2.1 Introduction

Theoretical aspects of various structural, electrical and magnetic parameters

relevant to the present study are briefly discussed in this chapter. Structural

parameters provide crystallographic information, lattice dimensions, site preferences

in the spinel lattice sites and surface microstructures such as grain size, impurities,

segregations at the grain boundary and composition related data.

The saturation magnetisation of ferrites reflects on the distribution of various

metallic cations over A-and B-sites. Curie temperature tells about the higher limit of

temperature at which the material would work safely in a device. The understanding

of permeability and magnetic losses enables to estimate the performance of the

material at a given frequency of operation. Electrical properties such as resistivity,

dielectric constant and dielectric loss give information on electrical losses, grain

structure, activation energies and distribution of ions in the spinel structure. They

also provide information related to suitability of the materials for work at a given

frequency of operation, in the present case at microwave frequencies.

2.2 Structural aspects

The structural aspects to be dealt in the study of the present materials include

crystal structure of the ferrite, distribution of the metallic cations in the

crystallographic sites of the ferrite, factors influencing the distribution of cations,

determination of crystal structure and its associated parameters like lattice constant,

density, porosity, etc. The microstructures, elemental compositions of the

investigated ferrites and their cation distributions will also be covered in this section

2.2.1 Spinel ferrite structure

The spinel structure was first determined by Bragg and Nishikava [1]. In the

ideal structure of a spinel, the anions form a face centered cubic (fcc) close packing

in which the cations partly occupy the tetrahedral and octahedral interstices. The unit

cell contains 32 anions forming 64 tetrahedral interstices and 32 octahedral

interstices; of these 8 tetrahedral and 16 octahedral sites are occupied by cations.

These are called A- and B- sites, respectively. The general formula of compounds

with spinel structure is AB2O4, and the space group is Fd3m. The unit cell of the

spinel crystal structure, and the tetrahedral (A-site) and the octahedral (B-site)

interstices are shown in Fig.1.

In the A-site, the interstitial is in the centre of a tetrahedron formed by four

lattice atoms. Three anions, touching each other, are in plane; the fourth anion sits in

the symmetrical position on the top at the center of the three anions. The cation is at

the center of the void created by these four anions. In this configuration, four anions

are occupied at the four corners of a cube and the cation occupying the body center

of the cube. However in the B-site, the interstitial is at the center of an octahedron

formed by 6 lattice anions. Four anions touching each other are in plane, the other

two anions sites in the symmetrical position above and below the center of the plane

formed by four anions. Cation occupies the void created by six anions forming an

octahedral structure. This configuration shows that six anions occupy the face

centers of a cube and cation occupies the body center of the cube. Each A-site cation

is surrounded by 4 nearest neighbor oxygen ions and 12 next nearest neighbor B-site

cations. Similarly, each B

neighbor oxygen ions and 6 next nearest neighbor A

Figure 2.1: Unit cell of the cubic spinel alongwith the tetrahedral and o

interstices

The interstices available in an ideal close packed structure of rigid oxygen

anions can incorporate only those metal ions with radius

sites and only those ions with radius

accommodate larger cations such as Co, Cu, Mn, Mg, Ni and Zn with ionic radii in

the range from 0.56 to 0.83 Å the lattice has to be expanded. Thus, the ideal

could not be realized, and the oxygen anions in the spinel structures are generally not

located at the exact positions of the fcc sub lattice. Accordingly, the difference in the

expansion of the tetrahedral and octahedral sites is characterized b

called oxygen parameter

cations. Similarly, each B-site cation in the lattice is surrounded by 6 nearest

neighbor oxygen ions and 6 next nearest neighbor A-site cations.

ell of the cubic spinel alongwith the tetrahedral and o

The interstices available in an ideal close packed structure of rigid oxygen

anions can incorporate only those metal ions with radius r tetra ≤

sites and only those ions with radius rocta ≤ 0.55 Å in octahedral sites. So in order to

accommodate larger cations such as Co, Cu, Mn, Mg, Ni and Zn with ionic radii in

the range from 0.56 to 0.83 Å the lattice has to be expanded. Thus, the ideal

could not be realized, and the oxygen anions in the spinel structures are generally not

located at the exact positions of the fcc sub lattice. Accordingly, the difference in the

expansion of the tetrahedral and octahedral sites is characterized b

called oxygen parameter (u).

site cation in the lattice is surrounded by 6 nearest

site cations.

ell of the cubic spinel alongwith the tetrahedral and octahedral

The interstices available in an ideal close packed structure of rigid oxygen

≤ 0.30 Å in tetrahedral

Å in octahedral sites. So in order to

accommodate larger cations such as Co, Cu, Mn, Mg, Ni and Zn with ionic radii in

the range from 0.56 to 0.83 Å the lattice has to be expanded. Thus, the ideal situation

could not be realized, and the oxygen anions in the spinel structures are generally not

located at the exact positions of the fcc sub lattice. Accordingly, the difference in the

expansion of the tetrahedral and octahedral sites is characterized by a parameter

2.2.2 Distribution of metal ions

An interesting property of ferrites to be extracted with the spinel crystal

structure is the so-called cation distribution. Depending up on the distribution of

metallic cations over the A- and B-sites in the spinel lattice, the ferrites again can

have three different configurations.

1) Normal

2) Inverse

3) Mixed

In normal ferrites, the divalent cations occupy the A-sites and only trivalent

cations occupy the B-sites are. The distribution in the normal ferrites is represented

by the formula (M2+)A [Me3+]B O2-4. Those cations enclosed by paranthesis ( )

occupy A-sites and comprise the tetrahedral sublattice, while the cations enclosed by

square brackets [ ] occupy B-sites and comprise the octahedral sublattice. A typical

example of normal spinel ferrite is bulk ZnFe2O4 [2].

In inverse ferrites, half of the trivalent ions occupy A-sites and the other half

of the trivalent cations and total divalent cations randomly occupy B-sites. These

ferrites are represented by the formula (Me3+)A [M2+Me3+]B O2-4. A typical example

of inverse spinel ferrite is bulk magnetite, Fe3O4, in which all the divalent cations of

Fe occupy only the B- sites [2].

Spinel ferrites with ionic distribution, intermediate between the normal and

the inverse configurations is known as mixed ferrites. The distribution in this case is

represented by the formula (M2+xMe3+

1-x)A [M2+1-xMe3+

1+x]B O2-4. Examples of having

mixed cation distributions are manganese-zinc and nickel-zinc ferrites [3].

In the assignment of cation distributions in ferrites, there is another parameter

of paramount importance in the literature is the degree of inversion. The inversion

parameter, δ, could be defined as the fraction of total number of divalent ions

occupying B-sites [3]. The magnitude of δ depends on several factors including the

method of preparation and nature of the constituents of the ferrites. For normal spinel

ferrites δ = 0 since all the divalent cations for such a configuration reside in A-sites

only, and for inverse spinel ferrites δ = 1 as all the divalent ions occupy B-sites only.

For mixed spinel ferrites, the δ ranges between these two extreme values.

2.2.3 Factors influencing the cation distribution

The useful structural, magnetic and electrical properties of spinels depend not

only on the kinds of cations in the lattice, but also on their distribution over the

available crystal lattice sites. It is thus of major importance to understand the factors

that contribute to the total lattice energy in spinels, i.e., (i) Elastic energy (ii)

Electrostatic (Made lung) energy (iii) Crystal field stabilization energy (iv) d-orbital

splitting and (v) polarization effects.

The elastic energy refers to the degree of distortion of the crystal structure due

to the difference in ionic radii assuming that ions adopt a spherical shape. Smaller

cations, with ionic radii of 0.2-0.4Å should occupy A-sites, while cations of radii

0.4-0.7 Å should enter B-sites. This distribution leads to a minimum in lattice strain.

Since trivalent cations are usually smaller than divalent ones, they can be assumed to

force a situation where the larger sized divalent cations migrate to B-sites, and hence

a tendency towards the inverse arrangement would be expected.

The Madelung constant of the spinel structure has been calculated by Verwey

et a1.[4] as a function of oxygen parameter u and the charge distribution among A-

and B- sites. Their results showed that this energy is dependent on the u-parameter.

For u>0.379, the normal distribution is more stable, while for lower u values the

inverse arrangement possesses a higher Madelung constant. It therefore implies the

presence of two kinds of cations in B-sites in inverse spinels leads to an additional

contribution to the Madelung energy. The critical u value then becomes 0.38142.

Madelung energy is higher for the normal spinel if u >0.381 and the inverse ordered

spinel is more stable for u < 0.381.

Crystal field factors used to help account for the site preferences in spinels.

Romeijn [5] was the first one who suggested the application of the crystal field

theory to understand the cation site preference in spinels. Dunitz and Orgel [6] has

calculated the octahedral site preference energies of transition metal ions in oxides

using crystal field theory (CFT) The data show that the systems with d5 and d10

configurations (Fe3+, Mn2+ and Zn2+) have no CFSE and hence no site preference.

The d3 system (Cr3+) has the highest octahedral site preference energy. The d4 and

d9 (Mn3+ and Cu2+) ions can be further stabilized by Jahn-Teller distortion.

Another factor which plays a role in cation distribution is d-orbital splitting

energy. Although, the CFSE contribution to the total bonding energy of a system is

only about 5-10%; it may be the deciding factor when other contributions are

reasonably constant. When the various factors are counter balancing, there can be a

completely random arrangement of metal ions among the 8 tetrahedral and 16

octahedral sites.

2.2.4 Crystal structure determination

Phase identification shall be done by comparing the observed structural

parameters with the standard values reported. The number and position of the peaks

in X-ray diffractin (XRD) pattern confirms whether the materials exhibits single-

phase spinel structure or not. From the experimental values of θ, distance between

the atomic planes 'd' can be calculated using the Bragg relation.

2d sinθ = nλ

where λ is the wavelength of incident radiation (X- rays) and θ is the diffraction

angle.

To find the crystal structure, we need to determine the lattice constant. It is

related to the inter planar spacing for the hkl planes in the cubic structure as

2 2 2hkld h k l a+ + = (1)

where hkl is the plane of reflection. From the above equation, one can get the value

of lattice parameter 'a'.

2.2.5 Lattice constant

A parameter defining the unit cell of a crystal lattice is the length of one of the

edges of the cell or an angle between edges. It is also known as lattice parameters or

lattice constant. Lattice constant refers to the constant distance between the lattice

points. It is calculated using equation (1). Lattice in three dimensions generally has

three lattice constants, referred as a, b, c. However, in cubical structures, all these

three constants are equal and we only refer to “a”.

2.2.6 Density

Density related properties have no connection with an external factor and so

are not mechanical. They must, however, be considered first before any other

property of the materials can be studied.

Measured density is an intrinsic property of materials that denote the

relationship between its mass and unit volume. It is used as an index property or an

independent variable to predict other properties of the materials and is difficult to

characterize because this parameter can be affected by temperature, pressure and the

amount of substitution of different elements. The measured density is determined

using the following formula

2m

m

r hρ

π=

(2)

where m is the mass, r is the radius and h is the height of the sample

2.2.7 X- Ray Density

The X- ray density of the prepared samples of ferrites is calculated by the

relation

3

8X ray

M

Naρ − =

(3)

where M is the molecular weight of the samples, N is the Avogadro’s number and

a is the lattice constant. Each cell has 8 formula units.

2.2.8 Porosity

The storage capacity of any material is referred to as the porosity of a material

depending on the shape, size of grains and on the degree of their storing and packing.

The porosity P of the samples is then determined using the relation

1 m

X ray

Pρ

ρ −

= − (4)

where mρ and X rayρ − are the measured and X-ray densities, respectively

2.2.9 Scanning electron microscopy

Scanning electron microscope (SEM) plays a vital role in the characterization

of materials. SEM can provide internal and surface imaging of all materials with the

magnification, 10,000 x plus. which is not possible in optical microscopes. Atomic

scale resolution can be, achieved at such higher magnifications.

Electron Microscopes use a beam of highly energetic electrons to examine

objects on a very fine scale. The electrons emitted from the source are accelerated

and shaped by the condenser lenses and objective lens. This lens-system facilitates

changing the current in the electron beam that bombards the sample and controls the

beam convergence. The interaction of the electrons with the matter in the sample

produces different signals (electrons, photons from the infrared through visible till

the X-ray range, etc.) These signals can be used for imaging different sample

characteristics, like surface morphology or distribution of electrically active crystal

defects or local composition, etc. The measured characteristics can be presented

qualitatively and interpreted intuitively or can be presented quantitatively.

SEM can mainly yield the information related to topography, which include

surface features of an object and how it looks, and morphology representing shape

and size of the grains making up the object.

2.2.10 Energy Dispersive X-ray Spectroscopy

Energy dispersive X-ray spectroscopy (EDS) is a technique used for the

compositional analysis of a material. It characterizes a sample through the

interactions between electromagnetic radiation and matter and analyzing the X-rays

emitted by matter in this particular case. As each element of the periodic table has a

unique electronic structure so it has a specific response to electromagnetic waves. In

this technique, the spectroscopic data are plotted as a graph of counts vs energy.

A beam of electrons or photons is bombarded to the sample which is to be

characterized. When an atom is at rest within the sample, it contains ground state

electrons in discrete energy levels around the nucleus. The incident beam may excite

an electron from the inner shell, and results in the formation of an electron-hole

within the atom’s electronic structure. An electron from an outer shell then fills the

hole, and the excess energy of that electron is released in the form of an X-ray. The

X-rays released in this way, create spectral lines that are highly specific to individual

elements. In this manner, the X-rays emission data are analyzed to characterize the

sample.

2.2.11 Fourier Transform Infrared Spectroscopy

Fourier transform infrared (FTIR) spectroscopy studies have been carried out

for the ferrite samples under present investigation to understand their spinel

structures. The FTIR spectroscopy is also highly relevant for surface analysis

because it allows the investigation of the chemical composition and the nature of the

chemical bonds on the outermost layer.

When the molecule is excited with IR radiation, the frequency of the incident

radiation coincides with the vibrational frequency of some part of the molecule.

Hence, resonance occurs and energy is absorbed. This absorbed energy transforms

the vibrational energy of the molecules due to change in dipole moment. When the

molecule returns to the ground state from its excited state, the absorbed energy is

released resulting in distinct IR peaks in the spectrum.

2.3 Electrical Properties

The electrical properties of the investigated spinel ferrites deal with dc

resistivity as a function of composition and temperature, activations energies for

conduction, dielectric constant and dielectric loss tangent as a function of

composition and frequency. It also gives an estimate of the performance as a

microwave magnetic material.

2.3.1 DC resistivity

Ferrites should have high electrical resistivities to eliminate eddy current

losses and allow full penetration of electromagnetic fields through out the solid.

High resistivity is obtained only when a cation has one valance in one lattice site.

The electronic conduction in ferrites is mainly due to hopping of electrons between

the ions of the same elements present in more than one valence state, distributed

randomly over crystallographically equivalent lattice sites [7]. High sintering

temperatures and reducing atmospheres tend to produce mixed valance cations of an

element in the equivalent lattice sites in an otherwise stoichiometric ferrite, thereby

reducing the resistivity.

Ferrites are ionic solids and their electrical resistivity (ρ) drops exponentially

with rising temperature (T), according to the relation [8]

ρ = ρ0 exp (E / KT)

where ρ0 is the resistivity at 0°C

E is the activation energy, i.e., energy needed for an electron to

jump from one ion to the neighbouring ion of same element

and K is the Boltzmann constant

From the plots of log ρ versus 1/T, activation energies are obtained which

explain the involved conduction mechanisms in ferrite [9, 10].

2.3.2 Dielectric constant

The dielectric properties of polycrystalline materials are generally determined

by a combination of various factors like method of preparation, sintering temperature,

grain size, substitutions used, the ratio of Fe3+/Fe2+ ions, and ac conductivity [11,12]

In the present work the dielectric properties of Ni-Cu and Ni-Mg substituted

Lithium ferrites have been measured by using capacitance bridge method. Air-dried

silver epoxy electrical contacts were deposited on the flat surfaces of sintered pellets,

and the dielectric constant ('ε ) and the dielectric loss tangent (tanδ ) were calculated

using the formula

'

o

Cd

Aε

ε=

Where ‘C ’ is the measured capacitance

‘d’ is the thickness of the sample

‘ A’ is the area of the capacitor’s plate

‘ oε ’ is the permittivity of free space and its value is -128.85×10 F/m

It has been known since the investigations of Blechschmidt [10] in 1938, that

ferrites have high dielectric constants at low frequencies falling to very low values of

ten to twenty at microwave frequencies. The low and high frequency values of

dielectric constant and the dispersion behaviour at frequencies smaller than 1010 This

cannot be explained by atomic polarization mechanism. Koops [13] found that the

dielectric constant of ferrite followed much the same course with changing

frequency as did the electrical resistance. To interpret the frequency response of

dielectric constant in ferrite materials, Koops suggested a theory by assuming that

the grains and grain boundaries of ferrite material can be represented the behaviour

of an in homogeneous dielectric structure, as discussed by Maxwell [14] and Wagner

[15], assuming that the same current flows through both the grain and the grain

boundaries.

As there are evident contradictions between Koops theory and experiments,

Krausse [16] developed a new model on the basis of Schottky barrier layer theory in

semiconductors [17]. According to this model, the ferrite structure can be

represented as a series of barrier layer and undisturbed grain material, i.e.,

electrically a series of combination of RC elements. This is analogous to Koops

model, however the poorly conducting layers are not chemical in nature but

correspond to Schottky barrier layers.

2.4 Magnetic Properties of spinel ferrites

The magnetic properties chosen for a theoretical treatment in this section

include saturation magnetization, coercivity, initial permeability and magnetic loss.

The experimentally obtained and theoretically calculated (as per the Neel’s

molecular field theory) magnetic moments of the materials provide valuable

information related to the distribution of cations among the tetrahedral and

octahedral lattice sites of the ferrites investigated. Followed by the complex

permeability measurements on the toroidal ferrite samples up to a few GHz, an

attempt has been made to characterize the material in terms of the Snoek’s product

for its applicability at microwave frequencies.

2.4.1 Saturation Magnetization

Saturation magnetization (Ms) of a material is the resultant dipole moment

per unit volume when the dipole moments associated with all the molecules are

aligned in the direction of the applied magnetic field, and is given by

Ms = N µ m . . . 2.3

where N is number of dipoles, and µ m is the dipole moment.

A magnetic ferrite crystal has domain structure similar to that of

ferromagnetic metals. The magnetization of each domain is associated to the

magnetic moment from the individual ions present in the ferrite, the free oxygen

atom has a partially unfilled 2p sub-shell which gets filled on acquisition of two

electrons when the free oxygen converts to O2− negative ion. The oxygen ions

therefore have no magnetic moment and so make no direct contribution to the

magnetic moment of the domain. The magnetic moments of the cations arise from

unfilled sub-shells. As in ferromagnetic metals and in paramagnetic solids the

magnetic moment due to the orbital motion of the electrons is quenched by internal

fields. The ionic magnetic moments are therefore due to parallel uncompensated

electron spins in the ions. In case of an atom of iron there are four uncompensated

spins in the 3d sub-shell, so that the iron atom has magnetic moment of 4 µB (µB

being the Bohr magneton) due to electron spins. The divalent iron ion having lost

two electrons from the 4s shell has a magnetic moment of 4 µB . On becoming

trivalent it loses as additional electron from the 3d sub-shell increasing the

uncompensated electron spins to 5 and the resultant magnetic moment becomes 5 µB

.

Between the spinning electrons in the neighbouring metal ions strong

quantum mechanical forces of interaction occur. In ferrites, according to Neel [18],

there exist three kinds of exchange interactions: the interaction between the various

magnetic ions located at A-site (A-A interaction), the interaction between the various

magnetic ions located at B-site (b-B interaction), and the interaction of magnetic ions

at A-sites with those at B sites (A-B interaction); of those A-B interaction

predominates in strength over A-A and B-B interactions. These interactions align all

the magnetic spins at A-site in one direction comprising A-sub lattice and those at B-

site in the opposite direction comprising B-sub lattice. The net magnetic moment of

the lattice is therefore the difference between the magnetic moments of B and A sub

lattices, i.e., M = M MB A− .

An interesting aspect is exhibited by the mixed zinc ferrites in which the ionic

distribution is given by [19].

( ) [ ]Zn Fe M Fe Ox x x x

2

1

3

1

2

1

3

4

2+−+

−+

++ −

assuming A-B interaction to be the predominant and that M2+ ions have a magnetic

moment of m units and Fe3+ ions have a theoretical magnetic moment of 5 µB , the

resultant moment per formula unit would be

M = M MB A−

= m (1 - x) + 5(1 + x) - 5(1 - x)

= 10x + m (1 - x)

this relation is shown by the broken lines in fig. 2.1. Experimentally observed

decrease in magnetic moment for x values greater than 0.5 is due to the fact that for

larger amount of zinc, the A-sublattice gets so much diluted by the diamagnetic zinc

that A-B interaction weakens and B-B interaction becomes important. The

magnetization measurements help us in knowing the cation distribution over A-and

B-sites and the role of exchange interactions.

2.4.2 Curie temperature

Magnetic properties of materials arise form unpaired electron spins, that result

in a net magnetic moment. Perfect alignment of magnetic spins of all molecules is

possible only at zero Kelvin. Randomizing effects of thermal motion cause net

magnetic moment to decrease as temperature increases from zero Kelvin; this

decrease becomes more rapid as the temperature nears the Curie point. The Curie

point is the temperature at which the aligning effect of the exchange interaction is

cancelled by the disordering effect of random thermal motion.

E j = E thermal

that is, - 2l j Z S2 = - 9S µB H eff = 3K Tc

where H eff = effective magnetic field,

E j = exchange energy,

Z = number of near neighbours, and

S = spin quantum number.

From the above equation it is obvious that the Curie temperature is

proportional to the exchange energy and the number of nearest magnetic neighbours.

The Curie temperature increases as the number of spins per atom increases, and this

increased spin momentum in turn results in a stronger interaction between the

neighbouring magnetic ions. The idea of Curie temperature gives an estimate of the

temperature at which the device can be operated safely.

2.4.3 Initial permeability

Like all ferromagnetic substances ferrites exhibit hysteresis. The ratio of

magnetic induction (B) to applied magnetic field (H) at a point on the hysteresis

loop, that represents the state of magnetisation of the material, is called the

permeability of the material and is written as

µ = B/H

The value of permeability for an initially unmagnetized ferrite specimen in the

presence of an infinitesimally small magnetic field is called as initial permeability (

µ j ).

Since ferrites are used at high frequencies, it is interesting to see how athe

permeability varies with frequency. According to Snoek [20], ferromagnetic

resonance frequency, fres is connected to the permeability, µ rot and the saturation

magnetization, Ms by the following relation

f res(µ rot -1) = 4/3 γ Ms

Where γ is a constant, called gyromagnetic ratio. For a certain value of

maximum application frequency (fres ), the highest permeability is obtained for the

material with the highest saturation magnetization. Another parameter that

determines µ rot and f res is magnetic anisotropy, K1 ; µ rot increases in inverse

proportion and fres proportionally to the anisotropy.

Polder in his exhaustive work [21] had subsequently separated the complex

permeability into real and imaginary components and established the importance of

imaginary permeability in determining the magnetic losses in ferrite materials.

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