The Absorbed Dose Rate from Both the Uncollided and Once-Collided Flux of Gamma Rays at the Center of a Spherical Shell Shield Bombarded by an Isotropic Flux Spectrum of Protons

A sevenfold integral expression is derived for the absorbed dose rate from the uncollided flux of gamma rays at the center of a spherical shell shield bombarded by an omnidirectional flux spectrum of protons. The general formulation is reduced to a fourfold integral on the basis of simplifying assumptions. This simpler formulation assumes that the gamma rays are produced isotropically by an isotropic proton flux, that protons penetrating the shell are not deflected from their original direction of incidence, that the spectrum and yield of photons are independent of proton bombarding energy, and that both the incident proton spectrum and the range-energy relation for protons in matter have power-law representations. A sixfold intergral expression is derived for the absorbed dose rate from the once-collided flux of gamma rays at the center of a spherical shell shield bombarded by an isotropic flux spectrum of protons. The once-collided differential (in energy) flux of photons at the shell center is given by a fivefold integral expression.