The distribution of emergent flight path lengths from homogeneous spheres

Empirical as well as formal arguments are used to show that the probability, P(x) dx, for particles born uniformly and emitted isotropically throughout a sphere of radius a, to have an escape flight path length L which lies in the interval adx about ax is given by P(x) dx=(3/4)[1-(x/2)2] dx where 0≤x≤2. This distribution may be used directly in the algebraic formulation of a number of interesting physical problems or else conveniently sampled numerically according to the expression x=-4 cos[2π-cos-1(ξ))/3], where ξ is a random number between 0 and 1. As an example of how the analytical form may be used a general expression for the energy spectrum of α-particles emerging from small, homogeneous, spherical particles of α-emitter is derived. (orig.)