Heisenberg's 1939 reactor theory, Serber's 1943 Los Alamos Primer, and Heisenberg's 1945 Farm Hall critical mass calculation

There is disagreement over how much Werner Heisenberg knew about reactors and the bomb 1939–1945. A critical paper, declassified and returned to the Germans in 1971, has been inadequately understood in the controversy. Heisenberg calculated the critical radius for a spherical reactor with a tamper but the same formula applies whether spherical reactor or spherical bomb because his result is independent of the reactor's neutron source at criticality (a bomb has no external neutron source). He didn't emphasize the source-independence of the critical radius but instead misinterpreted it. Under captivity at Farm Hall in England in 1945 with eight other leading German nuclear researchers and Max v. Laue, Heisenberg recalled his 1939 theory from memory and made the same critical radius calculation for a bomb with a tamper that Serber had made in 1943 at Los Alamos. Heisenberg's 1945 formula for a bomb is identical to his 1939 critical radius formula derived for a reactor. His 1945 predictions for the numerical values of the critical radius are close to Serber's, except that Heisenberg only knew the fast fission cross section within two bounds, 0.5 and 2.5 barn. Serber's fast fission cross section was in the middle at 1.5 barn. This places Heisenberg's 1939 and 1945 theory and calculations, finally, in the correct historic context.