Moulson A.J., Herbert J.M. Electroceramics: Materials, Properties, Applications

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calculated for any system of currents in vacuo by the vector summation of

induction elements dB, arising from current elements Idl (Fig. 9.2), where

dB ¼



Idl  r



ð9:3Þ

The magnitude of dB is ðm

=4pÞðIdl sin yÞ=r

in which m

, which is termed the

permeability of a vacuum, has the value 4p  10

7

An additional ﬁeld vector, the magnetic ﬁeld intensity H measured in ampe

res

per metre, is deﬁned so that in a vacuum H ¼ B=m

. Therefore, whereas B

depends upon the medium surrounding the wire (a vacuum in the present case),

H depends only upon the current.

In Chapter 2 (Eq. (2.74)) an important relation between D, E and P was

derived by considering the eﬀects of polarization in the dielectric of a parallel-

plate capacitor. An analogous relationship is now derived by considering the

magnetization of a material of a wound toroid of cross-section A and mean

circumference l (Fig. 9.3(a)). Because there is no break in the toroid, and so no

free poles and consequent demagnetizing ﬁelds (see Section 9.1.3), H in the

material must be due to the real currents only. The material becomes magnetized,

i.e. it acquires a magnetic moment per unit volume or magnetization M, and the

472 MAGNETIC CERAMICS

Fig. 9.3 Eﬀects arising from the presence of a magnetic material; I

is the amperian current

per unit length of toroid.

Fig. 9.2 Magnetic induction arising from a current element.

TEAMFLY

Team-Fly

magnetic moment of a volume element arises from microscopic currents, the

‘amperian’ currents.

Application of Ampe

re’s circuit theorem

B  dl ¼ m

total

to the dotted path

(Fig. 9.3(a)) around the coil wound on non-magnetic material gives B ¼ m

nI,

where I is the current and n is the number of turns per unit length. However, if

the coil is wound on magnetic material then, because amperian currents are now

also involved,

B ¼ m

nI þ m

ð9:4Þ

Furthermore, from Fig. 9.3(b), because I

dlA ¼ MdlA, it follows that

B ¼ m

ðH þ MÞð9:5Þ

and, if the magnetization is assumed to be proportional to the magnetizing ﬁeld,

B ¼ m

ðH þ w

HÞð9:6Þ

where w

is the volume magnetic susceptibility. Therefore

B ¼ m

ð1 þ w

ÞH ¼ m

H or mH ð9:7Þ

where m is the permeability and m

¼ 1 þw

is the relative permeability of the

material (the latter is a dimensionless quantity).

In general, the permeability and susceptibility are tensors; an implicit

assumption made throughout the remainder of the chapter is that the vectors

are collinear when the vector notation is dropped.

9.1.3 Shape anisotropy: demagnetization

Measurements of permeability and associated magnetic properties are usually

made on toroids of uniform section when, to a close approximation, the ﬂux

density B is uniform throughout the material and lies entirely within it. In most

practical applications the magnetic circuit is more complex, and variations in

component section and permeability give rise to variations in ﬂux density.

Important eﬀects arise from air gaps, which may be intentionally introduced or

may be cracks or porosity.

The eﬀect of the shape of a specimen on its magnetic behaviour, ‘shape

anisotropy’, is expressed by a demagnetization factor N

. A ﬁeld H

applied to a

solid of arbitrary shape is reduced by a factor proportional to its magnetization

M, so that the eﬀective ﬁeld H

within the body is given by

¼ H

 N

M ð9:8Þ

It follows from Eq. (9.5) that

MAGNETIC CERAMICS: BASIC CONCEPTS 473

¼ H

 N



 H



Multiplying this equation through by m

=B and putting m

¼ B=m

, where m

is the eﬀective relative permeability, gives

þ N



1 



ð9:9Þ

can only be calculated simply if the ﬁeld within the magnetized body is

uniform, and it can be shown that this is the case only when the body is

ellipsoidal. In ferromagnetic or ferrimagnetic bodies the permeability and

magnetization vary markedly with ﬁeld strength (cf. Section 9.1.10), so that M in

Eq. (9.8) will vary in diﬀerent parts of a body if the ﬁeld is not uniform. For a

sphere an exact calculation gives N

¼1/3. A long rod with a length-to-diameter

ratio of r can be approximated by a prolate ellipsoid with a large ratio r between

the major and minor axes. Then



lnð2rÞ1

ð9:10Þ

If r ¼ 10, N

¼ 0:02, so that for m

¼ 1000, m

 50, while for r ¼ 100,

 700.

The magnetic ﬁeld which emanates from regions near the ends of magnets

leads to the concept of magnetic poles which is useful in providing an

approximate model in complex magnetic structures.

A thin ﬂat disc magnetized normally to its plane will have a demagnetization

factor close to unity so that its real permeability must be very high if its eﬀective

permeability is to be appreciable. Minor structural defects, such as ﬁne cracks

normal to the direction of magnetization, set up demagnetizing ﬁelds which can

also markedly reduce eﬀective permeability. The eﬀect of such a crack can be

calculated by considering a toroid of overall length l containing an air gap of

length al, as shown in Fig. 9.4; al is assumed to be very small so that the eﬀective

cross-section A

of the gap, is approximately equal to the cross-section A

of the

toroid. If the toroid carries nI ampe

re turns, then, by Ampe

re’s theorem,

474 MAGNETIC CERAMICS

Fig. 9.4 The eﬀect of an air gap in a toroid.

nI ¼

Hdl ¼ H

lð1  aÞþH

al ¼ H

l ð9:11Þ

where H

and H

are the ﬁelds in the material and the gap respectively and H

the eﬀective ﬁeld. Under the assumption that the total ﬂux through the circuit

remains constant, i.e. B

¼ B

, and that A

¼ A

, where the

subscripts have the same meanings as before, it follows that

þ a



1 



ð9:12Þ

From a comparison of Eqs (9.9) and (9.12) it can be seen that a corresponds to

and that, for m

to be close to m

a must be small compared with 1/m

. For

example, suppose that, for a given material, m

¼1000 and m

is required not to

be less than 0.9m

. Then the maximum permitted crack width in a toroid of length

10 cm, say, is 10 mm. It is evident that if high-permeability ceramics are to be fully

exploited they must be free from cracks normal to the intended ﬂux direction.

9.1.4 Magnetic materials in alternating ¢elds

The opening comments made in the corresponding section on dielectric materials

(Section 2.7.2) apply equally here. Magnetic materials are used extensively as

inductor and transformer cores where they are subjected to alternating magnetic

ﬁelds. There are also many instances where an electromagnetic wave interacts

with a magnetic material, as in microwave devices of various types. Under these

conditions energy is dissipated in the material by various mechanisms and,

analogously to the dielectric case, overall behaviour can be described with the

help of a complex permeability m* ¼ m

 jm@, in which m

and m@ are respectively

the real and imaginary parts of m*. The complex relative permeability is

¼ m

 jm@

Important relationships can be derived by reference to the wound toroid (Fig.

9.3(a)) where the current I is now regarded as sinusoidal and represented by

exp(jot). The appropriate phasor diagram is shown in Fig. 9.5. If the

instantaneous current is I, the corresponding ﬁeld is nI and the ﬂux linking the

circuit is m

IAl. For a current varying sinusoidally the e.m.f. is

U ¼

ðm*n

AlIÞ

¼ jom*n

AlI

¼ jAln

oIm

þ Aln

oIm@ ð9:13Þ

Since m@ is small, from Fig. 9.5 d

ð¼ tan

1

ðm@=m

ÞÞ is also small and therefore the

voltage leads the current by nearly 908.

MAGNETIC CERAMICS: BASIC CONCEPTS 475

The mean power



PP dissipated over a cycle of period t is the mean value of the

product of the current and the ‘in-phase’ voltage component U

, i.e.



PP ¼

dt ¼

lom@ ð9:14Þ



om@ ¼

tan d

ð9:15Þ

where V is the volume of the toroid. m

tan d

is a material constant proportional

to the heat generated per unit volume in a magnetic body in an alternating ﬁeld.

It is not of great practical use unless the ﬁeld at which it is measured is also

speciﬁed, since both m

and tan d

are strongly ﬁeld dependent except at very low

ﬁelds. The behaviour is similar to that of ferroelectric dielectrics with respect to

electric ﬁelds.

In the pot-cores used in inductors (see Section 9.5.1) for LC ﬁlter circuits, the

ﬁelds are low enough for m

and tan d

to vary only gradually with the ﬁeld and

the ratio (tan d

)/m

is sensibly constant. These cores are designed with small

gaps of variable width so that their eﬀective permeability can be adjusted in

accordance with Eq. (9.12). If m

 1 Eq. (9.12) can be expressed in complex

form to take account of losses and rearranged to give

1 þ am*

ð9:16Þ

If m*

and m*

are put in the complex form

¼ m

ð1  j tan d

Þð9:17Þ

and correspondingly for m*

, Eq. (9.16) becomes

ð1  j tan d

Þ¼

f1 þ am

ð1 þ tan

Þj tan d

1 þ 2am

þ a

ð1 þ tan

)

ð9:18Þ

476 MAGNETIC CERAMICS

Fig. 9.5 Phasor diagram showing the components of the voltage U in phase and out of phase

with the driving current I.

Since low loss is essential in pot-core applications, tan

 1 and can be

neglected. Separating real and imaginary parts then gives

1 þ am

tan d

ð1 þ am

ð9:19Þ

so that

tan d

ð9:20Þ

Eq. (9.20) shows that, for a small gap and at low ﬁeld strengths, the ratio (tan d)/

m of a high-permeability low-loss material is independent of the gap width and is

therefore a useful material constant in the pot-core context. It is often referred to

as the magnetic loss factor but is clearly quite diﬀerent in character from the

dielectric loss factor (see Section 2.7.2). It indicates that a reduction in

permeability due to the introduction or enlargement of a gap is accompanied

by a proportionate reduction in tan d

(or increase in Q

9.1.5 Classi¢cation of magnetic materials

There are various types of magnetic material classiﬁed by their magnetic

susceptibilities w

. Most materials are diamagnetic and have very small negative

susceptibilities (about 10

). Examples are the inert gases, hydrogen, many

metals, most non-metals and many organic compounds. In these instances the

electron motions are such that they produce zero net magnetic moment. When a

magnetic ﬁeld is applied to a diamagnetic substance the electron motions are

modiﬁed and a small net magnetization is induced in a sense opposing the

applied ﬁeld. As already mentioned, the eﬀect is very small and of no practical

signiﬁcance in the present context, and is therefore disregarded.

Paramagnetics are those materials in which the atoms have a permanent

magnetic moment arising from spinning and orbiting electrons. An applied ﬁeld

tends to orient the moments and so a resultant is induced in the same sense as

that of the applied ﬁeld. The susceptibilities are therefore positive but again

small, usually in the range 10

–10

. An important feature of many

paramagnetics is that they obey Curie’s law w

/ 1=T, reﬂecting the ordering

eﬀect of the applied ﬁeld opposed by the disordering eﬀect of thermal energy.

The most strongly paramagnetic substances are compounds containing

transition metal or rare earth ions and ferromagnetics and ferrites above their

Curie temperatures.

MAGNETIC CERAMICS: BASIC CONCEPTS 477

Ferromagnetic materials are spontaneously magnetized below a temperature

termed the Curie point or Curie temperature (the two terms are synonymous).

The spontaneous magnetization is not apparent in materials which have not been

exposed to an external ﬁeld because of the formation of small volumes (domains)

of material each having its own direction of magnetization (see Section 9.1.9). In

their lowest energy state the domains are so arranged that their magnetizations

cancel. When a ﬁeld is applied the domains in which the magnetization is more

nearly parallel to the ﬁeld grow at the expense of those with more nearly

antiparallel magnetizations. Since the spontaneous magnetization may be several

orders of magnitude greater than the applied ﬁeld, ferromagnetic materials have

very high permeabilities. When the applied ﬁeld is removed some part of the

induced domain alignment remains so that the body is now a ‘magnet’ in the

ordinary sense of the term. The overall relation between ﬁeld strength and

magnetization is the familiar hysteresis loop as illustrated in Fig. 9.10 below.

Spontaneous magnetization is due to the alignment of uncompensated electron

spins by the strong quantum-mechanical ‘exchange’ forces. It is a relatively rare

phenomenon conﬁned to the elements iron, cobalt, nickel and gadolinium and

certain alloys. One or two ferromagnetic oxides are known, in particular CrO

which is used in recording tapes. These ferromagnetic oxides show metallic-type

conduction and the mechanism underlying their magnetic behaviour is probably

similar to that of magnetic metals.

In antiferromagnetic materials the uncompensated electron spins associated

with neighbouring cations orient themselves, below a temperature known as the

el point, in such a way that their magnetizations neutralize one another so that

the overall magnetization is zero. Metallic manganese and chromium and many

transition metal oxides belong to this class. Their susceptibilities are low (about

) except when the temperature is close to the Ne

el point when the

antiferromagnetic coupling breaks down and the materials become paramagnetic.

Finally, there are the important ferrimagnetic materials, the subject of much of

this text. In these there is antiferromagnetic coupling between cations occupying

crystallographically diﬀerent sites, and the magnetization of one sublattice is

antiparallel to that of another sublattice. Because the two magnetizations are of

unequal strength there is a net spontaneous magnetization. As the temperature is

increased from 0 K the magnetization decreases, reaching zero at what strictly

speaking is the Ne

el point (see Fig. 9.7 below); however, it is commonly called

the Curie point because of the generally close similarity of ferrimagnetic to

ferromagnetic behaviour.

9.1. 6 The paramagnetic e¡ect and spontaneous magnetization

In the interests of simplicity, attention is conﬁned to a crystal consisting of N

atoms per unit volume for which the magnetic moment per atom arises from a

478 MAGNETIC CERAMICS

spinning electron only and the values resolved in the direction of an applied

magnetic ﬁeld are +1 m

. Some of the moments will be directed parallel and

some antiparallel to the ﬁeld and, because there is an energy diﬀerence 2m

between the two states, the relative populations in them will depend upon

temperature in accordance with Boltzmann statistics.

If n" denotes the number of atoms per unit volume with moments in the

direction of B and n# the number in the antiparallel sense, the net induced

magnetization is M ¼ðn"n#Þm

. It follows that

¼ exp







ð9:21Þ

and, since N ¼ n"þn#, it is straightforward to show that

M ¼ Nm

tanh





ð9:22Þ

Under most practical conditions m

B  kT, and then (using the approxima-

tion e

 1 þ x for x  1) it follows that

M 

or w

ð9:23Þ

which is Curie’s law. It should be noted that in the derivation B has been equated

with m

H since m

 1.

In 1907 Weiss suggested that the magnetic ﬁeld H

‘seen’ by an individual

dipole in a solid was the macroscopic internal ﬁeld H modiﬁed by the presence of

dipoles in the neighbourhood of the individual. This idea was expressed by

¼ H þ wM, (c.f. Eq. (2.84)) where w is known as the ‘molecular ﬁeld

constant’. Under this assumption Eq. (9.22) is modiﬁed to

M ¼ Nm

tanh



ðH þ wMÞ



ð9:24Þ

MAGNETIC CERAMICS: BASIC CONCEPTS 479

Fig. 9.6 Graphs illustrating the possibility of spontaneous magnetization below a critical

temperature y

Spontaneous magnetization implies a value for M when H ¼ 0, and the practical

feasibility of this can be explored by putting H ¼ 0 in Eq. (9.24) and then

examining whether the equation can be satisﬁed by non-trivial values for M.

Therefore, putting Nm

¼ M

, where M

is the saturation magnetization, we can

write Eq. (9.24) as

¼ tanh





or, more conveniently,

¼ tanh x ð9:25Þ

where

x ¼

ð9:26Þ

It follows from Eq. (9.26) that

x ¼

x ð9:27Þ

in which y

has the dimensions of temperature and is the Curie temperature.

Figure 9.6 shows the graphs of Eqs (9.25) and (9.27).

It is clear that, for spontaneous magnetization to be possible in principle, Eqs

(9.25) and (9.27) must be consistent, i.e. both must be simultaneously satisﬁed for

particular values of M/M

and x. Also, for small values of x , tanh x  x and so,

near to the origin, M=M

¼ x; therefore for T ¼ y

the equations have the same

slope at the origin. This deﬁnes a temperature y

such that if T4y

(slope greater

than unity) there is no intersection at non-zero values of x and, in consequence,

no spontaneous magnetization is possible. For T5y

there is an intersection and

spontaneous magnetization is a possibility. It follows that, for temperatures up

480 MAGNETIC CERAMICS

Fig. 9.7 The variation of the ratio of saturation magnetization at T K to that at 0 K with the

ratio of the temperature to the Curie temperature.

to T ¼ y

, the M/M

value corresponding to the intersection can be found,

leading to Fig. 9.7 which shows good agreement with experiment for iron, cobalt

and nickel.

According to the Weiss theory just outlined, ferromagnetism is caused by a

strong internal magnetic ﬁeld aligning the magnetic moments on individual ions.

The physical origin of the internal ﬁeld is now known to be quantum mechanical

in nature and to involve the ‘exchange forces’ which determine the relative

orientation of the spins on adjacent electrons. The exchange energy arising from

exchange forces plays a dominating role in determining the nature of the

important magnetic materials. In some instances, e.g. the metals iron, cobalt and

nickel, the exchange energy is minimized for parallel spins; this is exceptional

since, generally, the exchange energy is minimized when adjacent spins are

antiparallel. In the non-metallic antiferromagnetics and ferrites the ordering is

antiparallel through superexchange forces, so-called because they act between

the spins of neighbouring cations with the involvement of an intermediate ion

which, in the case of the ferrites, is oxygen.

9.1.7 Magnetocrystalline anisotrophy

A spinning electron, free from any restraints, can be aligned by an inﬁnitely small

ﬁeld, implying an inﬁnite permeability. A restraint leading to ﬁnite permeabilities

in magnetic materials is caused by a coupling between the spins and the crystal

lattice through the agency of the orbital motion of the electron. This spin–orbit

lattice coupling results in orientation of the spins relative to the crystal lattice in a

minimum energy direction, the so-called ‘easy direction’ of magnetization.

Aligning the spins in any other direction leads to an increase in energy, the

anisotropy energy E

. For a cubic lattice, such as a spinel, E

is related to two

anisotropy constants K

and K

¼ K

ða

þ a

ÞþK

 ð9:28Þ

where a

, a

and a

are the direction cosines of the magnetization vector relative

to the crystallographic axes.

The approximate expression for E

(Eq. (9.28)) leaves out all but two of the

terms of an inﬁnite power series, and even the term involving K

can often be

safely neglected. If it is assumed that K

is negligible and K

is positive, then E

has a minimum value of zero if any two of the direction cosines are zero, i.e. the

anisotropy energy is a minimum along all three crystal axes and these are

therefore the ‘easy’ directions. If K

is negative, the minimum occurs for

¼ a

¼ 1=

3, i.e. the body diagonal is the ‘easy’ direction.

The anisotropy constants listed in Table 9.1, which anticipate the discussion of

ferrites (Section 9.2 et seq.), indicate that the easy directions for cubic crystals are

[111] except for those containing cobalt, in which case they are [100]. For

MAGNETIC CERAMICS: BASIC CONCEPTS 481