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

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 vₚ, below the electron thermal velocity vₜₕ, and on solutions nonlinear in the coupling parameter scrZ=Zeff/(n₀{{λ}}D³$) between the heavy-ion projectile with effective charge ${Z}eff$ and the plasma with electron density ${n}₀$ and Debye length ${{λ}}D. Concerning the stopping power in the low-velocity regime relevant for the Bragg peak at the end of the ion range, results on the friction term dE/dx∝{v}ₚ$ 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 ${v}ₚ$>${v}ₜₕ$, but also describes the limit ${v}ₚ→0 correctly.For{v}ₚvₜₕ, nonlinear results are found to contribute to the stopping power with terms {∝}scrZ5/2 for positive ions and terms {∝}scrZ³ for negative ions in addition to the basic scrZ² term; they are derived from a low-velocity expansion of a form-factor representation of dE/dx. Concerning high velocities vₚ>vₜₕ, the relevant coupling parameter is scrZ(vₜₕ/vₚ{)}³, and nonlinear corrections to the stopping power ∝{scrZ}³$/${v}ₚ⁵$ 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, Zeff is assumed to be constant; the physics determining Zeff is treated in a subsequent article [Peter and Meyer-ter-Vehn, following paper, Phys. Rev. A 43, 2015 (1991)].