orientation and the different damage scenarios are averaged. As shown in Appendix \[app1\], the average intensity can be written in the form $$\langle I(\textbf{q}) \rangle = r_e^2 P(\textbf{q}) d\Omega I_0 \left[ N A(q) + 2 B(q) \sum_{i=1}^N \sum_{j=1}^{i-1} \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \right] \;,
\label{eq:Iav}$$ where we have $$\langle A_{i}(q) \rangle = A(q) \equiv I_0 \int_0^T \langle |f(q,t)|^2 \rangle  dt 
\label{eq:Aav}$$ and $$\langle B_{ij}(\textbf{q}) \rangle = B(q) \cos[ 2 \pi \textbf{q} \cdot (\textbf{R}_i - \textbf{R}_j)] \;,
\label{eq:Bijav}$$ where $$B(q) \equiv \int_0^T  \langle f(q,t) \rangle^2 e^{-4\pi^2 q^2 \overline{\bm{\epsilon}}(t)^2} dt$$ and $\overline{\bm{\epsilon}}(t)$ is the root mean square (rms) displacement of an ion as a function of time.

If the analysis is restricted to damage processes that are random and spatially uncorrelated, then we can treat the terms $A_{i}(q)$ and $B_{ij}(\textbf{q})$ as random variables and study the effect of damage statistically. We also treat the initial atomic positions $\textbf{R}_i$ as random with a uniform probability distribution, as is done in crystallography to analyse the statistics of Bragg intensities (Wilson statistics) at high scattering angles ($q > 0.33 $nm$^{-1}$) [@Huldt2003]. Both ionization and ion diffusion can be treated within this framework and, as we will show, are both involved in 