Asymptotic Derivation of the Fermi Pencil-Beam Approximation

The Fermi pencil-beam approximation describes the broadening of a monoenergetic, nearly monodirectional particle beam in an optically thick system in which the mean scattering angle is small and large-angle scattering is negligible. This physical problem has applications in such diverse fields as astrophysics, materials science, electron microscopy, and radiation cancer therapy. The Fermi equation is derived two different ways: as an asymptotic limit of the Fokker-Planck equation for σ_tr → 0 and as an asymptotic limit of the linear Boltzmann equation for σ_tr→ 0 and σ_t → ∞. Some numerical results illustrating the Fermi approximation are also given.