Structure Factor

The structure factor describes how incident radiation scatters from material.The scattering of X-ray radiation can be described using the partial structure factors \(S_{\alpha \beta}\), which were introduced by Faber and Zimann. Using this formalism the partial structure factor in dependence of the magnitude of the scattering vector \(Q\)

\[S_{\alpha \beta}(Q) = 1 + 4\pi \rho \int _{0}^{\infty} \mathrm{d}r r^2 \frac{\sin{Qr}}{Qr} (g_{\alpha \beta}(r)-1).\]

can be calculated requiring only knowledge of the radial distribution function \(g_{\alpha \beta }(r)\) and the particle density \(\rho\). The total structure factor \(S(Q)\) is calculated using a weighted sum of the partial structure factors as described in Tovey et. al. and Keen

\[S(Q) = \sum _{\alpha \beta} w_{\alpha \beta}^{x} S_{\alpha \beta} (Q).\]

The weights are determined using the atomic form factors \(f_i(Q)\) of species i and the molar fraction \(c_i\)

\[w_{\alpha \beta}^{x} = c_{\alpha} c_{\beta} \frac{f_{\alpha}(Q)f_{\beta}(Q)}{\sum _{i=1}^{n} c_i f_i(Q)}.\]

The atomic form factors depend on the type and charge of the element and are approximated by a sum of gaussians, whose coeffients are taken from TU Graz. The atomic form factors are valid for a range of \(0 < Q < 25 \, \text{Å} ^{-1}\). In order for the calculation to work the element names and charges need to be set in the species dictionary as shown in the sample script structure_factor.py .