Multiple Scattering of Fast Charged Particles

The theory of multiple scattering of charged particles has been extended in the small-angle approximation valid for thin foils and fast particles. The extension consists of an exact solution of the integral diffusion equation for the correlated probabilities of lateral and angular displacements, and the numerical integration of the resulting expression for the angular distribution. The projection of the scattering on a plane has been used for simplicity. The results are expressed in terms of dimensionless variables $\frac{\ensuremath{\eta}}{{\ensuremath{\eta}}_{0}}$ and $\frac{z}{\ensuremath{\lambda}}$ representing respectively the deflection angle in terms of a small unit determined by the screening, and the foil thickness in terms of the mean free path for scattering. Numerical calculations for values of $\frac{z}{\ensuremath{\lambda}}$ from 100 to 84,000, and for an adequate range of $\frac{\ensuremath{\eta}}{{\ensuremath{\eta}}_{0}}$, have been carried out, and tables have been made available. Curves are presented for a few values of $\frac{z}{\ensuremath{\lambda}}$. The matching is shown between the approximately Gaussian result for small angles and the Rutherford single scattering result valid for large angles. The deviations of the new results from each of these limiting values is quite large over a wide range of angle. An explicit asymptotic formula for large $\ensuremath{\eta}$ is given, showing correction terms to the single scattering formula.