Moments Method Solution of the Boltzmann Transport Equation Using Pade Approximants.

The moments method is a semi-analytical technique for solving the Boltzmann transport equation. This method was also the first deterministic technique to be successfully applied to the Boltzmann transport equation for solutions useful to reactor shielding. The moments method was used for many years (since about 1950) as the benchmark for calculating the total flux (particles/cm^2 -sec). However, to this day, the ability to calculate the angular flux (particles/cm^2 -sec by discrete angles) has not been successfully accomplished. A knowledge of all the moments would enable, in principle, both the total and the angular fluxes to be reconstructed exactly. However, because the series expansions are truncated, there are an infinite number of functions with moments that correspond to the finite number of moments calculated. Therefore, Pade approximants were used to accelerate convergence with the limited number of moments calculated. The results of this work show that at low scattering angles, the Pade approximant solution and the solution using the current methodology provide results similar to those obtained from MCNP 4A. The Pade approximant solution estimated the flux to be slightly less than two times the flux calculated from MCNP at large scattering angles. At these same scattering angles, the current method estimated the fluxes to be over ten times the calculated fluxes from MCNP. Additional improvement should be possible by first smoothing the angular distributions prior to reconstructing the angular flux. This research has not only provided closure to the moments method, but has also demonstrated the expanded application of the Pade approximant procedure to practical problems. Through the use of Pade approximants, not only can the angular flux be determined but the solution, as it should be, is invariant. The same results were obtained when the polynomials used to expand the spatial variable were changed from Laguerre polynomials to the U & V polynomials, when the number of moments was decreased from eight to seven, and when the angular quadrature was changed from Deltaomega = 0.1 to 0.05. In addition, the total flux can now be determined at deeper penetrations than before.