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pressure up to 2.5 GPa in a wide temperature range

from 4 K up to 400 K. In the vicinity of room tem-

perature, accurate measurements of the pressures

applied inside the cells are usually performed by

Manganin pressure sensors (Bridgman 1958). A high-

pressure diamond anvil cell (DAC) has been modiﬁed

for magnetization and magneto-optical measure-

ments in high and steady as well as in pulsed mag-

netic ﬁelds (Yamamoto et al. 1991). Extremely high

pressures of up to 100 GPa can be reached using the

DAC; however, the sample space in the DAC is ex-

tremely small, up to 10

3

. Detailed information

on the high-pressure techniques and methods of

measurements of material properties under pressure

can be found in Bridgman (1958), Sherman and

Stadtmuller (1987), or Eremets (1996).

2. Magnetic Properties Under Pressure

Extended experimental data on magnetic properties

of solids under pressure are presented in several re-

views (Bloch and Pavlovic 1969, Schilling 1981,

Wohlfarth 1981). In the tables and ﬁgures below,

typical and characteristic examples of the pressure

changes of the magnetization, magnetocrystalline an-

isotropy, and temperatures of the magnetic transi-

tions are presented.

2.1 Magnetic Moment

Direct measurements of the effect of high pressure on

atomic moments m have been very rare. The magne-

tostriction experiments and Eqns. (2) and (3) have

been frequently used to determine the pressure pa-

rameter dlnm

/dp at low temperatures. Using SQUID

magnetometers and the pressure cells mentioned

above, accurate values of the saturation magnetiza-

tion M

at low temperature and under high pressure

can be obtained (see, e.g., Mikulina et al. 1999). The

data presented in Table 1 provide evidence of general

trends.

In accordance with the discussion above, all three

ferromagnetic transition elements iron, cobalt, and

nickel with b.c.c., h.c.p., and f.c.c. crystal structures,

respectively, exhibit a very stable magnetic moment

with respect to the atomic volume. This is in good

agreement with ab initio calculations (Moruzzi and

Marcus 1988, Entel et al. 2000). On the other hand, a

very pronounced decrease in the Fe-moment was ob-

served under pressure in the Invar alloys Fe

1x

(0.7XxX0.6) with f.c.c. crystal structure (see Table 1).

This effect belongs to the Invar anomalies together

with zero thermal expansion and the large spontane-

ous magnetostriction below T

. This resolving effect

of the crystal structure on the pressure-induced de-

crease of the Fe-moment correlates well with the the-

oretically described moment–volume instabilities in

the iron-based alloys with f.c.c. crystal structure (for

more details see Invar Materials: Phenomena).

The Invar-like magnetovolume phenomena were

also observed in iron-rich RE–Fe intermetallics of the

types RE

and REFe

TM, where RE is yttrium

or a rare earth element and TM is a transition metal,

e.g., titanium, vanadium, or molybdenum (Givord

and Lemaire 1974, Buschow 1988). A much higher

value of the pressure parameter dlnm

/dp in Y

than in YFe

Ti was predicted theoretically (Sabirya-

nov and Jaswal 1998) and observed experimentally

(Mikulina et al. 1999) (see Table 1). The Y

compound has a rhombohedral crystal structure, but

the nearest neighbors of the iron atoms (and the cal-

culated density of state curve) exhibit features of

f.c.c.-iron. Thus, not only global but also local prop-

erties play a role in the magnetovolume effects. The

pressure-induced decreases of Fe-moment in the in-

dividual RE

intermetallics at 5 K differ from

each other depending on the RE elements (see Fig. 2).

However, a tendency of the parameter dlnM/dp to

diverge in the vicinity of T

due to the relation in

Table 1

Magnetic moments m

of transition metals TM, rare-earth RE, and uranium atoms and the magnetocrystalline

anisotropy constant K

in selected ferromagnetic materials under pressure.

Material Moment m

/at.(TM,RE,U) dlnm

/dp GPa

1

77K

K/f.u. dlnK

/dp GPa

1

0.61 0.24 10

2

0.06 2 10

2

1.72 0.22 10

2

0.34 3.5 10

2

2.22 0.25 10

2

0.05 5 10

2

1.73 4.7 10

2

——

YFe

1.73 3.4 10

2

21.5  3.7 10

2

2.15 8.2 10

2

46  17 10

2

7.63 0.11 10

2

0.2 —

10.2 0.15 10

2

4.3 —

UGe

1.4 2.4 10

2

——

UGa

2.4 16 10

2

——

aBloch and Pavlovic (1969). bHayashi and Mori (1981). cMikulina et al. (1999). dFranse et al. (1985).

700

Magnetic Sys tems: External Pressure-induced Phenomena

Eqn. (3) smoothes down these differences with in-

creasing temperature (see Fig. 2) (Kamarad et al.

1999).

The localized moments in rare-earth elements have

been observed to be pressure-independent. The pres-

sure effect on the contribution of 5d-electrons to the

RE-moment can be considered to be responsible for

the very low values of dlnm

/dp in gadolinium and

dysprosium, presented in Table 1 (Bloch and Pavlovic

1969). The 5f electrons responsible for magnetism in

the actinide metals are localized in the heavy actinide

elements only. Their further delocalization can be

enhanced by the effect of relatively moderate external

pressures. Not only the magnetic but also the struc-

tural state of actinide metals and compounds are thus

strongly affected by pressure (see Sechovsky and

Havela 1988 and references therein). As an example, a

very pronounced decrease in the U-moment in UGe

and UGa

intermetallics is presented in Table 1. Both

the 5f–5f overlap and the 5f–ligand hybridization can

be enhanced by external pressure. This leads to a very

complex response in the magnetic properties of the

uranium-based intermetallics—see the large initial

decrease of the U-moment in UGa

accompanied by

an increase in T

under pressure (Franse et al. 1985)

in Table 2.

2.2 Magnetocrystalline Anisotropy

In magnetically ordered solids, atomic moments and

the resulting magnetization M are preferentially ori-

ented into specific crystal lattice directions—easy

magnetization directions (EMD)—due to magneto-

crystalline anisotropy (see also Localized 4f and 5f

Moments: Magnetism). Changes in the orientation

of M relative to the EMD by an external magnetic

ﬁeld are connected with relevant changes in the

magnetocrystalline anisotropy energy. This energy

can be expressed in uniaxial (hexagonal or te-

tragonal) crystals as

¼ K

sin

y þ K

sin

y þ ? ð11Þ

where y is the angle between M and the c axis and K

, y are the magnetocrystalline anisotropy con-

stants. The materials listed in Table 1 represent

ferromagnets with both axial (EMD along the c-axis,

40) and planar (EMD in the plane perpendicular

to the c-axis, K

o0) magnetocrystalline anisotropy.

A decrease in 7K

7 under pressure was observed for

both types of magnetocrystalline anisotropy (Bloch

and Pavlovic 1969, Mikulina et al. 1999). The effect

of pressure on the magnetocrystalline anisotropy

constant K

in the Y

and the YFe

Ti inter-

metallics (K

) reﬂects the different local nearest-

neighbor environment of the iron atoms and the

difference in the parameters dlnm

/dp in both inter-

metallics (see Fig. 3).

2.3 Magnetic Phase Transitions

The effect of pressure on the (isotropic) exchange in-

teraction plays a dominant role in the pressure de-

pendence of the Curie or Ne

el temperatures of

magnetic systems with localized magnetic moments.

An increase in J with pressure was derived from

measurements of the Kondo effect in ‘‘single impurity

systems’’ as rare-earth elements or manganese diluted

in noble metals (Schilling 1981). The Kondo temper-

ature T

in such systems increases under pressure

with an almost constant Gru

neisen coefﬁcient

GB10/3. The values of G presented in Table 2 for

antiferromagnetic FeO and the ferrimagnetic garnet

were derived using Eqn. (6). They are

very close to the characteristic values of G for the

single impurity systems and they conﬁrm the local-

ized character of the magnetic moments in these

materials.

The very small pressure-induced changes of T

presented in Table 2 verify the above mentioned sta-

bility of the magnetic state in pure iron, cobalt, and

nickel metals in the b.c.c., h.c.p., and f.c.c. crystal

structures, respectively. However, a relatively low

critical pressure P

B10 GPa for the disappearance of

magnetism can be derived, Eqn. (4), for the Fe-based

alloys with the f.c.c. crystal structure or the iron-

based intermetallics with the f.c.c.-like near-neighbor

arrangement of atoms. The compressibility values

300 K

in Table 2 correspond to the ferro- or ferri-

magnetic compressibility k

in materials with

4300 K and, hence, they comprise the magnetic

contributions, k

, proportional to the spontaneous

magnetostriction o

, see Eqn. (5). The enhanced val-

ues of k

in the iron-based alloy and intermetallics are

almost twice those in pure iron and they reﬂect the

pronounced magnetovolume phenomena in these

Figure 2

Temperature dependence of the pressure-induced

changes of the Fe-moment m

in the R

intermetallic compounds.

701

Magnetic Sys tems: External Pressure-induced Phenomena

materials. The relevant anomalous decrease in T

under pressure belongs to the Invar characteristics

(see Invar Materials: Phenomena).

The high negative values of the pressure parameter

/dp, the spontaneous magnetostriction o

, and

the compressibility k

in the Fe

and Fe

alloys and in the Y

intermetallics presented in

Table 2 are typical for Invar materials.

The model of ‘‘strong itinerant ferromagnets’’ was

successfully used to describe the experimental pressure

data for nickel and for many crystalline nickel-based

alloys with a low content of Cu, Cr, Fe, V, Pd, and

Rh, except for the NiMn alloy. The increase of T

with pressure, i.e., the pressure parameter dT

/dp,is

proportional to T

, as seen in Fig. 4. Using Eqn. (9)

the value of GBl is close to Heine’s result l ¼5/3

(Heine 1967). The same value of G was derived also

for UGa

. However, the very pronounced decrease in

its magnetic moment under pressure is inconsistent

with the discussed model.

3. Concluding Remarks

The development of high-pressure experimental tech-

niques extends the basic thermodynamic framework

to the study of pressure-induced phenomena in mag-

netic systems. New miniature nonmagnetic, non-

metallic pressure cells will allow experiments to be

carried out on bulk single crystals under multi-

extreme conditions—in very high magnetic ﬁelds

(including pulsed ﬁelds up to 100 T), under high

Figure 4

The dependence of dT

/dp on T

in the nickel-based

alloys, as expressed in Eqn. (9).

Figure 3

Temperature dependence of anisotropy constant K

YFe

Ti and Y

intermetallics at ambient pressure

(open symbols) and under high pressure (0.8 GPa)

(full symbols).

Table 2

Curie and Ne

el temperatures T

and T

, G ¼dlnT

C,N

/dlnV, compressibility k, and spontaneous magnetostriction o

of selected magnetic materials under high pressure.

Material T

C,N

(K) dT

C,N

/dp (KGPa

1

) G k

300K

(GPa

1

) o

627 þ3.6 1.1 0.54 10

2

0.03 10

2

1400 0 0 0.52 10

2

—

1043 0.3 0.05 0.59 10

2

0.14 10

2

505  43 9 0.94 10

2

2.2 10

2

378  36 9.7 0.98 10

2

1.6 10

2

302  38 12 1.05 10

2

1.7 10

2

FeO (antiferro)

188 þ6.5 5.3 0.65 10

2

0.36 10

2

(ferri)

564 þ12.8 3.4 0.68 10

2

—

294  14 1.8 2.61 10

2

—

UGe

52  12.5 20 1.2 10

2

—

UGa

120 þ2.3 1.6 1.5 10

2

—

aBloch and Pavlovic (1969). bOomi and Mori (1981). cMikulina et al. (1999). dFranse et al. (1985).

702

Magnetic Sys tems: External Pressure-induced Phenomena

pressure (up to 100 GPa), and at low temperatures

(down to mK range).

Studies of pressure-induced phenomena in mag-

netic systems have become an integral part of re-

search into magnetism. Extraordinary efforts will

undoubtedly be made to ﬁnd a solution to the ‘‘tra-

ditional’’ problems and systems: the f- and d-electron

systems, magnetic instabilities in mixed-valence sys-

tems (see Intermediate Valence Systems), and prop-

erties of two-dimensional magnetic structures (see

Bulk Magnetic Materials: Low-dimensional Systems).

There has been an enormous increase in the number

of reports on high-pressure research, especially in the

relatively new ﬁelds of study of the heavy fermions

(see Heavy-fermion Systems) and non-Fermi liquid

systems (see Non-Fermi Liquid Behavior: Quantum

Phase Transitions).

Bibliography

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high pressure. In: Bradley R S (ed.) Advances in High Pres-

sure Research. Academic Press, London, pp. 41–154

Bridgman P W 1958 The Physics of High Pressure. Bell and

Sons, London

Buschow K H J 1988 Magnetovolume effects in ternary com-

pounds of novel RFe

Si (R ¼Nd, Sm, Tb, Ho, Er, Y) in-

termetallic compounds. J. Less-Common Metals. 144, 65–9

Entel P, Herper H C, Hoffmann E, Nepecks G, Wassermann E

F, Acet M, Crisnan V, Akai H 2000 Understanding iron and

its alloys from ﬁrst principles. Phil. Mag. B 80, 141–53

Eremets M I 1996 High Pressure Experimental Methods. Oxford

University Press, Oxford, UK

Franse J J M, Frings P H, Menovsky A, de Visser A 1985

Magnetism of uranium intermetallics under high pressure.

Physica B 130, 180–8

Garﬁnkel M, Mapother D E 1961 Pressure effect on supercon-

ducting lead. Phys. Rev. 122, 459–68

Givord D, Lemaire R 1974 Magnetic transition and anomalous

thermal expansion in R

compounds. IEEE Trans. Mag.

MAG. 10, 109–13

Hayashi K, Mori N 1981 Pressure effect on magnetization of

Fe–Ni and Fe–Pt Invar alloys. Solid State Commun. 38,

1057–9

Heine V 1967 s-d Interaction in transition metals. Phys. Rev.

153, 673–82

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R 1999 Magnetization and magnetocrystalline anisotropy

of R

intermetallics under pressure. J. Appl. Phys. 85,

4874–6

Kanamori J 1963 Electron correlation and ferromagnetism of

transition metals. Progr. Theor. Phys. 30, 275–89

Lang N D, Ehrenreich H 1968 Itinerant-electron theory of

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and NiCu alloys. Phys. Rev. 168, 605–22

Mikulina O, Kamarad J, Arnold Z, Garcia-Landa B, Algarabel

P A, Ibarra M R 1999 Invar behaviour of Y

and

YFe

Ti single crystals: magnetic moment of Fe under pres-

sure. J. Magn. Magn. Mater. 196–197, 649–50

Moruzzi V L, Marcus P M 1988 Magnetism in bcc 3d transition

metals: onset and approach to the Hund’s-rule limit. Phys.

Rev. B 38, 1613–20

Oomi G, Mori N 1981 Pressure effect on the spontaneous

volume magnetostriction of Fe–Ni and Fe–Pt Invar alloys.

J. Phys. Soc. Jpn. 50, 2924–30

Sabiryanov R F, Jaswal S S 1998 Electronic structure and

magnetic properties of Y–Fe compounds. Phys. Rev. B 57,

7767–72

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pressure. In: Schilling J S, Shelton R N (eds.) Physics of

Solids under High Pressure. North-Holland, Amsterdam,

pp. 345–56

Sechovsky V, Havela L 1988 Intermetallic compounds of ac-

tinides. In: Wohlfarth E P, Buschow K H J (eds.) Ferromag-

netic Materials. North-Holland, Amsterdam, Vol. 4, pp. 309–

491

Sherman W F, Stadtmuller A A 1987 Experimental Techniques

in High Pressure Research. Wiley, Chichester, UK

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High Pressure. North-Holland, Amsterdam, pp. 175–80

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J. Kamara

Academy of Sciences of the Czech Republic, Prague

Czech Republic

Magnetic Systems: Lattice

Geometry-originated Frustration

Frustration of exchange interactions occurs in many

magnetic systems owing to various microscopic

mechanisms:

(i) Competing long-range exchange interactions

(i.e., RKKY interactions) are known to stabilize he-

licoidal structures in several rare earth compounds.

(ii) Any crystallographic structure containing tri-

angles (f.c.c., triangular lattices, etc.) is frustrated

in the presence of nearest-neighbor antiferromag-

netic interactions; the two-dimensional kagome

lat-

tice (Fig. 1) and its three-dimensional analogue, the

pyrochlore lattice (Fig. 2), are considered as the

‘‘most frustrated’’ structures among existing materi-

als (Liebmann 1986).

(iii) In disordered magnetic systems competition

between random ferro- and antiferromagnetic inter-

actions leads to spin-glass (SG) behavior (Mydosh

1993).

Frustration in nondisordered systems has been in-

tensively studied and there are many indications,

both theoretical and experimental, that new types of

ground states may occur in these systems owing to

the large ground-state degeneracy. The main purpose

of this article is to give an overview of the different

possibilities.

703

Magnetic Systems: Lattice Geometry-originated Frustration

In addition to the lattice geometry, the range and

sign of the interactions and the number of degrees of

freedom of local moments are important factors in

frustrated systems. These systems are very sensitive to

their microscopic characteristics: different behaviors

are expected for metallic or insulating systems, as in

3d metallic systems where the size of magnetic mo-

ments can change, giving rise to an additional degree

of freedom; other important parameters are the value

of the spin (S) and the strength of magnetic aniso-

tropy. For example, the triangular lattice with clas-

sical (equivalent to inﬁnite S) or quantum S ¼1/2

spins is ordered, but with Ising spins it remains dis-

ordered with a ﬁnite entropy per site at T ¼0K.

Frustration can be measured for a given model by

several parameters: e.g., when the entropy per site

does not vanish at T ¼0 K (which contradicts the

third law of thermodynamics (Ramirez et al. 1999)),

its value is a measure of the number of degenerate

ground states; another measure is the constraint

function, F, deﬁned as the ratio between the ground-

state energy, E

, and the energy of the system if it

would be possible to satisfy all pair interactions, E

i.e., F ¼E

. For a nonfrustrated system, F ¼1,

while in presence of frustration 1oFo1.

Many frustrated compounds order at low temper-

ature and an empirical measure of frustration can be

deﬁned from experiments (Ramirez 1994) as the ratio

of the paramagnetic Curie–Weiss temperature, y

which measures the strength of exchange interactions,

to the ordering temperature, T

(which can be any

cooperative transition temperature, e.g., Ne

el or SG

transition): f ¼7y

7/T

. This definition is useful for

classifying the frustrated compounds which exhibit

any kind of transition; they all correspond to f41.

However, it is not appropriate to a disordered ground

state. Values of f larger than 100 are often observed

in frustrated compounds. Clearly, a mean ﬁeld ap-

proach cannot be valid for compounds with such

large f values.

1. Localized Moments Systems (Gaulin 1994,

Harris and Zinkin 1996, Schiffer and Ramirez 1996,

Reimers et al. 1991)

1.1 Frustrated Classical Spins Systems

The ground state of localized spin systems is usually

well described by minimizing the classical exchange

energy. For a Bravais lattice with one magnetic site

per unit cell this energy can be written:

E ¼

i;j

JðR

R

ÞS

¼

Jðq

ÞS

q

ð1Þ

where Jðq

Þ is the Fourier transform of the exchange

interactions JðR

 R

Þ. Thus the ground state is de-

termined by the q

value(s) which maximize Jðq

Þ. For

non-Bravais lattices this procedure can be generalized

in the following way. For n magnetic sites per unit

cell, the ground state is determined by the largest

eigenvalue, l( q

), of a n n matrix. However, on the

‘‘fully frustrated’’ pyrochlore (Reimers et al. 1991) or

kagome

(Harris et al. 1992) lattices, the largest

eigenvalue, l( q

), is independent of q

over all the

Brillouin zone. In these lattices the magnetic atoms

are located on triangles (kagome

) or tetrahedrons

(pyrochlore), in such a way that one site is shared by

only two elementary clusters (triangles or tetrahe-

drons). All ground states satisfy the local constraint:

cluster

¼ 0 ð2Þ

Figure 1

The kagome

lattice: a triangular lattice (generated by the

translation vectors a and b) with a triangular basis

(indicated by the shaded triangles).

Figure 2

The pyrochlore lattice (f.c.c. lattice with tetrahedral

basis).

704

Magnetic Sys tems: Lattice Geometry-originated Frustration

for any elementary cluster. However, this constraint

can be satisﬁed throughout the lattice without induc-

ing any long-range magnetic order. In such a situa-

tion, the number of classical ground states increases

as the number of sites, N.

It is important to point out that the energy land-

scape is quite different from that which occurs in an

SG state, where the degenerate ground-state conﬁg-

urations are separated by energy barriers. Here, the

energy landscape is ﬂat, without any energy barrier

between the different states. The thermodynamic

properties of such systems are expected to be very

different from both conventional ordered magnetic

systems (where the ground state is nondegenerate)

and SG. The absence of ordering in frustrated lattices

is also different from that found in one- and two-

dimensional systems, where magnetic correlations

develop below the mean ﬁeld transition temperature

but no ordering occurs (or only at T ¼0 K) due to

entropy effects.

Finally, in some lattices the highest eigenvalue is

independent of q

only along some lines of the Brill-

ouin zone called ‘‘degeneration lines.’’ In this case the

number of ground states increases only as N

1/3

and

ﬂuctuations beyond mean ﬁeld solution may induce a

phase transition; it has been proposed that such a

scenario applies for b-oxygen.

1.2 Possible Ground States in Fully Frustrated

Systems

In the mean ﬁeld approximation, even at T ¼0K, no

conventional magnetic long-range order occurs in

fully frustrated lattices. Several mechanisms have

been proposed which could remove the degeneracy:

The ﬂuctuations around a particular mean ﬁeld

state contribute to the free energy beyond the mean

ﬁeld approximation. This contribution may change

the energy landscape by introducing ‘‘valleys.’’ In this

case, one ordered state will be selected among all the

degenerate states. This phenomenon, called ‘‘order by

disorder,’’ may occur owing to thermal disorder or to

quantum zero point ﬂuctuations or in the presence of

a small concentration of impurities.

Any additional interaction, such as anisotropy,

deformations, and dipolar or further-neighbor ex-

change (Harris et al. 1992, Chandra and Coleman

1995), can play an important role in these systems

and induce ordering, even if they are much smaller in

magnitude than exchange. For quantum spins, more

exotic ground states have been proposed, such as

dimer, resonant valence bond, nematic, or chiral or-

dering (Chandra and Coleman 1995). No evidence

for such ground states has been found.

Probably, the ground state is a spin-liquid (SL)

state, i.e., a singlet state with short-range correlations

and a nonzero spin gap. However, the excitation

spectrum of these SL states would be different from

nonfrustrated ones (e.g., one-dimensional integer

spin chains). In the present case, the singlet–singlet

excitations form a continuum starting at zero energy,

while in the integer spin chains, the lowest energy

excitations have a nonzero gap corresponding to sin-

glet–triplet transitions.

1.3 Examples of Fully Frustrated Compounds

A few kagome

compounds have been studied:

In SrCr

8x

4 þx

(SCGO) (Schiffer and

Ramirez 1996), the Cr

3 þ

ions (S ¼3/2) are located

both on kagome

planes and on interplane triangular

layers. This compound does not order despite its high

negative paramagnetic Curie–Weiss temperature,

E500 K. It exhibits an SG behavior below

¼5 K, which could be due to a partial covering

of the kagome

planes.

The chromium and iron jarosites (with S ¼3/2

and 5/2, respectively) are also two-dimensional

kagome

compounds (Wills et al. 1998). Both systems

order (antiferromagnetic or SG state) at temperatures

quite low compared with the Curie–Weiss tempera-

tures, T

E0.03 for chromium and 0.08 for iron

jarosites. Moreover, below T

magnetization is

strongly reduced by ﬂuctuations.

The only example of an integer spin (S ¼1)

kagome

lattice is the organic antiferromagnet

m-MPYNN.BF

, which might have an SL ground

state (Wada et al. 1997).

There are many more compounds exhibiting the

three-dimensional pyrochlore structure: the oxides

with general formula A

; the spinels AB

and

, where both A and B can be magnetic (rare

earth or transition metal) ions; and the ﬂuorides

ABCF

. Among these compounds, some exhibit non-

collinear magnetic ordering which results from the

competition between frustrated interactions and an-

isotropy. Examples are FeF

(Ferey et al. 1986),

l-MnO

,Nd

, and Mn

(Gaulin 1994),

but many of them remain nonmagnetic and have un-

conventional low-temperature magnetic properties:

SG-like behavior is observed in many pyro-

chlores in the absence of any structural disorder

(Schiffer and Ramirez 1996).

Strong, short-range magnetic correlations, as

expected in an SL system, are observed in CsCrNiF

(Harris and Zinkin 1996) and Li

A ‘‘spin-ice’’ ground state has been proposed for

and Dy

(Ramirez et al. 1999), where

the exchange interactions are ferromagnetic but the

presence of strong local Ising-like anisotropy leads

to a disordered ground state analogous to common

water ice.

Finally, the gadolinium gallium garnet Gd

(GGG) (Schiffer and Ramirez 1996) belongs to the

same class of compounds: the Gd

3 þ

ions are located

on two interpenetrating corner-sharing triangular

705

Magnetic Systems: Lattice Geometry-originated Frustration

sublattices. It exhibits a complex ﬁeld-dependent

magnetic phase diagram, but there are indications

that the zero-ﬁeld ground state is a SL one (Petrenko

et al. 1998). Dipolar interactions may also play a role

in this compound.

Many of these compounds exhibit an SG-like be-

havior observed in magnetic response, but specific

heat, which probes the low-energy spectrum, is dif-

ferent from normal SG (T

behavior instead of T). It

is not yet clear whether this behavior is intrinsic to SL

states or if it is due to a very small (even undetectable)

amount of disorder.

2. Metallic Compounds

2.1 Mixed Magnetic Phases

(Ballou 1998, Yoshimura et al. 1986)

In itinerant electron antiferromagnets, the amplitude

of the magnetic moments can change with respect to

external parameters (pressure, temperature, magnetic

ﬁeld, etc.) or to local environment (coordination, in-

teratomic distance, etc.). This additional degree of

freedom may lead to nontrivial magnetic phases.

Close to a magnetic–nonmagnetic (M–NM) insta-

bility, such amplitude variations occur more easily:

frustration can then be lowered or even completely

suppressed if some of the magnetic moments vanish.

Such mixed phases, where magnetic and nonmag-

netic sites coexist, have been observed in several

compounds:

The RMn

compounds crystallize in either the

C15 cubic (for R ¼Y, Sm, Gd, Tb, Dy, or Ho) or the

C14 hexagonal Laves phase, the ﬁrst one being sim-

ilar to the pyrochlore structure (shown in Fig. 2). The

manganese magnetic moment depends strongly on

the interatomic Mn–Mn distance. This allows stab-

ilization of mixed magnetic phases in DyMn

(Yoshimura et al. 1986), TbMn

(Ballou et al.

1992), and ThMn

, with the concentration of mag-

netic manganese atoms equal to 1/4 (dysprosium and

terbium compounds) and 3/4 (hexagonal thorium

compound). The stability of these mixed phases as

well as their behavior with temperature or applied

ﬁelds are well understood in the framework of an

itinerant frustrated model close to an M–NM insta-

bility (see also Alloys of 4f (R) and 3d Elements (T):

Magnetism).

Mixed structures are also observed in a few ce-

rium and uranium compounds. In CeSb (Rossat-Mig-

nod et al. 1983) a large number of magnetic phases

have been observed and several of them can be de-

scribed by a stacking of ferromagnetic and non-

magnetic cerium planes. In the triangular UNi

(Mentink et al. 1994) or kagome

-like CePdAl (Ballou

1998) compounds, the magnetic atoms form chains,

one-third of which are nonmagnetic. These com-

pounds are also close to a M–NM instability in-

duced by the Kondo effect. Their observed magnetic

structures can be well explained by a frustrated

Kondo lattice model with intersite RKKY exchange

interactions of the same order as the Kondo con-

densation energy T

A different kind of mixed phase occurs when the

instability of the local moments is due to the crystal

ﬁeld scheme in non-Kramers ions. This occurs in

TbRu

owing to the small energy difference be-

tween the nonmagnetic singlet ground state and the

magnetic excited states (Lacroix et al. 1998).

All these mixed phases can be modeled within the

same pseudo-moment Hamiltonian which includes, in

addition to the exchange interaction between the

magnetic sites, the energy, D, necessary for stabilizing

a magnetic moment. In TbRu

, D is related to

crystal ﬁeld splitting; in the Kondo compounds,

DET

, while in the manganese Laves phase, D is the

stabilization energy of a magnetic moment in an itin-

erant 3d band. Mixed structures are observed when D

is of the order of the exchange.

2.2 Itinerant Spin Liquids

In some itinerant systems the absence of magnetic

ordering can be attributed to frustration, as in local-

ized frustrated systems, rather than to the instability

of band magnetism. This is the case for the cubic

Laves phase compound Y

0.97

0.03

, which re-

mains paramagnetic at low temperature, and shows

strong spin ﬂuctuations with short-range magnetic

correlations of the order of the interatomic distance.

The observed spectrum can be described completely

by a localized spin model on a pyrochlore lattice

(Canals and Lacroix 1998) indicating that it is the

crystal structure that plays the main role in the ab-

sence of ordering. Moreover, the signature of a sin-

glet ground state has been observed directly in

neutron experiments. The vanishing of the signal in

the ﬁrst Brillouin zone implies that:

wðq

¼ 0Þp

¼ 0 ð3Þ

The same behavior occurs in b-manganese, which has

a complicated frustrated crystallographic structure

formed by corner-sharing triangles and remains par-

amagnetic at low temperatures (Nakamura et al.

1997).

These frustrated metallic systems exhibit large

specific heat coefﬁcients, g (70 mJK

2

mol

1

for b-

manganese, 180 mJK

2

mol

1

for Y

0.97

0.03

and they are considered as ‘‘3d heavy fermions.’’

Much larger values have been measured in the spinel

compound LiV

where g ¼420 mJK

2

mol

1

(Kondo et al. 1997). However, unlike the usual 4f or

5f heavy fermion systems, the Kondo effect is not ob-

served in these 3d materials. Thus, in the absence of

any Kondo effect, such large g values indicate that the

706

Magnetic Sys tems: Lattice Geometry-originated Frustration

density of low-energy excitations is large in these itin-

erant SL states (see Heavy-fermion Systems).

3. Summary

Strongly frustrated antiferromagnets may exhibit

various unconventional ground states. An SL state

can be stabilized in both localized and itinerant sys-

tems with frustrated crystallographic structure:

The understanding of localized SL states has

increased considerably, but several aspects need fur-

ther study, especially the low-temperature dynamics

and the difference with conventional SG states. It has

been proposed that these systems belong to a new

class of ‘‘topological spin glasses.’’

For metallic SL states, static properties can be

described satisfactorily within a localized model, but

the dynamics are certainly different (‘‘3d heavy fer-

mions’’). Also, in various materials close to a mag-

netic instability, mixed magnetic structures are

stabilized.

See also: Bulk Magnetic Materials: Low-dimensional

Systems; Electron Systems: Strong Correlations;

Magnetic Systems: Disordered; Heavy-fermion Sys-

tems; Spin Fluctuations

Bibliography

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1997 Strong antiferromagnetic spin ﬂuctuations and the

quantum spin liquid state in geometrically frustrated b-Mn

and the transition to a spin glass state caused by non mag-

netic impurity. J. Phys.: Condens. Matter 9, 4701–28

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Laboratoire Louis Ne

el, CNRS, Grenoble, France

Magnetic Systems: Specific Heat

Historically, specific heat was deﬁned as the heat (dQ)

required to rise the temperature (T) of a unit of mass

of a substance in a unit of degree, C

¼ dQ=dT,

where y indicates the control parameters kept con-

stant. This elementary definition worked hand in

hand with the development of the thermodynamics

because it is directly related to those parameters to be

taken into account to describe the state of any phys-

ical system, i.e., its energy and temperature.

707

Magnetic Systems: Speciﬁc Heat

From the First Law, the heat absorbed by a system

is distributed as, dQ ¼ dU þ dW, where U is the in-

ternal energy and W the work done by the eventual

expansion (compression), or by the change in the

magnetization (M) under magnetic ﬁeld (h). If only

the thermal component of the system is taken into

account (i.e., dV ¼0), the simple expression C

¼dU/

dT is obtained. However, the measurements are nor-

mally performed under vacuum (adiabatic) condi-

tions, that implies at constant pressure (C

Therefore, C

¼dH/dT is actually measured, where

H ¼U þpV is the enthalpy of the system, which in-

cludes the expansion of the system during the heating

process. The difference between C

and C

is given by

C

¼TVa

/k, where a is the thermal expansion

and k the compressibility (e.g., see Goodstein 1985).

Provided that a and k are nearly temperature inde-

pendent, this difference becomes negligible at low

temperatures. But such is not strictly the case around

phase transitions or when T is comparable with the

characteristic energy of the system. The same pre-

cautions have to be taken with other relations, like

the Gru

neissen parameter, GBka=C

, when k and/or

a have significant T dependence and consequently

(T) differs from the measured C

(T).

The Second Law refers to the progressive access to

the excited states with the increase of temperature. The

measure of that process is evaluated by the entropy,

S(T), which is related to dQ through, dS ¼dQ/T. Then,

specific heat can be expressed as, C

¼T(dS/dT)

, and,

therefore, the experimental knowledge of C

(T)pro-

vides a direct measure of the entropy evolution of the

system, S(T) ¼

/T dT, i.e., how the system accesses

to its thermally excited levels. From this point of view,

the specific heat works like a thermal spectrometer,

where the excitation is given by dQ and the response by

dT. One of the most important aspects of the knowl-

edge of S(T) is that it provides information on the ﬁrst

term of the free energy variation (dG ¼SdT þ

VdpMdh). This is the reason why the access to the

evolution of C

/T(T) constitutes one of the basic tools

for the characterization of any new system.

1. Contributions to the Specific Heat

Since all allowed states of a system are equally likely,

and each component contributes with its own entropy

to the total value (i.e., S(T) ¼S

(T)), the specific

heat provides simultaneous information on the full

spectrum of excitations. Then from C

/T ¼dS/dT it is

possible to compute the contribution of all the com-

ponents of the system. To extract the information re-

lated to each of them requires ﬁnding at which range

of temperature a specific contribution becomes dom-

inant. According to their respective temperature range

of importance, the following typical contributions

can be enumerated: (i) nuclear for To0.5 K which

are usually recognized by a C

2

dependence, (ii)

conduction electrons within the 0.5 KoTo4 K range,

(iii) when present, heavy fermion (HF) quasi-particles

dominate the scene up to about 7 K, (iv) phonons for

about T4y

/30, where y

is the Debye temperature,

(v) magnons, which are related to some magnetic

transition and/or anomalies, are dominant up to

about 20 K, and (vi) Schottky anomalies are well ob-

served when their respective characteristic energies (or

associated gaps, D) are about k

Do20 K. Each of

these contributions is related to different statistics,

which describe the thermal population of the excited

states. For example, Maxwell–Boltzman statistics

applies to Schottky-type anomalies (like those origi-

nated by nuclear or crystalline electric ﬁeld (CF)

splitting), Fermi–Dirac statistics to the electrons

in the band, and Bose–Einstein statistics to phonons,

magnons, and other quasi-particles. Since many of

these contributions are extensively treated in the

literature (e.g., see Gopal 1966) only recent develop-

ments related to magnetic properties will be consider-

ed in this article.

Conduction electrons can be described as free elec-

trons and their linear contribution to the specific heat,

¼gT, is directly proportional to the density of

states at the Fermi level, N(E

) (e.g., see Goodstein

1985). The factor gpN(E

) is the Sommerfeld coefﬁ-

cient, which is constant within the experimental range

of temperature. Interaction of band electrons with

some quasi-particles (namely phonons, l

or magn-

ons, l

) dresses the measured g value, which is then

corrected as, g ¼ g

(1 þl

þl

). Experimental values

of g are extracted from a C

/T versus T

representa-

tion, as shown in Fig. 1, with g being the value at T-

0. In pure metals, it ranges between 0.19 mJ mol

1

2

and 10 mJ mol

1

2

for beryllium and yttrium, re-

spectively (Gopal 1966). These values are strongly

enhanced by electron–electron interactions or mag-

netic correlations, as it occurs in intermediate valent

compounds with 10ogo80 mJ mol

1

2

(e.g., CeIr

Sereni 1991) or the system showing spin ﬂuctuations

with 50ogo200 mJmol

1

2

(e.g., UAl

, Stewart

1984). A further increase of this electronic term is ob-

served in heavy fermions, where the g term becomes

up to three orders of magnitude larger than that of a

simple metal (e.g., CeCu

,Lo

hneysen 1995).

Phonon contribution, C

, is extracted from the

slope of a C

/T versus T

representation (see Fig. 1)

because C

/T ¼C

/T þC

/T ¼g þbT

, with b ¼

12p

N/5y

. This formula is valid up to T ¼y

24. Some misuse of bpN/y

causes an incorrect

evaluation of y

as it is strictly valid for pure ele-

ments or N atoms of equal mass. This occurs, for

example, when it is applied to systems involving

components with different masses, where false vari-

ation of y

(T) is extracted. A similar situation ap-

pears in solids containing a variable number of

interstitial atoms without a concomitant change in

the measured b. If it were included in N a false vari-

ation of y

(x) would be obtained.

708

Magnetic Sys tems: Speciﬁc Heat

Magnetic contributions, C

, include a large variety

of contributions to C

(T), which are evaluated as

¼C

(C

þC

). In ﬁrst glance, contributions

related to a magnetically ordered phase and those

arising from excitations which do not develop long-

range order (LRO) can be distinguished. Others, like

nearly or weak magnetic systems, are sometimes dif-

ﬁcult to deal with because they appear at the verge of

the ordered phase. Section 2 is devoted to magnetic

contributions directly related to the ordering transi-

tion, followed by one describing the ordered phase

itself. Afterwards, other types of magnetic transfor-

mations and excitations will be considered.

2. Specific Heat at the Ordered Phase Boundary

As the second derivative of the free energy (C

¼T@

), the specific heat deﬁnes the order of the phase

transition at the critical temperature (T ¼T

ord

). In the

case of a ﬁrst-order transition the discontinuity in S(T)

is reﬂected in C

(T) as a divergence at T

ord

, and the

latent heat (L ¼TDS) can be evaluated as,

(T)dT.

Here, L accounts the change of internal energy from

one magnetic phase to the other, plus the eventual

volume and/or magnetization changes (LpDH). The

Clausius–Clapeyron relationships, dT

ord

/dp ¼DV/DS

and dT

ord

/dh ¼DM/DS give valuable information

about the evolution of the phase transition as a func-

tion of the external control parameters.

In solids, most of these transitions are related to a

sudden change of volume due to magnetostrictive ef-

fects or crystalline symmetry modiﬁcations, driven by

some strong spin-phonon coupling. Examples of ﬁrst-

order transitions are those observed in dimerized sys-

tems, spin-Peirels chains, or quadrupolar splitting.

Unfortunately, due to the limited knowledge of the

elastic properties of these materials, a detailed de-

scription of the process involving a ﬁrst-order tran-

sition can become quite difﬁcult (Stanley 1971). The

temperature dependence of C

(T) close to the tran-

sition provides useful information about the micro-

scopic nature of the transition (e.g., dimensionality).

The exponent describing the temperature dependence

of C

(T) (with t ¼7TT

ord

7/T

ord

) follows a power

law, t

Z

, where Z is the critical exponent with a value

close to zero (e.g., see Stanley 1971).

In a second-order transition a jump in C

(T), DC

(like the one shown in Fig. 2) is expected at T ¼T

ord

because of the change of the dS/dT slope. According

to the Ginsburg–Landau (GL) theory, an order pa-

rameter (z) sets on at T ¼T

ord

, with inﬁnite initial

slope and develops continuously till T ¼0. Within

the scope of the GL theory, the free energy (G(F))

shows a minimum of G at F ¼z and then the var-

iation of z(T) is reﬂected in @

G/@T

as a disconti-

nuity or jump in the specific heat. Taking advantage

that in ferromagnets z corresponds to the sponta-

neous magnetization M

(T), a direct observation of

that parameter can be done as shown in Fig. 2. Fur-

thermore, as DU

then C

(T)pdM

/dT is

obtained. In broadened transitions, T

ord

can be

experimentally determined as the temperature where

Figure 1

Electronic (g) and phonon (b) contributions in a C

=T ¼

g þ bT

representation for different systems.

Figure 2

Ferromagnetic example for DC

ord

) and C

(T),

(T) and M

pz(T) evolutions. Inset: Arrenius

representation for the magnon gap evaluation.

709

Magnetic Systems: Speciﬁc Heat