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 [P. Abbamonte et al., Phys. Rev. Lett. 92, 237401 (2004)] 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 $\text{Å}$-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.

Figure 1

Plot of the periodic array of potential wells described by Eqs. (18, 19). For suitable choice of well depth and spacing, this system can be described in the tight-binding approximation as a set of coupled harmonic oscillators.

Figure 2

(Color online) Contour plot of the prefactor ${\psi }_0\left(x_1\right){\psi }_0\left(x_2\right)$ in Eq. (25).

Figure 3

(Color online) Plots of the function $f_n\left(t\right)$ for values of $n$ out to fourth neighbors. Energy rapidly spreads from the first Wigner-Seitz cell $\left(n=0\right)$ into neighboring cells.

Figure 4

(Color online) (a) Contour plot of $\chi \left(x_1,x_2,t\right)$ for $x_2=0$, i.e., for the source located in the center of an “atom”. The disturbance created is symmetric around the source point and propagates at the average group velocity of the band. (b) Same plot, but with $x_2=a/2$, i.e., for the source half way in between two atoms. The disturbance is now asymmetric, and no longer oscillates with a well-defined period. (c) Sections through the plots in (a) and (b) at $t=1000$ as on a linear plot, showing that the magnitude of the response in (a) is larger by $\sim 100$ times.

Figure 5

(Color online) $\chi \left(x_1,x_2,t\right)$ averaged over all source locations, $x_2$. This is the image that would result by inverting IXS data. Note the close similarity to Fig. 4a.

Figure 6

(Color online) $\chi \left(x,y,t\right)$ for a real IXS experiment from a single crystal of graphite. The sixfold symmetry is a result of the underlying honeycomb symmetry of the graphene sheets. This image can be thought of as a superposition of two sources, one on the center of a C atom in each of the crystalline sublattices (see text).