E!ect of the crystalline electric "eld on the NeHel temperatures of RCu compounds Faculty of Physics, Center for Materials Science, National University of Hanoi, 334 Nguyen Trai, Hanoi,
Trang 1* Corresponding author Fax: #84-4-858-94-96.
E-mail address: luong@cms.edu.vn (N.H Luong).
E!ect of the crystalline electric "eld on the NeHel temperatures
of RCu compounds
Faculty of Physics, Center for Materials Science, National University of Hanoi, 334 Nguyen Trai, Hanoi, Viet Nam
Van der Waals-Zeeman Laboratorium, Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, Netherlands
Abstract
Values for the NeHel temperature of the RCu compounds (R"Tb}Tm) have been calculated using a molecular-"eld model including crystalline-electric-"eld e!ect as presented by Noakes and Shenoy The calculated results show that the unusual behavior, at the NeHel temperatures, of these compounds can be explained on the basis of this model 2001 Published by Elsevier Science B.V.
Keywords: Crystal-"eld e!ect; NeHel temperature; Rare-earth intermetallic compounds
The RCu (R"rare-earth) compounds
crystal-lize in the orthorhombic CeCu structure The
magnetic properties of these compounds were "rst
studied by Sherwood et al [1] as early as in 1964
Most of the RCu order antiferromagnetically with
values for the NeHel temperature, ¹,, below 55K
Hashimoto et al [2] performed magnetization and
magnetic susceptibility measurements on
single-crystalline RCu samples A large magnetic
anisot-ropy was found to be present The magnetic
properties of the RCu compounds have been
extensively studied in recent years (see a review
by Luong and Franse [3] and references
therein)
One of the features of the RCu compounds is
that the values of the NeHel temperature for these
compounds are not simply proportional to the de
Gennes factor (gH!1)J(J#1) and reach a
max-imum for TbCu This fact suggests that the Ruder-man}Kittel}Kasuya}Yosida (RKKY) interaction alone is not su$cient to fully understand the mag-netic interactions in the RCu compounds In spite
of a substantial progress in the study of the mag-netic properties of the RCu compounds, the above-mentioned exception to the de Gennes rule remained unexplained
In this paper, we report on our calculations that have been performed in order to explain the anom-alous behavior, at the NeHel temperature, of the RCu compounds (R"Tb}Tm)
Our calculations are based on the model of Noakes and Shenoy [4] When considering only the exchange Hamiltonian, the de Gennes rule can
be derived in the simple molecular-"eld model: the rare-earth-atom Hamiltonian
(where the z-axis is de"ned to be the
ordered-moment axis andC the exchange parameter) leads
to an implicit equation for1JX2, and the ordering
0304-8853/01/$ - see front matter 2001 Published by Elsevier Science B.V.
PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 1 3 9 0 - 1
Trang 2Table 1 Values for the crystalline-electric-"eld parameters and the NeHel temperatures in the heavy RCu compounds
R B (K) B (K) ¹,exp (K) ¹,cal (K)
41 [1]
42 [6]
53.5 [2]
48.5 [5]
48 [14]
31.4 [2]
26.7 [5]
27 [7]
9.8 [2]
9.6 [5]
11 [19]
Er ! 0.28 [7] ! 0.22 [7] 11 [1] 9.1, 11.7
13.5 [2]
11.5 [5]
Tm ! 0.94 [8] ! 1.23 [8] 6.3 [20] 4.3, 6.7
NeHel temperature predicted by the full CEF Hamiltonian with
the B KL sets from Refs [7,8] for ErCu and TmCu, respectively.
temperature is obtained from the small 1JX2 limit
as
¹"2C (gH!1)J(J#1)/3. (2)
When the crystalline-electric-"eld (CEF) e!ects
are signi"cant, the de Gennes behavior is not to be
expected In this case, CEF Hamiltonian should be
added to the H This leads to the following
expression for the ordering temperature:
¹+"2C (gH!1)1JX(¹+)2!#$, (3)
where 1JX(¹+)2!#$ is the expectation value of
J X under the in#uence of CEF Hamiltonian alone
at the temperature ¹+ The exchange parameter, C,
can be evaluated from the ordering temperature of
the Gd compound when modeling a series of
rare-earth compounds, because Gd, an ¸"0 ion, is
essentially una!ected by CEF
For calculating values of the NeHel temperatures,
¹,, of the RCu compounds, expression (3) is used
in which ¹+ stands for ¹, For evaluating C in
these compounds, we take ¹,(GdCu)"¹"
41 K [1,5,6]
In the coordinate system of b"z, c"x and
a"y, the orthorhombic CEF Hamiltonian of
a CeCu-type of structure is given by
H!#$"BO#BO#BO#BO
#B O#BO#BO
#B O#BO, (4)
where B KL are the CEF parameters and OKL the
Stevens equivalent operators As pointed out by
Luong and Franse [3], information about the
cry-stalline-electric-"eld interaction in RCu is not
complete Until now, the full set of CEF parameters
is available only for NdCu, ErCu and TmCu
Nevertheless, the two lowest-order CEF
para-meters B and B have been derived for most of the
RCu compounds (see Ref [3] and references
therein) In our calculations, we "rst used the two
lowest-order terms in the CEF Hamiltonian
Values for B and B were taken for TbCu,
DyCu and HoCu from Ref [2], for ErCu from
Ref [7] and for TmCu from Ref [8]
In TbCu, DyCu and HoCu the magnetic
moments lie along the a-axis, whereas in ErCu
and TmCu the magnetic moments are oriented
along the b-direction [2,9}17] The ¹, values for
ErCu and TmCu were calculated directly using the CEF Hamiltonian (4) with only the two lowest-order terms For the RCu compounds with R"Tb, Dy and Ho, we used the CEF Hamiltonian transformed in the new coordinate system of
a"z, b"x, c"y as follows [18]:
H!#$"()(!B!B)O#()(3B!B)O (5)
The calculated values of ¹, are compared with experimental data in Table 1 and also in Fig 1 As one can see from Table 1 and Fig 1, addition of CEF interaction enhances ¹, over the de Gennes values in the RCu compounds Moreover, calcu-lations predict that TbCu has a highest NeHel tem-perature, in good agreement with experiments The calculated values for the NeHel temperatures across the series are in good agreement with experimental ones Calculations gave the value of ¹, for HoCu
Trang 3Fig 1 Comparison of experimental and calculated NeHel
tem-peratures for the RCu compounds The open circles represent
experimental data The solid circles (solid line) represent
calcu-lations using a CEF Hamiltonian with two lowest-order terms,
and the solid squares * calculations with the full CEF
Hamil-tonian as discussed in the text The dashed line represents the de
Gennes rule.
to be somewhat higher than the one obtained from
experiments
We tried also to derive the values for the NeHel
temperature of ErCu and TmCu with the full
B KL set taken from Refs [7,8], respectively The
results of these calculations using the full CEF
Hamiltonian (4) are also given in Table 1 and in
Fig 1 As it can be seen, the use of the full CEF
Hamiltonian gives better results than the use of the
two lowest-order CEF terms only
In conclusion, the magnetic ordering
temper-atures in the RCu compounds can be explained by
a combination of the RKKY interaction and
cry-stalline-electric-"eld e!ects
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