Frisch's likely path to the 2.1 cm critical radius in the 1940 Frisch–Peierls Memorandum

Abstract

Two striking, physically wrong assumptions are asserted in the 1940 Frisch-Peierls' Memorandum: The Memorandum ignores the known large scattering cross section while inventing a hypothetical fission cross section that is twenty times larger than the known fission cross section in 1940. The point of this paper is to show how Frisch used Peierls' approximate solution to his nonstandard chain reaction–diffusion equation along with the two wrong cross-section assumptions to make the Memorandum's famous critical radius prediction. Frisch's two 'vastly optimistic' cross sections were essential for getting the neutron multiplication number ν ≈ 2.3 (the average number of neutrons per fission event) into the right place in Peierls' partial differential equation. If we simply translate Frisch's assumptions as stated in the 1940 Memorandum directly into equations, then the prediction that R = 0.8 L with a correspondingly too small mean free path L = 2.6 m falls out from Peierls' approximate solution of his partial differential equation. Under these assumptions, the mean free path is identical with the fission mean free path. Summarizing, following the shortest path that Frisch could have taken leads us directly to R = 0.8 L and the end of the mystery of how he could have predicted the tiny critical radius.