Diffusion of High Energy Gamma-Rays through Matter. III. Refinement of the Solution of the Diffusion Equation

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Abstract
In Part I of the present series of papers an approximate equation was derived governing the diffusion of high energy gamma-rays through matter. In Part II an approximate solution of this diffusion equation was obtained in the energy region where the total gamma-ray cross section was substantially independent of energy. In the present paper, by consideration of the methods employed in obtaining the solution in II, a refinement of the solution is carried out which reduces the errors introduced by the approximations made both in the energy distribution and the angular distribution of the multiply-scattered gamma-rays. The solution is also modified to take into partial account the effect of small variations of the total gamma-ray cross section with energy. An upper and lower bound on the solution is obtained when the cross section is independent of energy.