Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access May 8, 2014

Angular dependence of resonant inelastic x-ray scattering: a spherical tensor expansion

Amélie Juhin, Christian Brouder and Frank Groot
From the journal Open Physics

Abstract

A spherical tensor expansion is carried out to express the resonant inelastic scattering cross-section as a sum of products of fundamental spectra with tensors involving wavevectors and polarization vectors of incident and scattered photons. The expression presented in this paper differs from that of the influential article by Carra et al. (Phys. Rev. Lett. 74, 3700, 1995) because it does not omit interference terms between electric dipole and quadrupole contributions when coupling each photon to itself. Some specific cases of the spherical tensor expansion are discussed. For example the case of isotropic samples is considered and the cross-section is expressed as a combination of only three fundamental spectra for the situation where electric dipole or electric quadrupole transitions in the absorption process are followed by electric dipole transitions in the emission. This situation includes the case of untextured powder samples, which corresponds to the most frequent situation met experimentally. Finally, it is predicted that some circular dichroism may be observed on isotropic samples provided that the circular polarization of the scattered beam can be detected.

[1] L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, J. van den Brink, Rev. Mod. Phys 83, 705 (2011) http://dx.doi.org/10.1103/RevModPhys.83.70510.1103/RevModPhys.83.705Search in Google Scholar

[2] J.-Ph. Schillé, J.-P. Kappler, Ph. Sainctavit, C. Cartier dit Moulin, Ch. Brouder, G. Krill, Phys. Rev. B 48, 9491 (1993) http://dx.doi.org/10.1103/PhysRevB.48.949110.1103/PhysRevB.48.9491Search in Google Scholar PubMed

[3] M. Blume, D. Gibbs, Phys. Rev. B 37, 1779–1789 (1988) http://dx.doi.org/10.1103/PhysRevB.37.177910.1103/PhysRevB.37.1779Search in Google Scholar

[4] B. T. Thole, P. Carra, F. Sette, G. van der Laan, Phys. Rev. Lett. 68, 1943–1946 (1992) http://dx.doi.org/10.1103/PhysRevLett.68.194310.1103/PhysRevLett.68.1943Search in Google Scholar PubMed

[5] B. T. Thole, G. van der Laan, M. Fabrizio, Phys. Rev. B 50, 11466 (1994) http://dx.doi.org/10.1103/PhysRevB.50.1146610.1103/PhysRevB.50.11466Search in Google Scholar

[6] B. T. Thole, G. van der Laan, Phys. Rev. B 44, 12424 (1991) http://dx.doi.org/10.1103/PhysRevB.44.1242410.1103/PhysRevB.44.12424Search in Google Scholar PubMed

[7] P. Carra, B. T. Thole, M. Altarelli, X. D. Wang, Phys. Rev. Lett. 70, 694–697 (1993). http://dx.doi.org/10.1103/PhysRevLett.70.69410.1103/PhysRevLett.70.694Search in Google Scholar PubMed

[8] G. van der Laan, B. T. Thole, Phys. Rev. B 48, 210 (1993) http://dx.doi.org/10.1103/PhysRevB.48.21010.1103/PhysRevB.48.210Search in Google Scholar

[9] B. T. Thole, G. van der Laan, Phys. Rev. B 49, 9613 (1994) http://dx.doi.org/10.1103/PhysRevB.49.961310.1103/PhysRevB.49.9613Search in Google Scholar

[10] G. van der Laan, B. T. Thole, Phys. Rev. B 52, 15355 (1995) http://dx.doi.org/10.1103/PhysRevB.52.1535510.1103/PhysRevB.52.15355Search in Google Scholar

[11] G. van der Laan, B. T. Thole, J. Phys. Condens. Matter 7, 9947 (1995) http://dx.doi.org/10.1088/0953-8984/7/50/02810.1088/0953-8984/7/50/028Search in Google Scholar

[12] Ch. Brouder, A. Juhin, A. Bordage, M.-A. Arrio, J. Phys.: Condens. Matt. 20, 455205 (2009) 10.1088/0953-8984/20/45/455205Search in Google Scholar

[13] A. P. Jucys, I. B. Levinson, V. V. Vanagas, Mathematical Apparatus of the Angular Momentum Theory, (Gordon and Breach, New York, 1963) Search in Google Scholar

[14] Z. Rudzikas, Theoretical Atomic Spectroscopy, (Cambridge University Press, Cambridge, 1997) http://dx.doi.org/10.1017/CBO978051152455410.1017/CBO9780511524554Search in Google Scholar

[15] H. A. Kramers, W. Heisenberg, Zeit. f. Physik 31, 681 (1925) http://dx.doi.org/10.1007/BF0298062410.1007/BF02980624Search in Google Scholar

[16] H. A. Kramers, W. Heisenberg, On the scattering of radiation by atoms, In: D. ter Haar (Ed.) Master of Modern Physics, Princeton Series in Physics (Princeton University Press, Princeton, 1998), 121 10.1515/9780691219523-009Search in Google Scholar

[17] L. A. Wray, Wanli Yang, H. Eisaki, Z. Hussain, Y.-D. Chuang, Phys. Rev. B 86, 195130 (2012) http://dx.doi.org/10.1103/PhysRevB.86.19513010.1103/PhysRevB.86.195130Search in Google Scholar

[18] S. Rossano, Ch. Brouder, M. Alouani, M.-A. Arrio, Phys. Chem. Minerals, 27, 170 (2000) http://dx.doi.org/10.1007/s00269005000510.1007/s002690050005Search in Google Scholar

[19] Ch. Brouder, J. Phys.: Condens. Matter 2, 701 (1990) 10.1088/0953-8984/2/3/018Search in Google Scholar

[20] M. van Veenendaal, R. Benoist, Phys. Rev. B 58, 3741 (1998) http://dx.doi.org/10.1103/PhysRevB.58.374110.1103/PhysRevB.58.3741Search in Google Scholar

[21] P. Ferriani, Ph.D. thesis, University of Modena (Modena, Italy, 2003) Search in Google Scholar

[22] I. Marri, Ph.D. thesis, University of Modena (Modena, Italy, 2003) Search in Google Scholar

[23] I. Marri, P. Carra, Phys. Rev. B 69, 113101 (2004) http://dx.doi.org/10.1103/PhysRevB.69.11310110.1103/PhysRevB.69.113101Search in Google Scholar

[24] I. Marri, C. M. Bertoni, P. Ferriani, Y. Joly, Phys. Rev. B 74, 212410 (2006) http://dx.doi.org/10.1103/PhysRevB.74.21241010.1103/PhysRevB.74.212410Search in Google Scholar

[25] P. Ferriani, C. M. Bertoni, J. Magn. Magn. Mater. 272, 1613 (2004) http://dx.doi.org/10.1016/j.jmmm.2003.12.107110.1016/j.jmmm.2003.12.1071Search in Google Scholar

[26] L. Braicovich, G. van der Laan, G. Ghiringhelli, A. Tagliaferri, N. B. Brookes, Phys. Rev. B 66, 174435 (2002) http://dx.doi.org/10.1103/PhysRevB.66.17443510.1103/PhysRevB.66.174435Search in Google Scholar

[27] L. Braicovich et al., Phys. Rev. Lett. 82, 1566 (1999) http://dx.doi.org/10.1103/PhysRevLett.82.156610.1103/PhysRevLett.82.1566Search in Google Scholar

[28] K. Fukui, H. Ogasawara, A. Kotani, T. Iwazumi, H. Shoji, T. Nakamura, J. Phys. Soc. Japan 70, 1230 (2001) http://dx.doi.org/10.1143/JPSJ.70.123010.1143/JPSJ.70.1230Search in Google Scholar

[29] K. Fukui, H. Ogasawara, A. Kotani, T. Iwazumi, H. Shoji, T. Nakamura, J. Phys. Soc. Japan 70, 3457(2001) http://dx.doi.org/10.1143/JPSJ.70.345710.1143/JPSJ.70.3457Search in Google Scholar

[30] H. Ogasawara, K. Fukui, M. Matsubara, J. Elect. Spect. Relat. Phenom. 136, 161 (2004) http://dx.doi.org/10.1016/j.elspec.2004.02.14310.1016/j.elspec.2004.02.143Search in Google Scholar

[31] P. Carra, M. Fabrizio, B. T. Thole, Phys. Rev. Lett. 74, 3700 (1995) http://dx.doi.org/10.1103/PhysRevLett.74.370010.1103/PhysRevLett.74.3700Search in Google Scholar PubMed

[32] L. Alagna et al., Phys. Rev. Lett 80, 4799 (1998) http://dx.doi.org/10.1103/PhysRevLett.80.479910.1103/PhysRevLett.80.4799Search in Google Scholar

[33] C.R. Natoli, Ch. Brouder, Ph. Sainctavit, J. Goulon, Ch. Goulon-Ginet, A. Rogalev, Europ. Phys. J. B 4, 1 (1998) http://dx.doi.org/10.1007/s10051005034410.1007/s100510050344Search in Google Scholar

[34] P. A. M. Dirac, Proc. R. Soc. Lond. A 114, 710 (1927) http://dx.doi.org/10.1098/rspa.1927.007110.1098/rspa.1927.0071Search in Google Scholar

[35] J. Tulkki and T. Åberg, J. Phys. B: Mol. Phys. 15, L435 (1982) http://dx.doi.org/10.1088/0022-3700/15/13/00410.1088/0022-3700/15/13/004Search in Google Scholar

[36] S. M. Blinder, Foundations of Quantum Dynamics, (Academic Press, London, 1974) Search in Google Scholar

[37] M. van Veenendaal, P. Carra, B. T. Thole, Phys. Rev. B 54, 16010 (1996). http://dx.doi.org/10.1103/PhysRevB.54.1601010.1103/PhysRevB.54.16010Search in Google Scholar PubMed

[38] J. Goulon et al., Phys. Rev. Lett. 85, 4385 (2000) http://dx.doi.org/10.1103/PhysRevLett.85.438510.1103/PhysRevLett.85.4385Search in Google Scholar PubMed

[39] J. Goulon et al., Phys. Rev. Lett. 88, 237401 (2002) http://dx.doi.org/10.1103/PhysRevLett.88.23740110.1103/PhysRevLett.88.237401Search in Google Scholar PubMed

[40] M. van Veenendaal, J. B. Goedkoop, B. T. Thole, Phys. Rev. Lett. 77, 1508 (1996) http://dx.doi.org/10.1103/PhysRevLett.77.150810.1103/PhysRevLett.77.1508Search in Google Scholar PubMed

[41] P. Carra. In: D. S. Gemmell, S. H. Southworth, R. W. Dunford, E. P. Kanter, L. Young (Eds.), X-ray and Inner-Shell Processes: 18th International Conference, (AIP Conference Proceedings, AIP, 2000), 506, 273 Search in Google Scholar

[42] K. Ishii et al., Phys. Rev. B 83, 241101 (2011) http://dx.doi.org/10.1103/PhysRevB.83.24110110.1103/PhysRevB.83.241101Search in Google Scholar

[43] D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum, (World Scientific, Singapore, 1988). http://dx.doi.org/10.1142/027010.1142/0270Search in Google Scholar

[44] H. Weyl, The Classical Groups. Their Invariants and Representations, (Princeton University Press, Princeton, 1946). 10.1515/9781400883905Search in Google Scholar

[45] D. R. Lehman, W. C. Parke, J. Math. Phys. 30, 2797 (1989) http://dx.doi.org/10.1063/1.52851510.1063/1.528515Search in Google Scholar

Published Online: 2014-5-8
Published in Print: 2014-5-1

© 2014 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Scroll Up Arrow