Multiple scattering

Scattering is normally thought of in the single-scattering limit. I.e. one imagines that incident x-rays pass through the material, and can be scattered into exit rays; one ignores the possibility of a scattered ray being scattered again. This is a reasonable approximation in many cases for x-ray scattering (and neutron scattering) since the scattering cross section is quite weak. However this is often a poor approximation for electron scattering since the interaction is so strong.

In the limit of weak scattering, one can model the effect by assuming that the incident field is not perturbed by the scattering elements. This is known as the Born approximation. The field throughout the entire sample is assumed to simply be the incident field.

However in principle any scattered ray can scatter again. I.e. the field in the material becomes progressively more complex as one sums the contributions from all possible scattering entities. In practice, one typically models multiple scattering by assuming the Born approximation and adding additional terms to account for multiple scattering effects. In GISAXS, a popular method is the Distorted Wave Born Approximation (DWBA).

For strongly-scattering samples, multiple scattering must be taken into account. For example, imagine a material that generates a large amount of small-angle diffuse scattering. A thin piece of this material will yield the direct beam surrounded by some SAXS scattering. For a thicker sample, the diffuse scattered rays will be scattered again, leading to further broadening of the diffuse scattering. For materials (e.g. crystals), the strong scattering of diffraction spots can scatter off the crystal lattice again, giving rise to scattering features that at first glance seem to be impossible (c.f. structured diffuse scattering). For instance one can observe diffuse lines interconnecting between diffraction peaks.

Kikuchi patterns

Sharp intersecting lines are sometimes observed in diffraction data for single-crystal or poly-crystalline materials, depending on the experimental geometry. In electron microscopy, multiple-scattering events create these Kikuchi patterns, while in x-ray scattering they may be called Kossel lines. These lines arise due to multiple-scattering effect, e.g. when the x-ray source has high divergence.

Using a high-flux x-ray source (synchrotron), one can observe a related effect purely from the diffuse scattering undergoing multiple scattering. 'Multiple diffuse scattering' (DMS) can thus give rise to weak lines in a scattering image for crystalline materials (e.g. if one's sample is supported on a silicon substrate).

See Also

• Diffuse scattering
  • structured diffuse scattering
• W. G. Morris Crystal orientation and lattice parameters from Kossel lines J. Appl. Phys. 1968, 39, 1813–1823. doi: 10.1063/1.1656436
• G. Nolze, C. Grosse and A. Winkelmann Kikuchi pattern analysis of noncentrosymmetric crystals J. Appl. Cryst. 2015, 48. doi: 10.1107/S1600576715014016
• A.G.A. Nisbet, G. Beutier, F. Fabrizi, B. Mosera and S. P. Collins Diffuse multiple scattering Acta Cryst. A 2015, 20-25. doi: 10.1107/S2053273314026515
• Y. Ohmasa and A. Chiba Diffuse + Bragg double scattering and specular reflection observed in the small-angle X-ray scattering from highly oriented pyrolytic graphite J. Appl. Cryst. 2019, 52. doi: 10.1107/S1600576719005648
• G. Nolze, T. Tokarski, G. Cios and A. Winkelmann Manual measurement of angles in backscattered and transmission Kikuchi diffraction patterns J. Appl. Cryst. 2020. doi: 10.1107/S1600576720000692