Energy loss of heavy ions in dense plasma. I. Linear and nonlinear Vlasov theory for the stopping power

Abstract

The plasma physics of heavy-ion stopping in fully ionized matter is developed on the basis of the Vlasov-Poisson equations with particular emphasis on small ion velocities ${\mathit{v}}_{\mathit{p}}$, below the electron thermal velocity ${\mathit{v}}_{\mathrm{th}}$, and on solutions nonlinear in the coupling parameter scrZ=${\mathit{Z}}_{\mathrm{eff}}$/(${\mathit{n}}_{0}$${\ensuremath{\lambda}}_{\mathit{D}}^{3}$) between the heavy-ion projectile with effective charge ${\mathit{Z}}_{\mathrm{eff}}$ and the plasma with electron density ${\mathit{n}}_{0}$ and Debye length ${\ensuremath{\lambda}}_{\mathit{D}}$. Concerning the stopping power in the low-velocity regime relevant for the Bragg peak at the end of the ion range, results on the friction term dE/dx\ensuremath{\propto}${\mathit{v}}_{\mathit{p}}$ are presented, and an improved dE/dx formula for plasma is derived in closed form and readily applicable for stopping-power calculations; it is identical to the standard result for ${\mathit{v}}_{\mathit{p}}$>${\mathit{v}}_{\mathrm{th}}$, but also describes the limit ${\mathit{v}}_{\mathit{p}}$\ensuremath{\rightarrow}0 correctly.For ${\mathit{v}}_{\mathit{p}}$${\mathit{v}}_{\mathrm{th}}$, nonlinear results are found to contribute to the stopping power with terms \ensuremath{\propto}${\mathit{scrZ}}^{5/2}$ for positive ions and terms \ensuremath{\propto}${\mathit{scrZ}}^{3}$ for negative ions in addition to the basic ${\mathit{scrZ}}^{2}$ term; they are derived from a low-velocity expansion of a form-factor representation of dE/dx. Concerning high velocities ${\mathit{v}}_{\mathit{p}}$>${\mathit{v}}_{\mathrm{th}}$, the relevant coupling parameter is scrZ(${\mathit{v}}_{\mathrm{th}}$/${\mathit{v}}_{\mathit{p}}$${)}^{3}$, and nonlinear corrections to the stopping power \ensuremath{\propto}${\mathit{scrZ}}^{3}$/${\mathit{v}}_{\mathit{p}}^{5}$ are obtained by extending the work of Ashley, Ritchie, and Brandt [Phys. Rev. B 5, 2393 (1972)] to the plasma case. An interpolation between the low- and the high-velocity results is given; taking, e.g., parameters characteristic for heavy-ion beam inertial fusion the nonlinear corrections further enhance dE/dx up to 10% in the Bragg peak region. An application of the present results to heavy-ion energy loss in an electron-cooling line is also discussed. In the present paper, ${\mathit{Z}}_{\mathrm{eff}}$ is assumed to be constant; the physics determining ${\mathit{Z}}_{\mathrm{eff}}$ is treated in a subsequent article [Peter and Meyer-ter-Vehn, following paper, Phys. Rev. A 43, 2015 (1991)].