechanics‖ with intriguing successful application to high-

temperatire superconductivity[14] and draw the attendant conclusions.

1.1 Statement of the Problem

The problem of setting up the potential energy function for a many-body

quantum mechanical system has been well known from the late 1920s when it

arose in the determination of structure of many-electron atoms in the Periodic

Table. In the physics of metals, we are similarly dealing with a large number of

particles – ions and valence electrons. An ion consists of the atomic nucleus and

the core electrons tightly bound to it. The valence electrons, more or less, form a

Fremi gas. On a macroscopic scale the net charge of the electron gas completely

neutralizes that of the ions so that the solid is electrically neutral. However, on a

microscopic scale, each valence electron feels the field of the ions and the

fluctuations in the density of the gas of other valence electrons on which the

electronic properties of the material depend. A description of such microscopic

processes requires setting up the potential felt by a valence electron in the

material and solving the appropriate wave equation for the energy eigenvalues or

bands and the wave functions of the valence electron, subject to suitable periodic

boundary conditions. Although the interaction between any pair of electrons or

between an electron and a proton is known to be Coulomb in character, an exact

solution would be extremely complicated for the 1023

electrons per cubic

centimeter in a normal metal. In fact, one has to solve an N-electron equation:

Err

eRrV

m

pH

i ji jii

i

i

2

21

,

2

,2

(1.1.1)

),....,,,( 321 Nrrrr

(1.1.2)

where ii rip

/ is the momentum operator of the ith electron, RrV i

, is its

interaction with the th ion which needs to be set up, and the term with 1

ji rr

represents the interaction of the ith and jth electrons.

The Hartree and/or Hartree-Fock self-consistent field methods which have

been very successful in setting up the potential due to the ions consist of writing

(1.1.2) as the symmetrized product of one-electron wave functions, k :

N

i

ik r

1

)(

(1.1.3)

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so that the energy, E , is then the sum of one-electron energies, kE ; the last term

in (1.1.1) represents now the exchange and correlation interaction of the electron

gas (Slater[15]) which is frequently denote simply by . The long-range character

of the exchange and correlation interactions, leads to an ―infinite self-energy‖

which is the classic problem of quantum electrodynamics and quantum field

theory.

We shall base this review of fifty years of pseudopotential theory on the

following formulation of the valence electron problem in terms of the concepts of

―bare‖ and ―effective‖ potentials. The eigenstates k and the energy eigenvalues

kE are given by the solution of the wave equation:

kkkH EVm

2

2

2

(1.1.4)

where HV is the Hartree potential and is a mass operator or ―self-energy‖

representing the exchange and correlation interactions. The total density of the

valence electrons is given by

)()()()( * rrknr kk

k

(1.1.5)

)(kn being an occupation number and the summation over spin indices being

understood. The procedure in recent years is then to consider HV , and )(r

in

(1.1.4) and (1.1.5) as the sum of three contributions due to

(i) the ions,

(ii) a uniform density of valence electrons )(0

r

neutralizing the

ionic charge, and

(iii) the change or fluctuation, in (1.1.2) from the constant value

0 as a result of choosing a ―test‖ valence electron in the system

(i) and (ii), and considering its interaction with the other electrons.

In other words, one imagines an array of positive ions immersed in a rigid

uniform jelly of valence electrons in (i) and (ii), and then includes the effect of

(iii) which is to ―unfreeze‖ the jelly. The potential due to (i) and (ii) is called the

potential of the ―bare‖ ions, or simply the ―bare‖ potential, bV (which is usually

represended by a pseudo or model potential) and the resulting potential after

taking into account the screening action of the electron gas arising from the

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density fluctuations (iii),is called the ―effective‖ or ―self-consistent‖ or ―screened‖

potential eff

V (or U) of the valence electron in the material.

2. THE POTENTIAL OF THE BARE IONS

2.1 Introduction

Considerable experience has been accumulated in the task of setting up the

bare potential, bV , from band structure calculations (see, for example, the 1964

review by Heine [5]). From the results of these calculations and experimental

studies of the shapes of the Fermi surface, (notably the experimental map of the

Fermi surface of lead by Gold[3] in 1958 and its theoretical extensions to

aluminium and other polyvalent metals by Harrison[4] over the period 1959-64),

it became clear that the valence electron in simple metals behaved effectively as if

it were nearly free: i.e., its wave function, , in the region between the atomic

cores is very close to a simple plane wave or a simple linear combination of plane

waves; in the region inside the atomic cores behaves like an atomic wave

function with several oscillations or sharp wiggles on account of the strong

attractive core potential in this region. But it has been shown, as we shall outline

presently, that these wiggles can be looked upon either as a manifestation of the

exclusion principle which requires the valence electron states to be orthogonal to

the core orbitals or as indicative of high kinetic energy of the electron

corresponding to several phase changes of 2 before it reaches the edge of the

strong attractive potential well. In either case, the effect is to cancel most of the

attractive potential resulting therefore in a weak net potential which is called the

pseudo-potential, ps

V , and which acts on a new wave function, . Thus (Fig.2.1)

in the region between the ion cores may be represented by a simple linear

combination of plane waves but in the region of the core it does not possess the

Fig.2.1: True wave function and Pseudowave function .

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sharp wiggles of the actual . In fact in the region inside the atomic cores may

be looked upon as a smooth extrapolation of the simple plane wave from the

region outside the cores into the core region.

2.2 Pseudopotential Transformation

A. The OPW-Pseudopotential

We now proceed to show how the cancellation of the strong negative

potential energy of the bare ions by the positive kinetic energy terms arising from

the oscillations of the actual comes about: this is the essence of the 1940s

orthogonalized plane wave (OPW) method of band structure calculation (Herring

[2]) and the pseudopotential theories of Phillips and Kleinman[1], Harrison[4],

Heine and others[5]. There are at least two ways of looking at the change from

to : (i) as a mathematical transformation and (ii) in terms of phase shifts. If we

mathematically transform the Schrodinger equation:

EVm

H

2

2

2

(2.3.1)

using

c

ccv (2.3.2)

where v

and c

are respectively the valence and core eigenfunctions of the

same Hamiltonian operator, H, in (2.3.1) and c

are arbitrary, the summation

being over the core states, then we obtain

EVm

ps

2

2

2

. (2.3.3)

As the c

are arbitrary, a variety of ps

V are obtainable from the various choices

of c

. For example, if we set, k

, a plane wav of wave vector k

, and set

kcc

(2.3.4)

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the matrix element of k

and c

, then we have made k

orthogonal to the

core states (Herring[2], Phillips and Kleinman[1]). Equation (2.3.2) takes the form

kQv

)1(

or more generally,

)1()1( QkQak

kv

(2.3.5)

where

c

ccQ (2.3.6)

is a projection operator which projects a function on the core states (Pick and

Sarma[5] ) and

kak

k

(2.3.7)

is the pseudowave function which is a simple linear combination of plane waves.

On setting up secular equation for band structure calculation (2.3.5) implies we

obtain rapid convergence of the secular equation on the basis of the nearly free

electron model. Equivalently, (2.3.3) and (2.3.6) imply

c

ccckps EEVV )( , (2.3.8)

where the second term may be interpreted as a repulsive potential, R

V (Phillips

and Kelienman[1]). As the energy of valence band k

E is always higher than those

of the core states c

E , it follows that R

V is always positive and consequently

cancels most of the negative attractive potential, V , in the first term. The

arbitrariness in ps

V is now apparent from the fact that if we add any arbitrary set

of core functions, the orthogonality condition (2.3.4) immediately subtracts it out

and we obtain exactly the same eigenvalue, E , in (2.3.1) and (2.3.3) for the new

-equation.

Various forms of ps

V have, therefore, been obtained by smoothing in

some sense. Cohen and Heine[5] invoked a minimum principle for the kinetic

8



energy term subject to all variations of : in contrast, Harrison[4] used a

maximum principle for the potential energy. Austin, Heine and Sham[5]

formulated a completely general form of ps

V which can be used for realistic

calculations and applies to the free atomic energy states as well as to the solid:

c

ccpsFVV (2.3.9)

where c

F are completely arbitrary core functions. The particular choice:

ccVF (2.3.10)

leads to the minimum principle in the sense of Cohen and Heine[5]. In actual

calculation, one starts by determining the atomic psudo-potential from (2.3.9)

written in the form

c

ccpsFVV )/( (2.3.11a)

on the basis of which the second term may be calculated from each angular

momentum state, s, p, d, etc by projecting out each state from the Hartree-Fock

equation for each atomic pseudo-state; and simplification emerges by expressing

it in the form:

l

l

l

RpsrVVV )( (2.3.11b)

where l

is a projection operator which picks out a particular angular momentum

state, l, and l

RV is the repulsive part of the pseudo-potential expanded according

to the spherical symmetry of )(rV . As each l

RV varies slightly with the energy,

the pseudo-potential in solids is obtained from the atomic one by smoothly

extrapolating the energy dependence in l

RV . A review of the current state of art

is available in the Cambridge-published 2004 book by Martin[16] entitled

Electronic Structure: Basic Theory and Practical Methods. Cambridge. UK.

B. The Heine-Abarenkov Model Potential

Another way of looking at the transformation from to is in terms of

the phase shifts, l

. In the Born approximation, a weak perturbing short-range

9



spherically symmetric potential. W. produces at a distance r, for an angular

momentum state, l, with energy mk 2/22 , a phase shift :

'')'()'(),(0

2

2

1 drrkrJrWmrkr

ll (2.3.12)

The cross-section. , for the scattering of a particle in W is given in terms of l

by

l

ll

k

2

2sin)12(

4 (2.3.13)

Consequently, phase shifts differing by integer multiples of produce the same

effect, implying in view of (2.3.12) that the weakest possible W producing exactly

the same phase shift is all that is require. In a very deep potential well the several

oscillations of the actual wave function correspond to several phase changes of

2 before the electron reaches the edge of the well. The elimination of these

wiggles by the use of a smooth is therefore equivalent to reducing these several

phase changes and amounts to choosing a weak scattering potential W or a

pseudo-potential ps

V from which all the nodes in in the region of the cores

have been eliminated. This may be achieved by ―relaxing‖ the condition on the

c in (2.3.1) and one obtains a pseudo-potential (sketched in Fig. 2.2) which, for

historical reasons, has been called the model potential (Heine and Abarenkov[6]):

psV or

M

l

Mll

M

Rrr

z

RrEA

V ,

,)(

(2.3.14)

Here, M

R is a model radius, z is the valence of the atom or ion under

consideration, l

is the projection operator defined already in (2.3.11b), and the

parameters )(EAl

are functions of the energy determined by the atomic

spectroscopic term values, E, for each l=0,1,2, etc.

10



In the solid, these atomicl

A are smoothly extrapolated in the spirit of the

quantum defect method (Ham[17]).Moreover, in the actual application and

determination of the model potential, the approximation that

2 ,2

lAAl

(2.3.15)

is also made, and consequently (2.3.14) becomes

M

M

M Rrr

z

RrAAAAA

V ,

,)()( 1210202

(2.3.16)

Apart from the complications in the idea of phase shifts introduced by the

Coulomb tail in the region M

Rr , the whole point of the ―relaxation‖ of the

condition on c

in (2.3.2) is to produce a model wave function M

having no

nodes in the region M

Rr 0 , that is, reproduces the phase shifts (modulo ).

We turn next to the question of representing the bare potential, bV of the

whole solid by the pseudo- or model potential. bV may be regarded as a

superposition of the atomic pseudo- or model potentials centered about each

atomic site: that is,

)(vj

bb RrV

(2.3.17)

where

MV

MR r

M

Fig. 2.2: The Heine-Abarenkov Model Potential

11



psVv b

or M

V (2.3.18)

is the pseudo- or model potential associated with each ion at position j

R

.

However, the use of the pseudo-wave function rather than the actual wave

function v

where

c

ccv , (2.3.19)

results in the reduction of the charge density of the valence electrons near the core

where 22

v

and a heaping up in the region between the ion cires. Thus

instead of )(0

r defined at Eq.(1.1.5) being equal to a constant everywhere,

where z is the valence, and 0 the atomic volume, it is now 0/)1( z

everywhere, being a small correction, together with an extra positive charge of

z electronic charges more or less uniformly spread over a sphere of radius C

R

equal to that of the ion core. This effect may be visualized by imagining a plane

wave turning into atomic-like oscillations inside the core, the mean value of 2

v

being replaced by a factor two (mean value of x2cos ); moreover, 2

v has to

drop at a radius a bit bigger than that of the core radius C

R because its outer node

has to come fairly far out in the main bump of the outer shell of the core states in

order to preserve orthogonality between them. Thus we expect,

aR/

Rn bigger tha

bit a radius

1 and

betweenfactor

C21

(2.3.20)

aR being the radius of a sphere of volume 0 . The value

3/

acRR (2.3.21)

was suggested by Heine and Abarenkov[6], the potential due to this correction

being called the ―orthogonalization correction‖

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C

C

CC

Rrr

z

RrR

r

Rz

,

,22

3

v3

2

OC

(2.3.22)

There is another small correction called the ―correlation correction‖ which affects

the charge density and gives rise to a correlation correction potential,

C

CC

Rr

Rr

0

),(v

0

CC

(2.3.23)

where )(0

C

is the local correlation potential depending on the electron

density 0

in the item (ii) of Sec 1. It is a contribution from the correlation part

(which we denoted by a suffix c) of the self-energy term due to 0

. Similar

contribution )(ion

C

where ion

is the density of the tightly bound core

electrons, has already been included exactly in the model potential of the bare ion.

In the approximation of Kohn and Sham[18] which is valid for non-transition

metals,

)(CC

(2.3.24)

being the total electron density, which in terms of the correlation energy (per

electron), C

E , is give by

d

dEEE

d

dC

CCC (2.3.25)

The last term is about 0.01 Ry which in taking the matrix element of CC

v is

reduced by the ratio of ionic to atomic volume (nearly ten times), and so,

CC

E (2.3.26)

will serve for calculations accurate to 0.01 Ry (see, Table 2.1)

13



Table 2.1: Values of C

E and C

for a uniform electron gas

In the range of metallic densities, C

E and C

vary slowly with density (see,

table 2.1) and in the limit of high density inside the ions they also vary slowly as

(Gell-Mann and Bruckner [19])

SC

rE ln0622.0 (2.3.27)

where S

r in atomic units is the usual radius giving the volume per electron. The

correction CC

v is seen to arise from the non-additivity of the correlation energy as

a function of density; a screened exchange part is additive and so doesn‘t

Sr

(atomic

units)

CE

(Ry) C

(Ry) S

r

(atomic

units)

CE

(Ry) C

(Ry)

1.6 -0.101 -0.108 3.4 -0.077 -0.090

1.8 -0.098 -0.106 3.6 -0.075 -0.088

2.0 -0.095 -0.104 3.8 -0.073 -0.085

2.2 -0.092 -0.102 4.0 -0.071 -0.083

2.4 -0.090 -0.100 4.2 -0.069 -0.081

2.6 -0.087 -0.098 4.4 -0.068 -0.079

2.8 -0.084 -0.096 4.6 -0.066 -0.076

3.0 -0.082 -0.094 4.8 -0.065 -0.074

3.2 -0.079 -0.092 5.0 -0.064 -0.072

14



contribute. On the other hand, the non-homogeneity of the valence electyron

density described as the orthogonalization correction above also affects the charge

density in the correlation and exchange hole in : it tends to cancel the

orthogonalization correction in the direct Hartree potential. As a compromise, in

place of in (2.3.22) we use

3

21

a

C

effR

R (2.3.28)

In view of these corrections, we have in place of (2.3.18)

CCOCM vvvv b (2.3.29)

The form (2.3.17) for the bare potential allows a very important

factorization of the matrix elements which is characteristic of diffraction theory

(Harrison [4], Sham and Ziman[20]). Fourier expansion of (2.3.17) takes the

form:

q

)r.q)exp(iqA(' const.)r(V

b (2.3.30)

where each Fourier component )qA(

may be factored as

kvqkS(q))qA( b

(2.3.31)

and ' excludes the term q=0. In (2.3.31)

j

i )R.qexp(N

1S(q)

j

(2.3.32)

is the structure factor which depends on the arrangement of the ions, but is quite

independent of the individual ionic potentials. The matrix element,

rdii 3bb r.kexp)r(vr.qkexp1

kvqk

(2.3.33)

is the Fourier component of the potential associated with a single ion and is

independent of the arrangement of the ions (except through th volume per ion).

Thus the factorization (2.3.31) includes solid metals, liquid metals, and metals

15



disturbed by a vacancy, dislocation or phonon. In the case of an alloy with several

atomic species, , we generalize (3.2.31) to

kvqk(q)S)qA( b

(2.3.34)

As a result of the factorization, the task of setting up the bare potential has

reduced to constructing a pseudo- or model potential for a single ion of a material.

2.3 The Effective Potential

This section is a review of the ideas about the effective potential which was

defined in Sec.1. It has been shown from the recent developments in many-body

theory (see, for example, Pines[21], Nozieres and Pines[22]) that the infinite self-

energy of the electron gas is a consequence of the long-range character of

electron-electron interactions, and may be removed by a process analogous to the

elimination of photon self-energy parts in the analysis of the S-matrix in quantum

electrodynamics. Bohm and Pines[23] introduced the concept of plasma

oscillations; Landau (see, e.g. Nozieres[24]) described in his theory of Fermi

liquids many of the properties of interacting Fermions in terms of a set of weakly

interacting elementary excitations called quasi-particles; finally, Gell-mann and

Bruckner[19], and Hubbard[25] formulated perturbation theory which could be

applied to a many-body system with long-range forces. It is argued that on

treating the Fermi gas of valence electrons in a metal as free, that is, suppose that

the background of ions just ensures electrical neutralioty of the gas at equilibrium,

then the effect of ―switching on‖ the electron-electron interaction in the system is

to induce density fluctuations ( ) which could be described as a polarization

cloud, P, carried by any chosen test electron. The polarization cloud screens the

bare interaction of the test electron with all other electrons of the system, the bare

interaction bv being reduced by a factor )1/(1 P . bv is then said to be

―renormalized‖ or ―screened‖ or we may say that such electron feels a net short-

range effective potential,

)1/(vv b Peff . (2.3.1)

P has been calculated in various approximations. For example, in the random

phase approximation (RPA), it was given for an electron gas by Lindhard [26]:

16



)2/()(4

)(2

2

FFkqgEN

q

eqP

(2.3.2)

where )(F

EN is the density of states at the Fermi surface and

x

x

x

xxg

1

1ln

1)(

2

41

21 (2.3.3)

For an electron-phonon field, if q

is the phonon frequency, q

the electron-

phonon matrix coupling matrix element, k

E the electron energy spectrum and

the normalization volume, then it is easy to show as in (2.3.2) that

kqkq

q

E

nnqP

k

2

E

)qk()k(12)(

(2.3.4)

where )k(

n is the Fermi distribution function. The function,

)(1)( qPq (2.3..5)

is the response or the dielectric function and has been related to the familiar

macrosciopic dielectric constant in an insulator by Nozieres and Pines[22]. It has

also been formulated by Ehreinreich and Cohen[27] in terms of linearized self-

consistent field approximation using density matrix; Kadanoff and Baym[28]

defined it in terms of the variational derivatives of Schwinger‘s action principle;

and there are several other equivalent formulations using for example the Fermi-

liquid theory of Landau of basically the same physical consequences. In the metal,

the treatment of screening in thr Hartree approximation was first used by Bardeen

[29] but has been formulated in terms of Lindhard[26] dielectric function by

Cohen and Phillips[30]. The latter authors argued that since the pseudo-potential

is weak, it could be treated as a perturbation of the free electron gas in the Hartree

approximation. In the perturbation expansion, the electron states are pseudo-

waves which are approximately plane waves apart from the ―orthogonality‖

correction to the screening electron densities mentioned earlier. Thus, in zero

order the charge density is constant and its contribution to the potential is

uninteresting. The first order screening potential contributes to the energy in the

second order, but the screening to higher order can be seen only to affect the

energy in the third order. Theefore, to second order the screening is required only

17



to first order and so each Fourier component of the bare potential (2.1.31) may be

screened independently so that the effective potential

)r(

U or )r.qexp((q)

)qA(' Const.)r(

iV

q

eff

(2.3.6)

The major effect of the screening is to cancel any long-range potentials arising

from the charge distribution of the system. This cancellation may be related to the

behavior of the dielectric function (q) at long wavelength 0)(q , as

discussed for example by Sham and Ziman[20], and by Heine and Abarenkov[6]

There are other effects due to the non-locality of the bare potential (2.1.31)

discussed by Animalu[31] as refinements for accuracy in quantitative calculations

of electronic structure.

3. MODEL POTENTIAL FOR TRANSITION AND RARE-EARTH METALS IN THE RESONANCE MODEL

3.1. Introduction

It is well-known (Loucks[9]) that the usual OPW-pseudopotential methods should

fail for the incomplete d-electron core state of the transition metals and the f-

electron core states of the rare-earth metals. Consequently, the d- and f- states are

usually described by the APW-pseudopotential method by introducing 2 and

3 phase-shifts of the resonance type. The main contributors to the

clarification of the resonance idea are J.M.Ziman[10], G. J. Morgan [32], V.

Heine [33] and J. Hubbard [34]. Heine especially related Ziman‘s formulation

based on the Korringa-Kohn-Rostaker (KKR) method to the parallel Linear

Combination of Atomic Orbitals (LCAO)-OPW interpolation scheme developed

by C. Hodges, H. Ehrenreich and N. D. Lang [35] and F.M. Mueller[36]. The

experimental evidence for the validity of the resonance model came subsequently

in the photoemission studies by N. V. Smith and co-workers[37] ealier analyzed

by his group in the framework of the LCAO-OPW interpolation scheme.

These works provided the motivation for the early efforts to extend the OPW-

pseudopotential method to the transition metals by Harrison[11] and Moriarty[38]

This was achieved by reformulating the pseudopotential method as an expansion

in an overcomplete set made up of plane waves, atomic core states and atomic d-

states. The final theory was unduly complicated by the special status given to the

treatment of the d-state and the resonance potential associated with it.

18



Consequently, a model potential method of Heine-Abarenkov-type, called

Transition-Matel Model Potential (TMMP) was designed by Animalu[12] to

simulate the 2 resonance in the transition metals and the 3 resonance in

the rare-earth metals through the corresponding model potential well-depths,

1

2

EEEA d and 1

3

EEEA f , where dE and fE are the

resonance energies. This means that the basic form of the Heine-Abarentkov

model potential for the simple metals is retained for all metals throughout the

Periodic Table. The distinction between simple, transition , and rare-earth metals

lies in the singular (mathematical) behavior of the model potential well-depths in

the transition-group and rear-earth-group of metals, which can be handled by

analytical continuation into the complex energy plane[39] or what is the same

thing, by the use of the T-matrix for the appropriate partial-wave states. This

approach is evidently more fundamental inasmuch as it unifies the treatment of all

metals in a comprehensive model potential theory.

In order to appreciate the formal structure of the model potential for the transition

and rare earth metals, we proceed to review Harrison‘s pseudopotential

transformation theory for the transition metals as the conceptual framework for

setting up an internally consistent computational TMMP method..

3.2 The Transition-Metal Pseudopotential Transformation

The objective of the pseudopotential or model potential method in simple metal

theory is to replace the one-electron wave equation for a Bloch electron in a

crystal,

kkk

EVm

2

2

2 (3.2..1)

by a pseudowave equation

kkkOPW EVm

2

2

2 (3.2.2)

where OPWV is understood to be weaker than the true potential V and the

pseudowave function k is generally a plane wave or a simple linear combination

of plane waves. The transformation that relates the true Bloch function k to the

pseudowave function k is

19



k

k

OPW

k E

V

1 (3.2.3)

for the class of pseudopotentials and model potentials of the form

c

ckOPW ccEEVV (3.2.4)

where the c ‘s are the ion-core states. For, if we take the partial derivative of

OPWV with respect to ~kE , and substitute for

~

/ kOPW EV in Eq. (3.2..3) we obtain

the standard result,

kk

P 1 (3.2.5)

where

c

ccP (3.2.6)

is the projection operator ( PP 2 ) that orthogonalizes ~k to the ion-core states

c . The depletion hole or orthogonalization correction associated with (3.2.3) is,

accordingly, given by

k

kk k

k

OPW

E

Vnz

1 (3.2.7)

where the summation is over the occupied states in the Fermi distribution

characterized by the distribution function k

n , and z is the chemical valence.

Now let us consider Harrison‘s generalization[11] of the OPW-

pseudopotential transformation (3.2.3) to the transition metals:

kkc d kdd

k PEE

ddddcc

~

11~

(3.2.8)

In a typical transition metal, such as vanadium, c runs through

the 62622 33221 pspss while d runs through the 33d , k characterizes the ionized

24s free-electron states; and dVdV is the hybridization potential

20



which generates the d -band resonance in the energy band structure. The

corresponding pseudowave equation turns out to be,

kkk

d kd

EEE

ddWm

)(2/ 22 (3.2.9)

where W is essentially the usual OPW-pseudopotential operator:

d

dkc

ckddddddEEccEEVW )( . (3.2.10)

What we call the transition-metal OPW-pseudopotential (including the resonance

term) is:

d kd

OPWEE

ddWV

. (3.2.11)

The generalization consists of the addition of the extra resonance term that is

proportional to 1

kd EE . The presence of this term implies, however, that the

depletion hole associated with the d-state is infinite, because the summation over

k in (3.2.7) is divergent whenever the domain includes the point dkEE .

Moreover, the resonance term leads to an expression for ~

/ kOPW EV , which is

not identical with an expression for ~

P in Eq. (3.2.8), i.e,

~

2

~

PEE

ddddcc

E

V

dkdc dk

OPW

. (3.2.12)

However, in ref.[39] it was shown how to remove the divergence in Harrison‘s

transformation and relate ~

/ kOPW EV to ~

P . It is necessary to allow a small

imaginary part, i.e. a resonance width, in Eq. (3.2.11), by making the replacement

~

d kd

OPWOPWiEE

ddWVV

(3.2.13)

and correspondingly

21



kkc d kdd

kkP

iEE

iddddcc

~

'1)(

1~ (3.2.14)

The replacement guarantees that near dk EE ~

, we have real part of P~

equals

dkdd

kOPWEE

ddccEV

22/

~

(3.2.15)

so that we generate a finite depletion hole. Moreover, if we define

c d kd

r

d

siEE

iddPddccP

)( , (3.2.16)

then, because 0 dd and 0cd , and if it is assumed that is nearly

constant over the core states so that 0 cd , we have

,2

SS PP ,02 rP ,rrS PPP 0Sr PP . (3.2.17a)

Thus

rSrSSrrSSrSrS PPPPPPPPPPPPP 2222 (3.2.17b)

This is the basic property ( )'~

'~ 2 PP of a valid pseudopotential transformation. In

the modified pseudopotential OPWV~

we in effect replaced the ordinary

pseudopotential by a T-matrix pseudopotential.

It was on the basis of this concept that Animalu[12] defined a model potential of

the Heine-Abarenkov type for the transition and rare-earth metals – called the

transition-metal model potential (TMMP) – by observing that s-d hybridization

could be incorporated in the framework of the model potential method through

2 well-depth of the form,

d

CA

3

2 , (3.2.18)

22



where C is a constant (independent of ). Starting from a modified quantum

defect relation in Eq. (3.2.19), one could write the true potential seen by a valence

electron in the state ,n in the presence of a positive transition-metal ion as

nnn rUrV 2

1 (3.2.19)

where the potentials are in atomic units, i.e., double rydberg, and n is in

rydberg. Thus the true radial wave equation for the electron is

rErrU

rdr

dnnnnn

2

122

2

. (3.2.20a)

It is as if the electron experiences a modified potential nU that is Coulombic at

large r (greater than the ion core radius) corresponding to a modified energy

spectrum of the form given by the old quantum defect method,

2

2

n

nnnn

zE

(3.2.20b)

We therefore replace nU by a model potential ,MV so as to reproduce the

modified term values n .

With this interpretation of the energy to which the model potential well-depths

A should refer in the transition metals, it was found that the TMMP had

precisely the same form as the simple-metal model potential (2.2.14) for 0 , 1

and includes an 2 resonance term through a well-depth of the form (3.2.18).

The well-depths A in the solid can therefore be determined from the observed

spectroscopic term values by extrapolating to the appropriate energy, FE say, of

an electron at the Fermi level in the spirit of the quantum defect method. In

practice, because of the scarcity of spectroscopic data for the transition-group

metals, the parameters of FE were obtained by correlating the transition metals

along row of the Period Table.

3.3 The Screened Transition-Metal Model Potential Form Factors.

So far we have emphasized the (mathematical) similarities between the simple-

metal model potential and the transition-metal model potentials. This means in

practice that the calculation of the screened model potential form factor for the

23



transition should follow the procedure outlined for the simple-metal model

potential in Sec. 2.3, with TMMP expressed in the explicit form:

M

M

M Rrr

z

RrCACACAC

V ,

,)()()( 121100

(3.3.1)

where C = )2,1,0(; lAR

zl

M

are the model potential parameters; and l (l = 0, 1,

2) are the projection operators of the lth angular momentum of an incident one-

electron wave-function. l For various applications involving scattering of

electrons near the Fermi surface, the effective potential may be calculated in the

local screening approximation in the form,

q

qkkFqBqkVk

, . (3.3.2)

where

ccc

c

effc

MMMM

qRqRqRqRq

z

q

E

qRq

zqRqRqR

qqB

cossin244

cos8

cossin8

32

0

3

0

2

0

3

0

(3.3.3)

For ,~~~kqk

cos20

cos12

)(]/)cos([2

,

2

3

1

2

22

3

0

120

2

11

3

0

1

2

00

3

0

PxjxjxjCAR

PxjxjxjCAR

xjxxxjCARqkkF

M

M

M

(3.3.4a)

where,

24



2

2

21cos

k

q ; coscos1 P , 1cos3

2

1cos 2

2 P ;

x

xxj

sin0 ,

x

x

x

xxj

cossin21 ,

xj

x

xjxj 0

12

3

xjx

xjxj 1

23

5 .

For ,kqk

'2

232322

0

2

3

'

1121222

0

1

3

010122

0

0

3

cos40

cos24

8,,

Pxjyyjyjxxjyx

CAR

Pxjyyjyjxxjyx

CAR

xjyyjyjxxjyx

CARqkkF

M

M

M

(3.3.4b)

where

MkRx , MRqky

, and xyqRyx M 2cos222' .

F

Fk

qE

q

zeqfq

23

2411

1

2

0

2 (3.3.5)

y

y

y

yy

1

1ln

1

4

1

2

12

; 222

2

2

1

SF kkq

qqf

.

Apart from the inclusion of the resonance term, the expressions are the same as

those for the simple metals.

We list in Table 3.1, all the parameters required to evaluate the form factor

qkVkqV F

for all the 27 transition metals and in Table3.2 the form

factors.

25



Table 3.1: Transition-Metal Model Potential (TMMP) Parameters for 27

Transition Metals

0A 1A 2A MR 0 z *m cR eff cE

Fk

P

element

0.250 0.400 0.215 2.200 79.400 1.000 1.000 1.814 0.157 0.086

8.930 0.921 7.655 Cu

0.223 0.400 0.218 2.600115.400 1.000 1.000 2.381 0.245 0.082

10.500 0.813 4.857 Ag

0.150 0.500 0.212 2.600114.600 1.000 1.000 2.589 0.317 0.082

19.280 0.815 3.609 Au

1.600 1.650 1.400 2.000168.700 3.000 1.000 1.531 0.045 0.090

2.990 0.716 18.678 Sc

0.750 1.300 1.100 2.00 223.100 3.000 1.000 1.739 0.049 0.087

4.480 0.653 11.539 Y

0.900 1.400 0.850 2.000252.200 3.000 1.000 2.154 0.083 0.096

6.170 0.626 8.698 La

2.300 2.500 2.100 2.000119.000 4.000 1.000 1.285 0.037 0.096

4.510 0.805 28.747 Ti

1.150 1.700 1.500 2.000157.000 4.000 1.000 1.493 0.044 0.095

6.510 0.734 18.136 Zr

1.300 1.800 1.350 2.000150.200 4.000 1.000 1.474 0.045 0.095

13.200 0.745 13.313 Hf

3.250 3.550 2.900 1.600 93.900 5.000 1.000 1.115 0.031 0.101

6.090 0.871 39.189 V

1.700 2.300 2.250 2.000121.300 5.000 1.000 1.304 0.038 0.100

8.580 0.800 25.558 Nb

1.750 2.350 2.250 2.000121.300 5.000 1.000 1.285 0.038 0.100

16.660 0.800 18.342 Ta

1.600 1.470 1.400 2.500 80.600 3.000 1.000 1.191 0.044 0.102

7.190 0.916 25.211 Cr

26



2.300 2.930 2.500 2.000105.500 6.000 1.000 1.323 0.046 0.100

10.220 0.838 32.310 Mo

2.300 2.850 2.500 2.000106.500 6.000 1.000 1.172 0.032 0.101

19.250 0.835 23.321 W

0.890 0.980 0.870 2.200 81.900 2.000 1.000 1.512 0.088 0.095

7.470 0.911 16.228 Mn

3.100 3.200 3.300 2.000 96.500 7.000 1.000 1.058 0.026 0.102

11.500 0.863 38.850 Tc

2.950 3.550 3.300 2.000 99.300 7.000 1.000 1.058 0.025 0.102

21.030 0.855 27.919 Re

1.600 1.470 1.400 2.000 79.800 3.000 1.000 1.400 0.072 0.090

7.870 0.919 24.339 Fe

1.150 1.700 1.500 2.000 91.900 4.000 1.000 1.266 0.046 0.098

12.360 0.877 22.486 Ru

1.300 1.800 1.350 2.000 94.800 4.000 1.000 1.304 0.049 0.098

22.580 0.868 16.127 Os

0.990 1.050 0.980 2.200 74.900 2.000 1.000 1.360 0.070 0.094

8.900 0.939 16.256 Co

0.750 1.300 1.100 2.000 92.600 3.000 1.000 1.285 0.048 0.096

12.360 0.875 16.737 Rh

1.300 1.800 1.350 2.000 95.500 4.000 1.000 1.285 0.047 0.098

22.550 0.866 16.020 Ir

0.990 1.050 0.980 2.200 73.600 2.000 1.000 1.304 0.063 0.093

8.910 0.944 16.534 Ni

0.890 0.750 0.870 2.600 99.300 2.000 1.000 1.512 0.073 0.091

12.000 0.855 10.560 Pd

0.970 1.110 0.850 2.600101.600 2.000 1.000 1.512 0.071 0.091

21.470 0.848 7.716 Pt

Note: All quantities are in atomic units except cE which is in rydberg; 0 and cR

are taken from C. Kittel‘s book [p. 38 of ref.[40] and the General Electric X-ray

27



periodic chart of elements respectively, and the parameter 2A applies to FEE .

is the density, Fk is the Fermi wave number, and P is the computed ion

plasma frequency (of interest for computation of phonon frequencies, see

Sec.3.4).

Table 3.2: Form Factors qkVkqV FF

)( of the Transition-Metal Model

Potential (TMMP)

Here, as in Table 8-4 of Harrison‘s book[ 8] the first column for each metal is

Fkq 2/ ; the second column is the form factor in rydbergs. For Fkq 2 these

correspond to initial and final states on the Fermi surface; for Fkq 2 these

correspond to initial state on the Fermi surface and initial and final states

antiparallel. These are reproduced from Fortran77 programme output adapted by

Essien from Animalu‘s mainframe Fortran 44 programme based on the analytical

expression for qkVkqV FF

)( in Eqs.(3.3.2), and sketched for Copper in

Fig. 3.1.

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1 1.5 2 2.5 3

Series1

)(qV

Fkq 2/

Fig.3.1. The Screened TMMP Form Factor for Copper (Cu)

28



Cu

Fkq 2/ )(qV

0.00 -0.34539

0.10 -0.33473

0.20 -0.30223

0.30 -0.25076

0.40 -0.18574

0.50 -0.11574

0.60 -0.04946

0.70 0.00695

0.80 0.05071

0.90 0.08178

1.00 0.24600

1.10 0.10490

1.20 0.09682

1.30 0.08305

1.40 0.06644

1.50 0.04907

1.60 0.03239

1.70 0.01741

1.80 0.00479

1.90 -0.00512

2.00 -0.01223

2.10 -0.01661

2.20 -0.01854

2.30 -0.01837

2.40 -0.01654

2.50 -0.01354

2.60 -0.00983

2.70 -0.00585

2.80 -0.00199

2.90 0.00144

3.00 0.00420

Ag

Fkq 2/ )(qV

0.00 -0.26919

0.10 -0.26176

0.20 -0.23880

0.30 -0.20157

0.40 -0.15275

0.50 -0.09761

0.60 -0.04233

0.70 0.00798

0.80 0.05041

0.90 0.08434

1.00 0.08594

1.10 0.11689

1.20 0.11032

1.30 0.09722

1.40 0.08062

1.50 0.06270

1.60 0.04505

1.70 0.02881

1.80 0.01473

1.90 0.00327

2.00 -0.00537

2.10 -0.01123

2.20 -0.01450

2.30 -0.01551

2.40 -0.01469

2.50 -0.01250

2.60 -0.00944

2.70 -0.00596

2.80 -0.00246

2.90 0.00071

3.00 0.00331

Au

Fkq 2/ )(qV

0.00 -0.27044

0.10 -0.26302

0.20 -0.23987

0.30 -0.20162

0.40 -0.15010

0.50 -0.08971

0.60 -0.02612

0.70 0.03572

0.80 0.09285

0.90 0.14471

1.00 0.15780

1.10 0.20509

1.20 0.19780

1.30 0.18011

1.40 0.15615

1.50 0.12895

1.60 0.10086

1.70 0.07366

1.80 0.04871

1.90 0.02696

2.00 0.00901

2.10 -0.00486

2.20 -0.01465

2.30 -0.02058

2.40 -0.02309

2.50 -0.02273

2.60 -0.02018

2.70 -0.01610

2.80 -0.01117

2.90 -0.00600

3.00 -0.00113

29



Sc

0.00 -0.43470

0.10 -0.42407

0.20 -0.39240

0.30 -0.34235

0.40 -0.27997

0.50 -0.21388

0.60 -0.15190

0.70 -0.09877

0.80 -0.05599

0.90 -0.02288

1.00 0.00509

1.10 0.01793

1.20 0.02691

1.30 0.03091

1.40 0.03136

1.50 0.02938

1.60 0.02586

1.70 0.02146

1.80 0.01671

1.90 0.01198

2.00 0.00756

2.10 0.00367

2.20 0.00042

2.30 -0.00211

2.40 -0.00391

2.50 -0.00500

2.60 -0.00546

2.70 -0.00536

2.80 -0.00483

2.90 -0.00397

3.00 -0.00292

Y

0.00 -0.36080

0.10 -0.34916

0.20 -0.31450

0.30 -0.26006

0.40 -0.19239

0.50 -0.12074

0.60 -0.05379

0.70 0.00285

0.80 0.04702

0.90 0.07907

1.00 0.10140

1.10 0.10641

1.20 0.10148

1.30 0.09073

1.40 0.07667

1.50 0.06112

1.60 0.04543

1.70 0.03057

1.80 0.01724

1.90 0.00591

2.00 -0.00318

2.10 -0.00993

2.20 -0.01441

2.30 -0.01679

2.40 -0.01732

2.50 -0.01634

2.60 -0.01419

2.70 -0.01125

2.80 -0.00787

2.90 -0.00439

3.00 -0.00108

La

0.00 -0.33249

0.10 -0.32339

0.20 -0.29594

0.30 -0.25208

0.40 -0.19625

0.50 -0.13544

0.60 -0.07684

0.70 -0.02554

0.80 0.01610

0.90 0.04796

1.00 0.05113

1.10 0.08112

1.20 0.08161

1.30 0.07648

1.40 0.06780

1.50 0.05712

1.60 0.04562

1.70 0.03415

1.80 0.02336

1.90 0.01370

2.00 0.00546

2.10 -0.00119

2.20 -0.00621

2.30 -0.00964

2.40 -0.01159

2.50 -0.01226

2.60 -0.01183

2.70 -0.01057

2.80 -0.00871

2.90 -0.00648

3.00 -0.00412

30



Ti

0.00 -0.66456

0.10 -0.64347

0.20 -0.58312

0.30 -0.49145

0.40 -0.38359

0.50 -0.27577

0.60 -0.17915

0.70 -0.09768

0.80 -0.03000

0.90 0.02786

1.00 0.05010

1.10 0.10120

1.20 0.10804

1.30 0.10505

1.40 0.09490

1.50 0.07996

1.60 0.06231

1.70 0.04382

1.80 0.02607

1.90 0.01034

2.00 -0.00245

2.10 -0.01177

2.20 -0.01743

2.30 -0.01962

2.40 -0.01882

2.50 -0.01571

2.60 -0.01110

2.70 -0.00583

2.80 -0.00067

2.90 0.00374

3.00 0.00695

Zr

0.00 -0.55245

0.10 -0.52769

0.20 -0.45524

0.30 -0.34779

0.40 -0.22429

0.50 -0.10567

0.60 -0.00700

0.70 0.06531

0.80 0.11149

0.90 0.13520

1.00 0.14189

1.10 0.12563

1.20 0.09995

1.30 0.07140

1.40 0.04376

1.50 0.01945

1.60 -0.00014

1.70 -0.01435

1.80 -0.02319

1.90 -0.02709

2.00 -0.02683

2.10 -0.02341

2.20 -0.01790

2.30 -0.01134

2.40 -0.00469

2.50 0.00129

2.60 0.00603

2.70 0.00921

2.80 0.01073

2.90 0.01068

3.00 0.00933

Hf

0.00 -0.56900

0.10 -0.54446

0.20 -0.47198

0.30 -0.36478

0.40 -0.24160

0.50 -0.12327

0.60 -0.02457

0.70 0.04835

0.80 0.09593

0.90 0.12198

1.00 0.13199

1.10 0.11871

1.20 0.09608

1.30 0.07030

1.40 0.04498

1.50 0.02247

1.60 0.00413

1.70 -0.00939

1.80 -0.01804

1.90 -0.02218

2.00 -0.02253

2.10 -0.01996

2.20 -0.01544

2.30 -0.00992

2.40 -0.00426

2.50 0.00084

2.60 0.00489

2.70 0.00760

2.80 0.00888

2.90 0.00882

3.00 0.00764

31



V

0.00 -0.90308

0.10 -0.86885

0.20 -0.77106

0.30 -0.62873

0.40 -0.46956

0.50 -0.31965

0.60 -0.19407

0.70 -0.09646

0.80 -0.02360

0.90 0.03034

1.00 0.07180

1.10 0.08656

1.20 0.08946

1.30 0.08448

1.40 0.07445

1.50 0.06155

1.60 0.04745

1.70 0.03343

1.80 0.02046

1.90 0.00922

2.00 0.00016

2.10 -0.00653

2.20 -0.01084

2.30 -0.01293

2.40 -0.01310

2.50 -0.01174

2.60 -0.00932

2.70 -0.00627

2.80 -0.00305

2.90 -0.00002

3.00 0.00251

Nb

0.00 -0.76137

0.10 -0.71983

0.20 -0.60176

0.30 -0.43427

0.40 -0.25366

0.50 -0.09329

0.60 0.02823

0.70 0.10728

0.80 0.14917

0.90 0.16243

1.00 0.15617

1.10 0.12266

1.20 0.08368

1.30 0.04691

1.40 0.01631

1.50 -0.00623

1.60 -0.02033

1.70 -0.02669

1.80 -0.02672

1.90 -0.02222

2.00 -0.01509

2.10 -0.00712

2.20 0.00025

2.30 0.00599

2.40 0.00952

2.50 0.01073

2.60 0.00988

2.70 0.00749

2.80 0.00423

2.90 0.00080

3.00 -0.00222

Ta

0.00 -0.76137

0.10 -0.72060

0.20 -0.60459

0.30 -0.43976

0.40 -0.26155

0.50 -0.10265

0.60 0.01869

0.70 0.09886

0.80 0.14308

0.90 0.15972

1.00 0.15783

1.10 0.12675

1.20 0.08946

1.30 0.05356

1.40 0.02305

1.50 -0.00006

1.60 -0.01524

1.70 -0.02297

1.80 -0.02447

1.90 -0.02136

2.00 -0.01539

2.10 -0.00826

2.20 -0.00138

2.30 0.00422

2.40 0.00791

2.50 0.00951

2.60 0.00916

2.70 0.00731

2.80 0.00452

2.90 0.00143

3.00 -0.00140

32



Cr

0.00 -0.71129

0.10 -0.68039

0.20 -0.60111

0.30 -0.48580

0.40 -0.36147

0.50 -0.24972

0.60 -0.16038

0.70 -0.09172

0.80 -0.03562

0.90 0.01773

1.00 0.07760

1.10 0.09416

1.20 0.09485

1.30 0.08365

1.40 0.06465

1.50 0.04200

1.60 0.01957

1.70 0.00053

1.80 -0.01302

1.90 -0.02016

2.00 -0.02119

2.10 -0.01736

2.20 -0.01055

2.30 -0.00283

2.40 0.00395

2.50 0.00849

2.60 0.01016

2.70 0.00909

2.80 0.00596

2.90 0.00183

3.00 -0.00218

Mo

0.00 -0.94360

0.10 -0.88754

0.20 -0.72701

0.30 -0.50794

0.40 -0.28216

0.50 -0.09251

0.60 0.04200

0.70 0.12248

0.80 0.16001

0.90 0.16793

1.00 0.15866

1.10 0.11746

1.20 0.07430

1.30 0.03681

1.40 0.00836

1.50 -0.01014

1.60 -0.01941

1.70 -0.02119

1.80 -0.01775

1.90 -0.01145

2.00 -0.00440

2.10 0.00181

2.20 0.00614

2.30 0.00818

2.40 0.00806

2.50 0.00628

2.60 0.00353

2.70 0.00058

2.80 -0.00194

2.90 -0.00359

3.00 -0.00418

W

0.00 -0.93768

0.10 -0.88073

0.20 -0.71821

0.30 -0.49739

0.40 -0.27135

0.50 -0.08338

0.60 0.04765

0.70 0.12310

0.80 0.15423

0.90 0.15441

1.00 0.13591

1.10 0.09441

1.20 0.05301

1.30 0.01872

1.40 -0.00567

1.50 -0.01979

1.60 -0.02485

1.70 -0.02301

1.80 -0.01682

1.90 -0.00876

2.00 -0.00093

2.10 0.00519

2.20 0.00880

2.30 0.00977

2.40 0.00847

2.50 0.00563

2.60 0.00214

2.70 -0.00119

2.80 -0.00370

2.90 -0.00501

3.00 -0.00506

33



Mn

0.00 -0.53706

0.10 -0.51850

0.20 -0.46434

0.30 -0.38205

0.40 -0.28480

0.50 -0.18810

0.60 -0.10381

0.70 -0.03732

0.80 0.01144

0.90 0.04559

1.00 0.10052

1.10 0.07356

1.20 0.06841

1.30 0.05830

1.40 0.04587

1.50 0.03297

1.60 0.02088

1.70 0.01041

1.80 0.00205

1.90 -0.00403

2.00 -0.00787

2.10 -0.00969

2.20 -0.00984

2.30 -0.00871

2.40 -0.00674

2.50 -0.00435

2.60 -0.00191

2.70 0.00028

2.80 0.00202

2.90 0.00318

3.00 0.00373

Tc

0.00 -1.10978

0.10 -1.03215

0.20 -0.81988

0.30 -0.54146

0.40 -0.27322

0.50 -0.06870

0.60 0.05518

0.70 0.10713

0.80 0.10586

0.90 0.07021

1.00 0.01505

1.10 -0.02116

1.20 -0.04538

1.30 -0.05539

1.40 -0.05279

1.50 -0.04123

1.60 -0.02513

1.70 -0.00866

1.80 0.00492

1.90 0.01367

2.00 0.01702

2.10 0.01557

2.20 0.01078

2.30 0.00444

2.40 -0.00168

2.50 -0.00623

2.60 -0.00846

2.70 -0.00828

2.80 -0.00615

2.90 -0.00290

3.00 0.00054

Re

0.00 -1.08882

0.10 -1.01717

0.20 -0.81991

0.30 -0.55761

0.40 -0.29866

0.50 -0.09209

0.60 0.04582

0.70 0.12240

0.80 0.15461

0.90 0.15993

1.00 0.15317

1.10 0.11107

1.20 0.06918

1.30 0.03432

1.40 0.00901

1.50 -0.00656

1.60 -0.01367

1.70 -0.01442

1.80 -0.01116

1.90 -0.00610

2.00 -0.00101

2.10 0.00291

2.20 0.00507

2.30 0.00545

2.40 0.00438

2.50 0.00246

2.60 0.00031

2.70 -0.00151

2.80 -0.00264

2.90 -0.00294

3.00 -0.00249

34



Fe

0.00 -0.71604

0.10 -0.68697

0.20 -0.60360

0.30 -0.48243

0.40 -0.34784

0.50 -0.22426

0.60 -0.12694

0.70 -0.06027

0.80 -0.02136

0.90 -0.00432

1.00 -0.00289

1.10 -0.00163

1.20 -0.00426

1.30 -0.00798

1.40 -0.01108

1.50 -0.01274

1.60 -0.01277

1.70 -0.01135

1.80 -0.00891

1.90 -0.00595

2.00 -0.00296

2.10 -0.00034

2.20 0.00166

2.30 0.00289

2.40 0.00335

2.50 0.00317

2.60 0.00250

2.70 0.00156

2.80 0.00057

2.90 -0.00031

3.00 -0.00095

Ru

0.00 -0.78950

0.10 -0.74293

0.20 -0.60802

0.30 -0.41958

0.40 -0.21928

0.50 -0.04526

0.60 0.08193

0.70 0.15890

0.80 0.19216

0.90 0.19155

1.00 0.16725

1.10 0.11930

1.20 0.06914

1.30 0.02534

1.40 -0.00810

1.50 -0.02979

1.60 -0.04020

1.70 -0.04106

1.80 -0.03485

1.90 -0.02436

2.00 -0.01226

2.10 -0.00085

2.20 0.00822

2.30 0.01397

2.40 0.01615

2.50 0.01509

2.60 0.01158

2.70 0.00663

2.80 0.00134

2.90 -0.00334

3.00 -0.00672

Os

0.00 -0.77332

0.10 -0.73125

0.20 -0.60732

0.30 -0.43343

0.40 -0.24646

0.50 -0.08130

0.60 0.04266

0.70 0.12174

0.80 0.16155

0.90 0.17103

1.00 0.14624

1.10 0.12185

1.20 0.07944

1.30 0.04047

1.40 0.00894

1.50 -0.01341

1.60 -0.02645

1.70 -0.03116

1.80 -0.02929

1.90 -0.02295

2.00 -0.01429

2.10 -0.00523

2.20 0.00268

2.30 0.00844

2.40 0.01157

2.50 0.01210

2.60 0.01043

2.70 0.00725

2.80 0.00336

2.90 -0.00048

3.00 -0.00361

35



Co

0.00 -0.57002

0.10 -0.55102

0.20 -0.49688

0.30 -0.41429

0.40 -0.31688

0.50 -0.21990

0.60 -0.13454

0.70 -0.06538

0.80 -0.01169

0.90 0.03008

1.00 0.06483

1.10 0.07579

1.20 0.07655

1.30 0.07056

1.40 0.06033

1.50 0.04781

1.60 0.03456

1.70 0.02179

1.80 0.01041

1.90 0.00107

2.00 -0.00589

2.10 -0.01035

2.20 -0.01244

2.30 -0.01246

2.40 -0.01087

2.50 -0.00817

2.60 -0.00492

2.70 -0.00163

2.80 0.00129

2.90 0.00352

3.00 0.00489

Rh

0.00 -0.64843

0.10 -0.61477

0.20 -0.51660

0.30 -0.37439

0.40 -0.21570

0.50 -0.06871

0.60 0.04849

0.70 0.12994

0.80 0.17797

0.90 0.19859

1.00 0.19948

1.10 0.16695

1.20 0.12512

1.30 0.08268

1.40 0.04437

1.50 0.01296

1.60 -0.01026

1.70 -0.02508

1.80 -0.03217

1.90 -0.03278

2.00 -0.02853

2.10 -0.02117

2.20 -0.01240

2.30 -0.00371

2.40 0.00376

2.50 0.00922

2.60 0.01232

2.70 0.01306

2.80 0.01175

2.90 0.00893

3.00 0.00523

Ir

0.00 -0.76953

0.10 -0.72776

0.20 -0.60472

0.30 -0.43199

0.40 -0.24615

0.50 -0.08184

0.60 0.04167

0.70 0.12064

0.80 0.16061

0.90 0.17042

1.00 0.15124

1.10 0.12208

1.20 0.07999

1.30 0.04118

1.40 0.00967

1.50 -0.01277

1.60 -0.02598

1.70 -0.03091

1.80 -0.02927

1.90 -0.02311

2.00 -0.01458

2.10 -0.00559

2.20 0.00234

2.30 0.00817

2.40 0.01140

2.50 0.01204

2.60 0.01048

2.70 0.00739

2.80 0.00355

2.90 -0.00028

3.00 -0.00345

36



Ni

0.00 -0.57671

0.10 -0.55718

0.20 -0.50162

0.30 -0.41714

0.40 -0.31792

0.50 -0.21959

0.60 -0.13343

0.70 -0.06390

0.80 -0.01009

0.90 0.03175

1.00 0.13078

1.10 0.07728

1.20 0.07766

1.30 0.07130

1.40 0.06073

1.50 0.04791

1.60 0.03443

1.70 0.02149

1.80 0.01003

1.90 0.00068

2.00 -0.00622

2.10 -0.01057

2.20 -0.01253

2.30 -0.01241

2.40 -0.01069

2.50 -0.00790

2.60 -0.00460

2.70 -0.00131

2.80 0.00155

2.90 0.00368

3.00 0.00492

Pd

0.00 -0.47233

0.10 -0.45415

0.20 -0.40386

0.30 -0.32860

0.40 -0.24306

0.50 -0.16299

0.60 -0.09977

0.70 -0.05790

0.80 -0.03636

0.90 -0.03121

1.00 0.03413

1.10 -0.03468

1.20 -0.03179

1.30 -0.02845

1.40 -0.02418

1.50 -0.01906

1.60 -0.01349

1.70 -0.00803

1.80 -0.00320

1.90 0.00059

2.00 0.00311

2.10 0.00433

2.20 0.00439

2.30 0.00355

2.40 0.00217

2.50 0.00060

2.60 -0.00085

2.70 -0.00193

2.80 -0.00252

2.90 -0.00257

3.00 -0.00216

Pt

0.00 -0.46517

0.10 -0.45458

0.20 -0.42421

0.30 -0.37460

0.40 -0.31066

0.50 -0.23902

0.60 -0.16532

0.70 -0.09212

0.80 -0.01880

0.90 0.05734

1.00 0.14114

1.10 0.16813

1.20 0.17500

1.30 0.16605

1.40 0.14518

1.50 0.11645

1.60 0.08384

1.70 0.05105

1.80 0.02126

1.90 -0.00315

2.00 -0.02070

2.10 -0.03086

2.20 -0.03399

2.30 -0.03120

2.40 -0.02413

2.50 -0.01464

2.60 -0.00461

2.70 0.00435

2.80 0.01104

2.90 0.01479

3.00 0.01549

37



3.4 Application of TMMP to Computation of Phonon Spectrum of Cobalt.

As an example of the application of the table of form factors, it is useful to

rehearse the computation of the phonon frequencies in a metal based on Toya‘s

self consistent field method[41]. One sets up the dynamical matrix,

erc DDDq )(D

, (3.4.1)

from which the corresponding phonon frequencies may be written as:

2222

erc (3.4.2)

where i

i DM 12 (i=c, r, e), and c labels the Coulombic part, r the repulsive

part; and e the electronic part. In the small ion core approximation, there is no

exchange-overlap interaction, and Eq. (3.4.2) reduces to:

222

ec (3.4.3)

Since the Coulombic contribution, 2

c is well known (see, for example ref.[13])

for cubic crystals, the phonon frequencies 2 defined by Eq. (3.4.3) can be

evaluated by computing 2

e from the expression(Cochran[42]:

)()()(0

2

2

2

22 HGH

HHHqG

Hq

HqHqq

H

p

H

pe

(3.4.4)

where H

is a reciprocal lattice vector, and the function )(qG

is given by:

)(1)(

1)()(v

)1(4)(

2

2

2

0

2

qfq

qq

q

zeqG beff

(3.4.5)

where FF

b kkqq

)vvv()(v ccocM is the form factor of the full bare ion

model-potential, eff is the orthogonalization charge linked with Voc; and (q) is

the usual Hubbard-Sham modification for exchange and correlation via the

function f(q) given by Eq.(3.3.5).

In the past three decades, the above TMMP formalism has been used for

computing the phonon frequencies for a number of transition metals[43] based on

mainframe Fortran 44 programme. Recently, however, this programme has been

38



modified by Essien[13] to Fortran77 version for laptops. For this reason, suffice

it to display in Fig. 3.2 the phonon frequencies of face-centered cubic cobalt along

the [001], [011] and [111] high symmetry directions computed from Eqs.(3.4.3)

and (3.4.4) by Essien[13] using the same laptop programme that we have used in

this paper to generate and cross-check our Table 3.2 of TMMP form factors with

the mainframe unpublished results obtained in 1973 by Animalu[12,13].

The good agreement between theory and experiment confirms the continued

effectiveness of the TMMP method for exploring the various aspects of the

electronic structure and properties of the transition and rare-earth metals.

X L ]111[ ]011[ ]001[

)2/( aq )2/( aq )2/( aq

39



4. EXPERIMENTAL VERIFICATION AND CRITIQUE OF THE

PSEUDOPOTENTIAL THEORY

4.1 Experimental Verification

In the Preface to his 1966-published book entitled Peudopotentials in the

Theory of Metals, Harrison[8] described pseudopotential theory as the ―single

point of view from which virtually all the properties of simple (non-transition)

metals may be studied‖. This statement was conifirmed at about the same time by

further applications of the model potential by Heine and co-workers[44] and

empirical pseudo potentials by M.L. Cohen and co-workers[45], to mention a few.

Our limited review in Sec. 3.4 of successful applications of the subsequent

generalization of the theory to the transiton and rare-earth metals as transition-

metal model potential (TMMP) was intended to show that the same verdict

applies not only to the simple metals but to all metals.

Nevertheless, there are shortcomings of the pseudopotential theory

especially the approximate of uniform electron gas density incorporated in the

―correlation correction‖ (2.3.23) which have dictated the need to extend the

principles Hartree-Fock method developed by Kohn and Sham[18] known as the

density functional theory (DFT) in a number of important areas, such as the

estimation of the band gaps of semiconductors and insulators[46]. A more serious

short-coming is associated with the ―orthogonalization correction‖(2.3.22) which

deserves further clarification to which we now turn.

4.2 Critique of Pseudopotential Theories and New Vistas

In spite of the experimental verification of the pseudopotential theory in

metal physics, the non-unitary character of pseudopotential transformation makes

the theory unsatisfactory as a quantum theory inasmuch as non-unitary theories

violate conservation of probability and hence conservation of electric charge,

which is the source of the ―orthogonalization correction‖, and it also violates to

some extent Pauli‘s exclusion principle. It is no wonder then that, beginning from

early 1990s, considerable effort [14] has gone into bringing the pseudopotential

theory in line with generalizations of non-unitary theories into iso-unitary ones in

the framework of the discipline of ―hadronic mechanics‖.

To highlight the main novel feature of the new vista that has emerged, consider a

pseudopotential of Austin-Heine-Sham form[5] arising from orthogonalization of

a pseudowave to an arbitrary functional of a singlet core function Rr

c e / and

a pseudowave function, )1( / Rre so that in Eq.(2.3.11a) we have:

40



)(

1 /

/

0 rVe

eVFVV HRr

Rrc

cAHS

(4.2.1)

where HV is a Huthen-type model potential. By using the formal expansion,

0

/

/1

1

k

Rkr

Rre

e (4.2.2)

one readily finds the Fourier transform[47]:

(4.2.3) .11

)(

)/(

1

)/(

1

q

0)/(q

q

)(

2

)1(2

2

02

2

02

0

0 0

)/(

0 0

)/(

0

/

0

q

R

qR

R

R

kqR

R

RkqRkq

Rkq

edrerdree

qV

qRkkk

k

rRkq

k

rRkqRkr

k

qr

H

Note, in contrast, that the Fourier transform of Coulomb ( r/1 )-potential is

proportional to 2/1 q .

If we expand Rre / to first order in the denominator, we get

Rr

eV

e

eVV

Rr

Rr

Rr

H/1

/

0/

/

0

(4.3.4)

i.e., an approximate Yukawa potential. And it happens that one can solve exactly

the Schrodinger equation for a two-particle bound state problem in the exact

Hulthen potential ( HV ) and find just a single low-lying bound level that permitted

Santilli[48] in 1978 to identify a compressed positronium )( ee atom with

the neutral pion ( 0 ) and the compressed hydrogen atom ( ) ep with the

(Rutherford-Santilli) neutron at inter-paticle separation of order cm10 13

0

r . A

consistent way of reformulating the underlying non-unitary pseudopotential

theory into an axiomatically consistent (iso-unitary) theory gave birth to the

discipline of ―hadronic mechanics‖[48]. The dividend for condensed matter

physics was realized when Animalu[14] developed a similar solution of a Hulthen

41



potential model of a Cooper pair[14], HV)( eCueCP z , in the high-

Tc superconducting cuprate materials, which correctly predicted the dependence

of Tc on the effective valence (z) of the copper ion ―trigger‖ in the framework of

―hadronic mechanics‖. An elebaoration of this theory to the iron pnictides by

Animalu, Akpojotor and Ironkwe[14] will be presented in this Seminar. But the

intriguing new vista is the extension of model (Hulthen) potential theory to the

wide range of interparticle separation, )(10)(10 138 cmrcm , of interest for

exploring the state of condensed matter from absolute zero to million degrees (K).

One of the outstanding problems (of high temperature superconductivity) in this

regime is to explain the coexistence of electron and proton superconductivity in

a neutron star (represented as condensed 2H system based on the self-explanatory

analogy sketched in Fig.3.3, in which the electron Cooper pair mediates proton

Cooper pairing.

(a)

Fig. 3.3(a): Attractive proton-proton interaction mediated by

virtual “phonon” exchange in the conventional BCS model; (b)

Attractive proton-proton pairing due to overlapping of proton

wave functions around electron pair HMeeCP ),(~

“trigger” in 1s2 -state envisaged in “hadronic” superconductivity

.model.

)( kp

)( kp

)'( kp

)'( kp

q

p

p

HMeeCP ),(~

(b)

p

p

42



5. SUMMARY AND CONCLUSION

We have reviewed the theoretical and practical aspects of pseudopotential

and model potential theories of simple and transition metals in the past fifty years,

in order to provide adequate details required for effective use of the method and

the tables of model potential form factors in contemporary computational solid

state physics. The review has also provided an opportunity to introduce an

unequivocal extension of pseudopotential (non-unitary) transformation theory

from its foundation in atomic/molecular structure calculations of 20th

C solid state

physics to nuclear/subnuclear structure problems of 21st Century theoretical

physics now in progress. Our conclusion is a re-affirmation of Harrison‘s dictum

cited earlier that pseudopotential theory remains the single point of view from

which virtually all the properties of condensed matter characterized by

interparticle separation in the range )(10)(10 138 cmrcm may be studied.

REFERENCES

[1] Philips, J.C. and Kleinman, L. Phys. Rev. 116, 287, 880 (1959).

[ 2] C. Herring, Phys. Rev. 57, 1169 (1940)

[3] A. V. Gold, Phil. Trans. Roy. Soc. London, A251, 85 (1958).

[4] W.A. Harrison, Fermi Surface of Aluminium Phys. Rev. 116, 555 (1959);

Band Structure of Aluminium Phys. Rev.118, 1182 (1960); Electronic

Structure of Polyvalent Metals, Phys. Rev. 118, 1190 (1960); Band

Structure and the Fermi Surface of Zinc, Phys. Rev. 126, 497 (1962);

Electronic Strucutre and the Properties of Metals: I. Formulation, Phys.

Rev. 129, 2603 (1963), II. Application to Zinc, Phys. Rev. 129, 2512

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http://www.i-b-r.org/Hadronic-Mechanics.htm

46



GENERALIZATION OF CONVENTIONAL BCS MODEL TO ISO-SUPERCONDUCTIVITY MODEL OF HIGH-TC

SUPERCONDUCTIVITY IN THE CUPRATES AND PNICTIDES‡

A. O. E. Animalu1, G. E. Akpojotor2, and P. I. Ironkwe3 1Department of Physics and Astronomy, University of Nigeria, Nsukka, Nigeria

2Department of Physics, Delta State University, Abraka 331001, Nigeria 3The Center for Superconductivity Technologies, Abuja FCT, Nigeria

e-mails: [email protected],

[email protected]

Abstract

After years of successful application of the pseudo and model potential

representation of electron-phonon interaction to conventional Bardeen-Cooper-

Schrieffer (BCS) theory of superconductivity, herein called the standard model,

we have developed a generalization (herein called isostandard or iso-

superconductivity model) that not only explains the differences between

conventional and high-Tc superconductivity in the cuprates but also permits, in

this paper, successful applications to the new high- Tc iron pnictides and toMgB2.

PACS numbers: 74.70.-b, 74.20.Mn,74.62.-c

1. INTRODUCTION

The discoveries by Berdnoz and Muller1 in 1986 at IBM Zurich of

superconducting phase transition in a family of ceramic oxide materials and by

Wu et al2 in 1987 in the 1-2-3 compound, (cuprates with structural formula,

xnmOCu (...) ) at rather high critical temperatures (Tc) of 35K and 95K

respectively, opened up the field of experimental and theoretical high-Tc

superconductivity research which has remained very active to date. In addition to

the 2001 discovery3

of high-TC of order 35K inMgB2, the more recent discovery

of superconductivity in the iron-based compounds4 - iron oxypnictides/single

layered LnOMPn (Ln = La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho and Y: M = Mn, Fe,

‡African Journal of Physics Vol. 2, pp.46-62, (2009)




47



Co and Ni: Pn = P and As)5; oxyfree-pnictides/single layered AMPn (A = LnO =

Li and Na: M = Mn, Fe, Co and Ni: Pn = P and As6; oxyfreepnictides/double

layered ALM2Pn2 (AL = Ba, Sr, Ca: M=Mn, Fe, Co and Ni: Pn = P and As7,8

and

chalcogen/nonlayered MCn (M = Mn, Fe, Co and Ni: Cn = S, Se and Te)9,10

has

heightened interest in developing a suitable generalization of the Bardeen-

Cooper-Schrieffer (BCS) model of superconductivity in simple metallic systems

for understanding the coexistence11

of superconductivity and magnetic order in

these solid compounds. The prospect of finding such a generalization is now

brighter than ever for the following reason: just as high-TC superconductivity in

the cuprates is known to be a two-band phenomenon involving the copper 3d and

oxygen 2p bands of the CuO2 planes, but was reduced to an effective single band

pairing problem by Anderson12

in 1987 via his doped resonant valence

bond(RVB) model and its generalization by Zhang and Rice13

in 1988 to the t – J

model, so also is high- Tc superconductivity in the iron-based compounds known

to involve multi-orbital effects of the Fe-3d with filling of approximately six

electrons per Fe-site in the pnictides, but has been shown in the 2009

selfconsistent fluctuation exchange (FLEX) model by Zhang et al14

to be

reducible to an orbital s(↑, ↓) coupling affair also known as the s state15

.

Our proposed generalization in this paper is based on the observation by

Animalu16

in 1991 and its elaboration17

in 1994 under the name iso-

superonductivity that the Cooper pair of the standard BCS model may have a

nonlocal nonhamiltonian structure CP = (e−↑, e− ↓) HM equivalent to the strong

interaction (‖hadronic‖ mechanics (HM)) structure of the neutral pion, as

compressed positronium atom, 0 = (e+↑, e−↓) HM , proposed by Santilli18

in

1978, i.e. an extended structure arising from the mutual overalpping/penetration

of the wavefunctions of the constituents of the pair. The result is an effective

generalization of the pseudo or model potential representation of electron-phonon

interaction in the standard BCS model in such a way that whereas the Feynman

graph for electron-electron scattering leading to net attraction, i.e., Cooper pairing

in the BCS model is mediated by virtual phonon exchange (as shown in Fig. 1a),

the situation in the iso-superconductivity model for a high-TC cuprate materials,

xnmOCu (...) , is that a zCu ion of effective valence, z ≡ 2(n − x)/m (or zFe for

appropriate value of z in the iron pnictides) provides a ‖trigger‖ for the

overlapping (i.e., ‖covalent‖ mixing) of electron wavefunctions to form a singlet

pair, (e−↓, e−↑)HM (see, Fig.1b). The main features of the transition from the BCS

model to the isosuperconductivity model and its prediction for TC will be

presented in Sec.2.

48



In models that propose to unite superconducting and antiferromagnetic phases in a

larger symmetry group, SO(2N), an outstanding problem has been their

inconclusive nature due to the inability to exactly diagonalize the model

Hamiltonians. However, in 1992 C.N. Animalu19

proposed an alternative exact

method for diagonalizing any second-quantized Hamiltonian model for arbitrary

N-electron system based on 2N×2

N- matrix representation of the electron creation

and annihilation operators suggested in 1961 by Thouless20

. Such a representation

(which we shall skip for simplicity) is of considerable interest because it provides

a 2N-dimensional spinor representation of the group, SO(2N), corresponding to

the groups, SO(8) for N = 4 and SO(10) for N = 5, the latter being realized for a

lattice of 32 × 32 sites that has been employed by Zhang et al14

in their 2009

solution of the FLEX equation on imaginary frequency axis. These SO(2N)

groups are among the groups that matter in string/superstring theories21

. In Sec. 3,

we shall recapitulate the experimental verification of the predictions of TC by the

isosuperconductivity theory in the cuprates and present, for the first time, the

corresponding verification for the pnictides, as well as for MgB2, leading as a

consequence, to a useful semi-empirical formula for designing superconductors

with Periodic Table-based maps and material databases in the current search for

room temperature superconductors. Conclusions will be drawn in Sec. 4.

FIG. 1: (a) Attractive electron-electron interaction mediated by virtual

phonon exchange in the conventional BCS model; (b) attractive electron-

electron pairing due to overlapping of electron wave functions around

Cuz+

or Fez+

ion ”trigger” in orbital s(↑,↓)–state envisaged in iso-

superconductivity model.

49



2. THE ISOSUPERCONDUCTIVITY MODEL

As a prelude to the definition and characterization of the isosuperconductivity

model, we begin in this section with a brief review of the conventional/non-

conventional models of superconductivity.

Review of Conventional and Non-conventional Models

Because the conventional/non-conventional quantum mechanics models of

superconductivity are based on second quantization formalism in both Bloch (k-

space) and Wannier (r-space) representations, it should be recalled that one can

formally transform the BCS model for superconductivity at low temperatures

given in the Bloch reprsentation by

,'

'''

kk

kkkkkk

k

kkkBCS ccccVccH

(2.1a)

into the Wannier representation

,)(,

ji iiiijjiij

W

BCS nnUcccctH (2.1b)

This is achieved by making the following substitutions:

j

j

i

k

j

j

i

k ceNcceNc jj

k.Rk.R2

1

2

1

;

k

i

kij

ij

jiij

k

kk eN

tcctn)1

; ij Rk.(R

(2.2)

where jiij ccn and by using the following approximations

,/ ;2 '' UUNVttt ijklkkijkkkk (2.3)

where imjijt )( 2

2

2

is the hopping matrix wlwmwnt while the

electron-electron interaction energy is derived from the general expression

ijkl

ilkjijkl ccccU2

1

where

50



)()()()()( 22

*

11

*

1 rrrrrrdrdr 212 lkeeijijkl VU

is the matrix element of the effective Coulomb interaction betweenWannier states

on different sites, )()( iRrr i . Finally, from the Hubbard-type Hamiltonian

form in Eq.(2.1b), the t − J model for high-TC close to half-filling is abstracted in

the form

ij

ji

ij

jjiiJt nnJchnccntH41

,

..11 ji .ss

(2.4)

2.2. Definition of the Isosuperconductivity Model

In its simplest non-relativistic form, the isostandard model of

superconductivity16,17

is a generalization of the Lurie-Cremer22

quasiparticle wave

equation,

13

2

2

1 ),,(),(

prr

mHtHt

ti (2.5)

via the non-unitary (‖isotopic lifting‖) transformation of the underlying ‖metric‖

(g),

gT

g

0

01ˆ

10

0133 (2.6)

which is characterized by a nonlocal integral (pseudopotential) opertar defined by

)'()'()()'(')( **3*rrrr

rrrdT (2.7)

where )(*r

and )(r

are the two spinor components of the quasi-particle

wavefunction )0,(r in the Nambu representation, mp 2/2 being the kinetic

energy operator (measured from the Fermi level) and is the pair potential energy. It

is apparent from Eq. (2.7) that when the overlap integrals or ‖orthogonalization

term‖

**3 )'()'('2

1

rrrdZ (2.8)

Is zero, T reduces to unity and we recover the standard (BCS) model exactly.

Since we may rewrite T in the form

51



*1

T (2.9a)

so that TT 2 if 0*

, the physical effect of T is that the charge on the

e reprsentated by the expectation value of T, i.e.,

ZT

1* (2.9b)

is ―depleted‖ by an amunt Z (called the ―orthogonalization charge‖) whereas the

charge on e appears to vanish, i.e.,

0*

T (2.9c)

In other words, e behaves like a neutral spin-21 quasiparticle (spinion) while

e behaves like a fractionally-charged quasiparticle (‖anyon‖).

Consequently, in the solid state where the wavefunction ),( tr to which

the nonlocal transformation in Eq.(2.6) is to be applied is related to the )(ri and

)(tck of the secondquantized formulation by

)()(),( rtctr ik (2.10)

the corresponding transformation of the corresponding creation and annihilation

operators,

kc and kc , into iso-creationand iso-annihilation operators, is defined

by

ikikikikikikik ccnncTc ),1(ˆ (2.11)

and similarly for ikc where for and vice versa. This has the effect

of transforming the hopping (kinetic energy) term exactly into

,,

).1()1(ˆˆij

jjii

ij

ji nccntcct (2.12)

as in Eq.(2.4) characterizing the t − J model. It follows that the difference

between the t − J model and the isosuperconductivity model lies in the

replacement of the U-term in Eq.(2.1b) by the J−term in the t − J model (with the

antiferromagnetic exchange constant UtJ /2 via second-order perturbation

theory). Typically, i(j) = d, p label electrons (bands) of Cu 3d and/or O 2p

characters whose wavefunctions may overlap and/or bands hybridize; and (i(j) =

1, 2, ...,N) in the nearest-neighbour electron transfer (hopping) integral.

52



By virtue of the transformation defined by Eq.(2.11), only single

occupancy per spin site is permitted but double occupancy of an orbital site is not

forbidden. Another feature of the second-quantized theory form of the iso-

creation and iso-annihilation operators is that the waveoverlapping is associated

with the coexistence of a non-zero antiferromagnetic spin wave state, 0

iicc

and Cooper pair state 0 ii

cc under Gor‘kov‘s factorization of the products of

three fermion creation and annihilation operators involved in the transformation

iiiiiiiiiiii

iiiiij

cccccccccccc

ccccT

)1(

(2.13)

In this (mean field) sense, one can derive from the isosuperconductivity model

one of the primary objectives of the t – J model which is to describe the

coexistence of superconductivity and antiferromagnetismin high-TC materials as a

function of band filling. The most important difference between the t−J model and

the isosuperconductivity model lies in the ability of the latter to predict TC from

an exact solution of the model, to which we now turn.

2.3. Prediction of TC

In conventional BCS model, the determination of the critical temperature for

superconductivity involves solving an integral equation for the energy gap. But

the beauty of the isosuperconductivity model is that instead of an integral

equation, the desired result comes from the self-consistent solution of the

conventional Schrodinger equation for one spin state,(

) say,

,2/2

EVmpH C (2.15a)

in the Coulomb field VC of the zCu ion ‖trigger‖ in Fig. 1b, and an iso-

Schrodinger equation

,2/2

EVmpHT H (2.15b)

for the opposite spin state(

), where T is the non-local (psuedopotential)

integral operator defined by Eq.(2.7). This has the effect of replacing the

Coulomb potential, VC , by an effective Hulthen potential, VH in Eq.(2.15b) for

(e− ↓, e− ↑) pairing in a singlet state:

53



HkrCC Ve

VE

VV

1

10

(2.15c)

where 0V is proportional to

. From the exact solution of Eq.(2.15b),

Animalu16,17

derived the following formula for the critical temperature having the

general form:

1)1

exp(NV

T J

C (2.16a)

where NV reprsents the dimensionless coupling constant while

)( DB

p

Jqdk

(2.16b)

is the ‖jellium‖ temperature, d = 1, 2, 3 being the effective dimensionality of the

system. and )( Dq the Hatree dielectric function evaluated at the Debye

wavenumber Dq . We observe that in the weak coupling limit NV < 1, we may

express the result in the BCS form:

)/1exp( NVT JC (2.17a)

But in the strong coupling limit, i.e. if NV > 1, we may expand the exponential in

the denominator of Eq.(2.16a) to first order in 1/NV to get

NVT JC (2.17b)

Our interest is to show how accurately these results agree with

experimental data to which we now turn.

3. EXPERIMENTAL VERIFICATION

3.1. The Cuprates

An explicit form of Eq.(2.17a) used in Ref. 16 for the verification

with experimental data in the cuprates with structural

formula xnmOCu (...) is:

)( 6.13

exp3.367

)6.13

exp( 0 Kzd

z

zT JC

(3.1)

54



where the effective valence z of the zCu ion is given bym

xnz

)(2 .

Table 1 Dependence of Tc on the effective valence z of zCu in the cuprates

and zFe in the iron pnictides.

55



FIG. 2: Predicted dependencies of the Jellium temperature J and the

superconducting transition temperature Tc on the effective valence

mxnz /)(2 of zCu ions in the family of compounds, xnmOCu (...) are

compared with experimental data (•) as discussed in the ref.[17]. The

experimental Debye temperatures of pure copper )(CuD and pure

vanadium )(VD are indicated.

56



As a further confirmation of the formula in Eq.(2.17a), we present in Tables 2-4

and Fig. 3 another set of experimental data on the high-Tc doped compounds23-25

,

yxx OMnCuYBa 32 and 7312 )( OMCuGdBa xx ,(M =Ni and Zn), in which the

effects of the substitution of Cu by transition- and non-transition-metal ions are

represented by the modification of the effective valance (z) on zCu indicated in

tables 3-4. Again reasonable agreement between theory and experiment is

obtained for z lying in the range, 4.61 ≥ z ≥ 4.21.

Table 2. yxx OMnCuYBa 32 (After Ref. 23)

Note: Tc(theory) is given by Eq.(3.1) where the effect of replacing

3Cu by xx MnCu 3 is obtained by replacing 3 by (3 − x) + 2x = 3 + x

which lowers the effective valence (z) on zCu ions to )3(

2

x

yz

.

57



.

FIG. 3: Predicted dependence of the superconducting transition temperature

Tc on the effective valence (z) of Cuz+ ions (continuous curve) given by Tc =

367.32exp(−13.6/z) in the doped 1:2:3 cuprates are compared with

experimental data as discussed in Tables 1,2,3 and the text

58



Table 3: 7312 )( ONiCuGdBa xx (after ref. 24)

Note: Tc (theory is given by Eq.(3.1) and )1(3

2

x

yz

as discussed in Table 2.

Table 4 : 7312 )( OMnCuGdBa xx (after ref.25)

Note: Tc (theory is given by Eq.(3.1) and )](31[3

22xx

yz

includes an extra

2x term for the non-transition metal Zn ion substitution in order to give a

reasonable phenomenological fit to the data.

3.2 The Iron pnictides

In order to compare with experimental data in the iron pnictides, we now

turn to a realization of the formula in Eq.(2.17b) in a similar form:

);(6.13

exp0.467 0 Kz

zTC

where 467.0 = D is the experimental Debye temperature of iron (see Table 1). It

is also plotted alongside the result for the cuprates in Fig. 4. There is good

59



agreement with the experimental data in the pnictide, 224.06.0 AsFeKBa from

neutron scattering7 (see, Fig. 5).

FIG. 4: (colour online) Predicted dependence of the transition

temperature Tc on the effective valence z for the cuprates in

Eq.(3.1) and the pnictides in Eq.(3.2)

60



3.3 MgB2

For MgB2, the corresponding prediction is

)(6.13

exp0.406 0 Kz

zTC

(3.3)

where 406.0 = D is the experimental Debye temperature of Mg. The results are

tabulated in Table 6 which shows that the observed TC of (39 K0 ) corresponds to

a value of z close to 3.75.

Fig. 5: (colour line) Predicted Tc for 224.06.0 AsFeKBa

61



Table 6: Dependence of Tc on the effective valence of zMg in 2MgB

4. DISCUSSION AND CONCLUSION

By the time the iso-superconductivity (iso-standard) model was proposed by

Animalu16,17

, the BCS model had lost predictive power for the available

experimental data in the cuprates. Subsequently, when the highest TC of 165K so

far in the cuprates was reported in 199425

, the result was also in agreement with

the iso-standard model prediction26

. The successful application of the iso-standard

model in this paper to the prediction of the recent data on the iron pnictides lends

further credence to the iso-standard model even though the effective valence (z)

on Fez+ has been treated as a phenomenological parameter. We are therefore led

to the conclusion that more serious studies of the foundation

of iso-superconductivity in the analogy between Santilli‘s model of the neutral

pion as a compressed postronium atom and the isostandard model of the Cooper

pair should be undertaken: this is the subject-matter of the isotopic branch of

‖hadronic‖ mechanics with far-reaching implications for quantum physics in the

21st C.

62



ACKNOWLEDGMENTS

We are grateful to the International Center for Basic Research, Abuja for its

support of this research project. GEA acknowledges that part of this work was

done at the Max Planck Institute for Physics of Complex Systems, Dresden,

Germany.

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2 M.K. Wu et al., Phys. Rev Lett. 58, 908 (1987).

3 J. Nagamatsu et al, Nature 410, 63 (2001).

4 Y. Kamihara et. al., J. Am. Chem. Soc. 130, 3296 (2008).

5 X. H. Chen et. al., Nature 453, 761 (2008).

6 J. H. Tapp et al., Phys. Rev. B 78, 060505 (R) (2008).

7 A. D. Christianson et. al., Nature 456, 930 (2008).

8 J. Zhao et al., Phys. Rev. Lett. 101, 167203 (2008).

9 A. Subedi et al., Phys. Rev. B 78, 134514 (2008).

10 U. Patel et al., App. Phys. Lett 94, 082508 (2009).

11 A. J. Drew et al., Phys. Rev. Lett. 101, 097010 (2008).

12 P. W. Anderson, Science 256, (1987).

13 F.C. Zhang and T.M. Rice, Phys. Rev. B 37 3759 (1988).

14 J. Zhang, arXiv:0903.4473

15 I. I. Mazin et al., Phys. Rev. Lett. 101 057003 (2008)

16 A.O.E. Animalu, Hadronic J. 14, 459 (1991).

17 A.O.E. Animalu, Hadronic J. 17, 349 (1994); A.O.E. Animalu, B. Ezekoye

and K.E.Essien, African J. Phys. Vol. 2, pp. 1-45 (2009). 18

R.M. Santilli, Hadronic J. 1, 574 (1978). See also Foundations of Hadronic

Chemistry with Applications to new Clean Energies and Fuels (Kluwer

Academic Publishers, dordrecht, Boston, London, 2001) 19

C. N. Animalu, Hadronic Journal Supplement 7, 287 (1992). 20

D.J. Thouless, The Quantum Mechanics of Many-Body Systems

(Academic Press New York and London, 1961). 21

N. Prakash, Mathematical Perspectives on Theoeretical Physics,

A Journey from Black Holes to Supersprings (Tata McGraw-Hill

Publishing Co Ltd, New Delhi, 2000) p. 774 22

D. Lurie and S. Cremer, Progr. Theor. Phys. 44, 300 (1970). 23

N. L. Saini et. al., Int. J. Mod. Phys. 6, 3515 (1992). 24

C. Lin et al., Phys. Rev. B 42, 2554 (1990). 25

L. Gao et al., Phys. Rev. B 50, 4260 (1994). 26

A.O.E. Animalu and R.M. Santilli, Int. J. Quantum Chemistry 29,175 (1995).

63



A SIMPLIFIED FORMULATION OF THE LANCZOS

TECHNIQUE FOR STRONGLY CORRELATED SYSTEMS§

S. Ehika1, E.O Igbinovia 2, J.O.A. Idiodi3 1Department of physics, Ambrose Alli University Ekpoma, Edo State,Nigeria

2Department of Basic Sciences, Benson Idahosa University, Benin City, Nigeria

3Department of Physics, University of Benin, Benin City, Nigeria

Abstract

The basic idea of the Lanczos method is to construct a special basis where the

Hamiltonian has a tridiagonal representation. Once in this form, the matrix can be

diagonalized easily using standard library subroutines. This approach is known as

the standard Lanczos technique (SLT) which still has the problem of

diagonalizing large matrix sizes emanating from increase of the size of the Hilbert

space with the size of the system. In this current presentation, we develop a

simplified formulation of the SLT and then use it to study the rapid convergence

to the ground state energy and wavefunction of some finite systems.

1. INTRODUCTION

The discovery of high- temperature superconductors has induced considerable

theoretical work on strongly correlated systems. Various analytical methods [2, 3]

as well as numerical techniques [4-10] have been employed to study these

systems. High accuracy studies can be achieved by using direct diagonalization

method [11] (the Lanczos algorithm in particular). In this method, special bases

are constructed which transform the Hamiltonian into a tridiagonal matrix. Once

in this form, the matrix can be diagonalized easily using standard library

subroutines. This approach is known as the standard Lanczos technique (SLT).

This method is still beset with the problem of diagonalizing large matrix, since the

size of the matrix grows like the size of the Hilbert space.

In this paper we study the rapid convergence to the ground state by

formulating a simplified version of (SLT). The contribution of this current

§ African Journal of Physics Vol.2, pp. 63-75, (2009)


64



presentation is the simplification of the algorithm use by Dagotto and other

researchers in this field. [5, 9] The approach use in this paper makes iterations

easier to carry out. It was also demonstrated that the rate of convergence is

dependent on the choice of the initial trial vector. Further details on this method

can be found in the cited references.

This method is demonstrated by the single band Hubbard Hamiltonian. [13]

This Hamiltonian reads

i

iiji

jI nnUCHCCtH

,,

).(

Where ji, denotes nearest–neighbour sites. )( ,, ii CC Creates (annihilates) an

electron at site i in the spin state σ = ↑ or ↓, t is the hopping term, U is the onsite

interaction term and ,in is the number operator.

The Layout of this paper is as follows. In section II we describe our algorithm.

Application of the algorithm to a case of two electrons on two, four and six sites

(in the subspace of zS total =0) are presented in sec. III. In section V, a

comparison of the results obtained with this method and that of variation results

by Enaibe and Idiodi [14] is given in sec IV. We summarize and conclude in sec

V

2: DESCRIPTION OF THE METHOD

In this section, we describe our method. As in the standard Lanczos technique

[10], and the modified Lanczos method [5-10], the method requires the selection

of an initial trial vector 0 (normalized to one). If H acts on o the result can

be written as

00

00

00

0

~

HH [5] (2)

Where 0

~ is a new state orthogonal to 0 . Since 0 is normalized, eqn. (2)

becomes

kHH 00000

~ (3)

The constant k ensures that 0 is normalized.

65



From eqn.3, we have that 1~~ 22

000

2

000 kHH

This gives

21

2

000

2

0

HHk (4)

Now, the action of H on 0

~ gives

kHHHH 0000

2

0

~ (5)

So that

kHHH2

900

2

000

~

k

HH12

12

000

2

0 (6)

From eqn.5, it can easily be shown that

00000

2

0000

~~~~ HHHHc (7)

If 00 H is denoted by 0a , 0b by k

1 then in the basis 0 and 0

~ , a 2x2 matrix

representation of H is given by

00

00

,cb

baH ji (8)

This 2x2 matrix can easily be diagonalized. Its lowest eigenvalue 1a is given by

2

4 2

000

2

0000

1

bcacacaa

(9)

Its corresponding eigenvector 1 is given by

00001

~ (10)

66



Where 2

10

2

0

2

00

aab

b

, and

2

10

2

0

2

100

aab

aa

1a and 1 are better approximations to 0 (true ground state energy) 0 (true

ground state wavefunction) than 0a and 0 respectively.

The method can be iterated by considering 1 as a new trial vector and

repeating the steps from eqn. 2 to 10. In each iteration, the orthogonal pairs

( 0 , 0

~ ), ( 1 , 1

~ ), etc are normalized.

3: APPLICATION OF THE METHOD TO FINITE SYSTEMS

TWO ELECTRONS ON TWO SITES

First, in this section, the modified Lanczos algorithm is applied to the case of

two electrons on two sites with periodic boundary condition. Also, in order to

demonstrate how the choice of initial trial wave vector affect the rate of

convergence to 0 and 0 , two different choices of wave vector are used.

The relevant electronic states of this system are:

2,14,2,13,2,22,1,11

The ground state wavefunction of this system is known to be a singlet state.

With this prior nowledge, the normalized trial vector 0

below can be

constructed

212

10

Following the steps outlined in section II (Eqn 2-10), we have that

Ua 0 , tb 20 , 00 c

67



342

1~0

02

2

t

tUH

The improved ground state energy gives

22

1 162

1tUUa ,

and the corresponding improved wavefunction is

342

212

001

22

2

0

16

16

2

1

Ut

UUt

and

22

2

16

16

2

1

Ut

UUt

It is obvious that our choice of trial wave vector immediately reproduces 0 and

0 in just one iteration

If a prior knowledge of the ground state is not known, a single vector from the

Hilbert space can be chosen. This will elucidate how the choice of trial wave

vector can affect the rate of convergence to the actual ground state energy and

wavefunction.

From the Hilbert space of the system above, let us consider 1,10

1ST

ITERATION

The results obtained at the end of the first iteration are given below

22

1 82

1tuua , 00001

~

0 and 0 were defined in section II

68



342

1

2

1143~0

t

uut

Obviously, 0 and 0 are not true representations of the ground state properties

of the System. Therefore, it is necessary to carry out more iterations in order to

get close to the 0 and 0 .

2ND

ITERATION

For the second iteration we have

2

111

2

11112 42

1bcacacaa , 11112

~

Where 2~

1

The results obtained for this system for iterations up to four are summarized in

table 1.0 in section V

4. TWO ELECTRONS ON FOUR SITES

In this section, we apply the modified Lanczos method to a system of two

electrons on four sites. The relevant electronic states of this system are

,4,310,4,39,3,28,3,27,2,16

,2,15,4,44,3,33,2,22,1,11

4,216

,4,215,3,114,3,113,4,112,4,111

The Hilbert space above can be reduced to 4 by the following four vectors

16151413

,12111098765,4321

T

SR,

It can be shown that

StTHTtRtSHRUStRH 2,44,2

69



Let our initial normalized trial vector be given by

R2

10 .

The results obtained for this system for iterations up to four are summarized in

table 2.0 in section V

TWO ELECTRONS ON SIX SITES

In this section, the algorithm is applied to a system of two electrons on six

sites. The possible electronic states of this system are shown below:

1,11 4,313 6,225

2,22 4,314 6,226

3,33 5,415 5,327

4,44 5,416

5,328

5,55 6,517

6,429

6,66 6,518

6,430

2,17 3,119

4,131

2,18 3,120

4,132

6,19

5,121

5,233

6,110

5,122

5,234

3,211

4,223

6,335

3,212

4,224

6,336

As before, it can be shown that

70



StTHRtTt

SHStQtRHQURtQH

2,24

,24,2

where

654321 Q

181716151413121110987 R

302928272625242322212019 S

363534333231 T

For this system, if the wavefunction obtained at the end of an (n-1)th iteration

is given by

TASARAQAn 43211 ,

where 1A , 2A , 3A and 4A are the electronic weights of the states Q, R, S and T

respectively,

the matrix elements emanating from the nth iteration can be shown to be

21

2

134423312211

1

242212221246

n

n

aBtAtBtBAtBtBAtBUBA

b

where

211 4tAUAB , 312 22 tAtAB , 423 22 tAtAB and 34 4tAB ;

and

344233122112

3

1 2422122212461

CtCtCtCCtCtCCtCUCCb

cn

where inii AaBC 1

Let the initial trial wavefunction be given by

Q6

10 .

The results obtained for this system for iterations up to eight are summarized in

table 2.0 in section VI

71



5. SUMMARY OF RESULTS

TWO ELECTRONS ON TWO SITES

Table 1.0.Ground-state energy of the Hubbard model with two electrons on two

sites as a function U (at t=1).Results are presented for four iterations, and a

comparison is made with variational results (Enaibe and Idiodi, 2003)

U 1a

2a 3a 4a variational

20.00 -0.09951 -0.19804 -0.19804 -0.19804 -0.19804

10.00 -0.19615 -0.38510 -0.38517 -0.38517 -0.38517

5.00 -0.37228 -0.70040 -0.70156 -0.70156 -0.70156

1.00 -1.00000 -1.52753 -1.55986 -1.56147 -1.56155

0.50 -1.18614 -1.71597 -1.76218 -1.76534 -1.76556

0.00 -1.41421 -1.93185 -1.99386 -1.99946 -2.00000

-0.05 -1.43943 -1.95507 -2.01868 -2.02457 -2.02516

-1.00 -2.00000 -2.45743 -2.54770 -2.55964 -2.56155

-5.00 -5.37228 -5.59143 -5.67505 -5.69540 -5.70156

-10.00 -10.31570 -10.36670 -10.36670 -10.38040 -10.38520

72



TWO ELECTRONS ON FOUR SITES

Table 2.0.Ground-state energy of the Hubbard model with two electrons on four

sites as a function U (at t=1).Results are presented for four iterations, and a

comparison is made with variational results. (Enaibe and Idiodi, 2003)

U 1a

2a 3a 4a variational

20.00 -0.39231 -3.00462 -3.00762 -3.000762 -3.00762

10.00 -0.74456 -3.13274 -3.14888 -3.14895 3.14895

5.00 -1.27492 -3.29366 -3.34702 -3.34788 -3.34789

1.00 -2.37228 -3.65661 -3.77674 -3.78471 -3.78526

0.50 -2.58945 -3.75003 -3.87384 -3.88349 -3.88428

0.00 -2.82843 -3.86370 -3.98772 -3.99892 -4.00000

-0.05 -2.85354 -3.87632 -4.00015 -4.01148 -4.01260

-1.00 -3.37228 -4.16541 -4.27836 -4.29131 -4.29295

-5.00 -6.274920 -6.48329 -6.50989 -6.51315 -6.51360

-10.00 -10.74460 -10.79260 -10.79550 -10.79560 -10.79570

73



TWO ELECTRONS ON SIX SITES

U 1a

2a 3a 4a 5a

6a

20.00 -0.39231 -2.18688 -3.24601 -3.38813 -3.45268 -3.48506

10.00 -0.74456 -2.34189 -3.27026 -3.46750 -3.53138 -3.55738

5.00 1.27492 -2.56901 -3.33006 -3.55584 -3.62208 -3.64394

1.00 -2.37228 -3.12394 -3.57481 -3.76387 -3.83233 -3.85570

0.50 -2.58945 -3.25748 -3.65302 -3.82512 -3.89031- -3.91356

0.00 -2.82843 -3.41420 -375329 -3.90501 3.96491 -3.98724

-0.05 -2.85354 -3.43126 -3.76471 -3.91423 -3.97348 -3.99568

-1.00 -3.37228 -3.80626 -4.03771 -4.14272 -4.18637 -4.15811

-5.00 -6.27492 -6.38078 -6.40803 -6.41504 -6.41684 -6.41730

-10.00 -10.7446 -10.7686 -10.7710 -10.7713 -10.77130 -10.7713

74



Table 3.0.Ground-state energy of the Hubbard model with two electrons on six

sites as a function of U (at t=1).Results are presented for eight iterations, and a

comparison is made with variational results. (Enaibe and Idiodi, 2003)

V1: CONCLUSIONS

In this paper we have studied the rapid convergence to the ground state

properties of strongly correlated finite system in a single band Hubbard model.

The analysis was done using a new version of (SLT) on small lattices. We

presented results for two electrons on two, four, and six sites. The results for

these systems obtained were compared with those obtained using variational

method by Enaibe and Idiodi, 2003 and were found to be in excellent agreement.

The algorithm in our new version of (SLT) is similar in fashion to that use by

Dagotto and other researchers in this field, but I consider our approach to be more

appealing and easier to apply because of its simplicity. It was also demonstrated

that the rate of convergence is dependent on the choice of the initial trial vector

U 7a

8a Variational

20.00 -3.50243 -3.51215 -3.52520

10.00 -3.56829 -3.57288 -3.57619

5.00 -3.65101 -3.65329 -3.65437

1.00 -3.86350 -3.86608 -3.86736

0.50 -3.92166 -3.92447 -3.92594

0.00 -3.99539 -3.99834 -4.00000

-0.05 -4.00381 -4.00676 -4.00844

-1.00 -4.21056 -4.21321 -4.21489

-5.00 -6.41742 -6.41745 -6.41746

-10.00 -10.77130 -10.77130 -10.77130

75



ACKNOWLEDGMENT

The authors would like to acknowledge Dr. G. Akpojotor for his useful

discussions and supply of relevant materials for this work. We also acknowledge

Mr. Philipp Hasmann in Max Planck Institute Germany for making it easy for us

to access recent materials relevant to this work.

REFERENCES

(1) J.G Bednorz and R.A. Muller Z. Phys. B 64, 188 (1986)

(2) Chen. L. and Mei .C. Calculation Phys. Res. B. 339, 9006 (1989).

(3) C. Kane, P. Lee and N. Reed, Phys. Rev, Lett. 39, 6880.

(4) K. Binder and P.W Heemann, , Monte Carlo Simulations in Statistical

Physics (1992)

(5) E. Dagotto and A. Moreo, Phys. Rev. B.38, 5087 (1985)

(6) E. Dagotto and A. Moreo, Phys. Rev. D 31, 865 (1985)

(7) E.Dagotto,etal,Phys.Rev.B.34,167(1986)

(8) R. Haydock, V. Heine and M.J. Kelly. J. Phys. C5, 2845. (1972)

(9) G. Gross and G. Pastori Parrav, Adv. Chem Phys. LA111, 137 (1985).

(10) E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).

(11) V.S. Viswanath and . G. Muller. The Recursion method applications to

many body dynamics volume M23 (Springer Verlag, New York, 1940)

(12) C. Lanczos J. Res. Nat. Bur. Stand, 45, 255 (1990)

(13) J. Hubbard, proc. R. Soc. London, Ser. A 276, 238 (1963)

(14) E.Edison and J.Idiodi.Variational approach to Study of highly correlated

systems.Thesis (2003)

76



FIRST-PRINCIPLES INVESTIGATION OF STRUCTURAL AND

ELECTRONIC PROPERTIES OF NEW ANTIPEROVSKITE-TYPE

SUPERCONDUCTOR ZnNNi3 IN COMPARISON WITH ZnCNi3**

C. M. I. Okoye

Department of Physics and Astronomy,

University of Nigeria, Nsukka, Nigeria

e-mails: [email protected] ; [email protected]

Abstract A theoretical study of the structural and electronic properties of a new

antiperovskite-type nitrogen-based superconductor ZnNyNi3, y = 1.012 ± 0.208

has been performed on the stiochiometric compound, ZnNNi3, using the

augmented plane waves plus local orbital (APW + lo) method within the

framework of density functional theory. This is compared with the isostructural

non-superconducting ZnCNi3. The optimized structural parameters were

determined using different exchange-correlation potentials. The calculated lattice

constants are within the usual accuracy range of such calculations although the

deviations of results obtained using the genaralized gradient approximation

proposed by Wu-Cohen (WC-GGA) are the least. The electronic band structures,

total, site and orbital decomposed densities of states (DOS) were obtained and

analysed. Our electronic structure results show that in ZnNNi3, states near the

Fermi energy are dominated by Ni d and N p states. This is also the case for

ZnCNi3. The peak in the DOS due to Ni dxz, dyz in ZnNNi3 is closest to the Fermi

energy, and is about 0.21eV away from the Fermi energy compared to an energy

distance of 0.09eV away of similar peak in ZnCNi3, resulting in decreased value

of Fermi level density of states in ZnNNi3. Our results show that the

stoichiometric ZnNNi3 and ZnCNi3 are very much alike in both structural and

elastic properties but differ in electronic properties. The agreement with available

theoretical and experimental data is reasonable.

PACS Numbers: 62.20.-x, 71.15.Ap, 71.15.Mb, 71.20.-h, 74.25.Jb, 78.20.Ci

** African Journal of Physics Vol. 2, pp. 76-88 (2009)




77



1. INTRODUCTION

For some time now, since the discovery of superconductivity near 8K in

MgCNi3[1], attention has been directed on the isostructural cubic antiperovskites,

with the general formula ACNi3, where A is a group II or III element, as possible

compounds with not only high superconducting transition temperature, but other

technologically important properties. This is probably because the ternary

carbides with the cubic antiperovskite structure are known to exhibit a variety of

interesting thermodynamic, chemical and physical properties[2]. Furthermore,

MgCNi3 contains large amount of ferromagnetic nickel, and it is known that alloy

BCS-type superconductors do not involve nickel. It is, therefore, important to

investigate the properties of these antiperovskites which might help elucidate the

nature of superconductivity in them. Interest in this type of compounds has

resulted in the synthesis of many more cubic antiperovskites, some of which

include ZnCNi3[3], AlCNi3[4], GaCNi3[5], CdCNi3[6] and InCNi3[7]. It is

noteworthy that antiperovskites with trivalent metals MIII

CNi3 (namely, AlCNi3

and GaCNi3) are nonsuperconducting while InCNi3 is magnetic and also

nonsuperconducting. Over the years, the understanding of several properties of

superconducting materials including antiperovskites were provided by the result

of first-principles calculations[8-23].

Very recently, a new superconducting antiperovskite ZnNyNi3(y = 1.012 ±

0.208) with Tc ~ 3K, which belongs to this class of materials, but with carbon

replaced by nitrogen was successfully synthesized[24] and some of its properties

have been investigated. The stoichiometric compound ZnNNi3, has the same

structure with ZnCNi3 where no superconductivity was found down to about

2K[3]. ZnNyNi3 occurs in simple cubic lattice with lattice constant a = 3.756°A

and nitrogen content 1.0124±0.208 respectively[24]. Experiments to properly

determine the nitrogen content via Rietveld analysis using RIETAN-2000

program and sample weight change before and after sintering, yield nitogen

content values of y= 1.012 and 0.98 respectively. These two values indicate that

y ~ 1. The space group is m3Pm (space group No. 221). Zn occupies the corner

position (1a), and nitrogen occupies the center of the cube(1b) while the three

nickel atoms reside on the face-centered sites labelled 3c. In this study we

assume that ZnNyNi3 is likely to be stoichometric, that is, y ~ 1. To our

knowledge, no theoretical investigation of ZnNNi3 has been done.

In this paper, we present the results of a systematic study of the structural

and electronic properties of the stoichiometric form (ZnNNi3) of the new

antiperovkite superconductor ZnNyNi3 alongside that of the isostructural ZnCNi3

by using density functional theory approach as embodied in the WIEN2k

package[25]. Our study will enable us investigate the structural and electronic

properties of ZnNNi3 for the first time and compare them with that of

78



isostructural ZnCNi3 with a view to revealing any key differences in there

electronic properties that may be used to account for the absence of

superconductivity in ZnCNi3 down to 2K.

The outline of the paper is as follows, In section II, we give a brief

description of the computational procedures used. In section III, the calculated

results of the structural and electronic properties are presented and

discussed.Conclusions are drawn in section IV.

2. THEORETICAL PROCEDURE

The crystal structure of the antiperovskites considered possess the cubic

space group, m3Pm ( No. 221). In this structure, the Zn ions are at the corners,

nitrogen(carbon) at the body center, and nickel at the face centers of the cube. The

atomic positions are Zn: 1a (0,0,0); N(C): 1b (21

21

21 ,, ); Ni: 3c (

21 ,

21 , 0). In this

structure, there are six Ni atoms at the face-centered positions of each unit cell

forming a three-dimensional network of Ni6 octahedron similar to oxygen

octahedron in CaTiO3. Each N or C atom is located in the body-centered cubic

position surrounded by Ni6-octahedron cage. In order to study the structural and

electronic properties of ZnNNi3 and ZnCNi3, first-principles calculations were

performed by employing a full-potential(linear) augmented plane wave plus local

orbital (FP-(L)APW +lo)[27-29] method, based on density functional

theory[30,31] and implemented in theWIEN2k package[25]. The generalized

gradient approximations(GGA) to exhange-correlation potential of Perdew, Burke

and Ernzerhoff (GGA-PBE)[32] and Wu and Cohen (GGA-WC)[33] as well as

local density approximation(LDA)[34] were used. In this method of calculation,

no shape approximations to the electronic charge density or potential is made.

Also, the unit cell is divided into non-overlapping muffin-tin spheres centered at

atomic sites separated by an interstitial region. In the atomic sphere, a linear

combination of radial functions times spherical harmonics is used and in the

interstitial region, the basis set consists of plane waves. The basis set inside each

muffin-tin sphere is split into core and valence subsets. The core states are treated

within the spherical part of the potential only, and are assumed to have a

spherically symmetric charge density confined within the muffin-tin spheres. The

valence part is treated with the potential extended into spherical harmonics up to

4 . Also the valence wave functions inside the muffin-tin spheres are

expanded up to 10 partial waves. In this study we treat the core electrons fully

relativistically, and the valence electrons semi-relativistically.

79



In the calculations, and the muffin-tin radii are chosen to be 2.3, 1.6, 1.6,

1.8 a.u for Zn, N, C, and Ni respectively. The basis functions are expanded up to

RMTKmax equal to 8.0 (where Kmax is the plane wave cut-off and RMT is the

smallest of all the muffin-tin sphere radii). The integrations over the Brillouin

zone are perfomed via the tetrahedron method with 56 k-points in the irreducible

part of the Brillouin zone. The self-consistent calculations were considered to be

converged when the difference in the total energy of the crystal did not exceed

0.1mRy as calculated at consecutive steps. The density of states (DOS) was

obtained using a modified tetrahedron method[35].

3. RESULTS AND DISCUSSION

A. Structural properties

In order to calculate the ground state properties of these compounds, the total

energies are calculated for different volumes around the experimental cell

volume. The calculated total energies are fitted to the Birch-Murnaghan equation

of state[35] to determine the ground state properties such as the equilibrium lattice

constant a0, the bulk modulus B0 and the pressure derivative of the bulk modulus

B′. It has been reported[36] that the new exchange-correlation functional

proposed by Wu and Cohen(GGA-WC)[33] is more accurate in predicting the

equilibrium lattice constant and bulk moduli for solids significantly over both

local-density approximation(LDA)[33] and Perdew-Burke-Ernzerhof(GGA-

PBE)[31] generalized gradient approximation. For this reason, the structural

properties have been calculated using the LDA, GGA-PBE and GGA-WC in

order to test the suitability of these exchange-potential approximations in studying

the structural properties of these antiperovskites. The results of the calculations of

the structural parameters are displayed in Table 1 together with experimental data

and results of previous theoretical studies.

It is seen that the equilibrium lattice constant determined by employing

different exchange-correlation functionals exhibit, to some extent, the expected

pattern(GGA generally overestimates the lattice parameter).

80



Table 1: Calculated lattice constants a0 (in A ) bulk modulus B0(in GPa) and its

pressure derivative '

0B for ZnNNi3 and ZnCNi3 at the theoretical equilibrium

volumn compared with available experimental data and other theoretical

calculations.

3Reference 24,

4Reference 22,

5Reference 13,

6Reference 14,

7Reference 6,

8Reference 3

However, it is noteworthy that for ZnNNi3, only GGA-PBE gives a value of

equilibrium lattice constant that is greater than the experimental value. For

ZnCNi3, the equilibrium lattice constant obtained using all the exchange-

correlation schemes, are larger than the experimental value. However, in both

compounds, the level of overestimation obtained, when the GGA-WC scheme

was used, is less that that resulting from the use of GGA-PBE scheme. The

observed overestimation in ZnCNi3, even when LDA scheme was used, has been

reported by some previous studies[10,14,22] and it was thought to be probably

due to underestimation of the experimental lattice data arising possibly from

carbon defeciency. Although the experimental lattice constant aexpt(ZnNNi3) >

aexpt(ZnCNi3), the theoretically calculated equilibrium lattice constants do not

81



show the same pattern since a0(ZnNNi3) < a0(ZnCNi3). It may well be that the

deviation of the investigated sample (ZnNNi3) from stoichiometry, may also have

some effect. On the whole, although the results of the lattice constants show that

the GGA-WC is better suited for proper description of these compounds, one

should not expect a perfect agreement between the experimental lattice constant

and the GGA-WC values. This is because experimental lattice constants are

usually measured at room temperature and the effect of thermal expansion and

zero-point quantum fluctuations, which will enlarge the calculated lattice

constant, are not included in density functional schemes[37].

Furthermore, it was observed that the bulk moduli of these materials

increase in the sequence B(ZnCNi3) < B(ZnNNi3), that is, in reverse sequence to

a0, in agreement with the well known relationship[37] between B and the lattice

constant (cell volume V0, as 1

0~ VB ). This trend, where a larger lattice constant

leads to a smaller bulk modulus, has been reported for various

antiperovskites[22,23].

B. Electronic Properties

The electronic band structure and density of states of states(DOS)

calculated using the Wu-Cohen generalized gradient approximation at the

equilibrium lattice constant are shown in Fig. 1 and Figs. (2-4) respectively. The

self-consistent calculations show that the materials, ZnNNi3 and ZnCNi3, in the

normal state are typical metallic compounds.The valence and conduction bands

overlap appreciably between R − and Z − M − symmetry lines in ZnNNi3 as

well as between R− and −X−Z−M− in ZnCNi3. At the moment, there exists

a number of reports on the electronic band structure calculation for ZnCNi3[22]

but non for ZnNNi3.

82



Figure 1: Calculated electronic band structures of (a) ZnNNi3 (b)ZnCNi3. The

valence band maximum is at zero.

The overall band profiles are found to be in fairly good agreement with previous

theoretical results[22]. The general features of the bands are nearly the same

except for a few differences. For instance, in energy range shown, the band

structure for ZnNNi3, is clearly divided into two broad groups. The lowest group

extends from about -9.0eV to about -5eV with a small gap separating it from the

other group of bands with higher energy that cross the Fermi level. The bands

crossing the Fermi level are predominantly of Zn d and N p states. No such gap is

present in ZnCNi3 (Fig. 1b) between -8eV and the Fermi level. However, there are

also two groups of bands in the bandstructure of ZnCNi3. The lowest group of

bands lies between -14eV and -11eV and are predominantly of C s character.

Similar band probably due to N s are not present in ZnNNi3 in the energy range

shown(Fig. 1a) but lies further down. This lowest lying C s band in ZnCNi3 is

separated from the rest of the valence bands by a gap of about 3eV (between -

11.0eV and -8eV). In both compounds, the upper valence band consists of

predominantly hybridized Ni d and N(C) p states while the conduction bands are

dominated by mixture of p states from all the constituent atoms.

The total densities of states as well as the site decomposed contributions

for the two compounds are displayed in the two upper panels of Figs. 2. Generally

as earlier noted in the case of their band structures, the general features of the

DOS for the two compounds are quite similar. It is interesting to note that there is

83



a sharp peak in the DOS of the two compounds clo se to the Fermi level. This

peak is associated with the quasiflat bands close to the Fermi level that are

predominantly due to Ni dxz, dyz states. This has been observed as a common

feature of all the Ni-based antiperovskites[14,22,23]. However, in ZnNNi3, this

DOS peak is a little farther away(~ 0.21eV), while that of ZnCNi3 is only about

0.09eV below the Fermi level. The effect of this is seen in the reduced value of

DOS at the Fermi level for ZnNNi3. This shift of the DOS peak towards the Fermi

level, EF , from the low energy side on replacing nitrogen with carbon, is similar

to what is observed with increasing the lattice parameter a, regardless of the kind

of element A in ACNi3 (A=Mg, Zn, Al, Ga[21]).

The peak in the DOS around -7eV is due mainly to Zn d states mixed with

some N(C) p as well as Ni p states. It arises from the nearly flat bands in the band

structures of both compounds around -7eV. The intensity of this peak is more in

ZnCNi3 than ZnNNi3 Around this intense peak, there are two small structures due

to N p states in ZnNNi3, these are less prominent in the DOS of ZnCNi3. It is

notewothy to report that there is far more carbon s contribution than nitrogen s

contribution at the Fermi level. Also, the structure due to the N p contribution

which crosses the Fermi level is fairly broad in ZnNNi3. This is in contrast to the

presence of a peak due to C p states which lies on the Fermi level and arises due

to the flat bands that lie almost on the Fermi level between X-M- high

symmetry points in Fig. 1a.

The lowest panel in Fig. 2 shows a comparison of the total density of

states of ZnNNi3 and ZnCNi3 within a smaller energy panel around the Fermi

level in order to bring out the differences especially in the region close to the

Fermi level. It clearly shows the shift of Ni dxz, dyz dominated peak farther away

from the Fermi level in ZnNNi3 than in ZnCNi3.

In Fig. 3, we have ploted the main contributions to the upper valence band

DOS of the two compounds. It arises mainly from hybridization of nitrogen or

carbon p and nickel d states. A comparison with the band structure of the

compounds show that whereas the C p band is very close to the Fermi level at the

M point and causing a peak in the DOS, the N p band is flat around a wider

region(from Z-M- ) causing a structure that is more like a hump. The d states of

the two compounds around the Fermi level look almost alike except for the larger

distance away from the Fermi level in the case of ZnNNi3. The differences arise

from the N p and C p contributions which are about 12% and 7 % in ZnNNi3 and

ZnCNi3 respectively.The calculated total density of states at the Fermi level,

N(EF ), for ZnNNi3 is about 55% that of ZnCNi3. The implication of the smaller

density of states at the EF for conductivity is probably that the electrical

conductivity in ZnNNi3 is lower than in ZnCNi3.

84



Finally in Fig. 4, we have tried to mimic the effect of pressure on the

electronic structure of ZnNNi3 by plotting in the top panel, the DOS of ZnNNi3 at

the experimental lattice constant (3.756°A) as well as at a smaller latice constant,

for example, the experimental lattice constant of ZnCNi3 (3.66°A). It is observed

that the density of states obtained using the two lattice constants are nearly the

same close to the Fermi level. More noticeable differences are observed lower

down in energy where the effect of pressure clearly shifts the peaks in the DOS

further down in energy. In the lower panel of this figure(Fig.4), the DOS of

ZnCNi3 is plotted at both the experimental and theoretical lattice constants. This

simulates the effect of expansion of the lattice by about 2%. The plots indicate

that the peaks far below the Fermi level are shifted to higher energies, This trend

is in agreement with the result of lattice contraction( in the case of ZnNNi3) which

shifted the peaks downward. This shift to higher energies decreases towards the

Fermi level and results to an increase in the magnitude of the density of states at

the Fermi energy(N(EF)). Our results indicate that application of pressure(lattice

contraction) decreases the density of electrons around the Fermi level which may

reduce the electrical conductivity of the material.

85



Figure 2: Total and site decomposed density of states of (a) ZnNNi3 and (b)

ZnNNi3 at their equilibrium lattice constants. (c) Comparison of the total density

of states of ZnNNi3 and ZnCNi3, within smaller energy panel, at their equilibrium

lattice constants.

86



4. CONCLUSIONS

We have studied the recently synthesized superconducting anti-perovskite nitride

ZnNyNi3 in the stoichiometric form, ZnNNi3 as well as isostructural ZnCNi3

using first-principles APW+lo method in order to compare their structural and

electronic properties.

Figure 3: Density of states calculated at the equilibrium lattice constant (a)

ZnNNi3 and (b)ZnCNi3 showing the nitogen p and nickel d contributions around

the Fermi level.

Figure 4: Total density of states for (a) ZnNNi3 at the experimental lattice

constant of ZnNNi3 (3.756°A) and ZnCNi3 (3.66°A) (b) Total density of states of

ZnCNi3 at the experimental(3.66°A) and theoretical(3.728°A) lattice constants.

Our calculations show that their structural properties are very similar. The

band structure plots also show a metallic character in ZnNNi3 as in ZnCNi3. The

87



width of the valence band in ZnNNi3 extends from about 4.0eV below the Fermi

level and this is smaller than a bandwidth of about 8eV observed in ZnCNi3. The

presence of C s states in the region between 11.0 and 14.0 eV below the Fermi

level in ZnCNi3 as well as the greater relative concentration of N p states in

comparison to C p states in ZnNNi3 and ZnCNi3 respectively at the Fermi level,

are some of the major differences in the electronic structure of these compounds.

The DOS at EF in the new superconducting antiperovskite is suppressed since it is

about half the value in ZnCNi3. This is probably due to the shift farther away from

below the Fermi level by the peak in the DOS of ZnNNi3. It is suggested that this

decrease in DOS at the Fermi level might also contribute to the relatively high Tc

that is observed in the newly synthesized first antiperovskite nitride

superconductor, ZnNNi3, in contrast to the analoguos compound, ZnCNi3, which

has not been found to be superconducting down to 2K.

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89



CdS THIN SOLID FILMS FOR PHOTOVOLTAIC

APPLICATION††

S.C. Ezugwu*, P.U. Asogwa, R.U. Osuji, F.I. Ezema, B.A. Ezekoye,

A.B.C. Ekwealor

Department of Physics and Astronomy, University of Nigeria, Nsukka

*E-mail: [email protected]

Abstract

Nanocrystalline ternary thin films composed of TlS-CdS and PbS-CdS were

deposited by a simple and inexpensive chemical bath deposition technique within

the pores of polyvinyl alcohol. The films were studied for possible application in

photovoltaic architecture. By characterizing the films using x-ray diffractometer,

scanning electron microscope and UV-VIS spectrophotometer the optical band

gap energy, calculated from the absorption spectra, was found to be in the desired

interval to be used as solar absorber material for photovoltaic fabrication.

Keywords: Band gap energy, CBD, photovoltaic, ternary thin film

1. INTRODUCTION

Solar energy is one of the most convenient non-conventional energy

resources to be considered for the power requirements of the 21st century. The

studies of semiconductor nanoparticles have shown that they exhibit novel optical

properties. These unique properties led to the appearance of many new application

areas, such as their use in solar cell, photodetectors, light-emitting diodes and

switches [1,4].

Energy conversion in solar cell consists of generation of electron-hole

pairs in semiconductors by the absorption of light and separation of electrons and

holes by an internal electric field. Charge carriers collected by two electrodes give

†† African Journal of Physics Vol. 2, pp. 89- 99 (2009)



90



rise to a photocurrent when the two terminals are connected externally. The

spectrum of solar light energy spreads from the ultraviolet region (300nm) to the

infrared region (3000nm). When the photon energy is less than band gap of the

semiconductor, the light is transmitted through the material, that is, the

semiconductor is transparent to the light. When the photon energy is larger than

band gap, the electrons in the valence band are excited to the conduction band. It

means that a photon is absorbed to create an electron-hole pair. This process is

called intrinsic transition or band-to-band transition.

A heterojunction is formed by joining two layers of semiconductors with

differing band-gap energies. When the layers have the same conductivity type an

isotype heterojunction is formed, whereas in an anisotype heterojunction, the

layer conductivity type differs. The requirement to get appropriate band-gap

energies for device application has led to the development of binary, ternary and

quaternary thin films [3-10].

Cadmium sulphide (CdS) is one of the most promising II-IV compound

materials because of its wide range of application in various optoelectronic, piezo-

electronic and semiconducting devices [11, 12]. High efficiency thin film solar

cells have been achieved using two types of structures: SnO2:F/CdS/CdTe and

ZnO/CdS/CuInSe2 [13]. In these devices, the systems SnO2:F/CdS and ZnO/CdS

act as optical windows and the CdTe and CuInS2 act as absorbent layers. The

highest efficiency in CdTe and Cu(InGa)Se2-based solar cells has been archived

using CdS films deposited by chemical bath deposition process[14, 15]

In this paper, we report the chemical bath deposition of ternary thin films

and the analysis of the band-gap energies and the optical transmission for possible

use in solar cells and other applications.

2. EXPERIMENTAL DETAILS

2.1 Preparation of TlS-CdS thin film

Thin film of TlS was deposited on clean microscope glass slide by using

5ml of 0.2M TlNO3, 4ml of 1M C3H4(OH)(COONa)32H2O, 4ml of 1M (NH2)2CS

and 34ml of PVA solution put in that order in 50ml beaker. The PVA solution

used in this work was prepared by adding 900ml of distilled water to 1.8g of solid

PVA (-C2H4O)n (where n=1700), and stirred by a magnetic stirrer at 90oC for

1hour. The homogenous solution was aged until the temperature drops to 25oC.

The deposition was allowed to proceed at room temperature for 90mins after

which the coated substrate was removed, washed well with distilled water and

91



allowed to dry. The glass-TlS system was used as the substrate for the deposition

of CdS film. The bath for the chemical deposition of CdS was composed of 3ml

of 1M CdCl2, 5ml of NH3 solution, 10ml of 1M (NH2)2CS and 35ml of PVA

solution. The deposition time was 360mins. The film was again rinsed thoroughly

with distilled water and allowed to dry. The deposited TlS-CdS thin film was

annealed in an oven at 1000C for 60mins.

2.2 Preparation of PbS-CdS thin film

The chemical bath deposition of PbS thin film on clean microscope glass

slide was achieved by using 10ml of 0.1M Pb(NO3)2, 5ml of 1M NaOH, 10ml of

1M (NH2)2CS and 25ml of PVA solution. The deposition proceeded at room

temperature and lasted for 60mins. The deposited glass substrate was then

removed, rinsed with distilled water and allowed to dry. The formation of PbS-

CdS thin film was achieved by using the procedure described in section 2.1

above. The deposited PbS-CdS thin film was again annealed in an oven at 1000C

for 60mins.

2.3 Thin film characterization

The samples were characterized with SEM, XRD and UV-VIS

Spectrophotometer. Optical properties of chemical bath deposited TlS-CdS and

PbS-CdS thin films were measured at room temperature by using a double beam

Perkin-Elmer UV-VIS Lambda 35 spectrometer. Optical band-gaps were

calculated from the absorption spectra. X-ray diffraction (XRD) is an efficient

tool for the structural analysis of crystalline materials. The XRD patterns for the

samples were recorded using D/max-2000 Rigaku powder X-ray diffractometer in

the 2θ range 200 - 80

0 using CuKα radiation of wavelength λ = 1.5408Ǻ. The grain

size of the deposited films was viewed by using scanning electron microscopy

(SEM) technique.

3. RESULT AND DISCUSSION

Figure 1a - b show the XRD patterns of TlS-CdS and PbS-CdS thin films

deposited in this work. The patterns were recorded after annealing the samples at

150oC for 1hr. This was done to improve upon the intensities of the

peaks/crystallinity of the films. The parameters of interest from XRD for both

samples are displayed in tables 1 and 2.

92



Table 1: Obtained result from XRD for PbS-CdS thin film

2 d-value FWHM I/Io

26.18 3.4011 0.071 75

26.28 3.3884 0.212 100

30.30 2.9474 0.259 88

43.36 2.0851 0.118 43

43.42 2.0824 0.118 46

Table 2: Obtained result from XRD for TlS-CdS thin film

2 d-value FWHM I/Io

21.50 4.1297 0.071 18

25.46 3.4956 0.071 20

25.58 3.4795 0.094 18

29.56 3.0194 0.259 100

39.20 2.2963 0.141 17

39.42 2.2839 0.165 20

44.74 2.0239 0.141 16

93



The peaks at 2 values of 26.28o and 44.74

o are attributed to cubic CdS (JCPD

card No 80-0019) [1], having lattice parameters a=b=c= 5.811Å. These were

assigned to the diffraction lines produced by (111) and (220) planes. However,

the additional peaks at an angle of 26.18o and 30.30

o are identified to be of PbS

(JCPD card No 78-1901), and assigned to the diffraction line produced by (111)

and (200) planes of the PbS cubic phase (galena) [2,3]. Similarly, the XRD

pattern at 2 values of 25.58o and 29.56

o are identified to be TlS (PDF No 43-

1067) [4]. These were assigned to the diffraction line produced by (022) and (202)

planes. These results suggest that each of the thin films deposited in this work is a

mixture of binary chalcogenides ( i.e. PbS-CdS and TlS-CdS)

The average crystallite size of the films was calculated from the recorded

XRD patterns using Scherrer formula:

D = 0.89 λ/β cos θ

Where D is the average crystallite size, λ is the wavelength of the incident X-ray,

β is the full width at half maximum of X-ray diffraction and θ is the Bragg‘s

angle.

Fig.1a: X-ray diffractogram of TlS-CdS thin film

0

50

100

150

200

250

300

350

400

450

500

10 20 30 40 50 60 70 80

94



Fig 1b. X-ray diffractogram of PbS-CdS thin film

0

50

100

150

200

250

300

350

10 20 30 40 50 60 70 80

The average crystallite size for the thin film of TlS-CdS and PbS-CdS were found

to be 11.3nm and 11.4nm respectively.

The scanning electron micrographs of TlS-CdS and PbS-CdS thin films

reported here are shown in figure 2a-b. From the micrographs, it is observed that

the films are uniform throughout all the regions: the films are without pinhole or

cracks. We clearly observe the small nanosized grains engaged in a flower-like

structure, which indicates the nanocrystalline nature of the films.

The optical absorption spectra of the films deposited onto glass substrate

were studied in the range of wavelengths 200 – 1100nm. The variation of

absorbance (A) and transmittance (%T) with wavelength for the two samples

Fig.2 SEM of (a) TlS-CdS thin film (left); (b) PbS-CdS thin film (right)

95



under study are shown in fig 3 and 4 respectively. Thin films of PbS-CdS show

good absorption in the visible spectrum and a lower absorbance values in IR

region of the solar spectrum. The plot in figure 3 also reveals that TlS-CdS thin

film has high absorbance values in the IR region and virtually non-absorbing in

the UV-VIS.

Fig.3. Absorbance vs. wavelength for TlS-CdS

& PbS-CdS thin films

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

200 400 600 800 1000Wavelength (nm)

Abs

orba

nce

TlS-CdS

PbS-CdS

96



The transmittance plot in figure 4 shows that the films transmit well in the

wavelength range opposite to that of the absorbance. In order words, films that

absorb well in the IR region transmit poorly in the same region. The spectral

absorbance and transmittance displayed in figs. 3 and 4 show that some of the

films deposited in this work could be used as spectrally selective window coatings

in cold climate to facilitate transmission of VIS and NIR while suppressing the

UV portion of solar radiation. The films can be used for coating eyeglasses for

protection from sunburn caused by UV radiations.

The details of the mathematical determination of the absorption coefficient

(α) can be found in literature [17, 18] while the plots of absorption coefficient

against photon energy is shown in fig. 5

Fig. 4. Transmittance VS. wavelength for TlS-CdS and PbS-CdS

97



Fig. 5. Absorption coefficient vs. photon

energy for TlS-CdS & PbS-CdS thin films

0

1

2

3

4

5

6

0 1 2 3 4 5Photon Energy (eV)

a x

10

6m

-1PbS-CdS

TlS-CdS

These absorption spectra, which are the most direct and perhaps the

simplest method for probing the band structure of semiconductors, are employed

in the determination of the energy gap, Eg. The Eg was calculated using the

following relation [17-19]: α = A(hν - Eg)n /hν,

Where A is a constant, hν is the photon energy and α is the absorption coefficient,

while n depends on the nature of the transition. For direct transitions n = ½ or ⅔,

while for indirect ones n = 2 or 3, depending on whether they are allowed or

forbidden, respectively. The usual difficulty in applying this concept to

polycrystalline thin films with nanometer-scale crystalline grains is the size

distribution of grains and consequent variation in the band gap due to quantum

confinement effects. Thus the straight-line portion may not extend beyond a few

tenths of an electronvolt, and hence value of the band gap could turn out to be

very subjective [20]. The best fit of the experimental curve to a band gap

semiconductor absorption function was obtained for n = ½. The calculated values

of the direct energy band gap, from fig.6 are 1.4eV and 1.2eV for TlS-CdS and

PbS-CdS respectively.

98



A material with a direct band gap of about 1.5eV and a high absorption

coefficient

Fig.6. (h)2 VS. h

0

5

10

15

20

25

30

0 1 2 3 4

h (eV)

( h

)2

PbS-CdS

(1.2eV)

TlS-CdS

(1.4eV)

of more than 104cm

-1 has been regarded as a promising absorber for thin film

photovoltaic applications [16]. The low band gap values exhibited by these films

together with high absorbance in the VIS make the films suitable for use as

absorber material in solar cell application. For laser diode application, the band

gap energies should essentially lie in the range of 0.9 to 1.5eV. While band-to-

band radiative recombination is favored in direct band gap materials, the band gap

energy controls the emission wavelength: λ ≈ 1.2 /Eg. [1]. Hence these films could

also be used for fabrication of laser diodes.

4.0 CONCLUSION

Chemical bath deposition technique has been successfully used to deposit

ternary thin films of TlS-CdS and PbS-CdS. Their optical band gaps, which lie

within 1.2 and 1.4eV, are in the desired interval to be used as absorber materials

for solar cell fabrication.

99



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2. H. weller, Chem. Int. Edn. Engl. 32 (1993) 41

3. M.T.N. Nair, Y. Para, J.Compos, V.M. Garcia, P.K. Nair; J. Electrochem.

Soc.145 (1998) 6

4. R. Suarez, P.K. Nair; J. of Solid State Chemistry 123 (1996) 296-300.

5. Y. Rodriguez-Lazcano, M.T.S. Nair, P.K. Nair; J of Crystal Growth 223

(2001) 399-406

6. A. Abu El-Fadl, Galal A. Mohamad, A.B. Abd El-moiz, M. Rashad; Physica

B 366 (2005) 44-54

7. F.I Ezema, R.U. Osuji: FIZIKA A(Zagreb)16 (2007)2, 107-116

8. F.I Ezema, R.U Osuji; J of Allied Sciences 6 (8) (2006) 1827-1832

9. S. Jana, R. Thapa, R Maity, K.K Chattopadhyay; Physica E (2008),

doi:10.1016/j.physe.2008.04.015

10. S.C Ezugwu, F.I. Ezema, R.U. Osuji, P.U. Asogwa, A.B.C. Ekwealor, B. A.

Ezekoye; Optoelectronics and Advanced Materials-Rapid Comm. 3

(2009)141-144.

11. Y.Iyechia; SPIE Opt. Comput. 88 (2000) 103

12. V. V. Stefko; Sov J. Commun. Techno. Electron 36 (1991)

13. C.S. Ferekids, D. Mariskiy, B. Tesali, D. Oman and D.L. Morel; 25th

IEEE

PVSC (Washington !996), P.751

14 M. Contreras, K. Ramanathan, F. Hason abd R. Nuofi; Progress in

Photovoltaic 7, 311 (1999)

15 J. Britt and C. Ferekids; App. Phys. Lett. 62, 2851 (1993)

16 Watanabe, M. Matsui; J. Appl. Phys. 38 (1999) 1379-1382

17 F.I. Ezema; Turk J. Phys. 29 (2005) 105

18 I.C. Ndukwe; Sol Ener. Cells 40 (1996) 123

19 V. Estrella, M.T.S. Nair, P.K. Nair; Semicond. SC. TEchnol.; 18 (2003) 190

20 V. Estrella, M.T.S. Nair, P.K. Nair; Thin Solid Films 414 (2002)289-295

100



SYNTHESIS AND CHARACTERIZATION POTASSIUM

PERCHLORATE SINGLE CRYSTAL IN SILICA GEL‡‡

Uchechukwu .V. Okpala

Emeagwali Centre for Research on Renewable, Energy and Material Science

(ERREMS). Anambra State University of Science & Technology, Uli,

Anambra State. E-mail: [email protected]

Abstract

In this work, Sol gel method of thin film growth/deposition is used in growing

crystals of potassium per chlorate (KClO4). Impurities of locally produced

materials were added to see how they permeate into the fabrics of the

aforementioned crystals and affect the optical properties. The spectral analysis of

the said growth was carried out to enable us determine their properties.

Key Words: Sol-gel, optical characterization, silica model, KClO4 crystals and

band gap.

1. INTRODUCTION.

Mobilizing Physical Science based enterprise is a collective responsibility

between the government and the governed. Object oriented projects should be

adequately funded. In order to carry object oriented project we must imitate the

nature. Under the sun, everything has a small beginning. The building of nature

from small beginning is conspicuously found in crystal growths/ depositions, with

little manipulation on the constitution and arrangements of the molecules of these

materials new dimensions emerge. In the recent times, Sol-gel growth/deposition

of thin films has been found a veritable asset to materials scientists and solid

state industries.In this work crystals of potassium perchlorates were grown with

the addition of impurities of a local material (Bamboo). One reason for adopting

‡‡ African Journal of Physics Vol. 2, pp.100-114, (2009)



101



sol-gel technique is the easy control over film deposition and easy fabrication of a

large area thin film with low cost [1]

Potassium Perchlorate Crystal

The crystals of potassium perchlorate (KClO4) are colourless rhombuses

which are slightly soluble in water [2]. The solubility at 0oc is 0.75gm per 100gm

of water and is less soluble in aqueous ethyl alcohol. It is used in the separation of

the former and acts as a reagent, oxidising agent, pyrotechnic ( i.e. the

manufacture of fire works) antipyretic ( i.e. a drug reliving fever), sedative (B.P)

and source oxygen.

Bamboo

Bamboo is one of the most marvellous plants in nature. Some giant species

of bamboo grow up to 1.22 meters in 24hrs. Bamboo is stronger than wood or

timber in tension and compression. The tensile strength of the fibres of vascular

bundles could be up to 12,000Kg/cm2 , almost that of steel[3]. Chemical analysis

reveals that bamboo has about 1.3% ash, 4.6% ethanol- toluene, 26.1% lignin,

49.7% cellulose, 27.7% pentosan [4]. In spite of the strength and hardness of the

giant bamboo culm wall, the culm can easily be cut in few minutes, even with a

stone axe if we know the exact place of the internode. In Hiroshima, Japan, the

only plant which survived the radiation of the atomic bomb in 1945 was a bamboo

plant [5]

Theroretical Considerations

Crystalline and amorphous semi-conductors, near the fundamental absorption age

there is the dependence of the absorption coefficient on the photon energy. In high

absorption regions the form of the absorption coefficient with photon energy was

given in more general term by [6,7] as

n

gEhAh )( (1.0)

Where is the frequency of the incident photon, h is the Planck‘s constant, A is a

constant, gE is the optical energy gap and variable n has the Value 2 for direct

allowed transition, ½ for indirect allowed.

102



When the linear portion of nh )( as a function of hf is extrapolated to α = 0, the

intercept gives the transition band gaps. For semi-conductors and insulators

(where K2 <<n

2) there exist a relationship between R and n given by [5,6]

22 )1/()1( nnR (1.1)

There is also a relationship between k and α given by [6,8].

4/k (1.2)

where α = absorption coefficient of the film, = wavelenght of electromagnetic

wave and k is the absorption coefficient. The relationship between E and K is

given by [6]

2)( iknEEE ir (1.3)

where rE and iE are real and imaginary parts of E respectively.

Optical conductivity 0 is given by [6,8]

4/0 nc (1.4)

where c is the speed of light.

2. EXPERIMENTAL DETAILS

2.1. Growth of Potassium Perchlorate Single Crystal in Silica Gel

The experiments were conducted in 100ml beakers. Twenty five millilitre

(25ml) of sodium silicate solution of pH greater than eleven and with specific

gravity 1.04 was placed in a test- tube some quantities of IN of perchloric acid

were added to the sodium silicate solution to form a gel. The pH of the mixture

was set at 5.0. The gel was allowed to set at room temperature for a period of 5,

15 and 25 days, after which a feed solution of potassium chloride (KCL) was

placed above the gels for crystallization, potassium chloride (KCL) of different

normalities (0.5N, 0.8N and 1.8N) was used.

The chemical reaction which took place in the gel medium is represented as

KCL + HCLO4 — KCLO4 + HCL ------2 .0

103



Immediately after the addition of the feed solution, some quantities of bamboo

were added as impurities and their effects investigated. The effect of

concentration of the feed solution was also investigated.

2.2. Drying

The samples were first treated with all glass distilled water to avoid

impurities and make it slurry before it was introduced into a buckner funnel

covered with filter paper then attached to a suction flask connected to the vacuum

pump through its nozzle. Once the pump is on it will create a vacuum that allows

for the absorption of H20 from the sample. The filter in the buckner funnel

prevents the solid from being sucked. The sample is then taken to the oven at an

appropriate temperature of 1040C for 30mins. After which it is placed inside the

desiccator to maintain dryness. CaCL2 was used as a desiccant.

3.0. RESULTS AND DISCUSSIONS.

The result of the spectral analysis of KClO4 grown by sol-gel method after 5, 15

and 25 days of ageing are shown in figures 1 to 6 below.

3.1. Analysis of the Effect of Impurity on KCLO4.

The plot of fig.1 shows that all the samples absorb poorly in both VIS and NIR of

the solar spectrum. It also shows that absorbance decreases with increase in the

concentration of bamboo.

Fig. 1 Absorbance vs. wavelength of KClO4 with different quantity of Bamboo

-0.05

0

0.05

0.1

0.15

0.2

200 300 400 500 600 700 800 900 1000 1100 1200

Wavelength (nm)

Ab

sorb

an

ce

A

B

C

Fig.1. Absorbance vs. wavelength in KClO4

104



The misbehaviour of C perhaps may be a function of growth/deposition

conditions.

In figure 2, all the films are transmitting well in both VIS and NIR. This

is an ideal property for solar control applications and could also be used in arctic

region to allow IR warm the rooms and reduce the cost of warming rooms by

conventional means.

Fig. 2. Transmittance vs. wavelength of KClO4 with different quantity of Bamboo

0

20

40

60

80

100

120

140

0 200 400 600 800 1000 1200Wavelength (nm)

Tra

nsm

itta

nce (

%)

T*100 A

T*100 B

T*100 C

Fig.2. Transmittance vs. wavelength of KClO4

Fig.3. Reflectance vs. wavelength of KClO4 with different quantity of Bamboo-2

0

2

4

6

8

10

12

14

16

0 200 400 600 800 1000 1200Wavelength (nm)

Refle

cta

nce (

%)

R*100 A

R*100 B

R*100 C

Fig.3. Reflectance vs wavelength of KClO4 with different

quantities of Bamboo.

105



The variation of Reflectance with wavelength for the sample under investigation

is shown in fig.3. All the three samples show very low reflectance in both VIS

and IR of solar spectrum. This makes the crystal ideal material to be used as anti-

reflection coating in solar cell architecture.

The plot of absorption coefficient vs. photon energy in fig 4 shows that the

absorption coefficient increases from 0.1 to 0.35 for samples A and B in the VIS

and decrease towards the NIS. Sample C has negative absorption coefficient. The

variation in the band gap plot against photon energy as in fig. 5 reveals that the

band gaps lie between 2.00 and 2.09eV. No band gap existed in C The plot of

refractive index (n) against photon energy as seen in fig.6 reveals that the

refractive index increases from VIS to NIR regions.

Fig.4. Absorption coefficient vs.photon energy for KClO4

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Photon energy (eV)

a*

10

6m

-1

a*106 A

a*106 B

a*106 C

Fig.4. Absorption coefficient vs. photon energy in KClO4

106



Fig.5. Band gap vs photon energy

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.5 1 1.5 2 2.5 3 3.5

hv (ev)

(ah

v)2

ahn^2 A

ahn^2 B

ahn^2 C

Fig.5. Band gap vs. photon energy

Fig 6: Refractive Index vs. photon energy for KClO4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Photon energy (eV)

Ref

ract

ive

ind

ex

A

B

C

Fig.6. Refractive index against photon energy for KClO4

107



4.2 Analysis of the Effect of the Concentration of the Feed Solution (KCL) on KCL04.

In fig. 1a, all the samples absorb poorly both VIS and NIR of the solar

spectrum. A comparison with the plot for bamboo doped KClO4 shows that the

samples without bamboo impurity have higher absorbance.

Fig.1 Plot of Absorbance vs wavelength (nm) in KClo4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 200 400 600 800 1000 1200

Wavelength (nm)

Ab

sorb

ance

D

E

F

Fig.1a the plot of absorbance vs. wavelength in KClO4

The plot of transmittance against photon is shown in fig 2a; all the

samples are highly transmitting in the VIS and NIR regions. However, the

transmittance increases with the wavelength. In the NIR, the least transmitting

sample has a transmittance of 70% and above. This indicates that the films can be

used for solar controlling coating. They would also allow good passage of infra

red and as such could be used in cold regions. Their use can also be employed in

coating brooder roofs to allow infra red warm the chicks.

108



Fig.2. Plot of transmittance vs.wavelength in KClo4

0

10

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200

wavelength (nm)

Tran

smit

tan

ce (

%)

D

E

F

Fig.2a. The plot of transmittance vs. wavelength in KClO4

The variation of reflectance with wavelength is shown in fig 3a. The reflectance

decreases with wavelength. It also decreases as the concentration KCl solution

increases. This shows that the materials have poor anti reflection capability.

109



Fig.3. Plot of Reflectance vs.wavelength in KClo4

0

5

10

15

20

25

0 200 400 600 800 1000 1200

wavelength (nm)

Re

fle

ctan

ce (

%)

D

E

F

Fig.3a. The plot of reflectance vs. wavelength in KClO4

The variation of band gap against photon energy is plotted in fig. 4a, the band gap

energy of the samples lie between 2.0 and 2.7. The band gap energy increases

with increase increasing concentration of KCl solution.

110



Fig.4. Band gap vs. photon energy on the effect of the conc.of KCl

on KClO4

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4

hv (eV)

(ah

v)2

D(0.5M)

E(0.8M)

F(1.8M)

Fig.4a. Band gap vs. photon energy on the effect of the concentration of KCl

on KCLO4

The plot of absorption coefficient (α) vs. photon energy (hv) is shown in fig 5a.

The absorption coefficient is high in the VIS region and decreased towards NIR.

It decreases with increase in the concentration of the KCl solution.

111



Fig.5. Absorption coefficient vs.photon energy in feed KClO4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Photon energy (eV)

a*1

06

m-1

D

E

F

Fig.5a. The plot of absorption coefficient vs. photon energy.

The variation of Refractive index vs. photon energy is shown in fig 6a. The

refractive index is high in the VIS region and decreased towards the NIR region.

112



Fig.6a. the refractive index vs. photon energy in KClO4

TABLE OF VALUES.

Table 1.

Specimen Amount of

sodium silicate

Na2Si03(g)

Amount of

KClO3

pH Concentration

of KCl(N)

D 25.0 Some

quantity

0.04 0.5

E 20.0 „‟ 0.8 0.6

F 20.0 „‟ 4.35 1.8

Fig.6. Refractive index vs.photon energy on feed KClO4

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Photon energy (hv)

Re

frac

tive

ind

ex

(n)

D

E

F

113



Table 2.

Bamboo Doped KClO4

Specimen Amount of PH

sodium silicate

NaS203(g)

Amount of

KClo3

pH Concentration

of Kcl(N)

A 25 Some

quantity

5.03 0.10

B 25 „‟ 5.06 0.15

C 20 „‟ 5.08 1.18

4.0. CONCLUSION

Sol-gel deposition technique has been successfully used to grow ternary films of

impurity doped potassium perchlorate crystals. Their optical band gaps lie

between 1.5 and 2.7. This shows that the films have wild band gaps and can be

used in high power, high temperature, and high frequency and short- wavelength

devices [10] in addition to their pyrotechnic and antipyretic functions [2].

114



REFERENCES

(1). C.J. Brinker and G.W. Scherer. Sol-Gel Science. Academic Press, New

York, (1990).

(2). A.R. Patel and A . V. Rao. Nucleation and Growth of the Potassium Per

chlorate (KCLO4) Single Crystal in Silica Gel. Journal of Crystal Growth,

New York, (1978).

(3) O.H. Lopez. Manual de Comctrauccion con Bamboo, CIBAM. Universidad

National de Colombia, (2001).

(4). Li, Xiaobo. Physical, Chemical and Mechanical Properties of Bamboo and

its Utilization Potentials for Fiberboard Manufacturing. M.Sc. Thesis, M.S.

Chinese Academy of Forsestry, (2004).

(5) D. Debore and K. Baries. Bamboo Building and Culture, the Architecture of

Simon Valez. Colombia, (2004).

(6). F.I. Ezeama and P.U Asogwa. Preparation and Optical Properties of

Chemical Bath Deposited Beryllium chloride ( BeCl2) Thin Film. Pacific

Journal of Science and Technology, 5(1) 33, (2004).

(7). M. Janar, D.D.Alfred, D.C. Booth and B.O. Seraphin. Optical Properties and

Structures of Amorphous Silicon Film Prepared by C.V.D. Sol. Ener. Mater,

(2003).

(8). J. I. Pankove. Optical Process in Semi Conductors, Prentice Hall, New

York, (1971).

(9). I.M Tsidilkovsk. Band structure of Semiconductors. Pergamon Press,

Oxford, (1982).

(10). B. G. Yacobi. Semiconductor Materials. Kluwer Academic Publishing, New

York, (2003).

115



DETERMINATION OF RADIONUCLIDES, CONCENTRATIONS, AND DIFFERENTIATING FACTORS FOR SOME BIOLOGICAL

SAMPLES BY NEUTRON ACTIVATION ANALYSIS (NAA)§§

S.O. Yunus a* S.A. Jonah b K.J. Oyewumi a

aDepartment of Physics, University of Ilorin, Ilorin-Nigeria.

bCenter for Energy Research and Training, Ahmadu Bello University, Zaria-Nigeria.

E-mail: [email protected]

Abstract

The analysis of some biological samples was performed by the use of Neutron

Activation Analysis (NAA). This research was carried out using Nigeria Research

Reactor one (NIRR-1) facility in the Center for Energy Research and Training,

Ahmadu Bello University, Zaria (CERT).The aims of the research are to

determine radionuclides, the concentrations, and to find the differentiating factors

for those analyzed samples. It was deduced from the result obtained that the

radionuclides of some elements with their concentrations were present in each of

the biological samples and from same result, we also concluded that the use of the

sample ZRS2A2 which is African processed locust beans (also known as irú)

should be adopted as food seasonings (i.e. spice) instead of the sample ZYCAB2

(i.e. seasonings) owing to what they contained. And again sample ZRS8A3 which

is known as Pumpkin leaf should be using as vegetable in food in place of the

sample ZRS6B3, that is, African spinach because of its significance.

Keywords: spices, vegetables, pumpkin leaf, processed locust beans, seasonings,

spinach, biological samples, Indian hemp, tobacco, and NAA.

1. INTRODUCTION

Neutron activation analysis (NAA) is a sensitive analytical technique useful for

performing both qualitative and quantitative multi-element analysis of major,

minor, and trace elements in samples from almost every conceivable field of

scientific or technological interest and it was discovered in 1936 by Hevesy and

§§ African Journal of Physics Vol. 2, pp 115 - 131 , (2009)



116



Levi, (Vértes et al. 1998, Soete et al. 1972, Das et al. 1989, Jonah 2001). For

many elements and applications, NAA offers sensitivities that are superior to

those attainable by other methods, on the order of parts per billion or better. In

addition, because of its accuracy and reliability, NAA is generally recognized as

the "referee method" of choice when new procedures are being developed or

when other methods yield results that do not agree. Worldwide application of

NAA is so widespread and it is estimated that approximately 100,000 samples

undergo analysis each year (Alfassi Z. B., 1998).

The basic essentials required to carry out an analysis of samples by NAA are a

source of neutrons, instrumentation suitable for detecting gamma rays, and a

detailed knowledge of the reactions that occur when neutrons interact with target

nuclei.

2. THE NAA METHOD

The most common type of nuclear reaction used for NAA is the neutron capture

or (n, γ) reaction, is illustrated in Figure 1. When a neutron interacts with the

target nucleus via a non-elastic collision, a compound nucleus forms in an excited

state. The excitation energy of the compound nucleus is due to the binding energy

of the neutron with the nucleus. The compound nucleus will almost

instantaneously de-excite into a more stable configuration through emission of

one or more characteristic prompt γ-rays. In many cases, this new configuration

yields a radioactive nucleus which also de-excites (or decays) by emission of one

or more characteristic delayed γ-rays, but at a much slower rate according to the

unique half-life of the radioactive nucleus (Pollard A. M., Heron C., 1996).

Depending upon the particular radioactive species, half-lives can range from

fractions of a second to several years.

117



3. INSTRUMENTATION AND METHODOLOGY

Nuclear reactors are the most important neutron sources because of the high stable

neutron fluxes and sample irradiation volumes available. More than 300 research

reactors with NAA capability are operational world-wide such as NIRR-

1(Nigeria) i.e. Nigeria Research Nuclear Reactor one (shown below) in which the

analysis was done provide suitable neutron fluxes (1x1010

– 1x1012

n.cm-2

.s-1

) for

most biological applications of NAA.

Figure 1: Diagram illustrating the process of neutron capture by a target

nucleus followed by the emission of γ-rays.

118



Reactor Vessel

Pool Water

Reactivity regulator

CR Fully Withdrawn

Shim Tray

Annular Be Reflector

Core with Fuel Pins

Bottom Be Reflector

Al Support Structure

Slant Tube

Typical NAA methodology is to irradiate samples (which can be solid or liquid)

and elemental standards (or a monitor in the Ko method) for a time determined by

the half life of the radionuclide or radiological considerations and the composition

of the sample. Unwanted short-lived nuclides are allowed to decay for a

predetermined period and the γ-ray spectra are recorded on a Ge detector (shown

in the diagram below) coupled to a computer (or a multichannel analyzer). Γ-ray

spectra of irradiated biological samples are typically complex (up to several

hundred γ-ray peaks); hence highly stable electronics and corrections for losses at

high count rates (Westphal, 1992) are required to achieve the required high energy

resolution (typically 1.5-2.0KeV at 1333KeV).

Figure 2: Research Reactor in NIRR-1 laboratory

119



.

Sample Preparation

Samples preparation took place in the NIRR-1 preparation of sample laboratory.

Apparatus:

The most important apparatus used are; Air blower; Disposable gloves; Polythene

bags; Vial (small container); Distilled water; Analytical balance;

Procedures: The preparation of sample started with crushing of sample from granulated form

to powdery form.

The following are the steps followed:

Figure 3: Detector coupled with multi-channel analyzer

120



Disposable glove was used first to avoid contamination of sample through

sweating;

Agate mortal was washed with water and then clean up with cotton wool

wet with acetone to remove any form of impurity;

Brush is then used to remove any particle in the sieve and clean with just

ordinary cotton wool;

After the precaution has been taking, each sample was crushed and little

quantity of the sample was used;

Each sample was then taking for weighing using analytical balance called

Mettlar EA 240 and simultaneously the weight of the samples were

recorded for further calculation;

After that, each sample was wrap up with polythene bags with the use of a

air blower for sealing (Filby R. H., 1995);

The samples were then pack into a vial and sealed which then ready for

irradiation.

Irradiation: This is the bombardment of target sample with flux of neutrons leading to

emission of γ – rays

Type of Irradiations

Typically two irradiations are performed using NIRR-1 facilities:

One to determine short-lived radionuclides, and

One for long-lived radionuclides

For activation analysis to produce short lived radionuclides, irradiation time is

set to 5mins. The radionuclides produced are shown in the result. Counting

period of 10mins (or 600s) is used to determine the elements present in the

sample.

Germanium γ-counting is done in two separate ways for short lived

irradiation:

First short counting, and

Second short counting.

.First short lived counting is the one which take place instantly after

sample irradiation and it is last for exactly 10mins in which some elements

121



will be detected while

Second short counting is the one that occur after 3 or 4 hours of sample

irradiation. This is done just because of interference in the peaks and by

the time the second count is taking, the elements that did not show in the

first counting will appear, (i.e. the elements having half lives of more than

10mins).

For activation analysis to produce long lived radionuclides, irradiation

time is set to at least 6hrs.

Γ-counting procedures for long-lived radionuclides are also divided into

two:

First long-lived counting which occur after 3 or 4 days of sample

irradiation. The counting time is exactly 1800s (i.e. 30mins) and these are

the radionuclides that half-life in days while

Second long-lived counting is done after a week of first counting (or after

10/11 days of irradiation) and the counting time exactly 3600s (or 1hr).

These are the radionuclides having their half-life in days, weeks, month

and years (Landsberger S., 1994).

The samples analyzed for the determination of radionuclides and

concentrations are presented below:

122



S/

N

Sample

ID Sample type

Sample

code

Irradiation

Type

1 052B31 African processed

locust beans ZRS2A2 Short

052B32 African processed

locust beans ZRS2A2 Long

2 052B33

Pumpkin

leaf ZRS8A3 Short

052B34

Pumpkin

leaf ZRS8A3 Long

3 052B35 Seasonings RYCAB2 Short

052B36 Seasonings RYCAB2 Long

4 052B37 African Spinach ZRS6B3 Short

052B38 African Spinach ZRS6B3 Long

5 052B39 Tobacco BNHLA1 Short

052B40 Tobacco BNHLA1 Long

6 052B41 Indian hemp JCDEB1 Short

052B42 Indian hemp JCDEB1 Long

123



These samples were obtained in the Northern part of Nigeria i.e.Zaria (precisely

samaru market)

4. RESULT

The results of the analysis of six (6) biological samples analyzed for both short

and long-lived irradiation are presented below.

The result of the irradiation of short-lived for the samples in the first and second

γ-counting are shown in the table 1 and 2 respectively.

Table 1 Result of the concentration of elements for first short-lived irradiation.

ELEMENT

B31, B32

ZRS2A2

B33, B34

ZRS8A3

B35, B36

RYCAB2

B37, B38

ZRS6B3

B39, B40

BNHLA1

B41, B42

JCDEB1

Mg (%) 0.30 ± 0.02

0.42 ±

0.04 NA 0.68 ± 0.05 0.37 ± 0.03 0.30 ± 0.02

Al (%)

0.060 ±

0.004

0.30 ±

0.02 NA 0.16 ± 0.01 0.050 ± 0.004

0.060 ±

0.004

Ca (%) 2.7 ± 0.3

1.14 ±

0.16 NA 4.15 ± 0.54 1.78 ± 0.24 2.8 ± 0.4

Ti (ppm) BDL 362 ± 72 NA BDL BDL BDL

V (ppm) BDL BDL NA BDL BDL BDL

124



Table 2 Result of the concentration of elements for second short-lived irradiation.

ELEMENT

B31, B32

ZRS2A2

B33, B34

ZRS8A3

B35, B36

RYCAB2

B37, B38

ZRS6B3

B39, B40

BNHLA1

B41, B42

JCDEB1

Mn (ppm) 157 ± 6 78 ± 3 BDL 37 ± 2 137 ± 5 137 ± 5

Eu (ppm) BDL BDL BDL BDL BDL BDL

Dy (ppm) BDL BDL BDL BDL BDL BDL

Na (%)

0.055 ±

0.004 0.14 ± 0.08 25.8 ± 1.3

0.23 ±

0.01

0.060 ±

0.005

0.020 ±

0.003

The result of the irradiation of long-lived for the samples in the first and second γ-

counting are shown in the table 3 and 4 respectively.

Table 3 Result of the concentration of elements for first long-lived radionuclides.

ELEMENT

B31, B32

ZRS2A2

B33, B34

ZRS8A3

B35, B36

RYCAB2

B37, B38

ZRS6B3

B39, B40

BNHLA1

B41, B42

JCDEB1

K (ppm) BDL BDL BDL BDL BDL BDL

As (ppm) BDL BDL BDL BDL BDL BDL

Br (ppm) 1.0 ± 0.2 5.3 ± 0.5 35.8 ± 3.5 10 ± 1 51 ± 3 13 ± 1

La (ppm) BDL 5.6 ± 0.4 BDL 2.3 ± 0.3 2.3 ± 0.3 2.9 ± 0.3

Sm (ppm) BDL BDL BDL BDL BDL BDL

Yb (ppm) BDL

0.47 ±

0.14 BDL BDL BDL BDL

U (ppm) BDL BDL BDL BDL BDL BDL

125



Table 4 Result of the concentration of elements for second long-lived

radionuclides.

ELEMENT

B31, B32

ZRS2A2

B33, B34

ZRS8A3

B35, B36

RYCAB2

B37, B38

ZRS6B3

B39, B40

BNHLA1

B41, B42

JCDEB1

Sc (ppm) BDL BDL BDL BDL BDL BDL

Cr (ppm) BDL 35 ± 5 BDL 17 ± 3 BDL 14.3 ± 2.6

Fe (%)

0.035 ±

0.001 BDL BDL 0.38 ± 0.04 0.10 ± 0.02 BDL

Co (ppm) BDL BDL BDL BDL BDL BDL

Zn (ppm) 10 ± 2 26 ± 6 BDL BDL BDL 20.5 ± 4.7

Rb (ppm) BDL BDL BDL BDL BDL BDL

Cs (ppm) BDL BDL BDL BDL BDL BDL

Ba (ppm) BDL BDL BDL 452 ± 65 BDL BDL

Eu (ppm) BDL BDL BDL BDL BDL BDL

Tb (ppm) NA NA NA NA NA NA

Lu (ppm) BDL BDL BDL BDL BDL BDL

Hf (ppm) BDL BDL BDL BDL BDL BDL

Ta (ppm) BDL BDL BDL BDL BDL BDL

Sb (ppm) BDL BDL BDL BDL BDL BDL

Th (ppm) BDL BDL BDL BDL BDL BDL

126



A SUMMARY OF ANALYTICAL RESULTS

ELEMENTS

B31,

B32

ZRS2A2

B33,

B34

ZRS8A3

B35,

B36

RYCA

B2

B37,

B38

ZRS6B3

B39, B40

BNHLA1

B41, B42

JCDEB1

Mg (%)

0.30 ±

0.02

0.42 ±

0.04 NA

0.68 ±

0.05

0.37 ±

0.03

0.30 ±

0.02

Al (%)

0.060 ±

0.004

0.30 ±

0.02 NA

0.16 ±

0.01

0.050 ±

0.004

0.060 ±

0.004

Ca (%) 2.7 ± 0.3

1.14 ±

0.16 NA

4.15 ±

0.54

1.78 ±

0.24 2.8 ± 0.4

Ti

(ppm) BDL 362 ± 72 NA BDL BDL BDL

V

(ppm) BDL BDL NA BDL BDL BDL

Mn

(ppm) 157 ± 6 78 ± 3 BDL 37 ± 2 137 ± 5 137 ± 5

Eu

(ppm) BDL BDL BDL BDL BDL BDL

Dy


Na (%)

0.055 ±

0.004

0.14 ±

0.08

25.8 ±

1.3

0.23 ±

0.01

0.060 ±

0.005

0.020 ±

0.003

K


127



As


Br

(ppm) 1.0 ± 0.2 5.3 ± 0.5

35.8 ±

3.5 10 ± 1 51 ± 3 13 ± 1

La

(ppm) BDL 5.6 ± 0.4 BDL 2.3 ± 0.3 2.3 ± 0.3 2.9 ± 0.3

Sm


Yb

(ppm) BDL

0.47 ±

0.14 BDL BDL BDL BDL

U


Sc


Cr

(ppm) BDL 35 ± 5 BDL 17 ± 3 BDL 14.3 ± 2.6

Fe (%)

0.035 ±

0.001 BDL BDL

0.38 ±

0.04

0.10 ±

0.02 BDL

Co


Zn

(ppm) 10 ± 2 26 ± 6 BDL BDL BDL 20.5 ± 4.7

Rb


Cs


128



Ba

(ppm) BDL BDL BDL 452 ± 65 BDL BDL

Eu


Lu


Hf


Ta


Sb


Th


Note: BDL: - Below Detection Limit; NA: - Not Analyzed

5. DISCUSSION

5.1 . Sample ZRS2A2 and ZRS2A2

As it has been in the tables above, the sample ZRS2A2 which is known as African

processed locust beans (also known as irú) is very rich in magnesium, aluminum,

calcium, manganese, iron, and zinc than those in sample RYCAB2 i.e.

seasonings. Even those elements in the spice (i.e. Mg, Al, and Ca) cannot be

analyzed simply because of its high dead time due to high dose of radiation during

the first γ-counting of short-lived irradiation. Then talking about Na and Br, there

is a very high concentrations of Na (i.e. 25.8% which equivalent to 258g) and Br

which is 35.8ppm (i.e. equivalent of 35.8mg) in the sample RYCAB2,

―seasonings‖ compared to African processed locust beans that has 0.055% and

129



1.0ppm of Na and Br respectively. The concentration of Na in the seasonings has

exceeded 2.4g which is the toxic level and likewise the concentration of Br that is

found to be outside the estimated median daily intake recommended by World

Health Organization (W.H.O.) (Taubes G., 1998, Nelson, 2000). Thus processed

locust beans is nutritious and even more advantageous than seasonings because

the elements obtain in processed locust beans are in appropriate proportion that

our body system need while the ―seasonings‖ is dangerous to our health in the

sense that it contains high concentration of Na and Br of values 25.8% and

35.8ppm respectively which can later cause high blood pressure, congestive heart

failure, cardiovascular disease, cirrhosis, or kidney disease (Denton D, 1995,

Curhan G. C. 1997, Chrysant G. S, 2000) due to the high intake of NaBr.

5.2. Sample ZRS8A3 and ZRS6B3

The analytical result of sample ZRS8A3 which denotes Pumpkin leaf (also known

as ugwu) and the sample ZRS6B3 which represents African spinach (amarathus

spp) can be used to show the concentrations present in each sample i.e. pumpkin

leaf & African spinach. The amounts (in concentrations) of elements; Al (0.30%

which equivalents 3,000mg), Cr (35ppm, equivalent to 35mg), and La (5.6ppm

which equivalent to 5.6mg) present in the pumpkin leaf are larger than the

amounts found in the African spinach. Elements like Ca (4.15% which equivalent

to 41.5g), Na (0.23%, equivalent to 2.3g), Br (10ppm which equivalent to 10mg),

Mg (0.68%, that is, 6.8g), and Fe (0.38% which equivalent to 3.8g) found in the

African spinach of higher concentrations happen to be minerals that are required

in small quantity in the body systems. The maximum daily intake also

recommended for the elements Ca, Na, Br, Mg, and Fe are 1.3g, 2.4g, 3mg,

420mg, and 18mg respectively (OSU Retrieved, 2008). Therefore high intake of

these elements may cause problem to our health (i.e. when tolerance level is

exceeded.).

Note: 10000ppm = 1 percent; 1ppm = 1mg.

6. CONCLUSIONS AND RECOMMENDATIONS

From the analytical results of neutron activation analysis of some of biological

samples using CERT (Center for Energy Research and Training, Ahmadu Bello

University, Zaria) facility, we have the following conclusions:

1. Using African processed locust beans (also known as irú) as food seasonings is

more nutritious than using seasonings due to the high concentration of sodium

bromide (NaBr) obtained in the seasonings which may affect our body systems

130



and we will like to recommend African processed locust beans for everybody

owing to its nutritious advantages and more so, its consumptions does not affect

our health.

2. Using pumpkin leaf (also known as ugwu) instead of African spinach as

vegetable is more important simply because it contains higher concentration of

some essential nutrients that play important roles in the body systems. Thus we

will like to also recommend this pumpkin leaf for every one.

REFERENCES

Alfassi Z. B. (1998): Instrumental Multi-Element Chemical Analysis. Khewer

academic Publishers, Dordrecht, Netherlands

Curhan G. C., Willett W. C, Speizer F. E, Spiegelman D, Stampfer M. J. (1997):

Comparison of dietary calcium and other nutrients as factors affecting the

risk for kidney stones in women. Ann Intern Med; 126(7): 497-504.

Chrysant G. S (2000): High salt intake and cardiovascular disease: is there a

connection? Nutrition; 16(7-8): 662-664.

Das, H., Faanhof, A. and Van der sloot, H.(1989): Radioanalysis in

Geochemistry, Elsevier, Amsterdam.

Denton D, Weisinger R, Mundy N. I, et al (1995): The effect of increased salt

intake on blood pressure of chimpanzees. Nat Med.; 1(10): 1009-1016.

Filby R. H. (1995): Pure & Appl. Chem., Vol. 67, No. 11, pp. 1929-1941.

Jonah S.A. (2001): Lecture Notes in Applied Nuclear Physics, A foundation

Postgraduate course in Theoretical Nuclear Physics, National Mathematical

Center, Abuja, Nigeria.

Landsberger S. (1994): Delayed Instrumentation Neutron Activation Analysis.

Department of Nuclear Engineering, University Of Illinois, 214 Nuclear

Engineering Laboratory, 103 South Goodwin Ave., Urbana, IL61801, USA.

Linus Pauling Institute at Oregon State University. Retrieved on 2008-11-29.

Nelson, David L, Michael M. Cox (2000-02-15), Lehninger Principle of

Biochemistry, Third Edition (3Har/Com ed). W.H. Freeman. pp.1200 ISBN

1572599316.

131



Pollard A. M., Heron C. (1996): Archaeological Chemistry. Cambridge, Royal Society of

Chemistry.

Soete, D., Gijbels, R. and Hoste, J. (1972): Neutron Activation Analysis, Wiley

Interscience, New York.

Taubes G. (1998): The political science of salt. Science; 281 (5379): 898-901/903-897.

Vértes, A., Nagy, S. and Suvegh, K. (1998): Nuclear methods in Mineralogy and

Geology, plenum press, New York.

Westphal, G.P., Josh K., Lipp B., and Schröder, P. (1992): J. Radioanal Nucl., 160, 395.

132



OXYGEN AND HAEMOGLOBIN PAIR MODEL FOR SICKLE

CELL ANEAMIA PATIENTS***

B.O. Oyelami

National Mathematical Centre Abuja, Nigeria

Email: [email protected]

Abstract

In this paper, we studied the concentration of oxygen in the haemoglobin of the sickle

cell patient using the Oxygen- heamoglobin pair (OHP) model which is an

impulsive Hill-Fokker-Planck equation .Using the B-transform of Oyelami and Ale

we determine the best concentration for oxygen or haemoglobin to support the patient

using life-supporting drug like nitric oxide providing drugs. Since the sickle nature of

the erythrocyte of the patient has the contributory factor to sickling problem and there

is the need to correct this defect and to enhance the haemoglobin affinity for oxygen

absorption, thereby, reducing the patient‘s physiological problem. Using Lagrangian

optimization method coupled with the application of simple calculus and B-

transform we found that nk1

)/11(c 5000.0 * 3m gives range of the

concentration of oxygen that is required to be absorbed by Haemoglobin of the

sickle cell anemia patient for effectively performance of the body. Using entropy

objective function, the Lagrange function is unbounded above and could not offer

much information on the optimal concentration of haemoglobin to support the

patient.

Keywords: Model, Sickle cell aneamia, B-transform, Oxygen and Haemoglobin.

44A20, 68W30, 35C20 & 34A37.

*** African Journal of Physics Vol. 2, pp. 132- 148 (2009)



133



1. INTRODUCTION

Sickle cell anemia is caused by a "defective" allele (mutant form) of the

gene coding for a sub-unit of the haemoglobin protein. Haemoglobin binds

oxygen within red blood cells, which then transport the oxygen to body tissues

where it is released from the haemoglobin molecule. The sickle haemoglobin (in a

person with a mutant allele) tends to precipitate, or "clump together", within the

red blood cell after releasing its oxygen. If the clumping is extensive, the red

blood cell assumes an abnormal "sickle" shape. These sickle red blood cells plug

the blood vessels, thus preventing normal red blood cell passage and,

consequently, depriving the tissue of needed oxygen and leading to a short lived

red cell survival ([5],[17]&[21]).This situation often lead to stroke as result of

sequestration of blood into the lung, liver or spleen and cerebral

vessels([13],[17]&[21]).

Sickle cell aneamia is a genetic disorder commonly found among the

black race especially American Negroes, Africans and the people of the

Mediterranean countries. It is a genetic mutation problem wherein the normal

haemoglobin N in the blood is replaced with a defective haemoglobin S (defective

allele). Haemoglobin S is found to be extremely inefficient in carrying oxygen

([4], [5] & [17]) as a result of heterozygote advantage against malaria, the

inherited heamoglobin disorders are the commonest monogenic disease

([13]).Acute pain crises may be caused by infection, dehydration, environmental

temperature change, or change PH level of the blood especially if it is too acidic.

Supportive therapy includes fluid hydration, analgesic, and antibiotic therapies

when infection is suspected ([17]).

Sickle cell aneamia is one of frequent child mortality in the sub-Saharan

Africa where children with this disease hardly survive beyond 5 years and very

few survive beyond 18 years. Sickle cell anemia is associated with a multitude of

medical complications ranging from acute painful crises caused by the damage to

the spleen, kidneys, lungs, heart, muscles and brain. Repeated hospitalization for

intravenous pain medication, antibiotic therapy and blood transfusions is

undertaken to treat medical problems as they arise. These patients often die early

of overwhelming infection or as a consequence of acute or chronic damage to the

body organs ([13], [17] & [21]).

Recent researches from experimental point of view have it that sickle cell

disease is the polymerization of deoxygenated sickle haemoglobin S, reducing red

134



blood cell sickling is to increase red blood cell in the Hbs affinity for

oxygen([2]). Moreover, research finding also indicated that low concentration of

nitric oxide with increase oxygen affinity and could serve as alternative

therapeutic model for studying sickle cell aneamia ([2],[6],[13]&[21]).

There are several mathematical models on sickle cell aneamia. There are

models built upon the Hardy-Weinberg laws ([8]&[13]) with fundamental

assumption that gene frequency does not change with time that is, fixed from

generation to generation. There are those models that are of stochastic origin like

the HW family but fundamental developed using the idea of the birth and death

processes. More recently, mathematical models using impulsive differential

equations are being applied to biophysics with special applications to sickle cell

aneamia modeling ([8]&[13]).

Impulsive differential equations are systems that are characterized by

short time perturbations in form of jumps, shocks, rapid structural changes that act

momentarily. This branch of knowledge was developed not quite long if we

compared it to other branches of dynamical systems. The (IDEs) has found many

applications in medicine, biotechnology, and phamakenetics and so on ([1],[3]).

We hope it will find useful applications in genetic engineering and computer

based simulation of biomedical systems ([8-13],[18]&[19]).

In ([8]&[13]) using geometric and impulsive theoretic we are able to

compute the blood pressure generated in the body of the sickle cell aneamia

patient and even established to some extent that some physiological problems of

the patients are directly or indirectly connected to the blood pressure infringed on

the blood vessels of the patients.

Furthermore, the sickle nature of the erythrocyte of the patient has a

contributory factor to the sickler‘s problem and there is the need to correct this

defect and enhance the haemoglobin affinity to absorb oxygen to reduce the

patient's physiological problems ([5], [8], [13], [15] & [22]). However, bone

marrow transplantation, an expensive, high-risk medical procedure, remains the

only known cure for this disease ([21]).

In the modern times, several optical methods are developed to measure

haemoglobin concentration of oxygen saturation and principal dyshaemoglobins

in vitro and in vivo. Amongst these methods are pulse oximeters, fiber optic

oximeter ,multiwavelengths haemoglobin photometers(co-oximeters) and infrared

spectroscopy kind of equipment([2]&[23]) .

The oxygen dissociation curve (ODC) of haemoglobin (Hb) and the Bohr

effect associated with the of ODC because of the shift of the curve to the right as

135



PH decreases has profound clinical importance, as it is being applied in numerous

health and disease situations .Areas of applications ODC are in the neonatal

period, haemoglobinopathies such as sickle cell disease and so on ([15-17]). The

ODC is sigmoid in shape with unique properties, that oxygen saturation (SaO2)

approaches a horizontal asymptote as the oxygen tension exceeds 70 mmHg,

while it declines precipitously down the steep slope toward a point of inflexion

when the oxygen tension falls off the ―shoulder‖ of the ODC below 60

mmHg(see[7]).

In this paper, we intend to study the blood pressure of the sickler using

oxygen-haemoglobin pair; determine the absorption potential (range of

absorption) of oxygen by the haemoglobin or the best concentration of oxygen in

the haemoglobin if the patient is to be on life supporting drugs like nitric oxide

providing drugs ([22]).

2. STATEMENT OF THE PROBLEM AND METHODS

2.1 The Model

We propose that the partial pressure exerted by the oxygen on the

haemoglobin of the patient is of the form

CKHCHP ),( (1)

where H is the concentration of the haemoglobin in the blood plasma; C is the

concentration of oxygen in the blood plasma; K is non-dimensional constant

which can be obtained experimentally; and are some dimensional constants

and by simple dimensional analysis we can show that .1 and 3/1

We consider the oxygenation of a haemoglobin( Hb) molecule as four

sequential steps, given that each of the four heme groups within the two -globin

and two -globin chains binds to a molecule of oxygen(O2) (see[7]) for detail

formulation. The reaction process is formulated in the figure 1 below:

Figure 1 oxygeneration of Hb by O2. Source [5]

8

6

3

4

2

4

26

24

2

22

1

2

HbO

HbO

HbO

HbO

kOHbO

kOHbO

kOHbO

kOHb

136



where ki ,i=1,2…4 in Figure 1 are association constants ,by Hill‘s mass action law

(see [5]&[13])

n

Hbo

n

Hbo

Hbook

ok

Hb

HboS

][1

][

][

][

2

22

2

2

2

2HboS the saturation 2Hbo ; 2Hbok the net association constant of 2 Hbo

,n is the Hill coefficient and

.)n(haemoglobi Total

)obin(oxyhaemogl Total(%) 2

2 Hb

HboSHbo , [.] is the concentration of (.).

The actual data on human Hb is 166 104.19.2 and 101.10.2

2

mSao for

lyrespectiven haemoglobi of )chain( - and )chain( kk ([7]).

2.2 B-Transform

The B-transform of the function )(tx with impulses at fixed

moments ,..2,1},{ ktk during the evolutionary process is [1, 8, 10 &11]

(q)x + (q)x = x(t)B 1cn/

(2)

where xc(q) and xI(q) are the Ic LL and components of the B-transform and are

defined as

0,1,2,... = k,t tx(t)dt,e = x(t)L = (q)x kqt/-

0

cc

n

/

(3)

))tI(x(t ke = x(t)L = (q)x

q/t-

t<t<t

11

n1

k

ko

(4)

where

n‘ = 0, 1, 2,... n; n‘ is the order of the transform. For sake of simplicity, we will

choose n/ = 1. The advantage of taking n‘= 1 lies in the derivation of the inverse

transform.

The inverse transform for components of xc(q) and xI(q) can be obtained

(see [8,10 & 11]) as follows:

137



dqe(q)xi2

1 = (t)x

sqc

i+v

i-vc

(5)

))tq)I(x(,t( = (t)x kk

t<t<t

1

ko

In ([8]&[13]),using B-transform method we obtained the pressure the sickle red

blood cell exerted on the blood vessels of the sickle cell patient as

)x,(fe +r)x,g(Aex4+

)u,,P,rP(ex - (r)Pe = x)P(r,

koxk2-

x<x<x

xr2-22

moorp-233

oxr2

ko

,0 (6)

where

uD

xC2 = )u,x,,PP( 2

mfno (7)

Cf = Coefficient of friction; D = Diameter of the vessel

2

mu = Mean square velocity of the blood plasma

a

Figure 2: Normal Cell and Sickle red blood cell

138



)1(00

rxeAA is the area cut-off as a result of sickle shape of the blood cell.

g )(),,( 0

200 rAex

dr

xdArxA rx is the Arial potential of the sickle red blood cell.

r is the radius of the sickle red blood cell; ρ is the density of the blood

x is the movement of the blood along the x-axis

It will be recalled (see [8]), that we stated that xk - xk+1 = f0(α, xk), k = 0, 1,

2,... and xk depends on the surface density α and f0(α, xk) is a piecewise

continuous function. It was also noted that xk is, in fact, impulsive because of

vibration and variation effect of the texture of the composition of the surface of

the sickle blood cell.

In figure 3 we try to replicate a typical sickle cell erythrocyte by a means

of simulation. The simulation is carried out by finding the equation that describes

the area cut-off from the normal red blood cell as a result of sickle shape of the

blood cell. We observed that as the thickness of the each parabola in figure 1

increases we have something that is similar to typical sickle red blood cells.

Figure 3: Typical sickle cell aneamia blood

cells as simulated by some functions

139



1.2 Oxygen-haemoglobin model (OHM)

Consider the Oxygen- haemoglobin model which is an impulsive Hill-Fokker-

Planck equation

k

kk

kkkk

n

n

o

ttttt

tHx

tHtHtH

gtCxttC

Ck

CkHCk

t

C

x

HDkHC

x

Hv

t

H

lim,...0

)(),0(

and )(),0(

conditions initial Subject to

)(),(

1

210

10

2

21

2

2

(8)

where

. Hbby o of absorption

result a as cell blood sickle ofmovement ofeffect impulsive eaccount thg and

vessel..blood along distanceat t at time Hb and O ofion concentrat

theare and ),(constant. sHill' is and Hbo ofn associatio

net theis k action; mass todue constants rate are and vessel;blood in the eerythrocyt

sickle theof distance theis ;conviction of velocity theis t;coefficiendiffusion theis

2

k

2

2

21

k

o

x

H(t,x)HxtCCn

kk

xvD

Remark 1

0, 1 kk The equation is the Fokker-Planck equation and HC accounts for the

mass-action for the oxygen and haemoglobin respectively. The equilibrium state

for the model can be found by setting constant),( and 0,0 1

xtC

t

C

t

Hk

for fixed x in the equation (8).

Therefore

140



0)),(1(

),(

)),(1(

),(

)),(1(

),(

And

)),(1(

),(

2

1

2

2

2

2

2

2

1

20

2

1

2

txCk

txCk

xD

txCk

txCk

txCk

txCk

xv

txCk

txCkH

n

n

n

n

n

n

n

n

(9 )

Set 4n ,v0 = -0.25, k2=0.5,k1=0.08,D= 0.008 and

z

zzF

75.0)1(:)(

in the above equation we have

0)1(

004.0)1(

5625.175.0

2

275.0

z

z

dz

d

z

z

dz

d

(10)

Therefore

0)(004.0)(5625.12

2

zF

dz

dzF

dz

d

(11)

and the solution is

)8

3125(

21)(

z

eCCzF where C1 and C2 are arbitrary constants.

But

2

75.0

25.0

)1(

)1(

75.0)(

z

z

zzdz

zdF

(12)

Therefore 0)(

lim,0)1( dz

zdFF

z which implies that C2 =0 and C1 =0.And

therefore F(z)=0 which implies that z=1 or infinity. It follows that the equilibrium

point is such that constants),( and 0,0 1 xtCCH k and for C to be at

infinity is not realistic.

141



2.3 Formulation of Optimization problem for the Oxygen Haemoglobin model

We intend to find the optimal concentration for the oxygen to be absorbed by the

haemoglobin for the sickle cell patient for effective physiological processes. In

other to achieve this we use the Lagrange multiplier method as follows:

),(1

),(),( subject to),,(),(),(min 3

1

xtkC

xtkCxtCxtHxtkpxtC

n

n

x

(13)

We define the Lagrange equation as

)),(1

),(),((),(),(

xtkC

xtkCxtCxtCCL

n

n

(14)

We will find ).L(C, minimises that 0,0such that and

L

C

Lx

2.4 Maximum Entropy Weights

We define Le as a negative entropy function

n

iine wwwwwL1

21 log),...,,( .Let 1 be convex hull of points nhhh ,....,, 21 which

containsn

hwn

ii

h

1 and this occurs almost surely for large n. We are now in the

position to find the weight to minimize Le as follows:

n

ih

n

i

iine whwwwwL11

21 1, subject to ),,...,,(min

Let nihi ,...1, are the concentration of haemoglobin at moment i and

Define the Langrage function for the above problem as

n

ii

n

i

iii hwwvhwL1

0)ln( .

We need to determine v and for which

142



0 and 0,0

L

v

L

h

L

i

.

3. RESULTS AND DISCUSSIONS

Using Maple 11 for Lagrange equation we obtained sufficient condition for

extremum as

0)1(

,0))1()1(

1(12

1221

n

n

n

n

n

n

kC

kCC

L

kC

nCk

kC

kC

C

L

(15)

It implies that

2

122

1

1

1

)1(11

1 ,01

n

n

n

n

nn

kC

nCk

kC

kCkCkC

(16)

The first equation in equation (16) has n-roots by fundamental theorem of algebra

and some of the roots are real and others occur in complex conjugate .To find the

solution in general, it is intractable but using Galois theory the solution can be

found using radical expression or we find the numerical approximation to the

roots. We simulated the model for 16104.2 and 4 mkn found that

.1 Where.01967.0996.0 and 01967.09996.0,02007.1,9807000.0 2

4321 iii

Therefore 3 5000.0),(min mCL this is the minimum concentration of

oxygen required to be absorbed by Hb for effectively performance of the body.

We found that

n

i

i

n

i

in

i

i

n

i

i

h

w

nnwh

1

1

11

,)1(

11)ln(2

and therefore,

i

143



n

i

i

n

i

ii

n

i

i

n

i

i

n

i

i

n

i

i

n

i

ii

h

hww

nnwwhhwL

1

11

1111

1

)1(

1)ln(2)ln(

i.e.,

.1 since

,)1(

1)ln(2

)ln(

1

1

11

1

n

i

n

i

n

ii

n

in

ii

w

h

hw

nn

h

hwL

Proposition 1

Give that nihi ,...1, are non-negative concentrations of haemoglobin at the

period i such that the weight wi are such that 1|| iw .Then

n

i nnhL1

)1(|)ln(|2|| and it is unbounded above as n .

Proof

Straight forward by estimating (majorizing) L and taking note

that )1(2

||1

nn

wn

i .

Remark 1

The clinical implication of proposition 1 is that we cannot say much about the

concentrations of haemoglobin as the coupling size n becomes large whether the

maximum or minimum concentration exist using entropy objective function

.Therefore, it is advisable to rely on the concentration of oxygen in the blood

plasma as obtained from the analysis of the equations (15&16).

144



3.2 Application of B-Transform

We assume that the solution of the equation (6) exists and continuously depend on

the initial data (see [12],[18-19]) then the B-transform can be applied to Oxygen-

haemoglobin pair model as follows:

Equation (8) becomes

n

n

kC

kKC

x

HkD

x

Hkv

t

Ck

t

Hk

12

2

101 (17)

Applying B-Transform we have

xxx

kkk

qx

kI

qx

c

qx

c

k

k gxCexxtCL

Hq

HHq

dqex

H

x

HL

Hq

Hdqex

H

x

HL

0

))((),(

11

1

/

10

/

0

2

2

2

2

0

/

0

Now let tkCxz n fixedfor 1)( then

0

/

)(

11

1dqe

szC

CL qs

n

n

c .

Therefore application of B-Transform to the equation (17) gives

1

//

1010101

0 0

))(()(

11

1

)()(

z xxx

kkk

qxqs

k

k gxCedqeszK

HkvkDHqkDHkDkqvt

Ck

t

Hk

(18)

But

),0(),(),,0(),(00

qCqtCdss

CqHqtHds

s

Htt

Therefore, taking the inverse B-Transform of equation (18) and after simplifying

the equation we get

145



, ))(())(

11(

i2

1

),()(),0( ),0(

),0(2

)),0(),((),0(),(

00x

1

/

0

1000

0

0

1

xx

kkkk

z

tqqs

t

xq

xq

k

gxC)(xdqdsesz

K

dsxsHkvkDdqexHqkDtxHkDt

dqeqHqi

ktvxCxtC

k

kxHxtH

(19)

where

dqei

xC

qxqx

kkk

/

2

1)(

.

We can use equation (1) to find the relation between as ),( and ),( xtpxtH

))((

))(

11(

i2

1 ),()(

)(2

),0( ),0(),0(

2

)),0()(),(),((),0(),(

0

0

31

31

x

1

/

0

10

0

00

0

0

1

0

xx

kkkk

z

tqqs

t

xq

k

gxC)(x

dqdsesz

KdsxsHkvkD

tui

xktDHxktDHdqeqHq

i

ktv

xHxpxtHxtpk

kxHxtH

(20)

.otherwise 0

0 if 1 where

xdqqeu(x) xq

In order to determine ),( xtH completely we need to solve the integral equation in

the equation (20) completely.

The following theorem shows how the concentration of Oxygen changes with that

of heamoglobin:

Theorem 1

The haemoglobin of sickle cell aneamia patient will have maximum potential

absorption if there exists a 0* c such that

146



n

knkdqdse

qskc

qsc

iz

tqqs

n

n1

0

)1

1(c and 1

),(1

),(

2

1*

1

1

/

*

1

(21)

Proof

By differentiating the equation (20) twice with respect to C we get

1

/

2

1

2

2

1

/1

1

0

0

)),(1(

),()),((

2

),(1

),(

2

z

tqqs

n

nn

z

tqqs

n

n

dqdseqskC

qsCknqsnkCn

i

kn

C

H

dqdseqskC

qsC

i

kn

k

k

C

H

By simple rule in calculus the proof follows immediately but we must note that if

n

k

1

)1

1(c* we have point of inflexion for which we cannot infer whether the

absorption is maximum or minimum.

Acknowledgements

The author is grateful to the National Mathematical Centre Abuja, Nigeria and to

the Kaduna State University Kaduna for their supports.

References

[1] Ale S O. and Oyelami B.O., B-Stability and its Applications to some

constant Delay Impulsive Control Models. NMC-COMSATS Proceedings

on International Conference on Mathematical Modeling of some Global

Challenges in the 21 st Century,2009, pp56-65.

http://nmcabuja.org/nmc_proceeding.html.

[2] Alvin Head C. et.al. Low Concentrations of Nitric oxide increase oxygen

Affinity of sickle erythrocytes in Vitro and in Vivo. J. Clin Invest.

American Society for Clinical Investigation Inc

Vol.100,.5,Sept.1997,1193-1198.

[3] Beltrami, E. Mathematics for Dynamic Modeling, Academy Press

London, 1987.

http://nmcabuja.org/nmc_proceeding.html

147



[4] Bob Williamson, Our Human Genome -How can it serve us well?

Bulletin of the World Health Organization, 2001, 79 (11), pp. 1005.

[5] Gill S.J, Skold R, Fall L, Shaeffer T, Spokane P, Wyman J.,

Aggregation Effects on Oxygen binding of Sickle Cell Haemoglobin.

Science 201: 362– 364, 1978.

[6] Mark T. Gladwin et.al. Nitric oxide donor proper properties of

hydroxyurea in patients with sickle cell disease. British Journal of

Haematology vol.116, pp436, feb.2002, doi:10.1046/j. 1365-

2141.2002.03274.x, issue2.

[7] Melvin Khee-Shing Leow, Configuration of the Haemoglobin Oxygen

Dissociation Curve Demystified: A Basic Mathematical Proof for Medical

and Biological Sciences Undergraduates. Advan Physiol Educ 31:198-201,

2007. doi:10.1152/Advan.00012.2007.

[8] Oyelami, B. O. and Ale, S. O. B-transform Method and its Applications,

in obtaining Solutions of some Impulsive Models. International Journal of

Mathematics, Education, Science and Technology, 2000, Vol. 31, No. 4,

pp 525-538.

[9] Oyelami, B. O., On Military Model for Impulsive Reinforcement

Functions using Exclusion and Marginalization Techniques, Nonlinear

Analysis 35 (1999), pp 947-958.

[10] Oyelami, B. O. and Ale, S. O., B-transform and its Applications to a Fish-

Hyacinth Model, International Journal of Mathematics, Education, Science

and Technology, 2002 Vol. 33, No. 4, pp 565 - 573.

[11] Oyelami, B. O., Ale, S. O., Onumanyi P., Ogidi J.A. Impulsive HIV-1

model in the presence of Antiretroviral Drugs using B-transform method.

Proceedings of African Mathematical Union, 2003, pp 62-76.

[12] Oyelami B.O. and Ale S.O. On Existence of Solution, Oscillation and

Non-Oscillation properties of Delay Equations containing

‗Maximum‘.Acta Applicandae Mathematicae Journal, 2008.

DOI:10.1007/s10440-008-9340-1.

[13] Oyelami B.O., Ale S.O., Onumanyi P.and Ogidi J.A. B-transform and

Applications to the Sickle Cell Models. The proceedings of International

Seminar on Theoretic Physics and National Development published in the

African Journal of Physics 2009, 202-220. http://sirius-

c.ncat.edu/asn/ajp/allissue/ajp-ISOTPAND/index.html.

http://sirius-c.ncat.edu/asn/ajp/allissue/ajp-ISOTPAND/index.html

http://sirius-c.ncat.edu/asn/ajp/allissue/ajp-ISOTPAND/index.html

148



[14] Pandit, S. G. and Deo Sudashiv H., Differential Systems Involving

Impulsive. Lecture Notes in Mathematics, Springer-Verlag, Berlin -

Heidelberg - New York, 1982.

[15] Robert K. Fitzgerald, MD; Alan Johnson, MD Pulse Oximetry in Sickle

Cell Anemia. Crit Care Med 2001 Vol. 29, No. 9

[16] Robert, M.B.V., Biology: A Functional Approval, Nelson Butler and

Tanner Ltd. Rome and London, 1971.

[17] Rosalie A. Dance & James T. Sandefur A Study of Malaria and Sickle Cell

Anemia: A Hands-on Mathematical Investigation, National Science

Foundation Publication. 4/15/99, 1998.

[18] Simeonov, P. S. and Bainov, D. D., Impulsive Differential Equation:

Asymptotic Properties of the Solutions. World Scientific Publication,

Singapore, 1989.

[19] Simeonov, P. S. and Bainov, D. D., Theory of Impulsive Differential

Equations: Periodic Solutions and Applications. Longman, Essex, 1993.

[20] Weatheral, D. J. and Clegg, J. B., Inherited haemoglobin disorders: An

increasing global Health Problem, Bulletin World Health Organization,

2001, 79 (8), pp 704-711.

[21] World Health Organization, Sickle-cell Aneamia Report by the Secretariat.

Fifty-ninth World Health Assembly A59/9.Provisional Agenda item 11.4

24 April 2006.

[22] Vandergrift KD, Bellelli A, Samaja M, Malavalli A, Brunori M,Winslow

RM. Kinetics of NO and O2 binding to a Maleimide Poly (Ethyleneglycol)

- conjugated Human Heamoglobin. Biochem J 382: 183–189, 2004.

[23] Zijlstra W.G.,Buurma and Van Assendelf W.O., Visible and near infrared

Absorption Spectra of Human and Animal haemoglobin

Determination and Application.,2000, xvi, 368 pages ISBN 90-6-164-

317.http://www.vsppub.com/books/lifes/bk-

isNeaInfSpeHumAniHaedetapp.html.

149



NONLINEAR MAGNETO-OPTICAL EFFECTS IN

DIELECTRICS EMBEDDED WITH FERROMAGNETIC

NANOPARTICLES†††

1Arlene .P. Maclin and 2M. M. Noel,

1Professor of Engineering & Director, Center for Academic Excellence Norfolk

State University, Norfolk , Virginia 23504

Email: [email protected]

&

2Assistant Professor of Engineering, Norfolk State University, Norfolk, Virginia

23504; Email: [email protected]

,

Abstract

Magneto-optic effects in transparent dielectric materials embedded with

ferromagnetic nanoparticles have been investigated through simulation of a

nonlinear wave equation. The possibility of generation of harmonics due to

magnetic saturation in ferromagnetic nanoparticles was studied. A simplified

nonlinear spring-mass system model that accounts for magnetic saturation and

harmonic generation is presented. The simplified model is analyzed using the

finite element and finite difference methods and results are compared with data

from simulation studies.

1. INTRODUCTION

Nonlinear optical effects occur due to the nonlinearity of constitutive

relationships [1] in Maxwell‘s equations: ( )D E E and ( )B H H Nonlinear

effects due to nonlinearity of the constitutive relation for the electric field have

been demonstrated through harmonic generation experiments. In particular

Franken et al [2] demonstrated the generation of ultraviolet light by passing a

††† African Journal of Physics Vol. 2, pp. 149- 156, (2009)




150



ruby laser beam through a quartz crystal. However due to difficulties in

manufacturing high frequency ferromagnetic materials nonlinear optical effects

occurring due to the nonlinearity of the magnetic field, constitutive relationship

have not been extensively studied. Recently the nonlinearity of ferromagnetic

resonances was used to generate second and higher harmonics in the microwave

region of the electromagnetic spectrum [3].

With the maturation of nanotechnology it has now become feasible to

manufacture high frequency ferromagnetic materials. This paper considers the

generation of second and higher harmonics due to nonlinear magnetic effects

resulting from embedding ferromagnetic nanoparticles in a transparent dielectric

matrix.

2. RESULTS

The Maxwell‘s equations are for electromagnetic wave propagation in linear

media. However the linear wave equations are only valid at low field values. At

high field values, the relative permeability is not constant but decreases

monotonically until the free space value is reached. The decrease in relative

permeability with increasing magnetic field value is due to the alignment of

magnetic domains. When all magnetic domains have been aligned the

permeability cannot increase any further resulting in a constant permeability for

very large field values. This effect is modeled by the nonlinear wave equation (2).

The nonlinear wave equation can be numerically solved using a finite

difference approximation scheme. First the partial derivatives are approximated

by finite difference approximations. These approximations are then substituted in

the nonlinear wave equation resulting in a difference equation that expresses the

field at any time t and location (x,y) in terms of the field values at previous times

(4).

2

2

22

2

0

22

2

(1)

sec ( )

( sec ( )) (2)

r

r

BB

t

a b h cB

BB a b h cB

t

151



2 2 22

2 2 2

2

2 2

2

2 2

2

2 2

2

, , 1 1, , 1, ,

( ) ( ) (3)

( , , ) 2 ( , , ) ( , , )

( , , ) 2 ( , , ) ( , , )

( , , ) 2 ( , , ) ( , , )

(x y t d x y t x y t x

u u uConsider c u

t x y

u u x x y t u x y t u x x y t

x x

u u x y y t u x y t u x y y t

y y

u u x y t t u x y t u x y t t

t t

u c u u u

, 1, 1, , , , , , 1 , ,

2

4 ) 2 (4)

( 1 )

y t x y t x y t x y t x y t

d

u u u u

c for convergence

The results of the finite difference approximation simulation are shown below.

152



.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Figure 2: shows a contour plot of linear mode shown in figure 1

01

23

45

67

0

2

4

6

8

-1

-0.5

0

0.5

1

Time=10 Color: u Height: u

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Figure 1: shows a normal mode for the linear wave equation. The

solution was obtained by finite element method with triangular

elements over a rectangular domain.

153



In order to obtain an analytic solution for model validation, we model the

nonlinear saturation behavior in terms of a nonlinear mass-spring system model

for the electron response.

Equation (5) can be solved using a singular perturbation method approximation.

To this end, we assume the electron displacement x(t) due to the electromagnetic

wave to be of the form:

2 3

0 1 2 3( ) ( ) ( ) ( ) ( )... (6)x t x t x t x t x t

cos( ) (5)e e mm x x kx q E t

00.5

11.5

22.5

33.5

0

1

2

3

4

-10

-5

0

5

10

15

Figure 3: Shows variation of the magnetic field with position after

100 iterations for the the nonlinear wave equation. The solution

was obtained using a finite difference approximation scheme.

154



Equation (6) can be substituted in equation (5) and coefficients of the parameter

are equated to obtain a sequence of approximations. shown below in Figures 4.

and 5.

-0.1 -0.05 0 0.05

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

position

velo

city

Fig. 4: Phase space plot for linear response.

-5 0 5 10 15

x 10-3

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

position

velo

city

Fig. 5: Phase space plot for nonlinear response.

155



3. CONCLUSION

Based on these preliminary results, we believe that embedding

ferromagnetic nanoparticles in a dielectric medium can produce nonlinear

magnetic effects due to hysteresis and saturation effects. Nonlinear magnetic

effects induced by addition of ferromagnetic nanoparticles can be used for

harmonic generation. Due to the inverse relationship between wave velocity and

permeability, the refractive index can be increased due to the addition of

ferromagnetic nanoparticles [4-7]. Future experiments are needed to investigate

the effects the addition of ferromagnetic nanoparticles can make on optical

properties in dielectric media.

ACKNOWLEDGEMENT

Several technical discussions about this work were held with Dr. Vitaliy

Lomakin from the University of California @ San Diego These discussions are

gratefully acknowledged. This research was supported by the NSF funded

Engineering Research Center for Integrated Access Networks ( 081207) with the

University of Arizona as the lead institution. Dr. Maclin gratefully acknowledges

the invitation by the organizers of the ISOTPAND09 to participate in this very

important meeting for the dissemination of these research results to scientists and

engineers in Nigeria. It was a real pleasure to meet so many emerging African

scientists interested in pursuing graduate work in physics and engineering

REFERENCES

[1] Y. R. Shen, Principles of nonlinear optics, New York, Wiley-Interscience,

1984.

[2] P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of

optical harmonnics," Phys. Rev. Lett. 7, 118 - 119, 1961.

[3] G. P. Rodrigue, "A generation of microwave ferrite devices," Proceedings of

the IEEE, Volume 76, Issue 2, Feb 1988 Page(s):121 - 137.

[4] C A F Vaz et al, "Ferromagnetic nanorings," in the Journal of Condensed

Matter Physics, 2007.

156



[5] E. M. Brunsman et al, "Magnetic properties of carbon-coated, ferromagnetic

nanoparticles produced by a carbon-arc method," in the Journal of Applied

Physics, May 1994.

[6] L. Berger et al, "Ferromagnetic nanoparticles with strong surface anisotropy:

Spin structures and magnetization processes," in Physics Review, Volume 77,

March 2008.

[7] J. Nogues, "Exchange bias in ferromagnetic nanoparticles embedded in an

antiferromagnetic matrix," in the International Journal of Nanotechnology,

Volume 2, April 2005.

157



ANALYSIS OF ELECTROMAGNETIC BEAM PROPAGATION

IN A DIELECTRIC THIN FILM MATERIAL USING CLASSICAL

PERTURBATION TECHNIQUE‡‡‡

A.N. Nwachukwu1 and J.U. Ugwuanyi

2

Department of Physics, University of Port Harcourt, Port Harcourt, Rivers,

Nigeria.

Department of physics, University of Agriculture, Makurdi, Benue, Nigeria

1Email: [email protected]

Abstract

In this work, we investigated the propagation of electromagnetic wave through a

dielectric thin film material, which was considered to be fairly absorbing, strongly

absorbing and non-absorbing. The wave propagation obeyed a wave equation,

which was solved by classical perturbation technique. The plot of the wave

function, φ(r), against the period of the wave, T, showed that the magnitude of the

wave function decreased with increase in the dielectric perturbation, εp(r). This

meant that the dielectric perturbation, εp(r) modulated the wave exponential

decay.

1. INTRODUCTION

In Quantum mechanics, perturbation theory is a set of approximation schemes for

describing a complicated quantum system is terms of a simpler one. The idea is to

start with a simple system and gradually turn on additional interaction term

representing a weak disturbance to the system. If the disturbance is too large, the

‡‡‡ African Journal of Physics Vol. 2, pp. 157 -- 168, (2009)



158



various physical quantities associated with the perturbed system (e.g. its energy

levels and eigenstates), will be continuously generated from those of the simple

system. We can therefore study the former based on our known ledge of the later

(Wikipedia, the free encyclopedia).

A precise approximation is not a contradiction in terms but rather an

approximation with an error which is understood and is controllable. There are

two methods listed in Numerical Methods and Analytic Methods (Lighthill, 1949).

for obtaining precise approximations to solutions of an equation: This project is

about the later, i.e. the analytic approximation which is obtained when some

parametes of the problem is small, hence the name perturbation methods. The

perturbation and numerical methods are not however in competition, but rather

complement one another (E.J. Hinch, 1991).

2. BACKGROUND INFORMATION

Classically, the passage of light through a medium result in reflection

depending on the nature of the material. In some cases attenuation occurs. When

light passes through a vacuum, the disturbance (perturbation) produced (e.g

electric field vector) is characterized by the wave equation (1.0), provided that

there is no material propagation along the medium. (Gann, 1915; Brandson and

Joachin, 1989).

t)(r,E2t

2

2c

1t)(r,E2

(1.0)

However, thin film being a matter, when deposited on plane surface brings about

attenuation or absorption of electromagnetic radiation when it is allowed to pass

through it. In this case, thin film becomes the material through which wave

propagates. Therefore, the wave equation becomes.

0(r)ψr)p

εo

μo

ε2ωψ(r)2 (1.0a)

where is the angular frequency, 0 is the permittivity of the film medium and

o is the permeability of the film material.

With )()( rprefrp (1.0b)

Eqn. (1.0a) becomes

159



.0)()(2)(2 rrprefoor (1.0c)

It is important to note that the action of electromagnetic radiation on a plane

surface is different from its action when there is material propagation on such

surface and that is why in the above equation we have such parameters as

permittivity of the film o and permeability of the film, (see, Fig. 1.0):

3. THEORETICAL PROCEDURE

An electromagnetic wave was propagated through a dielectric thin film material

which were deposited on a glass substrate.

The following observations were made.

I. Radiation was absorbed strongly.

II. Radiation was fairly absorbed and,

III. Radiation was not absorbed.

A diagrammatic representation of the action of the wave on the dielectric thin film

material is shown in fig 2.0 and fig 3.0 below.

Electromagnetic

beam Thinfilm

deposited on

the plane

surface.

Fig. 1.0

Action of Electromagnetic beam on thin film deposited on the plane

surface

160



The expected behaviour is observed as seen in fig. 3.0 where we present the

relative amplitude of the field after transversing/passing through the film media.

Three different cases were investigated.

(a) A non-absorbing barrier p = ref

(b) A barrier with limited absorption

p = ref + p(r), Where p(r) is small

(c) A barrier with strong absorption

p = ref + p(r), p(r) is large

Fig 2.0: Plane wave impinging upon a dielectric barrier. The reference

medium ever corresponding to the fundamental level, whereas the

perturbation p describes the barrier. (Oliver et al, 1994).

O L Z

ref

p

o(z) = exp(ikrefz)

exp(ikrefz)

161



Fig. 3.0: The relative amplitude of field for different dielectric barriers as

investigated. The incoming wave is within the optical region with wavelength of

500nm – 700nm. (Oliver et al, 1994).

Statement of the Equation

The wave propagation obeys a wave equation which we solved by classical

perturbation technique. The statement of the wave equation is as follows.

2 (r) +

2 o o (r)(r) = 0 (1.1a)

(r) + ref + p(r) ( 1.1a)

ZAM

Dielectric

Harrier

Rel

ativ

e am

pli

tude

p (r) = 0

p = ref

p = re +

p(r) p(r) small

p(r) large p = ref +

p(r)

162



Method of Mathematical Solution

2 (r) +

2 o o (r)(r) = 0 (1.2)

The above equation has a solution similar to that of Schrödinger‘s wave equation;

and can be solved by classical perturbation method. The solution of the wave

equation may be assumed to have the form:

(r) = Aeikr

+ Be-ikr

(1.3)

where A and B are arbitrary constants . Then

ikrikr ikeBeikAer

)r(

= ik (Aeikr

– ikBe-kr

) (1.4)

krikr BekAek

r

r

22

2

2 )(

= - ik (Aeikr

+ k2 Be

-ikr) (1.5)

Putting (1.3) and (1.5) in Eqs (1.1a & b) gives

- k2 (Ae

ikr + k

2Be

-ikr) +

2 o µo r (Ae

ikr + k

2 Be

-kr) = 0

- k2 +

2 o µo r = 0

K2 =

2 o µo r (1.6)

Substituting equation (1.1) into (1.6) gives

K2 =

2 o µo r (ref + p (r)

K2 =

2 o µo ref +

2 o µo ref p (r) (1.7)

From Eq. (1.6), it follows that

163



oμoε2ω

2kε(r) (1.8)

Eq. (1.7) can be shortened as follows

K2 = ( + i) (1.9)

where = 2 o o ref

= 3 o µ o p(r)

Recalling that (r) = Aeike

+ Be-ikr

and substituting for k in the above equation

gives

(r) = Aei( + i)

+ Be-i( + i)r

It is important to note that the usual wave function is given in the form.

(r) = o(r) expi(kr - t) (2.0)

Assuming that the wave number, k has both real and complex parts, it can

therefore be expressed thus:

K = + i (2.1)

Putting this into Eq.(2.0) implies

(r) = o(r)expi[( + i)r - t]

= o(r)expi[(r + ir) - t]

= o(r)exp[i(r - r) - it]

= o(r)exp[(ir - r - it]

= o(r)exp[i(r - it) - r]

= o(r)exp - r expi(r - it) ( 2.2)

Substituting Eq.(1.9) into( 2.2) implies:

(r) = o(r)exp - 2 o o p(r) r expi(

2 o o refr - t)

= (r)exp-p(r) r expi(ref r - t) (2.3)

164



4. RESULT/DISCUSSION

The results of the equation are:

(r) = o(r)exp - r expi(r - t)

= o(r)exp - 2 o o p(r) r expi(

2o oref r - t)

= o(r)exp -p(r) r expi(ref r - t)

From equation (2.2) is the modulating factor since it is a function of the

dielectric perturbation, p(r). It, therefore, can be inferred that the larger the

dielectric perturbation, the larger the modulating factor, all other factors being

constant. It is given by:

= 2 o o p(r)

where = the angular frequency, = modulating factor, o = permittivity of the

film medium, o = permeability of the film medium, and p(r) = dielectric

perturbation. It therefore implies that the dielectric perturbation p(r) modulates

the wave propagation through the film.

Similarly, we infer that the magnitude of the relative amplitude depends on the

dielectric perturbation. Form the graphs when p(r) = (0.567 – 1) as in fig. 4.1

and 4.2, there was little effect on the magnitude of the wave function. When

p(r) = (3.5 -5.5), as in fig. 4.3 and 4.4, the effect became more pronounced on

the magnitude of the fig. 4.5, 4.6 and 4.7 respectively, the magnitude continued

decreasing. With further increase of p(r) from 80.5, 100.5 as in figs. 4.8 and 4.9,

the decrease in the magnitude of the relative of attenuation is attained when p(r)

= 600.5. Here the only thing observed was a straight line. It meant that the p(r)

was responsible for the wave exponential decay.

The plot of the wave function against the period of the wave was obtained

as shown in the graphs below. From the graphs, X – represents the wave function

(r) while T – represents time.

The equation used to plot the graph isas follows.

(r) = o(r)exp - p(r) r expi(ref r - t)

165



where o = amplitude of the wave, t = wave period, p(r)= dielectric perturbatio

ref = reference dielectric of the film medium.

5.5 Fig. 4.4 1

166



Fig. 4.5 Graph of ψ (r) against time for Δεp

(r) = 10.5

Fig. 4.6 Graph of ψ (r) against time for Δεp

(r) =22.5

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CONCLUSION

We conclude that perturbation is very useful in the analysis of a given wave

equation as its solution brings out clearly the properties of such wave on

incidence on a plane surface. Although the solutions produced by perturbation

theory are not exact, they are often fairly accurate. The result of the solution

indicates that when electromagnetic beam is propagated in a dielectric thin film

material, some part of the wave is absorbed, partially absorbed and non-absorbing

and that the dielectric perturbation p(r) is responsible for the exponential decay

of the wave.

168



REFERENCES

Gann, R. (1915). Propagation of light through an inhomogeneous medium.

Ann phys. 47, 709-736.

Brandson B.H and Joachin C.J (1987). Introduction to Quantum Mechanics.

K.F Riley. M.P. Hobson and S.J. Bence (1997). Mathematical methods for

physics and Engineering.

Lighthill, M.J. (1949). A technique for rendering approximation solutions to

physical problems informally valid. Phil. Mag. 40, 1179-1201.

Hinch, E.J. (1991). Perturbation methods.

Wikipedia, the free Encyclopedia.

169



THE THIRD REVOLUTION IN SCIENCE AND THE GLOBAL ENERGY

CHALLENGE§§§

Amagh Nduka

Department of Physics and Mathematics

Federal University of Technology, Owerri, Nigeria.

Email:[email protected]

Abstract: To effectively address today‘s energy challenges the world requires

scientific and technological leaps, strong policies, and large investments. In this

paper we explore the role future power plants using combinations of matter and

antimatter as raw materials can play in the resolution of the Global Energy

Challenge (GEC)

1. INTRODUCTION

Physics is concerned with, inter alia, the study of the nature of energy, its

various manifestations, its transformation from one form to another, and its

transmission from one place to another –indeed energy lies at the heart of physics.

For the non-physicist energy generally refers to the large-scale conversion of

stored energy into electricity, locomotion, or manufacture; and hence is seen as

the driver of society.

About two hundred years ago the ability to harness energy served as a

stimulus for the industrial revolution that was accompanied by an explosion of

agricultural productivity, human population, and economic growth. Human

society has, however, never before experienced energy consumption at today‘s

scales. The US Department of Energy (DOE) projects that the world‘s total

energy consumption will increase by 59% between 1999 and 2020, from 382

quads to 607 quads1 (a quad is one quadrillion of energy, which is equal to 3x10

11

kwh). The same report predicts a population increase from 6.0 to 7.5 billion

people on Earth. Thus, in order to provide sufficient energy for its population

come the year 2020 our world must find additional 225 quads of energy in just

over twenty years. If it is realized that the world total of 382 quads of energy was

achieved in over 200 years, finding the extra 225 quads in about 20 years is a

daunting task indeed. What sources will yield this extra energy?

§§§ African Journal of Physics Vol. 2, pp. 169-172, (2009)



170



Against this backdrop, policymakers, commercial enterprises, and

scientists have focused increased attention over the past few years on the issue of

energy. Further, the energy challenge has emerged in public discourse at a level

not seen since the oil shocks of the 1970s. The public debate has placed a

renewed focus on the role of energy technology and policy in meeting several

energy challenges. Prominent among these challenges are energy security,

provision of adequate and reliable electricity, and reduction of harmful emissions

in the face of increasing energy demands associated with economic growth.

2. CONVENTIONAL ENERGY SOURCES

To address this energy problem our world has focused almost entirely on

conventional energy sources which we group hereunder in an unconventional

way:

Fermion Energy Systems: These are quantum systems that have

Newtonian systems as classical analogue – fossil fuels (oil, natural gas,

coal), and fission considered conventionally to be non-renewables, and

renewables (biomass, hydrogen).

Force-particles (vector-boson) energy systems: These are quantum

systems that have Einsteinian systems as classical analogue – Renewables

(wind, hydroelectric, geothermal).

The above classification is in terms of the basic constituents of the energy

systems. The technologies for the extraction of energy from these sources are the

dividends of the first revolution (Isaac Newton), and second revolution (Albert

Einstein) in science. Fossil fuels provide about 85% of the worlds energy, and the

others only about 15%. According to conventional wisdom at some point in the

near future humankind must necessarily witness a declining annual availability of

fossil fuel resources, given their finite quantity and geological limitations to the

rate of their extraction. The disappearance of global resources that are so vital to

our lives is unprecedented and frightening. Our only choice therefore is to use

energy with great efficiency, reduce demand, and most importantly discover new

energy systems, or may be, employing a combination of these strategies.

If one accepts the conventional scenario, the world‘s energy resources

would be exhausted some day. The unavoidable conclusion to be drawn from this

is that our world faces a certain death – it is doomed to die, since a world without

energy is empty. That assumption that the quantity of each of nature‘s resources

is finite is a derivative of the quark model. According to the quark model, all the

furniture of the universe were created at the initial epoch of the universe derived

from the cataclysmic big bang scenario. Fortunately the big bang scenario is mere

figment of the imagination as quarks do not exist, as we shall now show.

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The basic assumption of the quark model is that matter is quark based; and

that prior to the big bang there existed quark-antiquark (and hence matter-

antimatter) asymmetry. With an asymmetry some (finite number) quarks

survived after the big bang and thereafter combined to constitute all the nucleons

and mesons of the universe. Thus matter-antimatter asymmetry of the fermion

world is at the very heart of the quark model; without it the quark idea is dead and

buried.

The works of P.A.M. Dirac; S.Weinberg, A.Salam, and S. Glashow

(WSG); and Amagh Nduka, however, show that there exists manifest matter-

antimatter symmetry in the fermion world, asymmetry in the vector-boson

world, and that there exists a third world, and hence a third independent energy

source, called fermion – vector boson (asymmetric) world2.

The conclusion to be drawn from these works is that nature‘s energy

sources are three in number, namely, matter, i.e. today‘s energy sources,

antimatter, and matter-antimatter sources; and that these resources are renewable

and hence inexhaustible. Matter and antimatter (fermion) resources are

symmetric under parity and hence are inter-changeable; while the matter-

antimatter resources are new and hence untapped – we call it matter-Antimatter

Energy System. There are just two types: fermion – antifermion resources,

with nucleons and antinucleons, or atom and antiatoms as raw materials; and

fermion – vector boson (fusion) resources, with nucleons, anti nucleons, and the

four Ws as raw materials.

The matter – antimatter energy resources are vast; they account for the

energy and fundamental particles from the Sun and Stars. Photovoltaics are

driven by these energy systems. The problem with photovoltaics, apart from their

very low efficiency, is that mankind has no control over these remote energy

sources. Since the science of matter, antimatter, and matter-antimatter processes

are perfectly understood today technological advances in these new area (e.g.

GEC) are now possible. Because these sources of nuclear energy will provide

almost unlimited supply of cheap and clean energy, and without the usual

problems associated with the fission plants, these future nuclear power plants

could help to satisfy the world‘s energy requirements. The technologies for the

extraction of energy from these sources would be the dividends of the third

revolution in science.

The developed world has correctly identified fusion as an untapped energy

resource. The USA, Europe, Russia, Japan, China, India, and South Korea have

even embarked on an international partnership to exploit it via the ITER project.

They averred that fusion is attractive as an energy source because the basic raw

materials are abundant. The problem with the ITER project is that the basic

nuclear reactions adopted for it is the 1939 Bethe process. That work is, however,

172



just a naïve calculation exercise rather than a theory of fusion. The fact that

nobody has achieved Bethe –type fusion in the laboratory in about 70 years

despite the generous infusion of funds should have convinced even unmitigated

optimist that the Bethe process is wrong! Fusion is actually a 3-body, not a 2-

body (Bethe-type), process mediated by strong nuclear interaction-fusion is the

only known 3-body process occurring in nature. Fusion, as has been noted, is

attractive and achievable now the science is known. It must, however, be noted

that fusion is what makes our Sun and the Stars to shine – its raw materials being

the nucleons and the Ws. Thus, once the threshold energy of the W particles

(about 1 Tev) is exceeded Global warming results. Fusion technologies must

therefore take cognizance of this important fact. The alternative fermion-

antifermion process is preferable because it is cheaper and operates below the

threshold energy of 1 Tev.

References

1. The report is available at http://www.eia.doe.gov/oiaf/ieo

2. A.Nduka, Quantum Geometrodynamics, J.NAMP, 13, 1 (2008).

http://www.eia.doe.gov/oiaf/ieo

173



ALTERNATIVE MEANS OF ENERGY SECTOR INVESTMENTS

IN NIGERIA****

Udochukwu. B. Akuru1 and A. O. E. Animalu 1,2

1International Centre for Basic Research, 20 Limpopo Street, FHA, Maitama, Abuja

2 Department of Physics and Astronomy, University of Nigeria, Nsukka

[email protected],

[email protected]

Abstract

The nature and extent of energy demand and utilization in a national economy are, to a large extent, indicative of its level of economic development. Access to energy services is critical to achieving economic and social development targets outlined in the National Economic Empowerment and Development Strategy (NEEDS) and the Millennium Development Goals (MDGs). For a productive economy and for rapid and secure economic advancement, the country must pay maximum attention to the optimal development and utilization of her energy resources and to the security of supply of her energy needs. To achieve this, a country therefore requires an efficient and productive energy sector investment. Nigeria over the years resorted to Federal Government (FGN) sponsored energy sector investment. The outcome which is a natural twist of the total amount of energy produced thereby. In the light of this development, the sole FG-sponsored investment in the energy sector is seen to be highly inefficient. The availability of alternative approaches in the form of Public Private Partnership (PPP), Private Sector Participation (PSP), and Renewable Energy (RE) investment option among other measures is thereby proposed as a good working alternative especially when the results from developed countries are brought to bear.

Index Terms—FGN, Investments, PSP and Renewable Energy.

**** African Journal of Physics Vol. 2, pp. 173 -183, (2009)




174



1. INTRODUCTION

One of the measures of industrial progress of any country is the degree of development of sources of its energy to accomplish useful work. The discovery of sources of energy in nature, the transportation of energy in its various forms from one place to another and the conversion of energy to a more serviceable form are the essential parts of an industrial economy. What really comes to mind when one talk about energy infrastructure in Nigeria is the Electric power, Oil and Gas systems. Energy has a major impact on every aspect of our socio-economic life. It plays a vital role in the economic, social and political development of our nation. Inadequate supply of energy restricts socio-economic activities, limits economic growth and adversely affects the quality of life. Improvements in standards of living are manifested in increased food production, increased industrial output, the provision of efficient transportation, adequate shelter, healthcare and other human services. These will require increased energy consumption. Thus, our future energy requirements will continue to grow with increase in living standards, industrialization and a host of other socio-economic factors. It is pertinent to note that the impact of energy goes beyond national boundaries. Therefore, energy supply can be used as an instrument of foreign policy in the promotion of international cooperation and development [1]. Nigeria is an energy resource rich country blessed with both fossil such as crude oil, natural gas, coal, and renewable energy resources like solar, wind, biomass, biogas, etc [2].

There is a large body of literature comprising of status report on the Federal Government investment into the energy sector already undertaken. However most of these studies undertaken came as public presentations to establish a case for Private Sector Participation in energy sector investment (Power Sector Reforms in Nigeria: Opportunities and Challenges by O.I. Okoro, P. Govender and E. Chikuni; Investment Opportunities in Nigeria by Baba Ijo O.O.O, 2008; Meeting Nigeria’s Power Demand by J.A. Tinubu, 2008; Investments in the Power Sector: Implications for Domestic Gas Sector by J.O. Makoju, 2007; Private Sector Participation in the Electric Power Sector – Risks and Incentives by Bisi Lamikanra, 2005; Power Sector Investment Needs in a selection of developing countries by Odd K. Ystgaard, 2005). These works also cover investments in unexplored energy resources as against conventional resources (Status of Renewable Energy in Nigeria by Francis S. Ikuponisi, 2004; Opportunities and Challenges of an Integrated Energy Policy for Nigeria – Prospective from a Competing Energy Product – Coal by Obi Timothy Nwasike, 2005, National Energy Policy by The Presidency, Energy Commission of Nigeria, 2003; National Energy Databank – a Compendium of Renewable Energy Systems deployed in Nigeria edited by Bala Adamu Azare and Bello Ayuba, 2007; Biogas Energy Use in Nigeria: Current Status, Future Prospects and Policy Implications by Akinbami, J. F. K., et al. 2001; Non-Conventional Energy Source: Development, Diffusion and Impact on Human Development Index in Nigeria by

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Chendo, M. A. C., 2001; Mobilizing Science-Based Enterprises for Energy, Water, and Medicines in Nigeria by NRC and NAS, 2007).

A preliminary assessment of available literature on energy sector investments in Nigeria is based strongly on unhealthy monopoly which demands an urgent implementation of the Energy Sector Reform that will encourage PSP include (Investments in the Power Sector: Implications for Domestic Gas Sector by J.O. Makoju, 2007 and Private Sector Participation in the Electric Power Sector – Risks and Incentives by Bisi Lamikanra, 2005). There have also been some efforts to assess the impacts of the perennial FG investment on the energy sector and its economical implication (Investments in the Power Sector: Implications for Domestic Gas Sector by J.O. Makoju, 2007). There is also extensive assessment on the level of impact which can come from renewable energy investments (Status of Renewable Energy in Nigeria by Francis S. Ikuponisi, 2004; Mobilizing Science-Based Enterprises for Energy, Water, and Medicines in Nigeria by NRC and NAS, 2007 and Solutions for Nigeria (Editorial) by R. R. Colwell, and M. Greene, 2008). Also necessary were the complementary works in comparing Nigeria’s energy sector with other countries by employing data from world energy indices and the automatic linkage between energy services and the MDGs (Energy and Resources – Nigeria (2003); International Energy Agency (IEA), Statistics Division 2006. Energy Balances of OECD Countries (2006 edition) and Energy Balances of Non-OECD Countries (2006 edition); Energy Supply and Economic Growth, World Energy Statistics and Balances, International Energy Agency, OECD FACTBOOK 2008; Key World Energy Statistics (2008); Energy and the Millennium Development Goals by Modi, V., S. McDade, D. Lallement, and J. Saghir, 2006; the Energy Challenge for Achieving the Millennium Development Goals by Mats Karlsson, 2005 and Energy and the Millennium Development Goals by M. Alan, 2004).

These facts are outstanding:

A number of recent studies have attempted to examine the socio-economic impact of Federal Government investment in the sector. Initial results from these studies seem to reveal that the sole-handling of the sector by the FG has done more harm than good [3].

Some analysts contend that reforms should and have produced some positive results in a few instances; there is some evidence that in majority of the case, reforms have led to a complete overhaul of the energy sector [4].

The analysts further argued that from the onset that the implementation of these reforms will not hold water until government monopoly totally hands-off in the sector; noting that these reforms were actually aimed at improving financial and technical efficiency of the energy sector, facilitate divestiture and guarantee future energy supply [5].

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A few ongoing or recently concluded assessment on energy sector investments reveals the need for the Nigerian Government to be keenly aware that sustaining democratic principles, enhancing security for life and property, and rebuilding and maintaining buoyant economy via multi-sector investments are necessary products of a working energy sector investment plan [6].

But the fact that many countries, Nigeria included, prior to reform had largely one mammoth state owned corporation carrying out all activities in the energy sector is primarily responsible for the all-government monopolistic sponsorship of the energy sector. Consequently, a paper written is thus quoted [7];

“In developing countries, at least in the least developed ones, investments will in most cases have to be by the host government or government owned utility with support of soft finance (donor assisted investments). This is because of the low ability to pay for electricity and the need for cross-subsidy in tariffs to facilitate electrification of the country – both factors that works counter to attracting private investors.”

Similarly, a presentation on ‗Financing the Energy Sector‘ [8] holds that for a good energy sector investment the following financial trends should be visible – the recognition of political risks, multilaterals, quality issuers, leverage and pricing and capital markets with a corresponding energy sector boom coming as rapid economic recovery, increased energy demand, industrial consolidation and investment security. Although findings from these studies are not fully conclusive, they do indicate that the sole FGN investment in the energy sector is ineffective and have resulted in a non-commensurate energy output with an attendant steady decline.

This paper, first and foremost, inquires from the perspective of the world, recent trends in the energy sector and the criticality of energy to realising the 8 MDGs. It also assesses the alternative to Federal Government investment and energy resources (especially on renewable) and uses the result to project the extent to which an operational investment plan can be structured. Furthermore, it proposes the options that could enhance Private Sector Participation (PSP) in the energy sector. Lastly, it still advises on the need for a comfortable government platform – a precursor on which any meaningful investment plan cannot be achieved.

The paper equally adds value to the limited but growing literature on energy sector investments in Nigeria. While past studies have mainly assessed the status and outcomes of energy sector investments by the Federal Government, this paper adds value by assessing whether the option is effective for furtherance. Moreover, this paper is one of the very few that have attempted to incorporate privately-driven investments in the energy sector within the context of alternative energy resources.

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2. ELECTRICITY SITUATION IN NIGERIA

Electricity is a form of energy, which enjoys considerable and diverse applications because of its flexibility and ease of transmission and distribution. Availability of electricity remains a major factor in the location of industries and a strong instrument of social development. Its supply is however still inadequate in the country. Commercial electricity is generated mainly from hydropower, steam plants and gas turbines in Nigeria.

2.1 FGN’s Energy Sector Investments (1999-2007)

Recent investments in the energy sector from 1999-2007 by the FGN were aimed at revamping the sector. These investments were meant to primarily increase generation, transmission and distribution of electricity. Between these

FG Amount Invested in Nbillion

6.697

49.784

70.927

44.196

5.207

54.494

70.31372.393

61.101

0

10

20

30

40

50

60

70

80

1 2 3 4 5 6 7 8 9

Year (1999 - 2007)

N b

illi

on

1999 2000 2001 2002 2003 2004 2005 2006 2007

Fig. 1: FG investment in the energy sector (1999 - 2007)

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periods billions of dollars was invested by the Obasanjo administration. The total amount approved by the National Assembly from 1999 to 2007 for the resuscitation of the sector was N575.872 billion, while the total amount released in the various warrants from the budget office was N527.878 billion and the cash back was N435.115 billion [9]. Fig. 1 is used to give the breakdown:

The emphasis of this work is that during the last eight years, the huge

investments made by the FGN into the energy sector have not yielded a tangible

result. These fundamental judgments become obvious: 1. That the political setting

of 1999 – 2007, in spite of huge investments, was no better than previous settings

with similar results; 2. Vis-à-vis, government funding of the energy sector over

the years is money down the drain; 3. The current energy crisis is a precursor of

bad energy investment strategies; and 4. Stating the obvious, there is an urgent

need to revive the ailing energy sector via efficient investment strategies.

2.2 Comparative Analysis with Other Economies

Not long ago, some parts of Europe and America experienced electricity blackouts, which made a lot of news and created a lot of anxiety in the developed countries. In contrast, in major parts of Africa and developing countries, electricity blackout may be considered a luxury and exclusive reserves for those that have access to electricity supply.

The method employed in this analysis was to use the data sourced on ‗Energy Consumption per GDP‘ [10] for the countries under study, as a factor of comparing the level of impact the amount of money invested into their energy sector has interpreted into concrete results of energy being made available for public utility.

Strong bases for analysis are that: “The energy sector is meant to be the bedrock of the nation's economy [1, 3, 4, 6, 11, 12, 13]” in addition to the fact that, the economic development and GDP finds a common factor to compare with a country’s energy output i.e., the expected result of any energy sector investment.; and, energy consumption has a strong link with national income [13].

3. ALTERNATIVE MEANS TO ENERGY SECTOR INVESTMENT

Going by the level of comatose that Nigeria‘s energy sector is currently operating in, there is, therefore, the urgent need to remedy the situation especially as it has to do with investments in the energy sector. This paper has proved so far that, up till the recent past, the FGN has been the sole-investor in the energy sector.

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Funding requirements for the entire energy sector is substantial. New investments are needed for exploration and exploitation activities. The required type of financing is long-term and involves both foreign and domestic financing resources. However, foreign investment capital, in addition to national foreign earnings provides the greater proportion of needed funds. This can be greatly facilitated by the provision of appropriate measures to encourage prospective investors in the energy sector.

Considering the risk element involved in energy projects, investments in the sector should be capable of yielding high rates of return and fast pay back periods in order to attract investors. Owing to other competing needs, government alone cannot continue to provide the major finance for the energy sector activities. Hence private sector participation is necessary and imperative. To attract foreign investments in the energy sector, certain necessary conditions would have to be met. These include:

Improvement in the financial performance of the energy supply companies in the country; and

Providing an environment conducive for investment that also protects our national interests.

By upgrading investment in the energy sector, it implies the adoption of those practices that encourages or utilizes a result-oriented investment approach. These will include:

Getting the investment in the right framework;

Deciding on the goals of restructuring and the ideal industry structure;

Preparing the players to participate in a competitive market;

Privatizing existing and new assets;

Ensuring that a competitive market is implemented properly;

Guarantee mechanisms to be put in place; and

Pursuing international financing within the ambit of the law.

The means by which investment in Nigeria‘s energy sector can be upgraded are innumerable as well as they are interdependent but uniquely defined. This paper has tried to narrow these operations to touch only those areas that are very crucial to the immediate resurrection of the country‘s ailing energy sector by exploration and exploitation, as it were, of the investments in alternative energy (RE) sources and the investments via private sector participation (PSP) respectively.

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3.2 The Need for Private Sector Participation

The shortage of public funds for the much needed energy infrastructure projects in Nigeria led to a strong drive towards restructuring of the sector in order to establish an enabling environment for private investments as a last resort to financing energy infrastructure development.

3.3 Renewable Energy Potentials

―Renewable electricity‖ refers to electric power obtained from energy sources whose utilization does not result in the depletion of the earth‘s resources. Renewable electricity also includes energy sources and technologies that have minimal environmental impacts, such as less intrusive hydro and certain biomass combustion. Hence, the reason it is also known as sustainable energy – the provision of energy such that it meets the needs of the present without compromising the ability of future generations to meet their needs. These sources of electricity normally will include solar energy, wind, biomass co-generation and gasification, hydro, geothermal, tide, wave and hydrogen energy. In the challenge towards Sustainable Development, energy will play a central role. Among the energy issues, the role of renewable energy is becoming increasingly relevant and important.

3.4 Solar Electricity: A RE Option

Going by the serious challenge facing the energy sector in Nigeria, energy experts believe the best energy source for the country is solar energy, especially being located in the tropics where there is so much sunshine to convert.

The works done by H. T. Abdulkarim [14] on Techno-Economic Analysis of Solar Energy for Electric Power Generation in Nigeria, the Committee on Creation of Science-Based Industries in Developing Countries Development, Security, and Cooperation Policy and Global Affairs [15] on Mobilizing Science-Based Enterprises for Energy, Water, and Medicines in Nigeria and, P. A. Ilenikhena [16] on Potential Areas of Solar Energy Application in Nigeria for National Development are masterpieces that will assist to advance solar power as a unbeatable RE option in this section of the paper and so, is referential to this general purpose of this paper.

4. ENERGY AND THE MILLENNIUM DEVELOPMENT GOALS

The world has an unprecedented opportunity to improve the lives of bil-lions of people by meeting the Millennium Development Goals (MDGs), the international community‘s time-bound and quantified targets for addressing extreme poverty in its many forms. Moreover, the Millennium Development Goals (MDGs) are the international community‘s bold commitment to halving poverty in

181



the world‘s poorest countries by 2015. While some of the world‘s poor countries have seen tremendous success in poverty reduction over the past decades and are on track to achieve the MDGs, many others are lagging.

A common finding of the ten Task Forces of the UN Millennium Project has been the urgent need to improve access to energy services as essential inputs for meeting each MDG [17]. Without increased investment in the energy sector, the MDGs will not be achieved in the Nigeria, locally and the world, globally. Therefore, the core message of the report is that energy services are essential to both social and economic development and that much wider and greater access to energy services is critical in achieving all of the MDGs. Strangely enough, energy was not mentioned in the MDG. However, the Energy-MDG link is generally proposed in ref.[13, 18].

5. CONCLUSION

In conclusion, having shown from our comprehensive analysis of the perennial inefficiency of the huge FGN investimate in the energy sector without commensurate result in recent times, it is our view that strong private sector participation should be encouraged especially in solar energy and other new and renewable energy technologies. This would require FGN to put appropriate legislation and security measures in place to attract and protect investors from the private sector and protect consumers in the years ahead.

.

REFERENCES

[1] The Presidency, ―Energy Commission of Nigeria‖, Federal Republic of

Nigeria – National Energy Policy, April 2003.

[2] Ikuponisi, F. S., ―Status of Renewable Energy in Nigeria‖, One

Sky/Energetic Solutions Conference, 2004.

[3] Lawal, L., ―Lights out for Oil-rich Nigeria‖, Fortune Magazine, December

2007.

[4] Director General, Bureau for Public Enterprises, ―Overview on the

Electric Power Sector Reform‖, a presentation at the National Workshop

on Electric Power Sector Reform, April 2005.

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[5] Mark, D. A. B., ―SENATE OF THE FEDERAL REPUBLIC OF NIGERIA

– Votes and Proceedings‖ 6TH

National Assembly, First Session, No. 23,

September, 2007.

[6] Okoro, O. I., Govender, P. and Chikuni, E., ―Power Sector Reforms in

Nigeria: Opportunities and Challenges‖, Proceeding for 10th

International

Conference on the Domestic Use of Energy, Cape Town/South Africa, pp.

29-34, 2006.

[7] Ystgaard, O. K., ―Power Sector Investment NEEDS‖, Power Sector

Taskforce Working Paper A in a selection of developing countries being

partners in Norwegian development cooperation, November 2005.

[8] Bilger, B. R., ―Financing the Energy Sector‖, May 2000.

[9] OJO, J., “Power sector probe: PHCN to refund $142m as CBN, AGF

others give conflicting figures‖, Daily Sun, Abuja, March 2008.

[10] International Energy Agency (IEA), Statistics Division 2006. Energy

Balances of OECD Countries (2006 edition) and Energy Balances of Non-

OECD Countries (2006 edition), Paris: IEA.

http://data.iea.org/ieastore/default.asp

[11] Federal Ministry of Power and Steel, ―Renewable Electricity Policy

Guidelines‖, ICEED Garki-Abuja, December 2006. www.iceednigeria.org

[12] Iloeje, O. C., ―Issues in the Regulation of Renewable Energy Based Power

Supply‖, NCERD Conference 2007, UNN, Nigeria.

[13] Mats Karlsson (2005), the Energy Challenge for Achieving the

Millennium Development Goals. http://esa.un.org/un-energy.

[14] Abdulkarim, H. T. ―Techno-Economic Analysis of Solar Energy for

Electric Power Generation in Nigeria‖, Department of

Electrical/Electronics College of Education, Minna, Niger State, Nigeria.

[15] Committee on Creation of Science-Based Industries in Developing

Countries Development, Security, and Cooperation Policy and Global

Affairs, ―Mobilizing Science-Based Enterprises for Energy, Water, and

Medicines in Nigeria‖, National Research Council of the National

Academies, Nigeria Academy of Science, The national Academies Press,

Washington, D.C., 2007.

[16] Ilenikhena, P. A. ―Potential Areas of Solar Energy Application in Nigeria

for National Development‖, DEPARTMENT OF PHYSICS, FACULTY

http://data.iea.org/ieastore/default.asp

http://www.iceednigeria.org/

http://esa.un.org/un-energy

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OF PHYSICAL SCIENCES, UNIVERSITY OF BENIN, BENIN CITY,

NIGERIA, July 2007.

[17] Modi, V., S. McDade, D. Lallement, and J. Saghir. 2006. Energy and the Millennium

Development Goals. New York: Energy Sector Management Assistance Programme, United

Nations Development Programme, UN Millennium Project, and World Bank.

[18] Kumar, S. Renewable Energy Policy and Planning for Sustainable

Development. Energy Field of Study, School of Environment, Resources

and Development, Asian Institute of Technology

184



SUSTAINABLE APPLICATION OF SOLAR ENERGY AS SMES

IN A DEVELOPING NATION††††

Udochukwu. B. Akuru1 and Ogbonnaya I. Okoro2

1International Centre for Basic Research, 20 Limpopo Street, FHA, Maitama, Abuja

2College of Engineering and Engineering Technology, Department of Electrical and

ElectronicEngineering, Michael Okpara University of Agriculture, Umudike, Abia State. [email protected],

[email protected]

Abstract

That: ―The energy sector is meant to be the bedrock of a nation's economy‖, in

addition to the fact that, ―economic development and GDP finds a common factor

to compare with a country‘s energy output‖, is not new. Therefore, for a

productive economy and for rapid and secure economic advancement, the country

must pay maximum attention to the optimal development and utilization of her

energy resources and to the security of supply of her energy needs. To achieve

this, a country therefore requires an efficient and productive energy sector

investment. Developing countries like Nigeria have over the years relied on

government-sponsored energy sector investments with a track record of

inefficiency. This paper presents a knowledge assessment report (curled from the

recent USA and Nigerian Academy of Sciences collaboration) on the feasibility of

science-based private-sector (PSP) driven enterprises by entrepreneurs for

delivery of solar-power for basic needs including water and health to homes and

communities in Nigeria at affordable cost in order to encourage government to put

in place policies that could engender privately-driven investments in SMEs.

Consequently, this form of investment in solar energy, a renewable energy source,

is seen to be sustainable and supports the attainment of the MDGs as against

much of the current energy supply and use, based, as it is, on limited resources of

†††† African Journal of Physics Vol. 2, pp. 184 --209, (2009)




185



fossil fuels, which is deemed to be environmentally unsustainable e.g. the

wasteful and harmful emission of greenhouse gases through gas flaring.

Index Terms—PSP, SME, Solar Energy/Power and Sustainable Energy.

I. INTRODUCTION

No country in modern times has substantially reduced poverty in the absence

of massive energy use, and countries with higher incomes and higher human

development indexes also tend to be those with higher energy consumption (UN-

Energy, 2007). Notable as well is the fact that, ―The energy sector is meant to be

the bedrock of the nation's economy‖ cited in National Energy Policy (2003),

Ikuponisi (2004), Okoro, Govender, Chikuni (2006), Annemarije (2005), ICEED

(2006), Iloeje (2007), Newberry (2005) and Mats (2005) in addition to the fact

that, ―the economic development and GDP finds a common factor to compare

with a country‘s energy output i.e., the expected result of any energy sector

investment.; and, energy consumption has a strong link with national income‖

Mats (2005). For instance, Table 1 is extracted to show the rate of growth of

electricity production and of GDP‡‡‡‡

(at constant US$ and also constant PPP$2)

and of electricity intensity (which is also the rate of growth of electricity

production less the rate of growth of GDP).

Therefore, energy supply can be used as an instrument of foreign policy in the

promotion of international cooperation and development (National Energy Policy,

2003).

One of the measures of industrial progress of any country is the degree of

development of sources of its energy to accomplish useful work. The discovery of

sources of energy in nature, the transportation of energy in its various forms from

one place to another and the conversion of energy to a more serviceable form are

the essential parts of an industrial economy.

Nigeria is an energy resource rich country blessed with both fossil such as crude

oil, natural gas, coal, and renewable energy resources like solar, wind, biomass,

biogas, etc (Ikuponisi, 2004). See Tables 2 and 3 below which shows various

1. Gross Domestic Product

2. Purchasing Power Parity Dollars

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conventional and non-conventional energy sources and their estimated reserves in

Nigeria

Table 1 Rates of growth of energy intensity 1986-8 to 1996-8 % p.a

Source: World Bank Development Indicators 2002 Resourced: Newberry (2005)

Country Electri

city

produc

tion

GDP

$95

PPP

$96

Electricity

prod/US$

(95)

Electricity

cons./$PPP

(96)

India 7.78% 5.84% 4.08

%

1.84% 1.59%

Pakistan 7.39% 4.59% 1.77

% 2.70% 2.47%

Bangladesh 7.90% 4.44% 2.70

% 3.34% 5.48%

Malaysia 12.38

%

8.57% 5.64

%

3.53% 4.46%

Sri Lanka 6.53% 4.75% 3.42

%

1.68% 1.35%

Nepal 8.10% 5.02% 2.61

% 2.92% 3.28%

Singapore 8.34% 8.81% 6.71

%

-0.44% -0.44%

China 8.52% 9.67% 5.91

%

-1.03% 0.42%

US 3.07% 2.91% 1.86

% 0.16% -0.03%

EU15 1.91% 2.10% -0.19%

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.

Table 2 Nigeria‟s conventional energy resources

Source: Chendo, 2001 Resourced: Ikuponisi, 2004

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Table 3 Nigeria‟s Non-conventional Energy Resources

Source: Chendo, 2001 Resourced: Ikuponisi, 2004

However, the country‘s overdependence on fossil fuels as its primary source

of generating energy proved not to be only highly depleting over the years but is

also bemoaned with the twin setbacks of perennial inefficient monopolistic FGN1-

sponsoring and severe environmental degradation through GHG§§§§

emissions and

gas flaring in very infamous cases with a resultant poor energy output. This severe

environmental degradation according to Adenuga, Kanayo and Friday (2003),

appears to be threatening the long-term development prospects of countries all

over the world, particularly the developing ones such as Nigeria while

Malumfashi (2007) says that, apart from being wastage of valuable resources, the

practice of gas flaring runs contrary to Nigeria‘s obligations to reduce GHG

emissions under, inter alia, the 1992 United Nations Framework Convention on

Climate Change and the 1997 Kyoto Protocol. On the other hand, Ikuponisi

1. Federal Government of Nigeria

2. Green House Gas

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(2004) holds that the reserve/production ratio or the depletion time of Nigeria‘s

crude oil if there is no further efforts to increase the reserve and if the production

still remains at the expected 1989 level, may be within the next 25-30 years.

The present energy situation in Nigeria is that generating plant availability is

low and the demand – supply gap is crippling. Poor services have forced most

industrial customers to install their own power generators, at high costs to

themselves and the Nigerian economy. Access to electricity services is low in

Nigeria. About 60 percent of the population – approximately 80 million people

are not served with electricity. Per capita consumption of electricity is

approximately 100kWh against 4500kWh, 1934 kWh and 1379 kWh in South

Africa, Brazil and China, respectively (ICEED, 2006). Under a business-as-usual

scenario, the proportion of Nigerians without access to electricity services will

continue to increase over time.

In the wake of these developments, it becomes practical for this study not to

delve into establishing a critique for branding the conventional FGN-sponsored

energy sector as inefficient and unproductive as there is a large body of literatures

to this effect coupled with a previous paper produced by the authors on

Alternative Means of Energy Sector Investments in Nigeria. But in working

along the line from the recent USA and Nigerian Academy of Sciences

collaboration on the feasibility of science-based private-sector driven enterprises

for delivery of solar-power for basic needs including water and health to homes

and communities in Nigeria at affordable cost, we will hence focus on

government-encouraged policies that could engender privately-driven investments

in Small and Medium-scale Enterprises (SMEs) in Renewable Energy

Technologies (RETs) – solar energy in this case. This measure is seen as a

palliative to cushion the effect of a lingering energy crisis.

The objective of this paper is to apply SMEs as privately-driven investment

instruments in mitigating the widening effect of the energy demand-supply gap in

Nigeria with a view to incorporating solar energy in the decision-making process

of sustainable availability of energy resources. This would ultimately reverse the

tendency to treat the teeming population with the harsh economic reality of erratic

or unavailable energy supply. Following this introduction as section one, the

paper is further divided into five sections. Section two examines what constitutes

the energy sector in Nigeria and reviews theoretical background. Section three

discusses Nigeria‘s overdependence on fossil fuels which have led to

environmental degradation and high depletion rate in Nigeria with regards to

GHG emissions while section four suggests solutions for energy availability by

adopting SMEs in solar energy with expected outcome of economic improvement

and environmental sustainability. Section five, based on background knowledge

190



from section four, then presents a knowledge assessment report (curled from the

recent USA and Nigerian Academy of Sciences collaboration) on the feasibility of

science-based private-sector (PSP) driven enterprises by entrepreneurs for

delivery of solar-power for basic needs. The paper later ends with some

concluding remarks and outlook in section six.

The core of the paper is formed by a description of online research findings by

relevant literatures. Finally, the propositions that have been formed in the initial

stage of the research will be discussed in the context of programmes and

initiatives on productive uses of SMEs to generate energy investments for

improving on the current energy system.

2. PRESENT STATUS OF NIGERIA‟S ENERGY SECTOR

Nigeria is blessed with abundant primary energy resources enough to meet its

present and future development requirements. Just for the records, the country

possesses the world‘s sixth largest reserve of crude oil. It is increasingly an

important gas province with proven reserves of nearly 5000 billion cubic meters.

Coal and lignite reserves are estimated to be 2.7billion tons, while tar sand

reserves represent 31 billion barrels of oil equivalent. Identified hydroelectricity

sites have an estimated capacity of about 14,250MW. Nigeria has significant

biomass resources to meet both traditional and modern energy uses, including

electricity generation. The country is exposed to a high solar radiation level with

an annual average of 3.5 – 7.0kWh/m2/day. Wind resources in Nigeria are

however poor - moderate, and efforts are yet to be made to test their commercial

competitiveness.

These include reserves of crude oil and natural gas, coal, tar sands and renewable

energy resources such as hydro, fuelwood, solar, wind and biomass. It is a current

fact that Nigeria, like many mostly rural developing countries, is not able to

provide all its population affordable electric power and that, two-thirds of

Nigerians, around 100 million people, lack household electricity (Collwell and

Greene, 2008).

In 2004, Nigeria‘s energy consumption mix was dominated by oil (58

percent), followed by natural gas (34 percent) and hydroelectricity (8 percent)

(Country Analysis Briefs – Nigeria, 2007). Coal, nuclear and other renewable are

currently not part of the country‘s energy consumption mix. Between 1984 and

2004, the share of oil in Nigeria‘s energy mix has decreased from 77 percent to 58

percent. Natural gas consumption increased from 18 percent to 34 percent.

Hydroelectricity has seen a slight increase as well from 5 percent to 8 percent.

191



However, what really comes to mind when one talk about energy

infrastructure in Nigeria is the Electric power, Oil and Gas systems. Therefore,

energy will imply electric power henceforth in this paper. Electricity is a form of

energy, which enjoys considerable and diverse applications because of its

flexibility and ease of transmission and distribution. Availability of electricity

remains a major factor in the location of industries and a strong instrument of

social development. Its supply is however still inadequate in the country.

The Nigerian power sector operates well below its estimated capacity, with

power outages being a frequent occurrence. To compensate for the power outages,

the commercial and industrial sectors are increasingly using privately operated

diesel generators to supply electricity with both economic and environmental

disadvantages. In 2004, total installed electricity capacity was 5.9 gigawatts (GW)

i.e. 5900 megawatts (MW). Total electricity generation during 2004 was 19

billion kilowatthours (Bkwh), while total consumption was 18 Bkwh.

Commercial electricity is generated mainly from hydropower, steam plants

and gas turbines in Nigeria. The installed capacity for electricity generation,

which is 98% owned by the Federal Government, increased by a factor of 6 over

the period 1968 to 1991 and by 1991, stood at 5881.6 MW. No further addition to

generating capacity was experienced over the subsequent decade. Over the years,

the availability varied from about 27% to 60% of installed capacity, while

transmission and distribution losses accounted for about 28% of electricity

generated. Despite endemic blackouts, customers are billed for services not

rendered, partially explaining Nigeria's widespread vandalism, power theft and

Power Company Holding of Nigeria‘s (PHCN) problems with payment collection.

Generating plant availability is low and the demand – supply gap is crippling.

Poor services have forced most industrial customers to install their own power

generators, at high costs to themselves and the Nigerian economy with great

environmental risks. PHCN‘s business operations are inefficient. The system

suffers from chronic under-investment, poor maintenance, and un-recorded

connections and under- billing arising from a preponderance of un-metered

connections. The utility‘s financial performance, as well as its ability to serve

customers satisfactorily has been consistently poor.

Electricity offers neat, flexible and variety in usage to the end –use services

that it is widely recognized as an energy form that drives economic development

and improves the quality of life. Its long -term benefit outweighs the cost of

extending it even to the poorest population. In Nigeria, improving electricity

remains a regular feature of political campaign agenda along with such laudable

programmes as employment generation, qualitative education, affordable housing

192



etc. However actualization through adequate funding and proper management has

consistently proved elusive. There is no commitment to well articulated

programme to ensure reliability of electricity. The relevance of electricity is

recognized only during power failure. Even at that, there is always the temptation

to embark on ad-hoc measures and these tend to aggravate the situation.

The political class has not shown much understanding in respect of investment

in the electricity infrastructure, management, capacity building and staff

motivation. Consumer understanding and cooperation on electricity issues are rare

to come by and many of the consumers are increasingly vocal about their

dissatisfaction with the performance of the electricity sector, not minding the fact

that they also contribute to the poor performance through illegal connections,

system overloading and other sharp practices. We therefore seek to arouse the

interest off all stakeholders, various tiers of government, the political class,

private investors, leaders and followers of thought as well as the entire populace

to the reality of inherent weakness in electricity infrastructure, which is

responsible for poor electricity services presently experienced due to past neglect

of the industry. Challenged by the present weakness, it calls for urgent, aggressive

and sustained investment to ensure reliability of electricity services.

3. ADVERSE EFFECTS OF OVERDEPENDENCE ON FOSSIL FUELS

Nigeria‘s energy sector is characterized by two major sub-sectors –

Renewable and Non-renewable energy sources. The country‘s installed capacity

comes primarily from the non-renewable energy resources (see Fig. 1 below).

Since the late 1960s, the economy has been solely dependent on the

exploitation of oil to meet its development expenditures. In 2001, oil revenue

alone accounted for about 98.7% of exports and 76.5% of total government

revenues. Ikuponisi (2004) opines that ―The National energy supply is at present

almost entirely dependent on fossil fuels and firewood (conventional energy

sources) which are depleting fast.

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Diane Abbot in her article entitled "Think Jamaica is bad? Try Nigeria" was

quoted in Malumfashi (2008) to have said that:

"Nigeria's greatest blessing has been oil; but it has also been its greatest curse. It

is the sixth biggest oil producer in the world. Oil accounts for 95 % of exports by

value and 80 % of government revenue amounting to billions and billions of

pounds. But the discovery of oil has been an ecological disaster for the Niger

Delta (one of the most populous parts of the country) where the oil is extracted.

Shell and other Western Oil companies have, in collusion with successive military

dictatorships, raped the region. Petrol contamination of the water table has made

local water undrinkable. Farming and fishing grounds have been ruined and gas

flaring in the Delta is cited as Africa's single biggest contribution to greenhouse

gas emissions. It is symbolical of the brutally exploitative nature of the oil

industry in Nigeria that the natural gas by-product (which other oil producers like

Trinidad liquefies and market) is simply burnt in giant flares which cause

incalculable environmental damage.‖

Also, coal (though not suggested as a major alternative in this paper) which used

to be the country‘s mainstay of energy before the discovery of crude, even with

abundant unexplored reserve deposits, has continued to suffer unattended neglect

over the years (Nwasike, 2003; Ministry of Solid Minerals Development, 2006).

Similarly, areas which used to be a great source of income and job creation (both

Fig.1. Nigeria‘s over-dependence on non-

renewable fossil fuel (adapted from Animalu (2007)

194



internally and externally) like coal mining and agriculture have suffered in no

small way because of the advent of the ―resource curse‖ – crude oil.

Consequently, overdependence on crude oil over the years has led the country to

focus primarily on it for its energy resources at the disadvantage of other relevant,

abundant and sustainable energy sources without any observed improvement in

energy utility. Other effects like corruption, dearth of local industries, loss of

livelihood, environmental pollution and more recently, militancy in the Niger

Deltas could all be visibly traced to the unwholesome habit of over depending on

oil as our major source of income.

If the Nigeria could shed off a good percentage of its dependence on fossil fuels

maybe other reliable and competitive energy sources will become visible as we

proffer in the next section of this paper.

4. SOLUTIONS FOR ENERGY AVAILABILITY

4.1 Need for Reforms

In Electricity Reforms, the first step involves passing an Electricity Law to allow

private investment, then establishing regulatory agencies to set tariffs, unbundling

the natural monopoly transmission and distribution businesses, and in some cases

privatising distribution companies and some generation assets. The typical form

of private participation has been by Independent Power Producers (IPPs) signing

long-term Power Purchase Agreements (PPAs) with the Single Buyer (normally

the incumbent power company).

The results of these reforms could be disappointing due to currency crises – for

instance the recent global economic recession – undermining the ability of the

Single Buyer to honour the PPAs, which were often largely denominated in

foreign currency. More generally, the tariffs needed to finance foreign direct

investment (given the perceived level of risk and the short term of most debt

finance) has led to high initial charges for electricity purchased from these IPPs.

The mismatch between the cost of these new PPAs, the average cost of existing

generation (with tariffs based on written down asset values and often under-priced

fuel), the lower average tariff of retail electricity, and the even lower average

revenue per unit generated, places the incumbent power company or its

counterparts under increasing financial stress.

195



At the beginning of July 2006, the take-off of 18 new successor companies from

the monopoly of PHCN (formerly NEPA*****

) commenced a winding down aimed

at ensuring that Nigeria has efficient, safe, affordable and cost-effective electricity

industry that will not only provide continuous electricity supply to consumers in

all geographical areas in Nigeria but will also support a more robust economic

growth in the country.

These objectives were hoped to be met largely through the implementation of the

following processes:

a. Vertical separation of NEPA into generating, transmission and/or dispatch

and distribution.

b. Establishment of a transmission company (TransysCo).

c. Horizontal unbundling of each of the functional segments into a number of

competing, successor companies (NBUs†††††

) as follows:

d. 6 Generating Companies (GenCos).

e. 11 Distribution Companies (DisCos).

f. The creation of a Special Purpose Entity to act as a financial vehicle to

take over NEPA legacy debts and stranded costs.

g. The establishment of a regulatory agency that will be called Nigerian

Electricity Regulatory Commission to oversee and monitor the activities

of the NBUs.

h. Creation and operation of a wholesale electricity market in Nigeria.

i. A Rural Electrification Agency to expand access to electricity to the rural

areas.

j. A Power Consumer Assistance Fund to subsidize the tariff for

underprivileged consumers.

k. Privatization of the newly established generation and distribution

companies as separate entities.

l. Management Contract for the Transmission Company.

The next section of this paper is buoyed in the light of the above-stated

implementation programmes for meeting the objectives of the electric power

sector reforms.

***** National Electric Power Authority ††††† Natioanl Business Units

196



4.2 Facts and Fears for SMEs

The only way to reduce poverty in a sustainable way is to promote economic

growth, through wealth and employment creation. In developing countries, SMEs

are the major source of income, a breeding ground for entrepreneurs and a

provider of employment.

-UNIDO, WSIS Report, February 2003

The fundamental issue in SME Development in Nigeria is the lack of total

commitment by government at all levels to developing the sector (Abugu, 2007).

It is very important to draw attention to this problem because of the thick cloud

surrounding the role of government in Enterprise Promotion in this country.

SMEs are important because on average, they comprise a great percent of the

economy. As of July 2006, close to 140 million SMEs in 130 countries employed

65 percent of the total labour force (World Bank, 2006). Moreover, SMEs are

attributed to be the driver of economic growth and innovation (Vadim, 2008). The

total number of SMEs in the economy depends on the rate of SME creation and

rate of SME destruction. Profitable market opportunities increase the rate of SME

creation. This increases the total number of SMEs in the country, which increases

job creation and income per capita. As people become wealthier, they will

increase their consumption, which in turn will open up new market opportunities

that will entice the creation of more SMEs.

Contrary to multinational corporations, the growth of SMEs directly benefits

the country because most SMEs are domestic firms. This reinforcing dynamic

generates economic growth. The reinforcing loop of innovation also drives

economic growth. As the number of SMEs increases, their knowledge of their

product and industry increases. Their knowledge allows them to innovate on the

product or process, which helps them form a competitive advantage to generate

more profits. Again, market opportunity as captured by the profitability of SMEs

will encourage more people to establish their own SMEs to capture the

opportunity. And the establishments of market zones enhance product availability

and affordability.

In addition, the development of SMEs can also help to achieve other

development goals. SMEs can either provide goods and services in areas critical

197



to development, such as health and education, or provide a source of income to

disadvantaged people. For example, efforts to develop women entrepreneurs help

increase gender equality by providing women with a source of income. In this

research paper, we also view the possibility of speedily bridging the existing

energy demand-supply gap at the point of proactive implementation of SMEs in

energy.

However, some fears exist that once the investment is sunk, it will not be

allowed to earn a remunerative return due to unfavourable government and

market policies such as credit schemes, interest rates and regulatory agencies. The

electricity sector is particularly problematic as private investors supply an

essential service directly to a large fraction of the voting population in

competition with under-priced supply from the state-owned sector. As prices will

have to rise to ensure that the investments are remunerative, the price rise will be

associated with the reforms that brought in private investors, and will be doubly

resisted on that account.

Also, many of the current beneficiaries of opaque accounting, cross-subsidies,

patronage in the appointment of regulators and senior management, etc., will have

an interest in preserving the status quo, including the low prices that deter

efficient commercial competition. The fact that external bodies such as the World

Bank are pressing for such reforms provides additional reasons for populist

resistance, for the price rises that are needed to ensure investment adequacy yield

current pain while the future benefit of improved quality of service may be some

way in the future, and beyond the politician‘s invariably short time horizon.

But understanding of the factors and underlying mechanisms that

influence the establishment and growth of small enterprises in energy could

alienate these fears. Also, research is needed to understand the linkages

between energy services and productive uses. Identifying factors that can

play a role in shaping these linkages can be a first step. Because these factors

can be expected to differ depending on the entrepreneurs themselves, the

energy supply characteristics, economic circumstances and other forces, is a

necessary second step to understand the mechanisms at play.

4.3. SMEs and RE: A Viable, Sustainable Alternative

With the exception of the upstream oil and gas sub-sectors, and to a smaller

extent the electricity sub-sector, government has been largely responsible for the

ownership and operation of the energy sector industries. In particular, investment

198



capital had been sourced from public funds, while the industries had relied on the

sense of public interest, within management, as the motivation for responsible and

transparent management of the industries.

The funds required for the maintenance and refurbishment of the energy

supply infrastructure, and for the expansion of capacity, are enormous. In the face

of increasing demands on government for investments in other areas of the

economy such as transport, health, education and security, government has been

unable to provide the funds needed by the energy sector. Efficient and transparent

management of the industries had also not been achieved. Consequently,

established facilities had progressively deteriorated while new capacity had not

been added, inspite of increasing demand. Furthermore, the funding and

management deficiencies had given rise to inadequate and unreliable supply,

especially of electricity and petroleum products, insecurity of the energy supply

system and loss of productivity the economy.

We strongly believe that increased private sector participation through the

establishment of SMEs in the energy sector will attract new investments to the

sector, while the profit motive will assist in solving much of the management

problems experienced under public ownership. The restructuring of the sector,

required to bring this about, which involves both deregulation and privatization

has already been achieved.

As a novel model, this paper, second only to the knowledge assessment report

emphasized in section 5, fronts the greater proportion of private investment funds

required by the sector to be indigenous capital that could come through the setting

up of SMEs for energy generation. But as it has been highlighted previously, this

SMEs may find it difficult catering for capital-intensive energy investments in

conventional energy resources, and so, the unexplored renewable energy (RE)

sector fits in as a viable alternative. Thus, the environment must be made

conducive to attract these types of investments to the sector.

Since RE ensures sustainable and environmentally friendly energy practices, it

is important to introduce its meaning at this point with respect to how it can be the

focus of SMEs in energy systems. But before that is done, the principle of energy

sustainability is explained by Wikipedia (2009) and IAEA (2005) as the provision

of energy such that it meets the needs of the present without compromising the

ability of future generations to meet their needs. A broader interpretation may

allow inclusion of fossil fuels and nuclear fission as transitional sources while

technology develops, as long as new sources are developed for future generations

to use. A narrower interpretation includes only energy sources which are not

expected to be depleted in a time frame relevant to the human race – in this case –

199



all renewable sources, such as biofuels, solar power, wind power, wave power,

geothermal power and tidal power. However, Mandil (2008) points out that

moving towards energy sustainability will require changes not only in the way

energy is supplied, but in the way it is used, and reducing the amount of energy

required to deliver various goods or services is essential.

―Renewable electricity‖ refers to electric power obtained from energy sources

whose utilization does not result in the depletion of the earth‘s resources (ICEED,

2006). Renewable electricity also includes energy sources and technologies that

have minimal environmental impacts, such as less intrusive hydro and certain

biomass combustion. These sources of electricity as earlier pointed, normally will

include solar energy, wind, biomass co-generation and gasification, hydro,

geothermal, tide, wave and hydrogen energy.

On the other hand, it is good to establish the fact that renewable energy has

been largely exploited through traditional firewood over the years but at a

disadvantage to the environment. Outside this, every other form of RE sources

supports sustainability.

Broadly speaking, renewable energy is derived from non-fossil and non-

nuclear sources in ways that can be replenished, are sustainable, and have no

harmful side effects. The ability of an energy source to be renewed also implies

that its harvesting, conversion and use occur in a sustainable manner, i.e. avoiding

negative impacts on the viability and rights of local communities and natural

ecosystems

Apparently, increased power generation from conventional sources and grid

extensions alone will not achieve electricity access expansion targets rapidly and

cost-effectively in a developing country like Nigeria. Accelerating rural

electrification coverage will require an aggressive deployment of multiple supply

options and business delivery systems e.g. SMEs. In line with the provisions of

the EPSR‡‡‡‡‡

Act, the Federal Government in seeking to meet national electricity

access targets through the following strategies according to ICEED (2006) could

be exploited for proactive small-scale investment strategies:

Grid-based extension for proximate areas;

Independent mini-grids for remote areas with concentrated loads where

grid service is not economic or will take many years to come; and

Standalone renewable electricity systems for remote areas with scattered

small loads.

‡‡‡‡‡ Electric Power Sector Reform

200



Non-conventional or renewable energy is a key element in the overall strategy

of the Federal Government in rapidly expanding access to electricity services in

the country. Beyond large hydropower, the current total contribution of renewable

energy in Nigeria‘s electricity industry is about 35MW composed of 30MW small

hydropower and 5MW solar PV. This represents about 0.6% of total nominal

electricity generating capacity in the country.

While most renewable energy projects and production is large-scale,

renewable technologies are also suited to small off-grid applications, sometimes

in rural and remote areas, where energy is often crucial in human development – a

critical factor this paper employs in recommending it for SMEs application; the

focus being on solar power investments.

5. EFFICIENT AND AVAILABLE ENERGY DELIVERY BY MEANS OF

SOLAR SMES

To set the stage for appreciating the basic issues, let us state the obvious fact

that the well-known example of a failed science-based enterprise in Nigeria is

the defunct NEPA which, until its scrapping was nicknamed, ―Never Expect

Power Always‖. There are other examples which we do not intend to name,

because though the enterprises may be dead they have not yet been buried. Our

objective is to make the case and set the stage for a new methodology for

creating science-based enterprises in Nigeria for meeting the MDGs; whose

achievement is critical on access to energy services (Modi, McDade, Lallement,

and Saghir, 2006).

5.1. Historical Perspective

In 1957 when the former Soviet Union launched an artificial satellite

(―sputnik‖) into orbit round the Earth, the USA, leading the Western

democracies, responded by launching a mission-oriented space research

programme whose target was to land man on the moon and bring him safely

back to Earth. It required the visionary leadership of President John F. Kennedy to

set the target and the political stability, economic strength, scientific

knowledge and technological know-how of the American people to meet the

target in 1968 when American astronauts landed on the moon and returned

safely back to Earth. Astronaut Amstrong called it ―a leap for man, a giant leap

for mankind‖.

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Nearer home, in 1962, the French exploded a nuclear bomb in the Sahara

Desert, thereby challenging the whole African continent. While the Nigerian

Government accepted the French assurance that it was safe, the Ghanaian

President Kwame Nkrumah, reacted by setting up the Ghanaian Atomic Energy

Commission and directing Ghanaian scientists to learn everything about the

French atomic blast and advise him accordingly. Today, despite its traumatic

problems, Ghana‘s Atomic Energy Commission is the clearing house for

nuclear science and technology in the West African sub-region.

In 1963 a disagreement in the Far East between the two socialist giants,

USSR & People‘s Republic of China, resulted in a sudden withdrawal of all

Soviet scientists and technicians from China. The Chinese Premier, Chairman

Mao Tse-Tung, picked up the challenge and in the following year, just before the

1964 Peking Symposium on Science, China displayed the progress made by its

scientists in science and technology by exploding the atomic bomb and

drawing up a Blue Print for the Development of Science. While one of the

authors was visiting China some years back, the China Daily of Thursday

December 9, 1999 carried a front-page picture of the Chinese President Jiang

Zemin shaking hands with the scientist, Qian Xuesen, a world-famous expert

on rockets and aerodynamics, in gratitude for his work towards China‘s

development of science and technology, especially the explosion of its first atom

bomb in1964, the launching of its first man-made satellite in 1970, and the

firing of its first transcontinental ballistic missile towards the Pacific in 1980.

In 1973, the Arab-Israeli War forced the Arabs to place an embargo on oil

supply to the Western World for supporting Israel. In response to the resulting

world-wide ―energy crisis‖, the USA directed its scientists and technologists

to embark on comprehensive energy development programme.

This brought to the market among other things the solar cell which was

invented during the space race of 1950s. As a token of one of the one of the

author‘s participation in Nixon‘s ―Project Independence‖ in Energy at Lincoln

Laboratory of Massachusetts Institute of Technology, USA, he could still boast of

having in his possession the solar cell device that he brought back to Nigeria

32 years ago. It was one of the features of the exhibitions of the Solar

Energy Society of Nigeria (founded 1980) in an attempt to create Nigerian

awareness and interest in solar energy.

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5.2. Nigeria’s Experience with the “Energy Crisis”

What was Nigeria‘s response to the ―energy crisis‖? It began on August 2-4,

1978, when the National Policy Development Centre (―THINK TANK‖) at the

Supreme Headquarters, Lagos organized an Energy Policy Conference at Jos,

capital of Plateau State. The result was Decree No. 62 of 1979 establishing the

Energy Commission of Nigeria which was signed into law by the Head of the

Federal Military Government at the time, and later the Civilian President

Olusegun Obasanjo. The Decree was one of the hand-over notes to the

succeeding Shehu Shagari Administration that took over the reigns of

government in 1979. However, it was not until 1988 that the Energy

Commission came into existence following the amendment of Decree No. 62 of

1979 by Decree No. 32 of 1988 and Decree No. 19 of 1989. This means that the

Energy Commission of Nigeria has been in place for just over 20 years despite the

fact that its vision was reasonably well articulated over 30 years ago.

On a positive note, on Tuesday, March 10, 1981, during the term of office of

the Shagari Administration, three Professors of Physics representing three of

the nation‘s older Universities (Ahmadu Bello University, Obafemi Awolowo

University (then University of Ife), and University of Nigeria, Nsukka) and

the Honourable Minister for the newly created Federal Ministry of Science

and Technology were invited to appear before the National Assembly

(Senate) Committee on Science and Technology. The Committee, under the

Chairmanship of distinguished (now late) Senator (Engr.) Garba Matta

(Panshin, Plateau State), wanted to ascertain from the practitioners in the field,

what progress had been made, vis-à-vis Mr. President‘s declaration while on tour

of USA that ―Nigeria was going nuclear‖. The highlight of the hearing was the

insertion of N15 million take-off grants in the budget of the Presidency for

―National Energy Research Projects‖. Eventually, only N10 million was released

and shared equally between the three Centres at ABU§§§§§

, OAU2 and UNN

3

while a matching grant was later made to Othman Dan Fodio University,

Sokoto, for the solar energy research centre there. The existence of these

Centres is our record of success. However, the question agitating many minds is

whether the Energy Commission supervising the National Energy Centres have

what it takes to nurture science-based enterprises in the energy field,

especially solar and renewable energy, in the same way that the NCC4 nurtured

the mobile phone revolution.

1. Ahmadu Bello University

2. Obafemi Awolowo University 3. University of Nigeria, Nsukka

4. Nigeria Communications Commission

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It seems to us that, in this country, all failures are blamed on the

leadership, without looking at the structure of the organization in which the

leadership is expected to perform. For this reason, during the 1999 visit to the

Chinese Academy of Science Secretariat at Beijing, China, one of the authors

posed a question to his Chinese hosts: how are the State Ministry of Science

and Technology and the Academy of Science of the People‘s Republic of

China organized for industrialization? An answer which was deduced from a

careful study of the brochures of the two institutions was that, by and

large, the organization of science and technology policies and activities

followed strategic planning perspective consisting of three levels, namely (1)

aims & objectives-level, (2) roles & responsibilities-level and (3) implementation

activities & targets-level. The author subsequently discussed all this in a book

entitled Education, Science and Technology Agenda for Nigeria in the

21st Century, (Nigerian Academy of Science, ISBN-978-2162-16-7, 2000).

That aside, it would appear an answer lies in knowledge assessment methodology

which is described next.

5.3. The Knowledge Assessment Methodology

The prospect for promotion of business and industrial activities in solar-

PV power systems and products is now much brighter than it was three decades

ago when the Nigerian Academy of Science was founded (1977) and the

Solar Energy Society of Nigeria was inaugurated (1980). However, in order

to overcome the difficulty lying in the shadowy landscape linking scientific and

technological development to national security, political stability and economic

strength which stalled our efforts throughout the 20th

Century, the authors wish

to simply reproduce the recommendation of the joint Nigerian Academy of

Science and USA National Academy of Science Knowledge Assessment

project titled, Mobilizing Science-Based Enterprises for Energy, Water, and

Medicines in Nigeria presented by the project chairman, Michael Greene,

and Rita R. Colwell of USA National Academy of Sciences as an ―editorial‖

in Science Vol. 319, 25 January 2008 (www.sciencemag.org) under the title,

SOLUTIONS FOR NIGERIA:

―Nigeria, like many mostly rural developing countries, is not able to

provide all its population with basic services such as safe potable piped water

and affordable electric power. The economics of extending the electric grid and

water distribution network into the countryside are daunting, and the

people who lack electricity, safe water, and effective medicines are usually

poor and clustered within extremely dense urban communities or live in highly

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dispersed rural communities with limited infrastructure. Two-thirds of Nigerians,

around 100 million people, lack household electricity, and about as many do not

have safe drinking water. Nigeria also has the world‘s largest burden of people

suffering from infectious diseases, mostly malaria, without effective treatment.

―Yet there are solutions. In Karnataka, India, the Solar Electric Light

Company (SELCO) sells, installs, and services solar home lighting systems

to tens of thousands of poor villagers—at a profit… These are sustainable

solutions in the sense that they do not depend on donor funds or ongoing financial

support from a government, because the profit comes from sales to consumers

alone. Can the private sector of a country such as Nigeria be mobilized to

provide basic services to the population that the government cannot

afford—at a profit? Many companies have developed business models that,

incorporated into a new approach to sustainability, can meet the needs of

marginal populations for electricity, safe water, and medicines, while

providing new sources of jobs and income. Their models include robust, but

not necessarily low-tech, products, customer training, microcredit, service

contracts, and franchising opportunities. As limiting as the conditions in Nigeria

seem to be, the great advantage to a company is the country‘s huge

number of potential clients. In India and other countries with large numbers of

poor people, companies aiming at the customer base at the wide bottom of the

economic pyramid have produced new, innovative products and services at

substantial profit to themselves as well as benefits to their customers. The

market in Nigeria for electric power, safe water, and effective malaria therapy

exceeds the total populations of all but a handful of countries.

―Mobilizing Science-Based Enterprises for Energy, Water, and Medicines

in Nigeria, a recent study issued by the U.S. National Academies and the

Nigerian Academy of Science, addresses the potential for a sustainable approach

to supplying these basic services to Nigeria‘s poor by encouraging private

companies to become involved. This study revolved around the findings of three

workshops that joined successful entrepreneurs from other countries,

including executives of SELCO, WaterHealth, and Potters for Peace, with

Nigerian business leaders and scientists. They prepared business models,

including cost estimates, adapted to the Nigerian market for companies to

manufacture, sell, and install solar photovoltaic units and water filtration

systems for the rural and urban poor, and… The study concludes that

businesses providing small-scale photovoltaic systems… could operate profitably

in Nigeria and in other countries of the region. But adoption of this approach

may require government incentives, educational campaigns, and a

corresponding shift in strategy by donor organizations and bilateral aid

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agencies. International aid programs may have to be reconfigured so that

they resemble venture capital companies with a diverse portfolio of

investments (taking into account that startup companies may not always

succeed) rather than discrete, one-of-a kind grants‖.

In taking a look at the actual report assessment, the focus of the Committee on

Creation of Science-Based Industries in Developing Countries Development,

Security, and Cooperation Policy and Global Affairs (2007) on solar photovoltaic

systems suitable for rural households, these systems usually consist of several

components (see Fig. 2). They include a PV module containing the silicon cells to

be mounted on the roof or another sunny spot, a battery for storing electrical

energy for use at night, a charge controller, wires and structural frames, and

outlets for lights and other appliances. Such a system can operate several

fluorescent lamps (often 4), a radio or television, perhaps a fan.

The system normally operates on 12 volts, direct current. Long lasting, deep-

cycle batteries, which can discharge 80% of their charge during extended overcast

weather, are best, but automobile batteries commonly available in Nigeria, also

could be used. The charge controller prevents damage to the system in the event

of overcharging by the solar module or prolonged battery discharge from overuse.

Other requirements are installation, are periodic battery replacement (once every

five years for a deep-cycle battery), and user training; they are often part of a

service contract for maintenance.

The cost of a 40-peak watt system is about $350-$500 worldwide, depending

largely on the input duties on solar panel, but this cost is beyond the reach of most

Nigerians. Further complicating the situation, kerosene, widely used for cooking

and lighting, was raised to 650 naira or $4.60 for 4 litres since 2006. Before then,

it was sold for only $0.78 for 4 litres in 2003. The alternative for many families

became firewood, and so Nigeria‘s forests were put at greater risk by the

government‘s increase in the price of kerosene. Therefore, they concluded that

that price will be a key indicator of the willingness of people to pay for solar

electric systems for their homes. They further considered a hypothetical enterprise

that will be directly charged with the production, sales and installation of solar

systems in Nigeria noting that a 40-watt system could be produced and sold for

about 75,000-80,000 naira. The opportunities for Small and Medium Enterprises

(SMEs) in production, sales and installation were also critically emphasized. The

challenge of initial capital for the SMEs, the Committee suggested, could come

from bank loans and government incentives.

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6. CONCLUSION

As a final point, underground plans are underway by the International

Centre for Basic Research (ICBR) – a research institute co-founded and presently

being operated by one of the authors – in line with the subject-matter, to

implement solar power SMEs under the private-public-partnership (PPP)

paradigm now in vogue. The ICBR, as the corporate driver of an Institute for

Energy, Environment and Health (based in the Department of Physics and

Astronomy of University of Nigeria, Nsukka) hopes to go into specific

modalities for setting up a ―Solar Energy R&D******

Company of Nigeria‖

as a vehicle for campaigning the delivery of solar power for basic needs

including water and health to homes and communities in Nigeria at affordable

cost. The range of efficiency of commercial PV modules or solar panels is

currently 10-15%, which means that, given standard solar radiation of 1000

Watts (1 kW) per square meter, it is technically feasible to derive 100-150

Watts per square meter of useful power. The Energy Commission of Nigeria, as

the Implementing Agency for the UNESCO††††††

–World Solar Programme, has

established many pilot PV plants in the country, including a 7.2 kWp

electrification plant at Kwalkwalawa Village, Sokoto State, a 1.87 kWp

electrification and communication plant at Iheakpu-Awka, Enugu State, a 1.5

1. Research and Development 2. United Nations Education and Scientific Children Organisation

Fig. 2: Block diagram of small-scale

photovoltaic system

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kWp water pumping facility at Nangere Village, Yobe State and many others. The

challenge now is to pass the technological know-how over to Small and

Medium-Scale Enterprises (SMEs) and individual entrepreneurs for further

innovation leading to marketable products. To this end, acquiring a module at the

cost of N1000.00 per peak Watt from a local vendor for powering security light in

the village is still very expensive.

It is high time the battle of control on who gets the highest share from the

national cake syndrome stops and useful energy re-channeled to bettering noble

research and technology in renewable and sustainable energy resources.

Our ultimate hope is that Small and Medium-Scale Enterprises (SMEs) can

be empowered to build their manufacturing capacity, attract foreign

investment, as well as export manufactured solar-PV goods to the

ECOWAS‡‡‡‡‡‡

market. The reality of ―these‖ is only constrained by time, but not

chance.

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Presentation)‖, International Conference on Financial System Strategy

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June, 2007

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Colwell, R. R. and Greene, M. (2008) ―Solutions for Nigeria USA National

Academy of Sciences as an ―editorial‖ in Science Vol. 319, 25 January

2008 (www.sciencemag.org)

‡‡‡‡‡‡ Economic Community of West African States

http://www.sciencemag.org/

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Country Analysis Briefs – Nigeria (April 2007) www.eia.doe.gov

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methodologies — Vienna: International Atomic Energy Agency, 2005.

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Supply‖, NCERD Conference 2007, UNN, Nigeria

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Prospects of a Multi-Win Project (Review of the Regulatory,

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Petroleum and Mineral Law and Policy (CEPMLP), University of Dundee,

Scotland, United Kingdom.

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(Information Memorandum), Federal Republic of Nigeria

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Goals‖, New York: Energy Sector Management Assistance Programme, United Nations

Development Programme, UN Millennium Project, and World Bank.

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April 2003. Federal Republic of Nigeria

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(SAFTA, June 2006)‖, Faculty of Economics, Cambridge

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for Nigeria – Perspective from a Competing Energy Product – Coal‖,

Chester Mead Associates, Port Harcourt, Nigeria. Presented at the SPE

Nigeria Council Annual Conference in Abuja in August 2003.

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Okoro, O. I., Govender, P. and Chikuni, E., ―Power Sector Reforms in Nigeria:

Opportunities and Challenges‖, Proceeding for 10th

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210



SOLAR PHOTOVOLTAIC POWER SYSTEM R&D INNOVATIONS§§§§§§

J.O. Inwelegbu1 and P.N. Okeke2

Centre for Basic Space Science, University of Nigeria, Nsukka.

1www.cbss-online.com

2email:[email protected]

Abstract

Engineers at the Centre for Basic Space Science (CBSS) Nsukka, have succeeded

in designing and fabricating a battery backed, solar photovoltaic power system,

featuring push-pull topology, modified-wave power inverter system acting also as

an off-line uninterruptible power system (UPS).It powers electronic equipment, in

areas where the utility power system is absent or unreliable. The design employs a

locally made transformer and inexpensive electronic components obtainable from

scrapped electronic printed circuit boards, thereby making it affordable. This

approach has successfully yielded a reliable and affordable solar photovoltaic

power inverter, which can be mass produced to alleviate the problem of power

supply in the country. Effective efficiency of over 70% DC to AC energy

conversion has been obtained under resistive load calculations from the machine.

A test case installation of the system has successfully powered home electronic

appliances, scientific equipment and hospital life-saving machines without a

breakdown for two years. The inverter offers over 200% cut in running cost, when

compared to running the common Tiger-650VA petrol generator very popular in

Nigerian homes, over a one month period. This paper reviews PV technology and

describes the design, construction and testing of a solar photovoltaic power

inverter featuring an innovative and effective overload circuitry, battery monitor

circuit, and a battery charge controller unit built from scrapped printed circuit

boards.

Keywords: Solar Photovoltaic power inverter, Push-pull topology, modified sine-

wave inverter, charge controller, uninterruptible power system

§§§§§§ African Journal of Physics Vol. 2, pp,210- 232, (2009)


http://www.cbss-online.com/

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1.0 INTRODUCTION.

Although global fossil fuel resources have not yet been exhausted, the negative

social, health, and environmental impacts of our current unsustainable patterns of

energy use are apparent (Keeling et al , 1997). In the future, large-scale alternative

methods of producing the vast quantities of energy needed to sustain and enhance

our standard of living are necessary (Flavin C., French V, 2000). Fortunately,

advances in science and technology have provided us with several alternative

means of producing energy on a sustainable level, such as wind, geothermal,

biomass, and solar (Turner J.A, 1999).These are the Renewable Energy sources.

Most of the renewable energy sources depend on the sun as their primary source

[1]. Solar energy is all the energy that reaches the earth directly from the sun

through radiation, having traveled about, 150 million kilometers of empty space

(Awe et al, 1992). It is silent, inexhaustible and non-polluting. If harnessed, it can

cater for the global current and future energy needs.

1.1 Objectives and Motivation.

In recent years, many places in the world have been experiencing continued

shortage of electric power or energy crisis due to their fast increasing demand [1]-

[2]. To solve this problem, significant efforts of research and development have

been given in two areas: Firstly, to improve the efficiency of present power

conversion and utilization system. Secondly, to develop efficient renewable

energy generation and conversion systems to supplement conventional fossil-fuel

based energy supply and eventually replace it.

The solar power system has the potential to become one of the main renewable

energy sources due to the commercial availability of semiconductor-based

photovoltaic devices, reduction in the system cost and rapid development of

power electronic and conversion technologies using faster and more powerful

Isolated Gate Bipolar Transistors (IGBT) and power MOSFETs. One of the

important tasks is to make solar power generation and conversion system more

affordable, efficient and more reliable. This work is geared towards contributing

to improve power availability in Nigeria by designing a cheap and reliable solar

photovoltaic inverter, able to power basic household electronics offering easy

manufacturability.

1.2 Inverter technologies

Various topologies exist for Sine wave inverters. Such topologies include:

a) Inverter using CVT (Constant Voltage Transformers).

b) Inverters with 50 Hz Toroidal or silicon steel, EI transformers with a variation.

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c) Inverter with High frequency isolated converter and separate secondary inverter

control bridge.

d) Inverter with more than one transformer configured in series or matrix

configuration.

e) Single switch step up converter

f) 3 to 8 stepped inverter

g) Quasi Sine inverter of various topologies

h) Controllable CVT inverters

i) PWM inverter

j) Utility interaction inverter

The above mentioned inverter topologies cover a wide range of inverters

manufactured today. Although ―new topologies‖ are developed and advertised

regularly, they will fit in with one of the above-mentioned topologies. Each

design topology has its advantage and disadvantages, which the design engineer

has to weigh prior to embarking on his design. Inverter literature contains the

topology, diagrammatic and circuit behavior of these topologies for selection by

the designer

2.0. LITERATURE SURVEY.

2.1 Photovoltaic Technology

‗Photovoltaic‘ is a marriage of two words: ‗photo‘, meaning light, and ‗voltaic‘,

meaning electricity. Photovoltaic (PV) technology is the technology that converts

sunlight directly into electricity, in complete absence of the machinery usually

associated with electricity generation. In its simplest form a photovoltaic device is

a solar-powered battery utilizing the sun that fuels it as the only consumable

source of energy. There are no moving parts; operation is environmentally benign.

The electricity is direct current and can be used that way, converted to alternating

current or stored for later use [3]. This phenomenon is called the photovoltaic

effect and was first observed in 1839 by the French scientist Becquerel (Hislop,

1992). Research was carried out over the years on host materials to increase

conversion efficiency. Selenium was reported by S. Bidwell, lead sulphide by E.

Adler, cuprous oxide by L. Bergman, P. Hallwachs, E.H. Kennard, L.O. Arnordal

and W. Schottky.

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2.2. How much power is available from the sun?

The sun‘s energy reaches the Earth‘s outer atmosphere at power of 1,367 watts

per square meter, defined as AM0, or ―air mass zero.‖ Atmospheric losses due to

collision with molecules reduce the sun‘s power to about 1000 W/m2, when the

sun is directly overhead on a cloudless day [4]. The actual usable radiation

component varies depending on geographical location, cloud cover, hours of

sunlight each day, etc. In reality, the solar flux density (same as power density)

varies between 250 and 2500 kilowatt hours per metre squared per year

(kWhm 2 /year). As might be expected the total solar radiation is highest at the

equator, especially in sunny, desert areas. It is estimated that the amount of solar

energy falling on earth in three days is equal to the known fossil fuel reserve of

the world (Eastop, 2004).

2.3. Converting sunlight to electricity

A typical photovoltaic cell consists of semiconductor material (usually silicon)

having a pn junction as shown in Figure 1. Sunlight striking the cell raises the

energy level of electrons and frees them from their atomic shells. The electric

field at the pn junction drives the electrons into the n region while positive

charges are driven to the p region. A metal grid on the surface of the cell collects

the electrons, while a metal back-plate collects the positive charges [5].

Figure 1. How solar cells work.

The electricity production from a Photovoltaic module, Ppv, can be expressed as

follows:

.................................(1)

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where, Pmax: Installed capacity [W]; Is: Solar radiation [W/m 2 ]; ISTC:

Radiation at standard conditions (1000 W/ m 2 ) [W/ m 2 ]; ¥s: Temperature

coefficient for efficiency [-]; Tcell: Operation cell temperature [°C]; TSTC: The

cell temperature at standard conditions (25°C) in [°C]

The operation cell temperature is calculated by the following formula (Antonio

Luque and Steven Hegedus, 2003):

.............................................(2)

where: Ta: Ambient temperature; NOCT: Nominal Operating Cell Temperature.

The power production at grid is calculated as:

Where misc, is miscellaneous PV array losses and other power conditioning

losses.

The solar cell is the basic building block of a PV system. Although photovoltaic

cells come in a variety of forms, the most common structure is a semiconductor

material into which a large-area diode, or p-n junction, has been formed. Silicon

polycrystalline thin film, amorphous silicon and thin films of other materials are

used these days for their production (Nwokoye, 2006).The fabrication processes is

the traditional semiconductor approaches: diffusion, ion implantation and so on.

Electrical current is taken from the device through grid contacts. Solar cells are

interconnected in series and in parallel to achieve the desired operating voltage

and current to make PV modules, which in turn are combined to create PV panels,

which can be added together to create a PV array. They are then protected by

encapsulation between glasses and housed in an aluminum frame to form a

module [6]. These modules, usually comprise of about 30 or more PV cells, form

the basic building block of a solar array. Modules may be connected in series or

parallel to increase the voltage and current, and thus achieve the required solar

array characteristics that will match the load. Typical module size is 50Watts,

12Volts and produces direct current electricity to charge batteries for lighting

loads and to power inverters (Hislop, 1992).

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2.4 Components Of A PV System

Photovoltaic systems are modular, and so their electrical power output can be

engineered for virtually any application, from low-powered consumer uses -

wristwatches, calculators and small battery chargers, to energy-significant

requirements such as generating power at the home or office. A typical PV system

home installation will comprise of the following system components:

PV modules, conduit, grounding circuit, fuses, safety disconnects, outlets, metal

structures for supporting Loads.

Balance of systems equipment (BOS).

Balance of systems equipment (BOS).

This includes the batteries, battery charge controllers, inverters (for loads

requiring alternating current), wires the modules, and any additional components

that are part of the PV system.

2.4.1. Batteries

In PV power systems, there is need to store the electric power for use when there

is no sun, so that power is available all the time. Rechargeable batteries are used

as a means of electrical energy storage. The storage battery like the solar panel is

a direct current (DC) producing machine. It converts chemical energy into

electrical energy. Batteries act as ‗buffers‘ between the solar array and the load,

supplying power to the load during periods of low sunlight, and accepting charge

from the array during periods of sunlight (Eastop et al, 2004). During the

discharge process, stored electrical energy is released from the battery, while the

energy is restored during the charging process. Solar panels are employed to

provide the required direct current (DC) voltage and current through a charge

controlling device, to efficiently charge the batteries. Batteries can be sub-divided

into the following types:

Primary cells or dry batteries which include (1) standard zinc-carbon and (2)

alkaline or heavy duty while Secondary cells or rechargeable batteries include (i)

Lead-acid (ii) vented lead-acid (iii) automotive (car) (iv) deep-discharge or

traction (v) stationary (vi) vented (vii) sealed batteries including deep cycle

battery for solar energy applications. Generally, Batteries should not be stored

where temperatures exceed 25°C. Deep cycle batteries were used in this design

since they can provide high AMO.

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2.4.2. Battery chemistry

(i) Discharge

When a battery is placed in use (connected to an electrical load), the stored

chemical energy is released in the form of DC electrical energy. During the

process, the internal components of the battery cells undergo a chemical change.

The sulphuric acid (H 2 SO 4 ) combines with the lead peroxide (PbO 2 ) of the

positive plates, and the sponge lead (Pb) of the negative plates and transforms

them to lead sulfate (PbSO4), with the release of electrical energy. The reversible

reaction is shown as follows:

Pb + PbO 2 + 2 H 2 SO 4 ----- discharge

Charge ------- 2 PbSO 4 + 2 H 2 O (E cell = 2.0V) ……………..........(3)

The chemical reactions of discharge converts the active components in the battery

plates, that is the lead in the negative plate, lead peroxide in positive plate, and the

sulphuric acid into free electrons water and lead sulphate.

(ii) Charging

Batteries must be charged in such a manner that the sulphates are eliminated by

recombining with water to re-form into an acid (sulphuric acid in this case)

without loosing the hydrogen and oxygen gasses that make up the water. During

charging, the chemical energy in the battery is restored, thereby reversing the

discharge reaction. Especially toward the end of charging, at charging voltage of

2.35Volts per cell, hydrogen and oxygen gas are produced with loss of water by

the secondary reaction known as electrolysis of water [7]. Chemically, the

reaction is:

………(4)

2.5 PV system configurations

Four major types of system configurations exist for PV powered lighting systems.

These include the direct coupled system, standalone system, grid/utility-connected

system, and the hybrid system [8], [9]. Direct coupled systems do not require

batteries and inverters, while the other configurations do require batteries, charge

controllers and inverters. Standalone PV systems are designed to operate

217



independently from the grid and to provide all of the electricity needed for homes,

pathways and parking lots. A grid-tied PV system generates electricity from the

PV system as well as from the grid. When your PV system is producing

electricity, your home will be powered by solar electricity. At night or during off-

PV periods, electricity will be supplied from the grid. Any excess electricity

produced by the system during PV operation can be fed back to the grid. It may

also require batteries. Depending on the type of load (dc or ac), a power-

conditioning unit or an ac load control centre is needed in grid-connected systems

to mix the input power from PV panels and the input power from the grid.

Optionally, some grid-connected systems can feed power back to the grid when

the PV panels generate more energy than needed. Hybrid systems are more

expensive, but more reliable and combine a number of electricity production and

storage elements like wind turbines and generators to meet the energy demand.

Figure 2. Block diagram of the solar PV power system.

2.6 The Inverter Device

An Inverter is an electrical machine that converts Direct Current (DC) to

Alternating Current (AC) and at the same time stepping it to a desired voltage and

frequency. (Bedford and Hoft, 1964).The inversion process can be achieved with

the help of transformers, transistors BJT (Bipolar Junction Transistors),

MOSFETs (Metal Oxide Silicon Field Effect Transistors), the Insulated Gate

Bipolar Transistor (IGBT), Silicon Controlled Rectifiers (SCRs) and tunnel

diodes, etc as electronic switching devices [10]. For low and mediun outputs,

transistorized inverters are suitable but for high power outputs, SCR inverters are

essentials (Theraja, 2002). Inverter types can be categorized by output waveform,

switch type, switching technology and frequency . When connected to the grid,

the application name termed to this form of power converters is Uninterrupted

Power Supply (UPS). In order to go from a constant DC voltage to an AC the

input DC voltage source, the battery is put through an oscillating circuit which

creates the output AC. The resultant voltage output waveform of the inverter can

be a square-wave, modified sine-wave, pulsed sine-wave or pure sine-wave. The

major application of power inverters are in solar photovoltaic power systems and

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emergency power supplies operated during mains power failures [10]. There are

four topologies of interest for the power stage of a single phase inverter, namely

the single ended; half bridge, full bridge and push pull topologies [11], as

depicted in figure 3.

Single ended topology Half-bridge topology

Full bridge topology Push-pull topology

Figure 3.Inverter topologies.

Some researchers spent efforts in developing PV inverter systems with grid

connection and active power filtering features using sensors to measure the load

current (Wu, 2005), (Kim, 1996), (Cheng, 1997) and (Kuo, 2001).

In this work, a thorough investigation was done on the state of current inverter

technologies. Due to the cost and complexity to design a sine-wave inverter, an

inverter producing simulated sine-wave AC output of 50/60Hz at 220Volts was

considered. This is adequate for rural application, where most critical and crucial

loads are resistive and capacitive loads; such as small compact fluorescent lights,

computers, notebooks, televisions, radios, VCR‘s etc. This work discusses the

design of a battery charge control device, low-battery cut off and an overload

circuitry; incorporated into a 1,500Watts solar photovoltaic inverter and UPS,

219



which was fabricated using electronic parts from scrapped printed circuit boards.

The 50Hz transformer inverter (push-pull) topology was used in this design.

(i) The 50Hz transformer inverter (push-pull) topology

Most power Inverters in the market are designed using this topology.

Figure 4. The 50Hz transformer inverter topology.

As shown in Figure 4, the principle is that an ―overrated‖ 1500Watts, 50Hz

transformer is used, which is made of EI, UI or Toroidal cores. All the control

and power switching is done on the primary side of the Transformer. A current

limit element i.e. inductor is placed in series with the output filter capacitor at the

secondary side to produce a modified AC sine-wave output voltage. Voltage

feedback to compensate for output voltage fluctuations are achieved using an

isolation transformer and electronic coupling, or direct, if isolation is not required.

The switching strategy is bi-polar Pulsed Width Modulation (PWM) and

frequency is 50/60Hz for EI-steel transformer laminations.

(ii) Advantages of 50Hz transformer inverter topology

• Only one energy conversion stage from DC (Battery) to AC.

• Relatively ―clean‖ quasi-sine-wave can be generated, with very simple and

limited control.

• Due to isolation and electronic simplicity with fewer switches, the

reliability is good.

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3. 0. INVERTER DESIGN.

The inverter design presented in this paper is able to deliver 1500Watts of electric

power at 220V AC, 50/60Hz. It comprises mainly of six circuit sections and the

heat sink, namely : ( a) A Pulse Width Modulation (PWM)/Oscillator control

section ( b) MOS driver section ( c) Overload protection section ( d) low batery

cut-off section ( e) Charge controller section (f) Mains charger section (g) Mains

change-over section (h) Power section heat sink.

3.1 PWM control section

The first step in designing the DC to AC inverter was to determine a useful way

of switching at the desired output frequency, 60 Hz. It was determined that this

switching could be implemented using pulse width modulation, PWM. The

SG3524 IC, was chosen as the desired pulse width modulator. The SG3524

provides 2 outputs with variable dead time. A timing resistor, RT, and capacitor,

CT, control the frequency of the two outputs, A and B. RT and CT were chosen to

be 15 k and 1 µF, respectively. The variable dead time is essential for proper

functioning of the inverter. The dead time allows for a small period of time when

the output voltage is zero. A resistor, R d , controls dead time. R d was set at 100

to provide 2.7 ms of dead time. The combination of these values for RT, CT,

and R d provided a switching period, T, of 17.1 ms. Frequency can be found using

Equation (5) below.

TFrequency

1 …………………………………………….(5)

With a period of T = 17.1 ms, the frequency of operation is 58.47 Hz. This

frequency is very near that of the desired 60 Hz. Normal household appliances

powered by this design will not notice this variance from 60 Hz.

3.2 MOS driver section:

The next step was to choose properly rated MOSFETs that will act as switches.

Switching devices on the primary side of an inverter handles very high current.

Hence the robust IRFZ44N MOSFETs were used, with RDS on = 24 mOHM, DC

current =20A at 100 °C and pulsed rain current at 100 °C is 120A. Its maximum

power dissipation is 35Watts, so it was mounted on aluminum heat sinks. Four

MOSFETs, in a push-pull topology provide the simulated sine-wave output.

Multiple MOSFET switches were used in parallel to boost power handling

capacity. Gating signals produced by the pulse width modulator IC-SG3524,

controls the voltage applied to each MOSFET. The logic supply voltage needed

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for the integrated circuits is 5Volts and 12olts respectively and is supplied by

LM7805 and LM7812 voltage regulators from the 24Volts battery source.

3.3 Overload protection section: In overload condition of the inverter output, an

―overload sensing voltage‖ is generated at the shunt connected at the MOSFET

source. Using operational amplifier circuits, this sensing voltage is converted into

a signal to glow an overload LED (Light Emitting Diode), sound a warning

buzzer and to shut down the PWM device and the inverter to avoid damage to any

component

.

3.4. Low battery voltage shutdown:

Batteries are high cost items in PV power systems. To prolong battery life, a low

battery voltage protection circuit was designed and incorporated. In permanent

PV installations the suggested low voltage cutout level is 11.4Volts for single

battery systems. The battery is then expected to recover up to at least 12.3Volts,

before the inverter will reconnect. This design is powered from 24V batteries

whose maximum voltage after full solar charge can reach 28Volts. We therefore

incorporated an electronic circuit that effectively shuts down the inverter when

battery discharges to about 22Volts to avoid damage to the MOSFETs and

associated circuitry.

3.5 Charge controller section

Since the brighter the sunlight, the more voltage the solar cells produce, the

excessive voltage could damage the batteries. A charge controller performs dual

function of maintaining the proper charging voltage on the batteries and during

low sunshine; it prevents the battery from discharging via the solar panel. The

charge regulatory function of a charge controller is governed by a 3 stage charge

cycle namely Bulk stage, Absorption stage and float stage [12],[13]. During the

Bulk phase, the voltage gradually rises to the Bulk level (usually 14.4 to 14.6

volts) while the batteries draw maximum current. When Bulk level voltage is

reached the absorption stage begins. During this phase the voltage is maintained

at Bulk voltage level for a specified time (usually an hour), while the current

gradually tapers off as the batteries charge up. After the absorption time passes

the charging enters the float stage. At this stage, the voltage is lowered to float

level (usually 13.4 to 13.7 volts) and the batteries draw a small maintenance

current until the next cycle.

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Figure 5. 3 stage charge cycle.

The Charge Controller we designed functions using this 3 stage charging strategy.

It was installed between each Solar Panel array and the battery it charges. The

charge controller designed consists of four basic elements namely:

(i) a stable reference voltage

(ii) a voltage sampling element

(iii) a voltage comparator, and

(iv) a power dissipating control device.

The PWM charging strategy was used for the charge controller circuit we

designed. Pulse Width Modulation (PWM) is the most effective means to achieve

constant voltage battery charging by switching the solar system controller‘s power

devices. When in PWM regulation, the current from the solar array tapers

according to the battery‘s condition and recharging needs. Unique benefits

derived from the PWM pulsing technique include:

(i.) Ability to recover lost battery capacity and de-sulphate a battery.

(ii.) Dramatically increase the charge acceptance of the battery.

(iii.) Maintain high average battery capacities (90% to 95%).

(iv) Equalize drifting battery cells.

(v) Reduce battery heating and gassing.

(vi) Automatically adjust for battery aging.

(vii) Self-regulate for voltage drops and temperature effects in solar systems.

3.6 Mains change-over section:

The changeover section is used to:

(i) Switch on the inverter when the AC (mains/grid) supply switches off and to

switch off the inverter when the AC mains supply returns.

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(ii) During the changeover, when the inverter receive AC mains supply, it stops

drawing power from the battery and the AC mains power is sent to the inverter

output socket.

3.7 Mains charger section

The Inverter system can also charge the batteries if it is connected to the AC

mains supply. When the inverter receives AC mains supply, inverter transformer

and MOSFET together work as a charger and charge the battery.

3.8 Power section heat sink

The heat sink is a vital component of the inverter. It serves to dissipate the heat

generated by electrical losses from the switching components and the transformer

to prevent their degradation or failure. The performance of the heat sink is crucial,

thus a material with high thermal conductivity characteristics must be employed

for the heat dissipation. Aluminum is commonly used. Aluminum was used in this

design and also a fan was attached to enhance the rate of heat dissipation from the

heat sink.

4.0 DESIGN CALCULATIONS

4.1.0. Battery selection

Battery Current output = 200A; Battery voltage = 24V DC (Input voltage)

The general function of the transformer in this inverter design is transformation of

alternating dc voltage pulses to alternating AC voltage pulses in step-up mode.

Transformer power output = 1.5KVA; Transformer voltage output = 220V (output

voltage).

From the transformer design parameters above, the primary current needed for the

required actual output is as follows:

Primary power (W) = secondary power (W)

2211 IVIV , VAVI 200024 1 , I 1 = 2000VA/24V = 83.3 Amps

The result show that the transformer secondary current should be at least 83.3A,

intended to power the full load rated at 1.5KVA for at least an hour; when there is

no mains or solar power. For an n hour operation, where n is the intended number

of hours,

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selected isbattery cycle deep AH 200 (AH)Amp/hour 166.66

A 83.3hours 2

Iruntimebattery Amphour

4.2 Transformer turns ratio

In a power inverter system, the switching devices (transistors) supply current from

the dc source (battery) to the next stage. An output transformer is commonly used.

Switching the devices changes the voltage applied to the primary. The secondary

of the output transformer thus produces a voltage by electromagnetic induction,

with its magnitude being determined by the transformer turns ratio. For secondary

and primary turns, using the turn‘s ratio formular;

NP/NS = EP/ES (7)

Where NP = primary turns , NS= secondary turns, EP=primary voltage,

ES=secondary voltage

Calculations:

Primary voltage (EP)=24V; Secondary voltage (ES)=200V/220V; Primary turns

(NP)=30T; Secondary turns (NS)=?

4.2.1. Secondary turns (NS)

NP/NS=EP/ES; NS 1 = N P E S /E P = 30 200/24 = 250 Turns

NS 2 = N P E S /E P = 30 220/24 = 275 Turns

4.2.2. Primary turns (NP)

N S E P /E S = 275 24/200 = 33; but 275 24/220 = 27.2; then 33 – 27 = 6/2 = 3

Therefore, the maximum number of turns balance for primary side windings

proportional to the secondary windings of N 1S and N 2S becomes 27+ 3 = 30 turns.

4.3 Transformer efficiency calculation

A transformer is a static piece of apparatus with two or more windings (primary

and secondary) which, by electromagnetic induction, transforms a system of

alternating voltage and current into another system of voltage and current of same

values or of different values, and at the same frequency for the purpose of

transferring electrical power. Heat losses, electromagnetic induction losses, eddy

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current losses and coil resistance losses reduce the efficiency of power

transformation.

Power input = IV = 83A 24Volts = 1992 Watts

Expected Power output = 1500Watts

Efficiency = Power output/power input 100/1 ………………………………(8)

1500/1992 100 = 75%.

4.4 Inverter Technical Specifications

Description/Heading Deduced Specification

Rated Capacity 1.5 KVA

Ambient operating temperature <= 45 C

Inverter Method

Overload protection

PWM (pulse width modulated)

Inverter

Electronic/Fuse

AC Input Frequency

No of phases

Voltage

Power factor

AC output Waveform

50/60Hz ± 5%

1

200V ± 5%

0.98

Simulated sine-wave

Overload Capacity 125% (2 min.), 150% (1min.)

Battery Backup time

Battery type

Input DC voltage range

≥ 10 hrs on half load

12V X 2, 200AH Sealed deep

cycle

20Volts – 28Volts DC

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5.0. TESTING OF THE BATTERY CHARGE CONTROLLER

Time Voltage(Volts)

9.20 11.89

9.50 12.02

10.20 12.09

10.50 12.22

11.20 12.33

11.50 12.79

12.20 12.99

12.50 13.02

13.20 13.10

13.50 13.18

14.20 13.56

14.50 13.97

15.20 14.02

Figure 6.Charge controller characteristics for 12V, 92AH deep-cycle battery.

We subjected the charge controller device developed to rigorous tests with a

SHARP - 85Watts solar panel having an open circuit voltage of 21Volts dc and a

short circuit current of 5A. The voltage reading of the deep-cycle 12Volts, 92AH

battery during charge on a sunny day, was then monitored in a 30minutes time

interval. The timing of the charging process was also monitored using a digital

clock. Voltage and time readings were recorded form 9.20am to 3.20pm when

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

11.5 12 12.5 13 13.5 14 14.5

Voltage (Volts)

Tim

e

Charging Graph of 28/04/09

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maximum sunlight was recorded on 28/04/2009. The graph in figure 6, was

finally generated using Excel spreadsheet. Result shows that the battery charge

controller performed satisfactorily, charging the battery from a discharge voltage

of 11.89Volts to nearly the bulk-level voltage (14.5Volts) of 14.02Volts at

3.20pm when maximum sunlight was recorded on that day. The slope of the graph

indicates a continuous voltage increase of the battery, until the termination of the

charging process.

6.0. COMPONENTS USED IN DESIGN.

Some of the Components used to realize this design are listed as follows:

(a) Transformers (b) Resistors (c) Diodes d) Capacitors (e) Relays f)

Transistors (g) Voltage-regulators (h) Integrated circuit ICs (g) Light emitting

diodes (h) Varistors (i) Solar panel

(j) Deep cycle storage batteries 12Volts, 200 AH X 2 (k) Switches (l)

Connectors, screws, plugs and (m) Buzzer.

6.1 Test Equipment.

The following laboratory equipment was utilised in the testing phase of the

system, the functionality of each block was tested and relationships were

investigated.

(a) Multimeter

(b) Voltmeter

(c) Ammeter

(d) Frequency meter

(e) Oscilloscope

(f) Digital logic probe

(g) Load – 20 Watts, CFL bulbs, 55Watts fan, 55Watts 14inch color TV,

200Watts laptop computer,

300Watts desktop computer, 200Watts medical equipment.

6.1.1. Multimeter

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A sensitive digital commercial multimeter ALDA DT-830D was used to measure

voltage, resistance, current, transistor type, transistor current gain (Beta) (ß) and

frequency at various stages of the design procedure. The measuring range of the

instrument is from 3000mV to 1000V for voltage and from 200µA to 10A for

current.

6.1.2. Voltmeter

A front panel mounted Tektronics AC voltmeter was used to monitor the Inverter

AC voltage output. The voltage measuring range is from 0 – 500V AC, with 2%

accuracy. Also a front panel mounted DC voltmeter was used to monitor the DC

battery voltage, with a voltage range of 0 – 50V DC and an accuracy of 2%.

6.1.3. Frequency meter

A front panel mounted Tektronics frequency meter was used to monitor the

frequency of the Inverter AC output. The frequency measuring range is from

45Hz to 100Hz, with 2% accuracy.

6.1.4. Oscilloscope (scope)

An oscilloscope creates a visible two-dimensional graph of electrical voltage

signals/quantities on the vertical axis, with respect to time on the horizontal axis.

Scopes fall into 2 categories: analog and digital. We used the bench mounted,

Tektronics 465 Cathode ray analog scope, to verify the output simulated sine-

wave AC voltage/time waveforms of our inverter design. It is a high quality, dual-

channel, 100Mhz bandwidth scope.

6.1.5. Digital logic probe

We used this instrument to monitor the logic states of the logic integrated circuits

we employed.

7.0. ADVANTAGES OF SOLAR PHOTOVOLTAIC POWER

SYSTEMS.

(a.) The fuel is free (sun).

(b.) There are no moving parts to wear out, break down or replace.

(c.) Only minimal maintenance is required to keep the system running.

229



(d.) The systems are modular and can be quickly installed/upgraded anywhere.

(e.) it produces no noise, harmful emissions or polluting gases.

Photovoltaic energy is unlike any other energy source that has ever been available

to utilities. PV generation requires a large initial expense, but the fuel costs are

zero. Coal or gas fired plants cost less to build initially (relative to their output)

but require continued fuel expense and maintenance. Fuel expenses fluctuate and

are difficult to predict due to the uncertainty of future environmental regulations.

Fossil fuel prices will rise over time, while the overall cost of PVs (and all

renewable energy resources) is expected to continue to drop, especially as their

environmental advantages are valued.

8.0. RESULTS AND DISCUSSIONS.

The inverter subsystems developed namely: (a) The Pulse Width Modulation

(PWM)/Oscillator control section (b) MOS driver section (c) Overload protection

section (d) low batery cut-off section (e) Charge controller section (f) Mains

charger section (g) Mains change-over sectio and (h) Power section heat sink ,

were tested individually and they performed satisfactorily. They were then

integrated to form the 1,500watts; 220V AC 50/60Hz modified sine-wave inverter,

pictured in figure 7. The inverter was tested with two fully charged 12V, 200AH

deep-cycle batteries, solar charged 2 units of the charge controller we developed.

Figure 6. shows the Charge controller charging characteristics for a 12V, 200AH

deep-cycle battery. It was found to perform satisfactorily, conforming to design

specifications and performance expectations as in the table of 4.4.0. The inverter

generated utility electric power without generating any harmful electromagnetic

interference. The simulated sine-wave output waveform of 200V ± 5%, 50/60Hz ±

5% obtained, is shown below in figure 7.

Figure 7. AC output waveform.

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The inverter powered an office desktop computer, laptop computer, 2 fans, mobile

phone chargers, clinical cardiac machine, Telephone PABX switching system,

one television and compact fluorescent lighting bulbs for more than 10 hours after

a 5 hours flat battery recharge from two units of 85watts-SHARP PV modules

using two units of the 14.5V, 10A capacity battery charge controller device we

developed.

The total cost of the project (about =N=165,000: One hundred and sixty-five

thousand naira only) is about half the cost of fueling a 650Watts portable gasoline

generator running for 10 hours daily for a year with the attendant maintenance

cost. (About =N=320,000: Three hundred and twenty thousand naira only). On

the contrary, it will cost nothing other than free sunshine, to run the solar power

system for a year. This proves that the product is grossly cost effective in the long

run.

Figure 8.The solar photovoltaic power system schematic and the developed

unit.

9.0. CONCLUSION

We have successfully designed, fabricated and tested a solar photovoltaic power

inverter developed using scrap electronic components from old PCBs. The

subunits of the system namely: (a) The Pulse Width Modulation

(PWM)/Oscillator control section ( b) MOS driver section

231



(c) Overload protection section (d) low batery cut-off section (e) Charge

controller section

(f) Mains charger section (g) Mains change-over section and (h) Power section

heat sink, were individually tested and integrated to form the 1,500watts, 220V;

50/60Hz simulated sine-wave inverter. Overall performance as depicted in table-1

above was satisfactory, yielding a DC to AC conversion efficiency of 75%.

10.0 FUTURE WORK

This work is a starting effort in our design of solar photovoltaic power systems.

Future designs shall address higher efficiency, better output waveform filtering,

better switching algorithms using the PIC microcontroller and software in a full-

bridge topology to achieve full sine-wave output and power stage component size

reduction using ferrite core transformers switching at 20khz, in a DC-DC step-up

converter and then DC-AC inverter configuration with a low pass filter at the AC

output. This approach will result in the development of portable higher power

small sized inverter units, with very low EMI (Electro Magnetic Interference) for

diverse PV power applications.

ACKNOWLEDGEMENT

The authors wish to acknowledge the Centre for Basic Space Science (CBSS)

Nsukka, and her engineers and scientists who have collectively ensured the

success of this work. We also thank the National Space Research and

Development Agency (NASRDA) and the Federal Ministry of Science and

Technology Nigeria, for their financial assistance in this project.

REFERENCES

[1]. Jie Chang, ―Advancement and Trends of Power Electronics for Industrial

Applications‖-IECON‘03. The 29th Annual Conference of the IEEE, Volume 3,

2003, pp. 3021-3022.

[2.] J. Chang, etc, ―Integrated AC-AC Converter and Potential Applications for

Renewable Energy Conversion‖, 2002 Power and Energy Systems Conference,

Marina del Rey, California, USA, May 13-15, 2002.

[3] M.A. Green, Solar Cells-Operating Principles, Technology, and System

Applications, Prentice-Hall, Englewood Cliffs, N.J. (1982).

232



[4] Jack L. Stone, ―Photovoltaics: Unlimited Electrical Energy From the Sun,‖

Physics Today, September 1993.

[5] Mark Hammonds, ―Getting Power From the Sun; Solar Power,‖ Chemistry

and Industry, no. 6, p. 219, March 16, 1998.

[6] http://www.ohioenergy.org/Solar%20Curriculum/5Lesson1IntrotoPV.pdf

[7] ―Battery service manual‖, http://www.bulldog-

battery.com/PDF%20Files/SVC.PDF

[8] U.S. Department of Energy, Office of Energy Efficiency and Renewable

Energy, ―PV in use:getting the job done with solar electricity.‖

www.eere.energy.gov/solar/pv_use.html (2004).

[9] Missouri Department of Natural Resources, ―Missouri‘s solar energy

resource,‖ www.dnr.state.mo.us/energy/renewables/solar9.htm (2004).

[10] ‗Power Electronics‘, Lander, C.W., McGraw-Hill, 1993, pp. 198-216.

[11] Power Electronics‘, Mohan, Undeland and Robbins, Wiley, 1993.

[12] http://www.freesunpower.com/chargecontrollers.php.

[13] Woodworth, J.R., Harrington, S.R., Dunlop, J.D., et al, "Evaluation of the

Batteries and Charge Controllers in Small Stand-alone Photovoltaic Systems",

First World Conference on Photovoltaic Energy Conversion, Hawaii, Dec. 1994.

233



EVALUATION OF SUPPLY RELIABILITY OF MICROGRIDS

INCLUDING PV AND WIND POWER*******

G. Ofualagba1, and E.U Ubeku

2,

1Department of Electrical and Electronics Engineering, Delta State Polytechnic,

Otefe, Oghara, Delta State.

2Department of Electrical and Electronics Engineering, University of Benin,

Benin City, Edo State.

1E-mail: [email protected]

Abstract

This paper presents a procedure of supply reliability evaluation for microgrids

including renewable energy sources such as wind power and photovoltaics.

Microgrid system can be used as a framework to flexibly introduce the renewable

energy sources. However, some renewable energy sources affect the power

quality negatively. Therefore, it is important to evaluate a microgrid system

adequately, and to discuss supply reliability evaluation. In this paper, the authors

introduce special reliability indices for microgrids.

Index Terms -- microgrid, supply reliability, interruption cost, demand and

supply ba1ance.

1. INTRODUCTION

DISTRIBUTED power generation of new energy sources is recently drawing

attention and increasing rapidly [1]. Various factors such as the rising costs of

fossil fuels, global warming, deregulation of power industry, and the proximity to

the demand are contributing to its rise.

******* African Journal of Physics Vol. 2, pp. 233 --250 , (2009)


234



However, the output of renewable energy sources such as photovoltaics and

wind farms are unstable, and their influence on power systems by the increased

penetration is concerned. Microgrids can be utilized as the framework of system

that can reduce the negative effects of power fluctuation on existing power

systems, simultaneously pursuing the coexistence of environment and supply, and

existing power system and distributed power generation. They can also be

considered to be a flexible load [2][3].

The dominant study of microgrids is on their control system for self-

sustained operation [4] and the modeling of control instrument. However, the

study of their reliability is still limited [5][6]. It is not clear that the performance

of such a new system including distributed power generation by renewable energy

sources can be evaluated by existing reliability evaluation indices. Therefore,

quantitatively evaluating the supply reliability will yield an important insight of

microgrid system development.

This paper proposes the indicators and procedure of supply reliability

evaluation for microgrids including the renewable energy sources such as wind

power and photovoltaic cells. In this paper, as well as an index to reflect the

impact of the power outages using Monte Carlo simulation, an additional index

intended for microgrids reliability evaluation is proposed based on the supply-

demand balance calculation.

The introduction of energy storage system will play an important role for the

maintenance of power quality in microgrids where many distributed power

generation of renewable energy sources are connected. Hence, in this paper, we

examine the effects of energy storage system on reliability.

2. EVALUATION OF SUPPLY RELIABILITY OF MICROGRIDS

2.1. Microgrids

Microgrids targeted in this study are independent areas having the power

demand of several hundred MW including gas turbines, wind turbines and

photovoltaic cells and serves

its own power demand. These microgrids are also connected to the external power

system by tie lines for reducing frequency/voltage fluctuation in the normal and of

emergency conditions. The system is based on the premise that it is operated

independently with zero tie-line flow under normal conditions.

235



2.2. Supply reliability evaluation indices

Supply reliability in utility power system is generally measured by the impact

of a single contingency (N-1 standard), and N-2 criteria may be used for

important equipment [7]. However, recently an increasing number of customers

require high supply reliability. Attempts to consider the interruption cost per each

customer is under way [8] because some customer‘s economic losses would be

tremendous. Therefore, for the supply reliability evaluation of microgrids in this

paper, the authors propose additional indices from the viewpoint of the customer

side in the microgrids.

For a supply reliability index in microgrids, we propose to use the following

four indices.

i) Expected Duration Not Supplied (EDNS): the annual expectation of total

hours not supplied [hours/year];

ii) Expected Energy Not Supplied (EENS): the annual expectation of energy

not supplied [kWh/year];

iii) Interruption Cost (IC): the amount of loss cost per unit quantity of energy

not supplied.

iv) Demand and Supply Balance Cost (DSBC): the sum of the deficit of power

in the microgrids purchased from the external systems.

Microgrids are basically aimed at autonomous control. However, when many

distributed power generation of renewable energy sources are introduced, control

only within microgrids become difficult. Therefore, microgrids keep the power

demand and supply balance by connecting to the external power system for

maintenance of frequency and voltage. Microgrid operators buy power through

tie-line from the utility company when power in the microgrids is deficient, and

sell power when the power generated is in excess. Hence, we define the cost due

to the imbalance of demand and supply calling it the demand and supply balance

cost.

2.3. Monte Carlo Simulation

We use Monte Carlo simulation to calculate supply reliability in this paper

[9]. Fig. 1 shows the flow chart of supply reliability calculation using Monte

Carlo simulation. The procedure of reliability analysis is described as follows:

At first supply reliability analysis initialize the state of system. Next, we

sample fault conditions that reflect the failure rate of all facilities. If some kind of

236



equipment failure occurred, we check if a system contingency would occur by

load flow analysis. When system contingency occurred, we try to solve the

problem by emergency operation. If we cannot solve it, we find the area of supply

interruption and the quantity of lost supply. We repeat the failure simulation until

the required accuracy is obtained, then calculate reliability indices as annual

average.

The load data include a stochastic deviation based on the normal distribution

added to an annual load curve of every one hour.

Fig. 1. Flow chart of supply reliability evaluation for microgrids.

2.4. Modeling of renewable energy sources

In the case of distributed power generator like a gas turbine that can control

the output, we can assume only the capacity as input data. However, in the case of

photovoltaics and wind

237



turbines, output is not constant; we therefore must use the data that reflects the

stochastic variation of output. Therefore, we use Weibull distribution [11] shown

in (1) to set the output

of the wind turbine.

where

f(V): incidence rate of wind velocity V; c: scale parameter; k: shape parameter.

An example of the wind turbine output based on Weibull distribution is shown

in Fig. 2.

Fig. 2. An example of wind turbine output.

The output of photovoltaics cells also change by the weather like the wind

turbines. In this paper, we have used the pattern of output that has been measured

by the field tests of photovoltaic cells such as shown in Fig.3. The average output

is assumed for daylight hours based on the weather conditions such as clear,

cloudy, or rain [12]. We add the stochastic variation based on normal distribution

to the average output.

238



Fig.3. Output of photovoltaics cells under different weather condition.

2.5. Calculation of interruption cost

We use the following equation to calculate the total interruption cost for the

reliability index on a consumer side.

where

IUC: unit cost of interruption at consumer m.

The unit cost of interruption needs to be investigated for each consumer. It is

known that the unit cost of interruption rises as the power outage sustains [10],

but we treat it as constant in this paper. Also, the damage caused by blackout is

substantially different between household consumers and commercial consumers.

In addition, among commercial consumers, there are consumers who require high

reliability, and the unit cost of interruption will be high for such consumers.

2.6. Calculation of the demand and supply balance costs

As mentioned above, the microgrid is assumed to run as a self-sustained

operation. However, in the case of imbalance of the demand and supply by the

influence of renewable energy sources, and the sudden change of power

generation output and/or of load such as at a time of accident, we need to buy and

sell the power from the external system. In such cases, we can calculate the

239



amount of cost that operator of microgrids will pay to the utility company for the

deficiency or surplus.

where

BP: amount of energy bought; BC: price of energy bought; SP: amount of

energy sold; SC: price of energy sold.

For the electric energy in deficiency or surplus, we accumulate the power flow

of tie line between the microgrid and the external system.

3. CASE STUDY

3.1. System configuration and data

We show the configuration of a radial microgrids model to study in this paper

in Fig. 4. This system includes six distributed power generation (four 3MW gas

turbines, one 1MW wind turbine, and one 300kW photovoltaics cells) and twelve

load for household (R) and commercial consumers (C). The network can be

reconfigured in looped network as shown in Fig. 5.

Fault condition of each facility is determined by the failure rate and recovery

time (mean time to repair) to be given as input. Failure rate and recovery time in

each facility is shown in Table 1.

240



Fig.4. Radial microgrid system.

Fig.5. Looped microgrid system.

In this paper, we set the unit cost of interruption as shown in Table 2. We also

assume the electric energy charges on which the microgrid would purchase power

from the electric utility company at 11 cents per kilowatthour [13].

241



TABLE I

FAILURE RATE AND REPAIR TIME.

TABLE II

INTERRUTION COST OF EACH DEMAND

3.2. System reliability evaluation

Fig. 6 and Fig. 7 show reliability indices of the microgrid with the radial and

looped networks, respectively. In the radial network, we find that interruption

time and expected energy not supplied at the end of the network is larger than that

of the looped network. This is because, when an accident occurred on the power

transmission line in a loop, the looped network is still capable of supplying power

to the loads. Also, when we compare the reliability cost of both systems, radial

network costs 11.17 [$1000/year] and looped network costs 10.42 [$1000/year].

From these results, as a policy to improve reliability when we design a

microgrid network, it is important to configure the system in a loop to reduce the

interruption cost by the power transmission line accidents.

3.3. Examination of the reliability improvement by electric energy storage

system

242



As a method of supply reliability improvement in micro grids, we can

consider installation of electric energy storage system [14]. The electric energy

storage system is expected to level output of renewable energy sources, and it can

function as backup at the time of generator failure. Therefore, we set up a

battery (BT) of 500 kW in load 5 that is assumed to be a particularly important

load. We have calculated the supply reliability, and compared the effect of electric

energy storage system in the looped system of micro grids as shown in Fig. 8.

Fig.6. Reliability index in the radial network.

Fig.7. Reliability index in the looped network.

243



Fig.8. Reliability index in the looped network with a battery.

When we compare Fig. 7 with Fig. 8, the battery supplements the deficient

power of load and improves the reliability as seen mainly on load 5. Also, the

interruption cost and demand and supply balance cost become lower when we

install the battery, and reliability cost is reduced to 8.54 [$1000/year] from 10.42

[$1000/year]. The above results have been obtained because we installed the

battery to load 5, and set the capacity at 500 kW. In general context, however, we

need more consideration whether sufficient capacity of batteries is installed for

the capacity of the renewable energy.

4. CONCLUSION

We have proposed a method of supply reliability evaluation for microgrids

including renewable energy sources in this paper. For the supply reliability

evaluation, in addition to a supply reliability index that is generally used for utility

power systems, we have introduced the interruption cost and the demand and

supply balance cost. We used Monte Carlo simulation for the calculation of

reliability indices. We have compared the reliability of radial network and looped

network as an example of calculation of reliability. In the future, we plan to

advance the more detailed modeling and simulation of microgrids and the

installation of electric energy storage systems.

244



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2001.

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Lasseter, R. N. Hatziargyriou, and T. Green, "Real-World Microgrids: An

Overview", 2007 IEEE International Conference on System of Systems

Engineering, San Antonio, USA, pp. 1 – 8, April 2007.

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2006), June 2006.

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and Expectations for Service Reliability‖, IEEE Trans Power Systems, Vol.

11, No.2, pp.989-995, 1996.

[11] S.H. Jangamshetti, and R. V. Guruprasada ―Normalized power curves as a

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245



[13] Tokyo Electric Power, Guideline of Extra High Voltage Supply, 2004.

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& Energy General Meeting, July 2008.

246



SUNSHINE HOURS VARIABILITY AND ITS RESPONSE TO

AEROSOLS IN LAGOS STATE NIGERIA†††††††

T. N. Obiekezie .

Dept of Physics and Industrial Physics Nnamdi Azikiwe University, Awka,

Anambra state , Nigeria. e-mail: [email protected] .

Abstract

This paper analyses the variability of sunshine hours for an equatorial region of

Lagos (06°35' N, 03°19' E) Nigeria. It is found that the total sunshine hours has

seasonal variability which maximizes in April and minimizes in July. Unlike a

decreasing trend in sunshine hours found in some countries the sunshine hours in

Lagos the busiest city in Nigeria is found to be increasing at a rate of 0.0449hrs/yr

for a period of nine years. The paper investigates the possibly effects of Aerosols

on the variability of the sunshine hours using Aerosol Index from Earth Probe

TOMS (Total Ozone Mapping Spectrometer) Version.8.

Key Words: Sunshine hours, Aerosols, clouds, Lagos, solar radiation,

1. INTRODUCTION

The energy transferred from the sun in the form of radiant energy to the

earth‘s surface is called solar radiation, Donald (1982). Solar radiation is an

important source of energy which plays a pivotal role in technological and social

development. In recent times, the cost of conventional fuel has been on the

increase and its availability is shrinking day by day. Realizing this, man has been

attempting for some time to make use of the sun‘s radiant energy as an alternative

source of energy. However, his success has been limited as the economic

utilization of solar energy requires a level of technological development that has

not yet been attained. Presently the world is witnessing very rapid technological

advances, as such it is expected that within a few years the cost of solar energy

tapped directly will become competitive with that of the conventional sources of

††††††† African Journal of Physics Vol. 2, pp.246-255, (2009)



247



energy. Solar radiation knowledge is essential in the study and design of systems

which use solar radiation data applications and should be thoroughly measured on

a continuous basis over long periods of time.

It has been observed that the network of stations measuring solar radiation

data is sparse in many countries of the world. In Nigeria, only few stations have

been measuring the daily solar radiation on a consistent basis (Chineke, 2007). It

therefore becomes necessary that solar irradiance has to be estimated from other

meteorological data (Zhou et al. 2005; Sabziparvar 2007; Wu et al. 2007; Bulut

and Buyukalaca 2007). Stanhill and Cohen 2005 recommended that Sunshine

duration is an equally reliable proxy for exploring changes in solar radiation

Solar irradiance measurements from various regions around the globe have

documented a steady decline in solar radiation across the globe from the mid -

1950s to the 1980s, a phenomenon that is generally termed as global dimming

(Stanhill and Cohen 2001; Liepert 2002; Alpert et al. 2005, Ohmura, 2006,

Roderick and Farquhar, 2002). From the mid 1980s an increasing trend in solar

irradiance has been observed around the globe which is termed solar brightening

(Wild et al., 2005).

A number of authors have discussed in detail the potential reasons for global

decrease in solar radiation and sunshine hours. This includes; Satheesh and

Moorthy (2005) who showed that wind speed quite significantly contributes to

global irradiative forces by influencing natural aerosol concentration, Liu et al

(2002) illustrated that increase in regional scale clouds as a result of increasing

anthropogenic emissions of aerosols could lead to reduction in sunshine hours,

while Cutforth and Judiesch (2007) suggested that increased cloudiness could

reduce the sunshine hours. Pinker et al. (2005) attributed the possible causes of

global dimming to changing cloud cover, increasing manmade aerosols and the

lowering of atmospheric transparency following explosive volcanic eruption .

Foukal et al., (2006) noted that both the solar dimming and brightening

cannot be explained by variations of the Sun‘s radiative output while Norris and

Wild (2007) suggested that it could be as a result of changing in atmospheric

transmittance caused by variations in aerosol concentrations.

Aerosols are known to affect climate in two ways, they influence the planetary

albedo by scattering and absorbing radiation (direct effect) and they modify the

physical and radiative properties of clouds by acting as cloud condensation nuclei

(indirect effect).

Although the exact magnitude of aerosol-induced radiative forcing is

uncertain, aerosols are thought to have a net cooling effect. Thus, they may mask

248



the warming effects of anthropogenic increases in greenhouse gases (Wigley,

1989; Kaufman et al., 1991;Charlson et al., 1992; Obiekezie and Okeke 2005;

Penner et al., 1992; Christopher et al., 1996; Kiehl, 1999, etc ).

Following the recommendation of Stanhill and Cohen (2005), this work is

set out to estimate the variations in sunshine hours for Lagos the busiest city in

Nigeria and to investigate the possibly effects of Aerosols on the variability of

this sunshine hours.

2. SOURCES OF DATA

Mean monthly sunshine hours, defined as the number of hours of bright sunlight

per day as measured by a sunshine recorder were collected for an equatorial

region of Lagos (06°35' N, 03°19' E) Nigeria from the archives of the Nigerian

Meteorological Agency, Federal Ministry of Aviation, Oshodi, Lagos, and were

available for nine consecutive years starting from 1997 to 2005. The Aerosol

index for Lagos was obtained from the Earth Probe TOMS (Total Ozone Mapping

Spectrometer) Version.8.


Fig.1. depicts the yearly variation in sunshine hours for the duration of nine

years from 1997- 2005. A positive linear trend of 0.0449hrs/yr is found which is

an indication that sunshine hours has not been decreasing but shows a minimal

growth or a steady rise (Solar brightening) over the years. This result is in

disagreement with the results obtained by (Tan 1999) in Southwest China , (Li

2000) in central China, (Yao and Wu 2002; Tang and Li 2003) Qinghai-Tibet

plateau, (Yang et al. 2004) North and Northeast China, (Zhang et al. 2004)

Eastern China , (Xu and Zhao2005 ) in the Yellow River basin, (Liu et al 2002) in

Taiwan and (Palle and Butler 2001) in Ireland all of which portrays a decreasing

trend in sunshine hours (Solar dimming).

From fig.1, 2004 is found to have the highest sunshine hours with a mean

value of about 5.6hrs/day while the minimum is found in 1998 with a mean

value of about 4.8 hrs/day. The general picture of how the sunshine hours varied

over the years shows that Lagos city has at least 5 hours of bright sunshine hours

on the average per day.

249



Trend = 0.0449

4.8

4.9

5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

1997 1998 1999 2000 2001 2002 2003 2004 2005

Year

Su

nsh

ine h

ou

rs

Fig. 1 Variation of sunshine hours with year

To thoroughly understand and isolate the driving forces behind the

variations in sunshine hours, average monthly sunshine hours for the nine year

period is plotted in Fig.2. average monthly sunshine hours is highest in April with

an a mean value of 6.3hrs/day and the lowest is found in July. The mean

sunshine-hours in April is found to be 1.67 times higher than the mean value in

July. Based on the prevailing seasons in the study area, i.e., The Long Rainy

Season (April –July), the Short Dry Season (August-September), the Short Rainy

Season (October-November), the Long Dry Season (December- March) the

average seasonal trends of sunshine hours are –0.84, 0.167, 0.48 and 0.15 hr,

respectively. These values suggest a decline in sunshine hours during the long

rainy season as compared to the other three seasons (short rainy season, short dry

season, and long dry season).

This result is consonance with the results of Ezekwe, (1988), Akpabio, and

Etuk, (2003).

250



0

1

2

3

4

5

6

7

Janu

ary

Febuar

y

Mar

chApr

il

May

June

July

Aug

ust

Sep

tem

ber

Octobe

r

Nove

mbe

r

Dece

mbe

r

Months

Su

nsh

ine H

ou

rs

Fig. 2. Variation of sunshine hours with months of the year

Fig 3 displays the monthly variation of TOMS aerosol index for Lagos

during the nine year period (TOMS aerosol index of less than 0.1 indicates a

crystal clear sky with maximum visibility, whereas a value of 4 indicates the

presence of aerosols so dense you would have difficulty seeing the mid-day sun).

From fig.3, the minimum index was found in August with a mean value of about

0.73 which indicates a clear sky free from aerosols while the highest value was

found in January with a mean value of about 3.18 indicating the presence of

moderate aerosols in the sky.

251



0

0.5

1

1.5

2

2.5

3

3.5

Janu

ary

Febuar

y

mar

chApr

il

may

june ju

ly

Aug

ust

sepe

mbe

r

octo

ber

nove

mbe

r

dece

mbe

r

Months

Aero

so

l In

dex

Fig. 3. Variation of Aerosol Index with months of the year

From fig. 3, the month of August is seen with a small value of aerosol

index indicating a very clear sky; in effect a high value in bright sunshine hours

is expected but when compared with fig.2, the month of August has less than 5hrs

of sunshine. Also, the month of January has high value of aerosol index implying

that aerosol is indicated in the sky; the hours of sunshine from fig. 2 was found to

be more than 5hrs. The linear regression analysis does not show a significant

positive correlation between sunshine hours and aerosol index both at monthly

level (r = 0.538214, r2

= 0.289674) and annual level (r = 0.3225, r2

= 0.1040). It

could be deduced from here that the direct effect of aerosols (absorption and

scattering of direct sunlight) does not influence sunshine hours in Lagos. The

result is in consonance with the results of Palle and Butler (2001) and in variance

with the results of Luo et al. (2000) and Guo and Ren (2006) who analyzed the

change of sunshine hours and effect of aerosol and concluded that a decrease in

visibility resulting from an increase of aerosol is the main reason of sunshine

duration decrease.

252



4. CONCLUSION

The results obtained here reveal that sunshine hours vary regularly; in years,

in months and in seasons. The yearly variation displays an increasing trend which

suggests solar brightening; the monthly variation is seen with a maximum

occurring in April which is about 1.67 times larger than the minimum which

occurred in July. The seasonal trend is seen to have a negative value for the long

rainy season implying that sunshine hours decreases in the long rainy seasons as

compared to the other three seasons (short rainy season, short dry season, and

long dry season).

The linear regression analysis result does not show a significant positive

correlation between sunshine hours and aerosol index. It is generally expected that

as the aerosols in the sky increases, the duration of sunshine hours should

decrease but this is not so for the city of Lagos, it is therefore concluded that

aerosols in the sky do not directly influence the duration of sunshine hours in

Lagos, Nigeria.

Consequently, it is suggested that a comprehensive statistical analysis of

other meteorological data could be very helpful in investigating the cause of this

observed variability in sunshine hours.

Acknowledgment

The Nigerian Meteorological Agency, Federal Ministry of Aviation, Oshodi,

Lagos and NASA/GSFC TOMS Processing Team are gratefully acknowledged

for providing the sun shine hours data and the Aerosol Index data.

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256



A STUDY OF THE MEASURED AERODYNAMIC DIAMETER

OF ATMOSPHERIC AEROSOLS IN NSUKKA, ENUGU STATE, NIGERIA‡‡‡‡‡‡‡

Okoro, E. C and 1Okeke, F. N

Department of Physics & Astronomy, University of Nigeria, Nsukka.

1email: [email protected]

Abstract

This work presents the ground-base measurement of aerodynamic diameter of

Atmospheric Aerosols captured in Nsukka, Enugu State during the harmattan

period of February to non-harmattan period of April 2004. Addition of harmful

substances such as aerosols to the atmosphere results in pollution of the

environment, bad effects on human health, and quality of life. These effects are

quantified when the aerodynamic diameter is determined. A comprehensive

research has been carried out on the particle sizes of the captured aerosols in

Nsukka Environ. Power-law and Statistical models were employed in the analysis

and some control strategies were considered.

1. INTRODUCTION

Each year in West Africa, from October to March, an area of several billion

hectares is subjected to a very dry, dust-laden atmosphere known as the

harmattan. The dust is uplifted from the Sahara desert and then carried Southward

by the North-East-trade wind. It is believed that with intense heating occurring

over the Sahara, dust is raised high into the atmosphere by the associated strong

convective activity. The fine particles of dust can rise as high as 300m above sea

level. Due to their small size and apparently, with continuing convective

instability they can remain suspended in the atmosphere for long periods of time

(Adetunji et al. 1991; Chukwuemeka 1990).

‡‡‡‡‡‡‡ African Journal of Physics Vol. 2, 256- 266, (2009)


257



The captured aerosols size in Nsukka environs was measured using an

optical microscope. Guided by this particle size analysis, one could easily

determine the atmospheric aerosol effects on health, weather, climate, vegetation,

materials, communication and atmospheric electricity. Aerosols range in size

from aggregates of only a few hundred molecules (~10-4

µm across) to particles of

about ten thousand times larger (~100 µm), as in cloud droplets (Rita and George

1981). Aerosols can be termed Aitken nuclei when the size is about 0.24 µm or

less large with size of 0.2 – 1.0 µm (McCartney 1976). Fine particles have

diameters less than four micrometers; they pose the greatest potential health

hazard and also contribute to scattering of light in the visible range, which as a

result of this scattering reduces air visibility (Strauss 1978; Dobbins 1979; Watta

1998). The largest and heaviest (coarse) aerosols have diameters greater than 4

µm and are deposited by gravity few minutes after being formed from their

sources, causing problems of laundry and windowsill soiling, deterioration of

materials etc (Murdoch 1975; Lynn 1976; Tsor 2003). Aerosols can condense and

can also grow; it can grow by absorbing a film of water from the atmosphere.

Increase in humidity also increases the average size of aerosols.

The equivalent aerodynamic diameter which is the diameter of a unit

density that has the same settling velocity in air as the particle of interest, is the

parameter frequently used in describing the aerosol size (Hesketh 1977; Strauss

1978; Stockham and Fochtman 1979). Fig 1 and fig. 2 depict the photomicrograph

of size measured aerosols. This particular physical property of aerosols has been

studied by Pope and Dockery (1999), Friedlander (1977), McCartney (1976), Tsor

(2003) and Ngadda (1991).

Fig.1

Fig. 2

The photomicrograph of captured aerosols: Fig.1: on 3/2/2004

(normal day); Fig. 2: on 12/4/2004 (after rainfall).

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2. DATA SOURCE

Aerosols were collected dry at six towns/sites in an open field by deposition

method. These towns were grouped under two stations because the data were too

numerous and those grouped together had indistinguishable similarities in their

data. Particles were collected daily for three months. The wind speed data for the

three months were obtained from the meteorological station at Enugu Airport.

3. DATA ANALYSIS

At STA I; the range of diameter data in the months of February, March and

April were 0.5 – 12.0 µm, 0.5 – 12.0 µm, and 0.5 – 11.0 µm respectively, while at

STA II, the diameter data ranges were 0.5 – 12.0 µm for February, 0.5 – 12.0 µm

for March and 0.5 – 12.0 µm for April.

The power-law size distribution function was developed by Junge (1963) and

described by Manson (1965) as;

…………………………………………….(a)

where C is the constant, D is the average diameter (particle size) and β is the

slope. Using Table 1 for clarity purpose and taking the logarithm of both sides

and plotting the power-law distribution curve we obtain fig. 3. (Note: size

distribution Table of any month could be used to achieve the aim).

259



Table 1: The daily captured aerosols size distribution for the month of February at

STA I.

Particle Size

Interval

(µm)

Mid

Size

(µm) D

Freq. of

Occurre

nce N

Cumulative

Frequency

of N (%)

ND Log D N log D dN/dlogD Log

(dN/dlogD)

0.5- 2.5

2.5 - 4.5

4.5 - 6.5

6.5 - 8.5

8.5 - 10.5

10.5- 12.5

TOTAL

1.5

3.5

5.5

7.5

9.5

11.5

391

320

103

33

19

4

870

44.9

81.7

93.6

97.4

99.5

100

586.5

1120.0

566.5

247.5

180.5

46.0

2747.0

0.1761

0.5441

0.7404

0.8751

0.9777

1.0607

68.85

174.10

76.26

28.88

18.58

4.24

370.91

2220

588

139

37

19

3

3.3464

2.7694

2.1430

1.5682

1.2788

0.4771

260



Fig 3: Junge Power-law size distribution curve for February at STA I

Table 2: The daily captured aerosols size distribution for the month of

March at STA I.

Particle Size

Interval

(µm)

Mid Size

(µm) D

Freq. of

Occurrence

N

Cumulative

Frequency

of N (%)

ND

0.5 - 2.5

2.5 - 4.5

4.5 - 6.5

6.5 - 8.5

8.5 - 10.5

10.5- 12.5

TOTAL

1.5

3.5

5.5

7.5

9.5

11.5

507

292

101

24

5

1

930

54.5

85.9

96.8

99.4

99.9

100.0

760.5

1022.0

555.5

180.0

47.5

11.5

2577.0

Table 3: The daily captured aerosols size distribution for the month of April

at STA I.

Particle Size

Interval

(µm)

Mid Size

(µm) D

Freq. of

Occurrence

N

Cumulative

Frequency

of N (%)

ND

0.5 - 2.5

2.5 - 4.5

4.5 - 6.5

6.5 - 8.5

8.5 - 10.5

10.5- 12.5

TOTAL

1.5

3.5

5.5

7.5

9.5

11.5

646

172

64

15

2

1

900

71.8

90.9

98.0

99.7

99.9

100.0

969.0

602.0

352.0

112.5

19.0

11.5

2066.0

261




February at STA II.

Particle Size

Interval

(µm)

Mid Size

(µm) D

Freq. of

Occurrence

N

Cumulative

Frequency

of N (%)

ND

0.5 - 2.5

2.5 - 4.5

4.5 - 6.5

6.5 - 8.5

8.5 - 10.5

10.5- 12.5

TOTAL

1.5

3.5

5.5

7.5

9.5

11.5

441

269

107

33

17

3

870

50.7

81.6

93.9

97.7

97.9

100.0

661.5

941.5

588.5

247.5

161.5

34.5

2635.0


March at STA II.

Particle Size

Interval

(µm)

Mid Size

(µm) D

Freq. of

Occurrence

N

Cumulative

Frequency

of N (%)

ND

0.5 - 2.5

2.5 - 4.5

4.5 - 6.5

6.5 - 8.5

8.5 - 10.5

10.5- 12.5

TOTAL

1.5

3.5

5.5

7.5

9.5

11.5

483

285

109

34

13

6

930

51.9

82.6

94.3

98.0

99.4

100.0

724.5

997.5

599.5

255.0

123.5

69.0

2769.0

262



Table 6: The daily captured aerosols size distribution for the month of April

at STA II.

Particle Size

Interval

(µm)

Mid Size

(µm) D

Freq. of

Occurrence

N

Cumulative

Frequency

of N (%)

ND

0.5 - 2.5

2.5 - 4.5

4.5 - 6.5

6.5 - 8.5

8.5 - 10.5

10.5- 12.5

TOTAL

1.5

3.5

5.5

7.5

9.5

11.5

608

175

85

25

5

2

900

67.6

87.0

96.4

99.2

99.8

100.0

912.0

612.5

467.5

187.5

47.5

23.0

2250.0

The slope (β) is derived from the graph as follows;

where s1…s5 are individual slopes at different points and β is the average slope. Β

is 3.8548 4; this indicates continental source of aerosols. The wind speed data

from February – April were 2.44m/s, 3.08m/s and 3.00m/s respectively.

4. DISCUSSION/CONCLUSION:

The average diameter of captured aerosols at STA I and STA II for the month

of February were greatest and the least was that captured in the month of April.

The higher wind speed, the lower the concentration of the particles and the

smaller the size. The sizes of aerosols obtained show that aerosols in Nsukka

environ can cause scattering of light and hence reduce visibility in the

atmosphere, thereby posing problems for aircrafts. This type can cause some

respiratory and cardiovascular diseases when inhaled and many more diseases if

highly concentrated.

263



They are under the fine particles classification. They obey the power-law size

distribution function. Also, from the power-law size distribution function, it

indicates that the source of aerosols obtained was continental.

Control Strategies:

The effects could be significantly reduced by controlling the emissions of

toxic substances like methane, black carbon from producing industries.

This could be achieved by not situating the industries in the proximity of

the residential areas and making sure that the height of chimney from the

ground level is high enough 9about 500m above sea level).

Young and elderly people that are already suffering from the respiratory

and cardiovascular diseases should avoid long-term exposures in an

aerosol filled environment.

Occupational hazards could be checked by the use of masks and protective

glasses. Uniforms should be provided for staff and laundered in a

moderately high temperature wash cycle.

The air could be primarily cleaned through filtration method. This is the

physical removal of particulate from air using filter fibers.

Government should endeavour to have all roads paved/tarred both in rural

and urban areas.

Human activities for instance bush burning, deforestation that cause land

surface changes, uses of insecticides and pest control should be

minimized. Farmland should be sited far away from habitations of human

beings and farm implements should be improved and refined rather than

using the local/old methods.

Government is encouraged to subsidize prices of conventional fuel like

gas, kerosene etc in order to minimize the use of fire wood.

Government should ensure that oil producing industries supply unleaded

fuel to the market. Irrespective of the importance of lead for the vehicle

engine, it is very poisonous to the lungs.

Cars having zero scrap value should be discarded from our towns and

cities because they are one of the biggest sources of aerosols.

There should be standard enforced law on ambient air pollution, defaulters

should be prosecuted.

264



Acknowledgement

The authors wish to thank Prof. E. U. Utah of the University of Jos for his

helpful discussions and assistance at various stages of this work, Prof. P. N.

Okeke of Centre for Basic Space Science for providing us the opportunity to

use his internet facilities during the research and Dr. F. I. Ezema for all his

helpful discussions.

REFERENCES

Adetunji, J. J. and Ibrahim, S. M. (1991): Measurement of Condensation

Nucleus Concentration in the Harmattan Haze. Nigerian Journal of Physics

Vol. 3 No. 4. p. 11 – 17.

Chukwuemeka, I. K. (1990): Aerosol Concentration and Air Composition.

M.Sc. Thesis, Dept. of Physics, University of Jos, Nigeria.

Dobbins, R. A. (1979): Atmospheric Motion and Air Pollution – An

Introduction for Students of Engineering and Science. John Wiley and Sons,

USA.

Friedlander, S. K. (1977): Smoke, Dust and Haze – Fundamentals of Aerosol

Behaviour. John Wiley and Sons, Inc., New York, USA.

Hansen, J. E. and Travis, L. D. (1974): Light Scattering in Planetary

Atmospheres. Space Science Rev., 16, 527 – 610.

Hesketh, H. E. (1977): Fine Particles in Gaseous Media. Ann Arbor Science

Publishers, Inc, Michigan, USA.

Junge, C. E. (1963): Air Chemistry and Radioactivity. Academic

Publishers, New York.

Lynn, D. A. (1976): Air Pollution – Threat and Response. Addison –

Wesley Publishing Company.

265



Manson, B. J. (1965): A case study of the vertical Distribution of

Atmospheric Ozone. Journal of Applied Meteorology. Vol. 4. p. 931

– 939.

McCartney, E. J. (1976): Optics of the Atmosphere – Scattering by

molecules and particles. John Wiley and Sons, New York.

Murdoch, W. W. (1975): Response, Pollution and Society. Sinauer

Associates Inc Publishers, Sunderland, Massachusetts.

Ngadda, I. A. (1991): Effects of Aerosol Size and Concentration on

Visibility in Jos Harmattan Air. M.Sc. Thesis, Dept. of Physics,

University of Jos, Nigeria.

Pope, C. A. III and Dockery, D. W. (1999): Epidemiology of Particle

Effects. Air pollution Health Vol. 81, p. 673 – 705.

Rita, G. L., and George, L. T. (1981): Encyclopedia of Physics. Addison-

Wesley, New York.

Stockham, J. D. and Fochtman, E. G. (1979): Particle Size Analysis. Ann

Arbor Science Publishers, Michigan, USA.

Strauss, W. (1978): Air Pollution Control Part III – Measuring and

Monitoring Air Pollutants. John Wiley and Sons, Inc, New York,

USA

Tsor, J. O. (2003): Characteristics of Aerosols in the Atmosphere during

the harmattan in three Nigerian cities. M.Sc. Thesis, Dept. of

Physics, University of Jos, Nigeria.

266



Watts, R. J. (1998): Sources, Pathways and Receptors. John Wiley and

Sons, Inc, USA.

267



DATABASE OF CO2 EMISSION IN NIGERIA: A PRELIMINARY

REPORT§§§§§§§

M. O. Ofomola and G. E. Akpojotor

Department of Physics, Delta State University, Abraka 331001, Nigeria

[email protected]; [email protected]

Abstract

The issue of global warming has become one of the hottest topics in our

world today and the reason is because of the increasing fear of its adverse

effects. There is a growing consensus today that the increased emission of

green house gases especially carbon dioxides (CO2) which is believed to

have increased by about 30% since the Industrial Revolution, are

responsible for the overall warming of our planet. The emission of CO2 is

attributed to the high rate of energy consumption from burning of fossil fuel

such as coal, oil and gas by the major industrialized nations. By global

indices of classification, Nigeria is not an industrialized country. However,

owing to its large population and the proliferation of alternative means of

energy from fuel combustion machines, Nigeria is feared to be a major

contributor to global warming. However, there are no reliable data to

ascertain the amount of CO2 that is emitted in the country. The purpose of

our study is to accumulate a database of CO2 emission in Nigeria especially

as there is already a report by scientists at the Lawrence Livermore

National Laboratory in the U.S that the effects of global warming are for

the first time, visible on a regional scale. In this preliminary report, we

demonstrated the theoretical and the experiment approaches of estimating

the amount of CO2 that is emitted in a place (or town) and the prospect of

using this methodology to obtain a database CO2 emission in Nigeria.

Keywords: Nigeria, Environment, global warming, Carbon dioxide emissions, database

§§§§§§§African Journal of Physics Vol. 2, pp. 267-279, (2009)


268



1. INTRODUCTION

Increase in the concentration of greenhouse gases has become one of

the most harzardous impacts on our environment it has resulted into an

increase in the temperature of the earth [1-5]. It is predicted that the global

average temperature will rise by about 1.60C - 6

0C by the year 2100 if

current trends of green house gases emission continue [1]. This increase in

the average temperature of the earth is termed global warming: it occurs

when certain gases commonly known as greenhouse gases trap the sun‘s

heat. When sunlight reaches the surface of the earth, some of it will be

absorbed by the earth‘s surface and this warms the earth. It is a case of heat

transfer since the earth‘s surface is much cooler than the sun and radiate

energy at much larger wavelengths than the sun. Some of the longer

wavelengths are absorbed by greenhouse gases in the atmosphere before it

can be lost to space. The absorption of this long wave radiant energy warms

the atmosphere. Greenhouse gases also emit long wave radiation both

upward to space and downward to the surface. The downward part of this

long wave radiation emitted by the atmosphere is the greenhouse effect [6].

The major greenhouse gas responsible for global warming is Carbon

dioxide (CO2). Atmospheric CO2 is derived from multiple natural sources

such as volcanic out gassing, the combination of organic matter and the

respiration processes of living anaerobic organisms. Apart from these

natural phenomena, man-made sources include the burning of various fossil

fuels for power generation in the industry, transportation. Agriculture, etc

[7]. The growing belief is that increase in the man-made sources of CO2

emission are responsible for the global warming hence it is also known as

anthropogenic climate change. These man-made sources depend on the

economies of the various countries. which in turn depends on their

developmental levels. This corroborate the report that the concentration of

CO2 has increased substantially since the industrial revolution and is

expected to continue to be so [2,3].

The adverse effects of our warming planet are also global such as

precipitation, depreciation of slow cover, glacier extent, etc, all of which

are believed to be responsible for the present drying up of some lakes, rise

in the sea levels leading to flooding, etc [8]. This flooding is expected to

affect African countries on the coastline such as the southern part of

269



Nigeria [9]. There also the fears of health related problems and uncertain

effects on the Agriculture [10]. Thus in general, the geological location of a

country will determine the level of the consequences of global warming on

it. This supports the report by scientists at the Lawrence Livermore

National Laboratory in the U.S that the effects of global warming are for

the first time visible on a local scale [3].

It is surprising to note that in Nigeria, there is still no clear

leadership in tackling the issue of climate change and the predicted

consequences. While it has been a national agenda in many countries even

with relatively high CO2 emission, it is only in June 2009 that a desk office

under the auspices of the Nigeria Climate Action Network (Nigeria CAN)

was inaugurated in the Fededal Ministry of Science and Technology,

Abuja.

As we pointed out in 2005 [5], though Nigeria is not considered by

global indices of classification an industrialised nation, the amount of CO2

pumped into the atmosphere in the country can be relatively alarming when

compared to that released in some many developing countries. The reason

being that the over 140 million population based on the 1991 census which

makes her the most populous country in Africa and the 8th most populous

country in the world, means a large transport potential which may translate

into a remarkable emission of CO2. This has been aggravated by the

epileptic power supply from the national grid and deforestation in the

tropical parts of the country [3,11].

It is worthy of note that from World Environment Statistics [12],

Nigeria is ranked. 51st among 178 countries in CO2 emission with 48,145.7

thousand metric tonnes, as against the United States with a CO2 emission

of 5,762,050 thousand metric tonnes which makes her the first. This total

emission of CO2 does not indicate the variation in emission level from one

place (or town) to another in the country. Therefore, it cannot be used for

proper environmental planning and policy making. The purpose of our

study here is to have a database of CO2 emission in Nigeria. In this

preliminary report, we have demonstrated how to obtain the total amount of

CO2 emission in any given place. The plan of our study is as follows. In sec.

II, we will show how to obtain the total emitted CO2 in any arbitrary

number of litres of petroleum products. Thereafter, we will show how to

obtain the total amount of litres of petroleum products consumed in one

270



place in sec. III. The results will be presented and discussed in sec. IV and

this will be followed by a conclusion.

2. AMOUNT OF CO2 EMISSION FROM N LITRES OF

PETROLEUM PRODUCTS

The first step in developing a database of the amount of emitted CO2

is to determine the amount of CO2 in a litre of the petroleum products. Each

of these various petroleum products, however, has various ratings

depending on the quality of the parent crude oil and the refining

technology. This variation in rating is not unconnected with the increase in

importation of petroleum products in recent years owing to the inability of

the four under producing refineries in the country to meet national

consumption. As reported by Nwachukwu and Bala-gbogbo [13], the

refineries in 2008 were only able to supply the market with 37,156 metric

tonnes (mt) of petrol which is far below the 8,909 million mt for the

national consumption of petrol and 169.088 mt of diesel lower than the

national consumption of 3,215 million mt. Therefore, more than 90 % of

the national consumption of petroleum products is imported. Since these

importations are from various refineries around the world, the CO2

emission from more than 90 % of the petroleum products varies from one

importation source to another. Thus for the purpose of the database

estimation here, we will show how to determine the amount of CO2 in a

litre of the various petroleum products. This will be done from theoretical

analysis using the stoichiometric equations and then by experiment.

Theoretical Analysis

Taking into account their stoichiometric equations and molecular

weights, it is easy to obtain the amount of CO2 in a litre of petrol, diesel and

kerosene:

For Petrol (C8H18 = 114)

Amount of CO2 in 1 litre of petrol = 1 x 0.74 x 0.84 x 3.6667

= 2.27kg of CO2 = 2270g of CO2

For Diesel (C6H34 = 226)

271



Amount of CO2 in 1 litre of diesel = 1 x 0.85 x 0.85 x 3.667

= 2.65kg = 2650g.

For Kerosene (C12H24 = 168)

Amount of CO2 in 1litre of kerosine = 1 x 0.817 x 0.85 x 3.6667

= 2.55kg = 2550g.

Thus for n litres of petrol, diesel and kerosine, the amount of CO2 is

given as 2.270n Kg, 2.650n Kg and 2.550n Kg respectively. It is worthy to

note that these values can also be written in terms of their specific gravities

[14]

nØ(3.6667).

where n = no. of litres consumed

= specific gravity (petrol = 0.74, Diesel = 0.85, Kerosene =

0.87)

Ø = Proportionality of carbon.

Experiment approach using an Eudiometer

The eudiometer looks like an autoclave built of thick glassy material

that is heat resistant. It contains two bowls, one containing the materials to

be combusted with the lighter (a resistor like device), which becomes hot

when the equipment is switched on, igniting the sample. The other bowl

contains a solution of calcium oxide which absorbs carbon (iv) oxide and

turns milky. It absorbs better if the solution is acidified.

The base of the equipment is bent to enable the steam generated by

combustion to be collected as water in a graduated container, when the

equipment cools down.

50ml of the sample is ignited inside the endiometer in the presence

of excess air, CO2 formed is absorbed by an acidified solution of calcium

hydroxide to form a milky solution of calcium trioxocarbonate (iv). The

amount of carbondioxide formed is estimated by standardising the calcium

trioxocarbonate (iv) solution using standard hydrochloric acid solution by

titrametic method.

272



Alternatively, the pressure within the eudiometer before and after the

spark can be determined by using gas equations, the volume of

carbondioxide can be calculated.

The result of combustion analysis using an eudiometer gives:

Sample A: Petrol:

Colour: Reddish

Specific gravity 0.737

50ml (36.856g) of sample A (Petrol) on combustion in the presence

of excess air yielded 113.65g of CO2 and 53g of H2O. The implication is

that 1 litre (1000ml) (737g) of petrol on combustion yielded 2,274g of CO2

and 1,040g of H2O.

Sample B: Diesel:

Colour: Brownish


50m/ (44.10g) of diesel on combustion in the presence of excess air

yielded 161.70g of CO2 and 69g of H2O. The implication is that 1 litre

(1000ml) of diesel on combustion yielded 3234g of CO2 and 1,380g of

H2O.

Sample C: Kerosine:

Colour: Light Bluish


50ml (40.85g) of kerosene in combustion in the presence of excess

air yielded 126.81g of CO2 and 56g of H2O. The implication in that 1 litre

(1000ml) of kerosene on combustion yielded 2.536g of CO2 and 1,127g of

H2O.

Therefore, from the practical approach, the amount of CO2 emission

from 1 litre of petrol, diesel and kerosene.

Observe that the amount of CO2 emitted from the petroleum products

samples used in the experiment are the same except for the slight variation

in that of diesel.

273



3. OBTAINING THE TOTAL AMOUNT OF CONSUMED

PETROLEUM PRODUCTS

It is now straightforward to develop the database of any given place once

we can obtain the total amount of petroleum products consumed over a

period of time there. Doing so for Nigeria would have been straightforward

if all the petroleum products consumed in the country are from our

refineries because the Petroleum Pricing and Marketing Company (PPMC)

of Nigeria will then be able to provide the amount of the petroleum

products supplied to various parts of the country. But as stated earlier, more

than 90% of the petroleum products consumed in the country are imported

and the importers sell their products to the oil marketers from various parts

of the country. The implication is that this arrangement makes it impossible

to get the total amount of the petroleum products consumed in most towns

from PPMC.

An alternative means to obtain estimate of the total amount of the

petroleum products consumed in any town is to get this information directly

from the fuel filling stations in that town. To demonstrate this approach, we

have used Abraka which is the host community to the Delta State

University, as a case study. Here we considered the possible relationship of

the sales of petroleum products and the possible amount of consumption of

energy resulting to the emission of CO2 to the environment. The sampling

was carried out in all the prominent filling stations in Abraka community;

Emole Nig. Limited, Texaco, Total, Buovo, Blue Point, SpringBeds, Acod

and Pellucid.

The samples were stratified to correspond to the amount of

petroleum products (Petrol, Diesel and Kerosine) consumed in the year

2008 by individuals, organizations, corporate firms, government parastatals

and transport workers in Abraka. In general, the random sample covers the

diversity of the environmental effect of the energy supply systems, since

95% of the sale of the product is utilised and consumed in Abraka.

Therefore, the amount of CO2 released from this 95% of petroleum products

gives a good estimates of the CO2 emission in Abraka.

Table 1 shows the sales of Petrol in thousand litres for 2008 in Abraka.

274



S/N Filling stations

Sales of Petrol (in thousand litres) for 2008

Jan Feb Mar Apr May Jun Jul Au

g

Sep Oct No

v

Dec

1 Emole Nig Ltd

180 180 210 210 180 210 240 240 240 270 270 180

2 Texaco 150 120 150 120 180 150 150 180 120 180 210 150

3 Total 150 150 180 120 150 120 120 180 150 120 150 120

4 Buovo 60 90 120 90 60 60 80 60 90 60 60 90

5 Blue Point 180 210 210 210 150 180 150 210 280 240 180 210

6 SpringBeds 210 240 240 210 240 180 240 270 230 240 230 250

7 Acod 120 150 150 90 120 120 60 90 150 150 150 150

8 Pellucid 60 30 60 30 40 50 60 30 45 60 70 60

TOTAL 1110 1170 1320 1080 1120 1070 1100 1260 1305 1320 1310 1210

Table 2 shows the sales of Diesel in thousand litres for 2008 in Abraka.

S/N

Filling stations

Sales of Diesel (in thousand litres) for 2008

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

275



1 Emole Nig Ltd

30 30 - 45 30 - 30 20 30 - 30 30

2 Texaco 30 45 - 30 30 - 45 30 30 20 30 30

3 Total 45 30 - 30 30 30 30 20 30 30 - 30

4 Buovo 30 45 45 - 30 30 30 25 30 - 30 30

5 Blue Point

30 30 60 45 40 45 40 30 60 35 40 30

6 SpringBeds

30 30 50 45 45 45 40 30 45 30 45 30

7 Acod 30 30 45 - 30 30 30 20 30 - 30 20

8 Pellucid 30 - 30 20 30 30 20 30 30 30 20 30

TOTAL 255 240 230 215 265 210 265 255 285 145 225 230

Table 3 shows the sales of Kerosene in thousand litres for 2008 in Abraka.

S/N

Filling stations

Sales of Kerosene (in thousand litres) for 2008

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

1 Emole Nig Ltd

26 26 39 39 39 26 13 13 26 26 26 39

276



2 Texaco 26 13 20 15 26 - - 39 39 26 26 13

3 Total 15 13 13 26 - - 26 26 13 13 26 26

4 Buovo 26 26 13 13 26 13 26 26 13 - - -

5 Blue Point 15 15 15 15 25 20 15 25 20 15 10 15

6 Spring

Beds

26 13 15 26 39 39 39 39 26 26 13 13

7 Acod 26 26 13 39 39 13 13 26 26 39 26 29

8 Pellucid 15 15 15 - - 15 15 15 15 15 15 15

TOTAL 175 147 143 173 194 126 147 209 178 160 142 149


The total consumption of petroleum products for year 2008 in Abraka is:

petrol, 14,375,000 litres; diesel, 2,820,000 litres and kerosene, 1,943,000

litres (see Tables 1, 2 and 3). Therefore the amount of CO2 emitted is as

follows:

14,375,000 litres of petrol will emit 3.2689x107 kg of CO2

2,820,000 litres of diesel will emit 9.2000x106 kg of CO2

1,943,000litres of kerosine will emit 4.927x106 kg of CO2

It follows that the average CO2 emission in Abraka per day will be

128,104 kg which is equivalent to an emission of 1,220 ppm. This is less

than the world health organisation (WHO) stipulated maximum of 20,000

ppm and therefore according to Greiner [15], this quantity is not high

enough to cause health hazard. However, the CO2 emission is about 3 times

higher the global concentration of atmospheric CO2 which has increased

from 280 ppm in 1700 to over 370 ppm today [4].

Ndoke et al [16] in their study on the contribution of vehicular

traffic to CO2 emissions in Kaduna and Abuja using CO2 measuring

277



gadgets shows that an average of 1160 – 1840 ppm of CO2 is emitted in the

area. Also, Ndoke and Jimoh [17] in an earlier similar study for Minna,

observe that the CO2 concentration in the area was as high as 5000 ppm.

With these results, it is expected that the CO2 emission in cities like Lagos

and Port-Harcourt with high population density and increase economic

activities may have reached alarming levels.

5. CONCLUSION

The major greenhouse gas is atmospheric CO2 which is believed to

have increase by about 30% since the industrial revolution. It is emitted

from the world major sources of energy, which is burning of fossil fuels

such as coal, oil and gas. The large population, poor power supply from the

national grid and deforestation makes Nigeria a potential high CO2 emitting

country. However, there is no database of CO2 emission to initiate actions

to study and predict the nature of the possible impacts of global warming at

a local scale within the country. This is the project we have embarked upon.

Getting data on consumption of petroleum products in various parts of the

country from government agencies have been a herculean task. The

methodology we have developed in this preliminary report is to get such

data from the oil marketers operating in our places of interest. It is therefore

hoped that we can use this methodology in all parts of the country and then

be able to develop a database of CO2 emission in the country which will

become very useful to researchers, town planners and policy makers.

Acknowledgement

We appreciate the kind assistance of the Abraka branch of the Oil

Marketers Association of Nigeria for providing us their monthly sales

output in 2008. We also appreciate the useful discussion with Dr Edmond

Atakpo. This work is supported in part by AFAHOSITECH.

278



REFERENCES

[1] Inter-governmental Panel on Climate Change (IPCC). Climate Change

2001: The Scientific Basis: Contribution of Working Group I to the

Third Assessment Report of the IPCC. Houghton J, Ding Y, Griggs

M, et al, eds. Cambridge, England: Cambridge University Press

(2001).

[2] Inter-governmental Panel on Climate Change, I.P.C.C Report (1995),

Atmosphere, Climate and Environmental Information Programme,

Climate Research Unit, University of East Anglia, Norwich, UK

[3] K Walter, The outlook is warming with measurable local effects,

Science and Technology Review, The Regent of the University of

California, pp4-12, May (2004).

[4] A. Parker, The Siren Call of the seas: Sequestering carbon dioxide,



[5] G. E. Akpojotor and T. Akporhonor, Global Warming: Methods to

sequester the increasing emitted CO2 in Nigeria. Proceedings of the

International Conference on Science and Technology, FUT, Akure,

Nigeria, pp 460-463, August (2005)

[6] D. Pearce, Report 3. Green Heat and Power. Eco-effective Energy

solutions in the 21st century, Available at: http://www.bellona.no

(1998).

[7] F. C. Albert, Man-made sources of carbon dioxide, Heineman

Educational Books Ltd, London. pp 10-12 (1987).

[8] See Effects of Global Warming,

http://environment.nationalgeographic.com/environment/global-

warming/gw-effects.html

[9] A. Raufu, Africa underwater: Nigeria's coastline is besieged by

Global Warming, The Environmental Magazine,

http://findarticles.com/p/articles/mi_m1594/is_2_13/ai_83667620/,

March-April, (2002)

http://www.bellona.no/

279



[10] A. Parker, Climate change and Agriculture: change begets change,



[11] K. A. Small. And C. Kazimi, On the costs of air pollution from

motor vehicles. J. Transp. Econ. Policy 29: 7 – 32 (1995)

[12] World Resources. World Resources Institute in collaboration with the

United Nations Environment Programme and the United Nations

Development programme Oxford University Press, New York, NY,

USA (1992).

[13] C. Nwachukwu and E. Bala-gbogbo, Global meltdown hitting us

where it hurts, Next, 234next.com. Timbuktu media (2009).

[14] A. Marion, Mathematical Approach to Carbondioxide Measurement

and Extraction (Unpublished)

[15] T. Greiner, Indoor air quality: carbon monoxide and carbon dioxide,

Iowa state University Extension Publication No. AEN-125 (1995).

[16] P.N, Ndoke, U.G. Akpan and M.E. Kato, Contribution of Vehicular

Traffic to Carbon Dioxide Emissions in Kaduna and Abuja, Northern

Nigeria. Leonardo Electronic Journal of Practices and Technologies,

5(9), pp 81-90 (2006).

[17] P.N, Ndoke, and O. D. Jimoh, Impact of Traffic Emission on Air

Quality in A Developing City of Nigeria.

www.journal.au.edu/au_techno/2005/.../vol8n04_abstract10.pdf

pp222-227 (2005)

http://www.journal.au.edu/au_techno/2005/.../vol8n04_abstract10.pdf%20pp222-227

http://www.journal.au.edu/au_techno/2005/.../vol8n04_abstract10.pdf%20pp222-227

280



TECHNO-ECONOMIC ANALYSIS OF A BIOGAS PLANT FOR AGRICULTURAL

APPLICATIONS: A CASE STUDY OF CONCORDIA FARMS LTD,

PORTHARCOURT********

L.M. S. Tobira

Department of Mechanical Engineering, University of Nigeria, Nsukka, Nigeria.

e-mail:[email protected]

Abstract

A techno-economic analysis of generating biogas using a fixed dome

digester coupled with a solar collector through a heat exchanger has been

studied for Concordia Farms Limited. This gas when generated from

organic waste on the farm could replace power-generating plant in the farm

and save the huge cost (in naira) consumed by the private power plant in

generating energy for the farm. Mathematical computations have been

made to optimize different analysis, namely; organic waste generating

capacity of the farms, volume of digester suitable for the farm, energy

requirements/needs of the farm, available energy sources of the farm and its

biogas generating potentials. The design criteria for thermal heating of an

active, fixed-dome type biogas plant is presented with the effects of heat

exchanger and collector panel incorporated in the thermal analysis.

Increasing the flow rate of the working fluid between the heat exchanger

and the collector loop can optimize the thermal efficiency. The economic

analysis takes into account, capital and maintenance costs, life of the

project, priced and unpriced benefits of owning a biogas plant. Priced

benefits involves cost valuation (in naira) of the various fuels used e.g. fuel

wood, kerosene, PMS, diesel and time and labour etc. which becomes the

******** African Journal of Physics Vol. 2, 280-299 (2009)


281



cost saved/avoided by owning a biogas plant. The benefit – cost ratio,

internal rate of returns and net present values, cost-payback and energy

payback of the investment are also computed to establish the viability of the

proposed biogas project.

1.0 INTRODUCTION

The energy crisis in the early 70‘s caused economic problems for many

countries that depend on imported oil and gases. With the high cost and instability

in the price, non-renewability of petroleum products as well as the growing

environmental concern (global warming) on burning of fossil fuels, the need for a

renewable and more environmentally friendly fuel has become imperative. The

exploitation of new energy sources and the adoption of new energy conversion

technologies became necessary towards reduction of enormous organic waste

generated especially in the integrated farms and providing an alternative,

environment compatible, cheap source of renewable energy for such farms – in

Nigeria. Huge quantities of organic waste running into several hundreds of tons

are generated in integrated farms a year. At the same time, these farms spent huge

sums of money on electricity bills, operating private power generating plants, fuel

wood, kerosene, etc. to meet energy needs of farm.

Biogas (also called ―Marsh gas‖), a by-product of anaerobic decomposition

of organic waste has been considered as an alternative source of energy. Wiley

(1996) noted that the common raw materials for biogas generation are often

defined as ―waste materials‖, e.g. animal manure, sewage sludge and vegetable

crop residues, all of which are rich in nutrients suitable for the growth of

anaerobic bacteria.

The interest in the present paper is therefore to produce biogas from animal

dungs generated on the this farm that can be used as a cheap, renewable source of

energy on the farm. It is also the aim of this work to compare the cost of owning a

owning a biogas plant by the farm with that of buying fossil fuels.

2. METHODOLOGIES AND MATERIALS

This project was conducted by using a triangulation method consisting of:

literature review, background research/case studies and direct interviews. The

literature review and background research provides an initial overview of biogas.

These sources described what biogas is, how it is produced, and how it could be

282



used. The literature review transcribed what studies have been done in reference

to biogas and current projects using biogas technology. To increase the validity of

the project, only recent journal articles were reviewed.

Background research and case studies were reviewed and will serve as a

comparison to the potential Concordia farms project and provide information as to

the size, capacity, and type of biogas plant that would best suit Concordia farms

limited. Interviews were conducted to several local farm workers about organic

wastes including farmers, farm manager, and farm equipment

operators/maintenance workers, and marketers. The verbal interview questions

were reviewed and passed by Concordia farms Office of Research Ethics.

Approval from the Research Ethics office was needed to interview the manager

and farmers. Interview participants were selected from criteria, which were based

on the proximity of the participants to the farm, and the volume of wastes that

could be generated. Maximum waste could be collected from such farm as

compared to slaughters‘ wastes. Participants were contacted directly.

Series of questions were asked regarding where the waste goes currently,

farms sources of energy, farm‘s cost on energy, energy needs of the farm and

whether they would be willing to donate their organic waste if a biogas plant is

built on the farm for biogas generation, and the sludge used as manure in

agriculture. The collected data was taken and assessed to determine extra amounts

of organic waste needed for the biogas plant. The economic feasibility of the

biogas plant was conducted with all data collected. This was be followed by a

discussion, recommendations and alternatives for the feasibility of this project.

This method of triangulation attempts to use the most recent and innovative

technologies to minimize potential operational and start-up problems. This

method also emphasizes the benefits a biogas operation would have on the local

community and Concordia farms limited

The farms have the following number of animals and poultry as sources of

dung generation for biogas; 3500 – Birds, 400 – Pigs, 200 – Sheep, 300 – Cows

The type, quantity, and cost of energy consumed per month by the farm are as

follows;

- Fuel wood 20,000kg/month N128, 000.00

- Kerosene 7000 litres/month N330, 750.00

- Diesel 10,000 litres/month N542, 750.00

- P.M.S 10,000 litres/month N514, 000.00

- Charcoal 3500kg/month N8, 750.00

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3. OBJECTIVE OF THE STUDY

To provide an alternative source of energy to the farm hence reduce its over

dependency on fossil fuels

-To produce a cheap, environmentally friendly energy for the farm use

To convert the huge organic waste generated on the farm into useful energy

hence enhancing good farm hygiene and reducing expenses on fossil fuels

To increase farm outputs and reduce inputs

To integrate biogas technology

4. LITERATURE SURVEY

Biogas is produced by decomposition of biomass and animal wastes, human

excreta, sewage sludge and vegetable residues and poultry wastes by decomposer

organisms like bacteria under anaerobic (airless) condition. This process is

favoured by warm, wet and dark conditions. This involves chemical and

biological processes known as “anaerobic fermentation”, but “digestion” is

often used in anaerobic conditions, that lead to methane production.

Biogas consists of 70% methane [CH4] and 29% carbon dioxide [CO2],

and 1% of hydrogen sulphide [H2S], nitrogen [N2], and some hydrogen [H2]. It

has a calorific value of 20Mj/m3. Biogas is generated from the slurry [50% water

and 50% dung] at an average temperature of about 35OC by chemical waste and

biological process called anaerobic fermentation. The optimum temperature for

maximum production of biogas from slurry is about 37oC. The quantity of gas

production depends on the nature of dung used. The optimum temperature of

maximum production is achieved after a number of days, referred to as retention

period, after feeding the slurry into the digester of the system. The production of

gas starts only after the retention period. Supplying thermal energy to the system

by external means, i.e. by heating slurry using either passive or active method,

can reduce the length of the retention period.

The anaerobic digestion of organic material is a very complicated

biochemical process, involving hundreds of possible intermediate compounds and

reactions, each of which is catalyzed by specific enzymes or catalysts. However,

the overall chemical reaction is often simplified to:

Organic matter anaerobic CH4 + CO2 + H2 + NH3 +H2S…………(1)

Digestion

In general, anaerobic digestion is considered to occur in the following stages:

284



The hydrolysis phase – liquefaction or polymer breakdown.

Acid formation phase

Methane formation.

In a biogas plant, all the three phases occur simultaneously and if only one

phase dominates, production of methane is seriously affected.

There are three main types of biogas plants suitable for integrated farms –

the fixed dome digester plant, the floating drum digester and the plastic covered

ditch. In this work, the fixed dome digester was used. Biogas has many

applications in integrated farms some of which are:

a). Biogas serves as a cooking fuel for farmers.

b). Biogas is used for lighting purposes on the farm.

c). Biogas lamps are use to warm birds and animals.

d). It is also possible to power an internal combustion (IC) engine that may

be found on the farms setting.

5. RESULTS AND DISCUSSIONS

5.1 Energy Audit and sizing of digester

The various forms of energy consumption per month and cost distribution account

were analyzed in this sub-section. The energy audit computed in table2 below is

based on fossil fuels used by the farms. From this table, Concordia farms utilize

1,218,111.25 Mj of fossil fuel per month (14,617,335 Mj per year) at a huge cost

of N1, 524,250.00 per month, (N18,291,000.00 per year).Table1 show heating

values of fuels used in the farm

Table 1: Heating vales of some fuels

Fuels Heating values (kj/kg) Heating values (Mj/kg)

Kerosene (paraffins) 46250 46.25Mj/kg

Fuel wood 12, 000 12 Mj/kg

Charcoal 9000 9 Mj/kg

Diesel (AGO) 46,000 46 Mj/kg

Motor petrol 46,800 46.8Mj/kg

[EASTop & McKonkey (1999)]

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Also, in order to design a digester of an appropriate and suitable capacity

of 681.3 M3 for the farm, the influent in (Kg) generated from livestock on the

farm were computed as seen in table3 and Fig.1 used in sizing the digester.

Table 3: Calculated Influent/day for Sizing the Farm Digester

Kinds Population Discharge per

day (kg)

TS value of

fresh

discharge

(% by wt)

Total influent

for each kinds

(Kg)

Cow 300 10 16 6000

Chicken 3,500 0.10 20 875

Pig 400 6 20 6000

Sheep 200 1.5 20 750

Total Influent generated on farm/day 13,625

With a hydraulic retention time (HRT) of 40 days, and Total influent (Q) of

13,625Kg, the digester volume was determined using the formulae

0.8V=Q HRT (1000Kg=1M3)……………………………………………(2)

From equation (1.2), the digester size is computed to be 681.3 M3

286



Table 4: Biogas Energy Audit of the farm

Material Quantity (kg) Gas yield per

day (M3)

Total yield of biogas/

per day (M3)

Cattle dungs 3000 0.36 1,080

Sheep wastes 300 0.10 30

Pig droppings 2400 0.25 600

Poultry droppings 350 0.0112 3.92

Total volume of biogas yield per day 1,713.92

With 6,050kg of organic waste, 1,713.92 (M3) of biogas will be yielded

per day. This indicates that in one month, a total of 1,713.92 x 30 = 51,417.6 M3

of biogas will be generated in the farms. Since 1M3 of biogas is equivalent to

0.4kg of diesel, 0.6kg of petrol, 3.5kg of fuel wood, and 0.8kg of charcoal and

0.5kg of kerosene. One can say that generating 51,417.6M3 of biogas in a month

is equivalent to buying.

Fig.1: Calculated Dimensions of the cylindrical shaped biogas digester

287



20,567.04kg of diesel/month = N1, 601,027.066 cost/month

30,850.56kg of petrol/ month = N2, 938,148.57

179,92.16kg of fuel wood/month = N1, 151,757.824

41,134.08kg of charcoal/ month = N102, 835.20

25,708.8kg of kerosene/ month = N2, 666,097.778

which is far more than the quantities Concordia farms purchase per month as

reflected in the energy analysis.

Also, a total of 51,417.6 m3 x 6,300 Kcal /m

3 = 323,930,880 kcal of

energy will be available to the farm in a month. 323,930,880 kcal = 3.2393088 x

1011

cal = 1.355909881 x 1012

joules = 1,355,909.881Mj of energy. This amount

of energy is far more than the calculated 1,218,111.25 Mj of energy consumed on

the farm per month.

So, the biogas generation prospects of the farm can actually meet the energy

needs of Concordia farms. This amount of energy can be utilize in cooking,

lightening, heating, warming etc. on the farm

5.2 Thermal analysis of the biogas plant

Result obtained from calculations reveal a thermal efficiency of 25 %.This

value shows poor efficiency of the heating system of the plant.It is obvious that

the various heat losses to the ambient and ground is responsible for the value

obtained.There is a significant decrease in thermal efficiency by the unglazing

effect due to reduced solar flux at the absorber of collector plate caused by

covection. The losses should be a minimum. The expression for thermal

efficiency is given below

)3......(..........)()(/)()/()exp(1

))(/)(

tITTCamUtFatNAat

tNAtITTCm

asossLc

csossst

288



0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

200

400

600

800

1000

1200

1400

1600

1800

2000

PTS [%]

Vb

[M3]

As seen in figure 2, the volume of biogas generated increases as percentage total

solid increase.In this research work,the total volume of biogas generated per day

stand at 1713.92 M3 and from the graph one can easily determine the average

PTS value to be 25.5 % .A marginal increase in PTS results in a geometrical

increase in the volume of biogas produced.

Figure 2: Graph of percentage total solid vs volume of biogas generated

289



0 2000 4000 6000 8000 10000 12000 140000

100

200

300

400

500

600

700

Q [Kg]

Vd

[M3]

10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

1400

1600

1800

HRT [days]

Vd

[M3]

Figure 3 Graph of digester volume Vd vs substrate Q at HRT of 40 days

Figure 4: Variation of Vd with HRT at substrate value of 13,625 Kg

290



291



292



As seen from result above, iron rod, iron wire and wood consume the highest

amount of construction cost of the digester plant representing 41.14%. Other

major costs are; sand & chippings, cement, and labour while digester accessories

gulp the list cost.

Cost payback time

Payback=capital cost/annual energy cost savings

Payback=5,954,100/18,214,175 = 0.33 years.

Energy payback time

Payback=1,218,111.25/1,355,909.88 = 0.9years

Figure 5: Cost Distribution of 681.3 m3 Biogas Plant

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5.3 Economic Valuation of Firewood and Charcoal

Use of firewood for cooking by a farm has negative effective on the

density of forest area in the locality, which in turn affects the microclimate of the

area and thus the society. Therefore, economic price of firewood has to be higher

for society than to an individual resulting into higher economic rate of returns on

the investment.

It is yet to be declared a single value for fuel wood that would reflect the

social cost or benefit of it. Some authorities have treated firewood as non-traded

goods and value it at lower than the financial price. Others value it at a percent

higher than the financial price. Still, other authorities have taken economic price

of firewood as 20 percent higher than the financial price.

5.4 Economic Valuation of Kerosene, p.m.s and Diesel

It is easier to arrive at the economic value of kerosene/PMS/Diesel as it is

readily marketed and the money value of subsidy in it can be calculated. In

Nigeria, petroleum products are refined locally and imported from oversees – for

imported goods; payment is made in US dollars. Assuming that the official

exchange rate between Nigeria‘s Naira and the US dollars would fully reflect the

true economic value of goods traded with these currencies, the border price paid

by Nigeria is taken as the economic price of these products, while the cost of

production is the economic price when locally refined. About 10 percent is added

to this price to reflect the economic cost involved in transportation and handling

of kerosene/diesel/PMS within the country.

5.5 Economic Valuation of Labour

The use of biogas results in the saving of unskilled labour time. A wage

rate for unskilled labour has to be reduced by a factor that would reflect the cost

of large-scale farming. Gautam used a factor of 0.65 to arrive at the economic

wage rate of an unskilled labour (Gautam, 1988).

5.6 Valuation of Slurry

Slurry is valued for its content of soil nutrients, particularly N.P.K. As all

chemical fertilizers in Nigeria is imported, the economic values of N, P and K are

calculated at the international market price of N. P and K fertilizers.

294



5.7 Investment Cost

The guarantee fee (if any) and service charge taken by biogas builders

should be deducted from the total investment, as they are only transfer of

payments. The subsidy (if any) should be included as part of the investment cost.

The total expenditure actually incurred for construction activities should be

reduced by a factor to reflect the true economic cost of materials and labor used in

construction. Gautam used the weighted average construction factor of 0.76 in the

case study referred above.

It is seen from the above that the economic cost of goods and services

used for biogas plant installation become lower than the costs used for financial

analysis. Also, the benefits of biogas use are valued at higher rate for economic

analysis than the financial analysis. Therefore, any plant that proves to be

financially viable to an individual user will still be viable at higher rate of return

from the economic or social point of view.

6.0 CONCLUSION

The choice of owning a biogas plant depends on; (1) the availability of

sufficient organic wastes which serves as raw material or input. (2) The energy

needs or requirements of the environment, it is to be installed. The volume of the

biogas plant will also depend on the amount of waste generated within the

locality and the amount of energy needed for consumption.

In the case study farm, we found out that the waste generation per day ran

into several thousands kilograms. This greatly influences the biogas digester

volume designed for these farms. Also, the amount of energy consumed per

month and resultantly per year ran from a million per day to several millions

mega joules per year.

In the thermal analysis, the instantaneous efficiency of the biogas plant

was used for the design of the active biogas system with a given heat capacity

(Ms Cs). From the economic point of view, the net cash flow of a 681.25m3 active

biogas plant without subsidy is positive in the first year. This indicates that

without subsidy, a user can still invest to get a positive return on investment. This

is not beyond the investment capacity for a commercial or large-scale or

mechanized farmer. Though there is still need for subsidy to encourage this

technology. Another factor noticed in the economic feasibility is the higher

benefit of biogas plant use in terms of petrol, diesel and kerosene saved. This

suggests that the biogas plant may not be viewed as profitable if these savings are

not used for generating more income by ploughing back these savings into the

295



farming business. Also, the biogas will be profitable if the labour saved is used for

generating income for the farm and the farm must attach values to all other

benefits of the biogas plant such as leisure, clean home stead/farm stead, and

better health.

Further more, the profitability of investment in biogas will increase with

the increase in the price of firewood, kerosene, diesel, etc. in the future. So far,

we have analyzed the organic waste generation of the case study farms, its energy

requirements, and we have compared its biogas generation prospects with energy

requirement. The economy studies also reveal the viability of a project of

installing an active biogas plant in Concordia farms Limited. Biogas is a potential

renewable energy source for rural Nigeria. Taking biogas generation as a farm

base activity, the energy requirements of these farms can be meet.

From these analyses, I come to the conclusion that the designed biogas

plant will be suitable for this farm.

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300



F-LAYER PEAK ELECTRON DENSITY VARIATIONS IN THE

IONOSPHERE††††††††

S.E.Onwuneme

Department of Physics University of Port Harcourt

Email:[email protected]

Abstract

F-layer peak electron density (NmF2) and height (hmF2) are important

ionospheric characteristics for planning and operation of navigation and

communication systems. These parameters were investigated in Port Harcourt

(4.500,7.00

0), during a year of high solar activity (2000) and a year of low solar

activity (2006).It was observed that NmF2 have two peaks one in May and the

other in October; its value is also enhanced during high solar activity. Generally it

was also noted that NmF2 is higher in value during the day time than night. hmF2

also exhibits variation with a peak occurring at 12 LT both for year of high and

low solar activity.

1. INTRODUCTION

The Earth‘s atmosphere is categorized into five regions at increasing height

from the Earth‘s surface, the troposphere, stratosphere, mesosphere, thermosphere

and exosphere. The two outermost layers of the Earth‘s atmosphere, the

thermosphere and exosphere, starting at about 75km from the Earth‘s surface are

sufficiently thin that ultraviolet radiation causes them to be ionized; electrons are

knocked out of atoms by photons, and the sparsity of the atmosphere allows them

to live free for some time before recombining with a nearby positive ion. This

plasma of free electrons and positively charged atomic ions is known as the

†††††††† African Journal of Physics Vol. 2, pp. 300-307, (2009)


301



ionosphere. The ionospheric characteristics exhibit significant variations with

solar cycle, season and local time, etc., which results from changes in the solar

extreme ultraviolet (EUV) and X-ray radiations, and from various chemical and

dynamic processes [e.g., Balan et al.,1994a, 1994b; Evans, 1965; Kane, 1992;

Kawamura et al., 2002; Richards et al., 1994b; Richards, 2001]. Studies of

variations of NmF2 and hmF2 are useful in investigating the physical processes

responsible for the ionospheric behavior .Moreover, the knowledge of NmF2 is

important in determining the ionospheric time delay of a radio wave in a satellite-

to-ground radio communication; Electron densities are useful because local

changes in terrestrial environment can induce variations in electron densities

which can produce inaccuracies in navigational satellites systems, global

positioning systems, satellite to ground receivers, Radio Astronomy, Orbit

Determination and other systems that operate within this region of space. A study

of how the NmF2 and hmF2 vary in the ionosphere will go along way in defining

the background ionosphere for radio wave propagation and other space weather

experiments. Electromagnetic waves, such as GPS signals, experience time delays

when traversing the ionosphere (Ratcliffe, 1959). Fluctuations in ionospheric

electron density can lead to an azimuth shift modulation in synthetic aperture

radar (SAR) imagery, which can be detected using satellite radar interferometry

SRI (Laurence 2000). For HF radio communication a good knowledge of the

heights and Electron density of the reflective layers of the ionosphere is critical

for continuous and high quality radio reception. HF communication is still of

great importance in many remote locations on our globe and for some specialized

military applications, Radio and television operators (satellite's communication).

In this work, I investigated the variations of the F-layer peak electron density

(NmF2) and height (hmF2) in Port Harcourt (4.500N,7.00

0E ).These variabilites

are important parameters for many space weather related application (spacecraft

design, internal/surface charging, sensor interference, satellite anomalies, loss of

navigation signal phase and amplitude lock) because an operator quite often not

only needs the most likely values of these Ionospheric parameters but also a

measure of how far the actual value can stray from the median prediction. Thomas

(1968), using solstice data from 1958, studied the north-south asymmetries in the

critical frequency foF2 and the height hmF2 of the F2 peak, derived partly from

N(h) analysis and also from the M3000 propagation factor scaled from ionograms.

In general foF2 is greater and hmF2 lower on the winter than on the summer side.

Annual and semiannual variations in the F2-layer, particularly the latter, still

present something of a problem. Their morphology has long been studied with the

aid of ionosonde data. Yonezawa (1967) used extensive data from two stations

within 1° of the geomagnetic dip equator but in opposite geographic hemispheres,

302



Huancayo (Peru, 12°S) and Kodaikanal (India, 10°N). He was largely concerned

with distinguishing `seasonal' and `non-seasonal differences in the annual

variation of NmF2, which need rather complicated formulas for their description.

He noted that semiannual variation is simpler, with consistent peaks in April and

October.

2. DATA SOURCE

The data used for this work was collected from the International Reference

Ionosphere (IRI).The International Reference Ionosphere (IRI) is a widely used

standard for the specification of ionospheric parameters and is recommended for

international use by the Committee on Space Research (COSPAR) and the

International Union of Radio Science (URSI). It data sources include worldwide

network of ionosonde stations that has monitored the ionosphere with varying

station density since the nineteen-thirties, rocket measurements and other

incoherent scatter (IS) radars, satellite data from in situ and topside sounder

instruments.

We collected data of NmF2 and hmF2 for Port Harcourt (4.500, 7.00

0) during

a year of high solar activity (2000) and during a year of low solar activity (2006);

the data was analyzed and the pattern of seasonal and diurnal variations of NmF2

& hmF2 were noted. We collected data for all the months of the years under

consideration.

3. RESULTS

In order to investigate the pattern of NmF2 variation in a year of high solar

activity some days were selected in the months of January, April, May, July, and

October of the year 2000, which is a year of high solar activity . The Graph is

represented in figure 1; while figure 2 represents that for a year of low solar

activity 2006.

Figures 3 and 4 represent the variations of hmF2 in a year of high and low

solar activity respectively.

In figure 5 we plotted the NmF2 in 2000 and 2006 to compare their pattern of

variations; while in figure 6 a plot of hmF2 was made in the same years.

303



0

500000

1000000

1500000

2000000

2500000

0 5 10 15 20 25 30

Local time (hrs)

Nm

F2

(m

-3) NmF2 01

NmF2 04

NmF2 07

NmF2 10

NmF2 05

Figure 1: Diurnal variation of maximum electron density in a year of high solar

activity (2000) (Jan, April, July, Oct and May)

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

0 5 10 15 20 25 30

Nm

F2

(m

-3)

Local time (hrs)

NmF2 01

NmF2 04

NmF2 07

NmF2 10

NmF2 05

Figure 2: Diurnal variation of maximum electron density in a year of low solar

activity (2006) (Jan, April, July, Oct and May)

304



0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25 30

Local time (hrs)

hm

F2

(k

m)

hmF2 01

hmF2 04

hmF2 07

hmF210

hmF2 05

Figure 3: Diurnal variation of F2 peak height in a year of high solar activity

(2000) (Jan, April, July, Oct and May)

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25 30

Local time (hrs)

hm

F2

( k

m) hmF2 01

hmF2 04

hmF2 07

hmF2 10

hmF2

305



Figure 4: Diurnal variation of F2 peak height in a year of low solar activity (2006)

(Jan, April, July, Oct and May)

0

500000

1000000

1500000

2000000

2500000

0 2 4 6 8 10 12 14

Local time (month)

Nm

F2

(m

-3)

NmF2 06

NmF2 00

Figure 5: Comparism of annual variation of maximum electron density in a year

of high/low solar activity

0

50

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14

Local time (month)

hm

F2

(k

m)

hmF2 2000

hmF2 2006

Figure 6: Comparism of annual variation of F2 peak height in a year of high/low

solar activity

306



4. DISCUSSION

A significant diurnal variation is observed in NmF2 for a year of high solar event

with the maximum occurring around 10 to 13 local time (LT) and minimum

around 5 (LT) as shown in figure 1.Dirunal variation is also observed in year of

low solar activity with maximum occurring at around 14 to 15LT as shown in

figure 2.Figure 5 shows the annual variation of NmF2 which has two maxima;

one in May which is more pronounced than the other in October, this is in

agreement with Mansilla et al (2005).As can be seen from figures 1,2 and 3 the

peak electron density is higher during the day than night and this is as a result of

the strong solar control of the ionospheric plasma. Generally NmF2 is higher in

2000 than 2006.

The F2 peak height exhibits a diurnal variation with a maximum around 12LT

both for years of high and low solar activities as shown in figures 3 and 4

respectively; Figure 6 shows the annual variation of hmF2 which is higher in

2000 than 2006.

4. CONCLUSION

Since Nigeria is getting involved in space research, all space weather experiments

or navigational systems that are intended for this part of Nigeria (4.500,7.00

0),

should be planned bearing in mind that NmF2 and hmF2 varies through out the

year , with maximum value of NmF2 occurring around May and October.

Diurnally due to the increased ionization in the Earth‘s ionosphere the value of

NmF2 is highest during the day time than night time, so radio frequencies that are

affected by high electron densities could be used for transmission at night when

NmF2 is low since dispersion and phase delay is a major concern for the

propagation of radio waves used for communication, navigation and observation

systems

REFERENCES

Balan, N., G. J. Bailey, B. Jenkins, P. B. Rao, and R. J. Moffett (1994a),

Variations of ionospheric ionization and related solar fluxes during an

intense solar cycle, J. Geophys. Res., 99(A2), 2243–2253.

Balan, N., G. J. Bailey, and R. J. Moffett (1994b), Modeling studies of

ionospheric variations during an intense solar cycle, J. Geophys. Res.,

99(A9), 17,467–17,475.

307



Danilov, A., N. Smirnova. Improving the 75 km to 300 km ion composition

model of the IRI. Adv. Space Res. 15 (2), 171–178, 1995.

Evans, J. V. (1965), Cause of the mid-latitude evening increase in foF2, J.

Geophys. Res., 70(5), 1175–1185.

Kane, R. P. (1992), Sunspots, solar radio noise, solar EUV and ionospheric foF2,

J. Atmos. Terr. Phys., 54, 463–466.

Kawamura, S., N. Balan, Y. Otsuka, and S. Fukao (2002), Annual and

semiannual variations of the midlatitude ionosphere under low solar

activity, J. Geophys. Res., 107(A8), 1166, doi: 10.1029/2001JA000267.

Laurence .A. Gray, Karim E. Mattar: Influence of Ionospheric Electron Density

Fluctuations on Satellite Radar Interferometry GEOPHYSICAL

RESEARCH LETTERS, VOL. 27, NO. 10, PAGES 1451-1454, MAY

15, 2000

Mansilla .G.A., M Mosert., R.G Ezquer: Seasonal variation of the total electron

content, maximum electron density and equivalent slab thickness at a

South-American station Journal of Atmospheric and Solar-Terrestrial

Physics 67 (2005) 1687–1690.

Richards, P. G., D. G. Torr, B. W. Reinisch, R. R. Gamache, and P. J. Wilkinson

(1994b), F2 peak electron density at Millstone Hill and Hobart:

Comparison of theory and measurement at solar maximum, J. Geophys.

Res., 99(A8), 15,005–15,016.hhnhn

Richards, P. G. (2001), Seasonal and solar cycle variations of the ionospheric

peak electron density: Comparison of measurement and models, J.

Geophys. Res., 106(A12), 12,803–12,819.

Ratcliffe, J.A. Magneto-ionic Theory and its Application to the Ionosphere.

Cambridge University Press, Cambridge, 1959.

Thomas, L., The F2-region equatorial anomaly during solstice periods at sunspot

maximum, J. Atmos. Terr. Phys., 30, 1631±1640, 1968.

Yonezawa, T., On the seasonal, non-seasonal and semi-annual variations in the

peak electron density of the F2-layer at noon in the equatorial zone, J.

Radio Res. Labs., 14, 1967.

308



EFFECT OF SOLAR CYCLE ON GEOMAGNETIC

STORMS‡‡‡‡‡‡‡‡

F.N. Okeke1 and E.A. Hanson

2

1. Department of Physics and Astronomy, University of Nigeria, Nsukka, Nigeria.

2. Centre for Basic Space Science, University of Nigeria, Nsukka, Nigeria.

2e-mail: [email protected]

Abstract

Geomagnetic activities have been studied for the solar years of 1991 to 2007. It

was found that the solar activity controls the intensity of geomagnetic storms. The

intensities of these storms are found to be more severe in solar maximum years

than in solar minimum years. The solar wind effect is dependent on the cycle, and

invariably both are well correlated with geomagnetic storm intensity. The effects

of disturbance ring current and large changes of interplanetary magnetic field

(IMF) Bz, both are responsible for the equatorial magnetic storm effects.

Keywords: geomagnetic storms; solar cycle; solar maximum; solar minimum;

intensity.

1. INTRODUCTION

Many workers have carried out investigations on the effects of solar activities on

geomagnetic storms at different latitudes. For example, Fuller-Powell, et al (2002)

carried out model studies of ionospheric electric disturbances at mid and low

latitudes associated with geomagnetic activity. Richmond et al. (2003) used the

Magnetosphere-Thermosphere-Ionosphere-Electrodynamics general circulation

Model (MTIEGM) of Peymirat et al. (1988) and found that three effects can be of

comparable importance on the equatorial electric field. These include global

winds driven by solar heating, direct penetration of polar cap electric fields to the

equator and disturbance winds driven by high-latitude Joule heating and plasma

‡‡‡‡‡‡‡‡ African Journal of Physics Vol. 2, 308- 314 (2009)



309



convection. They also discovered that the equatorial disturbance electric field

produced by disturbance winds tend to be opposite that produced by other effects.

Wu and Lepping (2006) found out from their study that the intensity of

geomagnetic storm is more severe in solar maximum than in solar minimum.

Tsurutani et al. (1988) and Gozalez et al. 1994 ascertained that the primary cause

of a magnetic storm is long duration, intense southward interplanetary magnetic

fields, which interconnect with the earth‘s magnetic field and allow solar wind

energy transport into the earth‘s magnetosphere. Burlaga et al. (1981) and Wilson,

(1990) stipulated that magnetic clouds are the major sources of long-lasting

Southward IMF Bz, and hence usually cause magnetic storm. Fairfield and Cahile

(1966) also noted that various changes in the IMF are well known to play a key

role in regulating geomagnetic activity.

This study investigates the geomagnetic activities of solar years of 1991 to 2007,

the dependence of geomagnetic storm intensities on solar activity. Furthermore,

the solar wind effect on the solar cycle is studied.

2. SOURCES OF DATA

The data used in this work were obtained from the World Data Center, Kyoto,

Japan and the Space Weather Prediction Center, USA. The Space Weather

Operation (SWO), in collaboration with the National Oceanic and Atmospheric

Administration (NOAA), USA, prepared the sunspot numbers (SSN), which were

used in generating Tables 1 and 2. On the other hand, the Dst values of 1989 were

derived from the dataset of World Data Center, Kyoto.

S1

0

50

100

150

200

250

SSN

200-250

150-200

100-150

50-100

0-5091

9395

9799

0103

0507Year

Fig. 1: Sunspot Number from 1991 to 2007

310



0 2 4 6 8 10 12150200250

SS

N '9

1

0 2 4 6 8 10 12100150200

SS

N '9

2

0 2 4 6 8 10 1250

100150

SS

N '9

3

0 2 4 6 8 10 12405060

SS

N '9

4

0 2 4 6 8 10 120

2040

SS

N '9

5

0 2 4 6 8 10 12101520

SS

N '9

6

0 2 4 6 8 10 120

50100

SS

N '9

7

0 2 4 6 8 10 1250

100150

SS

N '9

8

0 2 4 6 8 10 12100150200

SS

N '9

9

0 2 4 6 8 10 12160170180

SS

N '0

0

0 2 4 6 8 10 12160180200

SS

N '0

1

0 2 4 6 8 10 12100150200

SS

N '0

2

0 2 4 6 8 10 1250

100150

SS

N '0

3

0 2 4 6 8 10 126080

100

SS

N '0

4

0 2 4 6 8 10 12405060

SS

N '0

5

0 2 4 6 8 10 12203040

JAN - DEC

SS

N '0

6

0 2 4 6 8 10 120

1020

JAN - DEC

SS

N '0

7

Fig. 2: Monthly variation of Sunspot Number for years under study (1991 – 2007)

311



0

50

100

150

200

250

SSN

91 93 95 97 99 01 03 05 07

Year

Fig. 3: Averaged Sunspot Numbers for the 17-year period (1991 – 2007)

Fig. 4: Geomagnetic storm of October 29 -31, 2003. © World Data Index for Geomagnetism, Kyoto, Japan

Fig. 5: Geomagnetic storm of November 20 -21, 2003. © World Data Index for Geomagnetism, Kyoto,

Japan

312



Fig. 6: Geomagnetic storm of March 13 -14, 1989. © World Data Index for

Geomagnetism, Kyoto, Japan.

3. AND DISCUSSION OF RESULTS

The sunspot numbers (SSN) determine the extent of the solar activity, for

example figure 1 depicts the intensity variation of sunspot number for each year

under study. From figure 2, in the year 1991, the sunspot number is observed to

maintain high values between 221 and 194 from January to December. In 1992,

sunspot number (SSN) dropped to about 183 and 108 from January to December.

In 1993 the range suddenly dropped to ~ 105 and 58. It is observed that between

1994 and 1997, the decrease is seen to be very sharp and significant. The SSN has

maximum value in January of 1994, as 55.6 and least in December as 41.4, while

in 1995, in January, SSN registered 39.6, while in December it was 17.6. In 1996,

SSN in January registered 16.8 and in December 16.2. Its lowest value is

observed in the month of May (12.9).

In 1997, January value was recorded as 16.5. It continues to increase in value and

recorded 54.2 in December. This increment continued the following year (1998)

with 60.6 in January and 108.8 in December.

This increase was conspicuously noticed in the year 2000, that 168.0 in value of

SSN was recorded in January and 160.8 in December. It is very important to note

that in 2001 there was a drop in value of SSN in January, to 156.3 and increased

to 184.5 in December. The drop continued from May 2002 to December 2007.

Figure 3 depicts the averaged values of SSN over years, revealing the overall

trend for the 17-year period. It is evidently clear that the minimum value

coincides with the solar minimum year of 1996, while the maximum coincides

with the solar maximum year of 2000, which even extends to 2001 and 2002.

From the dst graph of March 13/14, 1989, a very severe storm was observed

(recorded) with a value of -650nT (see fig. 6), supporting the historical assertion

that most intense geomagnetic storms have occurred during the declining year of

the solar cycle. The same severe storms occurred on November 8, 1991, October

29 -312003 (see Fig. 4); November 20/21, 2003 (see Fig. 5). Several other storms

313



such as those of November 29-31, 2003, and November 8, 2004 (though not

shown on graphs) are well reported.

The intensity of these geomagnetic storms does not only depend on the solar

activity, but could also be seen to be highly dependent on strong solar wind

parameters (although not shown in this study). The major contributor of severe

storm is the intense solar activity that is normally observed during the solar

maximum years.

The severe storms such as that of March 13, 1989, could cause enhancement of

ionospheric irregularities, especially in their amplitudes and may result in signal

degradation, as Yizengaw et al. (2005) observed in their study using severe storm

of 31st March 2001.

4. CONCLUSION

Geomagnetic storms are generated by different solar dynamic activities.

Geomagnetic storms depend solely on sunspot number, which invariable

determine the extent of solar activity. The solar wind effect and the large changes

of IMF are related to severe equatorial magnetic storm effects.

5. RECOMMENDATIONS

The study of geomagnetic storm assumes great importance due to several safety

and economic implications. For instance, power grid is particularly vulnerable to

geomagnetic storms. Ground currents induced during geomagnetic storms can

actually melt the copper windings of transformers of many power distribution

systems. Sprawling power lines serve as antennas by picking the current and

spreading same over wide areas, this can result in power outage. The Quebec

geomagnetic power outage of March 1989 is a historical fact.

Geomagnetic storms account for huge economic loss to exploration and

communication firms. Sudden surge in signal strength creates complications in

magnetic surveying equipment of oil prospecting and exploration companies,

broken communication lines for airline and GSM operators, etc, resulting in

wastage of millions of dollars.

In view of all this, this work recommends that the Federal Government of Nigeria

unwaveringly support research on geomagnetism by way of offering scholarships,

funding projects, acquiring equipment and relevant materials for the advancement

of research on geomagnetism.

314



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(A3), 1118, d0i: 10. 1029/20002JA 009758.

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major magnetic storms near solar maximum (1978 - 19790), J.

Geophys. Res., 93, 8519 -8531.

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caused by interplanetary magnetic clouds, Ann. Geophys., 24, 3383 –

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magnetic storm using multi instrument measurement data. Annals

geophycae, 23, 707 -721.

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SPACE SCIENCE AND TECHNOLOGY IN NIGERIA: AN OVERVIEW

OF THE PROSPECTS AND PROBLEMSE§§§§§§§§

B. I. Okere1 and P. N. Okeke2.

Centre for Basic Space Science, University of Nigeria Nsukka.

E-mails. [email protected];

[email protected]

Abstract: It is an established fact, especially in recent decades, that human capital

development through training in science and technology has become the primary

driving force behind economic growth and prosperity of the successfully

developed economies. In this paper, we present the scope of Nigeria space policy,

planned missions and projects vis-à-vis their role in technology development. The

benefits of space science and technology were reviewed.

1. INTRODUCTION

Space is a term that is often treated as a category. Outer space (as commonly

used, the universe exclusive of Earth), is such an alien environment that

attempting to work with it leads inevitably to new leading edge techniques and

knowledge. New technologies originating with or accelerated by space-related

endeavors are often subsequently exploited in other economic activities. This has

been widely pointed to as beneficial spin-off by space advocates and enthusiasts

favoring the investment of public funds in space activities and programs. Political

opponents counter that it would be far cheaper to develop specific technologies

directly if they are beneficial and scoff at this justification for public expenditures

on space-related research.

Space science, body of scientific knowledge as it relates to space exploration;

it is sometimes also called astronautics. Space science draws on the conventional

sciences of physics, chemistry, biology, and engineering, as well as requiring

specific research of its own. The particular disciplines that are relevant depend on

the type of mission being planned. The problems that space science must deal

§§§§§§§§ African Journal of Physics Vol. 2, pp. 315-329, (2009)


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with include prediction and control of trajectories and orbits, telecommunications

between spacecraft and earth, spacecraft design and fabrication, and lifesupport

systems for human spaceflight.

On the other hand, Space Technology is the systematic application of

engineering and scientific disciplines to the exploration and utilization of outer

space. Space technology developed so that spacecraft and humans could function

in this environment that is so different from the Earth's surface. Conditions that

humans take for granted do not exist in outer space. Objects do not fall. There is

no atmosphere to breathe, to keep people warm in the shade, to transport heat by

convection, or to enable the burning of fuels. Stars do not twinkle. Liquids

evaporate very quickly and are deposited on nearby surfaces. The solar wind

sends electrons to charge the spacecraft, with lightninglike discharges that may

damage the craft. Cosmic rays and solar protons damage electronic circuits and

human flesh. The vast distances require reliable structures, electronics,

mechanisms, and software to enable the craft to perform when it gets to its goal –

and all of this with the design requirement that the spacecraft be the smallest and

lightest it can be while still operating as reliably as possible. All spacecraft

designs have some common features: structure and materials, electrical power and

storage, tracking and guidance, thermal control, and propulsion. The spacecraft

structure is designed to survive the forces of launching and ground handling. The

structure is made of metals (aluminum, beryllium, magnesium, titanium) or a

composite (boron/epoxy, graphite/epoxy). It must also fit the envelope of the

launcher. To maintain temperatures at acceptable limits, various active and

passive devices are used: coatings or surfaces with special absorptivities and

emissivities, numerous types of thermal insulation, such as multilayer insulation

and aerogel, mechanical louvers to vary the heat radiated to space, heat pipes,

electrical resistive heaters, or radioisotope heating units.

The ability to accomplish this results in technological advancement with its

attendant economic development.

2. BENEFITS OF BASIC SPACE SCIENCE AND TECHNOLOGY

Disaster Management

The recent tsunami disaster in the Indian Ocean demonstrated the extent

that space technologies can contribute to emergency response and disaster

reduction. The use of such technologies has been proven useful in the risk

assessment, mitigation and preparedness phases of disaster management. As the

global community learnt from the tsunami event, space technologies have also a

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central role to play in providing early warning to communities that are at risk. But

in order for developing countries to be able to incorporate the use of space

technology-based solutions there is a need to increase awareness, build national

capacity and also develop solutions that are customised and appropriate to the

needs of the developing world (http://www.nasa.gov).

Cordless Power Tools and Appliances

The technology that made cordless drill or shrub trimmer possible came

from NASA's Apollo program. Astronauts needed a way to drill down beneath the

moon's surface, as much as 10 feet, to collect core samples. Like everything else

that went to the moon, this drill had to be small, lightweight and battery-powered.

This technological advance made the battery-powered drill possible -- a computer

program was used to design the drill's motor to use as little power as possible.

That computer program, along with the knowledge and experience gained in

developing the drill, provided a strong technology base for developing battery

powered tools and appliances (http://www.nasangov).

Smoke Detectors

In the 1970's, NASA needed a smoke and fire detector for Skylab,

America's first space station. Honeywell, Inc. developed the unit for NASA.

Smoke detectors are now required by law to be placed in all new homes. They are

credited with saving countless lives (http://www.nasa.gov).

Clean Water For Home The H2OME Guardian filter technology developed by Western Water

International (WWI), combined with NASA technology during the Apollo

program, was developed to sterilize the astronauts' drinking water. This method

included the use of ions (an atom or group of atoms carrying a positive or

negative electrical charge) as part of the water filtering system. The H2OME

removes lead, chlorine, bad taste and odor, and other bad stuff. WWI also sells

filter units that can handle large volumes of water. These are used in large

buildings, and even by entire towns in countries where the water is contaminated

(http://www.nasa.gov).

"Cool" Laser Heart Surgery Until recently, heart bypass surgery, which replaces clogged blood vessels,

was the main treatment for serious cases. A new surgery method derived from

laser technology pioneered by NASA's Jet Propulsion Laboratory for remote

sensing of earth's ozone layer has been developed.

http://www.nasa.gov/



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The procedure involves threading a small catheter through coronary

arteries. The laser light is carried through fiber optic bundles within the catheter.

Another group of fibers shines a light at the tip to provide video pictures of the

inside of the artery. Watching the video pictures, the doctor can spot areas of

blockage and fire short bursts of laser beams to vaporize them. While other types

of lasers are too hot for delicate heart surgery, the excimer laser operates at a

"cool" 65° C, a temperature that human tissue can withstand


Space Telescope Looks for Cancer Breast examinations (mammographies) help in the detection of breast

cancer. Until recently, if a doctor saw a trouble spot on the x-ray he or she would

order a biopsy procedure. A biopsy required surgery to cut into the breast and

obtain a tissue sample. Now, however, with the help of Hubble Space Telescope

technology, biopsies can be performed with a needle instead of a scalpel

(http://www.nasa.gov). The development by Goddard Space Flight Center of an

advanced, supersensitive CCD to be installed in the Hubble Space Telescope in

1997, has been adopted breast biopsy system. The patient lies face down with one

breast protruding through an opening. The device images breast tissue more

clearly than conventional x-rays. This allows the system to pinpoint the area in

question. The doctor can then use the specially designed needle to extract a tiny

sample. The patient can then walk out of the office and resume normal activities.

Body Imaging

The digital image processing technology developed to allow computer

enhancement of Moon pictures, is now being used by doctors and hospitals to

record images of organs in the human body. Two of the most widely used body

imaging techniques are computer-aided tomography (CATScan) and magnetic

resonance imaging (MRI) (http://www.nasa.gov).

Light at the end of the Tunnel for Cancer Light emitting diodes (LED), used for plant experiments on the Space

Shuttle, are being used to perform surgery on patients with brain cancer.

Photodynamic Therapy, a MSFC Small Business Innovation Research (SBIR) and

Quantum Devices collaboration, uses LEDs to activate photosensitizers (light-

sensitive, tumor-treating drugs) that have been injected intravenously. Light

activation allows the drugs to destroy cancerous cells, leaving surrounding tissue

virtually untouched. Cancer treatment trials using the LEDs have thus far included

skin cancer and brain tumor patients, with promising results (http://www.nasa.gov).





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Pill-Sized Transmitter Monitors Fetus from Inside the Womb Medical personnel can monitor fetal activity from inside the womb using a

small pill-shaped transmitter developed at NASA's Ames Research Center in

Field, Calif. Pill transmitters are used to measure blood pressure and temperature

in astronauts aboard the International Space Station, but they are being studied for

a number of applications on Earth. The transmitters can be implanted in the

intrauterine to monitor fetal activity; swallowed to monitor intestinal activity; and

are being developed for use in monitoring athletes and high-stress professionals

such as firefighters and soldiers (http://www.nasa.gov).

Early Disease Prediction Through Chromosome Analysis

Technology used to study space probe photographs sent back to Earth is

now being used to analyze human chromosomes and could lead to disease

prediction in infants. The process, developed jointly by NASA's Jet Propulsion

Laboratory, in Pasadena, Calif., and NASA's Johnson Space Center, in Houston,

Texas, allows researchers to quickly and automatically arrange and analyze

human chromosomes and detect genetic abnormalities. The NASA technology

allows researchers to complete a job that once took hours in less than 10 minutes


Lightning Protection Bad weather is bad news for airplanes. One of the most unpredictable

elements of a storm is lightning. NASA's Langley Research Center played a

leading role in lightning investigations with its seven-year (1980-86) Storm

Hazards Research program. NASA found that lightning injects a large number of

electrical currents into an airplane. These currents can cause problems in the

plane's electronic systems, including incorrect instrument readings. The results of

this research led to the development of lightning protectors for Aircrafts


Global Communications TV signals are only one kind of data transmitted by satellites. Telephone

signals, computer data, and computer images are also beamed around the world

via satellite. The high-risk satellite data transmission technology developed and

tested in orbit by NASA in the 1960s and 1970s is being applied in areas such as

(http://www.nasa.gov):

Geosynchronous orbit (GEO) orbiting with the Earth so that the

satellite is always above a particular spot on the ground · satellite

stabilization

Keeping the satellite from wobbling in orbit ·




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Use of digital computers

Solid state high-power transmitters ·

Large antennae that provide high quality signals to small ground

receivers ·

Advanced materials (such as graphite composites) for building

satellites

Crop Management from Orbit Landsat satellites designed to observe the changing conditions on the

Earth‘s surface are used to manage the harvesting of fish in the world‘s oceans.

Landsat is also being used for crop harvest (http://www/nasa/gov).

How does this help farmers? The more potatoes there are, the less money

farmers will get for their crop. If the satellite data shows there are too many

potatoes, it means the price is going down, so sell your potatoes now! If fewer

potatoes are growing in the area, a farmer might decide to harvest early and get a

higher price.

Fishing from Orbit Landsats sense Earth‘s features from a remote location (from orbit) the

technology is called remote sensing. Remote sensing technology is used by

fishermen to locate fish. The satellite data tells them where in the ocean the

temperature is right for a particular type of fish (http://www.nasa.gov). Landsats

also provide weather information for a few square miles or four million square

miles. Satellite images allow ship captains to find the most favorable winds and

currents, to save time and fuel. Remote sensing technology is also used to map oil

spills.

Oil Spill Control The beeswax microcapsules designed so that water cannot get in, but oil

can is being applied in oil spill control. When the oil seeps through the shell, the

microorganisms inside release enzymes that digest the oil. When the balls get full

of digested oil, they explode. They release enzymes, carbon dioxide and water, all

environmentally safe. This mixture is even good fish food.

http://www/nasa/gov

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3. SCOPE OF NIGERIAN SPACE PROGRAM

The scope of the Nigerian Space Programme (NSP) (Da Costa, 2005) to

be implemented by the National Space Research and Development Agency

(NASRDA) include:

i. Basic Space Science and Technology to provide the understanding of

how the universe works and what its impact is on the world. This will

enable us to lay the foundation for deriving maximum benefits from

the nation‘s participation in the space enterprise.

ii. Remote Sensing to help Nigerians understand and manage our

environment and natural resources using space-acquired information.

This technology will enable us to better understand our land, air and

water resources and their associated problems.

iii. iii. Satellite Meteorology to study atmospheric and weather sciences

using satellite data to facilitate the effective management of our

environment.

iv. Communication and Information Technology to provide efficient and

reliable telecommunications services for Nigeria in order to enhance

the growth of the industrial, commercial and administrative sectors of

the economy.

v. Defense and Security: The Federal Government shall develop a

necessary Space Science Technology (SST) programme that will

address the national needs of Nigeria. For this purpose the government

shall establish a Defense Space Command in the Ministry of Defense.

The Command shall comprise representatives of the defense,

intelligence, security and law enforcement services and report through

the Ministry of Defense to the National Space Council.

3.1 Planned Missions

In furtherance of its space program, Nigeria through NARSDA has

planned the under listed missions:

i. NigeriaSat-2 - commenced November 6, 2006, due for launch in 2009.

This satellite is to replace NigeriaSat-1 whose life expired in 2008

(Borofice, 2006). This satellite shown below with higher resolution than

Nigeriasat-1 will be used for remote sensing.

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The purpose of this project is to:

a. Build upon the success of NigeriaSat-1

b. Acquire NigeriaSat-2 satellite manufacture know-How

c. Further develop skills in the design and development of satellites

d. Develop indigenous satellite manufacturing capability

ii. Launching of a Nigerian Astronaut into space in the next 15 years.

3.2 Plannned Projects

i. The 25m Nigeria Radio Telescope (NRT)

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This telescope will be a 25m fully steerable parabolic radio telescope with 15m

Aluminum inner surface and 10m wire mesh outer surface.

The Driving Science for the 25-m NRT (1) Pulsar and neutron star studies: very suitable for long-term precision

timing of radio pulsars while sky survey for new pulsars could be possible

with the state-of-the-art instrumentation.

(2) HI spectroscopy and continuum emission from galactic and extragalactic

objects (e.g. active galaxies, blazers and quasars).

(3) Study of astrophysical masers.

(4) Study of the sun (e.g. solar flares), stars, the supernova remnants and

Active Galactic Nuclei (AGNs).

(5) Geodetic studies

(6) VLBI: This facility will enable us pursue collaborations with leading

astronomical institutions in the world. A Nigerian radio telescope will be

vital in filling up the gap between the aggregates of observatories in the

far north and south in VLBI programs. It will certainly guarantee us a

place in most of the prestigious networks in the world.

Other collateral benefits include

(1) Science Education: A modern radio observatory in Nigeria will play

pivotal role in stimulating science education and public science awareness

in the country. This will enable more young people to take up careers in

sciences and engineering.

(2) Technological Development: Through developing, constructing and

operating a modern radio observatory in Nigeria, we would have built

capacity in antenna and radio receiver technology. This important

technology will find already market in the fast evolving communication

industry.

ii. Space weather Study equipments

Through one of its activity centres, NASRDA intends to install weather monitor

equipments at different locations in the six geopolitical zones of the country as

indicated in fig. (a) below. The equipments being installed include: weather link

Davis Pro-2 for microwave study, Campbell weather station for Nigeria

Environmental and Climatic Project (NECOP) for weather monitoring, GPS

Antenna/Receiver for Scintillation and Geodetic studies (fig. 3b) and SID Monitor

for Solar flare studies (fig. 3c).

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4. PROBLEMS AND PROSPECTS

4.1 Manpower Development

Many non-industrialised countries have developed some infrastructure for

the management of remote-sensing and telecommunications services. However,

these are just two of many end products offered by space technology. Without

indigenous capabilities, the rapid growth of these technologies rapidly renders the

acquired infrastructure obsolete and prolongs the dependence on the industrialised

countries. This fact has traditionally been ignored by policy makers and local

governments.

A trickle-down effect is produced when highly qualified scientists are

present to construct the required infrastructure by training other scientists,

engineers, technicians, and students. The buying of packaged ready-to-use

technology has failed as a model for technology transfer, with the expected

increase in knowledge not materialising. On the contrary, an increase in

dependence on the industrialised countries has been noted, and the accumulation

of costly and often obsolete equipment has been the end result.

4.2 Education and T raining in Basic Space Scienc e and Technology

Ever since human beings learnt to walk upright they have looked at the

sky and wondered. The sky has remained the same but human perception of it has

changed. First a divinity to be feared and appeased: then a phenomenon to be

observed and utilized: and finally a physics laboratory: the outer space over the

millennia has acquired a depth, in keeping with the changing patterns of

humankind's relationship with its cosmic environment. Basic space science today

is at the cutting edge of intellectual enquiry, and, at its most glamorous, a child of

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high technology. Hence the education that basic space science is bedrock of

technology.

In Nigeria there is a conspicuous lack of materials and infrastructure for

basic space technology education. The materials and infrastructure required

include:

1. Planetariums. A planetarium can be a powerful education aid. It

liberates sky viewing from the constraints of weather and seasons as well

as earth's rotation and sphericity. It can present cosmic objects and

phenomena realistically and dynamically. A sky show can create a visual

impact beyond the reach of a class-room lecture or the printed word. In

addition, a planetarium can play the role of an astronomical community

and education centre and a news room. The first and only planetarium to

be opened in Nigeria is the one at CBSS (Okeke, 2006). The reasons are

financial. The amount of money required to set up and maintain a

planetarium is outside the reach of most civic bodies or educational

institutions.

2. Popular books. A more permanent source of basic space science

information, especially for young readers, is science magazines, science

pages of newspapers and popular books. Very often the periodicals reprint

articles taken from the international press. Since the background level of

Nigerian readership is different, such reprints are of limited value. The

only book dealing with basic space science is the one written by Prof. P.

N. Okeke – ‗An Introduction to Astronomy and Astrophysics‘.

3. Schools. The Nigerian school system is characterized by heavy

centralization, obsession with examinations, severe paucity of funds, and

populism. Such a system does not have much of scope for hands-on

training. The emphasis is on teaching from textbooks written according to

a centrally prescribed syllabus. The respect for and fear of the text-book

could still be converted into an asset if the books were accurate, attractive

and user friendly. This unfortunately is not the case. Basic Space Science

and Technology has not been reflected in the schools‘ curriculum.

4. Colleges and universities. After spending 12 years in school. Nigerian

students have a number of options open to them. They can join a 4-5 year

course in engineering or medicine. (This is the current preference of the

brightest of the students). They can go to a college for a 4-year course

leading to a bachelor's degree in science, humanities, commerce or

management. Courses leading to master's degree in science are offered in

the universities which also offer bachelor-level honours courses in science.

The higher education system, like the school, is inflexible and examination

oriented. It is also heavily weighted against basic space science. The

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following are some of the features of the Nigerian higher education

system, which may be of all interest in a wider context.

i. At the B.Sc. level, basic space science is almost entirely absent from the

prescribed syllabus. The Universities offering courses in astronomy and

atmospheric science are University of Nigeria, Nsukka (UNN), Rivers State

University of Science and Technology (RUST), Nnamdi Azikiwe University,

Awka (UNIZIK), Federal University of Technology Akure, Federal University of

Technology Minna, Ebonyi State University Abakaliki, Imo State University

Owerri and Abia State University Uturu among others.

ii. Astronomy and Basic Space Science is offered up to MSc and PhD at UNN

only.

iii. There are only two observatories attached to two universities; both are still

under construction.

iv. The actual number of students studying for a master's degree in space science

(including astronomy) or offering space science as a special course for M.Sc. in

physics is very small.

v. Most of the teaching tends to be bookish rather than practical, which in turn

emphasizes "learning" rather than problem solving. More generally, the social

ambience that permeates the academic world also glamorizes brahmanical type of

studies as against dirty-hand experiments.

vi. There is almost total decoupling between fund-starved universities and rather

wellendowed research institutes. A student enrolling for Ph.D. in the latter has to

spend two years acquiring the necessary background knowledge.

4.3 Development of Basic Space Science

'Basic Space Sciences' have been defined as:

a. Astronomy and astrophysics

b. Solar-terrestrial interaction and its influence on terrestrial climate

c. Planetary and atmospheric studies

d. The origin of life and exobiology.

The development of basic science in non-industrialized countries is a key factor

for the economic prosperity of those countries. Scientific research is now being

recognized by the governments of non-industrialized countries as an essential

ingredient to guarantee economic growth through the establishment of a strong

infrastructure for the assimilation and use of high technologies.

Examples of space-technology applications for the benefit of non-

industrialized countries are already very well known: remote sensing,

telecommunications, monitoring of natural resources, environment and weather

conditions, etc. All of these applications have a direct influence on the economy.

The importance of developing basic space science has certainly been recognized

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in those countries that have sponsored the UN/ESA Workshops (Haubold, 1995),

i.e. India, Costa Rica, Colombia, Nigeria, and Egypt.

4.4 International Collaboration

In addition to the direct benefits of an international scientific conference, the

UN/ESA Workshops have generated a number of other supplementary activities

such as: the donation of a 15cm optical telescope by UNESCO to Nigeria, the

donation of a 3m radio astronomical telescope and planetarium by the Japanese

government to Nigeria. As well as the signing of MoU with HartRAO South

Africa and collaboration with University of Delaware USA (Okeke, 2006).

In March 1992, site-selection feasibility studies were initiated in Nsukka,

Nigeria, for a computerized optical telescope for detailed solar seismology

studies. This project is a collaborative effort by the University of Nigeria, Nsukka,

and the Zetetic Institute, Arizona, USA. Another project that originated in the

Workshops is the installation of an Internet node in Nigeria, with TPS support.

4.5 NARSDA Activity Centers

The establishment of the six activity centers by NARSDA to implement

the Nigerian space program has started yielding results:

Nigeria launched its first satellite, NigeriaSat 1, into orbit in September 2003,

after Nigerian experts underwent training in London. The Nigerian

communication satellite NigcomSat-1 was launched in China in May 2007.

The Centre for Transport and Propulsion, Lagos is preparing for the

launching of the first made in Nigeria satellite in 2015.

The Centre for Basic Space Science, Nsukka has embarked on massive

training of scientists and engineers, has signed MoU with South African Radio

and optical observatories, National Observatory of Japan and University of

Delaware for the development of basic space research, is about completing the

building of an optical observatory and has signed the contract for the building of a

25m Nigeria radio telescope (NRT) shown in the figure 2.

4.6 Collaborative efforts

Nigerian astronomical community should strive towards building research

capacity in wide fields of astronomy. One effective approach to this is through

systematic and sustained international cooperation. Already, a strong

collaboration has been established between the CBSS and the National

Astronomical Observatory of Japan (NAOJ) with a primary objective of building

research capacity in radio astronomy. In this regard it is important to point out

328



that a Memorandum of Understanding (MOU) has been signed between CBSS

and NAOJ towards the building of capacity in observational astronomy research

in Nigeria through a collaborative effort between the National Centre for Basic

Space Science and the National Astronomical Observatory of Japan. Under the

planned collaboration, NAOJ is expected to assist CBSS in establishing a modern

radio observatory in Nigeria. The level of participation of NAOJ in this project

will be agreed upon by the two organizations. However, the areas of telescope and

radio receiver construction and installation are likely to be covered under the

agreement. This, no doubt, will lead to the establishment of a Radio Astronomical

Observatory in Nigeria and a clear possibility of extending the CBSS/NAOJ

collaboration to other fields of astronomy (like the VLBI and solar radio

astronomy).

5. SUMMARY/CONCLUSION

The traditional problems of basic science (including isolation, brain drain,

lack of financial resources, ever-increasing gap with respect to industrialized

countries, lack of scientific tradition, and weak infrastructure) in non-

industrialized countries remain a major cause of concern. Interestingly, there are

new developments that may change the situation, and these developments deserve

serious consideration. The most important facts that have brought some hope for

the improvement of the situation of science in non-industrialized countries are:

first, national governments are understanding the need to recognize the intrinsic

value of basic space science and its importance as an essential component without

which the economy cannot grow; secondly, there is a new trend in the scientific

community to develop large international facilities, which may make use of the

climatological and geographic attributes of non-industrialized countries; thirdly,

the revolution in electronic communications allows close contact between

scientists and permits access to remote databases and computer power from any

part of the world. This technology helps ameliorate the isolation factor in

unprecedented ways. Lastly, global problems (i.e. environment) and the

consciousness that their solution must be global in nature has led to the inclusion

of non-industrialized countries in the science policies of the industrialized

countries.

The loss of researchers from non-industrialized countries via a 'brain drain'

is a serious problem that must be addressed. To reverse this process, efforts must

be made to promote awareness in non-industrialized countries of the importance

of space science and to ensure that space scientists have the basic resources

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necessary for their work. International efforts to ensure that scientists in non-

industrialized countries have adequate communication links to the international

scientific community, adequate access to the technical literature, more

opportunities to participate in international meetings, access to data and data-

processing facilities, and opportunities to participate in the planning, design and

use of space observatories, research programs and space missions, all help reduce

the feeling that scientists in non-industrialized countries have to move abroad in

order to produce good scientific work. The UN/ESA Workshops have been

organized to address these problems and to help identify solutions.

The objective of these Workshops is to strengthen basic space science in

non-industrialized countries by addressing ways and means by which the

following goals can be accomplished: to make scientists aware of current and

future scientific and technical aspects of basic space science, to enhance scientific

cooperation between developing countries, to explore avenues of education,

training and research in space-science subjects for the benefit of developing

countries, to create an international core group of scientists that will pursue the

objectives of the Workshop, to provide access to the most recent advances for

scientists from non-industrialised countries, to identify avenues that will facilitate

scientific cooperation, to create a forum for the discussion of problems and the

formulation of policies and recommendations.

It is better to catch students young. If basic space science were taught at

B.Sc./M.Sc. level, many students may discover that they have an aptitude for the

subject which they may decide to pursue. A positive, though small, step is the

conducting of summer schools where college and university students are given

lectures as well as, at times, practical training with optical and radio telescopes. It

is quite clear that a handful of purely-research institutes, decoupled from the

B.Sc./M.Sc. level students can only be of limited utility. If the culture of teaching

of, and research in, basic space science is to take roots and spread, the university

system at large would need to be activated by creating a core of inspiring teachers

and by providing rather rugged, easily repairable small observational facilities

under university auspices at a number of spread-out places.

It also recommended the need to explore the great scientific potential of

some non-industrialized countries due to their special attributes (i.e. climate,

geography, bio-diversity, etc.), which put them in a privileged position for the

development of certain fields of scientific research: geomagnetism studies,

electrojet currents, galactic mapping, solar photometry, astrometry and

environmental projects such as ozone mapping.

If the Federal government of Nigeria will go all out to develop research in

Basic Space Science, Nigeria will be on the road to technological development,

hence Economic development. This we can see in the so much emphasis the

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developed economies of the world such as America, Japan, Russia, etc put into

the development of Basic Space Science and technology.

REFERENCES

1. Boroffice,R.A. (2006): ―NICOMSAT-1 will Bridge Telecom Gap Between

Africa and the rest of the World‖, NASRDA News vol. 2 Issue 2.

2. Benefits of Space Program: http://www.nasa.gov/

3. Da Costa (2006): Nigeria Aggressively Pursues Space Program,

http://www.voanews.com/english/archive

4. Emmanuel: Use of NigeriaSat-1: www.columbusemmanuel.1warp.com

5. Haubold H.J. et al (1995: UN/ESA Workshop On Basic Space Science, ESA

Bulletin Nr.81

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THE OYIBO‟S GRAND UNIFICATION THEOREM WITH

REALIZATION OF SOME BASIC PHYSICAL PHENOMENA IN

GEOMETRIC OPTICS*********

M. W. Echenim and G. E. Akpojotor

Department of Physics, Delta State University, Abraka 331001, Nigeria

E-mail: [email protected]; [email protected]

Abstract

The kernel of the Oyibo`s formalism of the grand unified theorem is the generic

conservation equation from which he conjectured that all four known

force fields can be unified and that standard equations such as the Schwarzschild‘

solution of Einstein general relativity, Fermat principle for geometric optics as

well as the Schrödinger and Klein-Gordon wave equations can be obtained. This

work has been very controversial especially as it has involves some very

ambitious claims. However, it is widely believed that the mathematics is good.

The purpose of our presentation here is to review the Oyibo‘s formulation and

then use it to recover some of the aforementioned physical phenomena.

1. INTRODUCTION

In November 2004, the Nigerian born Professor of Mathematics of OFFPPIT

Institute of Technology, New York visited Nigeria on a lecture tour of his God

Almighty Grand Unification Theory (GAGUT) [1,2]. One intended benefits of

this visit packaged by the National University Commission (NUC) was to

popularized and probably attract researchers in the country to the GAGUT. About

half a decade now, this has not happened. To the best of our knowledge, it was

only Animalu who reviewed the work in Refs. [3,4] and a contributed chapter in

Ref. [5]. One of the reasons [6] for the lack of more studies of GAGUT in the

country may be due to the unconventional mathematical methodology introduced

by Oyibo. This methodology has been adopted from his experience at solving the

Navier Stokes equations in fluid mechanics using invariance of an arbitrary

function under a group of conformal transformations [1]. As pointed out by

********* African Journal of Physics Vol. 2, pp.331-345, (2009)


332



Animalu, the first problem in understanding GAGUT emanates from the lack of

direct relationship between this conformal transformations and the usual

characterization of conformal invariance or symmetry in analytical projective

space-time geometry as well as relativistic quantum field theory [4]. Therefore

Animalu provided this missing link by demonstrating how to realize other

definitions of the conformal group of transformations within the purview of

GAGUT. He was then led to the conclusion in both reviews [3,4] that just as the

Minkowskian geometry is the important approach to understanding the Einstein ‗s

special relativity theory, projective geometry [7] is the key approach to

understanding GAGUT.

Interestingly, Oyibo envisaged the problem of subjective interest on his

unconventional methodology and appealed that [1]:

Human experience seems to have demonstrated that under difficult circumstances

such as the ones that surround the search for the Unified Force Field Theory, it is

critical for one to be open-minded in one‘s investigation and analysis or even

expectations. This reminder to readers is provided to partially prepare them for

the coming presentation of the new methodology described in his book. The new

methodology would seem to be drastically or significantly different from … the

methodology that readers are familiar with or even expect to consider to be the

kind of methodologies that belong in this realm of research.

It is important to point out that introducing esoteric approaches or concepts has

been the best way to solve some difficult problems in most fields of studies. For

example, in the early development of relativistic quantum mechanics for the

electron, the Klein-Gordon theory was considered the best that could be achieved

by most contemporary researchers in this field even though there were

discrepancies between it and the general principle of quantum mechanics such as

its non-positive definite probability density and the presence of symmetry

between negative and positive energies. By introducing two valued quantities now

known as spinors to get away from tensors which he believed were inadequate

then to develop a relativistic quantum theory, Dirac obtained his celebrated theory

of relativistic electron [8]. According him [9],

Those people who were too familiar with tensors were not fitted to get away from

them and think up something more general, and I was able to do so only because I

was more attached to the general principle of quantum mechanics than to

tensors….One should always guard against getting too attached to one particular

line of thought

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In our opinion, therefore, it will be very necessary to consider the Oyibo‘s work

with open-mindedness within the general philosophy of a grand unified theorem

rather than its deviation from conventional methodologies. Then as the usual

practice, the first test of the validity of the theorem will be to reproduce previous

known results. In attempt to do so, Oyibo had argued without convincing and

direct proofs, that the Newton‘s universal gravitational and Einstein ‗s general

relativity gravitational force fields, Maxwell‘s electromagnetic equations, the

strong and weak force fields are likely present in his generic conservation

equations. While the need to undertake a more rigorous proofs for these claims as

partly done by Animalu [3,5] cannot be overemphasized, we have decided to use

it to obtain more simplified phenomena in geometric optics in order to bring

GAGUT within the philosophy of ISOTPAND which is to cover the frontiers of

physics with a pedagogical delivery when possible. Therefore, before

reproducing these simple phenomena from GAGUT in section 3, we will give a

brief review of the statement of the problem of GAGUT in section 2. There will

also be a summary and a conclusion after section 3.

2. BRIEF REVIEW OF THE STATEMENT OF THE PROBLEM OF GAGUT

The grand unification theories are proposed to unify all known forces in nature

such as the four major force fields namely gravitation, electromagnetism, strong

and weak forces [10-12] and other possibly unknown force fields [1,2]. In other

words, these theories can account for almost every known form of matter and

force and possibly the ones that are not yet known, conceivable and non-

conceivable. It is believed by some physicists that the successful achievement of

such a Theory of Everything (TOE) will lead to the end of physics or at least the

beginning of the end [11]. Scientifically, Einstein began the quest for a unified

force field theory when he attempted to incorporate electromagnetism into his

General Relativity Theory. As it is now well known in Textbooks, Einstein

mathematical framework for his Special Theory of Relativity is the Lorentz group

of linear coordinate transformations and by generalizing these transformations to

include non-linear cases, he was able to set up the mathematical framework for

the General Relativity Theory. It is therefore this methodology of general

coordinate transformations that Einstein attempted to unify electromagnetic and

gravitational force fields. Therefore, most other workers in the search for a unified

force field have adopted the Einstein methodology or modifications of it [12].

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The Oyibo methodology is esoteric as already stated and this is based on his

perception of some previous works in the quest and what he now conceives the

GUT to mean [1]:

A physically sound or credible set of mathematical equations from which to

determine or formulate the Grand Unified Force Field Theory comprising of the

four known forces in the universe which are the gravitational, the electromagnetic

and the nuclear forces of strong forces and weak forces as well as other forces

which may not have already been discovered.

To obtain such equations, Oyibo demanded that an arbitrary function G given by

)...,( 21 pYYYGG (2.1)

should be conformally invariant under the group of transformation:

),...(: 1 kyyfYT pii

k (2.2)

if kT is the group of transformations and

)...,( 21 pYYYGG = )...,).(,...( 211 pp yyykyyF (2.3)

where ),...( 1 kyyF p is a function of iy and k the single group parameter.

Now this group of transformations are to obey a new set of group laws and

possess a new form of group parameters [13,14]. The argument of Oyibo is that in

the final analysis, what establishes the integrity of this methodology is not so

much the group laws or group defined parameters but the end results or final

conclusion reached. With this conjection, Oyibo derived a set of conservative

equations

0)()()()( 3210 znynxntn GGGG , (2.4)

where n = 0, 1, 2, 3, 4.

Eq. (2.4) can be expressed in the Einstein-like form of conservative equations:

0mnG . (2.5)

This is the Oyibo generic (meaning the specific nature is determined by the

initial/boundary conditions and other physical constraint conditions) conservation

equation which is an arbitrary function of space and time coordinates (x,y,z,t),

velocities ),,( zyx , density )( , fluid or gas viscosity )( , temperature (T),

pressure (P), etc:

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,....).,,,,,,,,,,( PTzyxtzyxGG mnmn (2.6)

When the transformations in Eq. (2.2) is generalized to a system of partial

differential equations of order n given by

0)(

...)(

,...,,...,,,...,1

12121

np

qn

n

nqp

jx

y

x

yyyyxxxG (2.7)

and is conformally invariant under the transformations n

kT , then the generic

solutions to Eq. (2.4) is

1

3

1

2

1

1

1

0

n

n

n

n

n

n

n

nn zgygxgtg (2.8)

where n is the absolute invariant of the subgroup of transformations for the

independent coordinate variables and 210 ,, nnn ggg and 3ng are metric parameters.

The Oyibo`s generic equation in Eq. (2.5) can be recast into matrix form for

3,2,1,0, nm say,

33323130

23222120

13121110

03020100

GGGG

GGGG

GGGG

GGGG

(2.9)

The subgroup of transformation for the coordinate‘s variables is characterized by

the relationship . Therefore the hierarchy of the

Oyibo`s invariant solution for Eq.(2.9) has the following forms

zgygxgctg 302010000

(2.10a)

2

31

2

21

2

11

2

011 )( zgygxgctg

(2.10b)

3

32

3

22

3

12

3

022 )( zgygxgctg

(2.10c)

4

33

4

23

4

13

4

033 )( zgygxgctg

(2.10d)

In his review [3], Animalu demonstrated how to construct the realization of the

hierarchy of solutions of the generic equations for n = 0, 1, 2, 3, 4. Our goal here

is to explore the hierarchy of the invariant solution for the case where n = 0 and

try to recover some known standard equations of physics.

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2. OBTAINING THE SPACE-TIME INVARIANCE FOR THE GENERIC

SOLUTION FOR N = 0

McConnell states [15]

Let P be a point whose coordinates are and let Q be the neighboring point with

coordinate . If we donate the infinitesimal distance PQ by ds, which is

also called the element of the path, a 4-dimensional form for a physical metric is

stated as

(3.1)

where the denotes distinct variable that are used to denote a point in

space-time [1]

The space-time of a physical event can be described as a real and smooth

manifold with coordinates while is the infinitesimal

interval between two infinitesimal points on and which eventually

corresponds to the temporal and spatial world-line in the external world

(3.2)

where represent the space coordinates of world-

line in the manifold and represent the time part of the world-

line of the manifold

The challenge at hand is to be able to show that when n = 0, the invariant solution

( ) given by Eq. (2.10a) and are equivalent. In search of the

transformation law, we will be looking at the equation of a plane through the

point A with position vector a and perpendicular to a unit position vector (see

Fig. 1):

. 3.3a)

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This follows since the vector joining A to the general point R with the position

vector r is r-a and r will lie in the plane, if the vector is perpendicular to the

normal to the plane

. (3.3b)

Eq. (3.3a) can be recast into the form of

(3.3c)

where the unit normal to the plane is and is the

perpendicular distance of the plane from the origin.

The equation of a plane containing points a, b, c is

(3.3d)

A more symmetric form of the equation will be of the form [16]

(3.3e)

where .

Now let`s consider a curve r , parameterized by an Arc length s from some

point on the curve, if we write the length of the elemental path in the form

of equation

, (3.4a)

by including the time component into Eq.(3.4a), the resulting equation becomes

(3.4b)

where are constraining constants; if Eq. (3.4b)

becomes the familiar form of the equation of metric.

which can also be expressed as

(3.4c)

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where .

The expression for a general infinitesimal vector displacement is given by [17]

(3.5a)

where , and .

We note that a scalar product operation does not alter the geometric character of

the function to which it is applied, the scalar product of with will give us

=

(3.5b)

This is the infinitesimal change in going from , since r depends on

x, y, z such that defines a space curve, that is, the total derivative of

with respect to x, y, z along the curve is given by

(3.5c)

A careful inspection of Eq. (3.5c) shows that it is the differential form of the

spatial coordinates of equation (2.10a). We now will rewrite Eq. (3.5c) in the

form which will now include both the time and space component.

(3.5d)

From the earlier definition of

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(3.6)

By applying the conditions for Orthogonality [17]

(3.7)

It is easy to observe by comparing Eq. (3.4c) and (3.7) that the differential form of

the invariant solution is equivalent to

(3.8)

Fig. 1: A plane through the point A with position vector a and perpendicular to a

unit position vector .

3. RECOVERY OF RESULTS IN GEOMETRIC OPTICS

Geometric Optics gives only an approximation for small wavelength of

D`Alembert equation as Arnold Summerfield and Iris Runge demonstrated in

340



1911 but yield the exact propagation law of the electromagnetic waves front, quite

independently of the structure wave , a fundamental result which follows from the

theory of the characteristic manifold of the partial differential equation of the

second order. The law dictating electromagnetic radiation propagation in a

vacuum is the basic tenet of special relativity and the elemental path of the

travelling electromagnatic wave includes the results of geometric optics.

From Eq. (3.4c) and (3.7), we have showed that the metric of the square between

the neighboring points in space-time which is invariant is equivalent to the

differential form of the Oyibo`s invariant solution when n = 0 (Eq. (2.10a)), that

is,

(4.1)

Therefore, if we interprets Eq. (4.1) as an expression of space-time interval in

Minkowski manifold that would refers to a system of general coordinates, then

we can recover the results of geometric optics in a vacuum as it is described by

special relativity. This is the application we now turn to.

Fermat Principle in optics

The Fermat principle in optics states that [16]

If a ray of electromagnetic wave travelling through a medium of variable

refractive index follows a path such that the total optical path length is stationary,

then

Optical path length = physical path x refractive index

By enforcing and to be zero and and = 1 in Eq. (4.1), we obtain

(4.2)

From Eq. (4.2) the Optical path becomes

(4.3)

and

(4.4)

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From Fermat principle, the optical path is stationary

(4.5)

Therefore,

(using trig. identity) (4.6)

with and varying individually.

Case I

When and varies, then Eq. (4.6) can be expressed as

(4.7)

which is the Snell `s Law

Case II

When varies and remain constant, then Eq. (4.6) can be expressed as

(4.7). But for Eq. (4.6) to be true and , then

. (4.8)

So we can recover the law of refraction

(4.9)

4. SUMMARY AND CONCLUSION

It is the philosopher, John Dewey, who once asserted that, Every advance in

science has issued from a new audacity of imagination. In this work therefore, we

have argued for the need to consider the Oyibo‘s GAGUT with open mindedness

though his methodology is esoteric and some of his claims may appear

overambitious. The modeling philosophy of GAGUT is that [1,2]

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The most fundamental characteristics of the universe is motion. This fundamental

thing about the universe being motion can be basically derived from the fact that,

the material universe is made up of atoms consisting of electrons rotating around

the atomic nucleus perpetually, plus planets motions and solar systems motion

and the motion of galaxies, etc. This gives us the understanding that the universe

is basically characterized by motions. Therefore since motion can only be

provided by force, the universe could be viewed as a large force field.

Oyibo then represented the conservation of this large force field at a given space

time point in the universe by a set of generic equations, Gmn = 0. From this set of

equations, he obtained his generic solutions whose specific applications depend

on the initial/boundary conditions and other physical constraint conditions. An

important achievement of the Oyibo‘s methodology is that modeling with it is

reduced to algebraic operations rather differential equations for the most parts in

previous methodologies. With this understanding, we have been able to recover

both the Snell‘s law and the law of refraction from the generic solution for the

motion of wave, 0 . This is encouraging and therefore supports the possibility

that with more works, it may be possible to recover previous results from

GAGUT and also some of the predictions of Oyibo.

This conjecture is in line with a submission once made by Einstein [19] that

whether one observe a thing or not depends on the theory which one used. In

other words, it is the theory that decides what can be observed since observation

is the connection constructed from the phenomena and our realization. Einstein

even pointed out that his philosophy of abandoning absolute time (Galilean

transformations) and introducing only the time of special coordinate

transformations leading to his special theory of relativity may be wrong. His

reason being that any reasonable theory will besides all the things that can

immediately be observed from it, give the possibility of observing other things

more indirectly. We think this is the goal of the Oyibo‘s GAGUT.

Finally, one other controversial aspect of the GAGUT has to do with the claim by

Oyibo that it can be used to solve various man‘s problems including those in

health and economics. While this may seems overambitious, one must not lose

sight of the impact of the Einstein mass-energy equation, E = mc2. This world

most famous equation is believed to have [20] revolutionalized physics, redefine

strategic arms, and promises to transform our economy and environment with

plentiful, clean energy. It is therefore not naïve to postulate that if the extension of

this equation which is one of the salient conclusions from GAGUT that mass can

be transformed not only into energy but also into momentum is verified, then

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GAGUT may also hold the possibility of extending the promises of the Einstein

mass-energy equation.

Acknowledgement

We appreciate the very inspirational discussion with Professor A.O.E. Animalu

and also for making available to us his papers. We also acknowledge the useful

discussion with Professor Amagh Nduka. This work is supported in part by ICBR

and also by AFAHOSITECH.

REFERENCES

[1] G. A. Oyibo, ―Grand Unified Theorem; Discovery of the Theory of

Everything and the Building Block of Quantum Theory,‖ International

Journal of Mathematics, Game Theory and Algebra, Vol. 13, 281-354

(2003).

[2] G. A. Oyibo, ‗Grand Unified Theorem,‘ with subtitles: ―Representation of

the Unified Theory of Everything,‖ Nova Science Publishers, New York

(2001) and ―Discovery of the Theory of Everything and the Building

Block of Quantum Theory,‖ Nova Science Publishers, New York (2004).

[3] A. O. E. Animalu, ―A review of Oyibo‘s grand unified theorem with

realization of a hierarchy of Oyibo-Einstein relativities,‖ unpublished.

[4] A. O. E. Animalu, ―Realization of a new conformal symmetry group for

the grand unified theorem in projective space-time geometry,‖

unpublished.

[5] A. O. E. Animalu and P. N. Okeke, ―Theoretical High Energy Physics,

Astrophysics, Cosmology, Tensors Calculus, General Relativity and

Grand Unified Theorem (A foundation postgraduate course),‖ National

Mathematical Centre, Abuja (2005).

[6] Another reason for the lack of more research on GAGUT in the country

may be due to the poor access to the work: (1) reaching Oyibo has been

unnecessarily difficult and even when Oyibo came to the country, not very

many researchers could attend his lectures and (2) the price of $240 (=

N30,000.00 at $1 = N125.00) for each of the books (Ref. 2) may be too

exorbitant for individual researchers whose research works have no direct

relation with GAGUT.

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[7] Projective also know as Descriptive geometries means non-metrical

geometries which are different from the Minkowskian and Riemannian

geometries which are metrical. One important feature of projective

geometry is that one never measures anything, instead, one relates one set

of points to another by a projectivity. See H. S. M. Coxeter, ―Projective

Geometry,‖ (2nd Ed), Springer (2003).

[8] P. A. M. Dirac, ―Principle of Quantum Mechanics‖ (4th Ed), Oxford

University Press, New York (1958).

[9] P. A. M. Dirac, ―Methods in Theoretical Physics‖ in From my life in

physics, World Scientific Publishing Co. Pte. Ltd, Singapore, pp19 - 30

(1989).

[10] S. Shapin, ―The Scientific Revolution‖ University of Chicago Press.

Chicago (1996).

[11] A. E. E. Mckemzie, ―The Major Achievements of Science‖ (Vol.1),

Cambridge University Press (1969).

[12] P. G. Bergmann, ―Introduction to the Theory of Relativity,‖Dover

Publications, Inc., New York (1976).

[13] G. A. Oyibo, ―New Group Theory for Mathematical Physics,‖ Gas

Dynamics and Turbulence, Nova science Publishers, New York (1993).

[14] G. A. Oyibo, ―Generalized mathematical proof of Einstein‘s theory using

a new group theory,‖ Problems of Nonlinear Analysis in Eingeering

systems (An International Russian Journal) Vol. 2, pp 22-29 (1995):

International Journal of Mathematics, Game Theory and Algebra, Vol. 4,

1-24 (1996)

[15] J. B. Almeida, ―How much of the universe can be explained by geometry‖

arXiv:0801.4089 (2008).

[16] K. F. Riley et al., ―Mathematical Methods for Physics and Engineering,‖

Cambridge University Press (3rd

Ed) (2006).

[17] B. D. Gupta, ―Mathematical Physics‖ Vikas Publishing House (1986)

[18] A. Parker, ―At Livermore, audacious physics has thrived for 50 years,‖

Science and Technology Review, The Regents of the University of

Califonia, Califonia, pp16-21, May (2002).

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[19] W. Heisenberg, ―Theory, Criticism and a Philosophy‖ in From my life in

physics, World Scientific Publishing Co. Pte. Ltd, Singapore, pp32 - 55

(1989).

[20] A. Heller, ―How one equation changes the world,‖ Science and

Technology Review, The Regents of the University of Califonia,

Califonia, pp12-20, September (2005).

346



THE ATTITUDE OF FEMALE NIGERIAN UNIVERSITY STUDENTS

TOWARDS THE STUDY OF PHYSICS AND THEIR

PERFORMANCE†††††††††

Anita Edi and Otete Okobiah

Delta State University, Abraka, Delta State

Abstract

A review of the literature reveals that fewer female students register for

science courses, especially Physics than male students. A lot of factors ranging

from teaching strategies and classroom climates, parental emotional support,

content of the Physics course, textbooks used (Zhu, 2007) have been identified as

contributory factors. Other factors include socio-economic levels, previous

learning, methods of studying, attitudes, self-adequacy intelligence and teaching-

learning approaches (Guzel, 2004|). This study is an exploratory study to find out

the number of female students who enrolled in the Department of Physics, Delta

State University, Abraka, for four academic sessions, their attitude towards

Physics, and their performance. In conclusion a way-forward for the

improvement of the situation was suggested. Subsequent studies will look at

other aspects identified from literature.

1. INTRODUCTION

Education is wealth and female education, they say results in the nation‘s

education (Adedeji, 1989, UNECA, 1975)). Generally, female‘s enrolment in the

education sector is usually lower than male‘s enrolment especially in the

Sciences. Observations of the existing number of female students in core sciences

is low compared with that of male students. For instance, in engineering

especially mechanical, you hardly find a female student. In Physics class, the

female students are less than 25% of the total number of the class (four sessions –

††††††††† African Journal of Physics Vol. 2, pp. 346-356 , (2009)


347



12 out of 59, 9 out of 38, 24 out of 98 and 25 out of 141). This observation

corroborates Nnaka‘s (2005) and Aigbomian‘s (2002) findings. Aigbomian

(2002) found out that in a class of 153, 64 and 45, there were 11 (17.1%), 1

(1.5%) and 10(22.22%) female students respectively. These figures were all less

than 25%

Our present economies are driven by Science. So if women are to

contribute to the development of the nation, female students need to acquire

relevant scientific skills. Science courses/subjects are generally regarded as being

more difficult than humanities, social sciences education. Studies have indicated

that the attitudes of males and females toward science courses differ (Banya,

2004). Some reasons have been proffered for the low enrolment of female

students in the sciences as well as their low academic performance. The existence

of strong gender biases in science curricula and instruction (Okeke, 1990:

Erinogha, 1997, Njoku 2000, Olagunju 2001, Njoku 2006) is one of the factors.

Other reasons include child-rearing practices, which impede girls readiness for

scientific and technological studies (Njoku, 1993). The masculine nature of the

sciences deter girls from entering into the programmes (Mulemwa, 1999 and

Njoku 2005).

Globally, the concept of science being for males is the same. This

informed the formation of the body known as Girls in Science and Technology

(GIST) in England years ago. Girls who read sciences tend to be intimidated by

the kind of questions they are asked ―why do you study Physics? Why not

education where you will have time for your home. You are a woman‖. Our

female science students are also asked similar questions.

However, lots have been written about women‘s development and their

advantages to the family settings, political arena and the educational setting.

Government has given attention to the inclusion of women in governance. I have

heard it said many times that science rules the world. Educating women in the

sciences, therefore means that women will be in the areas of sciences to be able to

guide, direct and influence their children females inclusive. This will make the

children to have positive attitude towards science courses especially physics.

Training a woman means training a nation. Therefore, if there must be an

increase in the enrolment of females in the sciences, mothers should have more

knowledge in scientific skills to be able to expose their female children early

enough as they engage them in house chores, to scientific concepts. For example,

the steam from the pot of soup, kettle of boiling water could be related to

scientific/physics concepts. Pouring hot water in an insulated cup and non-

348



insulated cup can be related to the concepts of heat conductor in physics. Such

interactions can lead female students to develop interest in Physics and science

courses in general.

This study looks at the number of female students‘ enrolment in physics,

their performance, attitudes towards physics. Some suggestions were proffered as

a way forward for encouraging and stimulating girls‘ interest in Physics as well as

developing positive attitude towards physics.

2. METHODOLOGY

The subjects for the study were

The authors analyzed the performance of female students for four different

sessions in Physics Department of one of the Universities in Southern Nigeria.

Percentage was used to analyze the data collected from the University. There was

an open-ended question of the female students‘ attitude towards Physics. The

female students‘ expressed attitudes towards Physics were stated in Table 5.

3. RESEARCH QUESTION

1. Do the female students perform better than the male students in Physics

Department?

2. What are the attitudes of female students towards Physics?

3. ANALYSIS

Tables 1-4 indicate the results of the performance of the female students in

Physics in four different academic sessions. The number of female enrolment is

also indicated in the tables. Table 5 indicates the attitudes of the female students

towards Physics.

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4. RESULTS

Table 1: A table showing the performance of Female and male students in

2001/2002 and 2001/2002

From the above table, there are 12 (20.30%) female students as against 47 (79.63)

male students. There is no first class and pass grade among the males and the

females. 13.55%, 27.11% and 28.81% of the male students had 21, 2

2 and FRNS

respectively. 3.38% of the female students had 21, 2

2 and FRNS each. 10.16%

of the female students had 3rd

class as against 8.47% of the male students. One of

the male students withdrew while none of the female students withdrew. While

16.92% of the girls passed, 49.13% of the male students passed.

F M

No. % No. %

1st 0 0 0 0

21 2 3.38 8 13.55

22 2 3.38 16 27.11

3rd

6 10.16 5 8.47

Pass 0 0 0 0

Total % of

Passes

- 16.92 - 49.13

Withdrawn 0 0 1 1.69

FRNS 2 3.38 17 28.81

TOTAL 12 20.30 47 79.63

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Table 2: A table showing the performance of female and male Physics students

in

2003/2004 session.

F M

No. % No. %

1st 0 0 0 0

21 1 2.63 10 26.31

22 1 2.63 3 5.08

3rd

2 5.26 2 5.26

Pass 1 2.63 1 2.63

Total % of

Passes

13.15 31.28

Withdrawn 0 0 1 2.63

FRNS 4 10.52 12 31.57

TOTAL 9 23.67 29 73.48

From the above Table, none of the students made 1st class. While 2.63%

of the females made 21, 2

2 and pass, 26.31%, 5.08% and 2.63% of the males made

21, 2

2 and pass respectively. While no female withdrew, 2.63% of the males

withdrew. 10.52% of the females did not graduate as against 31.57% of the

males. On the whole, 31.28% of the males as against 13.15% of the girls passed.

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Table 3: A table showing the performance of female and male students in

2004/2005 session.

F M

No. % No. %

1st 1 1.02 0 0

21 6 6.12 5 5.10

22 7 7.14 31 31.63

3rd

6 6.12 15 15.30

Pass 0 0 7 7.14

Total % of

Passes

- 20.40 59.17 -

FRNS 4 4.08 16 16.32

TOTAL 24 24.44 74 75.49

Of the total 98 students, 24 (24.44%) are females while 74 (75.49%) are

males. 1.02% and 6.12% of the females as against 0% and 5.10% of the males

made first class and 21

respectively. 31.63%, 15.30%, 7.14% of the males as

against 7.14%, 6.12% and 0% of the females made 21

, 3rd

class and Pass degree.

Only 4.08% of the females did not graduate while 16.32% of the males did not

graduate.

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Table 4: A table showing the Female Students‘ Performance in 2007/2008

session

F M

No. % No. %

1st 0 0 1 0.70

21 3 2.23 7 4.96

22 8 5.97 19 13.47

3rd

5 3.73 33 23.40

Pass 0 0 7 4.96

Total % of

Passes

- 10.63 - 47.49

FRNS 9 6.38 49 34.75

TOTAL 25 17.73 111 82.24

From the above table 47.49% of the males as against 10.63% of the

females passed. However, only 6.38% of the females could not graduate while

34.75% of the males could not graduate. Of the total number of students that

enrolled, 82.24% are males while 17.73% are females. No female made 1st class

and the percentage of students‘ that passed in each grade was higher for the

males; 21 (4.96% vs 2.12%), 2

2 (13.47 vs 4.97%) 3

rd (23.40% vs 3.54%).

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Table 5: Attitudes of Female Physics Students towards Physics

Some Physics students from the Delta State University, Abraka were

asked to state their attitudes towards Physics. The following were their

expressed attitude.

1. Psychologically, female students do not like science subjects

because they are difficult

2. Most girls run away from Physics because of the mathematical,

quantum mechanics, classical mechanics.

3. Most people in the society believe that difficult courses/subjects

are meant for boys and not girls.

4. Girls are scared of physics.

5. Most adults in the society express surprise when they see female

science (physics) students. They say things like ―iron lady,

tomboy, strong lady‖.

6. Girls are meant for homes so they should prefer humanities,

education, social studies, management courses and not physics.

These courses will give more time to run their homes.

7. Physics is a hard course that requires good health, strong mind,

strong determination, man power, lot of energy. Girls appear not

to have these characteristics.

8. Girls that study Physics feel that they should corroborate the

saying that ―what a man can do, a woman can do it better‖. This

makes them to work very hard.

The expressed attitude of the female Physics students indicate that there are

socio-cultural, intelligence, emotional and =environmental factors, militating

against the female gender studying or who want to study Physics. This

corroborates Aigbomian (1985), Mulenwa (1999) and Njoku (2005).

5. DISCUSSION

The findings from this study indicate that female students studying

Physics for the four academic sessions are less than 25% of the students‘

population. In 2001/02, 2003/04, 2004/05 and 2007/08, the percentage of the

girls that enrolled is 20.30%, 23.67%, 24.44%, and 17.73% respectively. This

low percentage of female students enrolment in Physics corroborate the findings

of Njoku (2000, 2005, 2006), Olagunju (2001) and Aigbomian (2002). Total

number of passes for girls for the four academic sessions out of the 321 students

that enrolled is 5 (15.50%) while that of the males is 190 (57.75%). In another

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University whose detailed results could not be obtained, only 7 (8.24% of the

2007/08 class were females while 78 (91.76%) were males.

6. THE WAY FORWARD

Since training a woman means training a nation, and science rules the

world, and we have few female students enrolling for science courses, the

government and parents should ensure that:

1. There are awareness in the society to inform them that women can also

be scientist.

2. Teachers both at the primary, secondary and tertiary institutions should

expose pupils and students to the vocational relevance of the science

courses they are teaching.

3. There should be equipped laboratories at all educational levels to assist

students meaningful participation in the learning process.

4. Teachers should relate scientific concepts to the reality of life, and the

environment and to vocational opportunities..

5. Teachers should give assignments that at relevant to particular physics

courses, e.g. designing prototypes fans, machines (for light, light solar

panels, inverters, etc).

6. Students have days for exhibiting their scientific products to the

public, parents and Government.

6b. That students are given the opportunity to view themselves and their

future in relation to Physics.

7. Female students in Physics Departments should look beyond the

kitchen. We have influential women like Professor Grace Alele-

Williams, Prof Dora Akinluyi, Dr. Ngozi Okonji-Iweala, Dr. Oby

Nzekwesili, etc.

8. Teachers should take students on excursions to increase the level of

students‘ interest in Physics and increase the number of girls that will

choose physics.

9. Lecturers are dedicated and devoted so as to encourage all female

students to be focused and achieve better performance.

10. That the socio-cultural structures and processes that becloud and blur

female student‘s participation and science based programme are

removed and their scientific aspirations improve upon.

11. Teachers at all educational level, should always engage female pupils

in scientific activities. Such as making soap, distilled water, etc.

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12. Mothers should always expose their female daughters to physics

concepts early enough, e.g. utensils that are good and poor conductors

of heat, gas/vapour from boiling water, etc..

13. Industries, philanthropists, alumni and parents are encouraged to

provide scientific equipment to laboratories in schools (all levels of

institutions).

14. Teachers should employ the 5Es (exploration, experimentation,

explanation, extension and evaluation) while teaching Physics right

from the secondary school level.

7. CONCLUSION

Efforts have been made to find out the number of female student who

enrolled in Physics class. There was enough evidence to evince that there was

low female enrolment in Physics. The percentage of passes were also lower for

the females indicating that the males performed better. Some suggestions were

made if the number of our female students enrolling in Physics programmes will

be increased as well as increasing the level of their performances.

REFERENCES

Adedeji, A (1989) African Women Development:: Selected Statements. Addis

Ababa: UNECA

Aigbomian, D. O. (1985) Relationship Between Understanding of Physics

Concepts and Achievements in the West African School Certificate

Physics Examination. Unpublished Ph.D Thesis University of Nigeria,

Nsukka.

Aigbomian, D. O. (2002) Gender and Science, Technology and Mathematics in

Oriafo, S. O. and Ikponmwosa, O. I. (eds) Gender issues in Education for

National Development: 54-63

Banya, S. K. (2005) A Study of Factors Affecting The Attitudes of Youth Female

Students Toward Chemistry At the High School Level, Dissertation,

University of Southern Mississippi, USA.

356



Erinosha, S. Y. (1997) Female Participation in Science: an Analysis of

Secondary School Science Curriculum Materials in Nigeria: Abridged

Research Report No. 29, Nairobi.

Guzel, H (2004) The Relationship Between Students Success in Physics Lessons

and their Attitudes Towards Mathematics, Journal of Turkish Science

Education I (1)

Njoku, Z. C. (1993) Strategies for Improving the Enrolment and Achievement of

Girls in Science Technology and Mathematics at Secondary School level,

In Nworgu, B. E. (ed) Curriculum Development, Implementation and

Evaluation, A book of Reading Nsukka, APQEN Pub

Njoku, Z. C. (2000) Image of Females in Science: A Gender Analysis of Science

and Technology Activities in Nigerian Primary Science Textbooks.

Journal of Primary Education I (3-12)

Njoku, Z. C. (2006) Girls Disadvantages in Science and Technology Curriculum

and Instruction: Possible Explanation for Gender Difference in Pupils

Achievement and Interest in Primary Science: International Journal of

Educational Research I (1): 15-25

Nnaka, C. (2005) Women Participation in Science and Technology Education.

International Journal of Forum for African Women Educationalists

Nigeria I (3): 112-118

Olagunju, M. A. (2001) Increasing girls‘ Interest, Participation and Achievement

in Science: 42nd

Proceedings of the Science Teachers Association of

Nigeria: 52-58.

Okeke, E. A. C. 91990) Gender, Science and Technology in Africa: A Challenge

for Education. The Rama Mechta Lectute. Cambridge, Radcliff College.

Zhu, Z. (2007). Learning Content, Physics Self-Efficacy, and Female Students‘

Physics course taking. International Education Journal, 8 (2), 204-212

357



ENDOGENOUS ASTRONOMY OF PARTS OF NIGERIA‡‡‡‡‡‡‡‡‡

J.O. Urama1, E.N. Urama2 and A.O.E. Animalu1

1Dept. of Physics & Astronomy, University of Nigeria, Nsukka

2Dept. of English and Literary Studies, University of Nigeria, Nsukka

1e-mail: [email protected]

Abstract

Indigenous, endogenous, traditional, or cultural astronomy focuses on the many

ways that people and cultures interact with celestial bodies. In Nigeria, there is

very little awareness about modern astronomy. However, like ancient people

everywhere, Nigerians have always wondered at the sky and struggled to make

sense of it. Astronomical observations used by the ancient people of Nigeria, and

Africa in general, were developed out of the people's desire to have concrete

manifestations of their gods and religious beliefs as well as for time-keeping –

day, night and calendar for agricultural and festive seasons. Here, we discuss

some aspects of the culture and tradition of some of the ethnic groups in Nigeria

and the need to bridge the gap between cultural astronomy in Africa and modern

astronomy by providing scientific interpretation to such cosmogonies and ancient

astronomical practices. Through linking the traditional and the scientific, it is

believed that this would be used to create awareness and interest in modern

astronomy and sciences generally.

1. INTRODUCTION

A good understanding and application of the cosmological ideas of a

people are the basic prerequisites for achieving a balanced social, economic,

political and technological development. This is one of the greatest challenges of

our time – being able to revolutionize the thought pattern of the public;

‡‡‡‡‡‡‡‡‡African Journal of Physics Vol. 2, pp.357-371, (2009)



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―scientificating‖ our worldview and linking it with other worldviews; and

demystifying the ―mysterious heavenlies‖. The description of creation told by

followers of cabbala (a form of Jewish mysticism), for example, has been

interpreted along modern cosmological views[1]. The cabbalists developed a

theoretical system portraying God as having ten aspects, known in Hebrew as the

sephirot. Beyond the sephirot is Ein Sof, the unknowable aspect of God, from

which emanated a light that created the sephirot and the physical universe. Of the

ten sephirot the first three deal with creation, and they correspond fairly closely to

the concepts from the cosmological theories of inflation. In Nigeria, there are

hundreds of such theories about creation, life and living, etc that need to be

studied systematically.

As observed by Kunene[2], ―each society is concerned with its destiny

within the cosmic arena; without this perspective, the society can only be

stampeded into directions it does not fully comprehend or does not feel ready to

follow‖. He further argued that, the more fragmented, individualistic, and land-

alienated people are, the more they are inclined towards a fantasy that is outside

the earth. A scene from Kenya in ―The River Between‖[3] typifies this: ―There

was a general uniformity between all the houses that lay scattered over this ridge.

They consisted of round thatched huts standing in groups of three or four. A

natural hedge surrounded each household. Joshua‘s house was different. His was

a thin-roofed rectangular building standing quite distinctly by itself on the ridge.

The tin roof was already decaying and let in rain freely, so on top of the roof

could be seen little scraps of sacking that covered the very bad parts. The

building, standing so distinctly and defiantly was perhaps an indication that the

old isolation of Makuyu from the rest of world was being broken down.‖ The

picture created here is that of the changing and contrasting patterns of life. On the

one hand there is a communal living which is hinged on earth-centredness and a

world view which is cyclic and on the other hand an individualistic life based on

linear form.

In fact in many parts of West Africa and beyond, the world view is one of

dualities and relativity. This manifests in curvilinear (circular and spiral) forms of

thought and living as evident in the predominance of curvilinear forms in

traditional African artistic expressions such as the architecture, ornaments, ritual

dance, etc. The curvilinear forms are believed to be indicative of an ―intuitive

dialogue with nature‖ [4]. It had earlier been argued that, ―The natural world ... is

one of infinite varieties and complexities, a multidimensional world which

contains no straight lines or completely regular shapes, where things do not

happen in sequences but all together, a world where - as modern physics tells us -

even empty space is curved‖[5].

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In African traditional imagination, power is an aspect of beauty. It is the

expression of vitality mixed with mystery[6]. The sky entities manifest natural

and supernatural power and the feeling this power inspires is reverence and fear.

The sun, the moon and the stars are therefore perceived in their powerful, vital,

beneficial or harmful aspects. In Birago Diop‘s poem Omen, for example, the sun

is portrayed as a source of protection in all stages of human life.

Senanu and Vincent[7] argue that this poem is typical of Birago Diop, who is

preoccupied in many of his poems with that aspect of African culture which

emphasizes the importance of and the guiding spirit of our ancestors. This

creative work is influenced by Diop‘s experiences, his early life at Dakar in

Senegal through his life in France where he studied veterinary Science. The

colours of the sun symbolize stages of his life: the dawn of his life – his childhood

‗Omen‘ by Birago Diop

A naked sun - a yellow sun

A sun all naked at early dawn

Pours waves of gold over the bank

of the river of yellow

A naked sun – a white sun

A sun all naked, and white

Pour waves of silver

Over the river of white

A naked sun – a red sun

A sun all naked and red

Pours waves of red blood

Over the river of red

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symbolized by ‗a yellow sun‘, his youth by ‗a white sun‘ and his old age by ‗a red

sun‘. He strongly believes in the protection given him by the spirit of his

ancestors against all ills. No matter the colour of the river, it blends with the

colour of the naked sun; signifying that the spirits of his ancestors are always

available to protect and inspire him. The sun portrayed to be ‗a naked sun‘ shows

that there is no hidden agenda of the ancestral heritage of the Africans. Africans

are therefore encouraged to remember their roots. It is only by maintaining close

relationship with our ancestors that will make us to learn this wisdom and secure

our protection in all stages of our lives. His using the sun as a symbol of our

ancestors depicts the position of the sun in African traditional belief and culture

In this paper, we investigate some aspects of the culture and traditions of

some of the ethnic groups of Nigeria with special emphasis on their world view,

cosmogony and creation myths, indigenous lore of celestial bodies, calendars,

cycles, seasons and festivals. We shall attempt to re-interpret this body of

knowledge in the light of modern/ western astronomy.

2. BRIEF OVERVIEW OF MODERN ASTRONOMY/SPACE

SCIENCE IN NIGERIA

Astronomy is one of the oldest science disciplines. Modern astronomy is a

term used to refer to the scientific discipline in which we collect, correlate and

interpret data pertinent to our entire observable universe from our galaxy to the

farthest reaches of extragalactic space. It involves the study of structure, evolution

and origin of the universe and its constituent parts. Astronomy arose

independently in many parts of the world out of a practical human need of

calendar, telling time and direction finding.

Astronomy has been said to be, ―a science that has a universal appeal

because it encompasses all fields of human interest and endeavour‖[8]. It has been

argued that, ―Astronomy is more than the science of stars. It is intimately

connected to our ideas of ourselves, our purpose and place in the universe‖[9]. As

the science that provides the framework knowledge of where we, and the planet

on which we live, fit into the environment of the universe, astronomy is a vital

part of the culture of all mankind. A person deprived of the broad outlines of

astronomical knowledge is as culturally handicapped as one never exposed to

history, literature, music or art[10].

Modern astronomy came into Nigeria only four decades back. One of the

first major modern astronomy/space science activities in Nigeria was the setting

up of NASA‘s space tracking facility in Kano in the early 1960s for monitoring

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the missions of Gemini, Apollo and Skylab spacecraft. In South Africa, Australia

and other parts of the world, such NASA‘s space tracking facilities

metamorphosed into radio astronomy observatories but, unfortunately, were

dismantled in Nigeria. Many Universities and Research Centres in Nigeria have

also been involved with teaching and research in astronomy and space physics for

many years. The Departments of Mathematics and Physics & Astronomy of the

University of Nigeria, Nsukka, have been at the forefront of astronomy since

1962. Programs in basic space science have also been running at the Universities

of Ilorin; Bayero University, Kano; Federal Universities of Technology at Akure

and Minna for over 20 years. Nigeria is also a host to an international centre

involved with space science, the African Regional Centre for Space Science and

Technology Education in English Language (ARCSSTE-E), Ile-Ife, which was

inaugurated in November 1998. And two years later the National Centre for Basic

Space Science was set up at Nsukka with the mandate to conduct and carry out

active front-line research in Atmospheric Science and Astronomy. However,

astronomy remains largely an ―unwanted commodity‖ in Nigeria. It had been

pointed out that; ―the general feeling among Nigerians is that the idea of space

science and technology is beyond us as a developing country. This feeling

unfortunately is partly due to the belief that our problems are mundane and earth

bound and so the solutions to those problems must be sought on the ground with

classical and non-space based technology‖[11]. This could explain why there are

only about a dozen professional astronomers and very few amateur astronomers in

the country since it is seen to be of only esoteric interest and devoid of any

practical and economic value. In February 1990, there was a sudden

―disappearance‖ of the moon for about an hour. This eclipse, happening on the

eve of a scheduled visit to one of the cities in the north by the then President,

caused a stir in the city as this was seen as a bad omen and believed to have been

caused by ill-wishers of the President. Subsequently violent clashes erupted

between the self-proclaimed protagonists and the suspected antagonists of the

President, who were attacked in the belief that they had bewitched the moon. This

typifies the misconceptions and superstitious beliefs about even the commonest

astronomical event in modern day Nigeria.

It is, however, worthy of note that a Nigerian – Samuel Okoye – was the

first black African to obtain a doctorate in radio astronomy. His doctoral research

at the Mullard Radio Astronomy Observatory, University of Cambridge, led to the

discovery (with his supervisor, Prof. Anthony Hewish) of an extremely high

brightness temperature source in the Crab Nebula recognized as the first example

of a neutron star. This source later turned out to be none other than the famous

Crab Nebula Pulsar. The discovery was widely recognized of being of

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fundamental importance to basic physics and for which Prof. Hewish was

awarded the Nobel Prize in Physics in 1974.

He returned to Nigeria to set up a radio observatory – an ambition that

very well highlights the problems, challenges and prospects of doing modern

astronomy in this part of the world. According to Okeke [12], ―Professor Okoye‘s

initial efforts to set up a space research centre was frustrated by lack of funds until

in 1977 when the late Dr. Nnamdi Azikiwe made a handsome donation of the sum

of one hundred thousand Naira (about US $150,000 then) towards the setting up

of a space centre. With this initial assistance Professor Okoye and his group were

able to set up a 10-m dish operating at 327 MHz as the initial facility of the group.

This was done with a little assistance from University of California, and so a lot

of indigenous technology was built in. With this facility, the centre planned to

carry out the following projects: a two station pulsar observation with India, a

VLBI observation programme with Germany, ... All these experiments involve

signing of agreements with other governments and some financial support. There

was no support from the Nigerian government or from any other source, as a

result, all the above mentioned projects could not take of. Since no serious

activity was taking place around the telescope, which was cited in a remote corner

of the University as is usual with radio instruments, the dish and all the facilities

were vandalised.‖

However, with the recent restoration of democratic governance in Nigeria,

the National Centre for Basic Space Science, Nsukka, was created and it is hoped

that this would assist the nation in fulfilling aspects of the Draft National Policy

on Space Science and Technology and the dream of the first Nigerian President,

the late Rt. Hon. Dr Nnamdi Azikiwe. The Centre is currently working towards

the actualisation of a 25-m Nigerian Radio Telescope., and it has, also, has got

some small optical telescopes. And the Rivers State University of Science and

Technology, also, has got an optical telescope, which was recently installed and

already in use.

3. OVERVIEW OF CULTURAL ASTRONOMY OF PARTS OF

NIGERIA

Cultural astronomy has been said to be, ―the use of astronomical

knowledge, beliefs or theories to inspire, inform or influence social forms and

ideologies, or any aspect of human behaviour. Cultural astronomy also includes

the modern disciplines of ethnoastronomy and archeoastronomy‖[13]. The

cultural astronomy of the West African sub-region, and Nigeria in particular, is

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among the least investigated in the African continent. Ancient astronomical

practices of parts of Southern and Northern Africa have been relatively well

studied. Nigeria's endogenous astronomy is as diverse as its over 300 ethnic

groups. Most of the ethnic groups have astronomy-rich cultures. These ethno-

astronomical views are revealed in the folklore, traditional poetry and art works of

many of the ethnic groups. Some of these had names for star constellations as

well as some exciting star lore.

The Hausa speaking tribal nation of West Africa have a good number of

myths and folktales about the sun, moon and the stars. The stars are supposed to

visit each other and talk. They even have names for some constellations, a typical

example being a constellation that appears at the commencement of the rains

which is known as kaza Maiyaya (the Hen with Chickens). The morning-star in

harvest time (probably α-Aquila) is known as the eagle star [14]. In one of their

folktales, they have it that the moon and the sun were friendly until the sun gave

birth. Then the sun called the moon and asked him to hold her daughter while she

went and washed herself. The moon took the sun‘s daughter, but was not able to

hold it, for it burnt him, and he let it go, and it fell to the earth – that is why men

feel hot on earth. When the sun returned, she asked the moon where the daughter

was, and the moon replied, ―Your daughter was burning me so I let her go, and

she fell to earth.‖ Because of that the sun pursues the moon. Another variant is

that the moon‘s path is full of thorns, while that of the sun is sandy, and on that

account the moon cannot travel quickly, as does the sun. So when the moon can

proceed no farther, he gets on the sun‘s path, and the sun catches him. When the

sun has caught him the people take their drums and ask the sun to spare the

moon[15]. This ―catching-up‖ occurs during an eclipse of the sun – usually partial

or annular.

In Igala land, when an eclipse happens it is believed that the world wants

to come to an end, so the people would start beating the drums, buckets, plates

and bowls as a praise to their god to spare the world. And when the eclipse is over

they would start chanting, ―thanks be to our gods for they have heard our

prayers‖. Here, it is also believed that the moon has two wives – and these are the

brightest stars that stay very close to the moon when it appears in the night, the

most loved one staying closest to him [Joan Edime – private communication].

In Igbo land the supreme being Chukwu is commonly identified with the

sun (Anyanwu) so that the supreme being is often described as Anyanwu Eze

Chukwu Okike (The sun, the Lord God, the creator). In Nsukka area, almost every

household in those days had a shrine of anyanwu in his compund as a round

pottery dish sunk into the ground bottom upwards at the base of an ogbu tree.

There can be little doubt that this pottery dish is used as representing the sun‘s

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disc. Offerings are usually made at sunrise or sunset. In some cases the anyanwu

(sun-god) shrine is a mound of sand. Among the Jukun of the Benue basin there is

the same partial identification of the Sun with the Supreme Being and it is

noteworthy that the Jukun words for Sun and Supreme Being viz: Nyunu (Anu)

and Chi-do embody the same roots as the terms used by the Ibo. In the Okpoto

groups the Sun is called Enu and the Sun-god Olenu. It is perhaps not accidental

that Heliopolis the centre of Sun-worship in Ancient Egypt was known by the

Egyptians as Anu. It is also noteworthy that the use of mounds of sand in

connection with Sun-rites was common in Egypt and that among the Jukun today

the Sun-altars are two mounds of sand.[16]

The Yoruba cosmogony is, in a sense, the basis for their rituals, social

structure as well as their political activity. In the Yoruba cosmogony, Obatala

was issued with the task of building the Earth by Sky God Olorun, who gave him

blueprints, a handful of mud, a chain, a five-toed chicken, and detailed

Figure 1: A village shrine in Nsude, Enugu State of Nigeria. Photograph

was taken in the 1930s by the late G.I. Jones. (Source: Museum of

Archeology and Anthropolgy, University of Cambridge.)

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instructions. Unfortunately, on his way to perform this important task accidentally

gate crashed a God-party and spent the rest of the evening roaring, drunk on palm

wine. Seeing the chance for fame and glory, his younger brother Oduduwa

pinched the holy building materials and attempted to jerry-build the Earth himself.

Advised by a friendly chameleon, he lowered the chain over the edge of heaven,

climbed down, and tossed the lump of mud into the primeval sea. The chicken

hopped onto the mud and began scratching it in all directions. Pretty soon there

was a decent size landscape and thus was the Earth born [www.godchecker.com].

Figure 2: The Ooni of Ife, Oba Okunade Sijuade, appearing with the Are crown, the beaded

crown left behind by Oduduwa for Yoruba.

The Ooni wears the crown only once a year at the Olojo festival and it is believed to

have curative powers as any prayer said once the Ooni adorns the crown is said to be answered.

(

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Oduduwa, the first Ooni of Ife, is the father of the Yorubas and

progenitor of all Yoruba Oba's and the Oba of Benin. Oduduwa is believed to

have had 16 sons who later became powerful traditional rulers of Yoruba land,

most notably Alafin of Oyo, Ooni of Ife, Oragun of Ila, Owa of Ilesha, Alake of

Abeokuta and Osemawe of Ondo.

In Igboland, unlike the situation in Edo, Yoruba and Hausa, there is a

dearth of written material for the period before the 19th century. Although an

ideograph (sign-language), nsibidi, had been in use, literacy as it is understood

today, was introduced into Igboland after the first European visitors (John and

Richard Lander) travelled down the lower Niger down to the Niger Delta in 1830.

In many places in Igboland, the general life of the community still largely

hinges on the lunar calendar and the people look up to the king-priests who

determined agricultural seasons based on the lunar calendar. Here, some historic

annual, biannual and perennial events are not randomly fixed by mortal men;

rather some signs in the sky believed to be messages from the gods are used to

avoid the wrath of the gods and other calamities. Such festival like new yam

festival, cult or masquerade initiation, burial and funeral ceremonies, etc. are

therefore programmed on astronomical observations. These astronomical signs

include the appearing of the new moon, sunrise or sunset and the appearance of

specific stars. The respective significance of these signs is to the knowledge of the

high/chief priests who order the annunciation of dates to various activities.

In Chinua Achebe‘s Arrow of God[17], it is believed by the people of

Umuaro that the relationship they have with their religious and agricultural

existence is designed by the gods. Ezeulu, the protagonist of the novel and the

chief priest of Ulu is the custodian of the timetable of the events of the people.

This timetable of events depends on the moon. His hut is therefore built

differently from other men‘s hut so that it would be easier for him to do his sky

watching.

His obi was built differently from other men‘s hut. There was the usual,

long threshold in front but also a shorter one on the right as you

entered. The eaves on this additional entrance were cut back so that

sitting on the floor Ezeulu could watch that part of the sky where the

moon had its door.

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On sighting the moon, he announces it by beating the metal gong, then women

and children follow suit giving out shouts of joy welcoming the moon. The sky

entities are approached within African tradition with a mixture of feelings; as

shown in welcoming of the new moon by the people in Ezeulu‘s compound[18]:

The little children in Ezeulu‘s compound joined the rest in welcoming the moon

Obiageli‘s tiny voice stood out like a small ogene among drums and flutes. The

chief priest could also make out the voice of his youngest son, Nwafo. The

women too were in the open talking

The people utter their wishes to the moon for protection: ‗Moon may your face

meeting mine bring good fortune‘. This reflects the religious importance of the

moon in Igbo society. The Chief Priest of Ulu then enters his barn, take one yam

from the bambo platform built specially for the twelve sacred yams, roasts one

and eats it with no palm oil and also alone not giving anybody. These are for him

to order the annunciation of dates of the community‘s festivals which is the

significance of the moon to the community. Ezeulu as the Chief Priest is only a

messenger for the god – Ulu and he is supposed to be very careful to avoid the

wrath of the gods. The counting of the moon is therefore not just done in Igbo

society to fulfill all righteousness; rather it is an important timing event which is

to be taken serious by both the living and the dead.

‗Moo‘ ―I‖ said the senior wife Matefi, ‗May your face meeting mine bring

good fortune‘.

‗Where is it‘ asked Ugoye, the younger wife. ‗I don‘t see it or am I

blind?‘

‗Don‘t you beyond the top of the ukwa tree? Not there. Follow my

finger‘.

‗Oho, I see it. Moon, may your face meeting mine bring good

fortune. But how is it sitting I don‘t like its posture‘.

‗Why?‘ asked Matefi

‗I think it sits awkwardly - like an evil moon‘.

‗No‘, said Matefi. ‗A bad moon does not leave anyone in doubt.

Like the one under which Okuata died. Its legs were up in the air‘.

368



In Nsukka area these priests known as Atama are often the most influential

men in the towns as the keepers of the calendar for their communities. These

priests examined the motions of the sun, moon and planets, in some cases, to

come up with the calendar. Each lunar month has a name, a ritual associated with

it and also an economic activity specially connected with it. Each moon (month)

has seven market weeks (Izu asaa) and each izu is four days, that is Eke, Oye

(Orie), Afor and Nkwo. The Igbo year consists of twelve lunar months (354 days)

and as this falls short of the solar year by eleven days it is necessary to add a

thirteenth month from time to time so as to make the year correspond with the

seasons. The thirteenth month, when introduced is usually a ―nameless‖, ―void‖ or

―lost‖ month.

The calendar of Umuawulu (Anambra State) is as shown below[19]. The

yam-planting controls the timing of all the festivals in Umuawulu. The significant

thing is that Onwa mvu is the beginning of the farming season for all Ebeteghete.

It must be properly synchronised with the coming of the rains, or else the whole

clan would be ruined for the whole year.

Table 1: The traditional calendar of Umuawulu in Anambra State

Moon 1.1.1.1.1 Event 1.1.1.1.2 Calendar

Onwa Mvu (I) Ogugu Aho Afor before Moon

appears

May/June

Onwa Ibo (II) Onwa Ibo feast Afo ,June

Onwa Ito (III) Akwali Chukwu feast

Akwali Omufu Ritual

Iba na Akwu-Ozo

July

Onwa Ano (IV) Ikpa Unwu ritual August

369



Onwa Ise (V) Fejioku ritual

Okike-Onwa-Ise feast

Okpukpo Ngu

Oye

Eke

Oye, September

Onwa Ishi (VI) Nshi Ji

Ikpo Ngu

Afo

Oye, October

Onwa Isa (VII) Mgba Ajana

Egwu Aruja

Nkwo

Eke, November

Onwa Asato (VIII) Olili Kamanu

Onwa Asato feast

Ikpo Ngu Onwa Asato

Okike Onwa Asato

Izu n‘ese onwa

Afo (izu ato onwa)

Eke

Eke (izu ise onwa),

December

Onwa Iteghete (IX) Olili Onwa teghete

(women feast)

Afo, Izu n‘ato onwa

January

Onwa Ili (X) Okpukpo Oye feast Oye, Izu n‘ato onwa,

February

Di Okpala Onwa Ili (XI) Obele Ede (women) Eke, March

370



Onwa Uwhoro I (XII) Nnukwu Ede Afo, April

Onwa Uwhoro II (XIII) – do – May

In the old Nsukka Division, the Attama Ugwu-Eka of Amube (Enugwu-

Ezike) is reputed as the most note-worthy attama in the area[20]. His influence

extended to Iheaka, Iheakpu-Awka, Uhun‘Owerre, and Ovoko. As the keeper of

the calendar in the whole district, he worked in conjunction with the rain-maker

(Onyishi Igwe of Iheaka) to whom he gave secret instructions as to the date at

which rain-producing or rain-stopping rites might be performed with a reasonable

chance of success.

Figure 3: Uli sacred writing at Eha-Alumona of Nsukka by a 90 year old lady who learnt it

from her own mother. The Igbo sign of life, Onwa Zenke (the shining moon), Other moons,

planets, stars, etc are depicted. Source: J.A. Umeh [21]

371



The Igbo has a mystical symbolism understood and used by dibias. Some

of these mystical symbolisms have information on the solar and astral systems

buried in them as in the Uli writng shown in Fig. 3.

4. CONCLUSIONS

Nigeria's endogenous astronomy is as diverse as her over 300 ethnic groups. Most

of the ethnic groups have astronomy-rich cultures, hence there are hundreds of

ethnic cosmogonies and mythologies that need to be studied systematically. So,

while modern astronomy may be quite new and unpopular in Nigeria, ancient

architecture, folklore, myths, religion, calendars, etc. are quite rich in astronomy.

Part of our efforts in the African Cultural Astronomy Project is to unearth the

body of traditional knowledge of astronomy by peoples of the different ethnic

groups in Africa and to re-interpret this body of knowledge in the light of

modern/western astronomy. We hope that through this, we would be able to

understand the ways and degrees through which this knowledge and beliefs

shaped the lived realities of the people of Africa; and then add to our

understanding of African scientific practices, which can be used to augment

science education.

REFERENCES

[1] M. Wertheim: New Scientist 156(2101), (1997) p. 28.

[2] Kunene, M., ―The Relevance of African Cosmological Systems to African

Literature‖. (African Literature Today. Edited by D.J. Eldred) (Heinemann,

1980), p. 190.

[3] Wa Thiong‘o, N, ―The River Between‖ (Heinemann, 1965), p. 28.

[4] W.A. Umezinwa & A.O.E. Animalu: From African Symbols to Physics

(Nsukka, Ucheakonam Publ., 1988) p. 1.

[5] F. Capra: The Tao of Physics (Suffolk, Fontana/Collins, 1979)

[6] E. Obiechina: Culture, Tradition and Society in the West African Novel.

(London: Cambridge University Press, 1975) p. 47

[7] K.E. Senanu and T. Vincent: A Selection of African Poetry (Britain:

Longmans, 1976)

372



[8] C.P. Celebre & B.M. Soriano: Promotion of Astronomy in the Philippines‖.

IAU Commission 46 Newsletter 54 (2001) p. 7.

[9] N. Campion: Introduction: Cultural Astronomy in Astrology and the

Academy, ed. N. Campion (Bristol, Cinnabar Books, 2003) p. xv

[10] Bulletin of the Public Information Office, National Radio Astronomy

Observatory, Socorro.

[11] National paper of the Federal Republic of Nigeria presented at the third UN

Conference on the Peaceful uses of Outer Space (UNISPACE III) (1999).

[12] P.N. Okeke: 1999, Basic Space Science and Technology in Nigeria in the

21st Century (Preparations for a take-of). A public lecture organized by the

Nigerian Academy of Sciences.

[13] N. Campion: Editorial in Culture and Cosmos, Spring/Summer 1997, Vol. 1,

no 1, p.2

[14] A.J.N. Tremearne: Hausa Superstitions and Customs (Frank Cass & Co.,

London, 1970), p.114

[15] A.J.N. Treamearne: Ibid, p. 116 – 117.

[16] C.K. Meek: 1930, unpublished Ethnographical report on the peoples of

Nsukka Division, Onitsha province, (National Archives, Enugu), paragraphs 220

– 223.

[17] C. Achebe: Arrow of God (Ibadan: Heinemann, 1989) p. 1

[18] C. Achebe: Ibid, p. 2

[19] N. Nkala: Traditional Festivals in Umuawulu: A survey. M.A. Project

Report. Department of English, University of Nigeria, Nsukka, 1982.

[20] C.K. Meek: 1930, op cit. paragraphs 107 – 111.

[21] J.A. Umeh: After God is Dibia, Vol 2. (Karnak House, London, 1999) p. 209

373



ICHI LINGUISTIC GEOMETRY AND EVOLUTION§§§§§§§§§

A Commentary on They Lived Before Adam

A. O.E. Animalu1 and Catherine Acholonu2 1Department of Physcs & Astronomy and Institute of African Studies,

University of Nigeria, Nsukka, Nigeria. email: [email protected]

2Catherine Acholonu Research Centre, Abuja, Nigeria:

email: [email protected]

Abstract

In our continuing search for a unified world view for African renascence in the

21st C, we present a commentary on the recently published book entitled They

Lived Before Adam (TLBA) which has used inter alia esoteric literature to

characterize the linear linguistic writing codes (Mega-Igbo) and facial

scarification (―ichi‖) patterns of the African people that produced the prehistoric

Ikom monoliths and Igbo-Ukwu bronzes discovered by archealogists in

Soutrheastern Nigeria in the 20th

C. From the suggestion in TLBA based on the

theory of evolution that Eve‘s ―bloodline‖ must have been the homo-

erectus/descendants which predated the creation of homo-sapiens (Adam‘s

―bloodline‖) by the Nephilim (creating God), we abstract the geometric principles

underlying the ―ichi‖ linear linguistic writing code invented by ―Eve‖ and

preserved in the Ikom monoliths. We also relate the periodic hexagram structure

on the surface of the prehistoric Igbo-Ukwu bronze tyre – a torus – to the

Emeagwali ―hyper-ball‖ generated by solving a nonlinear wave equation by finite

element method with triangular elements over a rectangular domain to show how

close the pre-historic African world view was to Steven Hawking‘s 20th

C ―torus‖

model of a universe (without boundaries in space and time) derived by using

quantization to eliminate the ―black hole‖ singularity of Einstein‘s general

relativity theory of gravitation.

§§§§§§§§§ African Journal of Physics Vol. 2, pp. 373-393, (2009)




374



The artefact in this coverpage is Igbo-Ukwu bronze

cast of facial scarification (“ichi”) on a devine feminine

(“god-woman”) mother of the Nri (“god-man”) race of

never-been-ruled people that lived since 1.6 million B.C.

ago

375



1. INTRODUCTION

We wish in this article to develop ―Ichi Linguistic geometry and evolution‖ on

the basis of the assertion on p.128 of Acholonu et al book entitled They Lived

Before Adam (hereunder referred to as TLBA) that "Eve or Shi in [2-dimensional

(2D)] geometry meant the side of a square or triangle". In 3D the triangle is

replaced by the tetrahedron as the simplest of the 5 Pythagorean solids and its

primitive geometric elements (points: lines:planes) define an ichi ratio (4:6:4) of

symbolic ―Eve‖ (cf Dan Brown‘s The Da Vinci Code (5:8:5) for a pyramid).

Apparently, Ichi writing began with drawing lines to represent a (dual) pair of

―N‖ in three dimensional (3D) space whose nesting gives the tetrahedron as

shown in Fig.1.

In quantum physics, this geometry characterizes the process, (proton+antiproton =

light), discussed in chapter 4 of Animalu & Umezinwa‘s book, From African

Symbols to Physics (1988). It will be proved in Sec. 4 of this paper that the

evolution (motion in time) of this tetrahedron and its dual generates the X-ichi

scarification pattern (see coverpage) invented by the divine feminine (―Eve‖), a

heathen ―goddess‖ progenitor of an African race of ―dwarfs‖ ( Fig.2a).

Fig. 1:Nesting of two N‟s in 3D to form a tetrahedron (“light”).

(Goddess)

4:6:4

Fig. 2a: Tetrahedron-Ichi initiated by “Eve”

376



The celebrated Nigerian poet, Christopher Okigbo (1932-1967), in his poem

entitled Watermaid: And the Waves Escort Her and the artist, Obiora Udechukwu

in his homage to Okigbo capture the ―Eternal Day‖ of ―Eve‖ as shown in Fig. 2b.

Our task in this paper is to characterize this geometric framework of Ichi

writing and evolution into modern alphabet systems of various languages.

Fig. 2b: Watermaid

377



2. ICHI WRITING CODE

As stated on p. 155 of TLBD ―The Nag Hammadi says that ‗the entities Father,

Mother and Child exist as perceptible speech‘ having within it three ‗aspects,

three powers, and three names abiding in three NNN, three quadrangles, secretly

in ineffable silence‘ [of Ikom stone carving(?) or mathematically (Fig. 3a) in the

solid geometry of a (dual) pair of tetrahedron & symbolized (p.177 of TLBD) on

a Goddess (Fig.3b).

Fig. 3a: The 3D nesting of 3N‟s into a (dual) pair of tetrahedra with a

common triangular base having ichi geometric ratio (points:lines:planes)

=(5:9:6).

2

3

2

1

1

3

378



Evolution of the tetrahedron and its dual lead to creation of the pyramid

symbolized by its 2D ―shadow‖ (pentagon, in Fig.4a) or alternatively by its 3D

form (octahedron Fig.4b) equivalent to 3 rectangles & a cross as in Ichi.

Shishe goddess

4 3

2 1

4

3

1 2

Ikom Monolith

Fig. 3b: Relation of the tetrahedra to the three symbols, Palm frond

V-ichi Quadrangles & Double concentric circles (see Fig.9 below &

p.177, TLBA).

379



Fig. 4a: Construction of the pyramid from two tetrahedra (above)

&the pre-Renaissance Europe pentagon symbol of “Adam” as

“shadow” of the pyramid.

Fig. 4b: Building of octahedron from a pair of pyramids from 3

quadrangles (external) and a cross (within) transcending ichi (Umudioka)

design on Igbo-Ukwu Bronze (p.82 & 104-6, TLBA).

380



We have stated (as in Chapter 5 TLBA) that inasmuch as the lines of the solid

geometry of the tetrahedron and the octahedron (dual of a cube in Fig.5a)

generate perceptible speech symbols such as NNN, they provided the key to the

correlation of the linear form of stone column writing of Mega-igbo with the

Ogam code in the British Isles. The number of lines in a cluster and their positions

relative to the stem (as in Fig.5b) form the individual letters of the Ogam

Alphabet. Intriguingly in 3D geometry, the sample 20 Igbo column writing and

20 Ogam alphabets correspond to the number of the primitive elements

(points+lines+planes) = 5+9+6 =20 of the two tetrahedral on common base

(Fig.5a) ; for the octahedron (two pyramids on common base) the number is

6+12+8=26 which is the same for a cube (its dual in 3D), 8+12+6=26. These

provide mathematical closure test of alphabet systems!

Fig.5a: Association of number of Ogam Alphabets with number of

points, lines & planes of geometric objects.

Fig.5b( i) 20 Igbo column writing (excluding the single 5 strokes symbol);

and (ii) 20 Ogam Alphabet Codes(with inconsistent(?) inclusion of 5 strokes).

(ii)

(i) 5 strokes 5 strokes

381



It is not surprising, therefore, that Roman numerals, I, II, III, IV, V, VI, VII, VIII,

IX, X, … are made up of such strokes; and the English 26 alphabets can be

associated with the 26 primitive geometric elements of a cube as shown in Fig.6.

Moreover, it is of historical interest to note that the 1854 Igbo Standard alphabets

published by Lepsius had 20 alphabets which Rev. Samuel Crowther increased to

21 by adding N in 1857 for the British expedition to Igboland led by Captain

Bieke. The failure of the various additions to the Igbo autography to form a

geometrically closed system led to the 21st C autography initiated by Alex

Animalu and co-authors under the African Multlingual Project at the Institute of

African Studies of University of Ngeria, Nsukka (see, fig.7)

i j z

c

d

l

f

y

x

t n

p v

o

w

g e

m u k

r

h

s

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z.

Fig. 6: Association of the 26 English Alphabets system with 8 vertices (points),

12 edges (lines) and 6 phases (planes) of a cube, giving an ichi code (8:12:6).

Fig. 7: Two oppositely faced

English Alphabet Cubes

(above) define one (38)Igbo or

Hausa alphabets polyhedron

with ichi code (12:18:8)

x

q

A B C D E F G H I J K L M N O P

(Q) R S T U V W (X) Y Z; I N O U GB GH GW KP KW NW NY SH

38 IGBO ALPHABETS

A B C D E F G H I J K L M N O P

(Q) R S T U V W (X) Y Z; B D K Y FY GW GY KW KY KY SH TS

38 HAUSA ALPHABETS

382



3. APPLICATIONS OF THE ICHI LINGUISTIC GEOMETRY

An important aspect of the Ichi Linguistic Geometry is that it can be applied in

the manner illustrated in Figs. 8a to construct ichi mathematical equation and in

Fig. 8b & c to determine the ratios (points:lines:planes) for the alphabet system of

any language, such as the 28 alphabets in Arabic and Greek languages, and 30

ancient Egyptian aphabets (including the two centres of the two pair of pyramids).

Fig. 8a: In 3D, the tetrahedron defined by 4 points (WORD) = Tetrahedron

defined by 6 lines (CHRIST). This is equivalent to Ichi mathematical equation

for John 1 verse 1: “In the beginning was the Word, the Word was with God,

the Word was God(Christ)”, p.154 TLBA.

H

S T

C

R

I

W R

O

D

Fig. 8b: A pair of inverted tetrahedra define the 28 Arabic Alphabets,

with ichi geometric code 2x(4:6:4)=(8:12:8).

383



4. ZERO IN ICHI LINGUISTICS GEOMETRY

In Fig. 3a we have characterized the primeval (dual) pair of tetrahedra by three

sets of lines forming 3N‘s (Mother[Nne], Father[Nna], Son[Nwa]). In 2D space,

Fig. 8c: 30 Ancient Egyptian Alphabets

384



the geometric principle of duality allows us to represent a point in two ways,

either as a ―point‖ (per se) or ―intersection of two (non-parallel) lines‖, while a

line is a ―line‖ (per se) or ― join of two points‖. In the (2+1)D geometry of space-

time physics, a point is a ―point circle‖ (x2+y

2=t

2=0), i.e., a circle of ZERO

radius( t=0). This may be rewritten as intersection of two lines x = ±iy, with pure

imaginary slopes (i= 1 ) which (by principle of duality) define a triangle

circumscribed by a circle and inscribed in a circle: these are the two concentric

circles shown below in Fig.9.

One may, therefore, assert that Zero is a female. It is Mother. It is Eve. It is ever

in flux, appearing and disappearing from one tip of one tetrahedron to the tip of

the dual tetrahedron on a common base, generating a pair of triangles or

trigrams.

Inasmuch as ZERO time is represented in Fig. 9 by a line, it should not surprise

the non-mathematician that linear time (t) can be transformed into circular time

( ) (see, Fig. 10) via the Mobius transformation:

)1( ,

0

00

iit

itt

under which the line,

00 itit , becomes

a circle,

0 .

Fig.9: Representation of zero (O) as a “point” circle (at zero time) in the

complex xy-plane and generation of a pair of triangles or trigrams to which

we shall return in Sec.6.

O O O

x=+iy

x=-iy

t=0

385



We may, therefore, sum up by saying that, in the beginning (―Zero‖ time), there

was a single ―point-particle‖ (O) coincident with its ―dual point-particle‖ (O) and

a corresponding pair of tetrahedra (symbolizing the Mother goddess) each

inscribed in a sphere (physical wave that moves with the particle) and

circumscribing another sphere (phase wave that moves ahead of the particle); the

projecting (triangular) arms of the tetrahedra define a triagram and generate an X-

ichi pattern as O freely moves forward in time and O freely moves backward in

time.

This apparently completes the characterization (p.156 TLBA) of the Garden of

Eden, as a location belonging to ―Eternal Day‖ when God lived among men and

fed them God-substance!

planein Line t

0

0

planein Circle

Fig. 10:Transformation of line into circle with motion of O forward

and O backward in time generating the X-ichi.

386



5. SYMBOLIZATION OF THE CREATION AND FATE OF ADAM

Chapter 4 of TLBA described ―the creation of Adam as a work of genetic

engineering‖ that began with the conversation of the Nephilim (creating god):

―Let us make man in our own image‖. Of the 16 names of God in the Bible (see

Table below), only ―Elohim‖ refers to the eternal creation.

387



Our interest lies, however, in the startling evidence for ―genetically engineered

Adam‖ based on the correspondence between the pentagon symbol of Adam

given by the 2D ―image‖ of a pyramid (see, Fig. 4a) and the internal molecular

structure of DNA that carries the genetic code of life as shown in Fig.11.

How does this explain the sad fate of Adam – namely his short life? The creation

of Adam corresponds to the second level of evolution involving self-replication of

the tetrahedron with ichi code (4:6:4) and fusion with its replica tetrahedron into a

pyramid having ichi code (5:8:5) which is not one of the five Pythagorean solids

of 3D space. The third level involves a fusion of two pyramids to form the

octahedron with ichi code 2x(5:8:5) = (10:16:10) which (it so happens) is one of

the perfect Pythagorean solids and 10+16+10 = 36 is the number of the 20th

C

Igbo language polyhedron (excluding X and Q in Fig. 7). The fourth level

involves a fusion of two cubes (duals of the octahedron) into the polyhedron in

Fig.7 with ichi code (12:18:8) giving 12+18+8=38 as the number of the new 21st

C Igbo language alphabets.

Fig. 11: Replication of pentagon symbol of Adam in the

structure of DNA that carries the genetic code of life.

Adam DNA

388



6. THE MYSTERIOUS PREHISTORY OF THE INTERNET

One of the deepest geometric-linguistic-scientific messages (TLBA Chapter 32)

is the sketch of the mysterious prehistory of the Internet linking Igbo Ukwu

bronze ―enigmatic tyre‖ (or what mathematicians call a ―torus‖) and Philip

Emeagwali‘s model of the ―hyper-ball‖ (p.447 TLBA) shown in Fig. 12.

Fig. 12:Igbo-Ukwu bronze torus above; Emeagwali hyper-ball below

389



Emeagwali has claimed that his ―hyper-ball‖ inspiration came from observation

of the ―honey comb‖ which (see, Fig. 13a&b) represents

0 1 2 3 4 5 6 70

1

2

3

4

5

6

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Figure 13b: shows a contour plot of linear mode shown in figure 13a

having the “ichi” hexagram evolution pattern (see Maclin & Noel 2009)

01

23

45

67

0

2

4

6

8

-1

-0.5

0

0.5

1

Time=10 Color: u Height: u

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Figure 13a: shows a normal mode (“honey comb”) for a linear

wave equation. The solution was obtained by finite element

method with triangular elements over a rectangular domain.

390



the normal modes of a wave equation with triangular elements over a rectangular

domain (A.P. Maclin and M. Noel, 2009). One can see from this that the

interconnected system of hexagrams is a mathematical model of non-linear

communication wave propagation associated with a pyramid built from four

triangles and a square (base). A geometrical construction of Emeagwali‘s

hexagrams/pentagrams from the motion of the two triangles associated with O

and O characterized in Fig.9 is illustrated in Fig.14.

Indeed, Emeagwali‘s hyper-ball is equivalent to a torus formed by applying what

solid state physicists call periodic (cyclic) boundary condition on a square lattice

of such hexagrams to obtain ―quantization‖ of the wave vector of waves

propagating in such a lattice. Intriguingly, the ―torus‖ model of the universe was

derived by the celebrated British physicist – Steven Hawking – of Cavendish

Laboratory, University of Cambridge, U.K., by unifying Einstein‘s general

O

O

Fig. 14 : Construction of Emeagwali’s hexagrams from

the dual particle/wave motions characterized in fig.9.

391



relativity theory of gravitation and the quantum theory. The main result is that the

torus model (without boundary in space and time) eliminates the ―black-hole

singularity‖ usually associated with the ―Big Bang‖ theory of creation.

This gives meaning (in Fig. 15a) to the symbolic ―snake‖ totem of the ancient

civilization that produced the Igbo-Ukwu bronze: it is this totem that got trapped

in the rectangular box totem of Medieval Christian world view imposed on

African peoples by European imperialism as recounted by Chinua Achebe in the

Arrow of God (1969). The toroidal African world view is celebrated by women

through their ornaments (Fig.15b).

7. CONCLUSION

This paper has been propelled by the clarion call in TLBA for complementarity of

opposites as a universal dual principle of nature at the very foundation of creation

evident in the prevalence of the principle of duality in diverse cultures (see,

Fig.16). A similar appraoch is that of F. Capra‘s The Tao of Physics (1979).

Fig.15b: Celebration

of “toroidal” African

world view by an

African woman

through her

ornaments.

Hawking’s

torus

“Box” trap as

Quantization

Volume “box”

Achebe’s

snake in

a box

“Snake”

as Plane

Wave

Fig. 15a: Elaboration of analogy of thought

between Achebe “snake in a box” and “plane

wave quantized in a box” leads to an

abstraction from the “cyclic” African world view

of Hawking’s unified Einstein’s general

relativity and quantum mechanical view of the

universe as a torus (without boundary in space

and time) proposed by Hawking in his book, A

Brief History of Time (1990).

392



Our objective has been to eliminate linguistic ambiguities inherent in the two

dimensional nature of the written word by recasting the ideas in readily

understood two and three dimensional geometric forms which we have called ichi

linguistic geometry because it is the language used by the authors of the Ikom

monoliths and Igbo-Ukwu Bronzes thousands of years ago. We have made

evident the scientific underpinnings of the ichi linguistic geometry by unraveling

the mystery of the Igbo-Ukwu bronze ―torus‖, to show how close it is to our

modern view of the universe.

REFERENCES

Achebe, Chinua, Arrow of God (Heinemann Kenya, 1969).

Acholonu, Catherine: They Lived Before Adam (CARC Publication ISBN:

978=2579-51-10 (2009) referred to as TLBA.

Animalu, A.O.E. and Umezinwa, W. From African Symbols to Physics (Desk

Published by Ucheakonam Foundation (Nig.) Ltd) (1966).

Animalu, A.O.E., Achufusi, G.I., Umezinwa, W. and Jeff Unaegbu, Nelson

Mandela and Barack Obama African World Challenge (Ucheakonam Foundation

Nig. Ltd (2009) ISBN 978 34207-7-1.

(1) (2)

(3)

(4)

(5)

(1) (2)

(3)

(4)

(5)

Fig. 9c: Principle of Duality in different cultures

(1) Egyptian pyramid, (2) Chinese Tao, (3) Greek Janus,

(4) Jewish Star of David, and (5) Akuebilisi seed worn

by a dangerous masquerade in Igboland (in Africa).

393



Animalu A.O.E. and Azali, A.O. 21st C Igbo/English Primer; Animalu, A.O.E. ,

Otagburuagu, E.J., Abdullahi, S.Y., Lawal I. and Tivde, T. 21st C Igbo-Hausa-

Yoruba-Tiv/English Primer (Ucheakonam Foundation 2009).

Capra, F. The Tao of Physics (Fontana/Collins, 1979).

Dan Brown, The Da Vinci Code; see also by the same author, Angels & Demons

(Washinton Square Press available in paperback (2009)).

Hwaking, S., A Brief History of Time (Bantam Books, New York (1990)).

Maclin, A.P. and Noel, M. Nonlinear Magneto-optical Effects in Dielectrics

Embedded with Ferromagnetic Nanoparticles (in Proceedings of 2nd

International

Seminar on Theoretical Physics and National development, July 5-8, 2009, Abuja,

Nigeria: African Journal of Physics Vol. 2, 149-156, (2009).

394



A MODEL OF ECONOMIC GROWTH AND DEVELOPMENT IN

THE 21ST CENTURY**********

1Tertsegha Tivde and A.O.E. Animalu

Department of Physics & Astronomy and Institute of African Studies

University of Nigeria, Nsukka, Nigeria

1Email: [email protected]

Abstract

In this paper, we present a model based on population dynamics for correlating

national economic growth and development currently measured by the United

Nations Human Development Index (HDI) and Energy Consumption per capita as

a measure of manufacturing value added (MVA) capability of a nation. The

model explains why the 20th

C world energy and materials ―crises‖ caused by

over-dependence on non-renewable Earth‘s energy/materials resources and the

current state of MVA capability of both industrial and developing nations were

not only inevitable but have persisted and dovetailed into economic ―meltdown‖

into the 1st decade of the 21

st C with dire consequences for the entire world. To

cope with the exigencies of the 2nd

decade of the 21st C and beyond, the model

dictates that the world requires not only breakthrough(s) in scientific and

technologogical innovation (in industrial nations) and good governance (in

developing nations, especially Africa) but also investments in empowerment of

human capital (especially youths) for productivity through systematic

development of new and renewable energy resources, especially solar energy, as

well as reciprocal rather than exploitative socio-economic forces operating

within/among member states of the United Nations in general, and African Union

in particular.

**********African Journal of Physics Vol. 2, pp.394- 401, (2009)



395



1. INTRODUCTION

National development is usually understood in terms of development

economics. This was the subject-matter of a year-2000 published book entitled

Development as Freedom[1], written by Amartya Sen, the Indian born 1998

Nobel Laureate in the field of development economics. Sen is a Professor of

Economics and Master of Trinity College at Cambridge and his unrivalled

reputation among academics can be measured by the fact that he probably holds

the highest number of honrary degrees among his peers and he helped create

United Nation‘s Human Development Index (HDI) which has been the most

authoritative international source of welfare comparison between countries in the

20th

C. The three parameters of socio-economic life that measure HDI ( ) are

life expectancy index ( i ), education index ( i ) and per capita gross domestic

product, ( gdpi ), where )( i31

gdpii and 10 . For example, in 1998

and 1999, Nigeria had an overall HDI of 0.400 which placed her as 137th

out of a

total of 174 countries of the world in 1993. In terms of HDI, underdevelopment is

associated with low HDI and what Sen calls ―unfreedoms‖ such as poverty,

ignorance, hunger, ill-health, racial discrimination and gender discrimination, as

well as political, social and economic oppression, while high HDI such as 0.944

for USA is associated with ―expansion of freedom [as] both the primary end and

the primary means of development‖. For this reason, in Development as Freedom,

Sen holds the view that national economic growth ought to be measured less by

material output and more by the capacity and opportunities it enables people to

enjoy, or in other words, it gives people to do and to be. He laments that the

discipline of economics ―has tended to move away from focusing on the value of

freedoms to that of utilities, incomes and wealth‖. For this reason, he disagrees

with the so-called ―Lee Thesis‖ (after Lee Kuan Yew of Singapore) which holds

that authoritarian governments are able to promote faster growth than democratic

ones – a thesis that African military governments in the 20th

C bought to the

detriment of the ethical dimension of national life in African countries.

In the year 2000, HDI shifted to manufacturing value added (MVA)

capability of a nation, because it has been conclusively established that there is a

linkage between the standard of living of any people and the MVA potential per

capita; and MVA is, in turn, linked to technological capacity and hence energy

consumption per capita. The poverty in Nigeria can be seen from the fact that

whilst the average MVA per capita for sub-Saharan Africa was US$40 in 1997,

Nigeria‘s MVA was US$17 per capita. In 1997 the contribution of Nigeria to the

MVA of sub-Saharan Africa was 8.7% behind South Africa, Zimbabwe, Cote

d‘Ivoire and Cameroon.

396



In this paper, we wish to review models of the empirical relationship between

HDI and energy consumption per capita (in Sec. 2) as a prelude to a formulation

(in Sec. 3) of a model based on population dynamics for correlating economic

growth (measured by HDI) and MVA (measured by energy concumption per

capita) of both induatrial and developing nations. In Sec. 4 the model will be used

to explain why the 20th

C world energy and materials ―crises‖ caused by over-

dependence on non-renewable Earth‘s energy/materials resources and the current

state of MVA capability of both industrial and developing nations were not only

inevitable but have persisted and dovetailed into economic ―meltdown‖ in the 1st

decade of the 21st C with dire consequences for the entire world: these will lead

us, by way of conclusion, to recomendations for coping with these crises in the 2nd

decade of the 21st C and beyond.

2. REVIEW OF 20TH

C MODELS OF THE RELATIONSHIP

BETWEEN HDI AND ENERGY CONSUMPTION PER CAPITA

As a consequence of the 1973 world‘s ―energy crisis‖ alarm and its linkage to

the socio-economic landscape, the use of energy consumption per capita as a

measure of the HDI of the member states of the United Nations has been

undertaken in the 20th

C from various perspectives, such as global warming,

population growth and environmental degradation, sharply rising oil and gas

prices and rapid depletion of their supplies, armed conflicts in regions with major

oil deposits, higher energy costs to poor nations seeking to develop higher

standards of living, and a growing apprehension that American currency may be

undermined by a sudden lack of confidence brought on partially by instability of

world energy supplies. These studies showed that in 1979, the world annual per

capita energy consumption peaked (―Hubbert‘s Peak‖) and it has been decreasing

steadily thereafter (see, Fig.2.1). Apparently, the growth rate of population has

outstripped the growth rate of energy production, and this trend continues as both

world population and world energy production increase. The correlation of energy

consumption with MVA capability as represented by the world history of energy

production per capita is shown in Fig. 2.2.

397



Fig. 2.2: Correlation of Energy Consumption with world’s history of energy

production (i.e., Scientific & technological innovations, e.g. electric lighting

bulb., computer, etc). from Manuel Garcia, Jr (2006)[2].

Figure 2.1. World Annual Per Capita Energy Consumption 1950-1996

[BOE = barrel of oil equivalent], from Manuel Garcia, Jr (2006)[2]

398



During this first decade of the 21st century, the world‘s political economy

has been at the crest of the Hubbert‘s Peak; but today‘s efforts to maintain an

unprecedented rate of oil and energy production cannot be sustained for obvious

reasons. The world emerged from World War-II with over 90% of its oil still

untouched; oil depletion was about 10% in 1970; 50% in 2000 and is projected to

be 90% in 2030. It is expected that 80% of all proven world‘s oil reserve will be

used up within a sixty year span from 1970 to 2030. While the total amount of oil

on either side of Hubbert‘s Peak is identical, the downhill side will be a time of

scarcity and high prices because there are more people demanding this resource

than there were during the boom years of the run up. Today, the Chinese and

Indian economies are experiencing rapid growth, and their people are

experiencing a general rising of living standards though not uniformly distributed.

These populations represent over a quarter of humanity, and their combined thirst

for petroleum rivals that of the U.S.A. (with only 4.5% of the world's population).

A sharp downturn in oil production will result in a sharp downturn of the

per capita energy consumption E(y) as a function of the year (y). Inevitably, the

time constant in the E(y) exponential must fall from 300 years to something much

lower, like 30 years; and E(y) at 2030 might be like that of 1950. To visualize

what this might mean, consider the fact that in the U.S.A. about 10 calories of

petroleum-based energy is used to produce every calorie of food energy

consumed which poses the question: how would one adjust to a 50% cut in

available energy? Moreover, the most convenient form of energy is electricity

and the strong correlation between the availability of electricity and the level of

human social development has been known since at least 1895 with the

electrification of Niagara Falls with the then new polyphase alternating current

(AC) technology invented by Nikola Tesla. So, if "human development" as

measured by social and economic well-being depends on availability of

electricity, how sustainable will generation of electricity be in the future?

To answer these questions, it is necessary to construct a model based on

population dynamics that correlates the United Nations Human Development

Index (HDI) with energy consumption per capita to which we now turn.

3. MODEL OF ECONOMIC GROWTH AND DEVELOPMENT

In this section we proceed to set up a model of the relationship between HDI

and energy consumption per capita for understanding past and current trends and

for predicting future trends. We begin by observing that although HDI is defined

399



by the empirical formula )( i31

gdpii where 10 , each of the three

components ),,( i gdpii depends on the energy consumption per capita (E), i.e.

)(E . Since Earth‘s resource may be considered fixed, the rate of change of

with respect to E may be determined by the logistic model of population

dynamics, in which an unchecked consumption of non-renewable energy resource

grows exponentially with the population. However, since 1 its increase with

energy consumption per capita is limited in the form that may be expressed by

the Riccati equation (cf Tivde and Animalu[3]):

,1

RdE

d (3.1)

This model has a solution of the form:

.1

.(E) REBe

A

(3.2)

where , 0R and the constants A and B are determined by the lower and upper

bounds, )1/((0) 0 BA and 1)( AEM . For, minimum

,3.00 Eq.(3.2) takes the explicit form;

.67.21

1.(E)

REe (3.3)

This leads to an excellent agreement with the empirical data[4] as shown in Fig.

3.1 which we now proceed to discuss.

400



4. DISCUSSION AND CONCLUSION

In setting up the population dynamics-based model of )(E versus E , we

have exploited an analogy with the host-parasite system in a biological system in

which the way of life of the human population (parasites) depends on the

consumption of the energy resources of the Earth (host). This analogy naturally

explains why the 20th

C world energy and materials ―crises‖ caused by over-

dependence on non-renewable Earth‘s energy/materials, especially oil resources,

and the current state of MVA capability of both industrial and developing nations

were not only inevitable but have persisted and dovetailed into economic

―meltdown‖ in the 1st decade of the 21

st C with dire consequences for the entire

world. In 2002, the United Nations had indicated that the electricity consumption

per capita needed in order to support a society with a medium level of human

development was just over 1000 kilowatt-hours. From this point of view, an oil-

producing African country like Nigeria with energy consumption well below 1000

Fig. 3.1: Comparison of the empirical data on HDI (vertical axis) versus

Energy Consumption per capita (for R=1).

401



kilowatt-hours per capita remains poor, inasmuch as the marginal rise of her HDI

of 0.438 in 1988 through 0.438 in 1999 to 0.511 in 2007 ranked her (in 1999)

137th

out of a total of 174 countries of the world in 1993, and whereas in the first

15 years (1960-1975) of Nigeria‘s nationhood, the growth rate was positive

(9.4%), in the second fifteen years (1976-1990) it was negative (2.2%): these

trends were indicative of the past history of bloated foreign debt and high rate of

inflation (ascribable to exploitative neo-colonialism) which resulted in mass

unemployment especially of youths, HIV/AIDS scourge, deteriorated road

infrastructure, epileptic electric power supply, religious riots, ethnic militia, rural-

urban migration, insecurity of life, among other societal ills.

To cope with the exigencies of the 2nd

decade of the 21st C and beyond, our

model dictates that the world requires not only breakthrough(s) in scientific and

technologogical innovation (in industrial nations) and good governance (in

developing nations, especially Africa) but also investments in empowerment of

human capital (especially youths) for productivity through systematic

development of new and renewable energy resources, especially solar energy, as

well as reciprocal rather than exploitative socio-economic forces operating

within/among member states of the United Nations in general, and African Union

in particular.

REFRENCES

[1] Amartya Sen Development as Freedom

[2] Manuel Garcia, Jr (2006), An Introduction Linking Energy Use And

Human Development: EFHD_R_01.

[3] T. Tivde and A.O.E. Animalu, Riccati equation in Biophysics and other

Physical Phenomena, Afr. J. Phys. Vol. 1, p. 154-176 (2008).

402



TWO-PARTICLE CORRELATIONS IN THE ONE-DIMENSIONAL

EXTENDED HUBBARD MODEL: A GROUND-STATE PERTURBATIVE

SOLUTION††††††††††

O. R. Okanigbuan1

, Simon Ehika 1

, S. U. Azenobie 1

,

P. N. Okanigbuan 2

1Department of Physics, Ambrose Alli University, Ekpoma, Nigeria;

e-mail: [email protected];

2Department of Basic Sciences, Benson Idahosa University, Benin-City, Nigeria.

Abstract We present here a detailed study of the behaviour of 2 electrons in an infinite one-

dimensional lattice of the Extended Hubbard model, using perturbation method. It

is shown that for two electrons the results obtained gets better as the positive on-

site coulomb interaction (U) the nearest-neighbour interaction (V) and number of

sites N are increased, provided both the ratio

N

Uand

N

V are made small. In

other words the crucial parameter is not just U, but the ratio of the interaction

strength to the number of sites.

1.0 INTRODUCTION In this paper we present some results of the one-dimensional Extended

Hubbard model, defined by the Hamiltonian,

where ji, denotes nearest-neighbour sites, ii CC

, is the creation

(annihilation) operator with spin or at site i, and

ii

nnn1 where

†††††††††† African Journal of Physics Vol. 2, pp. 402-412 (2009)


i

ii

iii

ji

jiji nnVnnUCHCCtH 1

,

, .

1.1 1


403



. iii CCn The transfer integral jit , is written as tt ji , which means that all

hopping processes have the same probability[2]

. It is worth mentioning that in

principle, the parameters U and V are positive because they are direct coulomb

integrals. However, U and V could be negative if attractive indirect interactions,

through phonons or other bosonic excitations, are included and are stronger than

direct coulombic repulsions. The original Hubbard model corresponds to V=O in

eqn (1.1). The extended Hubbard model has at least two advantages over the

original Hubbard model (1). It has a rich phase diagram[3]

, containing e.g. charge

and spin density waves and various superconducting phases at half filling (II) It is

definitely more realistic; this has been pointed out already by Hubbard[4]

, who

argued for transition metals that the matrix element corresponding to nearest

neighbour coulomb repulsion is relatively large, so that its influence cannot

apriori be neglected.

The extended Hubbard model has been intensively studied previously,

both analytically (at weak[5]

and a strong[6,7]

coupling) and numerically (usually at

half[8,9]

or at quarter[10]

filling). These studies have been carried out primarily in

low dimensions, but some analytical results for the ground state in higher

dimensions are also available[8,9]

. As a result of this work, it is well known that

the ground state of the extended Hubbard model exhibits a transition between a

spin and a charge density wave at UV2

1 in all dimensions.

In this paper, we study the one-dimensional Extended Hubbard model

using Stationary perturbation theory. The validity and convenience of perturbation

theory in the context of the Hubbard model has been proved very recently for

small values of the ratio

.

11

N

U One important aspect of our study will be to

compare our results with those obtained from the correlated variational approach.

This paper is organised as follows. In the next section (2.0) we discuss the

fundamentals of the perturbative method. Results obtained are presented in

section (3.0). In section (4.0) we summarize and discuss the results.

2.0 PERTURBATION THEORY OF THE HUBBARD HAMILTONIAN

The Extended Hubbard Hamiltonian, eqn (1.1) is divided into 2 parts 13 , i.e.

1HHH o

404



ji

iiijo CHCCtH,

.. [1.2]

i i

iiiinnVnnUH 11 [1.3]

We perform a second order perturbation calculation in which the kinetic energy

term defined the starting point and the interaction term 1H is treated

perturbatively. Let us summarise our perturbation procedure.

Firstly, the one-electron Bloch wavefunction that diagonalize oH are

constructed.

j

j

R

j

Rik

k CeL

1

[1.4]

where jR runs over all the sites, L is the lattice size, and k are the allowed wave

vectors for the given lattice. Eqn (1.4) follows Bloch theorem which states the

eigenfunctions of the wave equation for a periodic potential are of the form of the

product of a plane the wave rike and a function rU k with the periodicity of the

crystal lattice, that is,

rk

rik

k Uer [1.5]

The subscript k indicates that the function rU k depends on the wave vector k.

Secondly, many body wave functions of the Hartree-fock type are

constructed,

N

n

kk Onn

1

[1.6]

k

n

nkk1

[1.7]

N

n

n

1

[1.8]

In this way many body wave function are classified according to both total wave

function are classified according to both total wave vector k and spin . In

principle, the number of wave functions increases as the lattice size increases, but

405



this is not true if this set is restricted to those wave functions that provide the

smallest kinetic energy oT for a determined number of electrons N

N

n

no kT1

[1.9]

The Z component of the spin is always fixed to O for an even number of

electrons and ½ in the case of an odd electrons number. Thirdly, wavefunctions

are classified according to their wave vector .k Finally, we construct a second

order Hamiltonian matrices within every k subspaces, this is the ground state

energy matrix,

oo

kkkk

kkokkTT

HHHTH [1.10]

The final form of the ground state wavefunction is

kksg DC. [1.11]

where zero-order coefficient C are obtained from diagonalization of the second

order Hamiltonian matrix eqn (1.10), whereas first order coefficients D are

given by

oo

kk

TT

CHD [1.12]

3.0 RESULTS

From the application of the perturbation method in section 2.0, the following

ground state energies are obtained for 2 electrons on N sites of a one-dimensional

lattice. For N=2,

VUtEg 2 [1.13]

and for ,2N

N

V

N

Ut 424 [1.14]

406



The ground state energy obtained using the correlated variational approach is

given by

22

162

1tVUVUEg [1.15]

for N=2, while for N>2, where N is even

2

1

22

2

1

2

0

1

1

2

28

k

k

i

io

k

oi

ii

g

XXXN

VNXUNXXXtN

E

iX are variational parameters, and .2

Nk

407



Table 1

Difference in the value of ground state energy obtained from perturbation

calculation (Ep ) and variational (Ev ) at U/t = 4 for 2electrons on 2 sites

V/t Perturbation

(Ep )

Variational

(Ev )

Ep-Ev U/N V/N

0 2.0000 -0.8284 2.8284 2.0 0

0.4 2.4000 -0.4907 2.8907 2.0 0.20

0.8 2.8000 -0.1612 2.9612 2.0 0.40

1.2 3.2000 0.1587 3.0413 2.0 0.60

1.6 3.6000 0.4676 3.1324 2.0 0.80

2.0 4.0000 0.7639 3.2361 2.0 1.00

2.4 4.4000 1.0459 3.3541 2.0 1.20

2.8 4.8000 1.3119 3.4881 2.0 1.40

3.2 5.2000 1.5604 3.6396 2.0 1.60

3.6 5.6000 1.7900 3.8100 2.0 1.80

4.0 6.0000 2.0000 4.0000 2.0 2.00

408



Table 2


calculation (Ep ) and variational (Ev ) at U/t = 4 for 2 electrons on 4 sites.

V/t Perturbation

(Ep )

Variational

(Ev )

Ep-Ev U/N V/N

0 -2.0000 -3.4186 1.4186 1.0 0

0.4 -1.6000 -3.2057 1.6057 1.0 0.10

0.8 -1.2000 -3.0046 1.8046 1.0 0.20

1.2 -0.8000 -2.8154 2.0154 1.0 0.30

1.6 -0.4000 -2.6379 2.2379 1.0 0.40

2.0 0 -2.4721 2.4721 1.0 0.50

2.4 0.4000 -2.3178 2.7178 1.0 0.60

2.8 0.8000 -2.1746 2.9746 1.0 0.70

3.2 1.2000 -2.0419 3.2419 1.0 0.80

3.6 1.6000 -1.9194 3.5194 1.0 0.90

4.0 2.0000 -1.8064 3.8064 1.0 1.00

409



Table 3


calculation (Ep) and variational (Ev) at U/t=4 for 2 electrons on 10 sites

V/t Perturbation

(Ep )

Variational

(Ev )

Ep-Ev U/N V/N

0 -3.2000 -3.8622 0.6622 0.40 0

0.4 -3.0400 -3.8340 0.7940 0.40 0.04

0.8 -2.8800 -3.8135 0.9335 0.40 0.08

1.2 -2.7200 -3.7979 1.0779 0.40 0.12

1.6 2.5600 -3.7859 1.2259 0.40 0.16

2.0 -2.4000 -3.7763 1.3763 0.40 0.20

2.4 -2.2400 -3.7685 1.5285 0.40 0.24

2.8 -2.0800 -3.7620 1.6820 0.40 0.28

3.2 -1.9200 -3.7566 1.8366 0.40 0.32

3.6 -1.7600 -3.7520 1.9920 0.40 0.36

4.0 -1.6000 -3.7480 2.1480 0.40 0.40

410



Table 4


calculation (Ep) and variational (Ev) at U/t = 4 for 2electrons on 80 sites

V/t Perturbation

(Ep )

Variational

(Ev )

Ep-Ev U/N V/N

0 -3.9000 -3.99706 0.0971 0.05 0

0.4 -3.8800 -3.99698 0.1170 0.05 0.005

0.8 -3.8600 -3.99693 0.1369 0.05 0.010

1.2 -3.8400 -3.99690 0.1569 0.05 0.015

1.6 -3.8200 -3.99688 0.1769 0.05 0.020

2.0 -3.8000 -3.99686 0.1969 0.05 0.025

2.4 -3.7800 -3.99685 0.2168 0.05 0.030

2.8 -3.7600 -3.99684 0.2368 0.05 0.035

3.2 -3.7400 -3.99683 0.2568 0.05 0.040

3.6 -3.7200 -3.99683 0.2768 0.05 0.045

4.0 -3.7000 -3.99682 0.2968 0.05 0.050

411



4.0 DISCUSSION OF RESULTS

We begin our discussion with the ground state energy obtained for 2

electrons on N lattice chain of the one-dimensional extended Hubbard model.

Even lattice sites, N = 2, 4, 10, and 80 where considered. There is a significant

improvement in the result obtained from perturbation method when N is very

large, say N=80, as indicated in Table 4 and fig. 1.

To summarize, we have shown in our study that accuracy of the results

obtained from the perturbation method is enhanced when the ratio

N

U and

N

V are small. This is a confirmation of the result obtained by Okanigbuan and

Idiodi,[11]

which says that the crucial parameter in the perturbation calculation for

the ground state energy is not just U, but the ratio of the interaction strength to the

number of sites.

Fig. 1: Difference in values of (E/t) as a function of (V/t) between perturbation method

and variational method for U/t=4

412



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Extended Hubbard Model at Half Filling – Physical Review Letters: Vol.

88, Number 5 56402-1-056402-4.

2. Vallejo, E. and Navarro, O. (2003). Two-Particle correlations in the one-

diemnsional Hubbard: a ground state analytical solution. REVISTA

MEXICAMA DE FISICA 49(3) 207-211.

3. Micnas R. Ranninger J. and Robaskiewiez, S. (1990). Rev. Mod. Phys 62, 113.

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5. Emery, V. J. (1979), in Highly conducting in One-Dimensional Solids, edited

by Devreese, J., Evrand, R. and Van Doren, V. Plenum, New York.

6. Bari, R. A. (1971) Phy. Rev B 49.

7. Dongen Van, P. G. J. (1994) Phys. Rev. B 49.

8. Hirsch, J. E. (1984) Phys Rev. lett 53, 2327.

9. Champel, D. K., Gammel Tinka J. and Loh, E. Y. (1990) Phys Rev B 42, 475.

10. Milla, F. and Zotos, X. (1993). EuroPhys. Lett 24, 133.

11. Okanigbuan O. R., and Idiodi, J. O. A. (2008). Ground state energy of the

Hubbard Hamiltonian: Perturbative results. J. Nig. Assoc. Math Phys: 37-

40.

12. Chen, L. and Mei, C. (1989) Exact Calculation of the two electron interaction

in the ground state of the Hubbard model. Phy. Rev B. 39. 9006-9011.

13. Galan, J. and Verges, J. A. (1991). Perturbation theory of the Hubbard

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