Multiple Scattering in a Semi-Infinite Medium

The Boltzmann equation for the multiple scattering of charged particles is solved for a semi-infinite medium by means of the Laplace transformation. The sources are isotropic and distributed throughout the medium, corresponding to a thick layer of electron emitting atoms. The solution is obtained as a series in spherical harmonics and is carried as far as the $P_2$ coefficient. In order to obtain a solution it is necessary to assume that the transport mean free path $\lambda$ is not a function of energy. As a particular evaluation of the solution the flux on the surface is examined and modified to correct for the energy dependence of $\lambda$ by comparing with the corresponding solution of the age diffusion equation. The resulting spectrum is compared with an experimental case of 301.3-kev photoelectrons produced in a thick thorium converter. Using the experimental spectrometer resolution of 1.5 percent the theoretical spectrum is integrated over the window curve and a shift of the peak from the edge of the primary spectrum of 4.3 kev is obtained. Considering the errors involved this compares favorably with the experimental peak shift of 3.5 kev for a converter thickness of 25 mg/${\mathrm{cm}}^2$.