An Alternative Method for Solving the Diffusion Equation under Approximation B in Electron-Photon Cascade Theory

概要

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A new analytical formulation is proposed to solve the diffusion equation under Approximation B in electron-photon cascade theory. The Suzuki-Trotter formula, analytical continuation of the hypergeometric function, and product integration are introduced. By using these methods the usual series solutions are obtained, and summation of the infinite series for arbitrary values of the energy E is performed by using the method of Prony's interpolation. As E 0, the infinite sum for the electron component turns out to be the function of K_1(s,-s) used in the usual cascade theory, and a logarithmic divergence arises for the photon component. Use of Prony's method makes it possible to derive the energy spectra as well as the track length distributions and the transition curves. Our numerical results agree well with previous authors' as expected. Our analytical approach provides a general framework for solving other diffusion equations containing non-commutative operators in different contexts.
理論物理学刊行会の論文
1998-03-25