dichroism 460 # 2001 International Union of Crystallography  Printed in Great Britain ± all rights reserved J. Synchrotron Rad. (2001). 8, 460±462 J. Synchrotron Rad. (1999). 6, 000–000 Natural linear dichroism in pyrite (FeS 2 ): experiments and calculations Delphine Cabaret, a * Christian Brouder, a Marie-Anne Arrio, a Philippe Sainctavit, a Yves Joly, b Andrei Rogalev c and Jos ´ e Goulon c a LMCP, UMR7590, case 115, 4 place Jussieu, 75252 Paris cedex 05 France, b Laboratoire de cristallographie, CNRS, B.P. 166, 38042 Grenoble cedex 9 France, and c ESRF, B.P. 220, 38043 Grenoble cedex France. E-mail: cabaret@lmcp.jussieu.fr Electric quadrupole transitions are revealed through the angular de- pendence of X-ray absorption spectra in a cubic crytal. Data col- lection was achieved at the iron K-edge in pyrite (FeS2) by using the angular moment method developed for X-ray Natural Circu- lar Dichoism (XNCD) measurements. The natural linear dichroism (XNLD) was found to be around 0.5 % of the edge jump. Experi- mental results are compared with monelectronic and multilectronic calculations. Calculations allow to quantitatively determine the pro- portion of quadrupolar transitions in the pre-edge structure and the anisotropy is explained in terms of structural and electronic param- eters. keywords : Electric quadrupole transitions, pyrite 1. Introduction Electric quadrupole transitions in X-ray absorption spectra have become quite important in the past few years because of their large contribution to X-ray Magnetic Circular Dichroism (Carra et al., 1993). More recently X-ray Natural Circular Dichroism has been measured in various gyrotropic crystals and has revealed interfer- ences between electric dipole and quadrupole transitions (Goulon et al., 2000, and references therein). In this paper, we investigate experimentically and theoretically an effect which is due to purely quadrupole transitions : the angular dependence of X-ray absorp- tion in a cubic crystal. Electric quadrupole transitions are simpler to measure in cubic crystals, because the electric dipole transitions are isotropic. How- ever the expected quadrupole dependence was not found for neither bcc-Fe, nor pyrite (Dr¨ ager et al., 1988). More recently the situation changed and a quite strong anisotropy (2 %) was detected in single crystals of NiO and FeO (Heumann et al., 1997), by still measuring the absorption in transmission mode at two angles. For this study, we chose the case of pyrite for several reasons. Pyrite crystallises in the cubic system, space group Pa ¯ 3 T 6 h . Iron sits in a slightly distorted octahedron of sulphur atoms (point group C3i ). In pyrite, iron is divalent and in low-spin states. Thus the t2g orbitals of the metal are completely filled. Consequently, in pyrite the quadrupole transitions involved at the iron K-edge are expected to be 1s 3d-eg . From a theoretical point of view, because of the strong covalence of iron with sulphur, we expect that the iron 3 d orbitals are not too strongly localized and that a one-electron model can work successfully and better than in transition metals oxides. Figure 1 Crystal structure of pyrite. Iron atoms are in C 3i site of multiplicity 4, oc- tahedrally coordinated to sulphur atoms. The octahedra are tilted from the crystallographic axis by about 23 . The orientation of the wave vec- tor k and the two orientations of the polarisation vector , leading to maximum dichroic signal, are represented. 2. Experiment and data analysis According to group theory, for a cubic crystal, the angular depen- dence of the X-ray absorption can be written as: ˆ ˆ k 0 2 x k 2 x 2 y k 2 y 2 z k 2 z 1 5 1 where 0 is the isotropic cross section and 1 is a purely quadrupo- lar transition. ˆ and ˆ k are the X-ray polarisation and wave unit vectors respectively. As was noticed by Dr¨ ager et al. (1988), the maximum variation may be obtained by turning the crystal around the [110] direction alined with k and by subtracting the absorption obtained with //[001] from the one obtained with //[ ¯ 110]. The crystal structure of pyrite and the orientation of the single crystal in the X-ray beam are illustrated in Fig. 1. The experimental data were collected at the ESRF beam line ID12-A, dedicated to polarisation dependent spectroscopies (Goulon et al., 1998). The absorption is measured by eight pho- todiode fluorescence detectors symmetrically set around the X-ray wave vector. A natural single crystal of pyrite, from Minas Victoria, Logro˜ no (Spain) was oriented, then placed on a rotating holder and turned around the [110] direction, parallel to the X-ray beam di- rection. Angle 0 corresponds to //[ ¯ 110]. The measurements were not carried out at two angles but, at each energy, the fluorescence intensity was measured by the eight detectors, for 121 angles from 0 to 360 , by step of 3 . For experiments carried out at the K-edge of transition metals in single crystals, it is necessary to be careful with the artefacts com- ing from the presence of diffraction peaks. Fig. 2 shows the inten- sity recorded by the eight fluorescence detectors for 120 angles at a fixed energy. The diffraction peaks can come from a diffracted ray that strikes a fluorescence detector or from diffraction that occurs inside the crystal. The presence of diffraction peaks may have a devastating effect on the measurements of absorption spectra. The fact that the data were collected at many angles makes possible to get rid of these artefacts and to use a filtering algorithm based on the angular momentum method developed by Goulon et al. (1999) Natural linear dichroism in pyrite (FeS 2 ): experiments and calculations " //   110 " // [001] k // [110]