Advances in Neutron Thermalization Theory. A Review

Parallel with further development of pulsed-neutral experiments and related techniques, thermalization theory has devoted much attention to eigenvalue problems of the transport equation, to problems of relaxation times and relaxation lengths, and to the related problem of neutron waves. The pictures of eigenvalue spectra are now reasonably clear, although not always rigorously established. In each of the three quoted cases part of the spectrum covers a continuum corresponding to singular eigenfunctions. Outside the continuum there may be a finite or sometimes infinite set of discrete eigenvalues, associated with regular eigenfunctions. These eigenvalues disappear one by one by reaching the edge of the continuum if a characteristic parameter of the system is varied in a certain sense. Thus all the space-eigenvalues disappear if too much absorber is added to the medium. Discrete time-eigenvalues cease to exist if the size of the system is made sufficiently small. And no regular neutron waves (associated with discrete wavelengths and attenuation lengths) are possible beyond a critical frequency. The puzzling phenomenon of so-called pseudo-discrete eigenvalues which in experiments with crystalline media have been found to invade the continuum has recently been clarified by careful analysis of a suitable model. Solutions of eigenvalue problems are needed for predicting the values of diffusion parameters. They are also a prerequisite for solving the initial-value problem by the Laplace transform technique. For the time-dependent Green's function this leads to a bi-linear form in terms of the eigenfunctions; to this a 'transient' term, in the form of an integral along the edge of the continuum, must be added. Unfortunately, many of the mathematical steps involved in this procedure are still tentative and the proofs are missing. In the development of methods of solution, problems with slab geometry have as usual been predominant. Approximate methods have often produced satisfactory results. Also the exact methods of Wiener and Hopf, and of Case, have been generalized to energy-dependent transport theory. However, practical success has in both cases been limited to simple models. A reciprocity property of the Green's function (a generalization of the detailed balance relation) has helped to obtain from diffusion theory easy asymptotic solutions of a few standard problems. Some methods have been exchanged with the kinetic theory of gases, where similar eigenvalue, initial-value and boundary-value problems have arisen. An occasional link to radiative transfer has also been established. (author)