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

Inelastic x-ray scattering (IXS) is now a widely used technique for studying the dynamics of electrons in condensed matter. We previously posed a solution to the phase problem for IXS that allows explicit reconstruction of the density propagator of a system. The propagator represents, physically, the response of the system to an idealized, point perturbation, so provides direct, real-time images of electron motion with attosecond time resolution and -scale spatial resolution. Here we show that the images generated by our procedure, as it was originally posed, are spatial averages over all source locations. Within an idealized, atomiclike model, we show that in most cases a simple relationship to the complete, unaveraged response can still be determined. We illustrate this concept for recent IXS measurements of single-crystal graphite.