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

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ferrites, but there are also important resonance eﬀects. Table 9.1 shows that the

magnetocrystalline anisotropy of the NiZn ferrites is higher than that of the

MnZn ferrites. An equivalent statement is that when the magnetization vectors

are in the ‘easy’ direction the electron spin moments in NiZn ferrite are in a

higher anisotropy ﬁeld H

than those in MnZn ferrite. In practice the applied

ﬁelds are usually small, and so H

is the dominant ﬁeld.

When excited by an applied alternating magnetic ﬁeld the magnetization

vector will precess around the anisotropy ﬁeld as discussed more fully later

(Section 9.3.4). Resonance occurs when the frequency of the applied ﬁeld

coincides with the natural precessional frequency, i.e. the Larmor frequency

¼ gm

, with the result that the permeability falls and losses increase, as

shown for a family of NiZn ferrites in Fig. 9.29. The onset of such ‘ferrimagnetic

resonances’ restricts the use of MnZn ferrites to frequencies of less than about

2 MHz. At higher frequencies, up to about 200 MHz, compositions from the

NiZn family are used.

There is another, quite distinct, resonance phenomenon concerned with

domain wall movements occurring at approximately one-tenth of the ferrimag-

netic resonance frequency. To understand this Bloch wall motion needs to be

502 MAGNETIC CERAMICS

Fig. 9.29 Magnetic properties of polycrystalline Ni

17d

as functions of d and

frequency: curve A, d ¼ 0: curve B, d ¼ 0.2; curve C, d ¼ 0.36; curve D, d ¼ 0.5; curve E,

d ¼ 0.64; F, d ¼ 0.7 (after [6]).

TEAMFLY

Team-Fly

considered in more detail. The initial part of the virgin magnetization curve is

associated with ‘reversible’ Bloch wall movements; when larger ﬁelds are applied

the movements are irreversible. This is best appreciated by considering the wall

intersecting a second-phase inclusion, say a pore, as indicated in Fig. 9.30.

The pore in Fig. 9.30(a) possesses a high magnetostatic energy by virtue of the

free poles at its surface. When a Bloch wall intersects the pore (Fig. 9.30(b)), this

energy can be reduced. There is therefore a tendency for walls to intersect as

many pores and other second-phase inclusions as possible. The wall is said to be

‘pinned’ at the inclusion, and a certain threshold ﬁeld must be applied to free it.

It is worth noting the close analogy that can be drawn between this situation and

grain boundary movement during sintering.

When a small external ﬁeld is applied the wall is displaced only slightly from

the energy minimum position and, on removal of the ﬁeld, the wall returns nearly

reversibly to its original conﬁguration. When larger ﬁelds are applied the wall

may be displaced over a potential energy barrier to a new distant minimum. On

removal of the ﬁeld, the wall cannot return to its original position because of the

intervening energy barrier. Such wall displacements are thus irreversible. Lattice

strain, lattice defects of all types and second-phase inclusions aﬀect the extent to

which ‘reversible’ wall motion can occur and, in consequence, the magnetic

hardness of the material.

From what has been said it follows that small displacements of a pinned

domain wall introduce restoring forces and, since the wall has inertia and its

movement is accompanied by energy dissipation, an equation of motion can be

written for a sinusoidal applied ﬁeld:

x þ b

x þ cx ¼ 2M

BðtÞð9:32Þ

where x is the displacement normal to the wall, m represents the inertia, b is a

damping coeﬃcient and c is a stiﬀness coeﬃcient. This equation describes

damped forced harmonic motion so that a resonance eﬀect will occur at a

characteristic frequency o ¼ðc=mÞ

1=2

if the damping is small.

PROPERTIES INFLUENCING MAGNETIC BEHAVIOUR 503

Fig. 9.30 Intersection of a Bloch wall by a pore.

Microstructure control can be used to reduce domain wall resonance eﬀects.

As the grain size is reduced the formation of domain walls becomes increasingly

energetically unfavourable (see Section 9.3.2) so that their total eﬀect at

resonance is greatly diminished. Figure 9.31 shows the frequency dependence of

the permeability for two NiZn ferrites with the same composition, resistivity and

density but diﬀering in grain size. The ‘normally’ sintered material has a grain

size of approximately 35 mm and the loss component m@ shows two maxima. The

lower-frequency maximum is due to wall resonance while the higher-frequency

maximum is a ferrimagnetic resonance eﬀect. The hot-pressed material, which

has a grain size of approximately 1 mm, shows no domain wall resonance, only

evidence of the higher-frequency ferrimagnetic resonance. Although the ﬁne-

grained material has a lower permeability, the loss factor (tan d)/m

is improved

over a wide frequency range.

A further resonance eﬀect depends upon the dimensions of the specimen and

so is referred to as ‘dimensional resonance’. The explanation is as follows. In a

loss-free medium the propagation velocity for electromagnetic waves is given by

v ¼ðmeÞ

1=2

¼ fl ð9:33Þ

where f is the frequency and l is the wavelength. For example, a MnZn ferrite

might have m

¼ 10

and e

¼ 5  10

and so, for a frequency of 1.5 MHz,

l ¼ 28 mm. The consequence of this is that if certain dimensions of the specimen

are close to l=2 then a standing-wave system can be established (c.f. Section

5.6.5). Under this resonance condition the net ﬂux in the specimen is zero and so

the eﬀective permeability falls to zero and, if the material is ‘lossy’ both m@ and e@

peak. The eﬀect is shown in Fig. 9.32 for two diﬀerent specimen sizes.

504 MAGNETIC CERAMICS

Fig. 9.31 Dependence of the permeability of the ferrite Ni

0.36

0.64

on frequency and

microstructure: S, normally sintered; HP, hot-pressed (after [9]).

9.3.2 Hard ferrites

Permanent magnetic materials are distinguished from the ‘soft’ variety by their

high coercivity H

, typically above 150 kA m

(see Fig. 9.11). The most common

engineering use to which permanent magnets are put is to establish a magnetic

ﬁeld in a particular region of space. The magnet must be permanent in the sense

that it is capable of withstanding the demagnetizing eﬀect both of its own internal

ﬁeld and from externally applied ﬁelds such as those encountered in d.c. motors.

A magnet is not a store of energy to be tapped as required until exhausted;

rather, its function is to set up a magnetic potential ﬁeld in much the same way as

the earth sets up its gravitational ﬁeld. Work is done as magnetic materials are

moved around in the ﬁeld, but the net energy change is zero.

The operating state of a permanent magnet lies in the second quadrant of

a B–H hysteresis loop since the magnet is always subject to its own

demagnetizing ﬁeld. Apart from its coercivity and remanence, a permanent

magnet material is rated by its ‘maximum energy product’ (BH)

max

The various parameters are now considered separately by reference to the

important hexaferrites.

Remanence (B

)

Remanence is determined partly by the saturation magnetization M

and partly

by the extent to which domain development disorients the magnetization

PROPERTIES INFLUENCING MAGNETIC BEHAVIOUR 505

Fig. 9.32 The eﬀect of dimensional resonance on permeability for two MnZn ferrite

components of diﬀerent sizes; (a) larger and (b) smaller. Permeability values are expressed

relative to those measured at 1 kHz.

vectors on removal of the saturating ﬁeld. There are two general points to be

made. First, because the magnetism in hexaferrites has its origin in

ferrimagnetic coupling and because a large proportion of the ions in the

crystal are non-magnetic (Ba, Sr, O), their saturation magnetization values (and

hence, in general, their B

values as well) are low compared with those of the

metallic magnets. Second, the hexaferrites have high uniaxial anisotropy and

thus B

can be improved by tailoring the microstructure so that the crystallites

are oriented to have their c axes (‘easy’ or ‘preferred’ direction) along a chosen

direction; this is also the case with the well-known metallic cobalt alloys

Columax, Alcomax, etc.

506 MAGNETIC CERAMICS

Fig. 9.33 Microstructure of an oriented barium hexaferrite. (Courtesy of Philips Technical

Review.)

5 mm

Although a single hexaferrite crystal might be anisotropic, a ceramic

comprising a very large number of randomly oriented microcrystals is normally

isotropic. Thus the ceramic has a lower remanent magnetization than that

oﬀered by the intrinsic properties of the material. The situation can be

signiﬁcantly improved by lining up the microcrystals so that their c axes all

point in the direction in which maximum magnetization is required. This is

achieved by subjecting the ferrite particles, usually in the form of a slurry, to a

strong magnetic ﬁeld during the ceramic forming process. Figure 9.33 is a

micrograph showing the alignment of particles and Fig. 9.34 shows the eﬀect on

remanence. As can be seen, it is necessary to distinguish between an oriented

anisotropic and an isotropic hexaferrite ceramic, even though both consist of

anisotropic crystallites.

Coercivity (H

)

The high coercivity of the hexaferrites depends basically on their high

magnetocrystalline anisotropy which results in anisotropy ﬁelds of approxi-

mately 1400 kA m

. However, the coercive ﬁelds achieved in practice are only a

fraction of this value, because smaller ﬁelds can cause Bloch wall movements. It

is found that the coercive ﬁeld is a maximum when the average grain size is

around 1 mm (Fig. 9.35). Reduction in particle size reduces the content of domain

walls because their existence becomes energetically unfavourable when the

volume they would occupy becomes an appreciable fraction of that of a particle

(the width of a domain wall is about 0.1 mm). Also, the small grain size indicates

an absence of grain growth during sintering, and it is during growth that the

grains free themselves from defects such as lattice irregularities, second-phase

PROPERTIES INFLUENCING MAGNETIC BEHAVIOUR 507

Fig. 9.34 Demagnetization curves for an oriented barium hexaferrite (curve A) and an

isotropic hexaferrite (curve B).

inclusions and porosity. Small grains are therefore likely to be highly defective so

that any domain walls are tightly pinned.

‘Maximum energy product’ (BH)

max

Consider a permanent magnet designed with the intention of setting up a

magnetic ﬁeld between its pole pieces (Fig. 9.36).

Applying Ampe

re’s circuit theorem to the magnetic circuit and noting that the

total ﬂux remains constant leads to

ðB

ÞV

¼ðB

ÞV

ð9:34Þ

where B

, B

and H

, H

are respectively the ﬂuxes and ﬁelds in the air gap and

the magnetic material (H

is the demagnetising ﬁeld), and V

and V

are the gap

and material volumes. Since the magnetic energy density is proportional to the

product of B and H, equation (9.34) states that, for a magnet of given volume

, the magnetic energy in the gap is a maximum when the product B

is a

maximum. Therefore (BH)

max

is a ﬁgure of merit for a permanent magnet

508 MAGNETIC CERAMICS

Fig. 9.35 The dependence of coercivity on the particle size of a barium hexaferrite powder.

Fig. 9.36 Permanent magnet with an air gap.

material. As is evident from Fig. 9.34, the energy product is also signiﬁcantly

improved by the particle-orienting fabrication process.

Figure 9.37 shows a typical demagnetization curve for a hard ferrite together

with the energy product curve. A horizontal line from the ( BH)

max

point

intersects the demagnetization curve to deﬁne the optimum working point P. The

magnetic-circuit designer can arrange for the magnet to operate at P by

appropriate design of magnet shape, since this determines the demagnetizing

ﬁeld.

Since at remanence B

¼ m

, where M

is the remanent magnetization, the

demagnetizing ﬁeld is given by

¼N

¼

ð9:35Þ

If H

exceeds a certain ﬁeld strength H

, it will reduce the remanent

magnetization permanently. For the magnet to retain its full strength it is

therefore necessary that H

, i.e. H

or m

. This relation

indicates that shapes with large demagnetizing factors may not retain their

remanent magnetization. As a rough guide H

can be replaced by H

, and it is one

of the advantages of ceramic permanent magnet materials that the value of

is close to unity so that they can be used in shapes with high

demagnetizing factors such as large thin discs. Ceramic magnets in the form of

discs are most eﬃcient when their thickness-to-diameter ratio is 1:2 since they

then operate near the point P in Fig. 9.37, but they can also be used eﬀectively

with much larger aspect ratios (i.e. ﬂatter discs). Ni–Al–Co magnets, however,

have a m

ratio of about 0.1, so that if they are in the form of cubes

 0.3), for example, the remanent magnetization would not be retained.

Such metal magnets operate most eﬃciently when in the form of rods with

length-to-diameter ratios of about 5.

PROPERTIES INFLUENCING MAGNETIC BEHAVIOUR 509

Fig. 9.37 Demagnetization and energy product curves.

510 MAGNETIC CERAMICS

Table 9.3

Typical properties of some important magnetic materials

r max

(tan d=

)/10

/A m

/T T

/8C

r=O m

Soft materials

150 5000

–

2.16 770 106

Fe+4%Si

2000 35 000

–

1.93 690 606

Mumetal

50 000 200 000

–

3.0

0.75 350 556

MnZn ferrites

500–10 000 – 1 at 10 kHz to

50 at 1000 kHz

5–100 0.35–0.50 90–280 0.01–1

NiZn ferrites

10–2000 – 20 at 100 kHz to

200 at 100 MHz

15–1600 0.10–0.40 90–500 10

–10

(BH)

max

/kJ m

/kA m

/8C

Hard materials

Carbon steel

1.6

4.4 0.9

770

Ticonal GX

1.35 860

360

800

1.4

310

Isotropic barium ferrite

6.0

200

0.3

450

Anisotropic (oriented) barium

ferrite

320

0.4

450

77Ni–16Fe–5Cu–2Cr.

Approximately 1% C.

14Ni, 8Al, 24Co and 3Cu.

9.3.3 Summary of properties

Table 9.3 summarizes typical values of the relevant properties of some of the

important magnetic material types.

9.3.4 Microwave ferrites

Conventional electronic circuits handling frequencies of up to, say, 300 MHz

(l  1 m) usually comprise resistors, capacitors, inductors, diodes and transistors

to control energy ﬂow. As the frequency increases so the wavelengths approach

the dimensions of the circuit and microwave techniques dominate. Although

there is no clear deﬁning lower-frequency limit for microwaves, typically they lie

in the range 1–300 GHz (wavelengths in the range 30 cm to 1 mm). In the context

of waveguide technology microwave frequency bands are designated code letters

as listed in Table 9.4.

Microwaves can be propagated down a waveguide, which is simply a metal

‘pipe’, and since the advent of microwave technology materials have been

introduced into waveguides to change their propagation characteristics; devices

such as isolators, gyrators, phase shifters and circulators are based on this

approach. Certain ferrites play an important role in these devices, principally

because their high electrical resistivity coupled with low magnetic losses lead to

what in this context is called ‘low insertion loss’. Therefore microwaves can pass

some considerable distance through the ferrites and, while doing so, are modiﬁed

in a predetermined way by interaction between the magnetic and electric ﬁeld

components of the wave and the magnetic and dielectric properties of the

material; when electromagnetic waves pass through non-magnetic dielectrics (cf.

Chapter 8) it is only the interaction between the electric ﬁeld component and the

material which is signiﬁcant.

When a magnetic ﬁeld H is applied to a spinning electron the angular

momentum vector is inclined at an angle to the ﬁeld direction. Because a

magnetic moment is associated with the angular momentum, the electron

experiences a torque and precesses around the ﬁeld direction with the Larmor

angular frequency o

¼ 2p f

¼ gm

H. This is analogous to the precessional

motion of a spinning top with its axis of rotation inclined to the gravitational

ﬁeld.

PROPERTIES INFLUENCING MAGNETIC BEHAVIOUR 511

Table 9.4 Microwave band designations

Frequency

range/GHz

1.12–

1.70

1.70–

2.60

2.60–

3.95

3.95–

5.85

5.85–

8.20

8.20–

12.4

12.4–

18.0

18.0–

26.5

26.5–

40.0

40.0–

60.0

60.0–

90.0

90.0–

140.0

140.0–

220.0

Designation L R S H C X Ku K Ka U E F G