Buschow K.H.J. (Ed.) Concise Encyclopedia of Magnetic and Superconducting Materials

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prehistory, a simple exponential time law

JðtÞ¼Jð0ÞþJ

ð1  e

t=t

Þð2Þ

with a single relaxation time t and some constant J

or a sum of such exponential terms or the logarithmic

law of Eqn. (1), etc., have been found. As an example,

the viscosity experiment on an aFe(C) sample

(Chikazumi 1997) can well be described by Eqn. (2),

and the temperature dependence of t is given by

t ¼ t

expðE=kTÞð3Þ

where k is the Boltzmann constant and t

and E

are constants (see Fig. 3). It was shown by H. A.

Kramers in 1940 that the Arrhenius formula (Eqn.

(3)) results from thermally activated species escaping

from a metastable state over an energy barrier E.

Obviously the diffusion of carbon in aFe(C) is con-

trolled by such an energy barrier.

2. Irreversible Magnetic Aftereffect

The irreversible aftereffect or ﬂuctuation aftereffect

occurs in all ferromagnetic substances and dominates

in permanent magnet materials. This effect was dis-

covered by H. Jordan in 1924 who found a ﬁeld- and

a frequency-independent contribution to the losses of

a coil with an iron core. The ﬁrst theoretical analysis

of this effect was done by F. Preisach in 1935 who

showed that the behavior of the losses is physically

equivalent to the logarithmic time law of Eqn. (1) for

the magnetization. Preisach also noticed that con-

trary to the reversible aftereffect, the irreversible

aftereffect does not fulﬁll a superposition principle

with respect to a set of succeeding ﬁeld jumps.

He suggested that the latter effect is based on

thermal ﬂuctuations resulting in irreversible changes

of magnetization and thus partially destroying the

memory of the magnetic prehistory of the sample.

Starting with the hypothesis that the irreversible af-

tereffect and hysteretic magnetization vs. ﬁeld curves

should be based on the same elementary magnetiza-

tion processes, Preisach introduced a simple model

considering ferromagnetic materials as collections of

bistable units each of them characterized by its indi-

vidual rectangular nonsymmetric hysteresis loop

(see Fig. 4). If the state of such an individual unit is

on its upper branch (j

) or its lower branch (j

) and

the applied ﬁeld, H, is near the critical value ba or

b þa (see Fig. 4), respectively, very small changes of

H will result in switching of that unit and, conse-

quently, will contribute to an irreversible change of

the total magnetization, J, along macroscopic hys-

teresis curves of the material. In such a critical state

of the bistable unit switching will also be caused

by small thermal ﬂuctuations, thus contributing to

the aftereffect, i.e., to a spontaneous change of J at

constant H. There has been a resurgence of interest in

Preisach-type models (Bertotti 1998) and they have

been widely used to describe hysteresis as well as re-

laxation phenomena.

The role of thermal ﬂuctuations for the irreversible

aftereffect was also emphasized by L. Ne

el in 1951

who introduced the phenomenological relationship

S ¼ m

irr

ð4Þ

where m is the permeability of vacuum. The irre-

versible part of the susceptibility, w

irr

, has to be

determined from the experimental quantities w (sus-

ceptibility) and w

rev

(reversible part of the susceptibil-

ity) by w

irr

¼ww

rev

, and the viscosity coefﬁcient, S

represents some intrinsic ﬂuctuation ﬁeld. The typical

example of Fig. 5 shows that, as expected, S

does not

vary much on a considered hysteresis curve, contrary

to the behavior of w

irr

and S. In a theoretical analysis

Street et al. (1987) found that the quantities in Eqn.

(4) depend on the demagnetization factor, D, which

characterizes the shape of the sample investigated.

Figure 3

Temperature dependence of the relaxation time t

(according to Eqn. (2)) for an aFe(C) sample (after

Chikazumi 1997).

Figure 4

Hysteresis loops of two individual bistable units of a

Preisach model with the parameters j

, a, and b: (a) b40,

a4b; (b) bo0, ao7b7.

720

Magnetic Vi scosity

Therefore, for D a 0, Eqn. (4) has to be replaced by

S ¼ S

irr

1  Dw

rev

ð5Þ

where now the viscosity coefﬁcient, S

, is independent

of D (in its range 0pDp1).

This has been experimentally veriﬁed for the

‘‘spring-magnet’’ material Nd

(Mu

ller et al.

1994). As can be seen from Fig. 6 the values of w

rev

and w

irr

of this material are very sensitive to D, but S

does not depend much on D if calculated by Eqn. (5).

The correlation of the irreversible aftereffect with

hysteresis was investigated earlier by J. C. Barbier

in 1954 who found a universal relationship between

and the coercive ﬁeld,

, for a large class of

materials in which both quantities vary over several

orders of magnitude (see Fig. 7). From these obser-

vations it can be concluded that the magnetic viscos-

ity, just as the coercivity, depends on details of the

chemical and topological microstructure of the ma-

terial considered and modeling of both quantities will

be difﬁcult because these microstructural features

have to be taken into account.

One of the few model systems for which the prob-

lem of slow magnetic relaxation has been exactly

solved is an assembly of uniformly oriented small,

spherical, noninteracting ferromagnetic particles of

equal size, with uniaxial magnetic anisotropy. The

particles are characterized, in the continuum theory

called micromagnetism, by their spontaneous magnet-

ization, J

, anisotropy constant, K, and exchange con-

stant, A. The only microstructural parameter of the

model is the particle volume, v (or the corresponding

diameter, D). For D less than a critical size

¼ cm

ðAKÞ

1=2

2

ð6Þ

where c is about 70, the particles are single domain.

For example, for Nd

B the value of D

is about

300 nm. Applying Brownian motion theory it has

been shown that for sufﬁciently low temperature, T,

and sufﬁciently large time, t, the relaxation of such a

Figure 5

Viscosity, S, and irreversible susceptibility, w

irr

,ofa

sintered NdFeB permanent magnet at 300 K. The

measurements are done on the major demagnetization

curve (i.e., starting with a high ﬁeld m

H ¼5 T). The

maximum of S and w

irr

is near the coercive ﬁeld of the

material.

Figure 7

Barbier plot showing the correlation between the

viscosity coefﬁcient, S

, and the coercive ﬁeld,

, for a

large class of magnetic materials. The straight line

corresponds to S

1:35

Figure 6

(a) Viscosity coefﬁcient, S

, (b) susceptibility w ¼@J/@H

and reversible part of the susceptibility, w

rev

, vs.

polarization, J, for a Nd

sample (same sample

as in Fig. 1), measured on major demagnetization curves

for two directions with the demagnetization factors

D ¼0.08 and 0.84. S

in (a) is calculated according to

Eqn. (5) (ﬁlled symbols) and Eqn. (4) (open symbols).

721

Magnetic Viscosity

system is well described by Eqns. (2) and (3) where

E ¼ Kv ð7Þ

¼ c



1=2

ð8Þ

where _ is Planck’s constant, m

the Bohr magneton,

and c

a number of the order of 1 (Brown 1979,

Aharoni 1996). For an Nd

B particle of few

nanometers in size the room temperature value of t

is about 10

11

s. It should be noted that in this ap-

proach T enters twice, explicitly in the Arrhenius

factor of Eqn. (3) and implicitly via the temperature

dependence of the material parameters J

, K, and A.

For real ferromagnetic materials the exponential

decay (Eqn. (2)) of magnetization is usually not ob-

served. An often cited example is shown in Fig. 8,

where the time dependence of the magnetization ob-

viously obeys Eqn. (1) and the viscosity, S, strongly

decreases with decreasing temperature. It is well

known that Eqn. (2) transforms into Eqn. (1) if a

relatively broad distribution, f(E), of activation en-

ergies has to be taken into account instead of a single

value, E (Street and Wooley 1949, Gaunt 1976). An

alternative explanation of the origin of logarithmic

relaxation observed in many complex systems has

been proposed by Chudnovsky and Tejada (1998).

They assume that the slow relaxation is governed by

some critical state of the system in which the energy

barriers in the Arrhenius relationship of Eqn. (3)

depend on the relaxing magnetization, J, e.g.,

EBðJ  J

Þð9Þ

If J exceeds the critical value J

the fast initial change

of J(t) stops and, as time goes on, the magnetization

grows logarithmically because the system arrives at

greater and greater barriers that are more and more

difﬁcult to overcome.

To analyze the magnetic viscosity of arbitrary

ferromagnetic materials the concept of an activa-

tion volume, v

, has been introduced by Street and

Woolley (1949):

¼ kT ð10Þ

According to Eqns. (3) and (7) the energy barrier

E ¼kT being relevant for the activation of a single-

domain particle at temperature T can be written as

K/J

. A comparison of this expression with Eqn.

(10) shows that the anisotropy ﬁeld, K/J

, is replaced

by Ne

el’s ﬂuctuation ﬁeld, S

, from Eqn. (5) and the

particle volume, v, by the activation volume, v

. The-

oretical and experimental analysis has revealed that

is representative for typical elementary magnetiza-

tion processes such as nucleation of reverse domains

or depinning of domain walls. Its value is typically

some hundreds of cubic nanometers and the phen-

omenological relationship v

B(A/K)

3/2

has been

found, where d is the domain wall width of the ma-

terial (Givord et al. 1987, Mu

ller et al. 1994).

3. Anomalous Aftereffect and Thermal

Remagnetization

If a viscosity experiment is dominated by the revers-

ible aftereffect the superposition principle is fulﬁlled

and, consequently, after a sequence of ﬁeld jumps,

(DH)

, with i ¼1, 2,ym, the rate, dJ(t)/dt, may have

a sign different from that of the last ﬁeld jump,

(DH)

,orJ(t) may even be a nonmonotonic function.

The ﬁrst observation of such a behavior, called the

anomalous magnetic aftereffect, was made by A.

Mitkevitch in 1936 who measured J(t)inaFe(C) after

two ﬁeld jumps (DH)

o0 and (DH)

40 had been

applied to the sample.

It had been believed that in viscosity experiments

based on the irreversible magnetic aftereffect, the re-

sponse of magnetization to a sudden change in the

applied magnetic ﬁeld is not very sensitive to the

magnetic prehistory of the samples investigated.

However, it was shown by Folks et al. (1994) by vis-

cosity experiments on recoil curves of a ﬁne-grained,

two-phase permanent magnet (similar to the sample

of Fig. 1) that some residual information on the

magnetic prehistory may be stored even if the super-

position principle is not valid. As shown in Fig. 9, the

viscosity, S, changes its sign if the starting coordinate

of the viscosity experiment is moved along the recoil

curve. It should be noted that although the two re-

laxation curves of Fig. 9 are totally different, the

magnetic prehistory of the sample is quite similar for

Figure 8

Time dependence of the magnetization of an Alnico

permanent magnet measured by Street and Wooley

(1949) for various temperatures.

722

Magnetic Vi scosity

both cases. The data of both Figs. 9(a) and (b) obey

Eqn. (1). However, the magnitude, 7S7, of the vis-

cosity is considerably smaller compared to the con-

ventional aftereffect experiment of Fig. 1(b). The ﬁt

yields very different effective waiting times, t

, al-

though all the measuring procedures were done on

the same timescale. Obviously the large value of t

Fig. 9(a) represents the more complicated magnetic

prehistory compared to that corresponding to Fig. 1.

If the coordinate on the recoil curve is appropriately

chosen the material of Fig. 1 even shows an anom-

alous magnetic aftereffect, i.e., the magnetization is a

nonmonotonic function of time.

The ﬁnal coordinate * on the recoil curve which

starts at the coordinate J in Fig. 1(a) represents the

d.c.-demagnetized sample. The time dependence of

the magnetization of such a sample is shown in

Fig. 10(a). As expected from Fig. 9(b) the sample re-

magnetizes spontaneously, i.e., without an external

ﬁeld. Seemingly the system moves away from the state

of thermal equilibrium where the sample magnetiza-

tion, J, is zero. As shown in Fig. 10(b), a much

stronger spontaneous increase of magnetization is ob-

served if the d.c. ﬁeld-demagnetized sample is heated

and subsequently cooled to the starting temperature.

This effect is called thermal remagnetization and was

discovered by Kavalerova et al. (1975) for SmCo

permanent magnets. In principle, the results of Figs. 9

and 10 are understandable for systems with different

relaxation times. If in such a system appropriate in-

itial conditions are realized by an appropriate mag-

netic history, the magnetization, J, may, in certain

periods of time, move away from its equilibrium value

and the time dependence, J(t), may be nonmonotonic.

However, as has been pointed out by Preisach and

has been proved for small particles (Brown 1979,

Aharoni 1996), a remarkable relaxation of J will only

be observed if the external ﬁeld, H, is close to the

coercive ﬁeld of the sample or to some switching ﬁeld

of a subsystem. If, for example, the Preisach units

of Fig. 4 are in the state j ¼j

, relaxation will only

occur if the applied ﬁeld, H, is close to (a þb), etc.

However, it is well known that, for general reasons of

symmetry, noninteracting subunits of a system (e.g.,

particles) have symmetric whysteresis loops. Conse-

quently, in such systems H40 will always result in

dJ/dt40 and Ho0indJ/dto0. Thus the results pre-

sented in Figs. 9 and 10 are due to interaction of

subunits. The coupling of the subunits can be caused

by exchange or by magnetostatic interaction and its

effect on slow magnetic relaxation will be particularly

pronounced if the Preisach subunits vary much in

their ‘‘hardness,’’ i.e., in the parameter a in Fig. 4. As

a typical example, the so-called spring-magnet mate-

rial for which the data of Figs. 1, 6, 9, and 10 have

been measured consists of a relatively hard and a

relatively soft magnetic phase and both phases are

assumed to be exchange coupled to each other. (See

Magnets: Remanence-enhanced)

Hence, if the hard units are predominantly in their

upper state þj

, typical soft units will be shifted to

the left-hand half of the Preisach plane (see Fig. 4(b))

and even at Hp0 (as in the coordinates m and * in

Fig. 1(a)) they may switch from j

to þj

thus con-

tributing to dJ/dt40 of the sample. The enhance-

ment of this effect by heating (Fig. 10(b)) can be

understood by the fact that the anisotropy constant,

K (and consequently the parameter a in Fig. 4), de-

creases more strongly with increasing temperature

than the strength of interaction (i.e., parameter b in

Fig. 4). Therefore, heating has a similar effect on the

subunit in Fig. 4(b) as an increase of the ﬁeld, H,

whereas the relatively hard subunits will not be as

much inﬂuenced by heating.

The relaxation behavior of a Preisach system

(Bertotti 1998) is demonstrated in Fig. 11. The ma-

terial is described by a probability distribution p(a, b)

for the parameters a and b of Fig. 4. In an applied

ﬁeld, H, all Preisach units with boH – a have the

magnetization þj

, those with b4H þa have the

magnetization j

. The state of the remaining units

depends on the prehistory of the system. The time

dependence of the assembly is determined by the

Figure 9

Magnetization, J, vs. time, t, for the sample of Fig. 1 after

two ﬁeld jumps characterized by the markers (a) Jand

* (in Fig. 1) and (b) Jand m (in Fig. 1) have been

applied. The parameters S and t

are ﬁtted to Eqn. (1).

Figure 10

Remagnetization phenomena in the sample of Fig. 1

after d.c. ﬁeld demagnetization characterized by the

markers J and * (in Fig. 1): (a) spontaneous

remagnetization at room temperature (where the sample

had been demagnetized); (b) thermal remagnetization.

723

Magnetic Viscosity

border line L( a, t), which separates the units with þj

from those with j

. This border line and p(a, b) de-

termine the magnetization of the assembly. The bor-

der line L(a, 0) for a sequence of ﬁeld jumps in the

applied ﬁeld from a large positive value of H to H

and then to H

, immediately after the jump to H

,is

shown in Fig. 11 (H

, H

corresponds to J in

Fig. 1). The equilibrium state is given by L(a,

N) ¼H

and is reached by a motion of the border

line owing to thermally activated reversal processes.

The evolution equation for L(a, t) has been derived

within generalized thermodynamics, treating the bor-

der line as an ‘‘internal degree of freedom’’ and using

an Arrhenius-type equation for the magnetization

reversal of the single units corresponding to the

points (a, b) near the border line. For the sequence of

ﬁeld jumps described above the two branches of the

border line L(a, 0) in Fig. 11 result in contributions of

different sign to the magnetization change in time.

The units with ao(H

H

)/2 (hatched region) have

smaller energy barriers than those with a4(H

H

and the typical time constants for the units within the

hatched region are smaller. Therefore, in the early

stages the units in the hatched region will change

more rapidly and the magnetization will ﬁrst move in

a positive direction and later it will decrease.

4. Quantum Tunneling of Magnetization

As discussed above, most experimental results on

viscosity experiments, e.g., those presented in Figs. 3

and 8, suggest that the reversible as well as the irre-

versible magnetic aftereffects are based on thermally

activated processes and, consequently, both effects

should disappear at very low temperatures. Contrary

to this prediction Lambeck (1971) found for various

materials that the magnetic viscosity, S, does not go

to zero but approaches a ﬁnite constant value at low

temperatures and it is assumed that the reason for

this observation is quantum tunneling through po-

tential barriers, as predicted by H. A. Kramers in

1940 for certain types of metastable states. It has been

proved theoretically (Caldeira and Leggett 1981) that

for quantum tunneling of macroscopic quantities,

such as the magnetization, the tunneling probability

may be enhanced by the dissipative coupling of the

macroscopic coordinates to the environment. How-

ever, it has been shown by Barbara and Gunther

(1993) that there are situations (e.g., particular dis-

tributions of energy barriers) where a plateau in S(T)

can also arise under purely classical thermal activa-

tion. In spite of this restriction a ﬁnite magnetic vis-

cosity at low temperatures caused by quantum

tunneling has been clearly demonstrated for various

materials (Chudnovsky and Tejada 1998). For exam-

ple, the magnetic relaxation in TbFe

ﬁlms is found to

be perfectly logarithmic and the magnetic viscosity,

S, deﬁned by Eqn. (1), indicates a crossover from

thermal to temperature-independent relaxation at

¼6 K (see Fig. 12).

See also: Coercivity Mechanisms; Magnetic Materi-

als: Hard; Magnetic Hysteresis

Figure 11

In the Preisach plane (a, b) the state of the collection of

units of Fig. 4 is described by the border line L(a, t)

separating the units with þj

(region with þsigns) from

those with j

(regions with – signs). The bold line

represents L(a, t ¼0) immediately after a sequence of

ﬁeld jumps H ¼N-H

-H

, where H

corresponds to the marker J in Fig. 1). L(a, N) is the

equilibrium line. )indicates the movement of L(a, t).

The units in the hatched region have lower energy

barriers than those near the branch of the border line

with positive slope.

Figure 12

Low-temperature dependence of the magnetic viscosity

of a TbFe

thin ﬁlm. T

is the crossover temperature

below which quantum tunneling of magnetization is

assumed to dominate. The solid lines are guides to the

eye (after Chudnovsky and Tejada 1998).

724

Magnetic Vi scosity

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Diego, CA

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IFW Dresden, Germany

Magnetism in Solids: General Introduction

Magnetism represents a class of fascinating physical

phenomena that is intimately connected with the

behavior of electrons in materials. Nuclear particles

with spin also give rise to magnetic phenomena, but

these are characterized by a much weaker (nearly

2000 times) magnetic response. This article is mostly

devoted to ‘‘electronic’’ magnetism.

Although properties of the lodestone (magnetite)

were already known to the Ancient Greeks by 800 bc ,

magnetism remains even nowadays a ﬁeld of perma-

nently growing interest. The interest is on the one

hand motivated by outstanding magnetic and related

electronic properties of novel materials which ﬁnd

frequent use in everyday modern life.

On the other hand, open questions about the elec-

tronic structure of materials and its projection in

magnetic, magnetoelastic (see Magnetoelastic Phe-

nomena), magnetotransport (see Magnetoresistance:

Magnetic and Nonmagnetic Intermetallic; Giant Mag-

netoresistance: Metamagnetic Transitions in Metallic

Antiferromagnets), and magnetocaloric (see Magne-

tocaloric Effect: From Theory to Practice) phenom-

ena represent a serious challenge both to theorists

and experimentalists active in research into con-

densed matter.

For studies of the relationship between magnetism

and electronic structure investigation of materials ex-

posed to extreme conditions, such as high magnetic

ﬁeld or high external pressure, is frequently helpful. In

the former, the application of a high magnetic ﬁelds

allows, for example, the determination of the energy

of magnetocrystalline anisotropy and exchange inter-

actions between magnetic moments or magnetic-ﬁeld-

induced changes of electronic structure connected,

e.g., with itinerant electron metamagnetism (see Mag-

netism: High-ﬁeld and Itinerant Electron Systems:

Magnetism (Ferromagnetism)). Application of exter-

nal pressure, on the other hand, facilitates experi-

ments studying the effects of varying interatomic

distances on exchange interactions and other charac-

teristic features of electronic structure (see Magnetic

Systems: External Pressure-induced Phenomena).

This article is devoted to a brief general introduc-

tion to the physics of magnetism in solids. In Sect. 1,

basic categories of magnetic behavior of materials are

presented phenomenologically. Sections 2 and 3 point

to microscopic mechanisms responsible for the bulk

magnetic properties. Basic information on magnetic

phase transitions and related phenomena is given in

Sect. 4, while Sect. 5 focuses on magnetocrystalline

anisotropy, its origin, and consequences. Basic mod-

els in magnetism are introduced in Sect. 6.

Where desirable, cross-references are given to ar-

ticles covering particular subjects. For detailed study

of various aspects of magnetism in solids several

725

Magnetism in Solids: General Introduction

literature sources are recommended. Besides the ex-

tended chapters on magnetism that can be found in

widely known text-books on solid-state physics (Kittel

1976, Ashcroft and Mermin 1988), several books con-

centrating on magnetism are recommended (Chika-

zumi 1964, Cracknel 1975, Barbara et al. 1988, Gilles

1994) including the theory of magnetism (Mattis 1981,

Yosida 1996). As a useful reference source the con-

tinuously updated series edited by Buschow (e.g.,

Buschow and Wohlfarth 1993) may be used. In the

books listed above information can also be found on

further aspects of magnetism in materials.

The most up-to-date information on the research

in the ﬁeld of magnetism can be obtained from

numerous journals (e.g., Journal of Magnetism and

Magnetic Materials, Physical Review B, Europhysics

Journal B (the follower of the Zeitschrift fu

r Physik B

and Journal de Physique), Journal of Physics: Con-

densed Matter and Journal of Japanese Physical So-

ciety) and at international conferences devoted to

physics of magnetism as, e.g., the triennial Interna-

tional Conference of Magnetism (ICM), which was

held in 2000 in Recife (Brazil) with proceedings pub-

lished in the Journal of Magnetism and Magnetic

Materials; the annual conference of the American

Institute of Physics—the Conference on Magnetism

and Magnetic Materials (MMM) which has proceed-

ings regularly published in the Journal of Applied

Physics (the last MMM was held in 2001 in San

Antonio). Since different units can still be found in

different literature, the reader is referred to Magnetic

Units for further information.

1. Basic Types of Magnet ic Behavior

Initial information on magnetic behavior of a partic-

ular material is usually obtained from measurements

of the magnetic response of a sample to an applied

magnetic ﬁeld H. The bulk magnetization M is stud-

ied by magnetometers that measure the force exerted

on a sample in an inhomogeneous external magnetic

ﬁeld or electrical current induced in pick-up coils

by moving through a magnetized sample (Cahn and

Lifshin 1993, Chikazumi 1964).

The rate of the magnetization change with the

magnetic ﬁeld

¼ w H

ð1Þ

determines the magnetic susceptibility w. Detailed

knowledge of the magnetic ﬁeld and temperature de-

pendence of magnetization M(H, T) and magnetic

susceptibility w(T), which are the basic magnetic char-

acteristics of a material, frequently allows the drawing

of conclusions about the type of magnetism, especial-

ly in simple systems. The basic categories of magnetic

behavior of materials are diamagnetism, paramagnet-

ism, ferromagnetism, antiferromagnetism, ferrimag-

netism, and metamagnetism.

1.1 Diamagnetism

Regarding the sign of magnetic susceptibility, a neg-

ative material response to an external magnetic ﬁeld

(wo0) is classiﬁed as diamagnetic. If a diamagnet is

exposed to an external magnetic ﬁeld a negative

magnetization (see Fig. 1) is induced due to a moving

electron charge in a manner described by Lenz’s law

of electromagnetism. Consequently a negative sus-

ceptibility

w ¼

NZe

S ð2Þ

is observed, where n is the number of atoms per unit

volume, Z is the number of electrons in ﬁlled shells

per atom, e and m

are the electronic charge and

mass, respectively, /r

S is the root mean square

atomic radius (typically 10

21

m), and m

is the per-

meability of the vacuum.

The susceptibility of a diamagnet is very weak and

essentially independent of temperature. Typical ex-

amples of diamagnets are copper, silver, gold, bis-

muth, beryllium, and germanium with values of

susceptibility ranging from 5.6 10

7

(Ge) to

2.7 10

6

(Au). Generally speaking, diamagnetism

is present in all substances under applied magnetic

ﬁeld due to inevitable electromagnetic induction in a

moving electron charge system. Nevertheless, it can

be well detected only in materials containing atoms

without permanent magnetic moments. Permanent

magnetic moments, which are associated with incom-

plete d- and f-electron shells (see Sect. 2 and Localized

4f and 5f Moments: Magnetism), yield a much strong-

er positive susceptibility.

A weak temperature-independent diamagnetic sus-

ceptibility (Landau diamagnetism) is produced by

conduction electrons in metals (Kittel 1976, Ashcroft

and Mermin 1988).

Figure 1

Typical magnetic ﬁeld dependence of magnetization for

a diamagnet (full line).

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Magnetism in Solids: General Introduction

Superconductors exhibit strong diamagnetism with

the susceptibility reaching w ¼1 (value for an ideal

diamagnet). The diamagnetic susceptibility of a su-

perconductor is strongly temperature-dependent and

vanishes at temperatures above the superconducting

temperature (Cahn and Lifshin 1993, also see Electro-

dynamics of Superconductors: Flux Properties).

1.2 Paramagnetism

Paramagnets are generally known as materials with

atoms (ions) carrying weakly interacting permanent

magnetic moments. As a consequence of negligible

exchange interactions between moments a true para-

magnet shows no sign of magnetic ordering down to

the lowest temperatures. In sufﬁciently low magnetic

ﬁelds and at not too low temperatures m

H/k

T51

is the Boltzmann constant and m

is the Bohr

magneton). The magnetization of a paramagnet var-

ies linearly with the applied magnetic ﬁeld yielding a

ﬁeld-independent susceptibility (see Fig. 2).

The magnetic susceptibility of a paramagnet is

positive and strongly dependent on temperature. As a

rule the susceptibility of a paramagnet follows the

Curie law

w ¼

ð3Þ

where C, the Curie constant, is proportional to the

sum of the squares of the effective moments, m

eff

,ina

unit volume. The effective magnetic moment of an

atom characterized by total-angular-momentum

quantum number J is:

eff

¼ g

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

JðJ þ 1Þ

ð4Þ

where g

is the Lande

factor (see Localized 4f and 5f

Moments: Magnetism). The Curie law implies that the

inverse susceptibility of a paramagnet is proportional

to temperature as can be seen in Fig. 2.

In higher ﬁelds and at lower temperatures the

magnetization is no longer a linear function of the

magnetic ﬁeld (and consequently the Curie law is no

longer valid) but in many cases can be expressed as:

M ¼ M

ðxÞ; ð5Þ

where

; ð6Þ

is the maximum attainable magnetization for a ma-

terial with one type of ‘‘magnetic’’ atoms character-

ized by J. The function B

(x) is the Brillouin

function of the argument

x ¼

ð7Þ

The graphical representation of B

(x) for several

typical values of J is shown in Fig. 3. For further

Figure 2

Typical magnetic ﬁeld dependence of magnetization and

temperature dependence of inverse susceptibility for a

paramagnet with localized magnetic electrons. The

randomly oriented magnetic moments in a paramagnet

are illustrated schematically.

Figure 3

The Brillouin function plotted for several typical values

of J.

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Magnetism in Solids: General Introduction

details see, e.g., Kittel (1976), Ashcroft and Mermin

(1988), or Gilles (1994).

In metals the Pauli paramagnetism that is charac-

terized by a nearly temperature-independent suscep-

tibility is also observed. The Pauli susceptibility is

proportional to the density of conduction–electron

states at the Fermi level E

. This behavior is a con-

sequence of the magnetic-ﬁeld-induced splitting of

the spin–up and spin–down conduction–electron en-

ergy sub-bands (Kittel (1976, Ashcroft and Mermin

(1988). The Pauli susceptibility can be strongly en-

hanced in systems with narrow bands crossed by E

at a high density of states (see also Itinerant Electron

Systems: Magnetism (Ferromagnetism) and Heavy-

fermion Systems).

1.3 Ferromagnetism

In a material with strongly interacting magnetic mo-

ments cooperative phenomena can be observed, yield-

ing long-range ordering of magnetic moments

(magnetic ordering) at temperatures below a charac-

teristic temperature. The most widely known type of

magnetic ordering is ferromagnetism. Ferromagnetic

ordering of magnetic moments means that they are

aligned parallel to one another (see Fig. 4), which is

accomplished by a ferromagnetic exchange interac-

tion (see Sect. 3). The characteristic temperature of

a ferromagnet is the Curie temperature (T

), at which

the susceptibility diverges. At temperatures below T

a spontaneous magnetization (i.e., a ﬁnite magnetiza-

tion existing in zero external magnetic ﬁeld) emerges.

Values of the Curie temperature of some typical

ferromagnets are given in Table 1.

At temperatures considerably higher than T

, the

magnetic moments behave as in a paramagnet and the

magnetic susceptibility follows the Curie–Weiss law:

w ¼

T Y

ð8Þ

Figure 4

Typical behavior of a ferromagnet: temperature dependence of inverse susceptibility at temperatures above the Curie

temperature (T4T

) and spontaneous magnetization at ToT

. The ferromagnetically coupled magnetic moments in

a ferromagnet ToT

are illustrated schematically.

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Magnetism in Solids: General Introduction

as can be seen in Fig. 4. The parameter Y

is called

the paramagnetic Curie temperature and in simple

ferromagnets Y

. Note in relation to the discus-

sion in Sect. 1.2 that in the paramagnetic regime at

temperatures close to T

and in sufﬁciently high

magnetic ﬁelds the magnetic ﬁeld dependence of mag-

netization becomes strongly nonlinear and the tem-

perature dependence of the susceptibility cannot be

described by the Curie–Weiss law.

At temperatures below T

ferromagnets show sev-

eral characteristic features. Principally there is the

appearance of a spontaneous magnetization (mag-

netization observed in zero magnetic ﬁeld) within one

domain (see Magnets, Soft and Hard: Domains). The

spontaneous magnetization increases with decreasing

temperature as shown in Fig. 4. A block of a ferro-

magnet contains, as a rule, a number of domains

whose spontaneous magnetizations compensate mu-

tually so that the total bulk magnetization is zero.

When a magnetic ﬁeld is applied, the magnetization

increases with increasing ﬁeld strength and in a suf-

ﬁciently high ﬁeld the magnetization saturates at the

value of saturated magnetization, M

, shown sche-

matically in Fig. 5. Ferromagnetic materials exhibit

a large variety of magnetization curves which fre-

quently show a hysteresis with respect to the history

of application of the magnetic ﬁeld (see Magnetic

Hysteresis; Soft Magnetic Materials: Basics; Hard

Magnetic Materials, Basic Principles of ). These fea-

tures of ferromagnets are particularly important in

some applications of magnetic materials.

1.4 Antiferromagnetism

In the case of an antiferromagnetic coupling between

the neighboring magnetic moments their antiparallel

ordering (see Fig. 6) is established at temperatures

lower than the Ne

el temperature T

. At temperatures

higher than T

, the magnetic moments behave as in a

paramagnet and the magnetic susceptibility follows

the Curie–Weiss law (Eqn. (5)). However, for ferro-

magnets Y

40, and the paramagnetic Curie temper-

ature for simple antiferromagnets is usually negative

o0). At low temperatures below T

, the suscep-

tibility decreases (inverse susceptibility increases, as

shown in Fig. 6). For simple antiferromagnets the

maximum on the w(T) curve is located at T

.In

many real systems T

does not coincide with the

susceptibility maximum but with the maximum in

@(Tw)/@T (Fisher 1962).

Table 1

Curie temperatures of several typical ferromagnets.

Material Curie temperature T

(K)

Iron (Fe) 1043

Cobalt (Co) 1403

Nickel (Ni) 631

Gadolinium (Gd) 293

GdNi

Alnico 1123

SmCo

993

B 585

Figure 5

Typical magnetization curve of a ferromagnet

(ferrimagnet).

Figure 6

Typical temperature dependence of inverse susceptibility

for an antiferromagnet. The antiferromagnetically

coupled magnetic moments in an antiferromagnet

ToT

are illustrated schematically.

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Magnetism in Solids: General Introduction