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

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the maximum slope of the measured C

(TCT

ord

)

anomaly is observed, or more precisely by the entro-

py compensation criterion around DC

ord

) (e.g.,

see Stewart 1984).

Once z(T) develops, any discontinuity implies a

jump in U

(T) and thus the occurrence of a ﬁrst-

order transition. Therefore, in magnetic systems pre-

senting further transformations within their ordered

phase, the second-order one is only expected at the

paramagnetic phase boundary and ﬁrst-order ones

at lower temperatures, (as observed in CeRu

Wilhelm et al. 1999). In some cases ﬁrst- and second-

order transitions occur simultaneously, as in the case

of magnetostrictive effects. There, the change of vol-

ume (ﬁrst-order) is triggered by the onset of magnetic

order (second-order transition).

2.1 Morphology of the Magnetic Phase Transitions

In a mean ﬁeld (MF) description of the magnetic

transitions the value of DC

ord

) depends on the

momentum J according to, DC

ord

) ¼2.5

R[(2J þ1)

1]/[(2J þ1)

þ1], with a related entropy

given by DS

¼R ln(2J þ1). For J ¼1/2 the respec-

tive values are, DC

ord

) ¼12.5 Jmol

1

and

DS ¼5.7 Jmol

1

, as shown in Fig. 2 for the ex-

emplary compound Ce(Pd

0.5

). Thus, a ﬁrst-order

transition can be recognized when DC

ord

) clearly

exceeds the predicted MF value, as in CeRu

(Wilhelm et al. 1999). Concerning the entropy, if it is

only due to a discontinuity in z(T), the total entropy

will not exceed DS

¼R ln(2J þ1). But, if the ﬁrst-

order transition corresponds to another contribution

(e.g., DVa0), there is an extra contribution to be

taken into account, because DS

tot

¼DS

þpDV/T

ord

2.2 Effects of Magnetic Field, Pressure and Alloying

External magnetic ﬁeld can be used to distinguish the

nature of the magnetic order. In a ferromagnet, the

presence of ﬁeld broadens the transition because of

the enhancement of magnetic correlations above T

ord

As a consequence z(T) (or M

(T)) does not set on

with inﬁnite but with ﬁnite slope and from higher

temperatures. In an antiferromagnet, the effect of h is

quite different because up to a critical ﬁeld (h

where a spin ﬂip is inducted by the applied ﬁeld, no

significant effect should occur and DC

ord

) is only

weakly shifted to lower temperatures. Above h

, the

system will behave like an induced ferromagnet and

(T) looks like a Schottky anomaly (see below),

which corresponds to a two-level system with the re-

spective parallel and antiparallel spin projection with

respect to h. Field effect becomes an important tool in

anisotropic systems, because T

ord

and DC

ord

) de-

pend on the relative direction between ﬁeld and mag-

netic axes.

In view of the mechanical energy involved, the ap-

plication of external pressure is not expected to pro-

duce drastic changes in a magnetic transition by itself,

aside from the increase of T

ord

, due to the reduction

of the interatomic distances. However, its effect can

be strongly enhanced by other mechanisms such as

the hybridization of the local and conduction electron

states related to the Kondo effect (e.g., see Sereni

1995). The variation of T

ord

with pressure is related

to DC

ord

) by the Ehrenfest relation, dT

ord

dp ¼VT

ord

Da/DC

The effect of alloying may produce more important

variations in a transition because the chemical po-

tential itself is affected in the substitution process.

Local structural pressure and disorder are also typical

effects of atomic substitution, with the obvious con-

sequence of the broadening of DC

ord

). Although

pressure and doping are expected to produce similar

effects, there are cases showing remarkable differenc-

es in C

(T). Such is the case of the alloyed compound

CeCu

(Si

1x

)

, where T

ord

decreases with x,

whereas the C

/T ratio is practically not affected.

On the contrary, the application of pressure on the

x ¼0.1 sample does not affect T

ord

, whereas C

changes significantly (Sparn et al. 1998). This differ-

ent dependence between pressure and doping indi-

cates that the magnetic interaction (i.e., T

ord

) does

not change hand in hand with the amount of con-

densed degrees of freedom. Knowing C

(T,p), it is

possible to relate this parameter with a through the

Maxwell relation, @S/@p ¼@V/@T ¼a, as veriﬁed

in CeCu

(Si

0.9

0.1

)

2.3 Low Dimensionality and Anisotropies

It is well known that 1D systems do not develop LRO

and their C

(T) is described as an anomaly. The ex-

act solutions for apparently elementary cases, like

those described by Ising (IS) or Heisenberg (HS)

models for chains, triangular, or quadratic cases, are

not simple and have to be presented as numerical

approximations. Nevertheless, the morphology of the

phase transition provides some useful information

regarding the characteristics of the ordered phase.

For example, a 2D-IS lattice is expected to exhibit a

logarithmic singularity, C

¼A ln t on both sides of

the transition, with the proportionality factor, A, de-

pending on the type of magnetic lattice (de Jongh and

Miedema 1974). The systems approaching the 2D-HS

magnetic conﬁguration can be recognized from the

evolution of their DS

(T) because it is predicted that

(ToT

ord

) EDS

(T4T

ord

). A good complement

is to also evaluate DU

(T) and to compare its dis-

tribution above and below T

ord

. For example, in a

3D-f.c.c. lattice with spins

, the values are,

S

ord

)/(R ln 2) ¼0.27 (HS) or 0.10 (IS) and

U

ord

)/U

¼0.44 (HS) or 0.15 (IS), respectively

(e.g., see Fisher 1967, Fig. 3).

710

Magnetic Sys tems: Speciﬁc Heat

In metallic systems, the anisotropy plays an im-

portant role in the magnetic behavior of rare earth

compounds because of the CF effect. In these cases

the DS

(T) evolution tends to resemble the one ex-

pected for systems with lower symmetry, though the

anisotropic z(T) develops in 3D. It is observed for

example, that C

/T versus T is symmetric with re-

spect to the temperature of the maximum of C

/T.

Such a L-type transition fulﬁlls the condition for a

2D-HS pattern, DS

mag

(ToT

ord

)EDS

mag

(T4T

ord

despite the 3D behavior which is expected in com-

pounds such as Ce

or CeCu

0.45

1.55

(see Fig. 3).

Indeed, for a strict definition of T

ord

, it must be taken

into account that in an antiferromagnet DU

(T)is

proportional to the magnetic correlation strength be-

tween spins, /s

eff

(the effective moment).

As DU

(T)pm

eff

pw(T)T, the following relation is

obtained, C

(T)pd(wT)/dT (de Jongh and Miedema

1974). Then, because T

ord

is deﬁned by the temper-

ature of the maximum variation of U

(T), it corre-

sponds to the maximum of C

(T), not to the

maximum slope (as in MF transitions).

2.4 Tails (T4T

ord

)

The ideal phase boundary between an LRO phase

and the paramagnetic (P) phase is not always well

deﬁned in real systems because magnetic ﬂuctuations

and/or correlations set on well above T

ord

. The region

where these precursors of the LRO dominate the

scenario can be considered as quasi-paramagnetic

(QP). From w(T), the boundary of such a QP region

can be recognized as a deviation from the Curie law,

wpT

1

. This criterion can be transferred to the spe-

cific heat by using the Maxwell relation, @S/@h ¼@M/

@T-g/@h ¼@

M/@T

¼h@

w/@T

(Stanley 1971),

from which a C

2

dependence is deduced for

the paramagnetic phase where @M/@h ¼wpT

1

Therefore, the C

¼B condition can be used to

deﬁne the boundary between P and QP regions. Some

selected examples of how the C

(T)T

reaches this

condition are shown in Fig. 4.

This way of approaching the P-phase can be used

to compare the strength of the correlations above

ord

in different systems. In Fig. 3, the temperature is

normalized to that at which C

(T) is 1/5 ( ¼20%) of

its maximum value. As shown in Fig. 3, the evolution

of quite different magnetic systems at high temper-

atures nearly coincides in their C

2

tendency,

though the way they reach their respective C

max

value

is very different.

In disordered or frustrated systems or those with

low lying spin excitations, LRO may not develop at

ﬁnite temperature. Nevertheless, the magnetic corre-

lations accounted by the C

(T) tails can be consider-

ed as precursors of a magnetic transition. However, if

it orders, the condensed degrees of freedom are taken

from the QP state, where magnetic correlations are

developing and then C

(T4T

ord

) can be used as a

reference function (C

). Frequently, these tails are

well described by a C

pT ln(T) function, which

can be independent of doping and pressure (e.g., see

Sereni 1998 and references therein).

3. The Magnetically Ordered Phase (ToT

ord

)

The description of the ordered state in a magnetic

system is given by the spin wave theory (e.g., see

Figure 3

Normalized C

max

versus the temperature at which

(T) ¼0.2C

max

, T/T

20%

, to compare the evolution of

magnetic correlations in 1D, 2D (s ¼1/2 Heissenberg)

and 3D magnetic systems.

Figure 4

Normalized evolution of different systems to their

respective paramagnetic states where C

-B, after

subtracting phonon and excited CF-levels contributions.

The Schottky type curve represents the C

(T)of

NiGa

. Inset: comparison of different anomalies

around C

max

711

Magnetic Systems: Speciﬁc Heat

Akhieza et al. 1968). Within this theory there are well

deﬁned C

(T) dependencies, as for example:

(i) ferro- and ferrimagnets with a quadratic (q ¼2)

magnon dispersion-relation, C

(T)pT

3/2

(ii) antiferromagnets with linear (q ¼1) magnon

dispersion-relation, C

(T)pT

(note that in these

two isotropic cases, the temperature exponent, T

,is

given by Z ¼D/q, see Gopal 1966)

(iii) under applied or anisotropic molecular ﬁelds,

an activation energy for the spin waves arises, and

then C

(T) increases exponentially at low tempera-

tures, with an associated gap (D

) in the magnon

spectrum. For h applied on the easy axis of anisot-

ropy of a ferromagnet C

(T)ispT

1/2

exp(D

/T),

and for one with easy plane and h perpendicular to it,

a T

1/2

exp(D

/T) dependence is expected

(iv) In the case of an antiferromagnet, with an

easy axis of anisotropy one also gets C

(T)pT

1/2

exp(D

/T), whereas for the ﬁeld perpendicular it is

Experimental examples of most of these cases can

be found in Sundstro

m (1978) and the inset of Fig. 2.

3.1 Microscopic Character of the Dispersion

Relation

The concept of specific heat as a thermal spectrome-

ter is conﬁrmed by its ability to recognize different

microscopical dispersion relations (including gaps) in

the magnon spectrum. Two illustrative examples are

Ce(Pd

0.9

0.1

) and CePd

. The ﬁrst shows anti-

ferromagnetic behavior in its w(T) macroscopic sig-

nal, despite the ferromagnetism of the CePd matrix.

Experimentally, C

(T)pT

3/2

indicates ferromagnetic

interactions, in agreement with the positive Curie–

Weiss temperature (y

40). In this case, the disorder

introduced by doping makes the reduced ferromag-

netic domains to order randomly among them, be-

having macroscopically as an antiferromagnet. A

mirror-like case is provided by CePd

, which or-

ders ferromagnetically at 6 K (Bauer et al. 1993) re-

gardless of its Kondo character (Coqblin et al. 1996).

Nevertheless, C

(T)pT

is observed for ToT

ord

in-

dicating the presence of antiferromagnetic interac-

tions, in agreement with the y

o0 value. In this case,

frustration effects due to the hexagonal symmetry of

the magnetic lattice may explain the difference be-

tween the microscopic interaction and the actual type

of order. These examples, and those related to the

gap detected in the magnon spectrum, conﬁrm the

skill of this technique in providing microscopic in-

formation within a range of energy of a fraction of

degrees (i.e., p1 10

5

eV).

4. Transformations and Related Anomalies

Every physical phenomenon has an associated char-

acteristic energy, which leads it to manifest within a

certain range of the control parameters. Once that

energy scale is identiﬁed, a comparative analysis with

other effects can be performed. For this purpose, a

distinction between phase transition and transforma-

tion has to be made. In the ﬁrst case, the ordering

energy (k

ord

) deﬁnes a critical value, whereas for

the second (e.g., k

in the Kondo screening or

in a spin glass) it only gives a scale of energy

where it competes with the thermal-energy k

T. This

implies that the condensation of the degrees of free-

dom occur within an extended range of temperature.

4.1 Kondo Effect

In a diluted magnetic system, where magnetic mo-

ments behave as single impurities, there is a spin-spin

interaction between the local moment ( f )andthe

conduction electrons (s). This s–f exchange is anti-

ferromagnetic and leads to the formation of a singlet

ground state (GS) due to the spin compensation below

the Kondo temperature T

. Such as process is not a

phase transition, but a continuous transformation oc-

curring in a logarithmic temperature scale. A conse-

quence of that interaction there is a significant increase

of the density of states N(E

) due to the formation of a

virtual-bound state, which is observed in thespecific

heat measurements as a strong enhancement of the g

coefﬁcient. The height of this resonant state (Bg)is

inversely proportional to its width and, therefore,

gp1/T

. The temperature dependence of the Kondo

contribution to the specific heat (C

) is linear at low

temperature, has a maximum at T/T

¼0.45 and

reaches the T

2

dependence at high temperature (e.g.,

see Bredl 1978). Once the GS is fully formed w(T-0)

becomes Pauli-like (i.e., BN(E

)) and, therefore, Bg.

The electronic scattering is also strongly enhanced,

producing the characteristic logarithmic increase of

the resistivity at low temperature. The problem of the

Kondo effect in transport properties is extensively

discussed in the chapter Kondo Systems and Heavy

Fermions: Transport Phenomena.

4.2 Spin Glasses

Magnetic interactions do not develop LRO when ran-

domness distribution inhibits the propagation of the

order parameter. In any case, the presence of a mean

molecular ﬁeld at each magnetic moment site is able to

freeze a magnetic conﬁguration at a certain tempera-

ture, T

, below which the system behaves in a co-

operative manner. This is a simpliﬁed picture for a

spin glass that becomes more complex when micro-

scopic aspects are considered (e.g., see Mydosh 1993).

Depending on the concentration (x) of the particles

carrying the magnetic moment, different regions can

be distinguished:

(i) In diluted magnetic systems, the specific heat

contribution (C

) and the characteristic temperature

712

Magnetic Sys tems: Speciﬁc Heat

can be scaled with the impurities concentration, lead-

ing to a universal C

/x vs. T

dependence. On the

ToT

range, C

(T) arises linearly with temperature,

with a T

dependence at T-0 (i.e., C

¼aT þbT

For TbT

the expected C

2

dependence is

veriﬁed (Sereni et al. 1979 and references therein).

Some of these features are illustrated by the

0.972

0.028

alloy in the inset of Fig. 4.

(ii) For higher concentrations short-range correla-

tions occur, leading the formation of magnetic clus-

ters. There, the scaling properties are lost and the

anomaly can be described by, C

¼ RðD

=TÞ

=ðe

 1Þ

, where D

represents the energy

gain in the cluster formation, which depends on the

magnetic moment, the volume of the cluster, and the

energy of anisotropy

(iii) when clusters interact among them a short

range order (within the clusters) coexists with the

glassy character of the mutual cluster interaction.

4.3 Schottky Anomalies

Another frequent anomaly observed in specific heat

measurements is the so-called Schottky anomaly,

which is related to the thermal excitations in a level

system split by D, like in nuclear or electronic GS,

where the degeneracy (g

) is removed by hyperﬁne

(nuclear) or crystalline (electric) ﬁelds. Because the

thermal population of the levels is governed by

Maxwell–Boltzman statistics, the specific heat is des-

cribed by C

Sch

¼R(g

)(D/T )

D/T

/[1 þg

D/T

]

For T5D one gets C

Sch

¼R(g

)(D/T )

D/T

and at

high temperatures the typical T

2

tail appears (Gopal

1966). Because the involved entropy is ﬁxed by the

relative degeneracy of the ground and excited levels,

DS ¼R ln(g

), the value of C

max

Sch

and its tempera-

ture depend on g

and D, respectively. In the case

where g

¼g

, one can write C

Sch

(T ) ¼R(D/

2T)

sech(D/2T )

. Note that a similar formula was

used for C

(T ), where the Bose statistics was applied

and thus C

(T ) ¼R(D/2T )

cosech(D/2T )

4.4 Ordering and Anomalies

Magnetic transitions may occur superimposed on

these anomalies. In such a case, the fraction of de-

grees of freedom involved are those not yet con-

densed into the GS, i.e., DS(T

ord

)/DS

. This

situation frequently occurs in Kondo systems, where

ord

) is reduced with respect to the expected

1.5R value (e.g., see Bredl et al. 1978), when the CF

excited levels are still thermally populated or two

magnetic conﬁgurations compete in the formation of

the ordered GS like in CeNi

4–x

. In the last two

cases a C

Sch

(T)-type function plays the role of a ref-

erence function, as shown in Fig. 4.

In order to distinguish between a broadened tran-

sition and a transformation, some thermodynamic

criteria are useful. In a broadened transition, DS(T)

should have a change of slope at T

ord

, reﬂected in a

sudden increase of the negative slope of DC

(T)by

cooling. Alternatively, in a transformation, the

(T) curvature (i.e., d

/dT

) changes monoto-

nously from negative to positive as T decreases.

5. Phases Involving Strongly Correlated Electrons

5.1 The Quasi-paramagnetic Region

In this QP region the magnetic correlations dominate

the scenario before the eventual formation of an

LRO. When such a situation persists down to very

low temperatures, divergencies in C

/T(T-0) are

observed because of the enhanced density of low

lying spin excitations. Such is the case of Ce-

0.9

0.1

shown in Fig. 1 (Lo

hneysen 1995). A gen-

eralized scaling function C

/t ¼F log(t) þE was

proposed for these systems, with t ¼T/T

)

and F ¼7.2 Jmol

1

as a universal constant. Besides

the logarithmic, algebraic divergencies of the type

=T ¼ g

 g

(with Z ¼1/2 or 2) were also pro-

posed. Nevertheless, a continuous spectrum of values

can be found experimentally in this non-Fermi-liquid

(NFL) region (e.g., see Sereni 1998).

Depending on whether the coupling parameter be-

tween the local and band electrons spins increases or

decreases, the respective low temperature boundary

of the QP region is the LRO or the nonmagnetic

phase. In the cases where these two phases converge

for T-0, the NFL region becomes narrow and a zero

temperature quantum critical point may occur with

/T showing some of the predicted T-divergencies

(see Lo

hneysen 1995 and Sparn et al . 1998). Howev-

er, in many cases such a region is quite extended

(mostly due to disorder effects) and then a variety of

/T(T-0) divergencies can be observed (Sereni

1998).

An alternative scenario for incipient magnetic in-

teractions is given by the itinerant magnetism or spin-

ﬂuctuations. Due to thermal excitations, spins ﬂuc-

tuate in orientation and amplitude with an associated

frequency (o

). When o

1

p k

is greater than the

thermal energy, the spins look nonmagnetic. On the

contrary, at high temperatures thermal ﬂuctuations

are more rapid than those of the spins and able to

‘‘see’’ the magnetic component. In this case it is pre-

dicted that C

/TpT

ln(T/T

), as shown in Fig. 1 by

the exemplary system UAl

(Stewart 1984).

5.2 The Nonmagnetic Phase: Heavy Fermions and

Intermediate Valent

In a periodic Kondo lattice a narrow electronic band

with heavy electrons is responsible for the large g

value. These heavy fermion (HF) quasi-particles

present a coherent state identiﬁed by a significant

713

Magnetic Systems: Speciﬁc Heat

reduction of the electrical resistivity (r ¼AT

) and a

strongly enhanced Pauli-like susceptibility (w). Be-

cause in an FL system g, w

and A

1/2

are proportional

to N(E

), all these parameters have to be propor-

tional between them. In Fig. 1, the C

(T) dependence

of the HF prototype CeCu

is shown, with g ¼

1.6 Jmol

1

2

. Like for Kondo impurities, gp1/T

and then its characteristic temperature is deduced to

be about 5 K. Indeed, g depends on temperature like

g(T) ¼C

(T)/T and, because the Kondo-lattice state

is built up from a doublet ground state, the related

entropy is also R ln2.

For large hybridization strength between local and

conduction states, both spin and charge components

of the wave function are mixed, leading to an inter-

mediate valent (IV) state. In that case, the width of

the energy levels becomes comparable to the CF

splitting, and then the system gains energy (when

XD) by going from a two-fold-degenerated GS

(with J ¼1/2) to the full spin-orbit six-fold-degener-

ated state (e.g., J ¼5/2 for Ce). With the consequent

increase of the effective level width (T

) and the re-

duction of g (compare CeCu

with CeIr

in Fig. 1). In

this IV state, the nonmagnetic character comes from

the disappearance of the moment due to the 4f-elec-

tron delocalization, whereas in the HF state only the

spin compensation is involved. Here the FL behavior

is fully veriﬁed through the Wilson (w

g ¼0.036 emuK

1

) and Kadowaki–Woods (A/

¼1 10

6

O cm (J Kmol

1

)

2

) ratios, in agreement

with the values predicted by the theory (e.g., see Se-

reni 1991 and references therein). Although the Wil-

son ratio is claimed to be applicable also for HF

systems, the lack of good quantitative determination

eff

of the doublet GS impedes a reliable experimental

veriﬁcation.

The transformation or cross-over from HF to IV

states by doping occurs at a cross-over concentration:

(Sereni 1995) where most of physical properties

(like V, g, y

, r, w) show a change of regime through

a change in the concentration dependence. Particu-

larly g(x), shows a maximum at x

. Although a de-

crease of g(x) should be expected when going from an

HF to an IV regime as T

increases, the experimental

observation indicates that the effective width of the

hybridized f level does not evolve monotonously (see

Coqblin et al. 1996 and references therein). There-

fore, the maximum observed in g( x) should be related

to the change (increase) of the GS degeneracy from

¼2 to 6 (c.f. J ¼5/2) as mentioned before for ce-

rium ions (Sereni 1995).

6. Concluding Remarks

This condensed overview on the specific heat of mag-

netic systems shows some examples of the wide range

of information that can be extracted from the exper-

imental study of this thermal parameter. Since it

covers macroscopic (thermodynamical) as well as mi-

croscopic (e.g., excitation spectrum) properties, it

represents one of the basic tools for the characteri-

zation of new systems (together with magnetic and

transport measurements). Alternatively, since it al-

lows a direct evaluation of the evolution of the en-

tropy with temperature, it gives access to many other

thermodynamical parameters through different Max-

well relations. Similar analysis of the specific heat

contributions can be applied to other physical phe-

nomena, such as elastic (structural) instabilities, par-

ticles diffusion, ferroelectricity, or superconducting

properties (see Gopal 1966).

See also: Crystal Field Effects in Intermetallic Com-

pounds: Inelastic Neutron Scattering Results; Elec-

tron Systems: Strong Correlations; Heavy-fermion

Systems; Intermediate Valence Systems; Magnetic

Excitations in Solids; Magnetic Systems: Disordered

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Sundstro

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Centro Atomico Bariloche (CNEA), San Carlos de

Bariloche, Argentina

Magnetic Units

The International System of Units, SI (Le Syste

International d’Unite

s), was adopted by the Con-

rence Ge

rale des Poids et Mesures in 1960. It is a

coherent system founded on seven base units (meter,

kilogram, second, ampere, kelvin, mole, and candela)

and its use is recommended by the International

Union of Pure and Applied Physics (Cohen and

Giacomo 1987). Nevertheless, a significant part of the

existing literature, particularly concerning magnet-

ism, is still expressed in terms of the electromagnetic

(c.g.s.) or the Gaussian systems. Hence, it might be

helpful to provide the relationships between the SI

and these systems, although it is not intended to

advance the use of the latter.

1. Relationship Between SI, c.g.s., and Gaussian

Systems

In contrast to the SI system, the c.g.s. system is based

on three base units only (centimeter, gram, and sec-

ond) being appropriate for mechanical problems. The

dual nature of the system emerged from the applica-

tion to electricity and magnetism, where all quantities

may be alternatively expressed in either electrostatic

units (esu) or electromagnetic units (emu). The elec-

trostatic system deﬁnes charge to be a derived quan-

tity based on Coulomb’s law for the force between

two charges, while the electromagnetic system deﬁnes

electric current as a derived quantity based on Am-

pere’s law for the force between two electric current

elements:

F ¼ k

ði

 rÞ

In the electrostatic system, k

¼1 is chosen and the

permittivity, e, is deﬁned to be dimensionless and

to adopt unity in free space. In the electromagnetic

system, k

¼1 is chosen and the permeability, m,is

deﬁned to be dimensionless and to adopt unity in free

space.

One of the main advantages of the SI system is the

uniﬁcation of magnetic quantities with practical elec-

tric units of ampere and volt. The unit for current in

esu is the statampere (3 10

esu s

1

’

¼1 A) while in

emu it is the abampere (1 abampere

’

¼10 A). For the

‘‘mixed’’ (or ‘‘symmetrized’’) c.g.s. or Gaussian sys-

tem, all electrical quantities are measured in esu and

all magnetic quantities in emu, hence statampere is

used here.

The duality is removed in the MKSA or practical

system, developed by Maxwell and Giorgi, by intro-

ducing the current as an additional base unit without

the attempt to express the ampere dimensionally in

length, mass, or time. This system is said to be ‘‘ra-

tionalized,’’ since the prefactors of Coulomb’s and

Ampere’s laws are set equal to 1/4p, while the choice

of these prefactors is unity in the c.g.s. and Gaussian

systems. This leads to the appearance of factors of 2p

and 4p in situations that involve plane geometry and

to their absence in situations with spherical or cylin-

drical symmetry, where these factors might normally

be expected in the c.g.s. and Gaussian systems. These

systems are therefore called ‘‘nonrationalized.’’ Even-

tually, the SI system was formulated on the basis of

the rationalized MKSA system. A comparison of the

basic equations (Maxwell’s and material equations)

of these two systems is given in Table 1, where the

quantities in the Gaussian system are indicated with

an asterisk and the symbols recommended by IUPAP

(Cohen and Giacomo 1987) are used (E, electric

ﬁeld; P, electric polarization; H, magnetic ﬁeld; B,

magnetic ﬂux density; M, magnetization; w

, w

magnetic or electric susceptibility; m, permeability; e,

permittivity).

As a result of the fourth base unit in the SI and

MKSA system, permeability m ¼m

and permittiv-

ity e ¼e

are dimensional physical quantities where

and m

must fulﬁll the condition e

¼1, in order

that electrostatics and electromagnetics are coherent.

This avoids the asymmetrical introduction of the fac-

tor c into the equations for electric and magnetic

Table 1

Maxwell’s and material equations.

Gaussian system with

three base quantities

SI system with four

base quantities

crE* ¼–@B*/@t rE ¼–@B/@t

crH* ¼4pj*–@D*/@t rH ¼j þ@D/@t

rD* ¼4pr* rD ¼r

rB* ¼0 rB ¼0

D* ¼e

E* ¼E* þ4pP* D ¼eE ¼e

E ¼e

E þP

B* ¼m

H* ¼H* þ4pM* B ¼mH ¼m

H ¼m

(H þM)

¼1 þ4pw

¼1 þw

¼1 þ4pw

¼1 þw

715

Magnetic Units

quantities. In the SI system the permeability of vac-

uum is deﬁned to have the value m

¼4p 10

7

2

¼4p 10

7

1

¼4p 10

7

VsA

1

The comparison of the corresponding equations in

both systems yield the following relationships:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4p=m

¼B*/B ¼M/M*;

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4pm

¼H*/H;4p ¼w/w*;

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4pe

¼E*/E¼r/r*¼j/j*¼P/P*;and

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4p=e

¼D*/D,

which are useful to convert an equation in Gaussian

units into the corresponding equation in the SI system

and vice versa.

For example, to transform an equation in Gauss-

ian units into SI, replace the quantities with an as-

terisk by those without an asterisk according to the

above relationships (e.g., B* )B

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4p=m

; w ¼

B H,

w* ¼(1/8p)B* H*). Quantities that involve only vol-

ume, force, energy, and length transform directly.

2. Definition of the Magnetic Quantit ies

The magnetic quantities are deﬁned in SI as follows

(International Organization for Standardization

1992). The magnetic ﬁeld strength, H, is a vector

quantity, the rotation (curl) of which is equal to the

sum of the electric current density and the time de-

rivative of the electric ﬂux density. The magnetic ﬂux

density or magnetic induction, B, is a vector quantity

such that the force exerted on an element of electric

current is equal to the vector product of this element

and the magnetic ﬂux density: F ¼IDs B. The mag-

netic moment, m, is a vector quantity, the vector

product of which with the magnetic ﬂux density of a

homogeneous ﬁeld is equal to the torque: m B ¼T.

The International Electrotechnical Commission

(IEC) deﬁnes also the magnetic dipole moment,

j ¼m

m. The magnetization as the magnetic moment

per unit volume is deﬁned by the relationship M ¼

B/m

H. The permeability is deﬁned by B ¼mH and

the relative permeability by m

¼m/m

, where the latter

is dimensionless. This is also the case for the magnetic

susceptibility deﬁned by w ¼m

1. The electromag-

netic ﬁeld energy divided by volume is given by

w ¼

E D þB H.

3. Analogies Between Electric and Magnetic

Field Quantities

According to the structure of Maxwell’s equations,

two types of analogies can be derived for the electric

and magnetic ﬁeld quantities:

(i) The distinction according to the structure of the

differential operators (curl and div) leads to the cor-

respondences E2H and D2B. This becomes also

evident from the integral formulation of Maxwell’s

relationships, where the electric and magnetic poten-

tial differences, U ¼

E ds and U

H ds, occur as

well as the electric ﬂux C ¼

(V)

D dA and the mag-

netic ﬂux F ¼

(V)

B dA. The electric and magnetic

energy densities w

E dD and w

H dB also

imply this correspondence. This indicates the primacy

of H based on ﬁctive magnetic surface poles (or ‘‘mag-

netic charges’’) traditionally used in magnetostatics by

analogy with electrostatics. However, this is not a val-

uable or helpful entry to electromagnetism (see below).

(ii) The distinction by homogeneous and inhomo-

geneous equations leads to the correspondences

E2B and D2H, which is also consequent with re-

spect to the inﬂuence of matter upon the ﬁeld quan-

tities: rD and rH are determined by the external

charge and current densities, respectively, and the

corresponding equations for rE and rB in

matter additionally contain polarization charge den-

sity r

¼rP and polarization and magnetization

current densities j

¼@P/@t and j

¼rM, respec-

tively. Thus, the magnetic induction or magnetic ﬂux

density, B, is the ﬁeld in matter.

4. The Primacy of B or H

The codiﬁcation of the ﬁeld quantities goes back to

Faraday, Maxwell, and their contemporaries. In this

tradition the primary magnetic ﬁeld is H as the

‘‘magnetizing force’’ now called ﬁeld strength. The

term ‘‘magnetic induction’’ for B points to the status

of a variable quantity that is determined by the sum

of H and M and corresponds to analogy (i) above.

Thus the ‘‘coefﬁcient of induced magnetization,’’ w,is

the ratio of M to H and the ‘‘coefﬁcient of induc-

tion,’’ m, is the ratio of B to H, named susceptibility

and permeability, respectively, by William Thomson

(Bennett et al. 1976, Goldfarb 1992).

However, as stated in (ii) above the ﬁeld within

magnetized matter is not H but B, since B is produced

by both the conduction and the Amperian (atomic)

currents, while H has its source in external conduction

currents only. Hence, analogy (ii) is more general and

those emphasizing the primacy of B have an excellent

viewpoint but are at a historical disadvantage, since

the magnetic pole approach has traditionally been

used in magnetostatics by analogy with electrostatics.

For the characterization of magnetic materials the

traditional approach is usually used because the de-

sired quantity is M(H,T). The relevant ﬁeld in a

magnetized sample is the internal ﬁeld, H, which oc-

curs in these equations and originates not only from

the external ﬁeld, H

ext

, but also from the stray- or

self-ﬁeld of the magnetized body. Thus, H

ext

has to be

corrected by the demagnetizing ﬁeld H

NM,

where N (or D) is the demagnetizing factor and de-

pends on the sample geometry:

H  H

int

¼ H

ext

þ H

¼ H

ext

 NM

Only spheres and ellipsoids have a uniform M in

a uniform H

ext

and can therefore be calculated in a

closed form while for cylinders and parallel-

epipeds only approximate N values can be given. The

716

Magnetic Uni ts

demagnetizing factors for the three orthogonal axes

of any uniformly magnetized body sum up to unity;

hence, N ¼1/3 for a sphere. For a review on demag-

netizing factors see, e.g., Chen et al. (1991) and

Goldfarb (1992).

5. SI System

In the SI system, the relationship between B, H, and

M is deﬁned as B ¼m

(H þM) and w ¼dM/dH. The

magnetic polarization J ¼m

M is also a recognized

quantity even though it is based on B ¼m

H þJ

which is not a deﬁning relationship in SI. The units of

B and J are tesla (T ¼Vsm

2

¼Wbm

2

) and those of

H and M are Am

1

; accordingly, w is dimensionless.

M and w are extensive quantities and are based on a

unit volume, i.e., the volume magnetization is deﬁned

as the magnetic moment (Am

) per unit volume,

which yields Am

1

. The volume, however, is usually

more difﬁcult to determine with the same precision as

the mass and is furthermore temperature dependent.

For technical applications, the volume magnetization

is the commonly used quantity. In condensed matter

physics, however, the temperature dependencies of

these quantities are of importance; the mass magnet-

ization (M or s) and the mass susceptibility (w

) are

preferred although the latter two are not recognized

as important SI quantities but are consistent with SI.

Their conversion into recognized units requires knowl-

edge of the density as a function of temperature, which

is usually not available, in particular for novel mate-

rials, at least not with the same precision as the meas-

urement of M(T)orw(T ). Furthermore, the mass

magnetization (Am

1

¼JT

1

) is preferred to

the volume magnetization since atomic quantities such

as the magnetic moment per atom, mole, or formula

unit (f.u.) can be derived directly. The number, n,of

Bohr magnetons per mole (m

mol

1

)issimply

obtained from the mass magnetization by Mm

mol

¼Mm

mol

/5.5849 ¼ n[m

mol

1

], where N

6.0221367 10

mol

1

, m

mol

is the molar mass, and

¼9.2740154  10

24

1

. The same holds also for

the effective moment, which is determined from the

Curie–Weiss law via w(T ) measurements.

6. Electromagnetic c.g.s. and Gaussian Systems

In the c.g.s. and Gaussian systems, the deﬁning rela-

tionship is B* ¼H* þ4pM*. Accordingly, one major

property of these systems is that B*andH* have the

same numerical value in empty space, considered as an

advantage by some and as a disadvantage by others

(e.g., Brown 1984). B* is given in gauss (G) and H*in

oersteds (Oe), which are dimensionally equivalent

(cm

1/2

1/2

1

). The term 4pM* can be considered as

the ﬁeld in gauss arising from the magnetization of the

material, whereas the magnetization, M*, considered

as the magnetic moment per unit volume, differs nu-

merically by the factor of 4p but is dimensionally the

same (emu cm

3

’

¼ergG

1

3

¼cm

1/2

1/2

1

¼G).

Table 2

Magnetic quantities and conversion from Gaussian to SI units.

Quantity Symbol Gaussian system

Conversion

factor SI system

Magnetic ﬂux F Mx, G cm

8

Wb, Vs

Magnetic ﬂux density,

magnetic induction

B G10

4

T, Wbm

2

Magnetic potential

difference,

magnetomotoric force

U, F Gb (gilbert) 10/4p A

Magnetic ﬁeld strength H Oe 1000/4p Am

1

Volume magnetization 4pM G 1000/4p Am

1

Volume magnetization M emu cm

3

, G 1000 Am

1

Magnetic polarization J emu cm

3

,G 4p 10

4

T, Wbm

2

Mass magnetization M, s emu g

1

,Gcm

1

1Am

1

,JT

1

Magnetic moment m emu, erg G

1

1/1000 Am

,JT

1

Magnetic dipole moment j emu, erg G

1

4p 10

10

Wbm, Vsm

Volume susceptibility w, k Dimensionless, emu cm

3

4p Dimensionless

Mass susceptibility w, k emu g

1

,cm

1

4p/1000 m

1

Molar susceptibility w

mol

emu mol

1

,cm

mol

1

4p 10

6

mol

1

Permeability, m ¼m

mm* ¼m

4p 10

7

1

, VsA

1

Relative permeability, m/m

Dimensionless 1 Dimensionless

Energy density, energy

product

w ergcm

3

1/10 Jm

3

Demagnetization factor N, D Dimensionless 1/4p Dimensionless

717

Magnetic Units

Note that emu is not a unit, it is merely an indication

that electromagnetic units are in use. This is also

evident from the units frequently used for the mass

magnetization (emu g

1

’

¼ergG

1

¼Gcm

1

)and

the mass susceptibility (emu g

1

¼cm

1

¼ emu g

1

). In terms of base units, the volume susceptibility

is dimensionless; however, w* is frequently expressed as

emu cm

’

¼emu Oe

1

3

’

¼erg Oe

1

3

¼1. Ac-

cordingly, the notation 4pw* for the c.g.s. volume sus-

ceptibility is numerically equal to the SI volume

susceptibility. The permeability is deﬁned in the Gauss-

ian system as the ratio B/H, hence the Gaussian per-

meability, m*, is identical to the relative permeability in

the SI system: m* ¼m

¼m/m

¼1 þw ¼1 þ4pw*(see

also the material equations in Table 1).

Table 2 gives numerical factors for the conversion

between the two unit systems: multiply the number

for the Gaussian quantity by the conversion factor

to obtain the number for the SI quantity (e.g.,

1Mx10

8

WbMx

1

¼10

8

Wb). Note that Gauss-

ian and electromagnetic c.g.s. units are the same for

magnetic quantities but are different for their electric

counterparts.

See also: Perturbed Angular Correlations (PAC)

Bibliography

Bennett L H, Page C H, Schwarzendruber L J 1976 Comments

on units in magnetism. In: Becker J J, Lander G H, Rhyne J J

(eds.) AIP Conf. Proc. American Institute of Physics, New

York, Vol. 29, pp. xix–xxi

Brown W F Jr 1984 Tutorial paper on dimensions and units.

IEEE Trans. Magn. 20, 112–7

Chen D-X, Brug J A, Goldfarb R B 1991 Demagnetizing fac-

tors for cylinders. IEEE. Trans. Magn. 27, 3601–19

Cohen E R, Giacomo P 1987 Symbols, units, nomenclature and

fundamental constants in physics. Physica A 146, 1–68

Goldfarb R B 1992 Demagnetizing factors. In: Evetts J (ed.)

Concise Encyclopedia of Magnetic and Superconducting Ma-

terials. Pergamon, Oxford, pp. 103–4

International Organization for Standardization 1992 Quantities

and Units: Part 5. Electricity and Magnetism, ISO 31-5:

1992(E). ISO, Geneva, Switzerland

G. Hilscher

Institut fu

r Experimentalphysik, TU Wien, Vienna

Austria

Magnetic Viscosity

Under the name of magnetic aftereffect (or time-lag

in magnetization or magnetic viscosity) everything is

included that is connected with the inﬂuence of time

on magnetization phenomena in ferromagnetic bod-

ies, with the following exceptions (a) the inﬂuence of

the sample inductance, (b) eddy currents, (c) the

irreversible relaxation of chemical or topological

microstructure, and (d) relaxation phenomena with

characteristic timescales less than 10

5

s. Magnetic

relaxation phenomena on shorter timescales are dom-

inated by excitation and damping of spin waves, fast

motion of domain walls, etc. These phenomena have

been extensively investigated by high-frequency res-

onance experiments (Haas and Callen 1963).

In a typical viscosity experiment the sample has

some given magnetic prehistory and the time depend-

ence of magnetization, J, is measured at constant

values of temperature, T, and magnetic ﬁeld, H.In

many cases the prehistory begins, as shown in Fig. 1,

at some large positive ﬁeld, H

, where the sample is in

the thermodynamic equilibrium state of magnetic

saturation. Then a negative ﬁeld, H, is applied as fast

as possible. The magnetization reacts rapidly (in less

than 10

5

s) and achieves some value, J(0), on the

major demagnetization curve (see Fig. 1(a)). Howev-

er, now the sample is not in thermal equilibrium and

the sudden change of magnetization is followed by a

slow relaxation (the aftereffect) on a timescale of

seconds, months, or even longer. The ﬁrst investiga-

tion of the magnetic aftereffect was done by J. A.

Ewing in the 1880s.

Experimental evidence distinguishes at least four

categories of magnetic aftereffects called (a) the re-

versible aftereffect, (b) the irreversible aftereffect, (c)

the anomalous aftereffect, and (d) quantum tunneling

of magnetization. Depending on the material inves-

tigated and the specific magnetic prehistory of the

samples, various types of the time dependence have

been observed for the slow relaxation of magnetiza-

tion. For a large class of experiments a logarithmic

time law has been found:

JðtÞ¼Jð0ÞþS lnð1 þ t=t

Þð1Þ

where S is the magnetic viscosity and t

is some ref-

erence time depending on the sample and on the

measuring procedure. The value of t

is usually ob-

tained by ﬁtting Eqn. (1) to the experimental data. If

is not appropriately chosen, J(t) will deviate from

logarithmic behavior, i.e., in plots such as that of Fig.

1(b), for values of t within an order of magnitude of

, the data points will deviate from a straight line.

This is not related to the above-mentioned deviations

expected for very small values of t (o10

5

s).

Obviously Eqn. (1) is also incorrect for t-N.

Thus, the magnitude of J(t) cannot exceed the value

of the saturation magnetization, J

, of the material

and, for example, at zero ﬁeld the thermal equilibri-

um value J ¼0 has to be achieved for t-N. How-

ever, the aftereffect is usually small and, in many

cases, deviations from Eqn. (1) have not been ob-

served even for times t as long as weeks or months. In

the typical example of Fig. 1(b) the relaxation of

magnetization over a period of some hours is only a

718

Magnetic Vi scosity

few percent. Similar results have been found for all

types of ferromagnets ranging from commercial per-

manent magnets (Givord et al. 1987) to the exotic,

purely organic soft-magnetic fullerene TDAE-C

(Dunsch et al. 1997). Slow relaxation processes sim-

ilar to the magnetic aftereffect are also known from

other ﬁelds of physics as the relaxation of remanent

magnetization in spin glasses (Huang 1985), the

low-frequency dielectric relaxation in ferroelectrics

(Jonscher 1983), the elastic aftereffect (found by

L. Boltzmann in 1876), or the ﬂux creep in super-

conductors (Yeshurun et al. 1996).

1. Reversible Magnetic Aftereffect

The reversible aftereffect, also called the diffusion or

Richter aftereffect, was discovered by G. Richter in

1937 in aFe(C) and interpreted by J. L. Snoek in

1941. This effect is associated with impurity atoms or

holes within the ferromagnetic lattice and is observed

only in certain substances. In the case of a Fe(C) the

carbon atoms occupy interstitial sites in the b.c.c.

lattice, locally modifying the magnetocrystalline an-

isotropy in a certain direction (see Fig. 2). When the

magnetization is quickly changed owing to a change

in the applied magnetic ﬁeld, the carbon atoms dif-

fuse relatively slowly to a new position. This results in

a slow change of the local magnetic anisotropy and,

consequently, in a sluggish further change of mag-

netization. The diffusion aftereffect also occurs if the

magnetization is reversibly changed by the magnetic

ﬁeld. If in such a case the ﬁeld is reversed to its former

value, the impurity atoms diffuse back to their former

position and the magnetization also relaxes to its

former value.

Therefore, this effect is called the reversible after-

effect (Kronmu

ller 1968). This effect, based on the

diffusion of small atoms such as hydrogen, carbon, or

nitrogen, has been observed in many crystalline

and amorphous alloys as well as intermetallic com-

pounds. An important property of the diffusion af-

tereffect is that a superposition principle is valid

concerning the response of the magnetization, J(t), to

previous ﬁeld jumps, (DH)

, with i ¼1, 2,y For vis-

cosity experiments being governed by the reversible

aftereffect, a variety of different time dependencies of

the relaxing magnetization, J (t), have been reported.

This is understandable because, owing to the validity

of the superposition principle, the function J(t) can

easily be inﬂuenced by the magnetic prehistory of the

samples.

Depending on the specific selection of the material

to be investigated and on the specific magnetic

Figure 2

Unit cell of the b.c.c. lattice of aFe(C) with one

additional carbon atom (*) which diffuses between

interstitial sites (*,J) thus changing the effective

anisotropy experienced by the local magnetization, J

Figure 1

(a) Hysteresis loop of the ferromagnetic material Nd

. The marker J is at the coordinate (H,J(0)) where the

experiment in (b) starts. Also shown are recoil loops beginning at the major demagnetization curve (left-hand branch

of the hysteresis loop). The markers *, m, and * characterize the coordinates where the experiments of Figs. 9 and 10

start. (b) Results of a typical viscosity experiment. The parameters S and t

are ﬁtted to Eqn. (1); J

is the

magnetization of the previously saturated sample.

719

Magnetic Viscosity