The most important assumption is that all neutrons are lumped into a single energy group, they are emitted and diffuse at thermal energy ( 0.025 eV). We will start with simple systems (planar) and increase complexity gradually. This is totally different from problems with external sources, which determine the absolute value of the neutron flux. This means that the absolute value of the neutron flux cannot possibly be deduced from the diffusion equation. Therefore if we find one solution of the equation, then any multiple is also a solution. This is very important, because such equation is a linear homogeneous equation in the flux. It must be noted that we will solve the diffusion equation without any external source. Р¤ denotes a reaction rate.В For example a fission of 235U by thermal neutron yields 2.43 neutrons. In this equation ОЅ is number of neutrons emitted in fission and ОЈ fВ is macroscopic cross-sectionВ of fission reaction. In various geometries that satisfy the boundary conditions. In this section, we will solve the following diffusion equation A вЂ“ a supercritical state B вЂ“ a critical state C вЂ“ a subcritical state This condition is known as the supercritical state. The number of neutrons is exponentially increasing in time (with the mean generation time). If the multiplication factor for a multiplying system is greater than 1.0, then the multiplying system produces more neutronsВ than are needed to be self-sustaining. It is obvious the infinite multiplication factor in a multiplying system is a measure of the change in the fission neutron population from one neutron generation to the subsequentВ generation. This can be expressed mathematically as shown below. The infinite multiplication factor is the ratio of the neutrons produced by fission in one neutron generation to the number of neutrons lost through absorption in the preceding neutron generation. This condition can be expressed conveniently in terms of the multiplication factor. Also the number of fissions occurring per unit time ( the reaction rate) within the system must be constant. The minimum condition is for each nucleus undergoing fission to produce, on the average, at least one neutron that causes fission of another nucleus. The required condition for a stable, self-sustained fission chain reaction in a multiplying system (in a nuclear reactor) is that exactly every fission initiate another fission.

Cylindrical coordinates equation