peierls' starting point was a paper by francis perrin, in which he had derived critical mass calculations in terms of nuclear constants. the physicists considered a sphere, which has the minimum surface area for a given volume. a critical mass occurs when the number of neutrons produced equals the number that escape. perrin assumed that the mean free path was much larger than the radius of the sphere. peierls did not agree, and commenced his own calculations. a key insight came from frisch, who wondered what would happen if, instead of natural uranium, someone gave you a sphere of the uranium-235 isotope, by definition, the mean free path is: