Solution of a non-isotropic random flight problem in the case of a non-isotropic point source

Abstract

In connection with the calculation of the perturbation caused by a thin circular detector in a thermal neutron flux, we have solved the following problem. Given is a point source whose strength is described by an isotropic distribution plus a non-isotropic cosine term. The single scattering law also contains one non-isotropic term. We have to calculate the neutron density and the distribution of the velocity directions in every point of space. Keeping the path probabilities arbitrary for the reason of generality, we have established exact recurrence relations between the defined probability distributions. Expanding these functions in series of spherical harmonics and making use of addition theorems involving Bessel functions and Legendre polynomials, it has been possible to transform the integral relations to simple linear equations. The final results are obtained in the form of integrals which can be calculated by numerical evaluation using computing machines. By introducing simplified functions under the integral signs, it is also possible to obtain good approximations for the neutron density and current. A simple example of this method is discussed in detail and the results are compared with those given by diffusion theory.