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. Math ematically, such a beam is described by a particle transport equation together with inflow boundary conditions. Under certain assumptions, t he transport equation can be well approximated by the Fokker-Planck eq uation. A further approximation, proposed by Fermi, leads to an explic itly solvable problem. In this paper, it is shown that without the Fer mi approximation, it is still possible to derive asymptotic expansions of certain moments of the transport and Fokker-Planck solutions in po wers of a small parameter which can be interpreted as the ratio of the transverse width of the beam and the slab thickness. Among the moment s of which expansions are derived in this paper, the most important on e is the transversely integrated scalar flux phi(0) i.e., the total am ount of mass per unit depth within the slab. The expansions of phi(0) are compared with results of Monte-Carlo simulations.