Milne's Problem for Thermal Neutrons in a Nonabsorbing Medium

The velocity dependent transport equation for a nonabsorbing semi-infinite medium with no sources is approximately solved by a variational method. A simple trial function, which takes into account the asymptotic behavior of the exact solution, is used to obtain an approximation for the extrapolation distance q_∞, and, by iteration, an approximation for the flux distribution. Numerical results are given for a monatomic gaseous medium with the atom mass equal to the neutron mass. The value q_∞ = 0.9345 l (∞) is obtained. The velocity distribution of the emerging neutrons shows a hardening effect, corresponding, in the average, to a 14% increase in the neutron temperature.