Stopping power of hot dense deuterium-tritium plasmas mixed with impurities to charged particles

In this work, we studied the stopping power of deuterium-tritium (DT) plasmas mixed with impurities to the injected charged particles. Based on the Brown-Preston-Singleton model, the analytical expression for the change ratio of stopping power (denoted by $\eta )$ induced by impurities in DT plasmas is developed, in which both classical short-distance collision part and quantum correction contribution are purely linear response to the impurity concentration ${\xi }_X$, while the classical long-range collision brings about higher-order nonlinear response to ${\xi }_X$. Furthermore, the expression for change ratio of deposition depth (denoted by $\chi )$ of charged particles induced by impurities in DT plasmas is also derived. As applications, we systemically investigated the energy loss of $\alpha$ particles deposited into a hot dense DT plasma mixed with impurity $X(X=\mathrm{C}$, Si, Ge), where the temperature and density of DT are smaller than 10 keV and 500 g/${\mathrm{cm}}^3$ and the concentration of $X{\xi }_X$ is less than $5\%$. The numerical results suggest that (i) for the case of C mixed into DT, both change ratios of stopping power and deposition depth of $\alpha$ particles (i.e., $\eta$ and $\chi )$ are linear response to the concentration of C ${\xi }_C$; (ii) for the case of Si mixed into DT, the second-order nonlinear response of $\eta$ and $\chi$ to ${\xi }_{\text{Si}}$ cannot be ignored when the densities of DT are larger than 200 g/${\mathrm{cm}}^3$; and (iii) for the case of Ge mixed into DT, the second- and third-order nonlinear response of $\eta$ and $\chi$ to ${\xi }_{\text{Ge}}$ are very remarkable because of the higher ionization degree and heavier atomic mass of Ge. The formulas and findings in this work may be helpful to the research of internal confinement fusion (ICF) related implosion physics and may provide useful theoretical guidance and data for the design of ICF target.

Figure 1

Overall coupling strength $g_p$ of the projected $\alpha$ particle to the $\mathrm{DT}+X(X$ = C, Si, and Ge) plasmas as a function of temperature $T$ (in unit of keV) and density ${\rho }_{\text{DT}}$ (in unit of g/${\mathrm{cm}}^3)$ of DT. Panels (a) and (b) are for the case of $\mathrm{DT}+\mathrm{C}$ with ${\xi }_C=2\%$ and ${\xi }_C=5\%$, panels (c) and (d) are for the case of $\mathrm{DT}+\mathrm{Si}$ with ${\xi }_{\text{Si}}=2\%$ and ${\xi }_{\text{Si}}=5\%$, and panels (e) and (f) are for the case of $\mathrm{DT}+\mathrm{Ge}$ with ${\xi }_{\text{Ge}}=2\%$ and ${\xi }_{\text{Ge}}=5\%$.

Figure 2

Change ratio of stopping power of DT plasmas mixed with carbon to $\alpha$ particle as a function of the incident energy of $\alpha$-particle $E_{\alpha }$. (a) For the total stopping power, (b) for the classical part of stopping power, and (c) for the quantum part of stopping power. Different data symbols are for different densities of T, i.e., black squares, red circles, green triangles, and blue stars correspond to ${\rho }_T=5$, 10, 50, and 100 g/${\mathrm{cm}}^3$, respectively. The temperature of plasma is chosen as $T=5$ keV, the carbon mixing concentration is ${\xi }_C=2\%$, and the density of D is chosen as ${\rho }_D={\rho }_T$.

Figure 3

Change ratio of stopping power of $\mathrm{DT}+\mathrm{C}$ plasmas to $\alpha$ particle as a function of the C-mixing concentration ${\xi }_C$. Various densities of DT plasmas are considered that ${\rho }_D={\rho }_T=1$, 5, 10, 50, 100, 200, 300, 400, and 500 g/${\mathrm{cm}}^3$, and the temperature of plasmas is chosen as $T=5$ keV in (a) and $T=10$ keV in (b), respectively. Discrete symbols are calculated via Eq. (32), while the color lines are calculated via Eq. (33) where just linear term $\sim {\alpha }_1{\xi }_C$ is taken into account. The initial energy of $\alpha$ particle is chosen as $E_0=3.54$ MeV.

Figure 4

Change ratio of deposition depth of $\alpha$ particle moving in the $\mathrm{DT}+\mathrm{C}$ plasmas as a function of the C-mixing concentration ${\xi }_C$. (a) for the temperature of plasmas $T=5$ keV, and (b) for $T=10$ keV, respectively. Discrete symbols are calculated via Eq. (42), while the color lines are calculated via Eqs. (43) and (44) where just linear term $\sim {\gamma }_1{\xi }_C$ is taken into account. Other parameters are the same as those used in Fig. 3.

Figure 5

Change ratio of stopping power of $\mathrm{DT}+\mathrm{Si}$ plasmas to $\alpha$ particle as a function of the Si-mixing concentration ${\xi }_{\text{Si}}$. All other parameters are the same as those used in Fig. 3.

Figure 6

Change ratio of deposition depth of $\alpha$ particle moving in $\mathrm{DT}+\mathrm{Si}$ plasmas as a function of the Si-mixing concentration ${\xi }_{\text{Si}}$. All other parameters are the same as those used in Fig. 4.

Figure 7

Change ratio of stopping power of $\mathrm{DT}+\mathrm{Ge}$ plasmas to $\alpha$ particle as a function of the Ge-mixing concentration ${\xi }_{\text{Ge}}$. All other parameters are the same as those used in Fig. 3.

Figure 8

Change ratio of deposition depth of $\alpha$-particle moving in $\mathrm{DT}+\mathrm{Ge}$ plasmas as a function of the Ge-mixing concentration ${\xi }_{\text{Ge}}$. Here we just show the results for three different densities ${\rho }_T=10$, 100, and 500 g/${\mathrm{cm}}^3$. All other parameters are the same as those used in Fig. 4.