Mean-Value Calculations for Projected Multiple Scattering

The multiple-scattering theories of Molière and Snyder-Scott are compared and the equivalence of the mathematical development stated. Using the preferable single-scattering probability of the Molière theory, results are quoted of interest to experimenters. Several mean-value quantities are given: mean arithmetic angle, median angle, half-width, $\frac{1}{e}$ width, angle $\frac{1}{P_0{\pi }^{\frac{1}{2}}}$ 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 $\Omega$ and the logarithm of $\Omega$, where $\Omega$ is the mean number of scatterings undergone by the particle in question. The linear relationships are good to 1 percent for $\Omega$ from ${10}^2$ to ${10}^5$. 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.