Analysis of density effects in plasmas and their influence on electron-impact cross sections

Density effects in plasmas are analyzed using a Thomas-Fermi approach for free electrons. First, scaling properties are determined for the free-electron potential and density. For hydrogen-like ions, the first two terms of an analytical expansion of this potential as a function of the plasma coupling parameter are obtained. In such ions, from these properties and numerical calculations, a simple analytical fit is proposed for the plasma potential, which holds for any electron density, temperature, and atomic number, at least assuming that Maxwell-Boltzmann statistics is applicable. This allows one to analyze perturbatively the influence of the plasma potential on energies, wave functions, transition rates, and electron-impact collision rates for single-electron ions. Second, plasmas with an arbitrary charge state are considered, using a modified version of the Flexible Atomic Code (FAC) package with a plasma potential based on a Thomas-Fermi approach. Various methods for the collision cross-section calculations are reviewed. The influence of plasma density on these cross sections is analyzed in detail. Moreover, it is demonstrated that, in a given transition, the radiative and collisional-excitation rates are differently affected by the plasma density. Some analytical expressions are proposed for hydrogen-like ions in the limit where the Born or Lotz approximation applies and are compared to the numerical results from the FAC.

Figure 1

Self-consistent free-electron density in H-like helium for various densities and temperatures. The local free-electron density $n_e\left(r\right)$ in units of the average density $N_e=3Z_f/4\pi R_0^3$ is plotted versus $r$ in units of the Wigner radius $R_0$ for various plasma-coupling parameters. See text for details.

Figure 2

Radial dependence of the self-consistent plasma potential in H-like aluminum, for three values of the coupling parameter, $R_1/R_0=0.01$, 1, and 100. The abscissa is the radius in $R_0$ units; the ordinate is the potential in $Z_f{e}^2/R_0$ units. The solid line is the numerically obtained self-consistent potential; the circles, crosses, and dashed line are quartic polynomial approximations obtained at three successive steps of the fit discussed in Sec. 4.

Figure 3

Variation of the reduced plasma potential divided by the coupling parameter $R_1/R_0$ in H-like helium. The scaled variation $\delta V_{\mathrm{pl}}^{\left(\mathrm{sc}\right)}\left(x\right)=(R_0/Z_f{e}^2)[V_{\mathrm{pl}}\left(r\right)-V_{\mathrm{pl}}^{\mathrm{UEGM}}\left(r\right)]/(R_1/R_0)$ is plotted versus $x=r/R_0$ for various $R_1/R_0$ and is compared to the analytical form $y=(3\pi /10){(1-x)}^2[5-x-{(1-x)}^2/4]$ as given by (15b). From bottom to top the various curves correspond to decreasing values of $R_1/R_0$, the topmost curve being the analytical form.

Figure 4

Influence of temperature on the binding energy of Al XIII for the $1s_{1/2}$ level with an average density $N_e={10}^{23}{\mathrm{cm}}^{-3}$.

Figure 5

Influence of density on the binding energy of Al XIII and XII for the $1s_{1/2}$ (upper group) and $1s^2{}^1{S}_0$ (lower group) levels.

Figure 6

Energy of helium-like Al relative to the level $1s4s{}^3{S}_1$ versus density for various levels of the configuration $1s4l$. Plasma density effects are accounted for using a Thomas-Fermi potential at 100 eV. The three curves associated with the ${}^3\mathrm{D}$ term are almost identical, with similar overlapping for the two curves associated with the ${}^3{P}_0$ and ${}^3{P}_1$ levels.

Figure 7

Dipolar radiative rates $1s4p{}^3{P}_J\to 1s4s{}^3{S}_1$ in Al XII versus the average electron density at $T_e=100\mathrm{eV}$. Gray (red) lines correspond to $J=1$; black lines, to $J=0$.

Figure 8

Dipolar radiative rates in Al XIII versus the average electron density at $T_e=500\mathrm{eV}$. Gray (red) curves represent the $2p_{3/2}\to 1s$ rates; black curves, the $2p_{1/2}\to 1s$ rates.

Figure 9

Comparison of excitation cross sections for the $1s–2p_{1/2}$ transition for Al XIII at several densities and $T=100\mathrm{eV}$.

Figure 10

Comparison of excitation cross sections for the transition between $1s4d{}^1{D}_2$ and $1s4p{}^1{P}_1$ for Al XII at several densities.

Figure 11

Excitation cross sections for the transition $1s_{1/2}–2p_{1/2}$ in Al XIII: comparison between the Born approximation and the Van Regemorter formula. Top: Cross sections. Bottom: Variations $\sigma (N_e=0)-\sigma (N_e>0)$ for both approximations. The plasma effect is accounted for within the UEGM. The Born data are shown in black; the Van Regemorter data, in gray (red).

Figure 12

Comparison of excitation cross sections for the transitions $1s–2p_{1/2}$ (dashed lines) and $1s_{1/2}–2s_{1/2}$ (solid line) for Al XIII using the Born approximation. When density effects are included, the Thomas-Fermi model is used, with a temperature of 500 eV. Top: Black (upper) curves represent cross sections without density effects, while gray (red; lower) curves represent cross sections at $N_e={10}^{25}{\mathrm{cm}}^{-3}$. Bottom: Variations $\sigma (N_e=0)-\sigma (N_e>0)$ for both transitions.

Figure 13

Comparison of ionization cross sections for the transition $1s^2$ to $1s$ for Al XII at $T=200\mathrm{eV}$ (and 50 eV) for $N_e={10}^{23}{\mathrm{cm}}^{-3}$. The free-electron density is obtained from the Thomas-Fermi model, and for the scattering process the BED and Lotz formalisms are compared. Grayscale version: the curves are in the same order in the graph and in the legend.

Figure 14

Modification of the ionization cross section due to the density effect for the transition $1s^2$ to $1s$ in Al XII obtained using the BED theory or Lotz formula. The plotted quantities are the cross sections at $N_e={10}^{23}{\mathrm{cm}}^{-3}$ and $T=200\mathrm{eV}$ minus the cross sections at $N_e=0{\mathrm{cm}}^{-3}$.

Figure 15

Influence of statistics on the self-consistent free-electron density and plasma potential for H-like aluminum at $T_e=1\mathrm{eV}$ and $N_e={10}^{24}{\mathrm{cm}}^{-3}$ or $0.148a_0^{-3}$. The density is in units of the average free-electron density $N_e=3Z_f/4\pi R_0^3$, the potential energy is in units of $Z_f{e}^2/R_0$, and the electronic distance to nucleus $r$ is in units of the Wigner sphere radius $R_0=2.684a_0$.