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Nuclear Science and Engineering | Volume 40 | Number 2 | May 1970 | Pages 224-238
Technical Paper | doi.org/10.13182/NSE70-A19684
Articles are hosted by Taylor and Francis Online.
A new method for the solution of the Boltzmann equation for time-dependent or stationary neutron transport is presented. The essentials of the method are a Fourier transform of the transport integral equation and the decomposition of the new kernel into a bilinear form. In the monoenergetic case, the transport problem is reduced to the solution of an (infinite) linear system of equations. It is important that the matrix elements in this linear system be determined only once for any geometry and that this be done in an analytical way. In the present paper, the matrix elements for plane, spherical, and cylindrical geometries are constructed explicitly and are presented in a form suitable for numerical applications. To illustrate the efficiency of the method, the critical dimensions for these geometries have been determined in various integral transform ITN approximations, N being the order of the linear system after truncation. For samples of a thickness up to ∼10 mean-free-paths, the results for the critical dimension in stationary problems, or for the fundamental time-decay constants in nonstationary problems, have in the IT4 approximation, at least, the accuracy of the corresponding S16 results. The convergence of the ITN results, with respect to the order N, is very fast. The results for N = 4 can be considered as exact within one or two units in the fifth digit. The convergence behavior of the SN and the PN method is decidedly slower. In contrast to the SN, PN or Case's method, the smaller the system, the better the accuracy of the ITN calculations for fixed N is. The increase in accuracy by increasing N requires less computational effort for the ITN approximation than for SN or PN approximation. Apart from providing an analytical standard for the check of numerical approximations, the present method is a simple and rigorous tool for the calculation of neutron distributions, particularly in small systems.
