Ab initio calculation of the ion feature in x-ray Thomson scattering

The spectrum of x-ray Thomson scattering is proportional to the dynamic structure factor. An important contribution is the ion feature which describes elastic scattering of x rays off electrons. We apply an ab initio method for the calculation of the form factor of bound electrons, the slope of the screening cloud of free electrons, and the ion-ion structure factor in warm dense beryllium. With the presented method we can calculate the ion feature from first principles. These results will facilitate a better understanding of x-ray scattering in warm dense matter and an accurate measurement of ion temperatures which would allow determining nonequilibrium conditions, e.g., along shock propagation.

Figure 1

Ion-ion structure factor for u-Be [14] for different temperatures. The circles at $k=0$ represent the values derived via the isothermal compressibility.

Figure 2

Same as in Fig. 1 but for c-Be [3].

Figure 3

Total form factor $N\left(k\right)$ (black), form factor of bound electrons $f\left(k\right)$ (red), and screening cloud of free electrons $q\left(k\right)$ (blue) for u-Be [14] at different temperatures. Red dash-dotted curve: Hartree-Fock form factor for ${\text{Be}}^{2+}$ ions [34].

Figure 4

Same as in Fig. 3 but for c-Be [3].

Figure 5

Ion feature for u-Be for different equilibrium temperatures. Thin lines are fits through the numerical data. The dots at $k=0$ represent the values calculated from the isothermal compressibility.

Figure 6

Same as in Fig. 5 but for c-Be.

Figure 7

Ion feature for u-Be for different equilibrium temperatures. Thin lines are fits through the numerical data. The dots at $k=0$ represent the values calculated from the isothermal compressibility. The dots with error bars represent experimental data (cyan) from Ref. [1] and a reanalyzed spectrum (green) given in Ref. [14].

Figure 8

Same as in Fig. 7 but for c-Be. The dots with error bars represent experimental data (cyan) from Refs. [3, 36] and reanalyzed spectra (green).

Figure 9

Different fits for the same experimental spectrum at $k=1.3 {\text{Å}}^{-1}$ for c-Be [3] leading to different calibration constant C [Eq. (13)]. Panel (a) shows the best fit to the scattering data, while panels (b) and (c) show slightly different fits but a strong derviation in the ion feature, see Table 1.

Figure 10

Normalized experimental scattering data for c-Be at $k=1.3 {\text{Å}}^{-1}$ (scattering angle $25 {}^{\circ }$) in red; in cyan (dashed) we show the original fit [3] and in green (solid) our new fit (a).

Figure 11

Ion feature for u-Be for nonequilibrium two-temperature systems. Thin lines are fits through the numerical data. The dots at $k=0$ represent the values calculated from the isothermal compressibility. The dots with error bars represent experimental data (cyan) from Ref. [1] and a reanalyzed spectrum (green) given in Ref. [14].

Figure 12

Same as in Fig. 11 but for c-Be. The dots with error bars represent experimental data (cyan) from Refs. [3, 36] and reanalyzed spectra (green).