Implicit spatial averaging in inversion of inelastic x-ray scattering data

Inelastic x-ray scattering (IXS) is usually said to measure the imaginary part of the dynamical density response of a material. However this is not rigorously true. The density response $\chi$, which describes the response of the system to a point charge source, is a function of {\it two} spatial coordinates and the time, i.e. $\chi = \chi(x_1,x_2,t)$. Its Fourier transform $\chi(k_1,k_2,\omega)$ is therefore a function of two, rather than just one, momenta. IXS does not probe this full response, but only its longitudinal or ``diagonal" part $Im[\chi(k,k,\omega)]$. In this talk I will show that recently developed IXS inversion algorithms [2], which have shown promise for imaging attosecond dynamics in real space, yield a specific spatial average of the response, i.e. $\chi(x_1,t) = \int dx' \chi(x_1,x_1+x',t)$. This can be thought of as an average over all possible source locations, a real space projection, or a specific type of Fourier space filtering. I will show, within a simple model, that the salient real space dynamics nonetheless survive, and that IXS inversion is still a useful and well-posed technique for imaging attosecond dynamics.