Mean-Value Calculations for Projected Multiple Scattering

The multiple-scattering theories of Moli\`ere and Snyder-Scott are compared and the equivalence of the mathematical development stated. Using the preferable single-scattering probability of the Moli\`ere theory, results are quoted of interest to experimenters. Several mean-value quantities are given: mean arithmetic angle, median angle, half-width, 1/e width, angle 1P₀π1/2 related to the zero-angle amplitude, and mean arithmetic angle with a cutoff at 4 times the mean. These quantities are given for both the projected tangent angle and projected chord angle distributions, in the form of linear relationships between the square of the angle divided by Ω and the logarithm of Ω, where Ω is the mean number of scatterings undergone by the particle in question. The linear relationships are good to 1 percent for Ω from 10² to 10⁵. Information is also given on smoothed-out distributions, and on an estimate of the error for the cut-off arithmetic mean angle. The scattering constant K is given for several methods of measurement, for Ilford G-5 emulsions.