Three-dimensional transport theory: Evaluation of analytical expressions of Williams and verification of MCNP

• An evaluation of 3-D neutron transport analytical expressions of Williams.
• Techniques for oscillating, singular and infinite integrals are applied.
• Disagreements with reported values are rare even at 5 significant figures.
• MCNP is verified against analytical results for several benchmarks.
• MCNP results generally agree with analytical results, except near singularities.

“Three-dimensional transport theory: an analytical solution of an internal beam searchlight problem, I”, Annals of Nuclear Energy, 36(8), 1256–1261 (2009) by Williams extends the range of analytical solutions, and the associated development of techniques, numerical results and analysis near singularities. The final integrals are not easy to evaluate as the integrands are highly oscillatory, singular and also on infinite range. We report here some further numerical evaluations of expressions of Williams, and also compare these with those of Williams and Ganapol and Kornreich. The numerical results compare very well. The disagreements are very rare, and even then in the fifth decimal place. We are also able to explore the nature of the results near singularities in conformity with the results of Williams. We also verify MCNP-5, the widely used Monte Carlo code against these analytical results. We have found that MCNP is easily able to provide results within 0.1% deviation from the “exact” results for most cases, and within 1% for almost all cases. It is challenged near the singularities, however, where the deviations are larger.