The transversely integrated scalar flux of a narrowly focused particle beam

The subject of this paper is a steady, narrow, nearly monodirectional beam of nuclear particles normally incident on a scattering slab. Mathematically, such a beam is described by a particle transport equation together with inflow boundary conditions. Under certain assumptions, the transport equation can be well approximated by the Fokker-Planck equation. A further approximation, proposed by Fermi, leads to an explicitly solvable problem. In this paper, it is shown that without the Fermi approximation, it is still possible to derive asymptotic expansions of certain moments of the transport and Fokker-Planck solutions in powers of a small parameter which can be interpreted as the ratio of the transverse width of the beam and the slab thickness. Among the moments of which expansions are derived in this paper, the most important one is the transversely integrated scalar flux φ0, i.e., the total amount of mass per unit depth within the slab. The expansions of φ0 are compared with results of Monte-Carlo simulations