Ionization potential depression and Pauli blocking in degenerate plasmas at extreme densities

New facilities explore warm dense matter (WDM) at conditions with extreme densities (exceeding ten times condensed matter densities) so that electrons are degenerate even at temperatures of $10–100$ eV. Whereas in the nondegenerate region correlation effects such as Debye screening are relevant for the ionization potential depression (IPD), new effects have to be considered in degenerate plasmas. In addition to the Fock shift of the self-energies, the bound-state Pauli blocking becomes important with increasing density. Standard approaches to IPD such as Stewart-Pyatt and widely used opacity tables (e.g., OPAL) do not contain Pauli blocking effects for bound states. The consideration of degeneracy effects leads to a reduction of the ionization potential and to a higher degree of ionization. As an example, we present calculations for the ionization degree of carbon plasmas at $T$ = 100 eV and extreme densities up to 40 $\mathrm{g}/{\mathrm{cm}}^3$, which are relevant to experiments that are currently scheduled at the National Ignition Facility.

Figure 1

Ionization potential depression (IPD) of ${\mathrm{C}}^{5+}$ as function of the free-electron density $n_e$ at fixed temperature $T=100$ eV. The Fock shift of the continuum edge (cont. Fock), Eq. (20) at $p=0$, together with the Stewart-Pyatt (SP) IPD (2) yields the shift of the continuum (cont. Fock+SP). For the ion ${\mathrm{C}}^{5+}$ in its ground state, the Pauli shift Eq. (32) (Pauli) and the bound-state Fock shift Eq. (30) (bound Fock) as well as the sum of both (bound Fock+Pauli) are presented. For comparison, the IPD obtained from the ionic structure factor shift of the continuum (SF, ions) as improvement of the SP model [20] is also shown. The Mott transition is predicted at $n_e=1.3\times {10}^{25}{\mathrm{cm}}^{-3}$. Upper scale: Carbon mass density.

Figure 2

Average ionization degree $\overline{Z}$ of carbon as function of the free-electron density $n_e$ for temperature $T=100$ eV. Ideal mixture with electron treated classically (ideal,class) and as Fermi gas (ideal, deg), OPAL and Stewart/Pyatt (BU,SP) are also shown. BU denotes the use of the Beth-Uhlenbeck Eqs. (15) and (18) for the intrinsic partition function, SP the Stewart-Pyatt contribution Eq. (2). In addition the SP Eq. (2), the full IPD (BU,SP,Pauli) contains the Fock shift of the continuum Eq. (20) at $p=0$ as well as the Fock shift of the bound-state Eq. (30) and the Pauli blocking Eq. (32).

Figure 3

Temperatures for the ionization degree of $\overline{Z}=5.9$ (isoionization lines) using different approximations for the IPD. $T_{5.9}^{\mathrm{ideal}}$ for the ideal mixture is compared with the ideal Beth-Uhlenbeck expression $T_{5.9}^{\mathrm{BU},\mathrm{ideal}}$ Eqs. (15) and (18), neglecting any medium effects. The account of the correlation shifts for the IPD, given by the Stewart-Pyatt model Eq. (2), yields the graph $T_{5.9}^{\mathrm{BU},\mathrm{SP},\mathrm{class}}$ for a classical free-electron gas or $T_{5.9}^{\mathrm{BU},\mathrm{SP},\mathrm{deg}}$ for the Fermi gas. In addition to the SP shift, the full IPD contains the Fock shift of the continuum Eq. (20), $p=0$, as well as the Fock shift of the bound-state Eq. (30) and the Pauli blocking Eq. (32). It gives the graph $T_{5.9}^{\mathrm{BU},\mathrm{SP},\mathrm{Pauli}}$.