Invariant Embedding and Propagation of Neutron Waves in a Semi-Infinite Medium

The experimental study of the propagation of neutron waves through a medium is of great interest in determining the characteristic of the medium and can often be applied in cases where the pulsed neutron experiments would be difficult or too long to perform. However, one of the main difficulties of the wave propagation method lies in the interpretation of the experimental results. The use of invariant embedding to compute the characteristics of the propagation allows one to overcome most of the difficulties involved in the theoretical prediction of numerical results. As a preliminary problem, the study of the propagation of neutron waves in a semi-infinite non multiplying medium has been performed, using invariant embedding. A degenerate kernel is used as the scattering kernel. Like the Milne problem, the method asks for solving first the equations giving the albedo of the semi-infinite medium. However, in the case of wave propagation, it is the Fourier transform of the time-dependent albedo which is of interest. Writing this Fourier transform, and solving the resulting set of integral equations, the asymptotic complex relaxation length can be easily found by solving an algebraic eigenvalue problem. It is easy to show that the complex relaxation length k(ω) also satisfies a system of integral equations. From these, the dispersion relation for the complex relaxation length was obtained. (author)