Calculation of critical mass for fission chain reaction

Neutrons at certain energies colliding with fissile nuclei may cause them to undergo fission reaction by splitting into fragments at different masses. Since new neutrons are produced at the end of each fission reaction, a certain amount of fissile material in any type of nuclear reactor core is required for a sustainable chain reaction by these neutrons. It is necessary to determine the amount of mass required for a sustainable fission chain reaction since the procedure is completely random for neutrons to strike target nuclei. It is known that the critical mass is the smallest amount of fissile material needed for such a sustainable nuclear chain reaction in any system of geometry. It is obvious that the critical mass of a fissionable material depends upon its nuclear properties such as its fission cross-section, density, shape, enrichment, purity, etc. In this study, the solutions of the steady-state one group diffusion equation are firstly analytically obtained and then compared with numerical results of the Monte Carlo Method coded to calculate the critical mass for three basic geometries. The survival fraction value, f_s (which is also called as the multiplication factor, k) is calculated for the criticality condition for these geometries and the results are tabulated for each one. Our numerical calculations by Monte Carlo Method show that the minimum critical mass is obtained in the case of spherical shape of fuel element for which the analytical solutions are also in aggrement.