Partial ionization in dense plasmas: Comparisons among average-atom density functional models

Nuclei interacting with electrons in dense plasmas acquire electronic bound states, modify continuum states, generate resonances and hopping electron states, and generate short-range ionic order. The mean ionization state (MIS), i.e, the mean charge $Z$ of an average ion in such plasmas, is a valuable concept: Pseudopotentials, pair-distribution functions, equations of state, transport properties, energy-relaxation rates, opacity, radiative processes, etc., can all be formulated using the MIS of the plasma more concisely than with an all-electron description. However, the MIS does not have a unique definition and is used and defined differently in different statistical models of plasmas. Here, using the MIS formulations of several average-atom models based on density functional theory, we compare numerical results for Be, Al, and Cu plasmas for conditions inclusive of incomplete atomic ionization and partial electron degeneracy. By contrasting modern orbital-based models with orbital-free Thomas-Fermi models, we quantify the effects of shell structure, continuum resonances, the role of exchange and correlation, and the effects of different choices of the fundamental cell and boundary conditions. Finally, the role of the MIS in plasma applications is illustrated in the context of x-ray Thomson scattering in warm dense matter.

Figure 1

Surface map of the Thomas-Fermi $\langle Z\rangle$ for Al as a function of temperature and relative density $\rho /{\rho }_0$ (top); surface map of $\Theta$, measuring quantum effects, for electrons in Al plasma, computed using the result from the top panel (middle); and surface map of the electron-ion coupling parameter for Al plasma computed using the result from the top panel (bottom).

Figure 2

Surface map of the ion-ion coupling parameter for Be (top), surface map of the ion-ion coupling parameter for Al (middle), and surface map of the ion-ion coupling parameter for Cu (bottom). In all cases, the TF $\langle Z\rangle$ definition of (19) was used.

Figure 3

Surface map of the absolute magnitude of the finite-temperature exchange-correlation interaction of Ref. [67], in units of eV, for the homogeneous electron gas. Contour lines denote (from left to right) $|u_{xc}|=0.1\text{,}1.0\text{,}\text{and}10$ eV.

Figure 4

Behavior of bound (negative-energy) and free (positive-energy) electron densities, in atomic units, within an IS, for Al at normal solid density and $T=30$ eV. There are fundamental differences between orbital-based and OFDFT models, here illustrated by muze and TF calculations. In particular, with orbitals, the bound electron density can extend beyond the sphere and the orthogonality of bound and continuum orbitals imposes obvious structure on the free-electron density near the nucleus.

Figure 5

Plots of mean ionization (top) and electron degeneracy ${\eta }_e$ (bottom) for Be at normal solid density, versus temperature, as determined by DFT models discussed in the text: Thomas-Fermi (TF), muze without exchange-correlation interactions (MUZE no XC), muze with exchange-correlation interactions (muze), and neutral pseudoatom (NPA).

Figure 6

Same as in Fig. 5, but for Al at normal solid density.

Figure 7

Same as in Fig. 5, but for Cu at normal solid density.

Figure 8

Mean ionization values $Z^{*}$ and $\langle Z\rangle$, as calculated by muze, for Al (top) and Cu (bottom) at $T=30$ eV, versus density. See the text for a discussion of abrupt jumps in $Z^{*}$.

Figure 9

Comparison of the three different MIS values defined in the muze model [(19), (27), and (31)], for Al at normal solid density, versus temperature (top) and comparison of $\langle Z\rangle$ values for Cu at normal solid density, as computed by different DFT models [(19), (31), and (38)], versus temperature (bottom).

Figure 10

Density dependence of Al $\langle Z\rangle$ calculated by muze, at two temperatures and with four different treatments of the exchange-correlation potential (including no XC).

Figure 11

Integrated fraction of the bound ($\epsilon <0$) electron density outside the radius $r=x{a}_i$ and the integrated fraction of the free ($\epsilon \ge 0$) electron density inside the radius $r=x{a}_i$, as computed by muze, for Al at $T=30$ eV and $\rho /{\rho }_0=1$ (top) and Al at $T=30$ eV and $\rho /{\rho }_0=10$ (bottom), with other details listed in Table .

Figure 12

Structure factors for Be at $\rho =1.85$g/cm${}^3$ are shown for the four temperatures: 5 eV (top pair, purple), 10 eV (first pair from top, green), 20 eV (second pair from bottom, red), and 40 eV (bottom pair, blue). The dashed lines correspond to the $Z^{*}$ definition, while the solid lines correspond to the $\langle Z\rangle$ definition, here calculated using the TF model for the ionization and a Yukawa structure calculation. Due to ionization effects when $\Theta <1$, the structure factors are very similar above $q=3$ and provide no ion temperature information over this broad temperature range; however, temperature information is available at longer wavelengths.