Unraveling the intrinsic atomic physics behind x-ray absorption line shifts in warm dense silicon plasmas

We present a free-energy density functional theory (DFT)-based methodology for optical property calculations of warm dense matter to cover a wide range of thermodynamic conditions and photon energies including the entire x-ray range. It uses Mermin-Kohn-Sham density functional theory with exchange-correlation (XC) thermal effects taken into account via a fully temperature dependent generalized gradient approximation XC functional. The methodology incorporates a combination of the ab initio molecular dynamics (AIMD) snapshotted Kubo-Greenwood optic data with a single atom in simulation cell calculations to close the photon energy gap between the $L$ and $K$ edges and extend the $K$-edge tail toward many-keV photon energies. This gap arises in the standard scheme due to a prohibitively large number of bands required for the Kubo-Greenwood calculations with AIMD snapshots. Kubo-Greenwood data on snapshots provide an accurate description of optic properties at low photon frequencies slightly beyond the $L$ edge and x-ray absorption near edges structure (XANES) spectra, while data from periodic calculations with single atoms cover the tail regions beyond the edges. To demonstrate its applicability to mid-$Z$ materials where the standard DFT-based approach is not computationally feasible, we have applied it to opacity calculations of warm dense silicon plasmas. These first-principles calculations revealed a very interesting phenomenon of redshift-to-blueshift in $K-L$ ($1s\to 2p$) and $K$-edge absorptions along both isotherm and isochore, which are absent in most continuum-lowering models of traditional plasma physics. This new physics phenomenon can be attributed to the underlying competition between the screening of deeply bound core electrons and the screening of outer-shell electrons caused by warm-dense-plasma conditions. We further demonstrate that the ratio of $1s\to 2p$ to the $K$-edge x-ray absorptions can be used to characterize warm-dense-plasma conditions. Eventually, based on our absorption calculations, we have established a first-principles opacity table (FPOT) for silicon in a wide range of material densities and temperatures.

Figure 1

(a) and (b) Absorption of silicon plasmas at $\rho =5.0\mathrm{g}/{\mathrm{cm}}^3$ and $T=250\mathrm{kK}$ from calculations for a single-atom-in-a-cell (black dashed curve), calculations along the MD trajectory (dashed-dotted blue curve), and final combined results (solid red curve). (c) Comparison between absorption calculated along the MD trajectory and data combined according to our methodology for $\rho =8.0\mathrm{g}/{\mathrm{cm}}^3$ and $T=1\mathrm{MK}$. (d) Comparison between the mass absorption coefficient calculated by the present method for $\rho =2.33\mathrm{g}/{\mathrm{cm}}^3$ and $T=300\mathrm{K}$ and NIST “cold solid” data for silicon (ambient conditions).

Figure 2

(a) Convergence of the electronic free-energy and electronic pressure with respect to the energy cutoff for an all-electron PAW data set with cutoff radius $r_{\mathrm{PAW}}=0.75a_0$ performed for a MD snapshot of Si using a 64-atom simulation supercell at ${\rho }_{\mathrm{Si}}=2.57\mathrm{g}/{\mathrm{cm}}^3$ and $T=2000\mathrm{K}$. (b) Convergence of the $L$-edge and $K$-edge tail absorption coefficient with respect to the PAW cutoff radius for Si single atom in simulation cell at ${\rho }_{\mathrm{Si}}=9.0\mathrm{g}/{\mathrm{cm}}^3$ and $T=62500\mathrm{K}$.

Figure 3

Convergence of the absorption coefficient with respect to the system size for silicon plasmas: (a) and (b) at $\rho =9.0\mathrm{g}/{\mathrm{cm}}^3$ and $T=0.5\mathrm{MK}$ and (c) and (d) at $\rho =6.0\mathrm{g}/{\mathrm{cm}}^3$ and $T=2.0\mathrm{MK}$.

Figure 4

(a) Mass absorption coefficient of silicon plasmas at $\rho =0.5\mathrm{g}/{\mathrm{cm}}^3$ and $T=250\mathrm{kK}$. (b) Corresponding density of states (solid blue curve), Fermi-Dirac occupations (dashed green curve), and integrated density of states (solid red curve). (c) and (d) Zoomed-in pictures of (a) and (b) near the $K$ edge and Fermi level respectively.

Figure 5

Mass absorption coefficient and density of states of silicon plasmas along the $T=500\mathrm{kK}$ isotherm.

Figure 6

Mass absorption coefficient and density of states of silicon plasmas along the $\rho =10\mathrm{g}/{\mathrm{cm}}^3$ isochore.

Figure 7

Mass absorption coefficient and density of states of silicon plasmas along the $\rho =35\mathrm{g}/{\mathrm{cm}}^3$ isochore.

Figure 8

(a) and (c) Pressure-induced effect on the $K$ edge and $1s\to 2p$ absorption lines in silicon plasmas along the $T=500\mathrm{kK}$ isotherm. (b) and (d) Temperature-induced effect on the $K$ edge and $1s\to 2p$ absorption line location in silicon plasmas along $\rho =35.0\mathrm{g}/{\mathrm{cm}}^3$ isochore. (e) The ratio of absorption coefficients between $\alpha \left(K\text{−}L\right)$ and $\alpha \left(K\text{edge}\right)$ with subtracted $L$-tail background absorption ${\alpha }_0$ along $T=500\mathrm{kK}$ isotherm. (f) Similar to (e) for the $\rho =35.0\mathrm{g}/{\mathrm{cm}}^3$ isochore.

Figure 9

Comparison of the 48-group mean Rosseland opacity between first-principles DFT and AOT as a function of the central photon energy in each group for silicon plasmas for $\rho =0.5$, 5, and $35\mathrm{g}/{\mathrm{cm}}^3$ and selected temperatures.

Figure 10

Comparison of the 48-group mean Rosseland opacity between first-principles DFT and AOT as a function of the central photon energy in each group for silicon plasmas for $\rho =100$, 250, and $250\mathrm{g}/{\mathrm{cm}}^3$ and selected temperatures.

Figure 11

Total Rosseland mean opacity of silicon at $\rho =5$, 35, and $250\mathrm{g}/{\mathrm{cm}}^3$ as a function of temperature as predicted by our first-principles DFT calculations and AOT model.