A spatial and angular moments analysis of the monoenergetic pencil beam problem

We consider the time-independent, monoenergetic, normally incident pencil beam (searchlight) problem for a homogeneous, source-free, finite thickness, purely scattering slab. By adopting a cylindrical coordinate system to describe this problem, exact equations are developed for the radial and azimuthal Fourier cosine moments of the angular flux. The analysis is carried out for both the full integral scattering desription as well as its Fokker-Planck differential approximation. Assuming that the beam remains nearly collimated as it passes through the slab, these moment equations are analysed using an additional (in this case asymptotic) moments method in the polar angle. The resulting set of coupled, one-dimensional (in the depth variable) ordinary differential equations are used to construct asymptotic power series representations for certain radial moments of the scalar flux. These one-dimensional series are then used to improve earlier descriptions, due to Fermi and Jette, of the radial and depth variations of the beam scalar flux