Theory of Multiple Scattering: Second Born Approximation and Corrections to Molière's Work

Abstract

The formula given by Moli\`ere for the scattering cross section of a charged particle by an atom, on which has been based the formula for the "screening angle" ${\ensuremath{\chi}}_{\ensuremath{\alpha}}$ in his theory of multiple scattering, has been examined and found to contain an inconsistent approximation in all orders of the parameter ${\ensuremath{\alpha}}_{1}=\frac{\mathrm{zZ}}{137\ensuremath{\beta}}$ except the lowest (the first Born approximation). In the present work, the correct expression of Dalitz is used for the single-scattering cross section of a relativistic Dirac particle by a screened atomic field up to the second Born approximation. It is found that the effect of the deviation from the first Born approximation on the screening angle is much smaller than Moli\`ere's expression for this quantity would lead one to believe. This is so because the deviation from the first Born approximation is very small at the small angles that go into the definition of the screening angle. In Moli\`ere's work, all the effect of the deviation from the first Born approximation on the distribution function $f(\ensuremath{\theta})$ for multiple scattering is contained in the quantity $B$ which depends only on ${\ensuremath{\chi}}_{\ensuremath{\alpha}}$. In the present work, it is shown that in a consistent treatment of terms of various orders in ${\ensuremath{\alpha}}_{1}$, there exist additional terms of order $\frac{\mathrm{zZ}}{137}$ in the distribution function. These terms, which represent the second Born approximation, become important at large angles. Calculations have been carried out for the scattering of 15.6-Mev electrons by Au and Be. The $\frac{1}{e}$ widths of the distribution function obtained are in good agreement with the experimental result of Hanson et al., whereas Moli\`ere's theory gives too great a width compared with the experimental value in the case of Be.