The Laplace Transform Method for the Albedo Boundary Conditions in Neutron Diffusion Eigenvalue Problems

Some heavy nuclei are fissile after having absorbed a neutron, i.e., they violently split into two unequal fragments, while at the same time ejecting two or three neutrons on average. This phenomenon is called fission. Neutrons ejected during one fission can, in turn, be absorbed by other neighboring fissile nuclei, thus creating a chain reaction. If this reaction is controlled and stabilized, one gets an energy source—this is what happens in a nuclear reactor [WF07]. Nuclear power is a proven technology and has the potential to generate virtually limitless energy with no significant greenhouse gas emissions. From a physimaterial understanding of criticality, it appears that any system containing fissile material could be made critical by arbitrarily varying the number of neutrons emitted in fission. It is well known that criticality calculations can often be best approached by solving eigenvalue problems. In elementary nuclear reactor theory, the dominant eigenvalue, i.e., the effective multiplication factor (keff), is thought of as the ratio between the numbers of neutrons generated in successive fission reactions. The eigenfunction corresponding to the dominant eigenvalue is proportional to the neutron flux within the reactor core. Furthermore, in most realistic reactor global calculations, it is necessary to consider an approximation of the energy-dependent eigenvalue problem in which the energy variable is discretized. The most common energy discretization method is the conventional multigroup approximation, in which the neutron energy range is divided into contiguous energy groups. In practice, multigroup diffusion theory has been applied extensively to nuclear reactor analyses and generally found to perform better than it theoretically has any right to, because it does not include the direction-of-motion variable [AlOd86]. Neutron fission events do not take place in the non-multiplying regions of nuclear re- actors, e.g., moderator, reflector, and structural core; therefore, we claim that we can improve the efficiency of nuclear reactor global calculations by eliminating the explicit numerical calculations within the non-multiplying regions around the active domain. In this chapter, we describe the application of the Laplace transform method in order to determine the energy-dependent albedo matrix that we use in the boundary conditions of multigroup neutron diffusion eigenvalue problems in slab geometry to substitute the explicit numerical calculations within the baffle–reflector system around a thermal nuclear reactor core. Albedo, the Latin word for “whiteness,” was defined by Lambert (1760) as the fraction of incident light reflected diffusely by a surface [Pa61]. This word has remained the usual scientific term in astronomy. Here, we extend it to the reflection of neutrons. At this point, an outline of the remainder of this chapter follows. In Section 28.2 we present the mathematical formulation. In Section 28.3 we present numerical results, while concluding remarks with suggestions for future work are given in Section 28.4.