## 4.3 Conclusion

We have expressed the XAS cross section using Green's function formalism and spherical tensors, which allows a tractable investigation of the different types of dichroism: on the one hand experimentally, by allowing the experimentalist to predict the smallest set of measurements required to recover the full spectroscopic information and therefore to optimize beamtime; on the other hand theoretically, using modern core level spectroscopy codes, by allowing a more efficient calculation rather than a point-by-point time-consuming treatment. In a forthcoming work, we intend to use a similar approach for resonant inelastic X-ray scattering (RIXS) spectroscopy, whose richness lies in the large number of possible spectra that can be obtained by varying the energy, direction, and polarization state of the incident and scattered beams [28]. But as a matter of fact, there is so much information in the spectra that it is difficult to know whether a specific set of experiments measures all potential information. In our opinion, the possibilities offered by angular and polarization dependent RIXS measurements have not yet been exploited to the best of their potential. Nevertheless, the technique has now become mature and popular and will benefit from the huge instrumentation progress achieved in the last years, which open new doors for the exploration of dichroisms in materials.

## References

1. C. Brouder, Angular dependence of X-ray absorption spectra. J. Phys.: Condens. Matter **2**, 701 (1990). https://doi.org/10.1088/0953-8984/2/3/018

2. A. Juhin, S.P. Collins, Y. Joly, M. Diaz-Lopez, K. Kvashnina, P. Glatzel, C. Brouder, F.M.F. de Groot, Measurement of *f* orbital hybridization in rare earths through electric dipole-octupole interference in X-ray absorption spectroscopy. Phys. Rev. Mater. **3**, 120801(R) (2019). https://doi.org/10.1103/PhysRevMaterials.3.120801

3. S. Sugano, T. Tanabe, H. Kamimura, *Multiplets of Transition Metal Ions in Crystals* (Academic, New York, 1970)

4. J.S. Griffith, *The Theory of Transition-Metal Ions* (Cambridge University Press, Cambridge, 1961)

5. E. König, S. Kremer, *Ligand Field Energy Diagrams* (Springer Science+Business Media, New York, 2013). https://doi.org/10.1007/978-1-4757-1529-3

6. R.D. Cowan, *The Theory of Atomic Structure and Spectra* (University of California Press, Berkeley, 1981)

7. M.W. Haverkort, Y. Lu, S. Macke, R. Green, M. Brass, S. Heinze, Quany – a quantum many body script language, http://www.quanty.org/doku.php?id(start&rev=1522759229

8. M. Baer, Multipole expansion package, https://github.com/maroba/multipoles

9. M.W. Haverkort, M. Zwierzycki, O.K. Andersen, Multiplet ligand-field theory using Wannier orbitals. Phys. Rev. B **85**, 165113 (2012). https://doi.org/10.1103/PhysRevB.85.165113

10. Y. Lu, M. Höppner, O. Gunnarsson, M.W. Haverkort, Efficient real-frequency solver for dynamical mean-field theory. Phys. Rev. B **90**, 085102 (2014). https://doi.org/10.1103/PhysRevB.90.085102

11. M.W. Haverkort, G. Sangiovanni, P. Hansmann, A. Toschi, Y. Lu, S. Macke, Bands, resonances, edge singularities and excitons in core level spectroscopy investigated within the dynamical
## 4.3 Conclusion

We have expressed the XAS cross section using Green's function formalism and spherical tensors, which allows a tractable investigation of the different types of dichroism: on the one hand experimentally, by allowing the experimentalist to predict the smallest set of measurements required to recover the full spectroscopic information and therefore to optimize beamtime; on the other hand theoretically, using modern core level spectroscopy codes, by allowing a more efficient calculation rather than a point-by-point time-consuming treatment. In a forthcoming work, we intend to use a similar approach for resonant inelastic X-ray scattering (RIXS) spectroscopy, whose richness lies in the large number of possible spectra that can be obtained by varying the energy, direction, and polarization state of the incident and scattered beams [28]. But as a matter of fact, there is so much information in the spectra that it is difficult to know whether a specific set of experiments measures all potential information. In our opinion, the possibilities offered by angular and polarization dependent RIXS measurements have not yet been exploited to the best of their potential. Nevertheless, the technique has now become mature and popular and will benefit from the huge instrumentation progress achieved in the last years, which open new doors for the exploration of dichroisms in materials.

## References

1. C. Brouder, Angular dependence of X-ray absorption spectra. J. Phys.: Condens. Matter **2**, 701 (1990). https://doi.org/10.1088/0953-8984/2/3/018

2. A. Juhin, S.P. Collins, Y. Joly, M. Diaz-Lopez, K. Kvashnina, P. Glatzel, C. Brouder, F.M.F. de Groot, Measurement of *f* orbital hybridization in rare earths through electric dipole-octupole interference in X-ray absorption spectroscopy. Phys. Rev. Mater. **3**, 120801(R) (2019). https://doi.org/10.1103/PhysRevMaterials.3.120801

3. S. Sugano, T. Tanabe, H. Kamimura, *Multiplets of Transition Metal Ions in Crystals* (Academic, New York, 1970)

4. J.S. Griffith, *The Theory of Transition-Metal Ions* (Cambridge University Press, Cambridge, 1961)

5. E. König, S. Kremer, *Ligand Field Energy Diagrams* (Springer Science+Business Media, New York, 2013). https://doi.org/10.1007/978-1-4757-1529-3

6. R.D. Cowan, *The Theory of Atomic Structure and Spectra* (University of California Press, Berkeley, 1981)

7. M.W. Haverkort, Y. Lu, S. Macke, R. Green, M. Brass, S. Heinze, Quany – a quantum many body script language, http://www.quanty.org/doku.php?id(start&rev=1522759229

8. M. Baer, Multipole expansion package, https://github.com/maroba/multipoles

9. M.W. Haverkort, M. Zwierzycki, O.K. Andersen, Multiplet ligand-field theory using Wannier orbitals. Phys. Rev. B **85**, 165113 (2012). https://doi.org/10.1103/PhysRevB.85.165113

10. Y. Lu, M. Höppner, O. Gunnarsson, M.W. Haverkort, Efficient real-frequency solver for dynamical mean-field theory. Phys. Rev. B **90**, 085102 (2014). https://doi.org/10.1103/PhysRevB.90.085102

11. M.W. Haverkort, G. Sangiovanni, P. Hansmann, A. Toschi, Y. Lu, S. Macke, Bands, resonances, edge singularities and excitons in core level spectroscopy investigated within the dynamical
mean-field theory. Europhys. Lett. **108**, 57004 (2014). https://doi.org/10.1209/0295-5075/108/57004

12. M.W. Haverkort, Quanty for core level spectroscopy - excitons, resonances and band excitations in time and frequency domain. J. Phys. Conf. Ser. **712**, 012001 (2016). https://doi.org/10.1088/1742-6596/712/1/012001

13. M. Retegan, Crispy: v0.7.3. https://doi.org/10.5281/zenodo.1008184

14. C. Brouder, A. Juhin, A. Bordage, M.-A. Arrio, Site symmetry and crystal symmetry: a spherical tensor analysis. J. Phys.: Condens. Matter **20**, 455205 (2008). https://doi.org/10.1088/0953-8984/20/45/455205

15. B.T. Thole, G. van der Laan, Spin polarization and magnetic dichroism in photoemission from core and valence states in localized magnetic systems. Phys. Rev. B **44**, 12424 (1991). https://doi.org/10.1103/PhysRevB.44.12424

16. G. van der Laan, B.T. Thole, Spin polarization and magnetic dichroism in photoemission from core and valence states in localized magnetic systems. II. Emission from open shells. Phys. Rev. B **48**, 210 (1993). https://doi.org/10.1103/PhysRevB.48.210

17. B.T. Thole, G. van der Laan, Spin polarization and magnetic dichroism in photoemission from core and valence states in localized magnetic systems. III. Angular distributions. Phys. Rev. B **49**, 9613 (1994). https://doi.org/10.1103/PhysRevB.49.9613

18. G. van der Laan, B.T. Thole, Spin polarization and magnetic dichroism in photoemission from core and valence states in localized magnetic systems. IV. Core-hole polarization in resonant photoemission. Phys. Rev. B **52**, 15355 (1995). https://doi.org/10.1103/PhysRevB.52.15355

19. G. van der Laan, B.T. Thole, Core hole polarization in resonant photoemission. J. Phys. Condens. Matter **7**, 9947 (1995). https://doi.org/10.1088/0953-8984/7/50/028

20. P. Carra, H. König, B.T. Thole, M. Altarelli, Magnetic X-ray dichroism: general features of dipolar and quadrupolar spectra. Physica B **192**, 182 (1993). https://doi.org/10.1016/0921-4526(93)90119-Q

21. J.-P. Schillé, J.-P. Kappler, P. Sainctavit, C. Cartier dit Moulin, C. Brouder, G. Krill, Experimental and calculated magnetic dichroism in the Ho 3d X-ray absorption of intermetallic HoCo₂. Phys. Rev. B **48**, 9491 (1993). https://doi.org/10.1103/PhysRevB.48.9491

22. B.T. Thole, G. van der Laan, M. Fabrizio, Magnetic ground-state properties and spectral distributions. I. X-ray-absorption spectra. Phys. Rev. B **50**, 11466 (1994). https://doi.org/10.1103/PhysRevB.50.11466

23. A. Juhin, C. Brouder, M.-A. Arrio, D. Cabaret, P. Sainctavit, E. Balan, A. Bordage, A.P. Seitsoenen, G. Calas, S.G. Eeckhout, P. Glatzel, X-ray linear dichroism in cubic compounds: the case of Cr³⁺ in MgAl₂O₄. Phys. Rev. B **78**, 195103 (2008). https://doi.org/10.1103/PhysRevB.78.195103

24. G. van der Laan, Magnetic linear X-ray dichroism as a probe of the magnetocrystalline anisotropy. Phys. Rev. Lett. **82**, 640 (1999). https://doi.org/10.1103/PhysRevLett.82.640

25. E. Arenholz, G. van der Laan, R.V. Chopdekar, Y. Suzuki, Anisotropic X-ray magnetic linear dichroism at the Fe L₂,₃ edges in Fe₃O₄. Phys. Rev. B **74**, 094407 (2006). https://doi.org/10.1103/PhysRevB.74.094407

26. G. Schütz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R. Frahm, G. Materlik, Absorption of circularly polarized X rays in iron. Phys. Rev. Lett. **58**, 737 (1987). https://doi.org/10.1103/PhysRevLett.58.737

27. P. Carra, B.T. Thole, M. Altarelli, X. Wang, X-ray circular dichroism and local magnetic fields. Phys. Rev. Lett. **70**, 694 (1993). https://doi.org/10.1103/PhysRevLett.70.694

28. A. Juhin, C. Brouder, F. de Groot, Angular dependence of resonant inelastic X-ray scattering: a spherical tensor expansion. Cent. Eur. J. Phys. **12**, 323 (2014). https://doi.org/10.2478/s11534-014-0450-2
mean-field theory. Europhys. Lett. **108**, 57004 (2014). https://doi.org/10.1209/0295-5075/108/57004

12. M.W. Haverkort, Quanty for core level spectroscopy - excitons, resonances and band excitations in time and frequency domain. J. Phys. Conf. Ser. **712**, 012001 (2016). https://doi.org/10.1088/1742-6596/712/1/012001

13. M. Retegan, Crispy: v0.7.3. https://doi.org/10.5281/zenodo.1008184

14. C. Brouder, A. Juhin, A. Bordage, M.-A. Arrio, Site symmetry and crystal symmetry: a spherical tensor analysis. J. Phys.: Condens. Matter **20**, 455205 (2008). https://doi.org/10.1088/0953-8984/20/45/455205

15. B.T. Thole, G. van der Laan, Spin polarization and magnetic dichroism in photoemission from core and valence states in localized magnetic systems. Phys. Rev. B **44**, 12424 (1991). https://doi.org/10.1103/PhysRevB.44.12424

16. G. van der Laan, B.T. Thole, Spin polarization and magnetic dichroism in photoemission from core and valence states in localized magnetic systems. II. Emission from open shells. Phys. Rev. B **48**, 210 (1993). https://doi.org/10.1103/PhysRevB.48.210

17. B.T. Thole, G. van der Laan, Spin polarization and magnetic dichroism in photoemission from core and valence states in localized magnetic systems. III. Angular distributions. Phys. Rev. B **49**, 9613 (1994). https://doi.org/10.1103/PhysRevB.49.9613

18. G. van der Laan, B.T. Thole, Spin polarization and magnetic dichroism in photoemission from core and valence states in localized magnetic systems. IV. Core-hole polarization in resonant photoemission. Phys. Rev. B **52**, 15355 (1995). https://doi.org/10.1103/PhysRevB.52.15355

19. G. van der Laan, B.T. Thole, Core hole polarization in resonant photoemission. J. Phys. Condens. Matter **7**, 9947 (1995). https://doi.org/10.1088/0953-8984/7/50/028

20. P. Carra, H. König, B.T. Thole, M. Altarelli, Magnetic X-ray dichroism: general features of dipolar and quadrupolar spectra. Physica B **192**, 182 (1993). https://doi.org/10.1016/0921-4526(93)90119-Q

21. J.-P. Schillé, J.-P. Kappler, P. Sainctavit, C. Cartier dit Moulin, C. Brouder, G. Krill, Experimental and calculated magnetic dichroism in the Ho 3d X-ray absorption of intermetallic HoCo₂. Phys. Rev. B **48**, 9491 (1993). https://doi.org/10.1103/PhysRevB.48.9491

22. B.T. Thole, G. van der Laan, M. Fabrizio, Magnetic ground-state properties and spectral distributions. I. X-ray-absorption spectra. Phys. Rev. B **50**, 11466 (1994). https://doi.org/10.1103/PhysRevB.50.11466

23. A. Juhin, C. Brouder, M.-A. Arrio, D. Cabaret, P. Sainctavit, E. Balan, A. Bordage, A.P. Seitsoenen, G. Calas, S.G. Eeckhout, P. Glatzel, X-ray linear dichroism in cubic compounds: the case of Cr³⁺ in MgAl₂O₄. Phys. Rev. B **78**, 195103 (2008). https://doi.org/10.1103/PhysRevB.78.195103

24. G. van der Laan, Magnetic linear X-ray dichroism as a probe of the magnetocrystalline anisotropy. Phys. Rev. Lett. **82**, 640 (1999). https://doi.org/10.1103/PhysRevLett.82.640

25. E. Arenholz, G. van der Laan, R.V. Chopdekar, Y. Suzuki, Anisotropic X-ray magnetic linear dichroism at the Fe L₂,₃ edges in Fe₃O₄. Phys. Rev. B **74**, 094407 (2006). https://doi.org/10.1103/PhysRevB.74.094407

26. G. Schütz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R. Frahm, G. Materlik, Absorption of circularly polarized X rays in iron. Phys. Rev. Lett. **58**, 737 (1987). https://doi.org/10.1103/PhysRevLett.58.737

27. P. Carra, B.T. Thole, M. Altarelli, X. Wang, X-ray circular dichroism and local magnetic fields. Phys. Rev. Lett. **70**, 694 (1993). https://doi.org/10.1103/PhysRevLett.70.694

28. A. Juhin, C. Brouder, F. de Groot, Angular dependence of resonant inelastic X-ray scattering: a spherical tensor expansion. Cent. Eur. J. Phys. **12**, 323 (2014). https://doi.org/10.2478/s11534-014-0450-2
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<figure><img src="image_25.png" /></figure>
**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

<figure><img src="image_25.png" /></figure>