Higher-order corrections to the stopping power proportional to $z^3$ are evaluated. Both close and distant collisions are considered. The energy-loss formula can be written $\frac{\mathrm{dE}}{\mathrm{dx}}=z^{2}I+z^3(J_c+J_d)$, where $I$ is the customary lowest-order energy loss and $J_c$ and $J_d$ are the close- and distant-collision parts of the $z^3$ term, respectively. The close-collision contribution $J_c$ is a relativistic effect, first estimated in unpublished work by Fermi. It has the simple form $J_c=\frac{\pi {\alpha }^C}{2\beta }$, where $C$ is the standard constant multiplying ${\beta }^{-2\mathrm{}}$ times the Bethe-Bloch logarithm in $I$ and $\alpha$ is the fine-structure constant. At high energies $J_c$ gives a constant-$z^3$ contribution to the energy loss and causes a range difference $\Delta R$ roughly proportional to the range $R$ for stopping particles of the same mass and energy, but opposite charge. For $2<\mathrm{}\frac{P}{\mathrm{Mc}}<20\mathrm{}$, $\frac{\Delta R}{R}$ changes by less than ±6% and depends only slightly on the stopping material, varying from 1.9 × ${10}^{-3\mathrm{}}$ for carbon to 2.5 × ${10}^{-3\mathrm{}}$ for lead for $z=\pm 1$. The distant-collision effect is important only at low velocities. The calculation of this contribution is patterned after a recent work of Ashley, Ritchie, and Brandt, but differs from it in detail. Using a statistical model for the atom it is found that at low velocities the relative $z^3$ contribution can be written $\frac{J_d}{I}=\frac{F\left(V\right)\mathrm{}}{{\mathrm{}\left(Z\right)\mathrm{}}^{\frac{1}{2}}}$, where $Z$ is the atomic number of the stopping medium and $F\left(V\right)\mathrm{}$ is a universal function of the reduced velocity variable $V=\frac{137\gamma \beta }{{\mathrm{}\left(Z\right)\mathrm{}}^{\frac{1}{2}}}$ In the region where $\frac{J_d}{I}$ is appreciable $\mathrm{}(1<V<\mathrm{}10\mathrm{})\mathrm{}$, $F\left(V\right)\mathrm{}$ varies as $V^{-n}$ with $n\simeq 2.0-2.5$. These results on the ${\mathcal{z}}^3$ effect at low velocities are in good agreement with available data on comparison of the energy loss of helium ions and protons of the same velocities. Range differences are calculated for carbon, copper, lead, and emulsion absorbers, including the effects of both close and distant collisions. The results are in rough agreement with data on slow-stopping pions and $\Sigma$ hyperons in emulsions and in good agreement with very recent measurements of fast positive and negative muons. The upper limit of the range of validity of the results is examined in some detail. It is found that the approximations begin to fail for dynamic reasons above $\gamma \simeq 20$ for muons, and presumably also for other heavy particles.

• Received 14 July 1972

DOI:https://doi.org/10.1103/PhysRevB.6.4131