Barkas effect in a dense medium: Stopping power and wake field

Abstract

A general expression has been derived for the polarization field set up by a heavy charged particle penetrating through a random medium. This expression includes contributions of first and second order in the projectile charge, and it determines the stopping power up to the third order. For a dilute gas, the stopping-power formula reduces to the one that can be derived directly from perturbation theory, applied to an isolated atom or molecule. The interaction between a charged particle and the medium is shown to be characterized by two functions, the well-known dielectric function ε( k, ω) and a function Y( k, ω; k′, ω′) depending on two sets of wave number and frequency variables. Explicit expressions for Y have been derived, (i) for a random medium of harmonic oscillators from the classical equation of motion, (ii) for a classical electron gas from Boltzmann's transport equation, (iii) for a system of Hartree atoms from quantal perturbation theory, and (iv) for the Fermi gas. Considerable care has been taken to ensure that different approaches yield identical results in those limits where such is to be expected. The Barkas (or Z 1 3) effect on the stopping power has been evaluated for a medium of classical oscillators with a resonance frequency ω 0 at an arbitrary density, characterized by a plasma frequency ω P . The isolated classical oscillator and the classical electron gas emerge from this as simple limiting cases. The results can be described well by one effective resonance frequency α 0 = (ω 0 2 + ω P 2) 1 2 , as is suggested by classical dispersion theory. The classical description applies to distant collisions only. Quantitative results have been derived for a quantal electron gas with the zero-point motion disregarded (static electron gas). The results compare well with recent experimental data on silicon. The Barkas correction to the spectrum of momentum transfer is positive for positively charged projectiles up to fairly large momenta, but it becomes negative near the edge. The Barkas correction to the stopping power is also mainly positive for positively charged projectiles, but it turns negative at low speed, in agreement with existing results for the harmonic osicllator. The second-order contribution to the induced field (or wake field) contains a higher harmonic, i.e., a field oscillating at twice the standard wave number. This alters the wake potential dramatically behind the penetrating particle at low velocities. This effect is possibly an explanation for observed anomalies in molecular-ion energy-loss spectra. The relative significance of the Barkas term in the stopping power and the wake field increases with increasing electron density in the medium.