DSpace at VNU: Crystalline-electric-field effect in some rare-earth intermetallic compounds

Physica B 319 2002 90–104Crystalline-electric-field effect in some rare-earth intermetallic compounds Nguyen Hoang Luong* Center for Materials Science, Faculty of Physics, College of Scie

Physica B 319 (2002) 90–104

Crystalline-electric-field effect in some rare-earth

intermetallic compounds Nguyen Hoang Luong*

Center for Materials Science, Faculty of Physics, College of Science, Vietnam National University, Hanoi,

334 Nguyen Trai, Hanoi, Viet Nam Received 12 March 2002; received in revised form 18 March 2002

Abstract

The results of research on the crystalline-electric-field (CEF) effect in RCu2, R2Fe14B and RFe11Ti compounds are presented In the study of the CEF effect in the RCu2compounds, attention is paid to the combined analysis of specific heat and thermal expansion An attempt has been undertaken to investigate the systematic behavior of CEF interactions by comparing different compounds with the same crystallographic structure From the analysis of spin-reorientation phenomena in R2Fe14B and RFe11Ti compounds the sets of CEF parameters are derived r 2002 Elsevier Science B.V All rights reserved

Keywords: Crystalline-electric-field effect; Rare-earth intermetallic compounds

1 Introduction

Rare-earth intermetallic compounds are in a

prominent situation not only from a fundamental

point of view but also because of the important

applications, in particular in the field of permanent

magnets Magnetic properties of rare-earth

inter-metallics result to a large extent from the interplay

of crystalline-electric-field (CEF) and exchange

interactions

The CEF removes the degeneracy of the ground

state multiplet of the rare-earth ion This results in

specific magnetic properties of the corresponding

compound The study of CEF effects is an

important subject in the field of magnetism and

magnetic materials

In this work, we present the results of research

on the CEF effect in some rare-earth intermetallic compounds Firstly, we discuss the RCu2(R=rare earth) compounds, the magnetic properties of which are largely affected by the CEF interactions

magnetic transition metal, 4f magnetism can be

disturbing the effects of the 3d magnetism An attempt has been undertaken to investigate the systematic behavior of CEF interactions by comparing different compounds with the same crystallographic structure Particular attention is paid to the CEF effect in ErCu2, in which the combined analysis of specific heat and thermal expansion is proved to be a valuable tool Then,

we discuss the CEF effect in the R2Fe14B and RFe11Ti compounds on which spin-reorientation phenomena are studied In these compounds, both

*Corresponding author Fax: +84-4-8589496.

E-mail address: luongnh@vnu.edu.vn (N.H Luong).

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 1 1 1 - 0

the rare-earth sublattice and the transition-metal

one are magnetic Spin-reorientation transitions

have been observed in many of these compounds

From the analysis of the spin-reorientation

phe-nomena, we derive sets of CEF parameters for the

R site It is shown that the study of

spin-reorientation phenomena in intermetallic

com-pounds is very useful in obtaining information

on the CEF parameters

2 The crystalline-electric-field effect

2.1 General formalism of the crystalline electric

field

The 4f electrons of a rare-earth ion in a solid,

being considered as well localized and separated

from other charges, experience an electrostatic

potential V ðrÞ that originates from the

surround-ing charge distribution The CEF Hamiltonian,

describing the electrostatic interaction of the

aspherical 4f charge distribution with the

asphe-rical electrostatic field arising from its

surround-ing, can be written as

i

Here, the summation is taken over all 4f electrons

The Hamiltonian may be expanded in spherical

harmonics Ym

n; since the charges of the CEF are

outside the shell of the 4f electrons:

HCEF¼XN

n¼0

Xn

m¼n

Amn X

i

rniYnmðyi; jiÞ; ð2Þ

where Am

n are coefficients of this expansion Their

values depend on the crystal structure considered

and determine the strength of the CEF interaction

The value of n in expression (2) is limited to np6

for the rare-earth series

The calculation of the matrix elements of the

Hamiltonian (2) can be performed by direct

integration However, the method called the

Stevens operator equivalent method is much more

convenient and is widely used This method of

Stevens [1] is described in detail by Hutchings [2]

In this method, the x; y; z coordinates of a

particular electron are replaced by the components

Jx; Jy; Jzof the multiplet J: The CEF Hamiltonian (2) then takes the form

HCEF ¼XN

n¼0

Xn m¼n

Here, the coefficients Bmn are called the CEF parameters and Om

n are the Stevens equivalent operators [1] The parameters Bm

n can be written as

Bmn ¼ yn/rn

4fSAm

In this expression, the factor related to the 4f ion,

yn/rn 4fS; and the factor related to the surrounding charges, Am

n; are separated The coefficients Am

n are known as the CEF coefficients yn is the appro-priate Stevens factor of order n which represents the proportionality between the operator functions

of x; y; z and the operator functions of Jx; Jy; Jz: The parameter ynis denoted as aJ; bJ; gJfor n ¼ 2;

4, and 6, respectively The sign of ynrepresents the type of asphericity associated with each Om

n term describing the angular distribution of the 4f-electron shell In particular, the factor aJ describes the ellipsoidal character of the 4f-electron distri-bution For aJ > 0; the electron distribution associated with Jz¼ J is prolate, i.e elongated along the moment direction whereas for aJo0 the 4f-electron-charge distribution is oblate, i.e ex-panded perpendicular to the moment direction For aJ ¼ 0 (which is the case of the Gd+3ion) the charge density has spherical symmetry /rn

4fS is the mean value of the nth power of the 4f radius Values for the average value /rn

4fS over the 4f wave function have been computed on the basis of Dirac–Fock studies of the electronic properties of the trivalent rare-earth ions by Freeman and Desclaux [3] Values for yn/rn

4fS have been collected by Franse and Radwa!nski [4]

The computation of the CEF coefficients, Amn; from microscopic, ab initio, calculations is a difficult problem A full band-structure calculation

of the charge distribution over the unit cell and, consequently, of the full set of CEF coefficients, is lacking for almost all compounds In some cases, the point-charge model, with electron charges centered at the ion positions in the lattice, can give the correct sign of the leading second-order CEF coefficients However, this model is ques-tioned, especially in metallic systems where the

contribution of valence electrons is expected to be

significant [5] This has been confirmed by

band-structure calculations by Coehoorn [6] who

con-cluded that the second-order CEF coefficient A0is

mainly determined by the asphericity of the

valence-shell electron density of the rare earth

under consideration

There is only a limited number of compounds

for which the CEF interactions have been

quanti-fied It is due to the lack of experimental

information as well as the complexity of the

crystallographic structures Discussions are still

going on, even for the best-known systems like the

cubic Laves-phase RT2compounds The situation

becomes more complex for the systems with a

lower crystal symmetry The orthorhombic RCu2

and the tetragonal R2Fe14B and RFe11Ti

com-pounds, which we are dealing with in this work,

belong to these latter cases

compounds, for instance) the CEF is described

by only two parameters B4and B6

HCEF¼ B4ðO04þ 5O44Þ þ B6ðO06 21O46Þ: ð5Þ

For tetragonal symmetry, five CEF parameters Bm

n

are needed

HCEF¼ B0

2O02þ B0

4O04þ B4

4O44þ B0

6O06þ B4

6O46 ð6Þ and for orthorhombic symmetry, nine CEF

para-meters Bmn are needed

HCEF¼ B02O02þ B22O22þ B04O04þ B24O24þ B44O44

þ B0

6O06þ B2

6O26þ B4

6O46þ B6

6O66: ð7Þ Usually the CEF parameters Bmn are evaluated

from the analysis of experimental data The

methods include the fitting of the magnetization

curves, inelastic neutron scattering, measurement

of the temperature dependence of the specific heat

and susceptibility, M.ossbauer spectroscopy, and

so on Below, we will discuss the methods of

analysis of experimental data that we use for

studying the CEF effects in RCu2, R2Fe14B and

RFe11Ti compounds These methods comprise the

Gr.uneisen analysis and the spin-reorientation

analysis

2.2 Gr.uneisen analysis

In the study of magnetic systems, the specific heat and the thermal expansion are very impor-tant The combined analysis of specific heat and the thermal expansion can give valuable informa-tion on the system under considerainforma-tion Here, we briefly describe a procedure that has successfully been applied to several different systems

The specific heat is written as the sum of electronic (ce), lattice (cph) and magnetic (cm) contributions

Similarly, the thermal expansion contains electro-nic (be), lattice (bph) and magnetic (bm) contribu-tions

In Eqs (8) and (9) we neglect a nuclear contribution The electronic part of the specific heat is written as

where g is called the electronic coefficient The electronic part of the thermal expansion is also a linear function of temperature, i.e

The phonon part of the specific heat (for a compound with r atoms per formula unit) is approximated by

cph¼ 9rRðT=yDÞ3

Z y D =T

0

x4ex

where yD is the Debye temperature, R the gas constant

The phonon contribution to the thermal expan-sion, like the contribution to the specific heat, is approximated by

bph¼ bðT =yDÞ3

Z y D =T

0

x4ex

An arbitrary contribution to the specific heat, ci;

is related to a corresponding contribution to the thermal expansion, bi; by a so-called Gr.uneisen relation

N.H Luong / Physica B 319 (2002) 90–104 92

Here V is the molar volume, k the compressibility

and Gi the appropriate Gr.uneisen parameter

For the electronic Gr.uneisen parameter we have

(see Eqs (10) and (11))

Using Eqs (12) and (13) for the lattice Gr.uneisen

parameter, we obtain

In treating the magnetic contributions to the

specific heat and to the thermal expansion, a

straightforward approach is to calculate an

effec-tive Gr.uneisen parameter, Geff; by the relation

and to follow its variation with temperature A

pronounced temperature dependence of GeffðT Þ

indicates the presence of several contributions

ErCu2can serve as an example, in which a change

in sign of the parameter Geff is observed upon

increasing the temperature [7] In this case, at least

two different contributions to cm and bm can be

distinguished Therefore, we write

cm¼ clrþ ccf and bm¼ blrþ bcf: ð18Þ

Here, cm and bm are split into two contributions,

the ‘long-range’ magnetic order contributions clr

and blr; and the contributions ccf and bcf

associated with the CEF splitting of the energy

levels Assuming that the contributions ci and bi

are related by Gr.uneisen parameters Gi; we have

Geff ¼X

i

Here fi¼ ci=cm: For any choice of Glrand Gcf; the

separated contributions can be calculated as

clr¼Geff Gcf

ccf ¼Glr Geff

blr¼1  Gcf=Geff

bcf¼ Gcf

Geff

1  Geff=Glr

This analysis and its application to ErCu2 is described in Refs [8,9] Brommer and Franse [10] have generalized this method, including Gr-.uneisen relations between specific-heat and linear-expansion contributions, as well as focusing attention to the criteria to decide whether the chosen Giparameters can be considered as genuine

Gr.uneisen parameters They successfully applied this analysis to a variety of materials (see Ref [10])

2.3 Spin-reorientation analysis For a uniaxial crystal, the magnetocrystalline anisotropy energy may be described by the phenomenological expression

Here y is the polar angle of the magnetization with respect to the c-axis K1and K2 are the anisotropy constants Minimizing anisotropy energy (24) with respect to y gives the orientation of the magnetiza-tion A sudden change from easy axis to easy plane may occur However, when, for K2> 0; K1changes sign at a certain temperature, a gradual spin reorientation will start at that temperature [11,12] The angle between the moment direction and the c-axis is given by

For a deeper understanding of the spin-reorienta-tion phenomena, one has to consider a micro-scopic model As mentioned in the introduction, the magnetic properties of 3d–4f compounds are governed by a combination of the 3d–4f exchange and CEF interactions The Hamiltonian of a 4f ion in 3d–4f compounds is usually given in the form

Here, HCEF and Hex are the CEF and exchange Hamiltonians, respectively The CEF Hamiltonian for a tetragonal structure is expressed by Eq (6) The exchange Hamiltonian is given by

where Bmis the molecular field acting on the rare-earth magnetic moment which is related to the

exchange field Bexacting on the rare-earth spin by

Bm¼ ½2ðg  1Þ=gBex: J and g are the total angular

momentum and the Land!e factor of the R3+

ion, respectively The field Bm is related to the

exchange constant nRT; which links the rare-earth

(R) and transition-metal (T) sublattices by

sublattice magnetization

The relation between KR

1 and Bm

n is [13]

K1R¼  ð3=2ÞB0

2/O0

2S  5B0

4/O0

4S

 ð21=2ÞB0

6/O0

Stevens operators

The ground state of the 4f ion is calculated by

diagonalizing Hamiltonian (26) The temperature

dependence of the rare-earth sublattice anisotropy,

KR

1ðxÞ; which normally dominates, is then

1ðxÞ ¼ ð1  xÞKR

1ð0Þ; where KR

1ðxÞ and KR

1ð0Þ are the rare-earth sublattice anisotropy constants for the

substituted and unsubstituted compounds,

respec-tively In the case of R2Fe14B compounds, in this

work KR

1 has been calculated as KR

1 ¼ E>c E8c; where E>c and E8c represent the ground state

energy of Hamiltonian (26) for Bm being

perpen-dicular and parallel to the c-axis, respectively

For calculating the temperature dependence of

the anisotropy energy of the rare-earth ion, the

Boltzmann distribution function is used The

temperature dependence of the transition-metal

sublattice anisotropy KT

1 is taken from the study of the isostructural compound in which R is

non-magnetic

In case the data on the temperature

depend-ence of the angle y in aligned powder samples

follows

For aligned powder samples of a material with

axial anisotropy, the crystallites (powder particles)

are oriented in such a way that their c-axis are

parallel to each other, whereas the a- and b-axis

are randomly distributed in the plane

perpendi-cular to the alignment direction Therefore, it is

appropriate to confine the exchange field to the

x2z plane [14] Such an approximation leads to

the following exchange Hamiltonian:

Hex¼ gmBBmðJzcos y þ Jysin yÞ: ð30Þ The rare-earth energy is obtained by diagonaliza-tion of Hamiltonian (26) and by calculating the partition function Zðy; TÞ: The rare-earth energy is given by

For a mixed system such as R1xRx 0Fe11Ti, the total free energy is expressed as

F ðy; TÞ ¼ ð1  xÞFRðy; T Þ þ xFR0ðy; T Þ

þ KT

1sin2y: ð32Þ Here the last term represents the contribution from the transition-metal sublattice For the calculation

of F ðy; TÞ; we need to know the CEF parameters

Bm

n and the molecular field Bm: The temperature dependence of the transition-metal sublattice anisotropy KT

1 is again taken from the study of

an isostructural compound in which R is non-magnetic The angular dependence of the total free energy was then calculated and the minimum in the free energy gives the orientation of the total magnetization vector In this way, the temper-ature dependence of the magnetic structure is determined

3 CEF effect in RCu2compounds

orthorhombic CeCu2 structure Early magnetiza-tion and magnetic susceptibility measurements performed by Hashimoto et al [15] and

Hashimo-to [16] demonstrated the importance of the CEF in these compounds During the last decade sub-stantial progress has been achieved in the study of the magnetic properties of the RCu2compounds

It is of interest to make an attempt to see some systematic behavior by comparing different com-pounds with the same crystallographic structure This attempt was undertaken in Ref [9] In this work, we focus our study to the CEF effect in the RCu2 compounds, taking into account the recent results For doing this, we recall also some works

by other authors

N.H Luong / Physica B 319 (2002) 90–104 94

CeCu2is a Kondo compound In Ref [9] some

results on the study of CEF effects are mentioned

Sugiyama et al [17] have measured the high-field

magnetization of CeCu2 in various temperatures

and have analyzed it on the basis of the

quadrupolar interaction and CEF PrCu2 shows

a nearly temperature-independent Van Vleck

paramagnetic behavior below 4.2 K and exhibits

cooperative nuclear antiferromagnetic order below

54 mK [18] From the measured paramagnetic

Curie temperatures, Hashimoto et al [19] have

estimated the following values for the

second-order CEF parameters for PrCu2: B0

2¼ 4:27 K and

B2

Hashimoto et al [19] have also calculated the

values of B0

2 and B2 and arrived at B0

2 cal ¼ 4:1 K and B2

2 cal¼ 3:48 K, in good agreement with

experiment The CEF and the metamagnetic

transition in PrCu2 has been studied in detail by

Ahmet et al [20] and Settai et al [21,22] We recall

which has been derived by Settai et al [22]:

B20¼ 4:93 K, B22 ¼ 3:50 K, B04 ¼ 5:08 102K,

B2

4¼ 5:02 102K, B44¼ 3:82 101K, B06¼

1:33 103K, B2

6¼ 1:80 102K, B4

6¼

2:98 102K, and B6

6 ¼ 4:97 102K

CEF parameters were first estimated by

Hashimo-to et al [15] from measurements of the

para-magnetic susceptibility in single-crystalline sample:

B0

B0

2 cal¼ 1:17 K and B2

2 cal¼ 1:01 K were obtained

by Hashimoto et al [15] by point-charge model

calculations A more detailed study of the CEF

effect in this compound has been carried out by

Gratz et al [23] These authors have derived the

following set of CEF parameters which best

describe the inelastic neutron-scattering data:

B2

0¼ 1:35 K, B2

2 ¼ 1:56 K, B0

4 ¼ 2:23 102K,

B2

4¼ 1:01 102K, B4

4¼ 1:96 102K, B0

6¼ 5:52 104K, B2

6¼ 1:35 104K, B4

6¼ 4:89

104K, and B6

6¼ 4:25 103K

Gratz et al [24] have shown that the coefficient

of thermal expansion of SmCu2 exhibits a

mini-mum at 45 K, caused by the CEF effect These

authors have used the position of the temperature

where the minimum occurs to estimate the splitting

energy between the ground-state doublet and the

first-excited state doublet (see, for instance, Ref [8]) They have derived a value of about 110 K for this CEF splitting

Turning to the heavy RCu2 compounds, first

we discuss TbCu2 Again from the values of

et al [19] have estimated the following values for the second-order CEF parameters: B02¼ 1:23 K and B22¼ 1:23 K These authors have also calcu-lated the second-order CEF parameters on the basis of the point-charge model Their calculated values are: B0

2 cal¼ 1:35 K and B2

2 cal¼ 1:12 K Experiments and calculations are in reasonable agreement with each other, indicating that the anisotropy observed in the paramagnetic state for the paramagnetic Curie temperatures along the crystallographic axes can be explained mainly by the CEF effect Measurements of the specific heat and thermal expansion were performed by Luong

et al [7] Apart from a l-type of anomaly at TN; apparently a broad anomaly is observed around

30 K This anomaly can be discussed in terms of

Gr.uneisen parameters For the DyCu2compound, also from the values of the paramagnetic Curie temperatures, Hashimoto et al [19] have estimated values for the second-order CEF parameters:

B0

2¼ 0:43 K and B2

2¼ 0:72 K A point-charge

B0

2 cal¼ 0:89 K and B2

2 cal¼ 0:71 K, in satisfactory agreement with the experimental values Specific heat and thermal expansion of DyCu2have been measured by Luong et al [7] Kimura et al [25] have calculated the specific heat of DyCu2in terms

of the molecular-field model including CEF inter-action They have obtained the temperature dependence of the specific heat which is similar

in trend with the experimental one of Luong et al [7] They have used three CEF parameters: B02 and

B2

2 experimentally obtained by Hashimoto et al [19] and B0

4equal to 2.37 103K By calculating the magnetic susceptibility and the magnetiza-tion and comparing these calculated properties with the experimental ones, Sugiyama et al [26] and Yoshida et al [27] have derived the full set of CEF parameters for DyCu2 The set of CEF parameters obtained in Ref [27] is the fol-lowing: B0¼ 0:708 K, B2¼ 0:99 K, B0¼ 0:28

104K, B2¼ 2:37 102K, B4¼ 0:26 105K,

6¼1:1 105K, B2¼ 0:79 105K, B4¼ 1:6

104K, B6¼ 1:7 104K

Like in other RCu2compounds, from the values

of the paramagnetic Curie temperatures,

Hashi-moto et al [19] have estimated the following

B02¼ 0:14 K and B2

B02 cal¼ 0:28 K and B2

2 cal¼ 0:23 K were obtained

by Hashimoto et al [19] on the basis of the

point-charge model The CEF effect in ErCu2 will be

discussed in detail below The CEF effects in

TmCu2have extensively been studied The values

for all nine CEF parameters obtained by different

methods and by combining the results of different

experiments for this compound are collected in

Ref [9] (see Table 4.1 of Ref [9])

It can be inferred from the above discussion and

from Ref [9] that information about the CEF

interaction in RCu2 is not complete Due to the

orthorhombic structure, nine CEF parameters are

needed to describe the CEF Hamiltonian In Table

1, we collect the second-order coefficients A02 and

A22for the RCu2compounds These coefficients are

related to the CEF parameters B02 and B22 by (see

Section 2)

A02¼ B02=aJ/r2

4fS;

A22¼ B2

2=aJ/r2

Here, the values for the quantity aJ/r2

4fS are taken from Franse and Radwa!nski [4] To our

knowledge, CEF parameters for SmCu2 are not

available Rather low values of A0

2 and A2

2 have

been obtained by Trump [31] for the Kondo compound CeCu2, in which anomalous properties are observed Except for CeCu2, as can be seen from Table 1, the coefficients A0 and A2 have the same sign and are of the same order of magnitude

We note that the same cannot be said for the higher-order CEF coefficients From the similarity

of the lowest-order CEF parameters, it seems that the CEF model can be used for describing the behavior of isostructural RCu2compounds At the same time, there is no reason why higher-order CEF parameters should be neglected The neces-sity to take these higher-order terms into account

is indicated in Ref [9] and below in a discussion of CEF effects in ErCu2 The importance of the higher-order CEF terms is also revealed from studies on substituted RCu2 compounds Analy-zing the data obtained on a Tb(Cu0.7Ni0.3)2 sample, Divis et al [32] have shown that the step-like appearance in the magnetization curves along the b-axis in this compound cannot be explained by using second-order terms in the CEF Hamiltonian only These authors have shown that

in order to account for all features of the magnetization data, the higher-order terms should

be included into the Hamiltonian Divis et al [33] have also used nine CEF parameters for describing the specific-heat data on Tm(Cu1xNix)2

We discuss below in detail the CEF effect in ErCu2 Luong et al [7] reported on the specific heat and thermal expansion of RCu2compounds and discussed the excess contributions of both quantities arising from magnetic ordering and CEF effects Apart from some sharp features indicating transitions between different types of antiferromagnetic order, and apart from a lambda type of anomaly characteristic for disordering the antiferromagnetic state, broad anomalies were observed for several compounds in the specific heat and thermal expansion In the case of ErCu2, Luong et al [7] could preliminarily analyze the excess contributions to the specific heat and thermal expansion by applying Gr.uneisen rela-tions This analysis has been applied and discussed

in more detail in Refs [8,9,34], showing that they consist of a ‘long-range’ magnetic and a CEF part (see above) The CEF term yields Schottky-type

Table 1

CEF coefficients, in units of Ka 2

0 ; for the RCu 2 compounds

N.H Luong / Physica B 319 (2002) 90–104 96

showed that the energy difference between the

ground-state doublet and the first excited doublet

amounts to 76 K Using the values for the two

lowest-order CEF parameters, B20¼ 0:35 K and

B22¼ 0:36 K, obtained by Hashimoto et al [19]

from an analysis of the paramagnetic

susceptibil-ity, the energy levels in ErCu2 have been

calcu-lated, and a splitting of 13 K between the two

lowest-order doublets has been derived [8]

Ap-parently, higher-order CEF terms have to be taken

into account in order to bring the splitting closer

to the experimental value of 76 K As pointed out

compounds, nine CEF parameters are needed to

describe the CEF of the rare-earth ion It is

difficult to derive the full set of CEF parameters A

combination of different techniques, experimental

and theoretical, is used in order to overcome this

difficulty Gubbens et al [28] have measured the

ErCu2 compound with 166Er M.ossbauer

spectro-scopy These authors also reported the results of

inelastic neutron scattering, which show a not yet

definitively determined level sequence of doublets

above TN at 0, 61, 78, 88, 124, 142, 148 and

160 K Gubbens et al [28] have determined a

tentative set of all nine CEF parameters This set

is: B2¼ 0:28 K, B2¼ 0:22 K, B0

4 ¼ 0:30

102K, B2¼ 0:14 102K, B4¼ 0:30 102K,

B0¼ 0:20 104K, B2¼ 0:47 104K, B4¼

0:97 104K, and B6 ¼ 2:96 104K

We have performed calculations of the CEF contribution to the specific heat of ErCu2 using CEF data obtained by Hashimoto et al [19] and Gubbens et al [28] Results of the calculations are shown in Fig 1, and are compared with the experimental data DCexp: On calculating DC1; using two lowest-order CEF parameters reported

by Hashimoto et al [19], we derived the following level scheme of doublets: 0, 13, 25, 33, 40, 49, 61 and 75 K As can be seen from Fig 1, the CEF contribution to the specific heat (curve DC1), given

by this energy scheme, disagrees with our experi-ments The curve DC2 in Fig 1 was obtained by taking into account all eight doublets reported by Gubbens et al [28] From this figure it can also be seen that the temperature dependence of DC2has a behavior similar to the experimental one, but the calculated value of DC2 is larger in the high temperature range This difference between the calculated DC2 and the experimental specific-heat curves has been discussed in more detail in Ref [34]

One of the reasons for the above-mentioned discrepancy could be an overestimation of the non-magnetic contribution to the specific heat In order to evaluate the magnetic contribution to the specific heat, non-magnetic (electronic and pho-non) contributions have to be subtracted from the total specific heat It is well known that a proper evaluation of the non-magnetic contribution of magnetic compounds is a difficult problem In the case of RCu2, the non-magnetic contribution to the specific heat is obtained from measuring the specific heat of YCu2 [35] We note that YCu2is taken as the non-f reference material instead of LaCu2because LaCu2possesses a different crystal-lographic structure and crystallizes in the hexago-nal AlB2-type of structure A value of 194 K for the Debye temperature of ErCu2, yD(ErCu2), was derived [34] Using the more sophisticated ap-proach of Bouvier et al [36], which accounts for the different molar masses of the components, a value of 197 K for yD(ErCu2) has been determined, i.e very close to the value 194 K obtained above Another possible reason of the discrepancy between the experimental curve DCexp and the calculated curve DC2 could be that higher energy levels do not substantially contribute to the

Fig 1 Calculated and experimental results for the CEF

contribution to the specific heat of ErCu 2 DC1and DC2curves

are calculated using CEF data in Refs [19,28], respectively.

DCexp (from Refs [7,8]) is obtained from specific heat and

thermal expansion measurements.

specific heat We have performed calculations of

the CEF contribution to the specific heat of ErCu2

taking into account only the four lowest doublets

reported by Gubbens et al [28] (see above) The

calculated curve is in close agreement with the

experimental one This result could suggest (in case

of a proper estimate of the phonon contribution to

the specific heat) that the energy spectrum in

ErCu2is divided into two groups The first group

consists of the four lowest doublets located below

88 K The second group, consisting of four higher

doublets, is separated from the first one We note,

as mentioned above, that the energy level scheme

in ErCu2 is not definitively determined and that

the set of CEF parameters for this compound is

not unique at the present stage of investigations

[28]

One of the features of the RCu2compounds is

that the values of the N!eel temperature for these

compounds are not simply proportional to the de

Gennes factor ðgJ 1Þ2JðJ þ 1Þ and reach a

maximum for TbCu2 This fact suggests that the

Ruderman–Kittel–Kasuya–Yosida (RKKY)

inter-action alone is not sufficient to fully understand

the magnetic interactions in the RCu2compounds

In spite of a substantial progress in the study of the

magnetic properties of the RCu2compounds, the

above-mentioned exception to the de Gennes rule

remained unexplained for a long time Luong et al

[37] have performed calculations in order to

explain the anomalous behavior of the N!eel

Tm) The calculations are based on the model of

Noakes and Shenoy [38] When considering only

the exchange Hamiltonian, the de Gennes rule can

be derived However, when the CEF effects are

significant, the de Gennes behavior is not to be

expected Adding the CEF Hamiltonian to Hexc

leads to the following expression for the ordering

temperature:

TM ¼ 2GðgJ 1Þ2/J2

zðTMÞSCEF; ð34Þ where /J2

zðTMÞSCEFis the expectation value of J2

z

under the influence of the CEF Hamiltonian alone,

at the temperature TM: The exchange parameter,

G; can be evaluated from the ordering temperature

of the Gd compound when modeling a series of

rare-earth compounds, because Gd, an L ¼ 0 ion,

is essentially unaffected by CEF For the calcula-tion of the N!eel temperatures, TN; of the RCu2 compounds expression (34) is used, in which TM

stands for TN: For evaluating G in these com-pounds, Luong et al [37] took TN(GdCu2)=41 K [7,39,40]

In the coordinate system of b ¼ z; c ¼ x and a ¼ y; the orthorhombic CEF Hamiltonian of a CeCu2 type of structure is given by Eq (7) In the calculations, Luong et al [37] first used the two lowest-order terms in the CEF Hamiltonian Values for B0

2 and B2

2 were taken for TbCu2, DyCu2 and HoCu2 from Ref [19], ErCu2 from Ref [28] and TmCu2from Ref [29]

In TbCu2, DyCu2 and HoCu2 the magnetic moments lie along the a-axis, whereas in ErCu2 and TmCu2 the magnetic moments are oriented along the b-direction (see Ref [37] and references therein) The TN values for ErCu2 and TmCu2 were calculated directly using the CEF Hamilto-nian (7) with only the two lowest-order terms For

used the CEF Hamiltonian transformed in the new coordinate system of a ¼ z; b ¼ x; c ¼ y as follows [41]:

HCEF ¼ ð1=2ÞðB0

2 B2

2ÞO0 2

The calculated values of TN are compared with experimental data in Table 2 and also in Fig 2 As one can see from Table 2 and Fig 2, addition of CEF interaction enhances TNover the de Gennes values in the RCu2compounds Moreover, calcu-lations predict that TbCu2 has the highest N!eel temperature, in good agreement with experiments The calculated values for the N!eel temperatures across the series are in good agreement with the experimental ones Calculations gave the value of

obtained from experiments Luong et al [37] tried also to derive the values for the N!eel temperature

of ErCu2 and TmCu2 with the full Bm

n set taken from Refs [28,29], respectively The results of these calculations, using the full CEF Hamiltonian (7), are also shown in Table 2 and Fig 2 As it can

be seen, the use of the full CEF Hamiltonian gives better results than the use of the two lowest-order

N.H Luong / Physica B 319 (2002) 90–104 98

CEF terms only Thus, the magnetic ordering

explained by a combination of the RKKY

inter-action and CEF effects

4 CEF effect in R2Fe14B compounds

In the tetragonal R2Fe14B compounds, the

earth sublattice consists of two inequivalent

rare-earth sites The anisotropy at room temperature

is uniaxial in the compounds with rare-earth

elements for which the Stevens factor aJ is

negative, while it is planar in the compounds with

rare-earth elements for which aJis positive [45] In

the compounds with Sm, Er, Tm, and Yb, for

which aJ is positive, a competition between

rare-earth and iron sublattice anisotropies is expected

and can lead to a spin reorientation Such

reorientations were observed in Er2Fe14B,

Tm2Fe14B [46] and Yb2Fe14B [47] In Sm2Fe14B

a spin reorientation has not been found Nd2Fe14B undergoes a different type of spin reorientation, namely from a conical to an axial arrangement with increasing temperature

The temperature dependence of the rare-earth contribution to the magnetocrystalline anisotropy energy of the R2Fe14B compounds (R=Nd, Sm,

Er, Tm, Yb) have been calculated [48,49] The results of these calculations in combination with the data on the iron sublattice anisotropy enable

us to derive the spin-reorientation temperatures in

R2Fe14B with R=Nd, Er, Tm, and Yb, and to show that a spin reorientation is not expected in

Sm2Fe14B

In the R2Fe14B compounds investigated the CEF parameter B0

2 is assumed to be dominant in

2 is considered here as a mean value for the two inequivalent rare-earth sites, as has been suggested

by several groups [50,51]

Thus, for calculating the rare-earth

Table 2

Values for the N !eel temperatures in the heavy RCu 2

compounds

41 [39]

42 [40]

53.5 [19]

48.5 [7]

48 [42]

31.4 [19]

26.7 [7]

27 [28]

9.8 [19]

9.6 [7]

11 [43]

13.5 [19]

11.5 [7]

a N!eel temperature predicted by the full CEF Hamiltonian

with the B m

n sets from Refs [28,29] for ErCu 2 and TmCu 2 ,

respectively.

Fig 2 Comparison of experimental and calculated N !eel temperatures for the RCu 2 compounds The open circles represent experimental data The solid circles (solid line) represent calculations using a CEF Hamiltonian with two lowest-order terms, and the solid squares represent calculations with the full CEF Hamiltonian as discussed in the text [37] The dashed line represents the de Gennes rule.