Unified description of interactions and energy loss of particles in dense matter and plasmas

In this work, we propose a unified model to evaluate processes of electronic interactions of charged particles with hot and dense matter, including energy losses, mean free paths, and thermalization ranges of protons or other light ions. To formulate this method, we introduce modifications to the extended-wave-packet method, which allows one to describe and evaluate the effects of ionization as well as changes in target density and temperature. The ionization of the target leads to the formation of a dense surrounding plasma with distinct energy-absorption properties. We use this unified method to evaluate the contributions of inner shells and free electrons (produced by the target ionization) to the energy loss of protons in Si, C, and Fe targets, on an extensive range of parameters that include low, intermediate, and high energies, with densities and temperatures going from normal laboratory conditions to very high values, such as those of interest for inertial fusion and astrophysical studies.

Figure 1

Stopping power values calculated with the present approach (solid red lines) and comparisons with reference values for a quantum plasma with $r_s=2$ and various temperatures. (a) Curve with solid circles: exact results calculated with Lindhard's dielectric function and with the AB theory (which coincides with the Lindhard function at $T=0$); open circles: results of Kaneko's WPM model for a degenerate FEG. (b)–(d) Results obtained with the present approach (solid red lines) and with the AB theory (solid circles). The dashed line in these panels shows the WPM results for $T=0$ only to illustrate the effect of the temperature with respect to these fixed values. The dotted curve shows the asymptotic limit given by the Bethe formula (which is independent of the temperature).

Figure 2

Stopping cross section for protons on a cold Si target as a function of the projectile velocity. Dashed blue line: WPM; full red line: EWPM; dotted red line: FEG model with $r_s=2.01$. Symbols: Experimental data extracted from [51].

Figure 3

Stopping cross section for protons on a cold C target as a function of the projectile velocity. Symbols: Experimental data extracted from [51].

Figure 4

Stopping cross section for protons on a cold Fe target as a function of the projectile velocity. Dash-dot-dotted blue line: WPM-1 calculated with the original parameters. Dashed blue line: WPM-2 calculated with a modified parameter for $d$ electrons as described in the text. Dash-dotted red line: EWPM-1 model calculated with the original parameters. Solid red line: EWPM-2 model calculated with a modified parameter for $d$ electrons as described in the text. Dotted red line: FEG with $r_s=2.12$. Symbols: Experimental data extracted from [51].

Figure 5

Stopping cross section for protons on three different cold targets for several fixed degrees of ionization $N$ as a function of the projectile velocity: (a) Si, (b) C, and (c) Fe.

Figure 6

Silicon ionization energies for levels $1s, 2s$, and $2p$ as a function of the number of ionized electrons $N$, according to Ref. [56].

Figure 7

Stopping cross-section contributions of levels $3s, 3p$, and $3d$ of Fe, as a function of the projectile velocity, for $T=0$ and for different numbers of ionized electrons $N$. Full thick line: total; full thin red line: FEG; dash-dotted line (only for $N=2$): $3d$; dashed line (for $N=2$ and 8): $3p$; dotted line: $3s$.

Figure 8

Stopping cross sections for different temperatures and a fixed value of target ionized electrons $N$ as a function of the projectile velocity: (a) Si, (b) C, and (c) Fe.

Figure 9

Stopping cross section for protons on a Si target with a fixed degree of ionization $N=4$ for different temperatures and densities, as a function of the projectile velocity. Black lines: normal density ${\rho }_0$; thick blue lines: $8{\rho }_0$; thin red lines: ${\rho }_0/8$. Full lines: $T=30$ eV; dashed lines: $T=60$ eV; dotted lines: $T=90$ eV.

Figure 10

(a) Mean ionization charge $\langle N\rangle$ as a function of temperature for a Si target with normal solid-state density according to the flychk code [59]. (b) Stopping cross section for protons traversing a Si target for different temperatures as a function of the projectile velocity.

Figure 11

Stopping cross section for protons traversing dense stellar regions with high temperatures, $T_1=45$ and $T_2=86\mathrm{keV}$, as a function of the particle velocity. Thin-solid and thin-dashed lines: relativistic corrections. Dash-dotted lines: analytical low-energy approximation of Eq. (23).

Figure 12

Stopping power, range ($R$), and inverse mean free path (IMFP) for protons, deuterons, and muons moving within solar regions, for densities and temperatures corresponding to the sun center ($n_e=8.4\times {10}^{25} {\mathrm{cm}}^{-3}, T_1=1.38\mathrm{keV}$), as a function of the projectile velocity.