Predictions of x-ray scattering spectra for warm dense matter

We present calculations of x-ray scattering spectra based on ionic and electronic structure factors that are computed from a new model for warm dense matter. In this model, which has no free parameters, the ionic structure is determined consistently with the electronic structure of the bound and free states. The x-ray scattering spectrum is thus fully determined by the plasma temperature, density and nuclear charge, and the experimental parameters. The combined model of warm dense matter and of the x-ray scattering theory is validated against an experiment on room-temperature, solid beryllium. It is then applied to experiments on warm dense beryllium and aluminum. Generally good agreement is found with the experiments. However, some significant discrepancies are revealed and appraised. Based on the strength of our model, we discuss the current state of x-ray scattering experiments on warm dense matter and their potential to determine plasma parameters, to discriminate among models, and to reveal interesting and difficult to model physics in dense plasmas.

Figure 1

Comparison of theoretical x-ray scattering spectra with an experiment on solid beryllium at ambient conditions [7]. “Total” corresponds to the sum of the bound-free and free-free terms, calculated with the Dirac exchange functional. The solid cyan (light gray) line is the total using Slater exchange in the calculation of bound and free electronic states. It is mostly indistinguishable from the calculation using Dirac exchange except near the bound-free threshold, which is shifted slightly to the right. Note that to match the presentation of the data in Ref. [7], the frequency axis is reversed from the convention used in WDM experiments and in Figs. 3, 4, 5 and 7 (here $\omega ={\omega }_0-{\omega }_1$).

Figure 2

$k$ dependence of the elastic component of the structure factor for beryllium at 13 eV and 5.55 g/cm${}^3$ as calculated with the WDM model of Ref. [12]. A quantum molecular dynamics calculation of $S_{ii}\left(k\right)$ is also shown [25]. The vertical dotted lines correspond the $k$ values of the experiment at ${25}^{\circ }$ ($k=0.720a_B^{-1}$) and ${90}^{\circ }$ ($k=2.35a_B^{-1}$). These correspond to scattering parameters of $\alpha =1.73$ and 0.53, respectively. The quantities shown on the vertical axis are dimensionless.

Figure 3

Comparison of synthetic spectra to an x-ray scattering experiment on warm dense beryllium [4] assuming a temperature of 13 eV and density of 5.55 g/cm${}^3$ (threefold compression). The experimental spectra are in arbitrary units. The two theoretical curves show the same calculation normalized to match the magnitude of either the right-hand peak (normalization A) or the left-hand peak (normalization B). The inset shows the spectrum of the incident x-ray source.

Figure 4

Same as Fig. 3 for warm dense Be but showing the separate contributions to the spectrum. Both the RPA and the Born-Mermin calculations of the free-free contribution are shown. The curve labeled “total” uses the BM result for the free-free component. The theoretical curves correspond to normalization B of Fig. 3.

Figure 5

Sensitivity of the calculated x-ray scattering spectrum to variations in density and temperature for the experiment on warm dense beryllium with $\theta ={90}^{\circ }$ [4]. The variations shown correspond to $\pm 30$% about the nominal values. The theoretical curves correspond to normalization B of Fig. 3. In the top panel the upper line corresponds to the highest density, and the lowest line to the lowest density. In the bottom panel the upper line corresponds to the lowest temperature, and the lowest line to the highest temperature.

Figure 6

$k$ dependence of the elastic component of the structure factor for aluminum at 10 eV and 8.1 g/cm${}^3$ as calculated with the WDM model of Ref. [12]. The dotted vertical lines correspond the the $k$ values of the experiment [6] at ${69}^{\circ }$ ($k=5.44a_B^{-1}$) and ${111}^{\circ }$ ($k=7.94a_B^{-1}$). The $S_{ii}\left(k\right)$ obtained from a quantum molecular dynamics calculation [29] is also shown for comparison. The small cross indicates the change in the position of the peak (solid curve) that results from replacing $S_{ii}\left(k\right)$ from our model with that of the QMD calculation in the calculation of the elastic feature. Points with error bars are $|f_I{\left(k\right)+q\left(k\right)|}^2{S}_{ii}\left(k\right)$ extracted from the experiment of Ref. [6]. For clarity, $f_I\left(k\right)$, $q\left(k\right)$, and $S_{ii}\left(k\right)$ have been scaled as indicated in the legend. The quantities on the vertical axis are dimensionless.

Figure 7

Comparison of synthetic x-ray scattering spectra to an experiment on warm dense aluminum [6], assuming a temperature of 10 eV and density of 8.1 g/cm${}^3$ (threefold compression). The vertical dotted line shows the position of the Compton edge. The curve labeled “total” uses the Born-Mermin (BM) calculation of the free-free component. All theoretical curves shown use the Dirac exchange potential except the solid purple (dark gray) line, which is the total spectrum obtained when using the Slater exchange potential. It overlaps the Dirac exchange calculation almost everywhere.

Figure 8

Comparison of ion-ion potentials used in calculations of the structure factor $S_{ii}\left(k\right)$ for warm dense Al at 10 eV and 8.1 g/cm${}^3$. For clarity, the product $rV\left(r\right)$ is shown. The potential calculated with a self-consistent model of WDM [12, 13] [$V_{ii}\left(r\right)$, black] is affected by the shell structure of the ion and is more repulsive than a Yukawa potential with screening wave vector $k_{\mathrm{scr}}=1.07a_B^{-1}$ [18, 21] [$V^Y\left(r\right)$, green (light gray)]. The dashed curve shows the pair distribution function $g\left(r\right)$ computed with the former potential. The ion sphere radius is indicated by the vertical dotted line. The two potentials are substantially different in a range of radii where $g\left(r\right)$ is not small.

Figure 9

Theoretical spectra for warm dense beryllium with the same conditions as in the warm dense Be experiment (5.55 g/cm${}^3$ and 13 eV; Figs. 3 and 4) and the same $k$ vectors and x-ray probe profile as for the experiment on cold Be [7] (Fig. 1). The curves are shown for calculations with the Dirac exchange functional. The solid cyan (light gray) line is the result using Slater exchange and is mostly indistinguishable from the Dirac exchange calculation except near the bound-free threshold. For ease of comparison, we show this figure with the same frequency axis as in Fig. 1, which is reversed from that of Figs. 3, 4, 5 and 4.

Figure 10

Theoretical bound-free and free-free contributions to the x-ray scattering spectrum of warm dense titanium at 5 eV and 4.51 g/cm${}^3$. The probe beam has an energy of ${\omega }_0=4.750$ keV and the angle is ${130}^{\circ }$. The spectrum is not convolved with a source spectrum. The black vertical dotted lines mark the energies of the $3s$ and $3p$ bound states. The green (light gray) vertical dotted lines mark these bound state energies plus the energy of the $3d$ resonant state ($11.7$ eV). Note that the frequency axis is reversed, as in Figs. 1 and 9.