ARTICLE IN PRESS

Physica B 345 (2004) 231–234

Interpreting magnetic X-ray scattering on Gd-compounds
using the McPhase simulation program
M. Rottera,b,*, A. Schneidewindb, M. Doerrb, M. Loewenhauptb, A.M. El
Massalamic, C. DetlefsdbaInstitut fur. Physikalische Chemie, Universitat. Wien, A-1090 Wien, Austria
Institut fur
Technische Universitat. Festkorperphysik,.
. Dresden, D-01062 Dresden, GermanycInstituto de Fisica-UFRJ, Caixa Postal 68528, 21945-970 Rio de Janeiro, BrazildEuropean Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France

Abstract
Magnetic X-ray scattering experiments on Gd-compounds provide valuable information about the magneticstructure, which is difﬁcult to obtain by neutron scattering due to the large absorption cross-section of Gd. Theinterpretation of the results is usually difﬁcult and sometimes ambiguous, and it would be desirable to have quantitative
calculations in order to verify or disprove theoretical models by the data of scattering experiments. Such model
calculations have been performed for the magnetic properties of the antiferromagnets GdNi2 B2 C and GdCu2 : The
results of these calculations are compared with experimental data from magnetic X-ray scattering experiments.
r 2003 Elsevier B.V. All rights reserved.
PACS: 75.25.+z; 75.30.Gw
Keywords: Magnetic X-ray diffraction; Gd compounds; GdNi2 B2 C; GdCu2

1. Introduction
The magnetic anisotropy of rare-earth compounds usually is caused by the crystal ﬁeld, unlessthe 4f angular momentum is zero ðL ¼ 0Þ such as
in the case of Gd3þ : The small but ﬁnite magnetic
anisotropy of L ¼ 0 rare-earth compounds isreﬂected in the orientation of the moments withrespect to the crystal lattice, which can be.
*Corresponding author. Institut fur. Festkorperphysik,Technische Universit.at Dresden, D-01062 Dresden, Germany. Tel.:+49-351-463-34460; fax: +49-351-463-34671.
E-mail address: rotter@physik.tu-dresden.de (M. Rotter).

determined by magnetic scattering experiments.
Because neutron diffraction on Gd compounds is
difﬁcult due to the large absorption cross-sectionof Gd, magnetic X-ray diffraction (MXD) usingsynchrotron light sources is a welcome alternative.
The moment direction can be investigated using
polarization analysis. The confrontation of thesedata with a quantitative model may lead to newinsights about the magnetic interactions. Here weassume a simple model based on the following
Hamiltonian:
()
iajX
1X
ba CD
H¼
JðijÞJi Jj þJi Jab ðijÞJj ; ð1Þ
2 ij
ab0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.physb.2003.11.061

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M. Rotter et al. / Physica B 345 (2004) 231–234

232

where the anisotropy enters only via the second
term which corresponds to the classical dipole
interaction
JCDab ðijÞ¼
ðgJ mB Þ23ðRai  Raj ÞðRbi  Rbj Þ  dab jRi  Rj j2jRi  Rj j5:

In these equations Ri denotes the lattice vectorof the ith Gd ion, Jia ða ¼ 1; 2; 3Þ the threecomponents of the total angular momentum, gJ
the Lande! factor, and mB the Bohr magneton. In
the following we report the interpretation of MXD
on two noncollinear antiferromagnets using amodel analysis based on the Hamiltonian (1).

2. GdCu2
GdCu2 crystallizes in the orthorhombic
CeCu2 type of structure (space group Imma).
At the Ne! el temperature of TN ¼ 41 K thespeciﬁc heat indicates an equal moment magneticstructure [1]. A noncollinear magnetic structure [2]was found by neutron diffraction. MXD [3] andmSR [4] agree with these results. The magnetostriction was interpreted quantitatively using correlation functions derived from the magnetic
structure [5].
We have used MXD with polarization analysis
to investigate the temperature dependence of themagnetic structure. In order to interpret our data amodel calculation based on the Hamiltonian (1)was performed using the McPhase program [6,www.mcphase.de]. From the diagonal, but anisotropic exchange parameters found for NdCu2 [7]only the isotropic part was taken and de Gennes
scaled in order to obtain the exchange interaction
for GdCu2
PNdCu2
ðijÞ
JGdCu2 ðijÞa¼a;b;c Jaa
¼
:
ð2Þ
22
GdNdðgJ  1Þ3ðgJ  1Þ
In addition, the classical dipolar exchange for1584 neighbors up to 3 nm was taken into accountin the McPhase calculation.
At low temperatures, the calculated magneticstructure differs by a small tilt of the magnetic
moments (8 in the AC plane) from the structuresuggested by the neutron scattering experiments

Fig. 1. GdCu2 —temperature-dependent measured MXD intensities on ð3:6667  1 1Þ: Open (full) symbols correspond topp ðpsÞ MXD intensities, respectively. Lines correspond to thecalculated temperature dependence of the squared FT of themagnetic moment conﬁguration.

[2]. Such a tilt cannot be deduced with the neutrondiffraction but was already suggested by mSR [4].
The temperature dependence of the Fourier transform (FT) of the magnetic moments MðtÞ has been
computed and is compared in Fig. 1 to thetemperature dependence of the MXD intensity
measured on the main propagation vector with abin the scattering plane. The temperature dependence of the ps intensity (sensitive to both M100and M010 ) agrees well with the calculated momentcomponent along a ðM100 Þ; because the component M010 is zero at all temperatures (both in thecalculation and as determined from the MXD
results in the ac scattering plane [8]). The calculation predicts M100 ¼ M001 at all temperatures.
Therefore it is difﬁcult to understand the pp data,
which indicate, that M001 decreases below 10 K:
Additional sources of anisotropy, other than thedipolar interaction have to be considered so as toaccount for this observation, which is also
supported by a sharp increase of the susceptibility
in c-direction below 10 K [3].

3. GdNi2 B2 C
GdNi2 B2 C crystallizes in a body centered
tetragonal structure (space group 14=mmm). TheARTICLE IN PRESS
M. Rotter et al. / Physica B 345 (2004) 231–234

Ne! el temperature is TN ¼ 19:5 K and a spin
reorientation has been reported at TR ¼ 13:5 K:
The magnetic structure has been investigated by
MXD [9]. Above TR the moments point along(010) with a propagation vector of t ¼ ð0:55 0 0Þ
while below TR a (0 0 1) component of the moment
appears. The propagation vector is only weakly
temperature dependent ðo0:5%Þ:For the McPhase calculation it is necessary toinclude 5 neighbors in order to get a global
maximum of the FT of the isotropic interaction
constants JðijÞ at t ¼ ð0:55 0 0Þ: We used for theneighbors at ð0 1 0Þ  12 meV; ð1 1 0Þ  27 meV;
ð0:5 0:5 0:5Þ  29 meV; ð2 2 0Þ þ 17 meV and at
ð0 0 1Þ þ 29 meV:The experimentally observed magnetic structures including the moment directions are well
reproduced by the calculation. The low-temperature modiﬁcation of the magnetic structure deserves some comments. By MXD a spiral and atransverse wave were suggested as possible magnetic structures [9]. The McPhase calculation
clearly stabilizes a spiral structure. Fig. 2 compares
MXD data with the calculated temperaturedependence of the FT of the magnetic moment
M: Note that the measured resonant intensity
corresponds to the component M001 (scaling withjM001 j2 ), which appears below the spin reorientation temperature in accordance with the experiment [9]. The measured non-resonant intensity has
to be compared to a linear combination of theM001 and M010 (see Fig. 2). The calculated
longitudinal component M100 is zero at alltemperatures. The factor of 1.986 in the linear
combination is derived from cos2 ðYÞ=sin2 ð2YÞ:The calculation underestimates the stability rangeof the high temperature collinear magnetic structure. This temperature range is determined by themagnitude of the dipolar anisotropy, which conﬁnes the moments to the (0 1 0) direction.
In conclusion, a numerical calculation based on
isotropic exchange and the dipolar interaction isable to predict the magnetic moment orientation
for both systems under investigation. The remaining quantitative inconsistencies with the MXD
experiment are attributed to anisotropic interactions, which are not taken into account in themodel (1).

233

Fig. 2. Measured temperature dependence of the resonant(open symbols) and the non-resonant (full symbols) intensity of
the ﬁrst harmonic ð1:447 0 0Þ magnetic reﬂection (data takenfrom [9] and scaled for comparison). Lines correspond to thecalculated temperature dependence of the squared FT of themoment components M010 and M001 for the ordering wave
vector t and the harmonic 3t:

Acknowledgements
Part of this work was performed within theprogram of the Sonderforschungsbereich 463(funded by the Deutsche Forschungsgemeinschaft).

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