Crystal structure and properties of ferrites 2.1 …shodhganga.inflibnet.ac.in/bitstream/10603/78861/7/07...[CHAPTER 2] A. A. Birajdar 20 2.1 Crystal structure of ferrite Ferrite is

[CHAPTER 2] A. A. Birajdar

20

2.1 Crystal structure of ferrite

Ferrite is a body-centered cubic (BCC) form of iron, in which a very

small amount (a maximum of 0.02% at 1333°F / 723°C) of carbon is

dissolved. This is far less carbon than can be dissolved in either austenite

or marten site, because the BCC structure has much less interstitial space

than the FCC structure. Ferrite is the component which gives steel and

cast iron their magnetic properties, and is the classic example of a

ferromagnetic material. This is also the reason that tool steel becomes

non-magnetic above the hardening temperature - all of the ferrite has

been converted to austenite. Most "mild" steels (plain carbon steels with

up to about 0.2 wt% C) consist mostly of ferrite, with increasing amounts

of cementite as the carbon content is increased, which together with

ferrite, form the mechanical mixture pearlite. Any iron-carbon alloy will

contain some amount of ferrite if it is allowed to reach equilibrium at

room temperature [1].

Crystal structure and properties of ferrites

CHAPTER 2


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Spinel ferrites crystallize into the spinel structure, which is named

after the mineral spinel, MgAl2O4. Primarily the oxygen ions lattice

determines the spinel crystal structure. The radii of the oxygen ions are

several times larger than the radii of the metallic ions in the compound.

Consequently, the crystal structure can be thought of as being made up of

the closest possible packing of layers of oxygen ions, with the metallic

ions fit in at the interstices A and B. A metallic ion located at the A site

has four nearest oxygen ion neighbors in tetrahedral coordination. The

metallic ion, which is situated at site B, has six nearest oxygen ion

neighbors in octahedral coordination.

Magnetic oxides, which are commonly known as ferrites are

ferrimagnetic in structure as originally proposed by Neel [2]. The ferrites

by virtue of their structure can accommodate a variety of cations at

different sites enabling a wide variation in properties. Further variation in

synthetic methods can bring about large changes in extrinsic properties.

A majority of them are high resistivity materials making them more

suitable for high frequency and low loss applications.

Spinel crystallizes in a close packed cubic structure and structure

was determined first by Bragg [3] and Nishikawa [4]. The unit cell

contains eight molecules and may thus be written as M8Fe16O32. The

crystal structure of spinel ferrite is shown in Fig. 2.1. The white circles in

this figure represent the oxygen ions and the black and hatched circles


22

represents the metal ions. The radius of the oxygen ion is about 1.32 Å,

which is much larger than that of metal ions (0.6-0.8 Å) hence the oxygen

ions in the lattice touch each other and form a close packed face-centered

cubic lattice. In this oxygen lattice the metal ions take interstitial position

which can be classified into two groups, one is a group of lattice sites

called tetrahedral sites or 8A sites, each of which is surrounded by four

oxygen ions as shown by the hatched circles in the Fig. 2.1.

Fig. 2.1: AB2O4 spinel (The red cubes are also contained in the back half

of the unit cell)

The other is a group of sites called octahedral or 16 B-sites, each of

which is surrounded by six oxygen ions as shown by the black circles.

These groups are called tetrahedral (A) sites and octahedral [B] sites.

From the point of view of valence, it seems reasonable to have M2+ ions

on A-sites and Fe3+ ions on B-sites, because of the number of oxygen ions


23

which surround A and B sites are in the ratio 2:3. There are ninety-six

interstitial sites in the unit cell size, 64 tetrahedral, 32 octahedral; of these

8 and 16 respectively are occupied by cations.

2.2 Types of spinel ferrites

The spinels are any of a class of minerals of general formulation

A2+B23+O4

2- which crystallise in the cubic (isometric) crystal system, with

the oxide anions arranged in a cubic close-packed lattice and the cations

A and B occupying some or all of the octahedral and tetrahedral sites in

the lattice. A and B can be divalent, trivalent, or quadrivalent cations,

including magnesium, zinc, iron, manganese, aluminium, chromium,

titanium, and silicon. Although the anion is normally oxide, structures

are also known for the rest of the chalcogenides. A and B can also be the

same metal under different charges, such as the case in Fe3O4 (as

Fe2+Fe23+O4

2-).

Members of the spinel group include [5]:

• Aluminium spinels:

o Spinel – MgAl2O4, after which this class of minerals is

named

o Gahnite - ZnAl2O4

o Hercynite - FeAl2O4


24

• Iron spinels:

o Cuprospinel - CuFe2O4

o Franklinite - (Fe,Mn,Zn)(Fe,Mn)2O4

o Jacobsite - MnFe2O4

o Magnetite - Fe3O4

o Trevorite - NiFe2O4

o Ulvöspinel - TiFe2O4

o Zinc ferrite - (Zn, Fe) Fe2O4

• Chromium spinels:

o Chromite - FeCr2O4

o Magnesiochromite - MgCr2O4

• Others with the spinel structure:

o Forsterite - Mg2SiO4

o Ringwoodite - (Mg,Fe)2SiO4, an abundant olivine polymorph

within the Earth's mantle from about 520 to 660 km depth,

and a rare mineral in meteorites

Cation disorder in multi-site oxides is quantified in terms of an

"inversion parameter" (δ). The spinel structure is cubic, with two distinct

cation sites characterized by different oxygen coordination (octahedral

and tetrahedral). There are twice as many octahedral sites as tetrahedral

sites.


25

Normal spinel: The cation disorder is defined in terms of a "normal"

spinel structure, such as that for ideal MgAl2O4, in which all the Mg

resides on sites tetrahedrally coordinated with oxygen, and all the AI

resides on sites octahedrally coordinated with oxygen. The inversion

parameter is defined relative to this configuration, and is the ratio of the

atomic fraction of AI on tetrahedral sites to the atomic fraction of AI on

octahedral sites. For a perfect normal spinel, the inversion parameter is

0.0. Normal spinel structures are usually cubic closed-packed oxides with

one octahedral and two tetrahedral sites per oxide. The tetrahedral points

are smaller than the octahedral points. B3+ ions occupy the octahedral

holes because of a charge factor, but can only occupy half of the

octahedral holes. A2+ ions occupy 1/8th of the tetrahedral holes. This

maximises the lattice energy if the ions are similar in size. A common

example of a normal spinel is MgAl2O4.

Inverse spinel: For an ideal "inverse" spinel structure (such as for

MgFe2O4), all of the Mg resides on octahedral sites, and the Fe is

distributed equally over the remaining octahedral sites and all of the

tetrahedral sites. In this case the inversion parameter would be 1.0.

Inverse spinel structures however are slightly different in that one must

take into account the crystal field stabilization energies (CFSE) of the

transition metals present. Some ions may have a distinct preference on


26

the octahedral site which is dependent on the d-electron count. If the A2+

ions have a strong preference for the octahedral site, they will force their

way into it and displace half of the B3+ ions from the octahedral sites to

the tetrahedral sites. If the B3+ ions have a low or zero octahedral site

stabilization energy (OSSE), then they have no preference and will adopt

the tetrahedral site. A common example of an inverse spinel is Fe3O4, if

the Fe2+ (A2+) ions are d6 high-spin and the Fe3+ (B3+) ions are d5 high-spin.

Random spinel: For a "random" spinel structure, the cations are equally

distributed over the two sites in ratios proportional to their stoichiometry

and the site ratios. A random spinel structure has an inversion parameter

of (2/3), or 0.667. In spinel ferrites if the divalent metal ions and trivalent

Fe3+ ions are distributed randomly over the tetrahedral and octahedral B-

sites, then the spinel ferrite is called random spinel.

A whole range of possible distribution is observed. This can be

represented in general terms by

−

δ+δ−δ−δ24

BIII1

II1

AIII1

IIO]FeMe[)FeMe( .

where the ions inside the bracket are located in octahedral sites and the

ions outside the brackets in tetrahedral sites.


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2.3 Oxygen parameter (U)

The interstices available in an ideal close packed structure of rigid

oxygen anions can incorporate in the tetrahedral sites, only metal ion

with a radius rtetra ≤ 0.30Å and in octahedral sites, only ions with a radius

roct ≤ 0.55Å. In order to accommodate cations like Co2+, Cu2+, Mg2+, Ni2+

and Zn2+ the lattice has to be expanded. The difference in the expansion

of octahedral and tetrahedral sites is characterized by a parameter called

oxygen parameter (u). In an ideal spinel, the tetrahedral and octahedral

sites are enlarged in the same ratio and accordingly the distance between

the tetrahedral is (0 0 0) and the oxygen site is 3/8 and hence uideal=3/8.

However the incorporation of divalent metal ions in tetrahedral sites

induces a larger expansion of the tetrahedral sites, leading to a large value

for ‘u’ than the ideal value.

The tetrahedral sites are expanded by an equal displacement of the

four oxygen ions onwards, along the body diagonals of the cube, still

occupying the corners of an expanded regular tetrahedron. The four

oxygen ions of the octahedral sites are shifted in such a way that this

oxygen tetrahedron shrink by the same amount as the first expands.

2.4 Magnetic properties

Magnetism is a property of materials that respond at an atomic or

subatomic level to an applied magnetic field. For example, the most well


28

known form of magnetism is ferromagnetism such that some

ferromagnetic materials produce their own persistent magnetic field.

However, all materials are influenced to greater or lesser degree by the

presence of a magnetic field. Some are attracted to a magnetic field

(paramagnetism); others are repulsed by a magnetic field

(diamagnetism); others have a much more complex relationship with an

applied magnetic field. Substances that are negligibly affected by

magnetic fields are known as non-magnetic substances. They include

copper, aluminium, gases, and plastic. The magnetic state (or phase) of a

material depends on temperature (and other variables such as pressure

and applied magnetic field) so that a material may exhibit more than one

form of magnetism depending on its temperature, etc.

The origin of magnetism lies in the orbital and spin motions of electrons

and how the electrons interact with one another. The best way to

introduce the different types of magnetism is to describe how materials

respond to magnetic fields. This may be surprising to some, but all matter

is magnetic. It's just that some materials are much more magnetic than

others. The main distinction is that in some materials there is no

collective interaction of atomic magnetic moments, whereas in other

materials there is a very strong interaction between atomic moments.

The magnetic behavior of materials can be classified into the

following five major groups:


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• Diamagnetism

• Paramagnetism

• Ferromagnetism

• Ferrimagnetism

• Antiferromagnetism

• Diamagnetism: Diamagnetism appears in all materials, and is the

tendency of a material to oppose an applied magnetic field, and therefore,

to be repelled by a magnetic field. However, in a material with

paramagnetic properties (that is, with a tendency to enhance an external

magnetic field), the paramagnetic behavior dominates [6]. Thus, despite

its universal occurrence, diamagnetic behavior is observed only in a

purely diamagnetic material. In a diamagnetic material, there are no

unpaired electrons, so the intrinsic electron magnetic moments cannot

produce any bulk effect. In these cases, the magnetization arises from the

electrons' orbital motions, which can be understood classically as follows:

When a material is put in a magnetic field, the electrons circling

the nucleus will experience, in addition to their Coulomb attraction to

the nucleus, a Lorentz force from the magnetic field. Depending on which

direction the electron is orbiting, this force may increase the centripetal

force on the electrons, pulling them in towards the nucleus, or it may

decrease the force, pulling them away from the nucleus. This effect


30

systematically increases the orbital magnetic moments that were aligned

opposite the field, and decreases the ones aligned parallel to the field (in

accordance with Lenz's law). This results in a small bulk magnetic

moment, with an opposite direction to the applied field.

Fig. 2.2: Hierarchy of types of magnetism [7]

• Paramagnetism: In a paramagnetic material there are unpaired

electrons, i.e. atomic or molecular orbitals with exactly one electron in

them. While paired electrons are required by the Pauli exclusion principle

to have their intrinsic ('spin') magnetic moments pointing in opposite

directions, causing their magnetic fields to cancel out, an unpaired

electron is free to align its magnetic moment in any direction. When an

external magnetic field is applied, these magnetic moments will tend to

align themselves in the same direction as the applied field, thus

reinforcing it.


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• Ferromagnetism: A ferromagnet, like a paramagnetic substance,

has unpaired electrons. However, in addition to the electrons' intrinsic

magnetic moment's tendency to be parallel to an applied field, there is

also in these materials a tendency for these magnetic moments to orient

parallel to each other to maintain a lowered energy state. Thus, even

when the applied field is removed, the electrons in the material maintain

a parallel orientation.

Every ferromagnetic substance has its own individual temperature,

called the Curie temperature, or Curie point, above which it loses its

ferromagnetic properties. This is because the thermal tendency to

disorder overwhelms the energy-lowering due to ferromagnetic order.

Some well-known ferromagnetic materials that exhibit easily

detectable magnetic properties (to form magnets) are nickel, iron, cobalt,

gadolinium and their alloys.

• Ferrimagnetism: Like ferromagnetism, ferrimagnets retain their

magnetization in the absence of a field. However, like antiferromagnets,

neighboring pairs of electron spins like to point in opposite directions.

These two properties are not contradictory, because in the optimal

geometrical arrangement, there is more magnetic moment from the

sublattice of electrons that point in one direction, than from the

sublattice that points in the opposite direction.

[CHAPTER 2]

The first discovered magnetic substance,

believed to be a ferromagnet;

the discovery of ferrimagnetism.

• Antiferromagnetism

ferromagnet, there is a tendency for the intrinsic magnetic moments of

neighboring valence electrons to point in

atoms are arranged in a substance so that each neighbor is 'anti

the substance is antiferromagnetic

magnetic moment, meaning no field is produced by them.

Antiferromagnets are less comm

behaviors, and are mostly observed at low temperatures. In varying

temperatures, antiferromagnets can be seen to exhibit diamagnetic and

ferrimagnetic properties.

In some materials, neighboring electrons want to point in opposite

directions, but there is no geometrical arrangement in which

A.

The first discovered magnetic substance, magnetite, was originally

believed to be a ferromagnet; Louis Néel disproved this, however, with

the discovery of ferrimagnetism.

Fig. 2.3: Ferrimagnetic ordering

Antiferromagnetism: In an antiferromagnet, unlike a


neighboring valence electrons to point in opposite directions. When all

atoms are arranged in a substance so that each neighbor is 'anti

antiferromagnetic. Antiferromagnets have a zero net


Antiferromagnets are less common compared to the other types of



ferrimagnetic properties.


directions, but there is no geometrical arrangement in which

A. A. Birajdar

32

, was originally

disproved this, however, with

In an antiferromagnet, unlike a


directions. When all

atoms are arranged in a substance so that each neighbor is 'anti-aligned',

. Antiferromagnets have a zero net


on compared to the other types of




directions, but there is no geometrical arrangement in which each pair of

[CHAPTER 2]

neighbors is anti-aligned. This is called a

geometrical frustration

Fig. 2.4:

• Magnetic domains:

ferromagnetic material cause them to behave something like tiny

permanent magnets. They stick together and align themselves into small

regions of more or less uniform alignment called

Weiss domains. Magnetic domains can be observed with a

microscope to reveal magnetic domain boundaries that resemble white

lines in the sketch. There are many scientific experiments that can

physically show magnetic fields.

When a domain contains too many molecules, it becomes unstable

and divides into two domains aligned in opposite directions so that they

stick together more stably as shown at the right.

magnetic field, the domain boundaries move so that the domains aligned

with the magnetic field grow and dominate the structure as shown at the

left. When the magnetizing field is removed, the domains may not return

A.

aligned. This is called a spin glass, and is an example of

geometrical frustration

Fig. 2.4: Antiferromagnetic ordering

Magnetic domains: The magnetic moment of atoms in a

material cause them to behave something like tiny


regions of more or less uniform alignment called magnetic domains

. Magnetic domains can be observed with a magnetic force

to reveal magnetic domain boundaries that resemble white


physically show magnetic fields.

hen a domain contains too many molecules, it becomes unstable


stick together more stably as shown at the right. When exposed to a




A. A. Birajdar

33

, and is an example of

The magnetic moment of atoms in a

material cause them to behave something like tiny


magnetic domains or

magnetic force

to reveal magnetic domain boundaries that resemble white


hen a domain contains too many molecules, it becomes unstable


When exposed to a





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to an unmagnetized state. This results in the ferromagnetic material's

being magnetized, forming a permanent magnet.

Fig. 2.5: Magnetic domains in ferromagnetic material.

When magnetized strongly enough that the prevailing domain

overruns all others to result in only one single domain, the material is

magnetically saturated. When a magnetized ferromagnetic material is

heated to the Curie point temperature, the molecules are agitated to the

point that the magnetic domains lose the organization and the magnetic

properties they cause cease. When the material is cooled, this domain

alignment structure spontaneously returns, in a manner roughly

analogous to how a liquid can freeze into a crystalline solid.


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Fig. 2.6: Effect of a magnet on the domains

2.4.1 Magnetic ordering and interactions

In ferrites, the metallic ions occupy two crystallographically

different sites, i.e. octahedral [B] and the tetrahedral (A) site. Three kind

of magnetic interactions are possible, between the metallic ions, through

the intermediate O2- ions, by super-exchange mechanism, namely, A-A

interaction, B-B interaction and A-B interaction.

It has been established experimentally that these interaction

energies are negative, and hence induce an anti-parallel orientation. In

general, the magnitude of the interaction energy between the magnetic

ion, MeI and MeII depends upon (i) the distances from these ions to the


36

oxygen which the interaction occurs and (ii) the angle MeI-O-MeII

represented by the term φ as shown in Fig. 2.7.

Fig. 2.7: Angle φ between MeI and MeII with oxygen ion.

An angle of 1800 will give rise to the greatest exchange energy and

the energy decreases very rapidly with increasing distances. The various

possible configurations of the ions pairs in spinel ferrites with favourable

distances and angle for an effective magnetic interaction as envisaged by

Gorter are given in Fig. 2.8.

Based on the values of the distance and the angle φ, it may be

concluded that, of the three interactions the A-B interaction is of the

greatest magnitude. The two configurations for A-B interaction have

small distances (p,q and q,r) and the values of the angle φ are fairly high.

Of the two configurations for the B-B interaction, only the first one will

be effective since in the second configuration, the distance ‘s’ is too large

for effective interaction. The A-A interaction is the weakest, as the

distance ‘r’ is large and the angle φ≈180 [8].


37

A-B interaction B-B interaction A-A interaction

φ = 1260 φ=1540 φ=900 φ=1250 φ=790

Fig. 2.8: Configuration of ion pairs in spinel ferrites with favourable

distance and angles for effective magnetic interaction.

Thus, with only A-B interaction predominating the spins of the A

and B site ions in ferrites will be oppositely magnetized sublattice A and B

with a resultant magnetic moment equal to the difference between those

of A and B site ions. In general, the value of saturation magnetic moment

for the B lattice (MB) is greater than that of the A lattice (MA), so that the

resultant saturation magnetization (Ms) may be written as

Ms=|MB-MA|

With this theory, Neel could satisfactorily explain the experimentally

observed magnetic susceptibility and magnetic saturation data obtained

for ferrites. Theoretically computed values of the saturation magnetic


38

moments per formula unit of the ferrites agree very well with the

experimental values as seen from Table 2.1.

2.4.2 Magnetization

The magnetization is a powerful tool to study the different

parameters such as domain wall rotation, anisotropy, magnetic hardness

or softness of material, magnetic ordering etc. Ferrites exhibit almost all

the properties similar to that of ferromagnetic materials. When the

magnetic field is applied to the ferromagnetic material, the magnetization

may vary from zero to saturation value. This behaviour is expressed by

Weiss [9] by introducing the idea of existence of domains. According to

Weiss, though each domain is spontaneously magnetized in the direction

of field, magnetization may vary from one domain to another domain. In

general, specimen consists of many domains, in domain configuration i.e.

a function of applied field. The magnetic moment of specimen is a vector

sum of magnetic moment of each domain. As a result the magnetization

or average magnetic moment per unit volume may have value between

zero to saturation.

Studies on magnetic hysteresis of ferrite provide useful information

of the magnetic parameter like saturation magnetization (Ms) coercive

force (HC) and remanence ratio (Mr/Ms). According to the values of these

parameters, the ferrites can be classified as soft and hard ferrites. The


39

ferrites with low coercive force are called soft ferrites and ferrites with

high Hc are called hard ferrites. Soft ferrites are those material which do

not retain permanent magnetism, which provide easy magnetic path.

Hard ferrites retain permanent magnetism and are difficult to magnetize

and demagnetize. According to Neel [2] the coercive force (HC) is related

to saturation magnetization, internal stress, porosity [10] and anisotropy

[11]. The Hysteresis properties are highly sensitive to crystal structure,

heat treatment, chemical composition, porosity and grain size.

2.5 Electrical properties

The ferrospinel compounds are well known for their high electrical

resistivity. Basically ferrites behave like semiconductors. For ferrites, the

resistivity at room temperature can vary, depending upon chemical

composition, between about 104 to 109 ohm-cm. It is known that low

resistivity is caused in particular by the simultaneous presence of ferrous

and ferric ions on equivalent lattice site (octahedral sites) [12]. For

example Fe3O4 at room temperature has resistivity approximately 7×10-3

ohms-cm and NiFe2O4 with some deficiency in iron and sintered in a

sufficiently oxidizing atmosphere so that the product contains no ferrous

ion, can have resistivity higher than 106 ohms-cm. Intermediate values of

resistivity have been given by Koops [13].


40

Ferrite behaves like semiconductors and their resistivity decreases

with increase in temperature according to the relation

)kT/Egexp(0 −ρ=ρ 2.1

where, Eg represent activation energy, which according to Verwey and De

Bore [14] is the energy needed to release an electron from the ion for a

jump to the neighbouring ion. According to Verwey and De Bore the

conductivity of high resistivity oxides can be increased by addition of

small amount of impurities to the structure. The substitutions of cations

of the low valance state give rise to p-type of conduction while the

substitution for cation of high valance state to n-type of conduction [15].

The presence of Fe2+ ions is sometimes desirable as it reduces magneto-

striction and resistivity. In many ferrite systems it is observed that the

slope of logρ vs 1000/T curve changes at the Curie point. The activation

energy increases on changing from ferrimagnetic to paramagnetic region,

this anomaly strongly supports the influence of magnetic ordering upon

the conductivity process in the ferrites.


41

References

[1] http://www.threeplanes.net/ferrite.html dated 16 Nov. 2010.

[2] L. Neel, Ann de Phys, 3 (1948) 137.

[3] W.H. Bragg, “Nature”, London 95, 561, Phil. Mag. 30 (1915) 305.

[4] S. Nishikawa, Proc. Tokyo Math. Phys. Soc . 8 (1915) 199.

[5] http://www.mindat.org/min-29156.html dated 16 Nov. 2010.

[6] C. Westbrook, C. Kaut, Carolyn Kaut-Roth (1998). MRI (Magnetic

Resonance Imaging) in practice (2 ed.). Wiley-Blackwell. p. 217.

[7] HP Meyers (1997). Introductory solid state physics 2 ed. CRC Press.

p. 322.

[8] B. Vishwanathan and V.R.K. Murthy, “Ferrite Material Science and

Technology” Narosa Publishing House, New Delhi (1990).

[9] P. Weiss, J. Phys. 6 (1907) 667.

[10] C.M. Srivastava, M.J. Patani, T.T. Srinivasan, J. Appl. Phys. 53 (1983)

2107.

[11] A.M. Alpr, “High temperature oxides” Academic Press, New York,

25 (1971).

[12] E.J.W. Verwey, P.W. Haaijman, F.R. Romeyn, G.W. Van.

Oostehout. Philips. Res. Rep. 5 (1950) 173

[13] C.G. Koops, Phys. Rev. 83 (1951) 121.

[14] E.J.W. Verwey and J.H. De Boer, Rec. Trav. Chim. Pay. Bas. 55

(1936) 531.

[15] S.L. Snoek, “New Development in ferromagnetic material” Elsevier

publishing co. New York, Amsterdam (1947).


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Table 2.1

Theoretically computed values of the saturation magnetic moments per

formula unit of the ferrites.

Ferrite Magnetic moment per molecule (µB)

Theoretical Experimental

Fe2O3 4 4.1

CoFe2O4 3 3.7

CuFe2O4 1 1.3

MnFe2O4 5 4.6

NiFe2O4 2 2.3

Documents

Crystal structure and properties of ferrites 2.1 …shodhganga.inflibnet.ac.in/bitstream/10603/78861/7/07...[CHAPTER 2] A. A. Birajdar 20 2.1 Crystal structure of ferrite Ferrite is