JOURNAL OF APPLIED PHYSICSVOLUME 91, NUMBER 1015 MAY 2002

Modeling magnetostriction in RCu2 compounds using McPhase
M. Rotter, M. Doerr, and M. Loewenhaupt
Institut für Angewandte Physik, TU Dresden, D-01062 Dresden, Germany

P. Svoboda
Department of Electron Systems, Charles University, 12116 Prague, The Czech Republic

In RCu2 compounds 共R⫽rare earth兲 magnetostriction results from two different sources, the single
ion crystal field contribution and the two ion exchange interaction. The crystal field contribution canalso be observed above the magnetic ordering temperature and leads to giant magnetostriction inhigh magnetic fields at some members of the series. The exchange contribution has to be consideredadditionally when describing the magnetostriction in the magnetically ordered state. A new featureof the mean field–Monte Carlo simulation program McPhase 共http://www.mcphase.de兲 allows to
calculate the magnetic phase diagram and model the exchange contribution. We present new data ofthermal expansion and the longitudinal and transversal magnetostriction of NdCu2 with magnetic
field applied along the a, b, and c direction of the orthorhombic single crystal. These data arecompared to the results of a McPhase simulation, which is based on exchange parameters derived
from measurements of the magnetic excitations. The magnetoelastic interaction is compared to thecase of GdCu2 . The analysis can be extended to other RCu2 compounds. © 2002 American
Institute of Physics. 关DOI: 10.1063/1.1449363兴

I. MAGNETOELASTIC INTERACTIONS IN RCu2COMPOUNDS

According to Ref. 1 the static and dynamic magnetic
properties of RCu2 compounds in the magnetically orderedstate can be described by a Hamiltonian, which is the sum ofcrystal field 共CF兲, exchange 共EX兲, and Zeeman contributions
and the elastic energy
H⫽HCF⫹HEX⫺

兺i g J ␮ B H•Ji⫹E el .

兺 B ml 共 ⑀ 兲 O ml 共 Ji兲 ,lm,i

HEX⫽⫺

1
J␣ J 共 ⑀ ,i⫺j兲 Jj␤ ,2 ij, ␣␤ i ␣␤兺

1⑀ ⑀c ␣␤␣ ⬘ ␤ ⬘ .
E el⫽
2 ␣␤␣ ⬘ ␤ ⬘ ␣␤ ␣ ⬘ ␤ ⬘兺EX⑀ ␣␣⫽1␤
关 K ␣␤ 具 J␤i J␤i 典 T,H⫹L ␣␤ 具 J␤i Ji⫹a/2⫹b/2⫹c/2
典 T,HN ␤i

兺i 关 A ␣ 具 O 02共 Ji兲 典 T,H⫹B ␣ 具 O 22共 Ji兲 典 T,H兴 ,共1.5兲

兺共1.6兲具 典 T,H denote thermal expectation values, i the position
vectors of the rare earth, a, b, and c the orthorhombic latticevectors.
Using a new version of the McPhase simulation program
it is possible to calculate the static correlation functions
␤具 Ji␤ Ji⬘ 典 T,H in Eq. 共1.6兲. The complete H and T dependence ofthe strain is determined by the expectation values 具 典 T,H .
The model used for the simulation of the magnetically ordered phases has been described in Ref. 6 for external fields
parallel to b. Here we extend it to fields in a and c using themagnetic exchange determined from the dispersion of the共1.1兲

共1.2兲

共1.3兲
TABLE I. Magnetoelastic parameters used in the evaluation of the strains by
Eq. 共1.5兲.

共1.4兲

From this model the diagonal components of the strain
tensor ⑀ can be derived. This is done for the special case oforthorhombic RCu2 compounds with ferromagnetic bc
planes 共such as NdCu2 兲 and considering the strain dependence of the CF for l⫽2 and of the EX up to the next nearest
bc plane. The following expressions are calculated using amean field model, expanding the CF and EX parameters linear in the strain tensor ⑀ and minimizing the free energy withrespect to ⑀ similar to4,5
CFEX⑀ ␣␣ ⫽ ⑀ ␣␣⫹ ⑀ ␣␣,0021-8979/2002/91(10)/8885/3/$19.00

1N␤⫹M ␣␤ 具 J␤i Ji⫹a典 T,H兴 .

In order to describe magnetoelastic properties the strain
dependence of the CF and EX part has to be considered 共forthe notation see Ref. 3兲:HCF⫽

CF⑀ ␣␣⫽

␣⫽abc

A ␣a
B ␣a
K ␣a
K ␣b
K ␣c
L ␣a
L ␣b
L ␣c
M ␣a
M ␣b
M ␣c

⫺5⫻10⫺5⫺5⫻10⫺5⫹4⫻10⫺4⫹5⫻10⫺6⫺1⫻10⫺4⫹7⫻10⫺5⫹3⫻10⫺4⫺1⫻10⫺4⫺6⫻10⫺5⫺4⫻10⫺4⫺8⫻10⫺6⫹2⫻10⫺4⫹2⫻10⫺5⫺4⫻10⫺5⫺1.8⫻10⫺4a8885⫹1.5⫻10⫺5⫹6⫻10⫺6⫺1.3⫻10⫺5⫺1.9⫻10⫺5

Values taken from Ref. 5.
© 2002 American Institute of Physics

8886

J. Appl. Phys., Vol. 91, No. 10, 15 May 2002

Rotter et al.

FIG. 1. Longitudinal and transversal magnetostriction of a NdCu2 single crystal. The symbols indicate the measured data. The solid lines 共—兲 correspond toCFresults of the McPhase calculation taking into account both the exchange contribution ⑀ EX
␣␣ and the CF contribution ⑀ ␣␣ to the strain. For comparison the CF
CFcontribution ⑀ CFisshown
separately
共longdashes–––兲.
Shortdashes
共---兲indicate⑀calculated
without
McPhase,i.e., considering only the paramagnetic␣␣␣␣
CF eigenstates 共see text兲.

CFmagnetic excitations.1 The CF contribution ⑀ ␣␣in Eq. 共1.5兲
was calculated using the mean fields calculated with
McPhase and the CF parameters given in Refs. 1 and 5.

II. RESULTS AND DISCUSSION

At low temperatures the contribution of phonons andelectrons to the strain is negligible and the magnetostrictionof NdCu2 can be calculated using the model 共1.5兲. Table I
shows the magnetoelastic parameters used in our evaluationof Eq. 共1.5兲. The values for A and B have been taken fromRef. 5. The magnetoelastic constants K,L, and M have been
determined to give a reasonable quantitative adaptation of
the calculated expressions 共1.5兲 to the experimental data. Interactions beyond the nearest neighbor 共i.e., L,M ⫽0兲 have
only been taken into account when necessary in orderto describe the experimental data. Magnetostriction data
GdCu
on GdCu2 共see Ref. 2兲 yield K a ␣ 2 ⫽⫹17⫻10⫺5 ,GdCu
GdCuK b ␣ 2 ⫽⫺1⫻10⫺5 , and K c ␣ 2 ⫽⫺1⫻10⫺5 ( ␣ ⫽a,b,c).
Comparing these values to NdCu2 we infer that in the heavy
RCu2 compounds the exchange contribution to the magnetostriction is of comparable magnitude. Note that in the case ofGdCu2 K ␣ a ⫽K ␣ b ⫽K ␣ c ( ␣ ⫽a,b,c) in contrast to NdCu2 .
Probably the orbital momentum (L⫽0) of the Nd3⫹ ionleads to the strong anisotropy in the derivatives of the exchange tensor J␣␤ with respect to the strain ⑀.
The results of the calculation are compared in Figs. 1and 2 to new experimental data measured by capacitancedilatometry on a single crystal. The magnetostriction data arein good agreement with a previous study of ⌬b/b for magnetic field in the b direction.7 The solid lines correspond toresults of the McPhase calculation taking into account both,EXCF
the exchange ( ⑀ ␣␣) and CF ( ⑀ ␣␣) contribution to the strain.

CFFor comparison the CF contribution ⑀ ␣␣is shown separately
共long dashes兲. Although magnetic order has been taken intoaccount, this contribution alone cannot describe our experimental data adequately. In order to show the influence ofCF
has also been calculated assuming thatmagnetic order, ⑀ ␣␣
the system stays paramagnetic down to zero temperature
关i.e., by putting all J␣␤ ( ⑀ ,i⫺j)⫽0 in Eq. 共1.3兲 and calculating the 具 O ml (Ji) 典 T,H in Eq. 共1.5兲 using the CF eigenstates兴.

FIG. 2. Thermal expansion of a NdCu2 single crystal in zero field in thecrystallographic a, b, and c direction. For explanations of symbols see Fig.
1 and text.

Rotter et al.

J. Appl. Phys., Vol. 91, No. 10, 15 May 2002

The results of this calculation are shown by short dashes inFigs. 1 and 2.
The onset of the axis conversion8 at around 8 T makes
the interpretation of magnetostriction for fields in the c direction difficult 共Fig. 1兲 and therefore values given for K ␣ c
are less reliable than the other magnetoelastic constants
given in Table I.
In conclusion, the present analysis shows that it is possible to model magnetostriction in compounds with very
complex magnetic phase diagrams using McPhase. Goodagreement between experiment and theory can be achieved,
if the parameters in the Hamiltonian are well determined.
Further improvements of ab initio calculations9 may eventually lead to estimates of the magnetoelastic parameters.
In the case of NdCu2 the CF contribution calculated according to Ref. 5 can only describe part of the experimental
data. In the magnetically ordered phase it is important toconsider the exchange contribution to the strain, which canbe calculated by McPhase. The remaining discrepancies
共e.g., the slope of ⌬b/b and ⌬c/c in high fields parallel a兲suggest that possibly magnetoelastic CF contributions forl⬎2 are important. Furthermore, the CF model for small
strains4 is not sufficient to describe the giant magnetostriction for high fields parallel c. This is probably due to theimportance of large quadrupolar interactions which leads to amagnetic axis conversion.88887

In contrast to the rare earth elements9 in the RCu2 series
the exchange contribution to the magnetostriction is of thesame order as the single ion contribution. In order to establish more details of the exchange contribution it is necessary
to extend the analysis of NdCu2 and GdCu2 to other RCu2
compounds.
ACKNOWLEDGMENTS

Part of this work was performed within the program of
the Sonderforschungsbereich 463 共funded by the DeutscheForschungsgemeinschaft兲. One of the authors, P.S. would
like to acknowledge the support of the Grant Agency ofCharles University 共Grant No. 165/01兲.
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