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

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-2

-1

5 T

9 T

0 T

150 Å

-4

-3

-2

-1

9 T

5 T

0 T

750 Å

0 50 100 150 200 250 300

-4

-3

-2

-1

9 T

5 T

0 T

1100 Å

T (K)

ρ (Ω cm)

-3

-2

-1

125 K

100 K

75 K

0 2 4 6 8

25.10

-3

25.10

-2

25.10

-1

100 K

125 K

50 K

75 K

Resistivity (Ω cm)

Magnetic field (T)

(a)

(b)

(c)

0 2 4 6 8

Magnetic field (T)

Figure 4

r(T) in different magnetic ﬁelds (0 T, 5 T, and 9 T) applied in the plane of the substrate for different PCMO ﬁlms on

STO: (a) 15 nm, (b) 75 nm, and (c) 110 nm. Arrows indicate the direction of the temperature variation. Experimental

points were taken with a stabilized temperature and a control of the sample ﬁlm itself. The crossing of the curves

ramping up and down in temperature is real. The insets of (b) and (c) depict the r(H) for a 75 nm and 110 nm ﬁlm,

respectively. Runs with increasing ﬁeld and decreasing ﬁeld are denoted by arrows. Reproduced from W. Prellier et al.

(2000) Thickness dependence of the stability of the charge-ordered state in Pr

0.5

MnO

thin ﬁlms. Phys. Rev. B 62,

R16337.

Colossal Magnetoresistance Effects: The Case of Charge-ordered Pr

0.5

MnO

Manganite Thin Films

compared to the bulk single crystal Pr

0.5

MnO

The hysteretic region grows when the temperature is

decreasing and rapidly vanishes at 8 T. In other

words, H

is smaller than 9 T at low temperature,

then H

is decreasing when the T is increasing. Above

50 K, H

begins to increase again to 8 T. In the mean

time, the hysteretic region is shrinking. Note that this

region is larger in the case of the 75 nm ﬁlm, indicat-

ing that the CO state is more stable as previously

seen from the r(T) curves (Fig. 3). In fact, the phase

diagram of the 110 nm ﬁlm has almost the same

proﬁle as a Pr

1x

MnO

single crystal with

0.35oxo0.45. This suggests that the whole electron-

ic structure of Pr

0.5

MnO

thin ﬁlms grown on

STO is similar, though the charge transfer is different

(x ¼0.5 instead of 0.4 in Pr

1x

MnO

). The inter-

esting point is that the modulation vector q is equal to

0.45 in PCMO on STO (Prellier et al. 2000), whereas

q is always equal to 0.5 in bulk Pr

1x

MnO

com-

pounds (0.35oxo0.45) (Barnabe

et al. 1998).

In summary, the melting magnetic ﬁeld of the in-

sulating charge-ordered state, in Pr

0.5

MnO

ﬁlms, is strongly sensitive to the substrate-induced

strain and thickness of the ﬁlm. When deposited on

SrTiO

(tensile stress), the application of magnetic

ﬁelds up to 9 T has almost no effect on the resistivity

of a very thin ﬁlm, but a 5 T magnetic ﬁeld can com-

pletely melt the CO state of a 110 nm thick ﬁlm.

When the ﬁlm is under compression (on LaAlO

), the

ﬁlms are insulating in a magnetic ﬁeld of 7 T. By

growing thin ﬁlms, the properties of the materials can

be modiﬁed compared to the bulk compounds. We

have studied the case of Pr

0.5

MnO

on two dif-

ferent substrates that are able to induce two different

types of strains (tensile on SrTiO

and compressive

on LaAlO

) and their effects on various parameters.

We observed that the lattice parameters of the ﬁlm,

the orientation of the ﬁlm with respect to the subst-

rate, the transport properties of the ﬁlm and the

stability of the charge/orbital ordered state are inﬂu-

enced by the strains suggesting some similarities be-

tween strain and pressure effects in oxides (Wang

et al. 1995).

Acknowledgements

The author acknowledges E. Rauwel Buzin, A. M.

Haghiri-Gosnet, B. Mercey, Ch. Simon, M. Hervieu,

and B. Raveau, who have also participated to this

work.

See also: Boltzmann Equation and Scattering Mech-

anisms; Giant Magnetoresistance: Metamagnetic

Transitions in Metallic Antiferromagnets; Inter-

metallic Compounds: Electrical Resistivity; Mag-

netoresistance, Anisotropic; Magnetoresistance:

Magnetic and Nonmagnetic Intermetallics; Magne-

toresistance in Transition Metal Oxides

Bibliography

Barnabe

A, Hervieu M, Martin C, Maignan A, Raveau B 1998

Role of the A-site size and oxygen stoichiometry in charge

ordering commensurability of Ln

0.5

MnO

manganites.

J. App. Phys. 84, 5506–14

Biswas A, Rajeswari M, Srivastava R C, Li Y H, Venkatesan T,

Greene R L, Millis A J 2000 Two-phase behavior in strained

thin ﬁlms of hole-doped manganites. Phys. Rev. B 61, 9665–8

Chahara K, Ohno T, Kasai M, Kozono Y 1993 Magnetore-

sistance in magnetic manganese oxide with intrinsic anti-

ferromagnetic spin structure. Appl. Phys. Lett. 63, 1990–2

Haghiri-Gosnet A M, Hervieu M, Simon Ch, Mercey B,

Raveau B 2000 Charge ordering in Pr

0.5

MnO

thin

ﬁlms: a new form initiated by strain effects of LaAlO

subst-

rate. J. Appl. Phys. 88, 3545–51

100

150

200

300

350

400

Charge-ordered

insulator

Temperature (K)

netic field (T)

100

150

200

250

300

02468

750Å

harge-ordered

insulator

Temperature (K)

Magnetic field (T)

Figure 5

Electronic phase diagram of a 110 nm PCMO/STO thin

ﬁlm determined by the critical ﬁelds. H

and H



are

taken as the inﬂexion points in r(H) for up and down

sweeps, respectively. In the dashed region, both CO

insulating and metallic states coexist. The inset shows

the phase diagram for a 75 nm PCMO ﬁlm.

Colossal Magnetoresistance Effects: The Case of Charge-ordered Pr

0.5

MnO

Manganite Thin Films

Jirak Z, KrupiImageka S, Simca Z, Dlouha

M, Vratislav S 1985

Neutron diffraction study of Pr

1x

MnO

perovskites. J.

Magn. Mat. 53, 153–66

Kuwahara H, Tomioka Y, Asamitsu A, Moritomo Y, Tokura

Y 1995 A ﬁrst-order phase transition induced by a magnetic

ﬁeld. Science 270, 936–61

McCormack M, Jin S, Tiefel T H, Fleming R M, Phillips J M,

Ramesh R 1994 Very large magnetoresistance in perovskite-

like La-Ca-Mn-O thin ﬁlms. App. Phys. Lett. 64, 3045–7

Prellier W, Biswas A, Rajeswari M, Venkatesan T, Greene R L

1999 Effect of substrate-induced strain on the charge-order-

ing transition in Nd

0.5

MnO

thin ﬁlms. Appl. Phys. Lett.

75, 397–9

Prellier W, Haghiri-Gosnet A M, Mercey B, Lecoeur Ph, Her-

vieu M, Simon Ch, Raveau B 2000 Spectacular decrease of

the melting magnetic ﬁeld in the charge-ordered state of

0.5

MnO

ﬁlms under tensile strain. Appl. Phys. Lett.

77, 1023–5

Prellier W, Lecoeur Ph, Mercey B 2001 Colossal magnetoresis-

tive manganite thin ﬁlms. J. Phys. Condens. Matter 13, R915–

Rao C N R, Arulraj A, Cheetham A K, Raveau B 2000 Charge

ordering in the rare earth manganates: the experimental sit-

uation. J. Phys.: Condens. Matter 12, R83–106

Tokunaga M, Miura N, Tomioka Y, Tokura Y 1998 High-

magnetic-ﬁeld study of the phase transitions of

1x

MnO

(R ¼Pr, Nd). Phys. Rev. B 57, 5259–64

Tomioka Y, Asamitsu A, Kuwahara H, Moritomo Y, Tokura

Y 1996 Magnetic-ﬁeld-induced metal-insulator phenomena

in Pr

1x

MnO

with controlled charge-ordering instabil-

ity. Phys. Rev. B 53, R1689–92

Von Helmolt R, Wecker J, Holzapfel B, Schultz L, Samwer K

1993 Giant negative magnetoresistance in perovskitelike La

2/3

1/3

MnO

ferromagnetic ﬁlms. Phys. Rev. Lett. 71, 2331–3

Wang H Y, Palstra T T M, Cheong S W, Batlogg B 1995

Pressure effects on the magnetoresistance in doped manga-

nese perovskites. Phys. Rev. B 52, 15046–9

W. Prellier

Laboratoire CRISMAT

CNRS UMR 6508 6 Boulevard du Marechal Juin

F-14050 Caen Cedex, France

Crystal Field and Magnetic Properties,

Relationship between

The crystal-ﬁeld (CF) model has been very successful

in the analysis of 4f

conﬁgurations of the triply ion-

ized rare-earth R ions in solids whose energy levels are

reproduced through a Hamiltonian which involves

both free-atom and CF operators (Morrison and

Leavitt 1982, Go

rller-Walrand and Binnemans 1996).

A satisfactory simulation of the experimental scheme

of energy levels can be achieved if the number and

quality of the operators is adequate. The procedure

also provides good quality wave functions associated

to the Stark levels, which can be used to simulate

other physical properties, depending on them and the

energies only. Thus, if there are no magnetic interac-

tions between the lanthanide ions, the effective

magnetic moment, the thermal evolution of the par-

amagnetic susceptibility w for any crystallographic

direction, as well as the g-values of the magnetic

splitting factor for each Kramers doublet, can be cal-

culated when the same L þg

S tensorial operator is

applied to the wave function of a level, according to

the Van Vleck formalism (Van Vleck 1932).

This article outlines the importance of the CF ef-

fects on the explanation of the magnetic properties in

a wide group of R-containing materials, prepared as

polycrystalline powders. In them, the one single cry-

stallographic position occupied by R has a point

symmetry ranging from a relatively high symmetry,

for both R

and Na

R(MoO

)

series, to a

very low symmetry, C

for R

and C

for

and MgR(BO

)

families (Cascales et al.

1995, 1999). The very good agreement found between

the calculated paramagnetic susceptibility curves, de-

rived through phenomenological and/or estimated

crystal ﬁeld parameters (CFPs), and the observed w

vs. T plots constitutes an excellent test of the ability

of the CF model to provide correct wave functions

for the 4f

conﬁgurations.

1. Crystal-ﬁeld Analysis and Simulation of the

Energy Level Schemes

In the development of a complete Hamiltonian for 4f

conﬁgurations, the central-ﬁeld approximation allows

considering separately the Hamiltonians correspond-

ing to the gaseous free-ion and to the CF interactions,

which arise when R is in a crystalline environment.

Thus the total Hamiltonian (Carnall et al.1989)con-

sists of two parts, H ¼H

þH

. The H

free-ion

part includes the spherically symmetrical one-electron

term of the Hamiltonian, the electrostatic interaction

between equivalent f electrons, the spin-orbit interac-

tion, and terms accounting for higher-order correc-

tions, as it is described in Carnall et al.1989.

In the presence of a crystalline electric ﬁeld the

degeneracy of each state of the free-ion will be lifted

according to the site symmetry of R in the crystal

lattice. Calculations are usually carried out within the

single-particle CF theory, and following Wybourne’s

formalism (Wybourne 1965), the CF H

Hamilton-

ian is expressed as a sum of products of spherical

harmonics and CFPs.

4;6

k¼2

q¼0

ðC

þð1Þ

q

ÞþiS

ðC

ð1Þ

q

Whereas for the S

symmetry of the R

3 þ

site in

the serial development of the CF potential

Crystal Fie ld and Magnetic Properties, Relationshi p between

involves ﬁve real B

and two complex S

parameters,

for the lowest C

symmetry found in MgR(BO

)

all

27 CFPs are kept nonzero. For R ions with low

point-group symmetries the constraints imposed by

group theory are weak and it becomes difﬁcult, with

polycrystalline samples but even with polarized op-

tical absorption and emission measurements on single

crystals, to unravel the optical spectra unambiguous-

ly. Even if this has been done, it is not straightfor-

ward to ﬁt the data with a total Hamiltonian. The

number of adjustable parameters are very large, and

the nonlinear least square ﬁts are unreliable, unless a

good initial set of CFPs is known. Besides the descent

of symmetry method (Wybourne 1965), various the-

oretical models for calculating the CF interactions

have been proposed (Garcia and Faucher 1995). The

simple overlap model SOM (Porcher et al. 1999), a

simpliﬁed semi-a priori calculation model used to ob-

tain the initial set of CFPs from the crystallographic

structure (Malta et al. 1991), was used for the limit

situations represented by the Na

R(MoO

)

series, and by the MgNd(BO

)

compound.

Finally, to verify the consistency and reliability of the

ﬁnal ﬁtted parameters, simultaneous simulations per-

formed for different 4f

conﬁgurations either in the

same or in an isostructural crystalline matrix, should

provide sets of CFPs with only smooth variations

(Morrison and Leavitt 1982).

2. Magnetic Sus ceptibility and Crystal-ﬁeld

Levels

The calculation of the magnetic susceptibility for

above R-compounds (Cascales et al. 1993, 1995,

1996, 1997, 1999) has been carried out from the con-

sistent sets of wave functions and energy levels pre-

viously determined (Cascales et al. 1990, 1992a, b, c,

1998), by diagonalizing the total Hamiltonian. This is

done using the Van Vleck formalism (Van Vleck

1932) the result of an application of the perturbation

theory,

ðiÞ

¼ Nb

7H7f

 2

7H7f

S/f

7H7f

 E

in which N is the Avogadro’s number, b the Bohr

magneton, k the Boltzmann constant, and E and f

the nonperturbed energy levels and wave functions,

respectively, described in the 7SLJM

S basis, and H

is the magnetic tensor operator L þg

S. The different

values of the tensor components (or combinations of

them) destroy the isotropy observed for the free-ion

or even for an ion in a cubic symmetry. For poly-

crystalline samples, the mean observed paramagnetic

susceptibility is w

¼(w

þw

)/3. The sums run

over thermally populated levels, according to the

Boltzmann population,

¼ expðE

ð0Þ

=kTÞ=

expðE

ð0Þ

=kTÞ:

In the expression, the matrix elements are calculated

using the Racah algebra rules.

The formula is the sum of a temperature-depend-

ent diagonal term and a temperature-independent

off-diagonal term, which is reminiscent of the classi-

cal Curie–Weiss law. The off-diagonal term, a result

of the second order perturbation, usually has little

importance, with the exception of the ground states

with J ¼0. The sum runs over all other states (baa).

Energy levels up to 5000 cm

1

and 10000 cm

1

, for

the diagonal and off-diagonal terms, respectively

(which are largely sufﬁcient to cover the thermal

population effect above 300K), were accounted in the

calculation along with the J mixing of the levels.

The simulation of Stark levels for 4f

conﬁgura-

tions as well as the calculations of the w vs. T curves

were performed by the programs REEL and IMAGE

(Porcher personal communication, 1989).

3. Results

The calculation of the thermal variation of the par-

amagnetic susceptibility (Cascales et al. 1993, 1995,

1996, 1997, 1999) has been carried out with two dif-

ferent collections of energy levels and associated wave

functions: (i) Those derived from CFPs obtained

through spectroscopic data. (ii) Calculated from

CFPs obtained by using SOM, through crystallo-

graphic data. Values of w have been found to be in-

dependent of the ﬁeld because of the relatively weak

applied ﬁeld. With the well-known exceptions of Eu

and Sm-compounds, w

1

vs. T curves at high tem-

peratures follow a Curie–Weiss-type behavior. Devi-

ations from linearity observed at low temperature can

be attributed to the splitting of the corresponding

free-ion ground state under the inﬂuence of the CF,

and the Weiss constants for these materials are en-

tirely due to CF effects since they do not undergo any

magnetic exchange interactions.

Figure 1 presents the comparison between exper-

imental and calculated average w

1

(or w for

R ¼Eu

3 þ

) vs. T curves for some selected compounds

of R

and MgR(BO

)

. The choice of these

series is related to the very different number of real

and complex parameters that the CF potential

involves in each case. As it can be seen, a very

satisfactory concordance is found between the exper-

imental and calculated curves, especially when these

last ones are from optical data. The observed slight

discrepancy between them when the temperature in-

creases is related with a variation of the CF, usually

assumed to be temperature-independent, since, in

fact, the network parameters smoothly vary with the

temperature.

Crystal Field and Magnetic Properties, Relationship between

4. Concluding Remarks

The good reproduction of experimental data, espe-

cially the observed deviation from the Curie–Weiss

behavior at low temperature in the w

1

vs. T plots,

underscores that the standard CF treatment of 4f

conﬁgurations can explain the magnetic properties of

these ions, whatever the number of parameters in-

volved in the simulation, with no need to consider

any sort of exchange interactions between them.

These calculations are extremely dependent on the

composition of the CF wave functions; hence the

comparison with experimental measurements consti-

tutes an excellent opportunity to test the ability of the

model to give adequate wave functions for the con-

sidered ion. Anyway, the correct simulation of phys-

ical properties can be considered as a proof of

obtaining ‘‘good wave functions’’ if it is done through

‘‘good CFPs,’’ i.e., those deduced from a high-quality

spectroscopically-observed energy level scheme for

the given 4f

conﬁguration. The oversimpliﬁed SOM

model seems to be useful at providing estimations of

CFPs, especially when the point symmetry of R in-

volves a reasonable number of nonzero CFPs. This

kind of calculation is of particular interest for opaque

materials, for which only a reduced number of Stark

levels can be observed from experimental techniques.

See also: Crystal Field Effects in Intermetallic

Compounds: Inelastic Neutron Scattering Results;

Localized 4f and 5f Moments: Magnetism.

Bibliography

Carnall W T, Goodman G L, Rajnak K, Rana R S 1989 A

systematic analysis of the spectra of the lanthanides doped

into single crystal LaF

. J. Chem. Phys. 90, 3443–57

Cascales C, Antic Fidancev E, Lemaitre Blaise M, Porcher P

1990 Spectroscopic properties of Eu

3 þ

in lanthanum chloro-

tungstates. J. Sol. State Chem. 89, 118–22

Cascales C, Antic Fidancev E, Lemaitre Blaise M, Porcher P

1992a Spectroscopic properties of Pr

3 þ

doped LaWO

Cl and

. Eur. J. Solid State Inorg. Chem. 29, 273–82

Cascales C, Antic Fidancev E, Lemaitre Blaise M, Porcher P

1992b Optical properties and crystal ﬁeld calculations of

europium tellurium oxide. J. Alloys Comp. 180, 111–6

Cascales C, Antic Fidancev E, Lemaitre Blaise M, Porcher P

1992c Spectroscopic properties and simulation of the energy

level scheme of Nd

3 þ

and Pr

3 þ

ions in rare earth tellurium

oxides. J. Phys.: Condens. Matter 4, 2721–34

Cascales C, Porcher P, Sa

ez Puche R 1993 Crystal ﬁeld effects

on the magnetic susceptibility of rare earth tellurium oxides.

J. Phys. Chem. Solids 54, 1471–4

Cascales C, Porcher P, Sa

ez Puche R 1996 Paramagnetic sus-

ceptibility simulations from crystal-ﬁeld effects on rare-earth

double molybdate and tungstate Na

RE(XO

)

. J. Phys.:

Condens. Matter 8, 6413–24

Cascales C, Porcher P, Sa

ez Puche R 1997 Paramagnetic sus-

ceptibility simulations from crystal-ﬁeld effects on rare-earth

antimonates R

. J. Alloys Comp. 250, 391–5

Cascales C, Sa

ez Puche R, Porcher P 1995 Crystal-ﬁeld effect

on the magnetic susceptibility of rare-earth (Pr, Nd, Eu)

mixed oxides. J. Sol. State Chem. 114, 52–6

Cascales C, Sa

ez Puche R, Porcher P 1998 Crystal-ﬁeld effects

and simulation of the energy level scheme of Nd

3 þ

in mag-

nesium borate MgNd(BO

)

. J. Alloys Comp. 277, 384–7

Cascales C, Sa

ez Puche R, Porcher P 1999 Paramagnetic sus-

ceptibility simulations from crystal-ﬁeld effects on Nd

3 þ

magnesium borate MgNd(BO

)

. Chem. Phys. 240, 291–301

Garcia D, Faucher M 1995 In: Gschneider K A, Eyring L (eds.)

Handbook on the Physics and Chemistry of Rare Earths, Vol.

21, Crystal Field in Non-metallic (Rare Earth) Compounds.

Elsevier, Amsterdam

rller-Walrand C, Binnemans K 1996 In: Gschneider K A,

Eyring L (eds.) Handbook on the Physics and Chemistry of

Rare Earths, Vol. 23, Rationalization of Crystal-ﬁeld Par-

ametrization. Elsevier, Amsterdam

Malta O L, Ribeiro S J L, Faucher M, Porcher P 1991 The-

oretical intensities of 4f–4f transitions between Stark levels of

the Eu

3 þ

ion in crystals. J. Phys. Chem. Solids 52, 587–93

Morrison C A, Leavitt R P 1982 In: Gschneider K A, Eyring L

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

Figure 1

Comparison between experimental (symbols) and

calculated (solid lines) w

1

(or w for the Eu

3 þ

compound) curves: (a) for selected R

compounds; (b) for MgNd(BO

)

(reproduced by

permission of Elsevier Science from J. Solid State

Chem., 1995, 114, 52–6; Chem. Phys., 1999, 240, 291–

301).

Crystal Fie ld and Magnetic Properties, Relationshi p between

Vol. 5, Spectroscopic Properties of the Triply Ionized Lan-

thanides in Transparent Host Crystals North-Holland,

Amsterdam

Porcher P, Dos Santos C M, Malta O 1999 Relationship be-

tween phenomenological crystal ﬁeld parameters and the

crystal structure: The simple overlap model. Phys. Chem.

Chem. Phys. 1, 397–405

Van Vleck J H 1932 The Theory of Electric and Magnetic Sus-

ceptibilities. Oxford University Press, London

Wybourne B G 1965 Spectroscopic Properties of Rare Earths.

Wiley, New York

C. Cascales

Instituto de Ciencia de Materiales de Madrid, Spain

R. Sa

ez-Puche

Universidad Complutense de Madrid, Spain

Crystal Field Effects in Intermetallic

Compounds: Inelastic Neutron

Scattering Results

The crystal ﬁeld (CF) interaction in rare-earth com-

pounds is responsible for an enormous variety of

magnetic phenomena. The electrostatic interaction

experienced by the rare-earth ion, arising from the

presence of near neighbor ions and outer valence

electrons, is much smaller than the spin–orbit cou-

pling, because of the minute spatial extent of the 4f-

wave functions. This CF interaction gives rise to

splittings of the ground-state multiplet of the order of

tens of meV for the lanthanides. A precise determi-

nation of CF parameters is a necessary precondition

for detailed analysis of bulk magnetic properties such

as the static susceptibility, magnetic contribution to

the specific heat, and complex magnetization proc-

esses and transitions in a large host of magnetic ma-

terials. The CF interaction intimately reﬂects the

crystallographic point symmetry of the lanthanide

ion and splits the (2J þ1)-fold degeneracy of the

ground state according to rules dictated by group

theory. In combination with an exchange interaction,

the result is a truly phenomenal array and variety of

magnetic properties. The situation pertaining to com-

pounds, such as those based on Ce, with an unstable

or delocalized 4f-shell leads to complications due to

the manner in which conduction electrons can couple

to 4f-electrons, giving rise to valence ﬂuctuating or

Kondo systems (see Intermediate Valence Systems;

Kondo Systems and Heavy Fermions: Transport Phe-

nomena). Similar circumstances are also pertinent for

5f-electron heavy-fermion systems.

The most direct experimental method for the de-

termination of CF eigenvalues and eigenfunctions in

crystalline rare-earth intermetallics is provided by in-

elastic neutron scattering (INS), a truly fundamental

and unique investigative probe of the microscopic

magnetization which yields the spatial and time de-

pendence of the dynamical magnetic susceptibility. In

the specific case of CF effects, the neutron induces

transitions between CF levels and these are most

easily observed by the INS technique. The investi-

gation of the CF interaction in more than 1000

rare-earth intermetallic compounds by INS and com-

plementary techniques has been reviewed in detail by

Moze (1998) whilst a briefer review of the INS tech-

nique, with specific reference to tetragonal REAg and

RNi

compounds, has been compiled by Blanco

(1998). The INS technique is also widely used for the

determination of CF parameters and associated line-

width of CF transitions in rare-earth-based high-

temperature superconductors (Furrer et al. 2001).

The CF interaction and INS for rare-earth ions in

icosahedral and amorphous environments have been

briefly summarized by Moze (1998).

1. Crystal Electric Field

The CF Hamiltonian is most commonly expressed as

m;n

ð1Þ

where B

and O

are, respectively, CF energy pa-

rameters (typically in meV) and Stevens operator

equivalents (Hutchings 1964). Alternatively, it can be

expressed in terms of CF coefﬁcients A

m;n

ð2Þ

where /r

S are radial averages over the 4f-electron

wave functions and y

are the Stevens coefﬁcients a

, g

for n ¼2, 4, 6, respectively. These expressions

are only valid for the CF split ground-state multiplet.

A well-known exception to the rule is the case of the

3 þ

ion. The number of CF parameters required to

characterize the interaction appropriate to the point

symmetry in which the rare-earth ion resides are tab-

ulated in Table 1. For triclinic and monoclinic crystal

point groups, where imaginary terms appear, the

relevant matrix elements have been derived by

Takegahara (2000).

The interaction represented in Eqn. (1) or (2) splits

the (2J þ 1)-fold degenerate ground-state multiplet

into a series of CF energy level eigenvalues with en-

ergy E

and corresponding eigenstate, labeled by its

irreducible representation G

. The uniqueness of the

INS technique in this regard is that CF eigenstates

can be obtained from the intensities of observed INS

transitions whilst eigenvalues are determined from

the energy positions of these transitions. It still

remains by far the most indispensable and direct

Crystal Field Effects in Intermetallic Compounds: Inelastic Neutron Scattering Results

experimental technique for investigating the ground-

state properties of a large class of modern magnetic

materials. These cover a wide range of interests, from

fundamental to technological, and in many cases

their magnetic properties are partially governed by

the CF interaction. It is hence of utmost importance

to be able to determine the characteristics of the CF,

for example, and for this purpose INS plays a very

prominent role in experimental magnetism.

2. Inelastic Neutron Scattering

The parameters of the CF Hamiltonian invariably

need to be determined by a number of different ex-

perimental techniques, of which INS is undoubtedly

the most powerful, since it is extremely sensitive to

the details of the CF. With no exchange interactions

between rare-earth ions, which is the case for a pure

CF system, the dynamic magnetic susceptibility is

simply related to the CF eigenstates

. The widely

used equation for the INS cross-section for transi-

tions within a purely CF split multiplet was ﬁrst de-

rived and presented by de Gennes (1963). The INS

master formula for CF transition intensities and en-

ergies is given by the double differential magnetic

scattering cross-section from N rare-earth ions, in the

dipole approximation, for neutrons scattered into an

element of solid angle dO and with a corresponding

energy change given by _o ¼ E

 E

(where E

is the

incident neutron energy and E

is the ﬁnal neutron

energy after scattering) as

dO dE

¼N

ðgr

2W

f ðQÞ





i;j







 dð_o  E

þ E

Þð3Þ

The terms in this equation are deﬁned by the

usual nomenclature: k

is the initial neutron wave-

vector, k

the ﬁnal neutron wavevector, g, the neutron

gyro-magnetic ratio, ¼1:913, r

¼ðe

Þ, the

classical electron radius, (0:2818  10

15

m), ex-

p(2W) is the Debye–Waller factor, representing

the thermal vibrational motion of atoms, g

the

Lande

g-factor, f ðQÞ the Fourier transform of the

magnetization density of unpaired electrons and

comprises both spin and orbital magnetization terms,

the CF thermal occupation factor, J

the compo-

nent of the angular momentum operator perpendic-

ular to the scattering vector Q. The term ðgr

corresponds to a magnetic cross-section of 0.2906

barn (1barn ¼ 10

28

). The subscripts i and j refer

to different CF eigenfunctions.

The delta function dð_o  E

þ E

Þ is appropriate

only for vanishing CF linewidths, but in practice the

linewidth always has an intrinsic temperature-de-

pendent CF component, with a distribution charac-

terized by a Lorentzian component Pð_o; TÞ. In such

cases, the delta function is replaced by

Pð_o; TÞ¼

þð_oÞ

ð4Þ

The Lorentzian full width at half height maximum

2G is inversely proportional to the relaxation rate of

the magnetization ﬂuctuation energy. Intrinsic CF

linewidths are usually associated with magnetic inter-

actions at low temperatures by a Korringa law. CF

excitations are nondispersive and are most conven-

iently studied with polycrystalline material. The rela-

tive intensities between CF levels, suitably averaged for

a powder, are given by the squared matrix elements:





þ G





þ 2 G





ð5Þ

As can be appreciated by inspection of Table 1, a

lowering of the point symmetry corresponds with an

increase in the number of parameters needed to com-

pletely specify the CF interaction. For systems with

cubic point symmetry there exists an ingenious and

Table 1

CF parameters relevant for the 32 point groups. Complex values are indicated by B

Symmetry Point group (Schoenﬂies) Point group (International) CF parameters

Triclinic C

1 B

Monoclinic C

22/mB

Orthorhombic D

222 2mm mmm B

Trigonal C

3 B

Trigonal C

3m 32 B

Hexagonal C

6m26mm 622 6 6/mmm 6/mB

Tetragonal D

4/mmm

42m 4mm 422 B

Tetragonal C

44/mB

Cubic TT

43mm3 432 m3mB

Crystal Fie ld Effects in Intermetallic Compounds: Inelastic Neutron Scattering Results

widely employed method (commonly referred to as

LLW) of parametrizing the eigenfunctions and eigen-

values of the CF Hamiltonian by setting the two pa-

rameters required for description of the CF in terms of

parameters x and W deﬁned by

Fð4Þ

; B

Wð1  x

jjÞ

Fð6Þ

ð6Þ

where 1rxr1, and listing all the eigenvalues and

eigenfunctions as functions of only one parameter x

(the factors F(4) and F(6) are functions only of J;Lea

et al. 1962). In the context of the INS cross-section for

CF excitations, the matrix elements for magnetic di-

pole transitions for cubic symmetry in terms of the

LLW parameter x have been conveniently tabulated

by Birgeneau (1972). Once the intensities of CF tran-

sitions have been determined, it is a relatively simple

matter to determine the two CF parameters using this

tabulation. With further reference to cubic symmetry,

an important point which has been recently brought to

light is a subtle difference in the CF eigenfunction

composition and INS transition probabilities between

point groups ð T

; TÞ and ðO

; O; T

Þ, respectively

(Takegahara et al.2001).Forsystemswithalower

point symmetry, no similar method of the same inge-

nuity and simplicity as the LLW method is available

for parametrizing the CF parameters.

CF transitions in the INS spectrum are usually

identiﬁed by their unique dependence on temperature

and the scattering vector Q. Analysis of the neutron

polarization can be used in specific circumstances when

it is necessary to isolate a particularly weak magnetic

scattering signal from scattering due to nonmagnetic

lattice vibrational modes (Janous

ava

et al. 2003).

3. Complimentary Techniques

When the crystal point group is progressively lowered,

the INS technique used on its own is generally not

sufﬁcient to give a unique set of parameters. In this

context, bulk techniques are most vital for veriﬁcation

of parameters determined by the INS technique for

rare-earth intermetallic compounds (Moze 1998). The

most widely used and significant of these are summa-

rized in brief in the following sections.

3.1 Magnetic Susceptibility

A limited amount of information about the CF in-

teraction can be obtained from the temperature de-

pendent magnetic susceptibility in the noninteracting

regime via the Van Vleck susceptibility

ðTÞ¼N

ðg

(

þ 2

n;m

jjG

hijj

 E

)

ð7Þ

with E

and jG

i eigenvalues and eigenfunctions, re-

spectively, of the CF Hamiltonian and Z the partition

function. There is an explicit dependence on the en-

ergy of the CF levels and the matrix elements con-

necting the eigenstates, but the limiting proviso is that

the sum is over all levels at any particular tempera-

ture and hence any information about one particular

level or set of levels is lost. However, a determination

of the three paramagnetic Curie temperatures y

, y

and y

, as measured along the three principal axes of

a single crystal, the CF parameters, B

and B

are

uniquely determined, viz.

¼

ð2J  1Þð2J þ 3Þ

¼

ðB

 B

Þð2J  1Þð2J þ 3Þ

¼

ðB

þ B

Þð2J  1Þð2J þ 3Þð8Þ

If such data are available, B

can be speciﬁed im-

mediately for tetragonal and hexagonal CFs, whilst

for an orthorhombic CF, B

and B

are similarly de-

termined (Wang 1971, Boutron 1973).

3.2 Magnetization

In the presence of combined CF, exchange and ap-

plied magnetic ﬁelds, the expectation value of the

magnetization can be calculated in the mean ﬁeld

approximation. The magneto-crystalline anisotropy

energy at any particular temperature is determined by

calculation of the variation of the magnetic moment

with the direction of the applied magnetic ﬁeld. The

thermal variation of the component aða ¼ x; y ; zÞ of

the magnetic moment (in Bohr magnetons) is given

hi¼

ðhÞ g

ðhÞ



expðbE

ðhÞÞ

expðbE

ðhÞÞ

ð9Þ

with b ¼ 1= k

T and

the eigenstates of the Ham-

iltonian if exchange and applied ﬁelds are included.

In the paramagnetic regime, rather high magnetic

ﬁelds are required to mix the different CF eigenstates.

In the ordered regime, when an exchange ﬁeld is

present, the ordered moment as a function of tem-

perature is calculated, in standard fashion, by ob-

taining self-consistent solutions which simultaneously

satisfy the above equation and the molecular ﬁeld

condition

h ¼ lNg

hi ð10Þ

The molecular ﬁeld constant l is determined by a

measurement of the ordering temperature. When the

CF is a small perturbation of a dominant rare-earth–

transition metal exchange interaction, simple rela-

tionships exist between the anisotropy constants and

Crystal Field Effects in Intermetallic Compounds: Inelastic Neutron Scattering Results

CF parameters (Rudowicz 1987). The transferibility

of CF parameters obtained from investigation of sys-

tems where the CF is the dominant interaction to

those where the exchange dominates must, however,

be treated with some caution (Moze and Buschow

1996).

3.3 Specific Heat Measurements

The magnetic specific heat is sensitive only to the

eigenvalues of the magnetic system (see Magnetic

Systems: Specific Heat). In the paramagnetic phase,

where CF effects are important, the CF levels are

evidenced via a Schottky-type contribution. The level

degeneracy can also be deduced and the technique is

now routinely used in analyses of systems with a sig-

nificant CF interaction. For a sequence of CF-only

eigenstates E

, at a temperature T, the magnetic part

of the specific heat is given by

CðTÞ¼

ðbE

expðbE

(



expðbE

;

ð11Þ

Limitations in the accuracy of this technique arise

from spin-phonon contributions and experience has

shown that several sets of CF parameters can equally

well ﬁt the specific heat data.

Raman scattering can be used to induce CF tran-

sitions in rare-earth intermetallics (Martinho et al.

2001), but the selection rules for transitions are more

restrictive than the INS technique. A recent novelty is

the joint use of experimental determinations of CF

parameters in rare-earth intermetallics and predic-

tions of the ground-state charge density using density

functional theory (DFT) (Liu et al. 1999, Stra

ssle

et al. 2003). This allows at least the low-order CF

parameters to be calculated approximately. The rel-

evant full set of CF parameters can then be reﬁned

from INS data. This innovative and important meth-

odology is expected to be widely employed in the

future.

4. Recent INS Results

Since the appearance of a review and compilation of

INS investigations concerning the CF interaction in

stable rare-earth intermetallic compounds (Moze

1998), further studies have been reported on a wide

variety of novel compounds which encompass an

equally wide variety of magnetic properties. Amongst

these are the following types of compounds: TmCu

(Kosaka et al. 1997), Nd

(Do

nni et al.

1998), PrPdSb (Adroja et al. 1999), RNiSb (Karla

et al. 1999), PrInAg

(Kelley et al. 2000), HoCr

(Moze et al. 2000), Tm

0.05

0.95

C (Sierks et al.

2000), ErNiAl (Javorsky´ et al. 2001), TmT

, with

T ¼Co, Ni, Cu (Harker et al. 2002), and ErAl

2x

(Stra

ssle et al. 2003). For Ce-based compounds

displaying valence ﬂuctuation, heavy-fermion, or

Kondo-type behavior, a broad INS spectrum is usu-

ally the norm, as the CF coexists with other magnetic

interactions (Christianson et al. 2002, Knafo et al.

2003) but well localized CF excitations have been re-

cently observed, for example, in CeTe

(Liu et al.

2003).

In order to illustrate briefly the unique ad-

vantadges of the INS technique, displayed in Fig. 1,

are the intensities and energies of dipole CF transi-

tions directly observed by INS in the hexagonal

compound NdAl

(Stra

ssle et al . 2003). At low tem-

peratures two transitions from the ground state are

clearly identiﬁed, G

ð1Þ

-G

and G

ð1Þ

-G

ð2Þ

respective-

ly, as well as an excited state transition G

ð1Þ

-G

ð2Þ

The pressure dependence of these transitions has also

been reported and their energy-level scheme at am-

bient and elevated pressure is displayed in Fig. 2,

together with the energy-level scheme determined for

PrAl

at ambient pressure. There are four CF pa-

rameters which need to be determined, as the appro-

priate point group is of hexagonal symmetry. DFT

predictions of the CF parameters at ambient and

elevated pressures for NdAl

were also compared

with those obtained from a careful ﬁtting of the INS

spectra. The parameters calculated within DFT pre-

dict the same signs and orders of magnitude (within

at least 50%) for all four CF terms: B

, B

The only parameter which is seriously underestimat-

ed is B

. Overall, this is highly encouraging given that

within a DFT approach applied to CF interactions

it is still difﬁcult to accurately treat highly correlated

4f-electron states.

Figure 1

Dipole CF transitions and energies for NdAl

observed by INS at ambient pressure and T ¼ 10 K

(reproduced by permission of Stra

ssle et al. (2003) from

J. Phys.: Condens. Matter 15, 3257–66; r IOP

Publishing).

Crystal Fie ld Effects in Intermetallic Compounds: Inelastic Neutron Scattering Results

5. Summary

The technique of neutron inelastic scattering is a

unique and hugely successful spectroscopic technique

which has been used with large effect for the deter-

mination of CF levels and level composition in a vast

number of rare-earth intermetallic compounds. An

intimate knowledge of the CF interaction allows for

detailed microscopic modeling and subsequent un-

derstanding of the principal mechanisms responsible

for complex magnetic structures and phase transi-

tions in systems composed of lanthanide intermetal-

lics. In systems where the R-ion resides in a site of

low symmetry, a determination of the large number

of CF parameters is an enormously difﬁcult task,

and the energy-level scheme and level composition

can only be determined by combined analysis of INS

and bulk measurements. On a more recent note, ex-

perimental INS data combined with DFT calcula-

tions of the ground-state charge density are a novel

and extremely promising methodology for an unam-

biguous determination of CF parameters. The ﬁeld is

very vibrant and rich and there are many systems,

relevant for either fundamental or technological rea-

sons, which will surely be investigated by INS in the

future.

See also: Crystal Field and Magnetic Properties,

Relationship between; Electronic Conﬁgurations of

3d,4f, and 5f Elements: Properties and Simulation;

Localized 4f and 5f Moments: Magnetism; Magnetic

Excitations in Solids

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Figure 2

Energy-level schemes, as determined by INS, for PrAl

at ambient pressure and for NdAl

at ambient and

elevated pressure. Arrows refer to the observed dipole

transitions whilst numbers refer to the INS transition

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permission of Stra

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Condens. Matter 15, 3257–66; r IOP Publishing).

Crystal Field Effects in Intermetallic Compounds: Inelastic Neutron Scattering Results