Quadrupolar excitation channels at the L 3 edge of rare-earth ions probed by resonant inelastic x-ray scattering Fernando Bartolome ´ Laboratoire de Cristallographie, CNRS, Boı ˆte Postale 166, 38042 Grenoble, France and Instituto de Ciencia de Materiales de Arago ´n, C.S.I.C., Universidad de Zaragoza, 50009 Zaragoza, Spain Michael H. Krisch European Synchrotron Radiation Facility, Boı ˆte Postale 220, 38043 Grenoble Cedex, France Denis Raoux and Jean-Marc Tonnerre Laboratoire de Cristallographie, CNRS, Boı ˆte Postale 166, 38042 Grenoble Cedex, France Received 6 April 1999; revised manuscript received 14 June 1999 A systematic resonant inelastic x-ray scattering study at the L 3 absorption edge of rare-earth ions in the R 2 Fe 14 B intermetallics is presented. The energy position and the relative strength of quadrupolar ( E 2) and dipolar ( E 1) intermediate-state excitation channels are extracted by decomposing the inelastic scattering spectra into their main contributions. For the cases in which theoretical calculations of the quadrupolar part are available, a good agreement for the spectral shape is found. With increasing atomic number, a monotonic increase of the energy difference and a decrease of the relative strength between E 1 and E 2 excitation channels is observed. This work provides important experimental data for the application of magneto-optical sum rules to the 5 d band of rare-earth systems, as well as for the validation of theoretical models aimed at a correct description of x-ray magnetic spectroscopies at the L edges of rare earths in metallic systems. S0163-18299900440-3 I. INTRODUCTION The interpretation of x-ray magnetic circular dichroism XMCD and x-ray resonant magnetic scattering XRMS recorded at the L 2,3 absorption edges of rare-earth RE ions has been a matter of debate since the first experimental re- sults were obtained. 1–4 The spectra, dominated by the strong dipolar 2 p →5 d electronic transition, are not straightfor- wardly connected to the spin polarization of the rare-earth 5 d band. Several aspects have to be included in a correct description of the experiments, such as the contribution from quadrupolar excitation channels in the pre-edge region, 5 the intra-atomic 4 f -5 d exchange interaction; 6 and the spin de- pendence of the radial matrix elements. 6,7 The first of these aspects, namely, the quadrupolar nature (2 p →4 f transi- tions of the pre-edge structures, has been well established in the last few years by several experimental techniques 8–17 and calculations. 18–21 Despite their weak absorption intensities, these quadrupolar excitations give rise to strong dichroic sig- nals at the L edges of rare-earth ions, as they access the 4 f shell, which is the magnetic moment carrier of the rare-earth ion. The determination of the energy position and the relative strength of the E 2 and E 1 excitation channels is therefore essential, not only to provide relevant information for the correct interpretation of XMCD data of RE compounds, but also to test and refine existing calculations, aiming at a com- plete understanding of rare-earth magnetism. In a previous publication, 17 we presented the preliminary results of a resonant inelastic x-ray scattering RIXS study at the L 3 absorption edge of rare-earth ions in the interme- tallic series R 2 Fe 14 B( R =Nd, Sm, and Gd to Tm. Weak resonances, ascribed to 2 p →4 f quadrupolar excitations, were systematically observed at energies below the L 3 white- line. The results evidenced the necessity of their inclusion for a correct interpretation of XMCD and XRMS experi- ments recorded at the L edges of rare-earth systems. In this paper, we present the whole set of experimental results completed with RIXS data recorded at the Yb L 3 edge in Yb 2 Fe 14 B). The data analysis is based on a decom- position of each inelastic scattering spectrum into its main final-state multiplet families. These multiplet families arise from the decay of dipolar or quadrupolar intermediate state excitation channels, which are resonantly enhanced as the incident photon energy is tuned through the absorption edge. The decomposition procedure allows a determination of the excitation energies and the relative strength of each absorp- tion channel. The paper is organized as follows: Sec. II pro- vides the theoretical background. The general characteristics of the RIXS cross section are illustrated by experimental results obtained at the Yb L 3 edge in Yb 2 Fe 14 B. Section III describes the experimental setup. In Sec. IV the experimental spectra are presented, followed by a detailed description of the analysis procedure. The extracted spectral shape for the final state multiplet arising from 2 p →4 fE 2 transitions is compared to theoretical results, whenever they are available. Finally, the quantitative results concerning the energy posi- tion and the relative strength of E 1 and E 2 excitations are discussed. The conclusions of the paper are presented in Sec. V. II. RIXS PROCESS The cross section for resonant inelastic x-ray scattering is given by the Kramers-Heisenberg formula, 22–25 and the in- PHYSICAL REVIEW B 15 NOVEMBER 1999-I VOLUME 60, NUMBER 19 PRB 60 0163-1829/99/6019/1349710/$15.00 13 497 ©1999 The American Physical Society