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2.2 Powder assumption
2.2 Powder assumption
The next stage in our derivation is important from a practical aspect. Total scattering is a powder diffraction technique. Measurements are made in 1D (or reduced from higher dimensions by integration to yield a 1D pattern). By ‘powder’ we do not necessarily mean a free flowing powder, but rather a ‘crystallographic’ powder meaning a sample in which there are crystallites in all possible orientations. This leads to an important simplification in the mathematics as we will see, but also will be important from the perspective of sample manufacture when seeking to employ the technique.
Spherical averaging
For a powder it was argued that the irradiated samples contains crystals in all possible orientations (a so called ‘Powder average’). As such, it is possible to spherically average the scattering function in Equation 2.4 over all space, to remove directionality. The full derivation of this is included in the Appendix, but the key results is that:
Separation of self-scattering terms
This equation is composed of a double sum that can be separated into two distinct components; self-scattering terms where i = j, and cross scattering terms where i ≠ j. We can therefore break the equation as:
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