Magnetic X-Ray Scattering

The first predictive calculation of the amplitude of X-rays elastically scattered by a magnetically ordered solid was carried out by Platzman and Tzoar in 1970 within the framework of the relativistic quantum theory. The need of a relativistic approach to access the relevant terms giving origin to magnetic term in the x-ray photon cross section is a testament to the weakness of the process.  Thanks to the heroic effort of De Bergevin and Brunel non resonant magnetic scattering (NRMS) was observed just two years later in NiO and reported in two seminal papers. The experiment was performed on a laboratory source with very long counting time and extreme attention to reducing any source of external noise. At that point NRMS was considered like a curiosity with little or no practical interest and that status only changed with the advent of third generation synchrotron. Even today it is very difficult to see non resonant magnetic scattering on a bending magnet beamline.      

The NRMS cross section is derived from resonant magnetic scattering when the incident photon energy is large compared with all the absorption edges present on the magnetically active ion. 
The advantage of NRMS on its resonant component is mainly related to the possibility of disentangling the spin and orbital component of the magnetization. This is due to the fact that the non-resonant cross section is different for the orbital and the spin component. Measuring several magnetic reflections corresponding to different sin(theta)/lambda both the spin and orbital magnetization can be reconstructed.    

Another advantage of the non resonant magnetic scattering is the relative freedom in choosing the incident photon energy. 

The choice of operating at high photon energy allows not only to ignore the resonant denominator, but also to sum over all the intermediate states probed by the virtual transition and it is particularly important for a correct evaluation of the orbital component. 

In practice, often results are consistent with the non resonant magnetic scattering approximation already going few tens of eV below the absorption K edge of all the magnetic ions present in the system. 

For this reason, the working energy in a NRMS experiment is often chosen to minimize the noise related to activating other resonant processes related to the absorption of a photon, like fluorescence, that in the contest of the study of magnetic ordering are only contributing to the noise and actually reduce the signal to noise ratio when in resonant condition. Another advantage of NRMS is that it can allow to suppress all the elastic resonant contribution of different origins that can interfere and combine with the magnetic contributions making it hard to isolate and interpret. 

NRMS is routinely used on I16 and led to several publications, a selection is listed below, including on very thin film of BiFeO₃, where NRMS at “low” energy was crucial in accessing reflections with large form factors and in reducing multiple scattering and other sources of noise.
 

The image above shows a typical experimental geometry for magnetic x-ray scattering, highlighting the polarisation of the beam. When not using phase-plates to change polarisation, the incident light is highly polarised horizontally. With the detector in the vertical scattering geometry (rotating Delta), the scattering plane is sigma-polarised, with a polarisation analyser rotating to provide sigma-sigma (Stokes 0) and sigma-pi (Stokes 90) channels.

With the detector in the horizontal geometry (rotating Gamma), the scatteirng plane is pi-polarised, with the polarisation analyser rotating to provide pi-sigma (Stokes 90) and pi-pi (Stokes 0) channels.

NB. By convention, sigma (σ) and pi (π) polarisations are the labels given to the components perpendicular and parallel to the scattering plane, respectivley. 

1. Platzman, P.M. & Tzoar, N., Physical Review B 2, (1970) 3556-3559.
2. de Bergevin, F. & Brunel, M., Phys. Letters A39, (1972) 141-142.
3. de Bergevin, F. & Brunel, M., Acta Cryst. A37, (1981) 314-331  
4. W. Neubeck, C. Vettier, F. de Bergevin, F. Yakhou, D. Mannix, L. Ranno, T. Chatterji, Journal of Physics and Chemistry of Solids, Volume 62, Issue 12 2001, Pages 2173-2180
5. J. Strempfer et al 1998 EPL 41 4
6. S. P. Collins, D. Laundy & A. J. Rollason (1992) Magnetic form factors of ferromagnetic iron by X-ray diffraction, Philosophical Magazine B, 65:1, 37-46, 
7. R. D. Johnson, P. Barone, A. Bombardi, R. J. Bean, S. Picozzi, P. G. Radaelli, Y. S. Oh, S.-W. Cheong, and L. C. Chapon Phys. Rev. Lett. 110, 217206

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