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

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 RCu2 compounds, 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 spinreorientation 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 intermetallics 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.

*Corresponding author. Fax: +84-4-8589496.
E-mail address: (N.H. Luong).

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.
Unlike the Rn Tm compounds, where T is a
magnetic transition metal, 4f magnetism can be
investigated in the RCu2 compounds without
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

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.
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N.H. Luong / Physica B 319 (2002) 90–104

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 phenomena, we derive sets of CEF parameters for the
R site. It is shown that the study of spinreorientation phenomena in intermetallic compounds 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 surrounding charge distribution. The CEF Hamiltonian,
describing the electrostatic interaction of the
aspherical 4f charge distribution with the aspherical electrostatic field arising from its surrounding, can be written as
X
HCEF ¼ À
eV ðri Þ:
ð1Þ
i

Here, the summation is taken over all 4f electrons.
The Hamiltonian may be expanded in spherical
harmonics Ynm ; since the charges of the CEF are
outside the shell of the 4f electrons:
N X
n
X
X
Am

rni Ynm ðyi ; ji Þ;
ð2Þ
HCEF ¼
n
n¼0 m¼Àn

i

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

91

Jx ; Jy ; Jz of the multiplet J: The CEF Hamiltonian
(2) then takes the form
N X
n
X

m
HCEF ¼
Bm
ð3Þ
n On ðJÞ:
n¼0 m¼Àn

Here, the coefficients Bm
n are called the CEF
parameters and Om
are
the Stevens equivalent
n
operators [1]. The parameters Bm
n can be written as
n
m
Bm
n ¼ yn /r4f SAn :

ð4Þ

In this expression, the factor related to the 4f ion,
yn /rn4f S; and the factor related to the surrounding
m
charges, Am
n ; are separated. The coefficients An are
known as the CEF coefficients. yn is the appropriate 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 yn is denoted as aJ ; bJ ; gJ for n ¼ 2;
4, and 6, respectively. The sign of yn represents the
type of asphericity associated with each Om
n term
describing the angular distribution of the 4felectron shell. In particular, the factor aJ describes
the ellipsoidal character of the 4f-electron distribution. For aJ > 0; the electron distribution
associated with Jz ¼ J is prolate, i.e. elongated
along the moment direction whereas for aJ o0 the
4f-electron-charge distribution is oblate, i.e. expanded perpendicular to the moment direction.
For aJ ¼ 0 (which is the case of the Gd+3 ion) the
charge density has spherical symmetry. /rn4f S is
the mean value of the nth power of the 4f radius.
Values for the average value /rn4f S 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 /rn4f S have been
!
collected by Franse and Radwanski
[4].
The computation of the CEF coefficients, Am
n;
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 questioned, especially in metallic systems where the


92

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

contribution of valence electrons is expected to be
significant [5]. This has been confirmed by bandstructure calculations by Coehoorn [6] who concluded that the second-order CEF coefficient A02 is
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 quantified. 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 RT2 compounds. The situation
becomes more complex for the systems with a
lower crystal symmetry. The orthorhombic RCu2
and the tetragonal R2Fe14B and RFe11Ti compounds, which we are dealing with in this work,
belong to these latter cases.
For cubic symmetry (in case of the RAl2
compounds, for instance) the CEF is described
by only two parameters B4 and B6
HCEF ¼ B4 ðO04 þ 5O44 Þ þ B6 ðO06 À 21O46 Þ:

ð5Þ

For tetragonal symmetry, five CEF parameters

are needed

Bm
n

HCEF ¼ B02 O02 þ B04 O04 þ B44 O44 þ B06 O06 þ B46 O46 ð6Þ
and for orthorhombic symmetry, nine CEF parameters Bm
n are needed
HCEF ¼ B02 O02 þ B22 O22 þ B04 O04 þ B24 O24 þ B44 O44
þ B06 O06 þ B26 O26 þ B46 O46 þ B66 O66 :

ð7Þ

Usually the CEF parameters Bm
n 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, Mossbauer
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
Gruneisen
.
analysis and the spin-reorientation
analysis.

2.2. Gruneisen
analysis
.
In the study of magnetic systems, the specific
heat and the thermal expansion are very important. The combined analysis of specific heat and
the thermal expansion can give valuable information on the system under consideration. 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
c ¼ ce þ cph þ cm :

ð8Þ

Similarly, the thermal expansion contains electronic (be ), lattice (bph ) and magnetic (bm ) contributions
b ¼ be þ bph þ bm :

ð9Þ

In Eqs. (8) and (9) we neglect a nuclear contribution.
The electronic part of the specific heat is written
as
ce ¼ gT;

ð10Þ

where g is called the electronic coefficient. The
electronic part of the thermal expansion is also a

linear function of temperature, i.e.
be ¼ aT:

ð11Þ

The phonon part of the specific heat (for a
compound with r atoms per formula unit) is
approximated by
Z yD =T
x 4 ex
cph ¼ 9rRðT=yD Þ3
dx;
ð12Þ
ðex À 1Þ2
0
where yD is the Debye temperature, R the gas
constant.
The phonon contribution to the thermal expansion, like the contribution to the specific heat, is
approximated by
Z yD =T
x4 ex
bph ¼ bðT=yD Þ3
dx:
ð13Þ
ðex À 1Þ2
0
An arbitrary contribution to the specific heat, ci ;
is related to a corresponding contribution to the
thermal expansion, bi ; by a so-called Gruneisen
.

relation
Gi ¼ V bi =kci :

ð14Þ


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

Here V is the molar volume, k the compressibility
and Gi the appropriate Gruneisen
.
parameter.
For the electronic Gruneisen
.
parameter we have
(see Eqs. (10) and (11))
Ge ¼ Va=kg:

ð15Þ

Using Eqs. (12) and (13) for the lattice Gruneisen
.
parameter, we obtain
Gph ¼ Vb=9rRk:

ð16Þ

In treating the magnetic contributions to the
specific heat and to the thermal expansion, a
straightforward approach is to calculate an effective Gruneisen

.
parameter, Geff ; by the relation
Geff ¼ V bm =kcm

ð17Þ

and to follow its variation with temperature. A
pronounced temperature dependence of Geff ðTÞ
indicates the presence of several contributions.
ErCu2 can 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
cm ¼ clr þ ccf

and

bm ¼ blr þ 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 Gruneisen
.
parameters Gi ; we have

X
Geff ¼
fi Gi ; i ¼ lr; cf:
ð19Þ
i

Here fi ¼ ci =cm : For any choice of Glr and Gcf ; the
separated contributions can be calculated as
Geff À Gcf
cm ;
Glr À Gcf

ð20Þ

ccf ¼

Glr À Geff
cm ;
Glr À Gcf

ð21Þ

blr ¼

1 À Gcf =Geff
b
1 À Gcf =Glr m

ð22Þ

Gcf 1 À Geff =Glr
bcf ¼
b :
Geff 1 À Gcf =Glr m

ð23Þ

clr ¼

93

This analysis and its application to ErCu2 is
described in Refs. [8,9]. Brommer and Franse
[10] have generalized this method, including Gruneisen
.
relations between specific-heat and linearexpansion contributions, as well as focusing
attention to the criteria to decide whether the
chosen Gi parameters can be considered as genuine
Gruneisen
.
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
E ¼ K1 sin2 y þ K2 sin4 y:

ð24Þ

Here y is the polar angle of the magnetization with
respect to the c-axis. K1 and K2 are the anisotropy
constants. Minimizing anisotropy energy (24) with
respect to y gives the orientation of the magnetization. A sudden change from easy axis to easy plane
may occur. However, when, for K2 > 0; K1 changes
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
sin2 y ¼ ÀK1 =2K2 :

ð25Þ

For a deeper understanding of the spin-reorientation phenomena, one has to consider a microscopic 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
HR ¼ HCEF þ Hex :

ð26Þ

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
Hex ¼ gmB JB m ¼ 2ðg À 1ÞmB JB ex ;

ð27Þ

where B m is the molecular field acting on the rareearth magnetic moment which is related to the


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

94

exchange field Bex acting on the rare-earth spin by
Bm ¼ ½2ðg À 1Þ=gŠB ex : J and g are the total angular
momentum and the Lande! 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
Bm ¼ ½2ðg À 1Þ=gŠnRT /M T S:

ð28Þ

Here /M T S is the average transition-metal
sublattice magnetization.
The relation between K1R and Bm
n is [13]
K1R ¼ À ð3=2ÞB02 /O02 S À 5B04 /O04 S
À ð21=2ÞB06 /O06 S:
/Om
nS

ð29Þ

are the thermal averages of the
Here
Stevens operators.
The ground state of the 4f ion is calculated by
diagonalizing Hamiltonian (26). The temperature
dependence of the rare-earth sublattice anisotropy,
K1R ðxÞ; which normally dominates, is then calculated according to Eq. (29) with K1R ðxÞ ¼
ð1 À xÞK1R ð0Þ; where K1R ðxÞ and K1R ð0Þ are the
rare-earth sublattice anisotropy constants for the
substituted and unsubstituted compounds, respectively. In the case of R2Fe14B compounds, in this
work K1R has been calculated as K1R ¼ E>c À E8c ;
where E>c and E8c represent the ground state
energy of Hamiltonian (26) for B m being perpendicular 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 K1T is taken from the study of
the isostructural compound in which R is nonmagnetic.
In case the data on the temperature dependence of the angle y in aligned powder samples
are available, the analysis is proceeded as
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 perpendicular 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:
Hex ¼ gmB Bm ðJ z cos y þ J y sin yÞ:

ð30Þ

The rare-earth energy is obtained by diagonalization of Hamiltonian (26) and by calculating the
partition function Zðy; TÞ: The rare-earth energy is
given by
FR ðy; TÞ ¼ ÀkB T ln Zðy; TÞ:

ð31Þ

For a mixed system such as R1ÀxRx0 Fe11Ti, the
total free energy is expressed as
F ðy; TÞ ¼ ð1 À xÞFR ðy; TÞ þ xFR0 ðy; TÞ
þ K1T sin2 y:

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 K1T is again taken from the study of
an isostructural compound in which R is nonmagnetic. 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 temperature dependence of the magnetic structure is

determined.

3. CEF effect in RCu2 compounds
The RCu2 intermetallic compounds have an
orthorhombic CeCu2 structure. Early magnetization and magnetic susceptibility measurements
performed by Hashimoto et al. [15] and Hashimoto [16] demonstrated the importance of the CEF in
these compounds. During the last decade substantial progress has been achieved in the study of
the magnetic properties of the RCu2 compounds.
It is of interest to make an attempt to see some
systematic behavior by comparing different compounds 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

CeCu2 is 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 secondorder CEF parameters for PrCu2: B02 ¼ 4:27 K and
B22 ¼ 2:97 K. By using a point-charge model,
Hashimoto et al. [19] have also calculated the
values of B02 and B22 and arrived at B02 cal ¼ 4:1 K
and B22 cal ¼ 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
here the set of CEF parameters for PrCu2,
which has been derived by Settai et al. [22]:
B20 ¼ 4:93 K, B22 ¼ 3:50 K, B04 ¼ 5:08 Â 10À2 K,
B24 ¼ 5:02 Â 10À2 K, B44 ¼ À3:82 Â 10À1 K, B06 ¼
À1:33 Â 10À3 K,
B26 ¼ À1:80 Â 10À2 K,
B46 ¼
À2
6
À2
À2:98 Â 10 K, and B6 ¼ À4:97 Â 10 K.
For the NdCu2 compound the second-order
CEF parameters were first estimated by Hashimoto et al. [15] from measurements of the paramagnetic susceptibility in single-crystalline sample:
B02 ¼ 0:8 K and B22 ¼ 1:1 K. The values of
B02 cal ¼ 1:17 K and B22 cal ¼ 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:
B20 ¼ 1:35 K, B22 ¼ 1:56 K, B04 ¼ 2:23 Â 10À2 K,

B24 ¼ 1:01 Â 10À2 K, B44 ¼ 1:96 Â 10À2 K, B06 ¼
5:52 Â 10À4 K, B26 ¼ 1:35 Â 10À4 K, B46 ¼ 4:89Â
10À4 K, and B66 ¼ 4:25 Â 10À3 K.
Gratz et al. [24] have shown that the coefficient
of thermal expansion of SmCu2 exhibits a minimum 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

95

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
paramagnetic Curie temperatures, Hashimoto
et al. [19] have estimated the following values for
the second-order CEF parameters: B02 ¼ 1:23 K
and B22 ¼ 1:23 K. These authors have also calculated the second-order CEF parameters on the
basis of the point-charge model. Their calculated
values are: B02 cal ¼ 1:35 K and B22 cal ¼ 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
Gruneisen
.
parameters. For the DyCu2 compound,
also from the values of the paramagnetic Curie
temperatures, Hashimoto et al. [19] have estimated
values for the second-order CEF parameters:
B02 ¼ 0:43 K and B22 ¼ 0:72 K. A point-charge
calculation by the same authors results in
B02 cal ¼ 0:89 K and B22 cal ¼ 0:71 K, in satisfactory
agreement with the experimental values. Specific
heat and thermal expansion of DyCu2 have been
measured by Luong et al. [7]. Kimura et al. [25]
have calculated the specific heat of DyCu2 in terms
of the molecular-field model including CEF interaction. 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
B22 experimentally obtained by Hashimoto et al.
[19] and B04 equal to 2.37 Â 10À3 K. By calculating
the magnetic susceptibility and the magnetization 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 following: B02 ¼ 0:708 K, B22 ¼ 0:99 K, B04 ¼ À0:28Â
10À4 K, B24 ¼ 2:37 Â 10À2 K, B44 ¼ 0:26 Â 10À5 K,


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

96

B06 ¼1:1 Â 10À5 K, B26 ¼ 0:79 Â 10À5 K, B46 ¼ 1:6Â
10À4 K, B66 ¼ À1:7 Â 10À4 K.
Like in other RCu2 compounds, from the values
of the paramagnetic Curie temperatures, Hashimoto et al. [19] have estimated the following
second-order CEF parameters for HoCu2:
B02 ¼ 0:14 K and B22 ¼ 0:12 K. The values of
B02 cal ¼ 0:28 K and B22 cal ¼ 0:23 K were obtained
by Hashimoto et al. [19] on the basis of the pointcharge model. The CEF effect in ErCu2 will be
discussed in detail below. The CEF effects in
TmCu2 have 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
A22 for the RCu2 compounds. These coefficients are
related to the CEF parameters B02 and B22 by (see
Section 2)
A02 ¼ B02 =aJ /r24f S;
A22 ¼ B22 =aJ /r24f S:

ð33Þ

Here, the values for the quantity aJ /r24f S are
!
taken from Franse and Radwanski
[4]. To our
knowledge, CEF parameters for SmCu2 are not
available. Rather low values of A02 and A22 have
Table 1
CEF coefficients, in units of KaÀ2
0 ; for the RCu2 compounds
R

A20

A22

Ref.

Pr

À168
À195.3
À112
À188
À148
À85.7
À141
À83.5
À190
À153

À134.7
À134.7

À117
À138.7
À154
À218
À148
À143.4
À197
À71.6
À194
À120
À186.4
À176.3

[16]
[22]
[16]
[23]
[19]
[19]
[27]
[19]
[19]
[28]
[30]
[29]

Nd

Tb
Dy
Ho
Er
Tm

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 A02 and A22 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 RCu2 compounds. At the
same time, there is no reason why higher-order
CEF parameters should be neglected. The necessity 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. Analyzing 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(Cu1ÀxNix)2.
We discuss below in detail the CEF effect in
ErCu2. Luong et al. [7] reported on the specific
heat and thermal expansion of RCu2 compounds
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 Gruneisen
.
relations. 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
DCexp curve, shown in Fig. 1. Luong et al. [7]


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

Fig. 1. Calculated and experimental results for the CEF
contribution to the specific heat of ErCu2. DC1 and DC2 curves
are calculated using CEF data in Refs. [19,28], respectively.

DCexp (from Refs. [7,8]) is obtained from specific heat and
thermal expansion measurements.

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, B20 ¼ À0:35 K and
B22 ¼ À0:36 K, obtained by Hashimoto et al. [19]
from an analysis of the paramagnetic susceptibility, the energy levels in ErCu2 have been calculated, and a splitting of 13 K between the two
lowest-order doublets has been derived [8]. Apparently, 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
earlier, due to the low symmetry in RCu2
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 Mossbauer
spectroscopy. 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: B20 ¼ À0:28 K, B22 ¼ À0:22 K, B04 ¼ À0:30Â
10À2 K, B24 ¼ À0:14 Â 10À2 K, B44 ¼ 0:30 Â 10À2 K,
B06 ¼ À0:20 Â 10À4 K, B26 ¼ À0:47 Â 10À4 K, B46 ¼

À0:97 Â 10À4 K, and B66 ¼ À2:96 Â 10À4 K.

97

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 experiments. 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 DC2 has 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 phonon) 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 YCu2 is
taken as the non-f reference material instead of
LaCu2 because LaCu2 possesses a different crystallographic structure and crystallizes in the hexagonal AlB2-type of structure. A value of 194 K for
the Debye temperature of ErCu2, yD(ErCu2), was
derived [34]. Using the more sophisticated approach 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


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

98

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
ErCu2 is 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 RCu2 compounds is
that the values of the Ne! el temperature for these
compounds are not simply proportional to the de
Gennes factor ðgJ À 1Þ2 JðJ þ 1Þ and reach a
maximum for TbCu2. This fact suggests that the
Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction alone is not sufficient to fully understand
the magnetic interactions in the RCu2 compounds.
In spite of a substantial progress in the study of the
magnetic properties of the RCu2 compounds, 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 Ne! el
temperature of the RCu2 compounds (R=Tb–
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 À 1Þ2 /Jz2 ðTM ÞSCEF ;
/Jz2 ðTM ÞSCEF

ð34Þ

where
is the expectation value of Jz2
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 calculation of the Ne! el temperatures, TN ; of the RCu2
compounds expression (34) is used, in which TM
stands for TN : For evaluating G in these compounds, 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 B02 and B22 were taken for TbCu2,
DyCu2 and HoCu2 from Ref. [19], ErCu2 from
Ref. [28] and TmCu2 from 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 Hamiltonian (7) with only the two lowest-order terms. For
the RCu2 compounds with R=Tb, Dy and Ho we
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=2ÞðÀB02 À B22 ÞO02
þ ð1=2Þð3B02 À B22 ÞO22 :

ð35Þ

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 TN over the de Gennes
values in the RCu2 compounds. Moreover, calculations predict that TbCu2 has the highest Ne! el
temperature, in good agreement with experiments.
The calculated values for the Ne! el temperatures
across the series are in good agreement with the
experimental ones. Calculations gave the value of
TN for HoCu2 somewhat higher than the one
obtained from experiments. Luong et al. [37] tried
also to derive the values for the Ne! el 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

99

Table 2
Values for the N!eel temperatures in the heavy RCu2
compounds
R

TN exp (K)

TN cal (K)

Gd

40 [7]
41 [39]
42 [40]

Tb

54 [39]
53.5 [19]
48.5 [7]
48 [42]

46.4

Dy

24 [39]
31.4 [19]
26.7 [7]
27 [28]

28.7

Ho

9 [39]
9.8 [19]
9.6 [7]
11 [43]

17.2

Er

11 [39]
13.5 [19]
11.5 [7]

9.1, 11.7a

Tm

6.3 [44]

4.3, 6.7a

a

N!eel temperature predicted by the full CEF Hamiltonian
with the Bm
n sets from Refs. [28,29] for ErCu2 and TmCu2,
respectively.

CEF terms only. Thus, the magnetic ordering
temperatures in the RCu2 compounds can be
explained by a combination of the RKKY interaction and CEF effects.

4. CEF effect in R2Fe14B compounds
In the tetragonal R2Fe14B compounds, the rareearth sublattice consists of two inequivalent rareearth 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 aJ is positive [45]. In
the compounds with Sm, Er, Tm, and Yb, for
which aJ is positive, a competition between rareearth and iron sublattice anisotropies is expected
and can lead to a spin reorientation. Such

Fig. 2. Comparison of experimental and calculated N!eel
temperatures for the RCu2 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.

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 B02 is assumed to be dominant in
the CEF Hamiltonian. The value of B02 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 magnetocrystalline anisotropy we use the following


100

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

Hamiltonian which is simplified from (26) as
H ¼ B02 O02 þ gmB J Á Bm :

ð36Þ

The procedure of analysis has been described in
Section 2. The temperature dependence of the iron
sublattice anisotropy, K1Fe ; is deduced from the
magnetocrystalline anisotropy study of Y2Fe14B
by Sagawa et al. [52]. Results of calculations are
presented in Figs. 3 and 4. In these figures we also
plot the temperature dependence of the experimental erbium and thullium sublattice anisotropies, KlEr and KlTm respectively, which we derived
from the study of Y2F14B [52] and of Er2Fe14B and
Tm2Fe14B [46]. From these figures, it is clear that
in the R2Fe14B compounds (R=Er, Tm, Yb) the
rare-earth and iron anisotropies are competing
and lead to spin reorientations. The deduced
spin-reorientation temperatures are presented in
Table 3. The experimental and calculated values of
TSR reported by several groups are also shown in
Table 3. By using a simple Hamiltonian (36) with
only a second-order CEF parameter, B02 we could
attain good fits to the experimental rare-earth
anisotropy constants K1R for Sm, Er, and Tm

Fig. 4. Temperature dependence of rare earth and iron
anisotropy coefficients K1 in Tm2Fe14B and Yb2Fe14B. K1Fe as
in Fig. 3. For the experimental K1Tm ðTÞ curve see text. - - - - experimental, —— calculated.

Table 3
CEF coefficients Am
n ; molecular-field parameters gmB Bm ; and
spin-reorientation temperatures TSR in the R2Fe14B compounds
A20 (K) A04 (K) gmB Bm (K) TSR (K)

Fig. 3. Temperature dependence of erbium and iron anisotropy
coefficients K1 in Er2Fe14B. K1Fe has been taken from Ref. [51].
For the experimental K1Er ðTÞ curve see text. - - - - - experimental,
—— calculated.

Nd2Fe14B

175

Er2Fe14B

175

65

91.36
67

Tm2Fe14B 130

55.8

Yb2Fe14B

56

47.9

Sm2Fe14B

310

290

124
122
320
316
316
350
310
310
360
380
118
115
210
—

Exp. [49]
Cal. [49]
Cal. [48]
Exp. [49]
Exp. [46]

Cal. [53]
Cal. [48]
Exp. [46]
Cal. [53]
Cal. [54]
Cal. [48]
Exp. [47]
Cal. [53]

compounds in a wide temperature range, and
reproduce the correct values of the spin-reorientation temperatures in the compounds with R=Er,
Tm and Yb. The CEF coefficients A02 deduced


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

from the parameters B02 for the compounds
investigated are presented in Table 3 together with
the molecular-field parameters. As can be seen
from this table, A02 in Tm2Fe14B is somewhat
smaller than that in Er2Fe14B, while A02 in
Yb2Fe14B is significantly small in comparison with
that in Er2Fe14B. Such a behavior of A02 in the
R2Fe14B series is also reported by Cadogan et al.
[51].
In the case of Nd, the following Hamiltonian
was employed [49]:
H¼

B02 O02

þ

B04 O04

þ gmB J Á B m :

ð37Þ

From the values of B02 and Bex obtained for
Er2Fe14B (see Table 3), we derived, after scaling to
the Nd case, the corresponding values for B02 and
the molecular-field parameter for Nd2Fe14B. The
value of B04 has been chosen in order to reproduce
the observed spin-reorientation temperature in
Nd2Fe14B. The CEF coefficients A02 and A04 for
this compound are presented in Table 3.

5. CEF effect in RFe11Ti compounds
We have studied the effect of Y substitution on
the spin reorientation in NdFe11Ti [55] by magnetization measurements on bulk and oriented
powder samples, and by measurements of the AC
susceptibility on bulk samples as a function of
temperature. The spin-reorientation temperatures,
TSR, were determined. No data for the temperature dependence of the magnetic structure are
available.
The method of analysis, based on the expression
for the anisotropy constant (Eq. (29)), was applied
to discuss the spin-reorientation phenomena in the
Nd1ÀxYxFe11Ti compounds. For calculating K1R

according to Eq. (29) we have chosen the set of
coefficients Am
n presented in Table 4. The used
molecular-field parameter gmB Bm value, expressed
as 161 K, was derived from the value of nRT given
by Hu et al. [56]. The temperature dependence of
the ion sublattice anisotropy K1Fe ð¼ K1T Þ is taken
from the study of YFe11Ti by the same authors
[56]. The calculated values of the spin-reorientation temperature are in good agreement with the
experimental ones (see Refs. [55,57] and Table 5).

101

Table 4
Àn
CEF coefficients Am
n ; in unit of Ka0 ; for the R1ÀxYxFe11Ti
compounds
R

A02

A04

A06

A44

A46

Nd
Tb
Dy

À57
À54.1
À32

À14.3
À0.9
À9

0
0
3.2

0
0
105

0
0
0

Table 5
Spin-reorientation temperature
R1ÀxYxFe11Ti compounds

TSR

in

the

series

of

R

x

exp
(K)
TSR

calc
TSR
(K)

Nd
Nd
Nd
Nd
Tb
Tb
Tb
Dy
Dy
Dy

Dy
Dy

0.0
0.2
0.4
0.6
0.0
0.2
0.4
0.0
0.2
0.4
0.6
0.8

187
140
114
70
325
235
120
195
171
141
107
73

180

152
120
75
328
235
70
196
172
142
110
76

For the Dy1ÀxYxFe11Ti compounds, TSR stands for the
temperature of the spin reorientation of the axis-to-cone type.

and
In
the
case
of
Tb1ÀxYxFe11Ti
Dy1ÀxYxFe11Ti compounds, we have measured
the temperature dependence of the cone angle y
[58,59]. The experimental results were analyzed
based on a method of calculating the free energy
(Eq. (32)). In our analysis of the Tb1ÀxYxFe11Ti
and Dy1ÀxYxFe11Ti data, the assumption is made
that the exchange interaction stabilizes the ferrimagnetic collinearity between the magnetic moments of the transition-metal and rare-earth
sublattices. In fact, analyzing the data of a
single-crystalline DyFe11Ti sample, Hu et al. [60]

have calculated the canting angle between the iron
and dysprosium sublattice magnetizations. These
authors have reported that the maximum value of
this canting angle in DyFe11Ti is about 2.51 for the
intermediate values of the cone angle. The CEF
coefficients Am
n used for fitting the experimental
results for Tb1ÀxYxFe11Ti and Dy1ÀxYxFe11Ti
compounds are presented in Table 4 [61]. Values


102

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

for the molecular-field parameter gmB Bm are
expressed as 128.7 and 85.6 K for Tb1ÀxYxFe11Ti
and Dy1ÀxYxFe11Ti, respectively. The temperature
dependence of the iron sublattice anisotropy, K1Fe ;
was again taken from the study of YFe11Ti [56].
The calculated temperature dependencies of the
angle y for the Dy1ÀxYxFe11Ti and Tb1ÀxYxFe11Ti compounds are shown together with the
experimental ones in Figs. 5 and 6, respectively.
As can be seen from Fig. 5, for the Dy1ÀxYxFe11Ti
compounds, good agreement between calculations
and experiments has been obtained. The calculated
values for the spin-reorientation temperature are
in good agreement with those obtained from the
experiments for all substituted compounds investigated (see Table 5). For the Tb1ÀxYxFe11Ti
compounds the calculated magnetic structure is

also in agreement with the experimental one (see
Fig. 6). The calculated values for the spin-reorientation temperature are basically in agreement
with the experimental ones for the substituted
compounds exhibiting a spin reorientation (see
Ref. [59] and Table 5).
Inspection of Table 4 shows that the leading
CEF coefficient A02 obtained for the three investigated systems has the same sign and is of the same
order of magnitude. However, for reproducing the
experimental results, higher-order CEF coefficients should also be included. From an analysis
of single-crystal data of DyFe11Ti, Hu et al. [60]

Fig. 5. Experimental and calculated thermal dependence of the
cone angle y between the magnetization direction and the c-axis
for the Dy1ÀxYxFe11Ti compounds [58].

Fig. 6. Experimental (symbols) and calculated (full curves)
thermal dependence of the cone angle y between the magnetization direction and the c-axis for the Tb1ÀxYxFe11Ti compounds
[59].

A04 ¼ À12:4 KaÀ4
deduced: A02 ¼ À32:3 KaÀ2
0 ;
0 ;
4
À4
0
À6
A4 ¼ 118 Ka0 ; A6 ¼ 2:56 Ka0
and
A46 ¼

0:64 KaÀ6
0 : We have used this set of CEF
coefficients to study the spin-reorientation phenomena in the systems considered. For the
Nd1ÀxYxFe11Ti compounds, the calculated values
for the spin-reorientation temperature, by using
this set of Am
n ; are somewhat different from those
obtained in our experiments [55]. Moreover, the
calculations do not predict a spin reorientation in
the Y-substituted compounds with x > 4; which is
not consistent with our experimental data. The set
of CEF coefficients given in Table 4 predicts that
a spin reorientation occurs in the compound
with x ¼ 0:6 (in agreement with experiments). No
spin reorientation is expected in the compound
with x ¼ 0:8 since for this compound the iron
anisotropy is dominant at all temperatures.
Measurements on Nd0.8Y0.2Fe11Ti support this
expectation, indeed, see Ref. [57]. For
Dy1ÀxYxFe11Ti compounds, calculations performed in the model described above, using the
set of Am
n values given by Hu et al. [60], gave spinreorientation temperatures which are somewhat
different from our experimental ones. In addition,
this set of Am
n of Hu et al. [60] does not reproduce
the spin-reorientation behavior in TbFe11Ti.
Ivanova et al. [61] have reported on the magnetic anisotropy and spin-reorientation transitions


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

103

personality. The author would like to thank Prof.
T.D. Hien, Prof. N.P. Thuy. Dr. L.T. Tai and
Dr. P.H. Quang for close collaboration throughout the course of this work. The author wishes to
thank Prof. K. Krop and Dr. D. Givord for
valuable comments, discussions and help, Prof. Y.
Onuki and Prof. K. Sugiyama for providing their
work and discussions. The author is grateful to
all members of the Cryogenic Laboratory and
Dr. F.F. Bekker for cooperation, Dr. P.E.
Brommer for reading and valuable comments on
the manuscript.
Fig. 7. Calculated temperature dependence of K1Tb (solid line)
of the terbium sublattice in TbFe11Ti. Experimental K1Fe values
quoted from Hu et al. [56] and experimental K1Tb (dashed line)
values extracted from Ivanova et al. [62] are also shown.

in TbFe11ÀxCoxTi single crystals. These authors
have derived the temperature dependence of K1 for
TbFe11Ti. From these results and the iron
sublattice anisotropy K1Fe derived by Hu et al.
[56], we are able to separate out the magnetic
anisotropy of the terbium sublattice K1Tb : The
temperature dependence of K1Tb deduced from
experiments in this manner is shown in Fig. 7.
Using the set of CEF parameters in Table 4 for
TbFe11Ti, we have calculated the temperature
dependence of K1Tb in this compound. The

calculated temperature dependence of K1Tb is also
plotted in Fig. 7. The agreement between the
calculated K1Tb ðTÞ curve and experimental data is
good in a wide temperature range.
Our results show that the study of spinreorientation phenomena in intermetallic compounds with substitution of a non-magnetic rare
earth for a magnetic one is very useful in obtaining
information on CEF parameters. This is particularly valuable when data on single crystal are
lacking.

Acknowledgements
The author is greatly indebted to Prof. J.J.M.
Franse for all his support and encouragement for
many years as well as throughout the course of this
work. The author deeply admires his physics and

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