Theory of Omega-d Modes

Abstract

A theory is presented on the omega-d modes which are eigenfunctions of a particular type of neutron diffusion operator. First, a general discussion is given on the implicit eigenvalue problem, reality, boundedness and distribution of the eigenvalues, order of magnitude and closure of the eigenfunctions. It is shown that the regular and adjoint eigenvalue problems reduce to an identical implicit eigenvalue problem. No one has in the past explicitly shown that the eigenvalue is real. The large variation in the order of magnitude of the eigenfunctions gives rise to a large variation in the mode coupling. This implies existence of weakly coupled modes that can be ignored. Next, these pro-perties are verified for a symmetric reflected slab reactor by numerical calculations. The results disprove the conjecture advanced by others that the neutron fluxes associated with a particular cluster of seven modes are approximately same. Finally, an equation is derived for the mode amplitudes which takes into account temperature feedback and external disturbance. Discussions are also given on the choice of appropriate modes for describing various kinds of dynamical processes.