Supersymmetric methods in neutron diffusion

We present the Witten and the double Darboux constructions as applied to the diffusion of thermal neutrons from an infinitely long line source. Supersymmetric one-dimensional quantum mechanics has been introduced in 1981 as a toy model for symmetry-breaking phenomena in quantum field theory [1]. With great speed its status has changed to a powerful research discipline as one can contemplate in the most recent review [2]. Recently, we obtained interesting results for various physical problems by using Witten's factorization and a more general supersymmetric double Darboux procedure [3]. The aim of this work is to apply the two supersymmetric methods to the theory of diffusion of thermal neutrons. We shall use the illustrative example of the neutron diffusion problem as presented in the textbook of Arfken [4]. The example is an ideal case in that it refers to an infinite line (Dirac delta) source of neutrons and actually provides the Green's function for this case. The steady state continuity equation for the neutrons is D∇ 2 φ − Σ a φ + S = 0 (1) where the first term represents the diffusion, the second stands for the absorption losses and the third is the source strength. The diffusion constant D is related to the neutron mean free path