Average-atom model combined with the hypernetted chain approximation applied to warm dense matter

We have combined the average-atom model with the hypernetted chain approximation (AAHNC) to describe the electronic and ionic structure in the warm dense matter regime. On the basis of the electronic and ionic structures, the x-ray Thomson scattering (XRTS) spectrum is calculated using the random-phase approximation. While the electronic structure is described within the average-atom model, the effects of other ions on the electronic structure are considered using an integral equation method of the theory of liquids, namely the hypernetted chain approximation. The ion-ion pair potential is calculated using the modified Gordon-Kim model based on the electronic density distribution. Finally, the electronic and ionic structures are determined self-consistently. The XRTS spectrum is calculated according to the Chihara formula, where the scattering contributions are divided into three components: elastic, bound-free, and free-free. Comparison of the present AAHNC results with other theoretical models and experimental data shows very good agreement. Thus the AAHNC model can give a reasonable description of the electronic and ionic structure in warm dense matter.

Figure 1

The ion-ion pair distribution functions $g\left(r\right)$ for Al at a temperature of 10 eV and a mass density of 2.7 g/${\mathrm{cm}}^3$. Black solid line: AAHNC result, red dashed line with circles: calculated by classical molecular-dynamics simulations using the ion-ion pair potential from AAHNC, blue dot-dashed line: QHNC result [35], green dashed line: QMD result [48]. The orange dashed line labels the distance of twice the ion-sphere radius.

Figure 2

Density dependence of the average ionization degree of Al at a temperature of 100 eV (left panel) and 10 eV (right panel). Black line with circles: AAHNC result, red line with squares: result of the AA model with broadening of the energy levels.

Figure 3

Elastic XRTS contribution for Al as a function of $k$ at a temperature of 10 eV and a density of 8.1 g/${\mathrm{cm}}^3$: AAHHC (black lines with circles), QMD (red lines with triangles) [61], and HNC (blue lines with squares) [38] results are compared with experimental results (points with error bars) [39]. The static structure factors $S_{\text{ii}}\left(k\right)$ are shown as dot-dashed lines, total form factors $|f_I\left(k\right)+q\left(k\right)|$ as dashed lines, and the ion features $|f_I{\left(k\right)+q\left(k\right)|}^2{S}_{\text{ii}}\left(k\right)$ as solid lines with the respective color code.

Figure 4

Calculated XRTS spectra for warm dense Al at a temperature of 10 eV and a density of 8.1 g/${\mathrm{cm}}^3$ compared with experimental results (blue dot-dashed line) [39] at scattering angles ${69}^{\circ }$ (left panel) and ${111}^{\circ }$ (right panel). The black and orange solid lines denote the total spectrum (high) and bound-free (bf) (low) contributions, respectively, calculated from the AAHNC method; the free-free contribution is calculated in the RPA. The red and green dashed lines are, respectively, the total spectrum (high) and the bound-free (low) contributions from Souza et al. [38].

Figure 5

Electron density distribution when two nuclei approach each other; see [57].

Figure 6

Flow chart for the self-consistent calculation of the electronic and ionic structure in the AAHNC method.