Ab initio approach to model x-ray diffraction in warm dense matter

It is demonstrated how the static electron-electron structure factor in warm dense matter can be obtained from density functional theory in combination with quantum Monte Carlo data. In contrast to theories assuming well-separated bound and free states, this ab initio approach yields also valid results for systems close to the Mott transition (pressure ionization), where bound states are strongly modified and merge with the continuum. The approach is applied to x-ray Thomson scattering and compared to predictions of the Chihara formula whereby we use the ion-ion and electron-ion structure from the same simulations. The results show significant deviations of the screening cloud from the often applied Debye-like form.

Figure 1

Top panel: Static structure factors for beryllium at $T=12\mathrm{eV}$ and $n=1.848{\mathrm{g}\mathrm{cm}}^{-3}$: the free-electron structure factor $S_{ee}^0$ from QMC is according to Ref. [28]. The dots for $k=0$ are the structure factors as obtained from the compressibility sum rule [1]. Bottom panel: All partial structure factors for plastic (CH) at $T=10\mathrm{eV}$ and $n=5.84{\mathrm{g}\mathrm{cm}}^{-3}$.

Figure 2

Top panel: Screening cloud in warm dense beryllium at $T=12\mathrm{eV}$ and $n=1.848{\mathrm{g}\mathrm{cm}}^{-3}$. The DFT result (solid red line), data from linear-response theory [Eq. (12)] (dotted green line), and empty core approximation [Eq. (13)] (dashed black line) [47]. Bottom panel: Generalized form factors in plastic at $T=10\mathrm{eV}$ and $n=5.84{\mathrm{g}\mathrm{cm}}^{-3}$ for carbon (solid red line) and hydrogen (dash-dotted blue line). Also shown are the form factors for a carbon atom (dashed brown line), for a ${\text{C}}^{4+}$ carbon ion (dotted brown line), a screening cloud model for hydrogen using Eq. (12), and $f+q$ for carbon.

Figure 3

The weight of the Rayleigh peak. Top panel: beryllium at $T=12\mathrm{eV}$ and $n=1.848{\mathrm{g}\mathrm{cm}}^{-3}$. The direct DFT result via Eq. (11) is compared to the results from Eq. (10) applying $S_{ei}$ and $S_{ii}$ from DFT-MD and results of Eq. (2) employing two models for the screening cloud. Additionally, the square of the form factor of an isolated ${\text{Be}}^{2+}$ ion is shown. Bottom panel: plastic (CH) at $T=10\mathrm{eV}$ and $n=5.84{\mathrm{g}\mathrm{cm}}^{-3}$. The direct DFT result via Eq. (11) and the decompositions according to Eqs. (10) and (2) using the empty core model for the screening cloud are compared. Furthermore, the square of the form factor of a ${\text{C}}^{4+}$ ion is shown.