Nonideal effects on ionization potential depression and ionization balance in dense Al and Au plasmas

For low-density plasmas, the ionization balance can be properly described by the normal Saha equation in the chemical picture. For dense plasmas, however, nonideal effects due to the interactions between the electrons and ions and among the electrons themselves affect the ionization potential depression and the ionization balance. With the increasing of plasma density, the pressure ionization starts to play a more obvious role and competes with the thermal ionization. Based on a local-density temperature-dependent ion-sphere model, we develop a unified and self-consistent theoretical formalism to simultaneously investigate the ionization potential depression, the ionization balance, and the charge states distributions of the dense plasmas. In this work, we choose Al and Au plasmas as examples as Al is a prototype light element and Au is an important heavy element in many research fields such as in the inertial confinement fusion. The nonideal effect of the free electrons in the plasmas is considered by the single-electron effective potential contributed by both the bound electrons of different charge states and the free electrons in the plasmas. For the Al plasmas, we can reconcile the results of two experiments on measuring the ionization potential depression, in which one experiment can be better explained by the Stewart-Pyatt model while the other fits better with the Ecker-Kröll model. For dense Au plasmas, the results show that the double peak structure of the charge state distribution appears to be a common phenomenon. In particular, the calculated ionization balance shows that the two- and three-peak structures can appear simultaneously for denser Au plasmas above $\sim 30\mathrm{g}/{\mathrm{cm}}^3$.

Figure 1

Predictions of the IPDs for solid-density Al plasmas by our model (this work 1) are compared with the experimental results (Expt.) [5, 6] and other available theoretical models of the dynamical ionic structure factor (SF) [24], the SP [12], the ion-sphere (IS) [53], the EK [13], and the Debye-Hückel (DH) [54] models. The electron temperature is taken to be 200 eV which is obtained from a theoretical simulation [6]. The dashed-dot-diamond line (this work 2) represents our theoretical results using an effective electron temperature of 155, 92, and 45 eV for ${\mathrm{Al}}^{7+}$, ${\mathrm{Al}}^{8+}$, and ${\mathrm{Al}}^{9+}$, respectively.

Figure 2

Ionization potential depressions of Al plasmas at the solid-density of 2.7 $\mathrm{g}/{\mathrm{cm}}^3$ and temperatures of 100 eV (a), 300 eV (b), and 600 eV (c). Predictions by our model are compared with the analytical models of SP [12], the ion-sphere (IS) [53], EK [13], and Debye-Hückel (DH) [54].

Figure 3

Ionization potential depressions of Al plasmas at the temperature of 200 eV and mass densities of 1.2 $\mathrm{g}/{\mathrm{cm}}^3$ (a), 2.5 $\mathrm{g}/{\mathrm{cm}}^3$ (b), and 4.0 $\mathrm{g}/{\mathrm{cm}}^3$ (c). Predictions by our model are compared with those by the other analytical models, including the SP [12], the IS [53], the EK [13], and the DH [54] models.

Figure 4

Ionization potential depressions of Al plasmas at the temperature of 550 eV and mass densities of 4.0 $\mathrm{g}/{\mathrm{cm}}^3$ (a), 5.5 $\mathrm{g}/{\mathrm{cm}}^3$ (b), and 9.0 $\mathrm{g}/{\mathrm{cm}}^3$ (c). Predictions by our model are compared with those by the analytical models of the SP [12], the IS [53], the EK [13], and the DH [54] models.

Figure 5

Comparison of our predicted charge state distribution of Al plasmas at the solid-density of 2.7 $\mathrm{g}/{\mathrm{cm}}^3$ and temperatures of 100 eV (a), 300 eV (b), and 600 eV (c) with those obtained by the analytical models of the SP [12], the IS [53], the EK [13], and the DH [54] models.

Figure 6

Comparison of our predicted charge state distribution in the Al plasmas at the temperature of 200 eV and mass densities of 1.2 $\mathrm{g}/{\mathrm{cm}}^3$ (a), 2.5 $\mathrm{g}/{\mathrm{cm}}^3$ (b), and 4.0 $\mathrm{g}/{\mathrm{cm}}^3$ (c) with those obtained by the analytical models of the SP [12], the IS [53], the EK [13], and the DH [54] models.

Figure 7

Comparison of our predicted charge state distribution in the Al plasmas at the temperature of 550 eV and mass densities of 4.0 $\mathrm{g}/{\mathrm{cm}}^3$ (a), 5.5 $\mathrm{g}/{\mathrm{cm}}^3$ (b), and 9.0 $\mathrm{g}/{\mathrm{cm}}^3$ (c) with those obtained by the theoretical models of the SF [24], the SP [12], the IS [53], the EK [13], and the DH [54] models. The black long-dashed line predicted by the SF model is almost completely overlapped by our results, making it nearly invisible in the plot. The label SFii represents the results obtained by the dynamical SF method contributed by the ionic part in the plasma in (b) at 5.5 $\mathrm{g}/{\mathrm{cm}}^3$.

Figure 8

Relative partition functions $b_i$ of the free electrons for different charge states in following Al plasma conditions of temperatures and densities: $[550\mathrm{eV},1.2\mathrm{g}/{\mathrm{cm}}^3]$, $[650\mathrm{eV},2.5\mathrm{g}/{\mathrm{cm}}^3]$, $[700\mathrm{eV},4.0\mathrm{g}/{\mathrm{cm}}^3]$, and $[700\mathrm{eV},9.0\mathrm{g}/{\mathrm{cm}}^3]$. The relative partition functions of the free electrons are evaluated relative to that of the uniform electron gas $[Z_e=2{\left(\frac{2\pi m_{e}kT}{h^2}\right)}^{3/2}]$ for each respective plasma condition.

Figure 9

Comparison of the charge state distribution of Al plasmas with (black solid lines) and without (red dashed lines) considering the NIC effect on the partition functions of free electrons at the following plasma temperature and density conditions: $[650\mathrm{eV},2.5\mathrm{g}/{\mathrm{cm}}^3]$ (a), $[700\mathrm{eV},4.0\mathrm{g}/{\mathrm{cm}}^3]$ (b), and $[700\mathrm{eV},9.0\mathrm{g}/{\mathrm{cm}}^3]$ (c).

Figure 10

Comparison of the theoretical emissivity with the experimental data [8] and the FLYCHK predictions using the SP [12] and the EK [13] models for the Al plasmas with (red dashed-dot lines) and without (blue dashed lines) considering the NIC effect on the partition functions of free electrons. The plasma temperature and density conditions are $[550\mathrm{eV},1.2\mathrm{g}/{\mathrm{cm}}^3]$, $[650\mathrm{eV},2.5\mathrm{g}/{\mathrm{cm}}^3]$, $[700\mathrm{eV},4.0\mathrm{g}/{\mathrm{cm}}^3]$, $[550\mathrm{eV},5.5\mathrm{g}/{\mathrm{cm}}^3]$, and $[700\mathrm{eV},9.0\mathrm{g}/{\mathrm{cm}}^3]$ from the bottom to the top. The SP model has an additional curve at the density of 11.6 $\mathrm{g}/{\mathrm{cm}}^3$.

Figure 11

Ionization potential depression of the dense Au plasmas at the temperature of 600 eV and mass densities of 10, 20, 30, and 40 $\mathrm{g}/{\mathrm{cm}}^3$.

Figure 12

IPD of dense Au plasmas at the temperature of 1000 eV and mass densities of 10, 20, 30, and 40 $\mathrm{g}/{\mathrm{cm}}^3$.

Figure 13

IPD of dense Au plasmas at temperatures of 400, 800, 1500, and 2000 eV and different mass densities of 40 $\mathrm{g}/{\mathrm{cm}}^3$ (a), 30 $\mathrm{g}/{\mathrm{cm}}^3$ (b), and 20 $\mathrm{g}/{\mathrm{cm}}^3$ (c).

Figure 14

Relative partition functions ($b_i$ in modified Saha [8]) of the free electrons for different charge states in dense Au plasmas at the temperature of 1000 eV and mass densities of 1.0, 10.0, 20,0, 30.0, and 40.0 $\mathrm{g}/{\mathrm{cm}}^3$, respectively. For each plasma condition, the partition functions are scaled relative to that of the uniform electron gas $[Z_e=2{\left(\frac{2\pi m_{e}kT}{h^2}\right)}^{3/2}]$.

Figure 15

Comparison of population fractions of the charge state distribution of dense Au plasmas with (solid lines) and without (dashed lines) considering the NIC effect of free electrons on the ionization balance at the mass densities of (a) 40.0 $\mathrm{g}/{\mathrm{cm}}^3$, (b) 30.0 $\mathrm{g}/{\mathrm{cm}}^3$, and (c) 5.0 $\mathrm{g}/{\mathrm{cm}}^3$ and at the temperatures of 400 eV (red solid and dashed lines), 600 eV (blue solid and dashed lines), 800 eV (brown solid and dashed lines), 1000 eV (yellow solid and dashed lines), 1500 eV (lavender solid and dashed lines), and 2000 eV (deep purple solid and dashed lines).

Figure 16

The two-peak structure of dense Au plasmas at the density of 5.0 $\mathrm{g}/{\mathrm{cm}}^3$ and different temperatures. The population fractions of the CSDs are given at the temperatures of 600, 650, 680, 700, 740, and 800 eV, at which plasma conditions the phenomenon of the two-peak structure can be found. The NIC effects have been considered in the determination of the ionization balance.

Figure 17

The two- and three-peak structures of dense Au plasmas plasma at the density of 40.0 $\mathrm{g}/{\mathrm{cm}}^3$ and different temperatures. The population fractions of different charge states are given at the temperatures of 150, 200, 230, 255, 280, and 300 eV (a); 400, 430, 450, 480, 500, and 570 eV (b); and 670, 750, 800, 900, and 1000 eV (c). At these plasma conditions the phenomenon of the two-peak structure can be found in a wide temperature range. The NIC effects have been considered in the determination of the ionization balance.

Figure 18

Average degree of ionization of Au plasmas as a function of plasma temperature at mass densities of 5.0, 20.0 and 40.0 $\mathrm{g}/{\mathrm{cm}}^3$. The NIC effects have been considered in the determination of ionization balance.