Joseph McCauley's recent paper on how Werner Heisenberg's 1939 analysis of criticality for a nuclear bomb/reactor captured the same physics as appears in Robert Serber's 1943 Los Alamos Primer helps clarify the often-contentious history of the wartime German nuclear program.1 

In this Letter, I draw attention to further background material on criticality calculations and the Manhattan Project, correct some erroneous statements that crept into McCauley's paper, and address his criticisms of my own work. The errors alluded to in no way impact his main conclusions, but readers unfamiliar with this history might be left puzzled upon consulting other sources.

A surprising aspect of early criticality calculations is that they were hardly secret: in the summer and fall of 1939, both Francis Perrin and Rudolf Peierls openly published analyses of the concept of critical mass; Perrin adopted a diffusion theory approach while Peierls developed an integral-equation method.2,3 Perrin drastically overestimated the critical mass of uranium as being tens of tons (he was thinking of natural-abundance uranium, not enriched); Peierls, who focused on the idea of a pure fissile isotope, did not substitute any numbers into his formulae as he lacked reliable cross section data, but his paper would play a role in the development of the secret Frisch–Peierls memorandum of March 1940 that initiated the British nuclear program. We don't know if Heisenberg saw these papers, but they certainly identified the relevant physics.

In his Abstract, McCauley claims that unlike a reactor, a bomb has no “external” neutron source. Strictly, this can be true: once a critical mass has been assembled, a stray neutron from a spontaneous fission or cosmic ray can initiate the explosion, but this would leave the triggering of a very expensive device to random chance. In the Manhattan Project bombs, neutron-generating “initiators” were developed to ensure detonation at the desired time. These operated by releasing a burst of 100 neutrons when the bombs' initiating chemical explosives were triggered; this was done by arranging for highly radioactive polonium to mix with beryllium at the right moment and having resulting (α,n) reactions initiate the nuclear chain reaction. The Manhattan polonium program was held in such secrecy that it was not mentioned at all in the 1945 Smyth report, and became more generally known only in the 1980s; for a history of this work, see Thomas.4 

In his section 2, McCauley states that 2 MeV neutrons move at near light-speed; they actually travel at about 0.07c.

Toward the end of his section 2, McCauley asserts that Fermi, Szilard, and Wigner had enough enriched uranium to produce a sustained chain reaction in 1942. This is not so; the CP-1 reactor in Chicago used natural-abundance uranium as construction of the enormous enrichment facilities at Oak Ridge had not yet begun. The X-10 reactor at Oak Ridge and the Hanford plutonium-production reactors were also natural-U fueled.

In the discussion in his section 5 of the fission cross section relating to Heisenberg's initially garbled random-walk model of criticality that he presented at Farm Hall, McCauley makes the strange comment that U-238 fissions without neutron production. This is not directly an issue in Heisenberg's (mis)calculation, but how can this possibly be? U-238 tends to capture most neutrons emitted in fissions, but some of the yield of the Hiroshima plutonium bomb was contributed by fissions in its U-238 tamper. U-238 does fission more readily when bombarded by higher-energy neutrons; as these are produced in abundance in thermonuclear weapons, 238 makes a substantial contribution to their yield.

McCauley criticizes other authors for taking Heisenberg's random-walk model too seriously at the cost of overlooking his more correct 1939 derivation. This is fair enough in that such emphases can be distorted to serve as a platform for criticizing Heisenberg. However, the model is part of the record, and should not be dismissed out of hand. I suggest we use it as a teachable moment for our students: explain why the model is flawed and point out that even Heisenberg could fall into an argument that sounds compelling at first but is later found to be flawed.

Near the start of his section 6, McCauley states that the Germans used uranium “with small U-235 enrichment” in their reactor experiments. This too is not so; the German program never achieved any appreciable enrichment, and modern analyses of German uranium reveals no evidence of any enrichment, depletion, or fission products.5 

In his section 7, McCauley points out that I and other authors were either not aware of Heisenberg's 1939 calculation or did not cover it in favor of looking at the random-walk model; in my case he references my book The Physics of the Manhattan Project. I plead guilty; I was unaware of the 1939 work. In my defense, my book is about the physics of nuclear weapons, not the German program. The passage regarding Heisenberg occupies a single page, and my motivation for including it was exactly the sort of pedagogical exercise alluded to above.

Finally, a very personal opinion with no supporting evidence regarding the conflict between Heisenberg's 1939 calculation and the erroneous random-walk model. There is a chasm of history between 1939 and 1945; in the calamity of the war, might he have simply forgotten what he did earlier? Who among us has not looked through old notes, only to be surprised at finding something we worked out years ago? Having been denounced by the SS, Heisenberg's situation was always potentially perilous; that he could keep his wits about him in such an environment is remarkable.

McCauley's paper reminds us that there is still much to learn regarding the Allied and German wartime nuclear programs and that there is much more to making a nuclear weapon than calculating a critical mass.

1.
J. L.
McCauley
, “
Heisenberg's 1939 reactor theory, Serber's 1943 Los Alamos Primer, and Heisenberg's 1945 Farm Hall critical mass calculation
,”
Am. J. Phys.
92
(
10
),
765
774
(
2024
).
2.
R.
Peierls
, “
Critical conditions in neutron multiplication
,”
Math. Proc. Cambridge Philos. Soc.
35
(
4
),
610
615
(
1939
).
3.
F.
Perrin
, “
Calcul relative aux conditions éventuelles de transmutation en chaine de l'uranium
,”
C. R.
208
,
1394
1396
(
1939
).
4.
L.
Thomas
,
Polonium in the Playhouse: The Manhattan Project's Secret Chemistry Work in Dayton, Ohio
(
Ohio State U.P.
,
Trillium
,
Columbus
,
OH
,
2017
).
5.
T.
Koeth
and
M.
Hiebert
, “
Tracking the journey of a uranium cube
,”
Phys. Today
72
(
5
),
36
43
(
2019
).