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

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410

Longitudinal Media: Fast Switching

Magnetic Anisotropy

‘‘Magnetic anisotropy’’ is an ambiguous term. Since

a magnetic ﬁeld is a vector quantity, it implies by

definition direction dependence. In 1821 Poisson as-

sumed an interplay of magnetism with crystallinity.

After 1850 the ‘‘magnetism of crystals’’ was experi-

mentally investigated. Since then it has been known

that diamagnetism, paramagnetism, and natural

ferromagnetism all depend on the properties of the

different materials, especially on the crystallographic

properties.

Any dependence of the magnetic properties of a

material on the direction can be termed magnetic

anisotropy. It can be described by a dimensionless

‘‘anisotropy function,’’ and it is closely related to

space-dependent magnetization patterns, yielding a

total (direction-dependent) magnetic free energy.

Usually, however, only the magnetic properties of

solids in the magnetically ordered state are charac-

terized by magnetic anisotropy.

Several different origins and manifestations of

magnetic anisotropy can be distinguished, namely

shape anisotropy, exchange anisotropy, stress aniso

tropy, surface anisotropy, anisotropy induced by a

magnetic ﬁeld, and, last but not least, magnetocrys-

talline anisotropy. The last is the most important for

modern hard magnetic materials and permanent

magnets.

1. Definitions of Magnetic Anisotropy

The term ‘‘magnetic anisotropy’’ is not a primary unit

(see Magnetic Units), although magnetic anisotropy is

of utmost importance for soft and especially hard

magnetic materials.

For soft magnetic materials (see Magnetic Losses)

the smallest possible coercivity

is a pri-

mary goal. It can be deduced from the hysteresis loop

in the M–H or B–H representation, respectively.

Since the magnetic anisotropy is linked to coercivity,

usually the magnetic anisotropy of any origin is

minimized in soft magnetic materials. Medium-soft

magnetic materials, especially magnetic recording

materials, rely on a sufﬁciently large (because of

long-term stability) but sufﬁciently small (because of

the necessity of frequent remagnetization) coercivity,

and is usually in the range of 10

–10

1

(ca. 100–

1000 Oe). For such applications materials with shape

anisotropy (e.g., Fe

needles), surface anisotropy

(thin ﬁlms), or magnetocrystalline anisotropy are

needed. A clear-cut distinction of the nature of mag-

netic anisotropy is rarely possible.

Shape anisotropy is nearly always present, since

the magnetic medium is never an ideal sphere. The

origin of shape anisotropy is the magnetostatic ener-

gy. Shape anisotropy can be used to prepare perma-

nent magnets. (In the 1950s elongated single domain

magnets were fabricated from pure cubic iron in the

shape of needles). Shape anisotropy also plays an

important role in the anisotropy of Alnico magnets

(see Alnicos and Hexaferrites).

Stress anisotropy is the counterpart to magneto-

striction. It is observed if an external stress, via

magnetoelastic coupling, favors energetically some

magnetization directions. The already mentioned sur-

face anisotropy is always present in thin layers or

if extended surfaces are present in the material, e.g.,

nanocrystalline samples (see, e.g., Amorphous and

Nanocrystalline Materials).

In this article mainly the subject of magnetocrys-

talline anisotropy (MCA) will be treated in more de-

tail, since it is the basis of modern permanent magnets.

2. Experimental Determination of Magnetic

Anisotropy

In principle the measurement of magnetic properties

(see Magnetic Measurements: Quasistatic and ac) can

be performed as a function of the crystallographic

directions of the sample and/or an applied magnetic

ﬁeld. From these results frequently information

about the magnetic anisotropy can be deduced. Some

methods especially suited to probe magnetic anisot-

ropy, however, are mentioned here. If a single crystal

of the magnetic material is available, the magnetic

anisotropy and the anisotropy coefﬁcients K, etc., are

obtained by, e.g., a torque magnetometer. An aligned

ensemble of powders can be used instead of a single

crystal but with reduced accuracy.

Due to the magnetic moment of the neutron, elastic

and inelastic neutron scattering (see Magnetic Ex-

citations in Solids) is a primary tool to establish the

spin structure of magnetic materials and also the

magnetic excitations. Since these measurements can

be performed at different magnetic ﬁelds and at

variable temperatures, many magnetic parameters,

including the magnetic anisotropy, can be deduced

from neutron experiments. By rotating a sample in a

magnetic ﬁeld and detecting the transverse suscepti-

bility the pinning ﬁeld, which is directly linked to

the magnetic anisotropy, can be measured (Elk and

Hermann 1992).

‘‘Free moving’’ particles studied for magnetization

at high (pulsed) ﬁelds allow an analysis of the cou-

pling of different magnetic sublattices and the asso-

ciated magnetic anisotropy (Franse and Radwanski

1996).

In their classical measurement of the magneti-

zation of a YCo

single crystal in two perpendicular

411

directions Hoffer and Strnat (1966) detected a huge

anisotropy ﬁeld of approximately 13 T (130 kOe) and

opened up the area of rare-earth permanent magnets

(see Rare Earth Magnets: Materials).

3. Magnetocrystalline Anisotropy

Disregarding the microscopic origin of MCA this

property can be measured and analyzed. It can be

described in the following manner (Kirchmayr 1991).

Since hexagonal and tetragonal crystal structures are

the most important structures for modern permanent

magnets, only these two are discussed here. The mean

magnetic moment of a ferro- or ferrimagnetic sub-

stance is directed, in the absence of an external mag-

netic ﬁeld, in the crystallographic direction in which

the absolute minimum free energy, F , is achieved.

This direction is called the ‘‘easy magnetization

direction’’ or ‘‘easy axis.’’ A part of the total free en-

ergy, therefore, depends on the direction of magnet-

ization with respect to the different crystallographic

directions and is called the magnetocrystalline free

energy, F

or the MCA. Usually, F

is expressed in

terms of simple sine or cosine functions. For hexa-

gonal crystals,

¼ K

sin

y þ K

sin

y þ K

sin

y þ K

sin

ycos6f

and for tetragonal crystals,

¼ K

sin

y þ K

sin

y þ K

sin

ycos4f

where y is the angle between the c axis and the vector

of magnetization, f is the angle between the projec-

tion of the magnetization vector in the basal plane of

the crystal and one of the a axes, and K

, K

, and

are the coefﬁcients of anisotropy, expressed in

3

or in GOe; 1 MGOe ¼7.9577 kJm

3

The anisotropy constants reﬂect the magnetic an-

isotropy only phenomenologically and have no direct

connection with the physical origin of magnetic an-

isotropy.

If an external magnetic ﬁeld H is applied to a crys-

tal, an additional term appears in the free energy, F,

which is given by

F ¼ F

 M

Hcosðy  bÞ

where M

is the saturation magnetization, b is the

angle between H and the c axis, and (yb) is the angle

between M

and H. In this equation the stray ﬁeld

energy, which depends on the size and the shape of

the sample, has been neglected. Equilibrium is

achieved when F reaches a minimum, hence

dF=dy ¼ dF

=dy þ m

Hsinðy  bÞ¼0

dF=dy ¼ m

Hsinðy  bÞ¼m

9M  H9

where m

9M H9 is the torque exerted by the crystal

anisotropy and is counterbalanced by the external

ﬁeld. For small deviations from the easy axis, Dy ¼0,

the term

ð1=m

Þ:ðd

=dy

Þ9

Db¼0

¼ H ¼ H

is equivalent to a ﬁeld, called the anisotropy ﬁeld,

As can be seen from the above discussion, H

deduced from small deviations from the easy axis and

is therefore deﬁned for small applied ﬁelds. It must be

distinguished from the anisotropy ﬁeld, H



, which

can be measured directly by the singular point de-

tection method, i.e., at high magnetic ﬁelds above

. This method has its merits because it may be

applied to polycrystalline samples and in pulsed ﬁelds

(see Magnetic Measurements: Pulsed Field ). It yields

directly the anisotropy ﬁeld H

The magnetic anisotropy can therefore be given

by the anisotropy constants K

, K

, and K

or by

the anisotropy ﬁeld H

resp. H

.(K

is usually ne-

glected.) The anisotropy ﬁeld can be measured

directly on polycrystalline samples, in contrast to

, K

, and K

which are usually not directly acces-

sible but can be determined on single crystals by

torque measurements. However, because of the prob-

lem of achieving well-developed single crystals with

different compositions and because of technical prob-

lems to measure the torque at different temperatures,

the classical determination of the magnetic anisotro-

py constants by means of torque measurements is of

limited practical use.

4. Microscopic Origin of Magnetocrystalline

Anisotropy

Long-range spin ordering over macroscopic distances

is the result of exchange, i. e., the combination of the

electrostatic coupling between electron orbitals and

the Pauli exclusion principle. The nature of this ex-

change is isotropic. Anisotropy, however, is the result

of the interaction of the electron orbitals with the

potentials of the lattice. The lattice symmetry is

therefore responsible for the MCA. A rigorous cal-

culation of the MCA from ﬁrst principles is at present

not possible. A simpliﬁed approach is the molecular

ﬁeld calculation. In the case of RE–3d (RE ¼rare-

earth) intermetallics a phenomenological separation

into the contributions of the RE and the 3d sublattice

is often meaningful. Micromagnetism tries to discuss

the behavior of magnetic materials without detailed

knowledge of the basic origin of magnetic ordering

and anisotropy. Much progress has been achieved by

this approach, especially by Kronmu

ller et al. (1994);

(see Coercivity Mechanisms).

There are three different types of permanent mag-

netic materials, namely RECo

and RE

, which

412

Magnetic Anisotropy

are both hexagonal, and RE

B which is tetrago-

nal (Kirchmayr 1996). The basis of the magnetic or-

dering and the magnetic anisotropy in metallic

compounds of RE metals with transition metals (T)

are the RE–RE, T–T, and RE–T interactions. (For a

review of the phases, crystal structures, and magnetic

properties of RE–3d intermetallics see Kirchmayr

and Burzo 1990). In the RE

B compounds the

situation is even more complex because of the pres-

ence of a third element. Naturally, in a metallic sys-

tem most of the interactions are mediated by the

electrons, especially the conduction electrons, which

means that the fullband structure has to be taken into

account. While the RE anisotropy is a single-ion an-

isotropy mainly governed by crystal ﬁeld effects, the

T anisotropy is complex and less well understood. All

these interactions plus sublattice anisotropies are re-

sponsible for various types of magnetic transitions

and peculiarities, such as ﬁrst-order magnetization

processes, critical ﬁelds, noncollinarity of ferrimag-

netic structures, etc. (see also Localized 4f and 5f

Moments: Magnetism and Alloys of 4f (R) and 3d

(T) Elements: Magnetism).

5. Magnetocrystalline Anisotropy as a Basis for

Modern Permanent Magnets

In Table 1 some basic data for hard magnetic ma-

terials are summarized (for the temperature depend-

ence of K

and K

of these compounds see also

Kirchmayr and Burzo 1990). A comparison of the

anisotropy ﬁeld H

and the coercivity

shows a

large discrepancy. H

serves as an upper limit for

and is a necessary prerequisite for a good mod-

ern permanent magnet with large saturation and co-

ercivity and therefore energy product (see Hard

Magnetic Materials, Basic Principles of ). In practice

the B–H representation of the hysteresis loop is used.

In this case

is sufﬁcient to achieve a linear

demagnetization curve in the second quadrant. Since

the coercivity decreases with increasing temperature,

even at the operating temperature

should

be valid. This condition is usually met by modern

permanent magnets. In the future a further increase

in the energy product seems therefore to be feasible

only by increasing the saturation, but not by increas-

ing the coercivity or the magnetic anisotropy.

6. Summary and Conclusions

Anisotropy is a basic property of all crystalline ma-

terials. Even in nanocrystals and amorphous solids,

e.g., metallic glasses, anisotropy is present on an

atomic level. Therefore magnetic anisotropy is an in-

trinsic property of magnetization in general.

In the last decades MCA made possible the devel-

opment of better and stronger permanent magnets,

characterized by a higher coercivity and an ever in-

creasing energy product. This fast development, how-

ever, may reach a natural boundary, because the

coercivity has reached a value which is sufﬁcient to

make full use of the possibilities of modern permanent

magnet materials. Any further progress in energy

product cannot be achieved by an increase in magnetic

anisotropy, but only by the development of materials

with larger saturation magnetization while maintaining

sufﬁciently high coercivity. The development of mate-

rials with large magnetocrystalline anisotropy attempts

to retain this property also at elevated temperature.

From the viewpoint of basic research, the under-

standing and calculation of magnetic anisotropy from

ﬁrst principles is still in its infancy. Therefore, studies

of the magnetic anisotropy will continue to be a fas-

cinating area of basic and applied research for the

time to come.

Bibliography

Coey J M D (ed.) 1996 Rare-earth Iron Permanent Magnets.

Clarendon Press, Oxford

Elk K, Hermann R 1992 The transverse susceptibility of text-

urized polycrystals. J. Magn. Magn. Mater. 117, 359–67

Franse J J M, Radwanski R F 1996 Intrinsic magnetic prop-

erties. In: Coey J M D (ed.) Rare-earth Iron Permanent

Magnets. Clarendon Press, Oxford

Gutﬂeisch O 2000 Controlling the properties of high energy

density permanent magnetic materials by different processing

routes. J. Phys. D: Appl. Phys. 33, R157–72

Hoffer G, Strnat K J 1966 Magnetocrystalline anisotropy of

YCo

and Y

. IEEE Trans. MAG. 2, 487–9

Kirchmayr H 1991 Determination of the anisotropy ﬁeld by the

singular point detection method. In: Long G J, Grandjean F

(eds.) Supermagnets, Hard Magnetic Materials. Kluwer

Academic Publishers, The Netherlands, pp. 449–60

Kirchmayr H R 1996 Permanent magnets and hard magnetic

materials. J. Phys. D: Appl. Phys. 29, 2763–78

Kirchmayr H R, Burzo E 1990 Magnetic properties of rare

earth compounds with 3d elements. In: Crystal and Solid

State Physics, Magnetic Properties of Metals; Subvolume d2.

Compounds Between Rare Earth Elements and 3d, 4d or 5d

Elements. Landolt–Bo

rnstein New Series Group III, Spring-

er, Berlin, vol. 19, Chap. 2.4, pp. 1–468

Kronmu

ller H, Schreﬂ T, Fischer R, Fidler J 1994 Micromag-

netic analysis of coercivity of rare earth–transition metal

intermetallics and interstitial compounds. Proc. 8th Symp. on

Magnetic Anisotropy and Coercivity in RE–TM Alloys,

Birmingham, pp. 1–20

H. Kirchmayr

Vienna University of Technology, Austria

Table 1

Basic data for hard magnetic materials.

(K)

Compound (K) (T) (MJ m

3

) (T)

SmCo

993 40 17 1.05

1100 6.5 3.3 1.30

B 585 6.7 5 1.60

Source: Gutﬂeisch (2000).

413

Magnetic Anisot ropy

Magnetic Excitations in Solids

This article is focused on the discussion of one im-

portant experimental method for measurements of

magnetic excitations in solids, i.e., inelastic neutron

scattering (INS): a method that has been in the past

and that will be in the future the method of choice in

this ﬁeld.

In the ﬁrst part, an overview is given on magnetic

neutron cross-sections, on the range of experimen-

tally accessible transfers of momentum and energy

((Q; o)-range), and on important types of neutron

spectrometers that are used for the measurements of

magnetic excitations.

In the second part, the hierarchy of relevant solid-

state interactions and the involved energy scales will

be discussed for compounds containing the mag-

netic elements with 3d-, 4f-, and 5f-electrons in open

atomic shells.

In the third and last part, elementary magnetic ex-

citations like those from spin waves, crystal-ﬁeld

transitions, inter-multiplet transitions, and spin ﬂuc-

tuations will be presented in relation to their occur-

rence in bulk magnetic materials containing the

elements with 3d-, 4f-, and 5f-electrons.

An early review on magnetic excitations was given

by (Stirling and McEwen 1986).

1. Magnetic Neutron Cross-sections

The interaction of thermal neutrons with matter is

twofold: the nuclear interaction with the atomic nu-

clei and the magnetic interaction of the neutron’s

magnetic dipole moment with the magnetic ﬁeld B

created by the unpaired electrons in the sample. A

comprehensive theoretical treatment is given in the

two volumes of the book by Lovesey (1984). Focus-

ing on magnetic scattering we give as an example the

scattering cross-section for unpolarized neutrons

(Bo

ni and Furrer (1998); for meaning of symbols

look up the cited article):

dO dE



mag

F QðÞ









Q; oðÞð1Þ

where S

Q; oðÞis the van Hove magnetic scattering

function:

Q; o

ðÞ

2p_

ðÞ



 exp iQ  r

0ðÞ



exp þiQ  r

tðÞ



 exp iotðÞdt ð2Þ

Q; oðÞcorresponds to the Fourier transform of

the magnetic pair correlation function that gives the

probability to ﬁnd a magnetic spin at position r

time t with component S

tðÞand a spin at position r

at time t ¼ 0 with a component S

0ðÞ. Therefore, the

magnetic cross-section depends on magnetic as well

as on lattice vibrational degrees of freedom. This is

usually taken into account by a Debye–Waller factor,

exp (2WðQÞ); in the following this factor has been

omitted.

In Eqn. (2), S

Q; o

ðÞ

is expressed in terms of

expectation values of time-dependent operators

(Heisenberg picture). Equivalently, the scattering

function can be written in terms of time-independent

operators (Schro

dinger picture) by replacing the in-

tegral over time by a d-function in energy obtaining

Q; oðÞ¼

exp iQ  r

 r



 l

d E

 E

þ _oðÞð3Þ

The magnetic scattering function is also directly

related to the imaginary part of the wavevector and

frequency-dependent susceptibility via the ﬂuctua-

tion–dissipation theorem by

Q; oðÞ¼

1  exp _o

TðÞ

Im w

Q; oðÞð4Þ

This theorem implies that the magnetic moment of

the neutron acts on the sample as a frequency- and

wavevector-dependent magnetic ﬁeld BQ; oðÞ; mon-

itoring the magnetic response MQ; oðÞof the sample

with

Q; oðÞ¼

Q; oðÞB

Q; oðÞ ð5Þ

where w

Q; oðÞis the generalized susceptibility ten-

sor. The ﬂuctuation–dissipation theorem allows di-

rect comparisons of w

as obtained by neutron

scattering with bulk measurements of w using the

Kramers–Kronig relation (see text books on solid-

state physics).

For the discussion of the polarization dependence

of the neutron cross-section, we refer the reader to

the book of (Williams 1988). Here it is sufﬁcient to

state that a polarization analysis allows us to separate

unambiguously nuclear and magnetic scattering

though a loss of overall intensity has to be taken

into account. Furthermore, it is possible to deter-

mine, for example, individual matrix elements of

the spin operators between crystal-ﬁeld states

ðj/l

Þ.

2. Accessible (Q; x)-range

The magnetic scattering functions S

ðQ; oÞ as given

in the previous section depend only on the transfer of

energy (_o ¼ E

 E

) and momentum (Q ¼ k

 k

)

414

Magnetic Excitations in Solids

from the neutron to the sample. E

) and E

)

refer to the initial and ﬁnal energy (wavevector) of

the neutron, respectively. The different formulations

of the scattering function (Eqns. (2)–(4)) allow for a

straightforward interpretation of the measured cross-

sections in meaningful theoretical concepts, i.e.,

spin–spin correlation functions (Eqn. (2)), transition

matrix elements (Eqn. (3)), or as generalized suscep-

tibility tensor (Eqn. (4)). The presently accessible

range of energy and momentum transfer by neutrons

is shown in Fig. 1 (ESS 2003). Also indicated by

arrows are its extension for neutron sources of the

next generation and the corresponding range of com-

plementary measuring techniques such as x-rays,

free-electron-lasers (FEL), NMR, etc.

3. Neutron Spectrometers

Since the ﬁrst neutron instrumentation at research

reactors in the 1950s and at accelerator-based neu-

tron spallation sources in the 1980s, a large variety of

neutron spectrometers has been developed. The two

main types of spectrometers for the investigation of

excitations in solids depend on the different way the

neutron wavevector is determined before and after

the scattering process: by reﬂection from a single

crystal or by measuring the neutron time-of-ﬂight.

In a triple-axis spectrometer (Brockhouse 1955,

Shirane et al. 2002), the reﬂection from a single crys-

tal (monochromator) selects a well-deﬁned wave-

vector k

from the white spectrum of the neutron

source (ﬁrst axis). The monochromatic beam is scat-

tered from the sample (scattering angle ¼second

axis). The scattered beam is reﬂected from the anal-

yzer crystal (third axis) onto the detector, thus deﬁning

the ﬁnal neutron wavevector k

and the energy trans-

fer _o as well. A schematic view is shown in Fig. 2

(IN1, ILL 2003). Data are taken point-by-point in

reciprocal space either with ﬁxed momentum transfer

Q (to obtain an energy spectrum) or with ﬁxed energy

transfer _o. This method is typically used for the

determination of dispersion relations using single

crystalline samples.

Time-of-ﬂight methods are preferentially used for

the investigation of polycrystalline samples and at

neutron spallation sources taking advantage of the

pulsed character of the neutron production in the

target. As an example the layout of the MERLIN

spectrometer at the ISIS Facility is shown in Fig. 3

(ISIS 2003). The pulsed beam is monochromatized by

the (Fermi-) chopper in front of the sample by phas-

ing the chopper opening with the neutron production

in the target (spallation target and moderator at up-

per right side, not shown in the ﬁgure). A background

chopper additionally suppresses unwanted neutrons

Figure 1

Accessible range of transfer in energy (_o in meV) and of the modulus of the momentum Q (in A

1

) for neutron

scattering in comparison to other experimental methods. Also indicated is the corresponding range in time (in ps) and

real space (in A

) (from ESS 2003).

415

Magnetic Excitations in Solids

from the source. A large array of detectors covers a

large solid angle and allows for the simultaneous

measurement of a large range in energy transfer and

of the modulus of the momentum transfer.

4. Magnetic Interactions in Solids

When elements with partly ﬁlled shells containing

3d-, 4f-, and 5f-electrons are condensed into the

solid state (either as pure elements, intermetallic

compounds, semiconductors, or insulators), they

often preserve their magnetic properties. As the

4f -electrons (elements of the Lanthanide series)

occupy inner spatial regions of the atom, they lose

only little of their magnetism when embedded in a

solid (see Localized 4f and 5f Moments: Magnetism).

The magnetic ground state is determined by taking

into account all three Hund’s rules (see textbooks

on solid-state physics). The interaction with the crys-

talline electric ﬁeld (CF) that surrounds the 4f -ion

(single-ion interaction) and the magnetic interaction

with the magnetic moments of neighboring atoms

(two-ion interaction) leads to a (CF-) splitting of the

ground-state multiplet and to a dispersion of the

magnetic excitations, respectively. Both interaction

strengths can be of the same order of magnitude

(1–100 meV). In compounds containing 3d-elements

the interaction leading to the third Hund’s rule (the

spin–orbit interaction) is much weaker than the CF-

interaction (see Magnetism in Solids: General Intro-

duction) resulting in a different scenario for the

description of the magnetic ground-state (band-type

electronic wave functions of e

and t

character). In

metallic systems this leads to a ‘‘quenching’’ of the

orbital moments resulting in a description of the

magnetic properties as spin-only moments dominated

by the collective magnetic interactions. 3d-Insulators

will not be considered here (a decoupling of spin,

charge, and orbital degrees of freedom is presently

discussed in connection with metal–insulator transi-

tions). The situation in systems with 5f -elements

(elements of the actinide series) is between that of

4f -systems (localized) and that of 3d-systems (itiner-

ant) and thus rather complicated.

Figure 2

Scheme of a neutron triple-axis spectrometer at a neutron reactor (IN1 at the Institute Laue-Langevin, ILL,

Grenoble) (from ILL 2003).

Figure 3

Schematic view of a neutron time-of-ﬂight spectrometer

at a neutron spallation source (MERLIN at the ISIS

Facility, Rutherford Appleton Laboratory) (from ISIS

2003).

416

Magnetic Excitations in Solids

5. Spin Waves, Crystal-ﬁeld and Inter-multiplet

Transitions

Spin waves (magnons) are the low-energy collective

excitations of a magnetically ordered system. For an

isotropic Heisenberg interaction of localized mag-

netic moments, the textbooks on solid-state physics

yield for a ferromagnet for small q

a quadratic dis-

persion in q

¼ Q  G

, with G being a reciprocal

lattice vector of the (ferromagnetic) lattice. For an

antiferromagnet the dispersion is linear in q

in the

neighborhood of the antiferromagnetic reciprocal

lattice vector. The magnetic excitations of the ele-

mentary magnets like Fe and Ni have been investi-

gated over many decades resulting in an isotropic and

gap-less spin wave-like spectrum in the low-energy

region but with drastic modiﬁcations when following

the dispersion towards the zone boundary. The latter

ﬁndings have been analyzed in the light of electronic

band structure effects or by saying that the sharp

collective spin waves interfere with single-particle

excitations of the 3d-electrons (exhibiting a Stoner

continuum) giving rise to strong damping effects

(Lovesey 1984, Mook 1988).

Spin waves in the presence of CF-interactions are

discussed experimentally and theoretically for the

rare earth elements by Jensen and Mackintosh

(1991). The excitation spectra of rare earths diluted

in nonmagnetic metals and intermetallic compounds

are discussed by Fulde and Loewenhaupt (1985,

1988) and for rare earth intermetallic compounds in-

cluding rare earth-transition metal compounds by

Moze (1998). The latter compounds are the base ma-

terials for novel permanent magnet production. The

types of excitations in a rare earth compound acces-

sible by inelastic neutron scattering are shown exem-

plarily in Fig. 4. In this case the hierarchy of energies

yields a spin wave spectrum in the very low energy

part (due to the low magnetic ordering temperature

of typical several K), transitions to excited CF-levels

(several meV; 1 meV corresponds to 11.6 K), and in-

ter-multiplet transitions (several hundredth of meV).

The advent of neutron spallation sources made it

possible to investigate also the high-energy region of

the magnetic excitation spectra, e.g., inter-multiplet

transitions, an area that was usually restricted to light

scattering (Hu

fner 1978). An early documentation of

the possibilities of neutron scattering in the ﬁeld of

investigating inter-multiplet transitions is given by

Osborn et al. (1991).

To determine the magnetic excitation spectra of an

intermetallic rare earth compound (as an example we

show the results for orthorhombic NdCu

), time-of-

ﬂight spectroscopy on a polycrystalline sample is

performed as a ﬁrst step. In Fig. 5 the CF-transitions

from the ground-state doublet to the four excited-

state doublets of Nd

3 þ

(having the angular momen-

tum quantum number J ¼9/2 and hence ﬁve doublets

due to the CF-interaction and Kramers degeneracy)

can clearly be identiﬁed. While the spectrum of Fig. 5

was taken in the magnetically disordered state at

10 K, the spectrum at 2 K in the magnetically ordered

state is shown in Fig. 6. The main difference is that in

the ordered state a molecular ﬁeld is present in ad-

dition to the CF. This yields a splitting of the dou-

blets and a modiﬁcation of the transition matrix

elements. The transitions within the split ground-

state doublet are shown with increased resolution in

the inset of Fig. 6. The structure of this part of the

excitation spectrum reﬂects the dispersive nature of

the splitting with its intensity being proportional to

the magnon density-of-states. To show the power of

inelastic neutron scattering on this very low energy

side of the range of excitations in 4f -systems, we

present in Fig. 7 the spin wave dispersion curves

measured on a single crystal of NdCu

in the ferro-

magnetically induced state in a ﬁeld of 3 T on a triple-

axis-spectrometer employing cold neutrons (Rotter

et al. 2000). The spin waves have been analyzed by a

mean-ﬁeld model in random-phase approximation

(MF-RPA; see, e.g., Jensen and Mackintosh 1991)

allowing for anisotropic effects due to the single-ion

CF-interaction and also for anisotropic two-ion ex-

change interactions. From the interpretation of the

magnetic-phase diagrams (Rotter 2003), the disper-

sion relations and the CF-excitation spectra of one

compound (NdCu

) predictions could be made for

the magnetic properties of the whole series RCu

with

R being any rare earth element.

In general, elementary excitations in solids such as

CF-transitions and lattice vibrations (phonons) are

considered decoupled and the determination and in-

terpretation of the measured spectra of the two phe-

nomena, i.e., the CF level schemes and the phonon

Figure 4

Typical splitting of levels in a rare earth system due to

spin–orbit (SO), crystal-ﬁeld (CF), and exchange

interaction (H

mol

). Scheme reproducing approximately

the situation of Nd

3 þ

ions in NdCu

417

Magnetic Excitations in Solids

dispersion relations, are performed independently of

each other. A survey on coupling effects between

both phenomena as investigated by INS is given by

Loewenhaupt and Witte (2003).

6. Spin Fluctuations

Spin wave theory in MF-RPA neglects spin quantum

ﬂuctuations. However, ﬂuctuations become important

Figure 6

CF-transitions and magnon density-of-states (shown on an enlarged scale in the inset) in the magnetically ordered

state of NdCu

at T ¼ 2KðT

¼ 6:5KÞ and zero external ﬁeld measured on time-of-ﬂight spectrometers (IN4 and

IN6, ILL) (from Gratz et al. 1991).

Figure 5

CF-transitions from the ground-state doublet to the four excited-state doublets in NdCu

measured on a time-of-ﬂight

spectrometer (IN4, ILL) in the magnetically disordered state at T ¼ 10 K and zero external ﬁeld. Inset shows the

CF-level scheme (from Gratz et al. 1991).

418

Magnetic Excitations in Solids

for low-dimensional systems and low spin- quantum

numbers (see Non-Fermi Liquid Behavior: Quantum

Phase Transitions). The new phenomena observed in

such systems are presently discussed in a wide context

from one-dimensional antiferromagnets (Cowley 1998,

2002, Mikeska 2002) via heavy-fermion superconduc-

tors (Bernhoeft et al. 1998, Thalmeier 2002) to high-

temperature superconductors (Bourges et al. 2000).

Classical examples of three-dimensional systems ex-

hibiting also strong spin ﬂuctuations are mixed-valent

and heavy-fermion compounds (see Intermediate

Valence Systems; Heavy-fermion Systems). Reviews

on the lanthanides (Loewenhaupt and Fischer 1993,

Aeppli and Broholm 1994) and actinides (Holland-

Moritz and Lander 1994) make the strong coupling

between the localized f -electrons and the delocalized

conduction electrons responsible for the observed

magnetic instabilities. Detailed INS measurements

were reported for single crystalline samples of the

series CeCu

6x

where the variation of x drives

the system from Fermi-liquid behavior (x ¼ 0) via a

quantum critical point (x ¼ 0:1) to a magnetically

ordered (x ¼ 1) system (Schro

der et al. 1997, Stockert

et al. 1997).

See also: Crystal Field Effects in Intermetallic Com-

pounds: Inelastic Neutron Scattering Results; Mag-

netism: Applications of Synchrotron Radiation;

Permanent Magnet Materials: Neutron Experiments

Bibliography

Aeppli G, Broholm C 1994 Magnetic correlations in heavy fer-

mion systems: neutron scattering from single crystals. In:

Gschneidner K A, Jr., Eyring L, Lander G H, Choppin G R

(eds.) Handbook on the Physics and Chemistry of Rare Earths.

Elsevier, Vol. 19, Chap. 131, pp. 123–75

Bernhoeft N, Sato N, Roessli B, Aso N, Hiess A, Lander G H,

Endoh Y, Komatsubara T 1998 Enhancement of magnetic

ﬂuctuations on passing below T

in the heavy fermion su-

perconductor UPd

. Phys. Rev. Lett. 81, 4244–7

ni P, Furrer A 1998 Introduction to neutron scattering. In:

Furrer A (ed.) Neutron Scattering in Layered Copper-oxide

Superconductors. Kluwer, Dordrecht, pp. 1–22

Figure 7

Spin waves in NdCu

measured in the ferromagnetically induced phase at T ¼ 2 K and with magnetic ﬁeld of 3 T

applied along the crystallographic b-direction measured on a triple-axis spectrometer (IN12, ILL) (from Rotter et al.

2000).

419

Magnetic Excitations in Solids