The number distribution of neutrons and gamma photons generated in a multiplying sample

The subject of this paper is an analytical derivation of the full probability distribution of the number of neutrons and photons generated in a sample with internal multiplication by one source emission event (spontaneous fission), and its comparison with Monte Carlo calculations. We derive recursive analytic expressions for the probability distributions P ( n ) up to values of n = N for which the cumulative probability ∑ n = 0 N P ( n ) is equal to or larger than 0.99. The derivation is achieved using the symbolic computation language Mathematica. With the introduction of a modified factorial moment of the number of neutrons and gamma photons generated in fission, the resulting expressions could be brought to a formally equivalent form with those for the factorial moments of the total number of neutrons and photons generated in the sample. The results were compared to Monte Carlo calculations, and excellent agreement was found between the analytical results and the simulations. The results show that the probability distributions change with increasing sample mass in such a way that the “bulk” of the distribution changes only slightly, but a tail develops for higher n values. This tail is the main reason for the increase of the factorial moments with increasing sample mass, an effect that was observed in earlier studies.