An elementary method to determine the critical mass of a sphere of fissile material based on a separation of neutron transport and nuclear reaction processes

TL;DR

This paper introduces a simple, pedagogical method to estimate the critical mass of a fissile sphere using elementary calculus and statistical arguments, separating neutron transport from nuclear reactions.

Contribution

It presents a novel, straightforward approach to criticality calculations that avoids solving the diffusion equation, suitable for educational purposes and quick estimations.

Findings

Method agrees with known critical masses within a few percent.

Applicable to impure and isotopically varied materials.

Can be used as a design guide or for quick Monte Carlo simulation checks.

Abstract

A simplified method to calculate the critical mass of a fissile material sphere is presented. This is a purely pedagogical study, in part to elucidate the historical evolution of criticality calculations. This method employs only elementary calculus and straightforward statistical arguments by formulating the problem in terms of the threshold condition that the number of neutrons in the sphere does not change with time; the average neutron path length in the material must be long enough to produce enough fission neutrons to balance losses by absorption due to nuclear reactions and leakage through the surface. This separates the nuclear reaction part of the problem from the geometry and mechanics of neutron transport, the only connection being the total path length which together with the distance between scatterings determines the sphere radius. This leads to an expression for the critical radius without the need to solve the diffusion equation. Comparison with known critical masses shows agreement at the few-percent level. The analysis can also be applied to impure materials, isotopically or otherwise, and can be extended to general neutronics estimations as a design guide or for order-of-magnitude checking of Monte Carlo N-Particle (MCNP) simulations. A comparison is made with the Oppenheimer-Bethe criticality formula, with the results of other calculations, and with the diffusion equation approach via a new treatment of the boundary conditions.