Bethe stopping theory for a harmonic oscillator and Bohr’s oscillator model of atomic stopping

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Abstract
Bethe’s expressions for the stopping cross section and the straggling parameter of a penetrating point charge have been evaluated for a spherical harmonic oscillator as a target. The results, which are rigorous except for the neglect of relativistic corrections and higher-order Born terms, are given in tabulated form as well as in the form of asymptotic shell-correction expansions to arbitrary order. Existing general expressions for the stopping cross section and straggling are confirmed as far as they go. The range of validity of various model theories of atomic stopping is tested. At velocities down to about the stopping maximum, the agreement is very good for the kinetic theory, while a Fermi gas with a properly chosen electron density yields a significantly different stopping maximum. The binary encounter theory underestimates the stopping power at all speeds. The dielectric or local-density approach does not reproduce the correct scaling behavior. None of the model theories reproduces the threshold behavior. For light projectiles, i.e., positrons and electrons, a Franck-Hertz-type structure is observed in the velocity dependence of the stopping cross section which is particularly pronounced near threshold. An equipartition rule is derived for contributions to the stopping cross section from low excitations, i.e., distant collisions, and close encounters. A straight extension of Bohr’s oscillator model of atomic stopping is proposed which is shown to reproduce leading shell corrections in both stopping power and straggling and which, when applied to hydrogen, accurately describes the stopping power at velocities around and above the stopping maximum.