electron thepry of metals

7/31/2019 Electron Thepry of Metals

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UNIT2

ELECTRON THEORY OF METALS

2.1 CONDUCTIVITY

2.1.1 Introduction

In an ionic solid, the atom forming a cation gives up its valence electron that is captured by theatom forming an anion. Here the electrons are strictly localized. However in metals, the coreelectrons are localized at the atoms but the valence electrons are delocalized and belong to theentire solid. The behaviour and energy states of these delocalized electrons determine manyproperties of these solids. We shall try to understand the behavious of these electrons and see howthey influence the property of the solid. The simplest approach is the classical free electron modelthat was successful in explaining some of the properties of metals. This was succeded by the more

sophisticated band theory of solids.

The general characteristics of electrical conductors are:

(i) The current density in the steady state is proportional to the electric field strength(Ohms law).

(ii) For pure specimens, the electrical conductivity () and the thermal conductivity (k)vary with temperature as follows:

T1 and k = constant (for T > D)

so that

T

TKis a constant independent of temperature (WiedemannFranz law);

T5 and K T2 (for T < D)

where D is the characteristic Debye temperature. The relation T5

is known as BlochGruneisen T5 law.

(iii) For metals that exhibit the phenomenon of superconductivity, their resistivity disappears

at temperature above 0 K and below the critical temperature (T c) for the superconductingphase transition (T

c= 4.15 k) for mercury.

31


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32 ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS

(iv) For metals containing small amounts of impurities, the electrical resistivity () may bewritten as

= 0 = p (T)

where 0 is a constant that increases with increasing unpurity content and (T) is thetemperature dependent part of the resistivity. This is known as Matthiessens rule.

(v) For most metals the electrical resistivity decreases with increase of pressure.

(vi) The resistivity of alloys that exhibit order-disorder transitions shows pronounced minimumcorresponding to the ordered phase.

Table 2.1: Electrical conductivity and concentration of electrons in some selected

metals

Metal Number of Concentration of Electrical Electrical

valence electrons conductivity conductivity

electrons (Z) in 1028/m3 at 100 K at T = 300K

in W1 m1

Copper 1 8.47 2.9 108 6.5 107

Lead 4 13.2 1.5 108 5.2 106

Zinc 2 13.2 6.2 108 1.8 107

Gold 1 5.90 1.6 108 5.0 107

Aluminium 3 18.1 2.1 108 4.0 107

Cadmium 2 9.27 4.3 108 1.5 107

Iron 2 17.0 8.0 108 1.1 107

2.1.2 Electron Drift in an Electric Field

When an electric field E is applied to a conductor an electric current is established in the conductor.The densityj of this current is given by

j = E

The constant of proportionality is called thespecific conductance or electrical conductivityof

the conductor. Its unit is ohm1 m1. The resistivity, =1

. The flow of current in a conductor

is an indication that the electrons in it, under the action of the applied field, move in a specificdirection. As a result the distribution function of the electrons in the conductor undergoes achange. This directional motion of the electrons is called a drift. The average velocity of this drift

motion is called drift velocity.

The force F acting on an electron in an electric field E is F = eE where e is the electroniccharge. The minus sign introduced because the charge on the electron is negative. The electrondrift is in a direction opposite to that of the applied field. Acted on by the applied field the electron

will have to accelerate continuously. But in its accelerated motion, the electron collides with thedefects in the lattice. As a result of the consequent scattering the electron loses the velocity it gainsfrom the field. The effect of the lattice may be reduced formally to the action of a retarding force.This force is proportional to the velocityv and mass m of the electron. Thus the retarding force maybe taken to be Kmv where K is a constant. We can then write the equation of motion of the electronas


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ELECTRON THEORY OF METALS 33

mdv

dt= eE Kmv ...(2.1.1)

After the field is applied the velocity of the electron rises till the retarding force which isproportional to the velocity equals the force due to applied field. When these forces become equalthe acceleration ceases. Thereafter the electron moves with the drift velocity. If this drift velocity is

vd, then

0 = eE K mvd

i.e., vd

= e

m

E

K...(2.1.2)

The ratio of the drift velocity to the applied field intensity is called carrier mobility.

Thus

Carrier mobility, m =e

mK...(2.1.3)

Suppose that as soon as the velocity of the directional motion of the electrons attains its steadyvalue, the field is cut off. Because of the collisions of the electrons with the lattice defects the velocitystarts decreasing. After some time the electron gas resumes its equilibrium condition. Such a process

which leads to the establishment of equilibrium in a system from which it was previously disturbedis called a relaxation process. When the applied filed is cut off the equation of motion of the electron

becomes:

mdv

dt= Kmv ...(2.1.4)

dv

dt

= Kv

This can be written as

dv

v= Kdt

Integrating

log v = Kt + C ...(2.1.4a)

Let us now find the value of integration constant C using the known conditions.

When t = 0, v = vd

and hence C = |log vd

|C = log vd. Now Eqn. 2.1.4a becomes

log v = Kt + log vd

or v = vd

exp ( Kt)

The time taken by an electron to reach a directional velocity which is1

eof the drift velocity

is called the relaxation time denoted by.

v

ed = v

dexp ( K)


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This is possible or this leads to the result K=1

. Thus Eqn. (2.1.2) becomes

e

m

E

or vd = emE ...(2.1.5)

V

A

Fig. 2.1.1 Drifting electrons

e1

Fig. 2.1.2 Scattering electron at ion cores

Let n be the density of electrons. The charge flowing per unit area per unit time is the currentdensity.

i.e., j = nevd

[Referring Eqn. 2.1.2]

or

j =ne

m

2E ...(2.1.6)

Defining the current density for unit electrial field as electrical conductivity, we get

j

E= or =

ne

m

2...(2.1.7)


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If we take the experimental value of s for copper as 6 107 (-m)1 and n = 8.5 1028/m3, we find t = 2 10-14 see. In discussing electrical conductivity it is convenient to introducethe mobility of charge carriers. The mobility m is the magnitude of the ratio of the drift velocity

to the applied electric field. Thus (Refer Eqn. 2.1.5).

=v e

m ne

d

E= =

...(2.1.7a)

The unit of ism m

sec volt= m2 V1 s1

The electrical resistivity, =1

2 =

m

ne

2.1.3 Temperature Dependence of Conductivity

In most solids the two main types of internal energy are (i) thevibrational energyof the atoms abouttheir mean lattice positions and (ii) the kinetic energyof the free electrons. If heat is supplied to thebody its temperature rises and the internal energy increases. The important thermal properties ofmatter such as heat capacity, thermal expansion and thermal conductivity depend upon the changes

in the energy of the atoms and free electrons. If an electrical field is applied to a solid the freeelectrons are accelerated. Their kinetic energy increases. Some of this kinetic energy is of course lostby collisions with the atoms in the lattice. The resulting current is proportional to the average

velocity of the free electrons. This velocity is determined by the applied electrical field and also thecollision frequency.

We should make a distinction here between the two different velocities associated with theelectrons. The velocity appearing in the Eqn. 2.1.5 is called the drift velocity (v

d) which is superimposed

on a much higher velocity called root mean square velocity (c

) due to the random motion of theelectron which is possible even in the absence of the electric field. The random motion, whichcontributes zero current, exists also in the presence of the field.

It is evident now that in the absence of an electric field the free electrons in a metal will bemoving about at random in all directions and will be in temperature equilibrium with it. If themass of an electron is m then at absolute temperature T it will possess an average random velocitygiven by the kinetic theory of gases. Thus

1

22mc =

3

2kT ...(2.1.8)

where k is theBoltzmann constant. Suppose an electric field E is applied. Under the influence of this

field the electron acquires a drift velocity. The resulting acceleration of the electron ise

m

Ewhere eE

is the force acting on the electron and m is the mass of the electron. The drift velocity is muchsmaller compared to random velocityc. Further the drift velocity is not retained after a collision

with an atom because of the relatively large mass. Hence just after a collision the drift velocity is

zero. If the mean free path is then the time that elapses before the next collision takes place is

c. Hence the drift velocity acquired just before the next collision takes place is


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u = acceleration time

=e

m c

EL

NM

O

QP

Thus the average drift velocity of the electron is

vd

=u e

m c2 2=

E

If the number of electrons per unit volume is n, then the number of electrons crossing unit areafor unit time is nothing but current density, j.

Thus j = nevd

i.e., j = nevd

= nee

mc

E

2

RSTUVW

with c

=3k

m

T

Thus j =ne

m

m

k

2

2 3

E

T

Thus electrical conductivity,

=j ne

mkE T=

2

12

=12

2

mk

ne

T

...(2.1.9)

These two expressions are independent of current. Hence at constant temperature the electrical

conductivity (i.e., the ratio of the current density to the applied field) is a constant. In other wordsat constant temperature the applied p.d is directly proportional to the current. This is Ohms law.

2.1.4 Impotentialness of Classical Free Electron Theory

Some of the drawbacks of classical free electron theory are listed below:

1. Classical theory failed to explain the variation of electronic specific heat at low temperatures,on the basis of Drudes model, it was concluded that the molar specific heat of electrongas for any metal was 3R (12.5 kJ/kmol). This value is many times the magnitude ofspecific heats found experimentally.

2. The mean free path of electrons in any metal, calculated on the basis of Drudes model

was ten times less than the experimentally observed value.

3. The Lorenz number, L calculated for many metals using the standard relationW

T=L =

22

3

k

e

FHG

IKJ is equal to 2.45 10

8 watt m/K2 comes to be a constant. However, for many


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metals, theLorentz number varies with temperature at low temperatures. This is due to thefact all the electrons may not be participating in conduction process. W is the thermalconductivity of the metal.

4. Drudes classical free electron theory totally failed to explain the conduction mechanism insemiconductors and insulators.

5. The classical model could not explain the origin of Paulis paramagnetism.

We shall now discuss one or two failures of classical free electron theory of metals withnecessary theory.

(i) Heat capacity of the electron gas: Classical free electron theory assumes that all thevalence electrons can absorb thermal energy, move to higher energy states and contribute to heatcapacity. According to the law of equipartition of energy, every free electron in a metal has an

average kinetic energy3

2KT. For a manovalent metal (copper) these will be NA(Avagadros number

of electrons = 6.02 1026) in one kmol. Hence the molar electronic specific heat is

[Cv]el

=d

d

U

T

with U =3

2NA kT

Thus [Cv]el

=d

d

U

T= 1.5 NA k = 1.5 Ru ...(2.1.10)

i.e., [Cv]el

= 1.5 R = 1.5 1.38 1023 6.02 1026

= 1.5 R or 12.5 103 J/kmol/k

This classical value of 1.5 R is about hundred times greater than the experimentallypredicted value.

(ii) Computation of mean free path: The microscopic expression for the resistivity(Eqn. 2.1.7) is

=m

ne2...(2.1.11)

The resistivity of the most useful metal copper at 20C is 1.69 108 ohm-m and theconcentration of free electrons in copper, n = 8.5 1028/m3. Thus

=m

ne 2

31

8 28 19 2

91 10

1 69 10 8 5 10 1 6 10=

.

. . .e j = 2.47 1014 sec

But =c

with =3km

T

= c = 2.47 1014 1.154 105

= 2.85 nm


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The experimentally found value for is about ten times above this value. Classical theorycould not explain this value.

2.1.5 Elementary Treatment of Quantum Free Electron Theory of MetalsThe general expression for the drift velocity is

vd = eE

m ...(2.1.12)

where is the average time elapsed after collision. Thus it is obvious that the average deift velocityof free electrons is proportional to force eE.

Therefore current density,

J = ne vne

mb =

2 E...(2.1.13)

But J = E

...(2.1.14)Comparing these two equations, one gets

=ne

m

2

Ifis the mean free paths andvF the speed of free electrons whose kinetic energy is equal to

Fermi energy since only electrons near Fermi level contribute to the conductivity. The average time between collisions is given by

=

vF

i.e., =ne

m v

2

F

...(2.1.15)

The drift velocity for unit electric field is called mobility.

i.e., =v e

m

e

m

d

E

E

E= =

Thus the electric conductivity,

=ne

mne

ne

mv

2 2

= =

F

...(2.1.16)

From expression (2.1.16) it is obvious at a given temperature the only factor which varies fromone metal to the other is densities of free electron.

One must also note that the energykT (where T is of the order of 300 k) can activate onlythe free electrons near the Fermi level to move to unoccupied states and contribute to specific heat.We may therefore require an energy EF called (very high compared with kT = 0.025 eV at 300 k)Fermi energy to make all the electrons to more to the unoccupied states corresponding to a

temperature TF called Fermi temperture. The unique relation connecting the various parametersin quantum theory of free electron is

EF =1

22mvF = kTF ...(2.1.17)


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2.1.6 Relation Between Thermal Conductivity and Electrical Conductivity

(Wiedemann-Franz Law)

Figure 2.1.3 shows the view of a copper rod of appreciable length with unit area of cross-sectionin the steady state.

Heat f low

Steam

A B C

E

T

T +dTdx T

dTdx

E +d Edx

E dEdx

Fig. 2.1.3 Flow of heat through a copper rod at the steady state

Let = AB = BC be the mean free path of the electron.

The excess of energy carried by an electron from A to B isd

dx

ELNM

OQP

. Hence the excess of energy

transported by the process of conduction through unit area in unit time at the middle layer B is

nc d

dx

6

ELNM

OQP

. Similarly the deficit of energy transported through B in the opposite direction is1

6nc

d

dx

ELNM

OQP

assuming1

6nc is the number of free electrons flowing in a given direction through unit area in unit

time.

Thus the net energy transported through unit area in unit time from A to B is:

1

6

1

6 3nc

d

dxnc

d

dx

nc d

d

d

dx

E E E

T

TLNM

OQP

LNM

OQP

=FHG

IKJ

FHG

IKJ

LNM

OQP

The general expression for the quantity of heat energy transported through unit area for unit

time is TTd

dx

FHG

IKJ. Equating the two equations, one gets

T =nc d

d

3

E

T

FHG

IKJ

Butd

d

E

Tis the energy required to raise the temperature by one degree and hence it is [ c

v]

el.

Now T =nc

cn c k

mv el

v el

3 3

3=

T...(2.1.18)


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But [cv]el

=3

2k with n = 1 electron

Thus T = n k kmn k k

m 3

32

32

3FHG IKJ =T T

The general expression for electrical conductivity by referring Eqn. 2.1.9.

=ne

mk

2

12

T

Thus

T =

n k k

m

mk

ne

2

3 122

T TRS|T|

UV|W|

T = 3

2k

e

L

N

MO

Q

P T ...(2.1.19)

This is called Wiedemann Franz law and the multiplying factor 3 (k/e)2 is called Lorentz number.

2.1.7 Heat Developed in Current Carrying Conductor (Joules Law)

The heat developed in a current carrying conductor is given by

H = I2 R =V

R

2

H =E

A

E Al

l

lb g2 2

=

i.e., H = E2 Al

Thus heat developed per unit volume per second is

W = E2

But current density,j = E

W =jE

E

2

= jE = E2 ...(2.1.20)

W will be found to be in watt/m3 (2.1.17)

In an isotropic medium, consider a particular electron which at the instant t = 0, has carried

out a collision with the lattice, and let the velocity components of the electron be vx, vy and vz.Now at the instant (t > 0), the electron has yet to collide with the lattice again, and assuming

an electric field of intensity E is applied along the negative x-direction, the velocity componentsof the electron are

ve

mtx +

FHG

IKJE


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The incremente

mtE

LNM

OQP

in velocity is due to the acceleration due to the field on the electron.

Thus the increase in energy of the electron over the field is

(W)t

=1

2

22m v

e

mt vx x+

RSTUVW

L

NMM

O

QPP

E

(W)t

=1

2

22

2 2me

mv t

e

mtx

LNM

OQP

+LNM

OQP

L

NMM

O

QPP

E E

The above expression may be averaged over a large number of electrons which have all livedthrough the period without having suffered a collision, but which presumbly have a random distributionof their velocities. Thus are finds

(W)t

=1

2 2

22 2

2 2

me

m

te t

m

F

HG I

KJ =E E ...(2.1.21)

If P (t) represents the probability that an electron moves for a ture t without suffering acollision, then P (t) = exp (t/

c). For isotropic scattering, the average time between collisions, is

equal to the relaxation time . Also, the probability that an electron will suffer a collision during

a time dt is given by

dt

c=

dt

So, the probability that the electron makes a collisions between t and t + dt is given by

expF

HGIKJ

LNM

OQPFHG

IKJ

t dt

Thus, the average energy increase of the electrons during the period between two collisionsis equal to

(W) = Wb gt

t

t dtexp

FHG

IKJ

=

z 0

=e

mt e dt

t2 22

02

E

LNMM

OQPP

zWe known that

x e dxn ax

z0

=

+

n

a

n 1

or W =e E

m

2 2 2

Ifn is the number of electrons present in unit volume, the total energy dissipated per unitvolume per second


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W =n

W

W = n em

2 2 2

2E E= ...(2.1.22)

This agrees with the experimental work.

2.1.8 Thermoelectric Effect

It is a phenomenon related to the thermal conductivity of electrons in that it too is a consequenceof the drift of electrons under a thermal gradient. These are two aspects to thermoelectricphenomena. The first is the Seeback effect in which a temperature between the two junctions of twodissimilar materials gives rise to an emf in the circuit. The second is the peltier effect in which, if

a current is circulated in a circuit consisting of two dissimilar materials, is liberated at one junctionand absorbed at the other. Both effects are the subject of considerable study because of their practicalapplications. The generation of electrical power using the Seeback effect offers the desirable freedom

from removing parts, and similar advantages are available in the use of the Peltier effect inrefrigeration. In either case one of the chief considerations is the size of the effect, which is aconsequence of the detailed electronic structure of the two materials, and a simple theoretical

treatment is not available. If emerges, however, that the parameters of the material which aresignificant in this respect are the same as for the conductivity, namely the effective mass of theelectrons and the relaxation time. Thermoelectric effects of either sign can be observed, dependingmainly on whether electron effects or hole effects are predominant. By careful selection of theparameters using doped semiconductors, thermoelectric power supplies now have efficienciesapproacting 20% and thermoelectric refrigerators have been built which can maintains a temperature

50F below room temperature.

The most familiar example of a metal-metal junction is the thermocouple, which is made ofa configuration of the type A-B-A. If a current is passed through such a combination, the temperature

of one of the functions is found to rise and that of the other to fall. This is called the Peltier effect.Conversely, if one junction is heated and the other cooled, a p.d develops across the combination.This is called Seeback effect.

Apart from their normal use as temperature sensitive elements, in recent years attention hasbeen paid to the possibility of using such devices as refrigerators and heat pumps (using the Peltiereffect) or as electrical generators (using the Seeback effect). With metal-metal combinations, botheffects are too small to be of any practical use (for example, the thermo-electric e.m.f developedfor a copper-constantan thermocouple is 40 V per Kelvin temperature difference), but since theadvant of semiconductors and the possibility of manufacturing metal-semiconductor junctions, the

constriction of practical devices based on these effects has become a reality.

2.1.9 Thermionic Emission

When metals are heated to sufficiently high temperatures, they emit electrons. This is known asthermionic emission. The free electrons in the metal should be supplied with a minimum amountenergy before they can escape from the metal by thermionic emission. This miniumum energy iscalled thermionic work function. The work function is different for different metals.

Richardson and Dushmans equation. Consider a pure metal situated in an evacuatedendosure. Let the metal be heated. It then emits electrons just as a heated liquid emits vapours. These


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electrons form a cloud over the metal just as a cloud of vapour is formed over an evaporating liquid.Just as in the case of vapour, the electrons in the electron clould will be returning into the metal.When equilibrium sets in, the number of electrons emitted from the metal per second will be equal

to the number of electrons returning to the metal per second.Consider one mole of electrons remembering that one mole of electrons will contain Avagadros

number of electrons (NA). Let the pressure exerted by these electrons on the metal be P. Let the

specific volumes of electrons in the metal be u and specific volume in the cloud byv. Let the heatenergy absorbed by the NA electrons when they escape from the metal be W. Let us assume thatClausius-Clapeyrons equation in thermodynamics can be applied to the electrons.

We then have

W = TP

T

d

dv ub g ...(2.1.23)

where T is the temperature of the metal and the enclosure. In reality, u


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i.e., + kT = TT

P

P

T

k d

d

2

or ddPT

= +FHG IKJk dTKTT

2

Integrating both sides, we get

loge

P = +

+z k d eTT T A2 log ...(2.1.27)i.e., log

eP =

d

ke e

T

TT A

2+ +z log log

d

ke e

T

TT A

2+ +z log log

where loge

A is a constant of integration

Let d

k

T

T2z = IThen log

eP = I + log

eT + log

eA

loge

P loge

T loge A = 1

logeP

AT

FHG

IKJ = I

P

AT= eI, or P = AT eI ...(2.1.28)

If these are n electrons per unit volume of the gas, we have from the following general

equation.

PV = RT = kA NA T

if V = 1, NA = n and hence

P = knT

n =P

T

AT

T

A Ik

e

k ke

I

= =

i.e., n =P

T

A Ik k

e= = BeI with B =A

k

According to kinetic theory of gases, we have

1

2

2mc =3

2

kT

According to Knudsens cosine law, the number of electrons n0 crossing unit area per secondis given by

n0 =nc

6


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=1

6

3

B

TIek

m

where m is the mass of the electron

n0 =1

6

3

KB T

meI

i.e., n0 = c eI T

where c =1

6

3

k

mB = constant

Richardson first assumed that the work function is independent of temperature, then

I = d

k k

T

T T2= z

n0 = c e(/KT)

Thus current density,

j = n0e = c e(/KT) T

j = D e/kT T ...(2.1.29)

where D = c e = constant

Equation (2.1.29) is the simplest form of Richardson and Dushman of themionic emission.

2.2 PROPERTIES AND APPLICATIONS OF METALS

As was pointed out in the discussion of the free electron theory of metals, the valence electrons ismetallic atoms, being loosdy bound, escape from the atom. These essentially free electrons providea medium of negative charge which helps to bind the positive ions. That this leads to a lower energystate which can be seen from the following arguments.

An electron in isolated atom is confined to a small volume around the nucleus. This confinement

gives rise to an uncertainty in the momentum p D

r, where r is the radius of the atoms.

Consequently, the electron has a fairly substantial amount of kinetic energy, of the order of several

eV. However, in the crystalline, metallic state, the electrons are essentially free to be anywhere inthe entire crystal. As a result, these is considerable reduction in kinetic energy. This is the sourceof metallic bonding. The bond between two metallic atoms is somewhat weaker than ionic orcovalent bonds. This leads to relatively low melting points, for example, 63C for K. However, the

cohesive energy of the metals is fairly large since each valence electron interacts with several ions.The metallic bonds are not directional where allows the planes of atoms to slide over each otherquite easily. Hence metals are found to be ductile and malleable rather than brittle. The existenceof essentially free electrons gives rise to high electrical and thermal conductivity for metals.

Because of these favourable properties metals are widely used in many areas mechanicalengineering and electrical engineering.


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2.2.1 Properties of Insulating Materials

Insulating materials are those which offer very high resistance to the flow of electric current. Theyrequire extremely high voltage of the order of kilovolt or megavolt to send a few milli-ampere of

current through them. They are non-metallic materials and are used to prevent leakage of currentfrom are conductor to another or to earth. Almost all insulating materials have negative temperature.Co-efficient of resistance and as such their resistivity is reduced with increase in temperature. The

function of any insulator is very important, for without which no electrical machines can work.Ideal insulators are those substances into which static electricfield penetrates uniformly. Primarilythey are used for storage of electrostatic energy as in capacitor which are known as dielectrics. Buthere we prefer to deal with that part of its function which prevents flow of electric current to anydesired conducting medium. The insulators provide mechanical strengths as well as insulation tothe conductors. Wood, paper, mica, oil, asbestos, porcelain are some examples of insulating materials.

Some important applications of insulators in the field of electrical engineering are:

(a) To isolate the overhead conductors from earth in the transmission and distribution ofelectrical energy

(b) In all rotary machines to isolate the winding from the body and other unwanted currentypath.

(c) In transformers to isolate the winding from the core or magnetic path.

(d) In industrial and house wiring to isolate the conductor from earthing.

(e) In all domestic, commercial electrical equipments to isolate the conductor from earth orbody.

Majority of the breakdown in the field of electrical engineering is due to the falure ofinsulation. The importance of insulating material is ever increasing day by day and there areinnumerable number of types of insulations are available in the market. The selection of the right

type of insulating materials for a particular job is very important because the life of equipment willdepend on the type of insulating materical used. All insulators when used should not only behave

as insulators over range of electrical voltage but also should be strong enough mechanically, shouldnot affected by heat, atmosphere, chemical and should be free from deformation due to aging.Therefore, the properties to be studied while selecting an insulator for a particular job are:

(a) Electrical properties

(b) Mechanical properties

(c) Visual properties

(d) Chemical properties

Some of the important electrical properties of insulators are

(a) Insulation resistance

(b) Volume and surface resistance

(c) Dielectric strength and dielectric loss(d) Breakdown voltage i.e., the voltage above which the material ceases to act as insulator.


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2.3 SUPERCONDUCTIVITY

2.3.1 Introduction

Superconductivityis a phenomenon observed in some materials: there is disappearance of the electricalresistivity at temperatures approaching 0K. The discovery of this phenomenon was not an attemptof any body; but by charm of face.

The phenomenon of superconductivity was discovered by Kamerlingh Onnes (1911), when hewas measuring the resistivity of mercury at low temperature. He observed that the electrical resistivity

of pure mercury drops abruptly to zero at about 4.2 K. He concluded that the mercury has passedinto a new state which is called the superconductivity state.

The materials that display this behaviour are called superconductors and the temperature atwhich they attain superconductivity is called critical temperature T

c.

The resistivity-temperature behaviours are contrasted in Fig. 2.3.1. The critical temperaturevaries from superconductor to superconductor but lies between less than 1 K and approximately

20 K for metals and alloys. Recently it has been demonstrated that some complex ceramics havecritical temperature approaching 100 K and above. Superconductivity is Adonis and the adolescenceis retained even after one hundred years. Four times Nobel Prize was awarded for discoveries andinventions in this area, and many more are aiming.

Superconductor Impure metal

Perfectlypure metal

0 T c T(in K)

o

Fig. 2.3.1 Variation of resistivity with temperature

The resistivity of metal falls with temperature T as per the law T when T > DT5 when T < D, D being theDebye temperature, but the experiments do not lead to the conclusion 0 as T 0. It is because any trace of impurity or crystal defects prevent the metals becomingperfect conductors and gives rise to a residual resistivity, 0. Then the question is whethersuperconductivity is related to purity of the metal. An emphatic no is the answer. Superconductivity

is a different phenomenon. If impurities are present in a superconductor, then fall of takes placein a very small region of temperature instead of sharply at Tc.

2.3.2 Some More Aspects of Superconductivity

The phenomenon superconductivity can be explained satisfactorily on the basis of wave mechanics.In an ordinary metal, the electrical resistance is the result of collisions of the conduction electrons


18/32


with the vibrating ions in the crystal lattice. In the superconductivity state, the electrons tend to bescattered in pairs rather than individually. This gives rise to an exchange force (similar to the forcebetween the atoms in a hydrogen molecule and the force between nucleans (in a nucleus), between

the electrons. The force is attractive; and is very strong if the electrons are ofopposite spins andmomenta. In the superconducting state, the forces of attraction between the conduction electronsexceed the forces ofelectrostatic repulsion. The entire system of conduction electrons then becomes

a bound system. No transfer of energy takes place from the system to the lattice ions. If an electricfield is established the bound system of electrons gain additional kinetic energy and give rise to acurrent. But they do not transfer this energy to the lattice, so that they do not get slowed down. Asa consequence of this, the substance does not possess any electrical resistivity. This theory was putforward, byJohn Bardeen, L. N. Cooper andJ. R. Schriffer in 1957. The bound pairs of electronsis called Cooper pairs.

Normally, the resistivity of a superconductor is measured by causing a current to flow in a ringshaped sample (one can start the current by induction after removing a magnetic flux linking thering), and observing the current as a function of time. If the sample is in a normal state, the current

damps out quickly because of the resistance of the ring. But if the ring has zero resistance, thecurrent, once set up, flows indefinitely without any decrease in value. According to experimentsconducted, the current remained constant in the ring even for several years.

2.3.3 Magnetic Properties of Superconductors

Effect of magnetic field: One should also know that the superconducting state can be broughtto the normal state by applying a magnetic field at which superconductivity vanishes is called criticalmagnetic field. Its value depends upon the nature of the material and its temperature.

H (0)c

H c

0 T T c

Normalstate

Superconductor

H (0)c

H c1

H cH c2

Superconductor

0 T c1 Tc2 T c

TTemperature

Notsuperconductor

Fig. 2.3.2 Variation of Hc

with temperature

The equation used in this connection is

Hc(T) = H TTc

c

0 12

2b g LNMM OQPP...(2.3.1)

2.3.4 Critical Currents

The readers should recognize that the magnetic field which destroy the superconducting propertyneed not be the external electrical field, it can be due to the current flowing through your


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superconductor. Hence the maximum current flowing through the specimen at which this propertyis destroyed is called critical current.

If a superconducting wire of radius r carries a current I, then as per

I

r

Magnet icl ines of force

Fig. 2.3.3 Superconducting wire carrying current

Amperes law,

H z dl = Ii.e., H2r = Iat H = H

c, I = I

c

Hence Ic

= 2rHc

...(2.3.2)

If I becomes Ic

superconductivity will be destroyed. If in addition to current, transversemagnetic field H is applied, the value of critical current decreases.

H

I

Fig. 2.3.4 Current carrying superconductor in transverse magnetic field

Now Hc

= HI + 2H where HI is the field due to current

HI = Hc 2H

HI =I

r

c

2= H

c 2H

Ic

= 2r (Hc 2H) ...(2.3.3)

This is called Silsbees rule.

2.3.5 The Meissner Effect

When superconducting material is cooled below its critical temperature, it not only becomes

resistanceless but perfectly diamagnetic also. That is to say that there is no magnetic field insidesuperconductor, whereas inside a merely resistanceless metal there may or may not be a magneticfield, depending on the circumstances.

This interesting observation, when superconductor placed inside a magnetic field cooledbelow its critical temperature, all the magnetic flux is expelled out of it, called Meisnner effect.


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T > Tc

Cooling

B 0 B = 0

B = 0

T < Tc

Fig. 2.3.5 Meisnner Effect

The perfect diamagnetism is an account of some special bulk magnetic property of thesuperconductor. If there is no magnetic field inside the superconductor, it can be said that its relative

permeabilityr

is zero. Here the mechanism of diamagnetism is not considered.

0H

M

M = H

Fig. 2.3.6 Versus H in a superconductor

The general equation connecting magnetic induction and magnetic field isor B = o (H + M)

H + M =B

o o= =

00 (under superconducting state B = 0)

and also H + M = 0

i.e., M = H

shows the magnetization curve for a superconductor.

The magnetic susceptibility is

M =M

H= 1 ...(2.3.4)

It must be noted that superconductivity is not only a strong diamagnetism but also a new typeof diamagnetism.

Critical field: Superconductivity can be destroyed by the application of a strong magneticfield called critical field, H

ceven at T < T

c. The critical field depends on temperature and decreases

as the temperature rises from T = 0 K to T = Tc.


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i.e., Hc(T) = H

T

Tc

c

0 1

2

b g FHGIKJ

L

NMM

O

QPP ...(2.3.5)

Figure 2.3.7 shows Hc

vs Tc

for some selected superconductors and the phase diagram.

Appliedm

agneticfield

strength

H c

0 T(K)

Superconduct ivity

PNormal

T c

Criticalm

agneticfield

0 1 33 4 87

T(K)

Al

Sn

Pb

6 104

3 104

2 104

1 104

(a) (b)

Fig. 2.3.7 (a) Phase diagram of a superconductor (b) Critical fields of superconductors

2.3.6 Superelectrons

According to Two-fluid Model (a model introduced for explaining the electrodynamic property ofsuperconductors), the conduction electrons in a superconducting substance fall into two classes:

Superelectrons and normal electrons: The superelectrons experience no scattering, have zeroentropy (perfect order), etc. The normal electrons behave in the usual fashion discussed in the freeelectron study chapter.

The number of superelectrons depends on the temperature as stated below:

ns

= nc

1

4

FHG

IKJ

L

NMM

O

QPP

T

T...(2.3.6)

1

n

ns

0

TT e

Fig. 2.3.8 Dependence of superelectrons on temperature


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This is plotted in Fig. 2.3.7. Thus at 0 K, all the electrons are superelectrons, but as T increases,the superelectrons decrease in number, and they become normal electrons at T = T

c.

2.3.7 Current Trends in SuperconductivityThe failure to raise the transition temperature (T

c) of a metallic system beyond 23.2 K forced

research to look into other kinds of systems for possible high temperature superconductivity. Thiseffort seems to be paying off with the recent discovery of ceramic oxides found to be superconducting

well beyond 100 K.

Superconductivity around 90 K (High temperature superconductors): In March1987, the Texas-Alabama group reported on the observation of stable and reproducablesuperconductivity between 93 K and 80 K in Y-Ba-Cu-O system. For the first time in history,

superconductivity above the liquid nitrogen boiling point (77 K) was discovered. Research groupsin India soon confirmed the existence of superconductivity in a system of normal composition Y1.2Ba0.8 Cu2 O4, in the 90 K range.

Superconductivity at 155 K: In Michigan, U.S., a research group synthesized fluorinatedY-Ba-Cu-O compounds of nominal composition Y1 Ba2 Cu3 F2 Oy (bulk material of multiphase) toachieve zero resistance at 155C.

Superconductivity above 200 K: Recently some researches have observed a drop in theelectrical dc resistance of their samples in the 200 K range. Unfortunately, in these cases repeatedcooling and warming (thermal cycling) appears to disrupt superconductivity which is perhapspresent only in tiny filaments embedded in a mostly non-superconducting sample.

Researches at the national physical laboratory, New Delhi and at Wayne State University,Detroit, U.S. have observed in multiphase samples of Y-Ba-Cu-O a drop in electrical resistance in

200 K range. It has become possible to fabricate superconducting wires from ceramic materials,but the mechanical properties such as ductility, flexibility may not be so good as aluminium.

A nightmare: Expressing concern over mounting transmission and distribution (T and D)

losses in the power sector, the former president of India Prof. A.P.J. Abdul Kalam asked the UnionPower Ministery to bring down the power loss from 31000 MW to 12000 MW in a couple of yearscomparing with that in developed countries. Mr. Kalam said if the country were to set up fresh

capacity to cover of this loss, it would cost over Rs. 76000 crores; this is the magnitude of theproblem we are faced with; he added while addressing a group of Engineers attached to CentralElectricity Authority.

Another method of solving this problem is summarized now:

The phenomenon of superconductivity was observed in mercury about 100 years ago (1911).The transition temperature of the metal aluminium used liberally in transmission lines on account

of their mechanical and electrical properties is 1.2 K. i.e., below this temperature electron-phonon-electron interaction becomes stronger and cooper pairs are formed. All of them have the same deBroglie wavelength with very small momentum and hence scattering of cooper pairs becomes

absent, mobility becomes infinity. This may brought down power loss almost nil. However this ispossible only at 1.2 K.

Huge amount of fund was made available in U.S. and in India for research in high temperature

superconductors hoping aluminium will be made superconductor in the wide range say 0C to 45Cby discovering new process, mechanism and techniques though the phenomenon was discoveredabout 100 years ago, but in vain. I hope we will succeed in this attempt in the years to come.


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2.3.8 Field Applications of Superconductivity

Superconductivity as it is, does not find much applications. This is on account of the difficultyinvolved in reaching a low temperature of 273C and maintaining the equipment in that state but

it is likely to find applications in the following main areas:

(a) Electrical machines: By using superconductivity it is possible to manufacture electricalgenerators and transformers in exceptionally small sizes, having an efficiency nearly

equal to 100%.

(b) Power cables:A 200 kV cable with superconducting material will enable transmissionof power over long distances using thin conductors without any significant power lossor voltage drop.

(c) Electromagnets: Superconducting solenoids have been made which do not produceany heat during operation. Superconductivity gets destroyed if the magnetic field exceeds

a critical value. In superconducting state, the magnetic field does not penetrate into thesuperconductor. It behaves as a substance with zero magnetic permeability. Therefore

the external magnetic field in the superconductor repel each other. This property can beutilized in developing frictionless bearings with magnetic lubrication for use in gyroscopesand electric machines.

Helium, an expensive gas is now used to attain a low temperature, required for

superconductivity. Efforts have already being made to develop compounds which are superconductorsat temperatures possible to obtain by easily available hydrogen gas.

2.3.9 Squid

Squid is an acronym for superconducting quantum interference device and the arrangement is as

shown in Fig. 2.3.9. All squids make use of the fact that the maximum current in superconductingring that contains a Josephson junction varies periodically as the magnetic flux through the ringchanges. This periodicity is interpreted as an interference effect involving the wave functions of the

cooper pairs. It consists of a ring superconducting material having two side arms A and B whichact as entrance and exist for the supercurrent respectively.

P

l1

l2

A

B

QJosephson

ju nct ion

B

Fig. 2.3.9 A squid


24/32


The insulating layers P and Q may have different thicknesses and let the currents throughthese layers be I1 and I2 respectively. The variations I1 and I2 versus the magnetic field is shownin Fig. 2.4.1. Both I1 and I2 vary periodically with the magnetic field, the periodicity of I1 being

greater than that of I2. The variation of I2 is an interference effect of the two junctions while I1is the diffraction effect that arises from the finite dimension of each junction. Since the current issensitive to very small changes in the magnetic field, the SQUID can be used as a very sensitive

galvanometer.

OBJECTIVE TYPE QUESTIONS

2.1 Which one of the following is a trivalent metal?

(a) Na (b) Cu

(c) Hg (d) Al

2.2 The temperature coefficient of resistivity is

(a) d

dT(b)

1

d

dT

(c) d

dT(d)

1

d

dT

2.3 If j is the current density through a conductor in a field E, then

(a) j 1

E(b) j E

(c) j E (d) j 12E

2.4 R1 is the resistance of a copper wire of length l cm and radius r cm. If R2 is the resistance of another

copper wire of length l2

cm and radius 2r cm. Now

R

R2

1is

(a)l

2(b)

l2

4

(c)l

4(d)

l

r

2.5 The length of copper wire is halved and diameter is doubled, the resistivity of copper

(a) reduces by 50% (b) increases by 25%

(c) decreases by 60% (d) remains the same

2.6 In the absence of an external field, the root mean square velocity c

follows:

(a)c T

(b)c

1

T

(c) c T2 (d) c 1

T4

2.7 The resistance of an aluminium wire of length l1 and radius r1 is R1 ohm. If the length is halved and

the radius doubled, the corresponding resistance R2 is


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(a) 0.5R1 (b) 4R1

(c)R18

(d) 16R 1

2.8 The mobility of electron in a metal is 0.00132 m2/V-sec subjected to a field 50 V/m. The drift velocityof the electron is

(a) 0.066 m/sec (b) 6.66 m/s

(c) 66 m/sec (d) 6.6 107 m/sec

2.9 At Fermi temperature the number of electrons excited from the occupied states to the unoccupied states

is

(a) 10% (b) 20%

(c) 50% (d) 100%

2.10 The formula for Fermi velocity is

(a) vF =k

m

TF2

(b) vF =2k

m

TF

(c) vF =kT

m

F

2

1 2LNM

OQP

(d) vF =m

k2 TF

2.11 If the mobility of the electrons in a metal increases, the resistivity

(a) decreases (b) increases

(c) first increases and then decreases (d) first decreases and then increases

2.12 In thermionic emission the work function for a given current

(a) is the same for all metals

(b) it varies exponentially

(c) it is different for different metals

(d) depending on the electronic structure of the metal

2.13 Depositing a layer of thorium on tungsten will

(a) lower the work function

(b) raise the work function

(c) no effect on the work function

2.14 The Fermi energy of a metal is 1.4 eV, the Fermi temperature of the metal is approximately

(a) 1.6 103 K (b) 1.6 104 K

(c) 1.6 105 K (d) 1.6 106 K

2.15 Superelectrons become normal electrons at

(a) zero degree kelvin (b) 0C

(c) the critical temperature (d) Debye temperature

2.16 If the temperature of a metal is reduced to a value below the critical temperature, the value of the

critical magnetic field will

(a) increase (b) decrease

(c) may increase or decrease (d) keeps a constant value


26/32


HINTS FOR SELECTED OBJECTIVE TYPE QUESTIONS

2.4 R1 =

lr2

and R 2 =

lr

lr

2

2

2

22 4b g

=

R

R2

1

=

l

r

r

l

l2

2

2

4 4

FHG

IKJ

=

R

R2

1

=l

4Ans.

2.6 P =1

32c if the volume is 1 m3

For molar volume

P =1

32m

cm

N

VA

PVm

=1

32m cN RTA

c2 =3 3RT

N

T

Am

k

m

c =

3k

m

T

Hence c

T

2.8 =vd

Ev

d = E = 0.00132 50

vd = 0.066 m/s

2.14 EF = kTF

EFk

= TF1 4 16 10

138 10

19

23

. .

.

TF = 1.62 104 K.

SHORT QUESTIONS WITH ANSWERS

2.1 Explain free electron gas model for metals.Ans. In most solids the two main types of internal energy are ( i) the vibrational energy of the atoms

about their mean lattice positions and (ii) the kinetic energy of the free electron. If heat is suppliedto a body its temperature rises and the internal energy increases. The important thermal propertiesof matter such as heat capacity, thermal expansion and thermal conductivity depend upon thechanges in the energy of the atom and free electrons. If an electric field is applied to a metal thefree electrons are accelerated. Their kinetic energy increases. Some of the K.E is of course lost by


27/32


collisions with the atoms in the lattice. The resulting current is proportional to the average velocityof the free electrons. The velocity is determined by the applied electric field and also the collision

frequency. Thus both thermal and electrical behaviour of matter depend on the free electrons in it.

2.2 Discuss some of the important factors which reduce electron mobility or scattering which in turn reducethe mobility.

Ans. (i) Thermal agitation: Increase in temperature increases the resistance or scattering which inturns reduce the mobility.

(ii) Plastic deformation: Internal stresses are caused by cold working which distort the energy

levels at which electron transfer takes place. This is the other reason.

(iii) Impurities: Lattice distortion and impurities cause irregularities and hence reduce the mean

free path of the electrons and thus reduce the mobility of the electron.

2.3 Distinguish between the two velocities associated with the electron in metals.

Ans. The velocity, vd

=e

m

Eis called drift velocity. This is super imposed on a much higher velocity

known as random velocity ( c ) due to the random motion of the electron (thermal motion). Just asin an ordimary gas, the electrons have random motion even in the absence of the field. This is dueto the fact that the electrons move about and occasionally scatter and change direction. The random

motion, which contributes zero current, exists also in the presence of the field, but in this case thereis an additional velocity opposite to the field.

2.4 Where metals and alloys are preferred?

Ans. Metals are chosen in places where high conductivity and large temperature coefficient of resistance

is desired. Alloys are chosen for heating purposes by I2R dissipation and for low temperaturecoefficient of resistance.

2.5 Why electrons and neutrons are diffracted easily by metallic crystals?

Ans. Both neutrons and electrons have wave characteristics and the wavelength is of the order of gratingelements in crystals.

2.6 Discuss briefly the important properties of superconducting materials.

Ans. (i) At room temperature, superconducting materials have greater resistivity than other elements.

(ii) The transition temperature Tc

is different for different isotopes of an element. If decreases

with increasing atomic weight of the isotopes.

(iii) The superconducting property of a superconducting element is not lost by impurities to it butthe critical temperature is lowered.

(iv) There is no change in the crystal structure as revealed by x-ray diffraction studies. This means

that superconductivity may be more concerned with the conduction electrons than with theatoms themselves.

(v) The thermal expansion and elastic properties do not change in the transition.

(vi) All thermoelectric effects disappear in superconducting state.

(vii) When a sufficiently strong magnetic field is applied to a superconductor below the critical

temperature, its superconducting property is destroyed. At any given temperature below T c,there is a critical magnetic field Hc such that the superconducting property is destroyed by theapplication of a magnetic field. The value of Hc decreases as the temperature increases.

2.7 How will you know that a metal has become a superconductor?

Ans. The simple method to study this problem is discussed below:

A superconducting metal is connected with a source of electricity. A voltmeter is connected acrossas shown in figure. When the material is in normal conducting state a p.d is maintained at the ends


28/32


of the wire or conductor. When the material is cooled below its critical temperature Tc, the potential

disappears as shown in the figure below:

V

V

(a )

V

V = 0

(b )

2.8 Show that the following two definitions of the resistivity are equivalent.

(a) resistance of a bar = length of bar

area of cross-section

(b) =electric field

current density

Ans. (a) =R a

l...(1)

(b) Ohms law:

V = IR

R =V

I

l

a

V =Il

a

E =V I

l aj

=E

jor =

j

E...(2)

Both equations are the same.

REVIEW QUESTIONS

2.1 Outline free electron model of metals. Derive an expression for the electrical conductivity of a metal

on the basis of classical free electron theory of metals.

2.2 What are the assumptions introduced by Drude-Lorentz to explain classical free electron theory of

metals? Discuss the achievements and failures of this model.

2.3 Discuss the failures of classical theory of free electrons with special reference to specific heat andmean free path.

2.4 Obtain Wiedemann-Franz law. Give the significance of Lorentz number.

2.5 Enumerate the heat developed in a current carrying conductor and get the expression for the totalenergy dissipated per unit volume per second.


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2.6 Discuss the properties of metals and insulators bringing in the important applications of them.

2.7 Give an account of the phenomenon of superconductivity.

2.8 Discuss the significance of critical temperature, critical magnetic field and critical current density in

superconductors.2.9 Distinguish between type I and type II superconductors with suitable diagrams.

2.10 List out the various applications of superconductors.

PROBLEMS AND SOLUTIONS

2.1 A copper wire of radius 1mm and length 10 metre carries a direct current of 5 ampere. Calculate the

drift velocity of electrons in copper if n = 5 1028/m3.

Sol. The expression for drift velocity is

vd

=e

m

E...(1)

and =ne

m

2

i.e., =m

ne2

ore

m

=

ne

Substituting this in Eqn. (1),

vd

=E

ne

E =V IR I

l l l

l

a

F

HG I

KJ

E =I

a

vd

=

ne a nea

FHG

IKJ

FHG

IKJ

I I

=5

5 10 16 10 1028 19 32

. e j

vd

= 1.99 104 m/s.

2.2 In the first case a copper wire of cross-sectional area 104 m2 and in which there is a current of 200

ampere is considered. If the free electron density of copper is 8.5 1025/m3, compute the drift velocity.

If a p.d. of 1 volt is applied across 20 metre of this wire, express the new drift velocity in terms of theone calculated in the previous case. Relaxation time is 1014 sec.

Sol. Case 1: If I is the current flowing, then

I = n Avd

e

vd

=I

An e

200

8 5 10 1 6 10 1028 19 4. .


30/32


vd

= 1.47 104 m/s

Case 2: E =1

20= 0.05 V/m

e

m

E= vd

119 14

31

16 10 0 05 10

9 1 10

. .

.= 8.8 105

v

v

d

d1

= 1.67 Ans.

2.3 Calculation of Fermi energy for some monovalent element yield the following results:

Metal Cu Li Rb Cs Ag K

EF

(eV) 7.04 4.72 1.82 1.53 5.51 2.12

If the Fermi velocity of the electron in one of the metals of the above series is 0.73 106 m/s, identify

the metal. Also compute its Fermi temperature.

Sol.1

22m vF = EF

i.e., EF =1

2

FHG

IKJ 9.1 10

31 (0.73 106)2 joule

= 4.55 1031 1012 0.732

=4 55 0 73 10

1 6 10

2 19

19

. .

.

EF = 1.5 eV Ans.

and EF = k TF

or TF =EkF 1 5 16 10

138 10

19

24. .

.

TF = 1.75 104 K Ans.

2.4 For a specimen of V3 Ga, the critical fields are respectively 1.4 105 and 4.2 105A/m for 14 K and

13 K. Calculate the transition temperature and critical fields at 0 K and 4.2 K.

Sol. The general formula is

Hc(T) = H

T

Tc

c

0 1

2

b gFHG

IKJ

L

NMM

O

QPP

1.4 10

5

=H

Tc c0 1

142

b g

F

HG

I

KJ

L

NMM

O

QPP

4.2 105 = HT

cc

0 113

2

b gFHG

IKJ

L

NMM

O

QPP


31/32


1

3=

T T

T T

c c

c c

2 2 2

2 2 2

14

13

e je j

3 Tc2 3 142 = Tc

2 132

2 Tc2 = 3 142 132

Tc

= 14.47 K

1.4 105 = Hc 0 114

14 47

2

2b gL

NMM

O

QPP.

= Hc (0) 0.16

Hc

=1 4 10

0 06

5.

.= 2.3 106.

2.5 The resistivity of a superconductor becomes zero. Consequently the flux density is zero due to this abrupt

change. Prove that the superconductor behaves as perfect diamagnetic.

Sol.

The general equation is

B = 0 (H + M)

Since B = 0

H = M orM

H= 1 or

H= 1 ...(1)

=M

H= (r 1)

Thus 1 = r 1or r = 0 ...(2)

Equations 1 and 2 show the possibility of perfect diamagnetism.

Some Important Tables

Table I: Critical field of some elements

Element BC (0) in 102 Wb/m2

Al 0.99

Cd 0.30

Ga 0.51

Pb 0.51

Hg () 4.13

Hg () 3.40

Ta 8.30

Sn 3.06


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Table II. Properties of some selected superconducting elements

Element Tc (K) H o (A/m) Type Element T cK HoA/m Type

Aluminium 1.196 0.79 104 1 Zinc 0.9 0.42 104 1

Cadmium 0.52 0.22 104 1 Tantalum 4.5 6.6 104 11

Gallium 1.09 0.4 104 1 Thlium 2.4 1.4 104 11

Indium 3.4 2.2 104 1 Niobium 9.3 15.6 104 11

Tin 3.72 2.4 104 1 Rhenium 1.7 1.6 104 11

Mercury 4.12 3.3 104 1 Thorium 1.4 1.3 104 11

Iridium 0.11 0.13 104 1 Zirconium 0.8 0.37 104 11

Lead 7.175 6.5 104 1 Osmium 0.7 0.5 104 11

ANSWERS TO OBJECTIVE TYPE QUESTIONS

2.1 (d) 2.2 (d) 2.3 (b) 2.4 (c) 2.5 (d)

2.6 (d) 2.7 (b) 2.8 (a) 2.9 (d) 2.10 (b)

2.11 (a) 2.12 (c) 2.13 (a) 2.14 (b) 2.15 (c)

2.16 (a)

GGG

electron thepry of metals

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