Relativistic theory of stopping for heavy ions

We calculate the electronic stopping power and the corresponding straggling for ions of arbitrary charge number, penetrating matter at any relativistic energy. The stopping powers are calculated by a simple method. Its starting point is the deviation of the precise theory from first-order quantum perturbation. We show that this deviation can be expressed in terms of the transport cross section, ${\mathrm{\sigma }}_{\mathrm{tr}}$, for scattering of a free electron by the ion. In the nonrelativistic case the deviation is precisely the Bloch correction to Bethe’s formula; we look into the nonrelativistic case in order to clarify both some features of our method and a seeming paradox in Rutherford scattering. The corresponding relativistic correction is obtained from ${\mathrm{\sigma }}_{\mathrm{tr}}$ for scattering of a Dirac electron in the ion potential. Here, the major practical advantage of the method shows up; we need not find the scattering distribution, but merely a single quantity, ${\mathrm{\sigma }}_{\mathrm{tr}}$, determined by differences of successive phase shifts. For a point nucleus our results improve and extend those of Ahlen. Our final results, however, are based on atomic nuclei with standard radii. Thereby, the stopping is changed substantially already for moderate values of γ=(1-${\mathit{v}}^2$/${\mathit{c}}^2$ ${)}^{\mathrm{-}1/2}$. An asymptotic saturation in stopping is obtained. Because of finite nuclear size, recoil corrections remain negligible at all energies. The average square fluctuation in energy loss is calculated as a simple fluctuation cross section for a free electron. The fluctuation in the relativistic case is generally larger than that of the perturbation formula, by a factor of ∼2–3 for heavy ions. But the finite nuclear radius leads to a strong reduction at high energies and the elimination of the factor ${\gamma }^2$ belonging to point nuclei. ©1996 The American Physical Society.