Bohr's Theory of Energy Losses of Moving Charged Particles

Abstract

The stopping power of a particle having charge ${Z}_{\mathrm{eff}}\ensuremath{\epsilon}$ and moving with velocity $v$ has been determined by Bohr for $\ensuremath{\kappa}=\frac{2{Z}_{\mathrm{eff}}{\ensuremath{\epsilon}}^{2}}{\ensuremath{\hbar}v}$ and ${\ensuremath{\eta}}_{s}=\frac{2v}{{u}_{s}}>2$, where ${u}_{s}$ is the velocity of the orbital electron that is effective in stopping the particle. Bohr's results are formulated differently for $\frac{\ensuremath{\kappa}}{{\ensuremath{\eta}}_{s}}<1$ and $\frac{\ensuremath{\kappa}}{{\ensuremath{\eta}}_{s}}>1$ and are based on an orbital picture of the processes involved. An alternative method has been developed in which the space surrounding the particle track has been subdivided into three regions: (a) the region of validity of the Rutherford formula (adjacent to the track), (b) the intermediate region involving large perturbations for which no adequate theory exists at the present time, and (c) the region of validity of the quantum perturbation theory (the most remote from the track). By extrapolating the formulas valid in regions (a) and (c) into the region (b), the contribution to the stopping power due to the region (b) has been estimated, and adding to it the contribution due to the regions (a) and (c) an expression has been derived from the total stopping power that is applicable whenever $\ensuremath{\kappa}>1$ and ${\ensuremath{\eta}}_{s}>1$. The expression has a wider region of applicability than the corresponding Bohr's formula since the latter can be used only for $\ensuremath{\kappa}\ensuremath{\ll}{{\ensuremath{\eta}}_{s}}^{3}$ if $\frac{\ensuremath{\kappa}}{{\ensuremath{\eta}}_{s}}>1$. The physical picture used for deriving this expression is somewhat similar to the one used by Bohr for $\frac{\ensuremath{\kappa}}{{\ensuremath{\eta}}_{s}}<1$ but is different from the one used by Bohr for $\frac{\ensuremath{\kappa}}{{\ensuremath{\eta}}_{s}}>1$.