Eigenvalue Formulations for the PN Approximation to the Neutron Transport Equation

The study of the eigenvalues of the neutron transport oper- ator yields an important insight into the physical features of the neutronic phenomena taking place in a nuclear reactor. Although the multiplication eigenvalue is the most popular because of its implication in the engineering design of multi- plying structures, alternative interesting formulations are pos- sible. In this paper the interest is focused on the multiplication, collision and time eigenvalues. The transport model is considered in the spherical harmonics approximation and the study is restricted to the one-dimensional plane geometry in the monokinetic case. The spectra of the different eigenvalues are investigated using a numerical code, validat- ing its performance against the results available in the litera- ture. The observation of the convergence trends allows to establish the performance of even- and odd-order approxima- tions. It is shown that in general even-order approximations yield slightly less accurate results, nevertheless they appear to converge to the reference values. The effect of the choice of the boundary conditions according to the methodologies proposed by either Mark or Marshak is also investigated. The analysis of all the results presented allows to characterize the convergence properties of the spherical harmonics approach to neutron transport. The spectrum of the time eigenvalues retains a very rich phys- ical meaning, as they are the actual time constants of the time-dependent solution of the transport problem. Therefore, in the last part of the paper the behavior of the pattern of the spectrum of the time eigenvalues when changing the scattering ratio and the order of the approximation is examined.