The Critical Mass

Jonathan Logan. American Scientist. Volume 84, Issue 3. May 1996.

Not far into the 200-odd pages of the recently declassified Farm Hall transcripts comes the extraordinary moment when Werner Heisenberg and the rune other German nuclear scientists being held at a country house in England hear that an Allied Atomic bomb has devastated the city of Hiroshima. The Allies’ atomic weapon, according to the BBC Home Service announcer, has delivered “as much explosive power as 2,000 of our great ten-tonners.” Hidden microphones conveyed the voices of the German “guests” to a nearby listening room at Farm Hall, and so we have in the reports a record of their astonished reaction to this news. Heisenberg responds by flatly rejecting the possibility that the bomb could have been a fission weapon. “Some dilettante in America who knows very little about it has bluffed them,” he says. “I don’t believe that it has anything to do with uranium.”

Even more informative, from a scientist’s point of view, are the intense technical discussions that consumed the subsequent hours and days as the captive scientists tried to puzzle out how their Allied counterparts could have managed to do what they had concluded was beyond reach. Of particular interest, and the main subject of this article, is Heisenberg’s informal estimate of the amount of uranium required for a bomb—the critical mass. Throughout the war Heisenberg seems to have believed that many tons of the rare isotope uranium-235 or of plutonium—an impossible quantity for any country to obtain—would be needed for a bomb, rather than the tens of kilograms actually required. “I consider it perfectly possible that they the Allies have about ten tons of enriched uranium,” Heisenberg says, “but not that they can have ten tons of pure U(235).”

This faulty estimate, apparently made early in the war and reiterated at Farm Hall, was an error of far-reaching consequence. With such an exaggerated notion of the task, the Germans ruled out the possibility that a bomb could be built during the war and instead focused on reactors, which could also be used to generate plutonium for an eventual bomb. A further effect of the mistake was to encourage a false sense of security, which deprived the German project of the kind of urgency that drove the Allies. Thus the chief of German research could report in July 1943 that “though the work will not lead in a short time towards the production of practical useful engines or explosives, it gives on the other hand the certainty that in this field the enemy powers cannot have any surprise in store for us.”

Because it was later asserted (beginning with Heisenberg’s own suggestion at the end of the war) that no bombs had been built in Germany as a conscious, principled choice, the factual question of what the Germans understood about the critical mass and other important requirements for nuclear fission acquired moral weight.

Troubled by news accounts that described them as bomb-builders who had failed—a description that was no more comfortable at home in Germany than abroad—the scientists at Farm Hall quickly drafted a statement, which described their work as entirely peaceful. No mention was made that their work was a sect military project, that their “peaceful” reactor program promised to provide fissionable material for an eventual bomb, or that atomic bombs had been an early goal—Carl Friedrich von Weizsaecker had apparently hoped that building a bomb would help him win the attention of Hitler—that the scientists had abandoned only when it came to appear impossibly difficult. Instead, as Heisenberg implied in several articles after the war, the Germans had considered bomb building but turned away from it on principle, occupying themselves instead with “important technical developments, with a peacetime application.”

Samuel Goudsmit, the physicist who led the Allies’ scientific investigating mission into Germany toward the end of the war, knew from his study of the records that the Germans’ reasons had been more practical than lofty; he found Heisenberg’s “half-truth,” as he called it, hard to leave unchallenged. The subsequent controversy was an unusual one, emotionally charged because of the notorious depravity of the regime for which the German scientists had labored, but hinging on subtle assessments of technical knowledge that could not easily be solved. The most germane evidence by far was to be found in the scientists’ own transcribed words at Farm Hall. Goudsmit, who had interned the scientists in the first place, had been among the first to read the transcripts, but immediately after the war they were placed under heavy secrecy. He was forbidden to quote from them or mention their existence, and soon access to them was denied to all.

In the space created by ambiguity in the available record, controversy has continued for 50 years. Most recently, the German project has been the subject of three well-publicized and seriously received books, which, though they reach more or less opposite conclusions, proceed from a common premise: that the German scientists indeed had achieved a working understanding of fission bombs, on the basis of which they reckoned the construction of atomic bombs to be feasible.

The Farm Hall reports were declassified in 1992. With their help and with information provided by the once-secret wartime progress reports in other archives, it is now possible to analyze the technical history of the German project in enough detail to understand the path it took. The Farm Hall reports bring us the scientists’ detailed reasoning on the subject in their own words.

To take advantage of this unusually direct source, however, we need a guide to the basics of fission-bomb physics as it was comprehended during the early years of the war. For this we now have The Los Alamos Primer (1992), an annotated compilation of Robert Serber’s lectures on fission-bomb physics at the start of the Manhattan Project in April 1943. The lucidity of Serber’s explanations contrasts startlingly with the confused speculations of the scientists at Farm Hall. The cumulative progress of Allied experiment and theory that made this clarity possible also throws into sharp relief the halting uncertainty of the German effort, which by 1942 had not managed to demonstrate the feasibility of any of the processes that would be needed to produce a bomb. It becomes clear that no heroic tale of internal sabotage nor convoluted theory of administrative prescience is required to explain why the Germans did not make a bomb.

The Discovery of Fission

The discovery of fission, announced by Otto Hahn and Fritz Strassmann in Berlin during the first week of January 1939, was understood by physicists everywhere as a possible opening into the vast energy concentrated in atomic nuclei. An estimate of that energy had first been obtained early in the century by measuring the heat evolved in the decay of radium. The news of its unexpected magnitude, reckoned by Ernest Rutherford and Frederick Soddy in 1903 as “at least twenty-thousand times and may be a million times as great as the energy of any molecular change,” inspired visions of bountiful energy and titanic weapons. In his prophetic 1914 novel, The World Set Free, H. G. Wells imagined a future atomic war in which Germany and Austria would battle England, France and possibly America. But prior to 1939 no means had been found to mobilize the nuclear energy so gradually and imperturbably released in the process of natural radioactivity, and by the 1930s such thoughtful minds as Rutherford and Niels Bohr had dismissed the idea as highly unlikely—”moonshine,” in Rutherford’s characteristically blunt phrase.

What Hahn and Strassmann had observed, as the physicists Lise Meitner and Otto Frisch were first to understand (and call “fission”), was the explosion of the uranium nucleus, triggered by bombardment with a neutron. Measurements by Frisch confirmed within a few weeks that the nuclear fragments propelled outward from this explosion carried the huge energies characteristic of the nucleus, ten million times larger than the energy of a chemical reaction.

Fission was therefore a way to liberate nuclear energy, but one that could have practical use only if a means could be found to release the energy in a sustained reaction, analogous to burning a fuel, or—the Second World War was about to begin—to the more rapid combustion responsible for the explosion of a bomb. One way to accomplish a sustained release, conceived in 1934 by Leo Szilard, would be a neutron chain reaction. If neutrons were created in the process of fission, as Bohr’s theory of the nucleus suggested they would be, and if their numbers were sufficient, these neutrons could in principle go on to fission other uranium nuclei, allowing an energy-liberating reaction to spread and escalate.

Within a few more heady weeks, by March of 1939, fission neutrons had been sought and found in several laboratories. During that spring, scientists on both sides of the impending hostilities reported to their governments on the possibility emerging for new weapons of war. On March 17, to a mildly interested group of military staff in Washington, Enrico Fermi, recently escaped from Fascist Italy, described fission experiments being prepared at Columbia University. More expansively, physical chemist Paul Harteck and his assistant at the University of Hamburg wrote to the research department of the Heere-swaffenant (German Army Weapons Department) of a possible new explosive that might be based on uranium fission, “many orders of magnitude more effective than the present one…. The country which first makes use of it,” they pointed out, “has an unsurpassable advantage over the others.”

It remained to be seen, however, whether such weapons could ever be made. On April 29, at the meeting of the American Physical Society in Washington, the possibility of a fission chain reaction was addressed in a public lecture by Bohr, who speculated that a neutron chain reaction in U(235) might produce an explosion, but hat isolating large quantities of the rare isotope would be so difficult as to border on the impossible. On the same day as Bohr’s talk, and long before any government action was taken in Britain or the United States, a secret uranium research project was established in Berlin under the auspices of the German Ministry of Education, and a ban was placed on all exports of uranium from the Reich.

The Search for the Chain Reaction

As can be seen from what they published, throughout 1939 the neutron chain action was very much on physicists’ minds. How best to achieve one remained unclear, but the general assumption evident in the papers was that low-speed neutrons would be the agents—Fermi had reported five years earlier that slow neutrons interacted more readily with nuclei than fast ones—and that tons of uranium would have to be involved.

In his April 29 talk, for example, Bohr had discussed an explosive chain reaction mediated by slow neutrons; Robert Oppenheimer had speculated, in a February 5 letter to a colleague, that a cube of uranium deuteride might “blow itself to hell,” pointing out that the deuterium would be needed “to slow the neutrons without capturing them.” Frederic Joliot, writing from Paris in Nature the same month, likewise invoked slow neutrons, predicting the likelihood of a chain reaction “if a sufficient amount of uranium were immersed in a suitable neutron-slowing moderator.” The prominent physicist Siegfried Fluegge, in a June Naturwissenschaften paper that he also worked into a popular article for the Deutsche Allgemeine Zeitung, argued that an explosive chain reaction might be set off in uranium, explaining that this would require a huge block of the metal several meters in diameter, an amount that would weigh hundreds of tons.

For reasons that are not hard to appreciate now, many ideas about explosive chain reactions in circulation during that first year of fission physics were mistaken. In reality, an explosive chain reaction cannot be produced with slow neutrons, despite their otherwise advantageous properties. Not tons of uranium, as most then supposed, but only tens of kilograms are needed to make a bomb. And, as some but not all physicists realized, this uranium must be nearly pure U(235) isotope. Before a clear picture of an atomic bomb could be envisioned, the curtain of wartime secrecy had begun to descend. Once it had, the images on each side were resolved in fatefully different ways.

Uranium as mined from the earth is a mixture of several isotopes, but mostly U(238) and U(235) in a ratio of about 139:1. Bohr took an important step toward understanding fission when early in 1939 he suggested that the fission properties of the two isotopes might be quite different. His “liquid drop” model of the nucleus predicted that only the rare 235 isotope would be easily enough destabilized to be fissioned by slow neutrons. The more common 238 isotope might be fissioned also, but only by very fast neutrons, whose high kinetic energy would contribute an additional disruptive effect. Thus the fission cross section of U(238) (see Figure 4), related to the ease with which the nucleus is fissioned, would be zero for neutrons below an energy of about one million electron volts (1 MeV) and then rise to a level Bohr’s theory could not accurately predict. A consequence of this energy dependence is that the fission observed when uranium is bombarded by slow neutrons must be due exclusively to the small percentage of U(235) in it. A measurement of slow neutron-induced fission in natural uranium that does not take this into account will underestimate the fissibility of U(235) by a factor of 140. (Figure 4 omitted)

Neutrons are emitted from fissioning uranium nuclei with a spectrum of velocities, and a substantial proportion of them are energetic enough to fission not only U(235) but also U(238). If all of the fission neutrons were emitted with these high energies, a chain reaction could spread through natural, isotopically mixed uranium, and atomic bombs could be made from ordinary uranium metal, which nowadays sells for a few dollars a pound on the international market. In actuality, as Serber points out in The Los ALamos Primer, only about three-fourths of the neutrons are emitted with this much energy, and of these only 25 percent manage to reach a U(238) nucleus before losing so much energy in unproductive collisions that they fall below the threshold. This leaves too few, by a modest margin, to propagate a chain reaction. So U(238) cannot contribute to the chain reaction in a bomb (although it can be used to surround a bomb core of U(235) to exploit the fast neutrons emerging from it and so increase the bomb’s yield).

Two facts about slow neutrons made them attractive. Since 1934, when Fermi first noticed the phenomenon, it had been known that, for basic quantum-mechanical reasons, slow neutrons interact far more readily with nuclei than do fast ones. As Serber’s diagram shows, a low-energy neutron is hundreds of times likelier to fission a U(235) nucleus than is a fast one of 1 MeV energy, simply because a collision is much more likely to occur. Equivalently stated, the average distance a neutron must travel before it collides with a nucleus, the mean free path, is much smaller for low-energy neutrons. Slowing the neutrons, by mixing the uranium with a neutron-retarding substance such as graphite, is therefore a way to greatly improve the efficiency of a chain reaction involving U(235). If U(238) is present, there is a further reason for slowing the neutrons, which is to avoid their “parasitic capture.” At certain intermediate speeds—slower than the initial ejection speed from a fission event but faster than the low “thermal” velocity to which neutrons are eventually slowed by a moderator—it happens that U(238) absorbs neutrons with extraordinary avidity; once absorbed, these neutrons cannot contribute to a chain reaction.

One can understand why it might have seemed to physicists in 1939 that whatever the choice of isotope, slowing the neutrons would improve the chances of obtaining the chain reaction needed for a nuclear explosion. That this is not the case only becomes obvious from analyzing the time course of an explosive chain reaction. Once the reaction has proceeded to the point of heating the uranium to vaporization, it must evolve very quickly, in a few ten-millionths of a second, before the now-gaseous uranium has dispersed to the point of halting the chain action. High-energy fission neutrons are emitted with speeds around 1.4 x 10(9) centimeters per second. Although this is about 5 percent of the speed of Light, a little calculation will show that it is a velocity just barely adequate to permit an explosive chain reaction to gather force in uranium before the material begins to fly apart. The chain reaction in the Hiroshima bomb, in fact, proceeded only far enough to fission 2 percent of the U(235) before the explosion halted it, scattering the remaining, unfissioned 98 percent into the fiery Japanese sky. With slow neutrons, no nuclear explosion is possible at all.

Heisenberg’s Early Ideas on Fission

Against the background of what was then known, or half-known, about neutron chain reactions, we have a sharp picture of Heisenberg’s ideas on the subject as of February 29, 1940. On that day he submitted the second, concluding portion of a comprehensive report to German Army Weapons, which by this time had established its own nuclear-fission project, with Heisenberg as its leading light. The two-part report, marked Geheim—”Secret”—by Heisenberg, was an overview of how a chain reaction might be produced and put to use in service of the war; in effect it was the founding document of the German fission program. In the reports (the first of which was completed in December 1939) he derives a set of differential equations describing the neutron diffusion, scattering and fission that would take place in a mixture of uranium and neutron moderator, as in a reactor. Although the general formalism is cogent, there are some fundamental mistakes. In such a mixture, Heisenberg concludes, the chain reaction will be inherently self-stabilizing: As the reactor was from fission-energy release, the chain reaction will become less efficient and automatically slow down until an equilibrium is reached. As was frighteningly demonstrated at Three Mile Island and Chornobyl (where because of the reactor’s design the chain reaction actually increased with temperature), this is unfortunately not true. Trusting this conclusion, Heisenberg did not generally design control mechanisms into his experimental reactors. Had any of them produced a self-sustaining reaction as he hoped, he and his collaborators might well have been killed.

Heisenberg goes on to argue that if the degree of isotope enrichment is increased, a sufficiently large reactor will produce an immense explosion—”the entire radiation energy of all the uranium atoms present would be freed in an instant.” If “almost pure” U(235) were used in the reactor, the minimum radius to produce such an explosion (the critical radius, R(c),) would be R(c) = 10-pi-l (where l is the mean free path), corresponding to some tens to hundreds of metric tons of U(235). He concludes that a compact energy-producing reactor could be produced by enriching uranium for U(235), and that this is also “the only method of producing explosive materials several orders of magnitude more powerful than the strongest explosives yet known.” he possibility of a slow neutron-mediated explosion is recapitulated in the concluding section of Heisenberg’s report to Army Weapons.

As the work of historian Paul Lawrence Rose has brought to light, although Heisenberg’s own understanding on this point later improved, the mistaken notion of a multi-ton “reactor bomb” nevertheless lived on to propagate confusion in subsequent German reports. The description of a uranium bomb in a May 1943 Gestapo report, for example, reads like Heisenberg’s 1939-1940 concept of a huge explosive reactor: “The uranium bomb can be realized … with neutrons … but their too-large initial speed must’ be slowed down….”

A further error of consequence is contained in Heisenberg’s 1940 report, in the incidental conclusion (mistaken but nonetheless dutifully confirmed by his assistants) that the properties of graphite make it unsuitable as a neutron moderator. This conclusion, and a mistaken measurement of the material’s neutron-absorption coefficient by Walther Bothe a year later, forced the Germans to depend on heavy water, an excellent moderator but extremely expensive and difficult to obtain.

At some point over the next two years Heisenberg seems to have realized that fast neutrons must be used in an atomic bomb. This would be a reasonable inference, anyway, from his description of a bomb as a mass of pure U(235), without moderator, in a February 26, 1942 presentation to high-ranking Nazi officials in Berlin. His discussion of reactors that day, however, again included the assertion that slow neutron chain reactions are inherently self-stabilizing, and again dismissed graphite. In the Berlin lecture Heisenberg did not state a specific figure for the amount of uranium that would be required for a bomb, but the magnitude he had in mind is suggested by his description of the expected energy release (Figure 6), which he puts at 15 trillion kilocalories per ton of U(235) in a bomb. As a criterion for the necessary mass Heisenberg stated—incorrectly, as we shall see—that neutron loss through its surface must be small compared to the increase inside. In a May 1942 lecture to Luftwaffe officials, according to a letter Heisenberg wrote to Goudsmit after the war, he made the same points. (Figure 6 omitted)

The detailed reasoning by which Heisenberg reached this conclusion, which effectively ruled out the possibility of building an atomic bomb, is not spelled out in the text of his February 1942 lecture, which was meant to be intelligible to military officials, and Heisenberg did not provide a calculation. On August 6, 1945, at Farm Hall, as we know, he repeated that a U(235) bomb would require many tons of uranium. There he did spell out his reasoning, which I shall examine a little further on.

The Critical Mass

Whatever the nature of the fission bomb one imagined in 1939, the immediate question that presented itself was, how much uranium would be required? Physicists understood, as a point too obvious to miss, that a chain reaction could occur only if a minimum quantity of uranium or other fissionable material were assembled in one place.

The fact that there is such a threshold amount, a critical mass, is a consequence of the fact that neutrons travel significant distances through matter. Because the nuclei of atoms are only tiny centers about one ten-thousandth the size of atoms as a whole, matter on a subatomic scale is mostly empty space. A neutron moving through a solid, even through dense metallic uranium, can therefore travel long distances before encountering a nucleus. Thus the average distance between collisions, the mean free path, is measured not in atomic diameters but in centimeters.

In a piece of uranium the size of a pea, almost every neutron that is released by the fission of a uranium nucleus will reach the surface of the uranium and escape into the surrounding air before it has had a chance to encounter another uranium nucleus. Very few induced fission events will occur, and a chain reaction cannot take place. In a mass of uranium the size of a boulder, on the other hand, almost all of the emitted neutrons will fission further uranium nuclei before leaving through the surface; so they can contribute to a growing chain reaction. The point between these two extremes at which a chain reaction is just possible is the critical mass. This is conventionally defined as the minimum mass of a sphere of fissionable material (the most efficient shape) for which a chain reaction can take place. The radius of such a sphere is the critical radius.

In May of 1939, Francis Perrin, working in Paris, published in the Comptes Rendus an explicit calculation of the critical mass, but one limited by his assumption that the radius of the uranium sphere must be much larger than the neutron mean free path. Perrin placed the critical mass for a self-sustaining neutron chain reaction in a solid sphere of uranium oxide at 40 tons. Perrin’s calculation was soon improved by Rudolf Peierls in England, who in December 1939 published a more general calculation for the critical mass, valid for all values of the mean free path. No numbers are included in the paper—only algebraic formulas—but a quick estimate had persuaded Peierls that many tons of uranium would be required to produce an explosive chain reaction, so that publication of the formula could be of no consequence in the impending war.

It is possible to fix exactly when clarity entered Allied thinking on the physics of atomic bombs, in the form of a memorandum submitted to British science advisor Henry Tizard on March 19th, 1940, by Otto Frisch and Rudolf Peierls. Emigres from Hitler’s Germany, both men had found temporary berths at the University of Birmingham, where Frisch was continuing his fundamental fission experiments. (Peierls, technically an enemy alien, was then doing unrelated theoretical work.)

To test Bohr’s hypothesis about the differing properties of U(235) and U(238), Frisch had been working on ways to separate the uranium isotopes, and this put him in a frame of mind to ponder what kind of chain reaction might occur with fast neutrons in a sphere of pure 235 isotope. No measurement of the U(235) fast-neutron fission cross section was yet available to insert into the calculation; so in order to make an estimate, “just sort of playfully,” Frisch took the value for neutron scattering by natural uranium (the total cross section), equivalent to assuming that every neutron colliding with a U(235) nucleus would result in a fission. With a few other mildly optimistic assumptions, and using Peierls’s critical-mass formula, Frisch computed the critical mass and found it astonishingly small. For an atomic bomb based on fast-neutron fission in U(235), “a possibility which seems to have been overlooked,” not tons or many kilograms, but 600 grams of uranium might suffice.

Together, Frisch and Peierls redid the calculations, now considering also the time scales involved (which were not treated in Peierls’s published paper), and estimated the efficiency of he explosion. Their conclusion, set out in a four-page memorandum of compelling clarity, was that 1 kilogram of U(235), an amount that could be obtained by a large but not unreasonable industrial effort, would be sufficient to make a weapon of overwhelming destructive power.

A second, more general memorandum noted that the startling possibility that had just opened before their eyes would presumably occur to German scientists as well—the war in Europe had begun several months before—and that no defense would be possible against such a weapon except deterrence. “If one works on the assumption that Germany is, or will be, in the possession of this weapon, it must be realised that no shelters are available…,” they wrote. “The most effective reply would be a counter-threat with a similar bomb.”

By sheer force of reason the Frisch-Peierls memorandum catalyzed what had been a tentative and skeptical inquiry into the possible wartime applications of fission, first in Britain and subsequently in the United States. Animated by the sudden plausibility of an atomic bomb, Tizard, in England, convened a fission advisory group whose final report, a year later, concluded that a uranium bomb of devastating power could be constructed from 11 kilograms of U(235). Wok on the atomic bomb, the MAUD report (the meaningless name having been chosen for concealment) urged, should “continue on the highest priority and on the increasing scale necessary to obtain the weapon in the shortest possible time.”

Progress on Fission, 1940-41

Before Allied authorities would commit major resources to a fission-bomb project, however, government and military leaders had to be convinced that the many assumptions on which this possibility was based could be demonstrated as fact. Fission, after all, had been discovered only two years before. Isotope separation would involve scaling technologies to the industrial level that had in some cases barely been accomplished on the microscopic. Prior to the spring of 1941 plutonium was a hypothetical element, (94)EkaOs(239), whose fissibility, neutron number and stability were all matters of conjecture. In a paper submitted at just the moment secrecy was imposed, Princeton physicist Louis Turner had given reasons to expect the element to be even more fissionable than U(235). But plutonium production for a bomb would be possible only after a self-sustaining chain reaction had been achieved, which at the start of 1941 was still a long time away. Nor had the fission parameters of U(235), on which the possibility and character of a nuclear explosion depend, been accurately measured.

Through 1941 these questions were systematically addressed in Britain and the U.S. Before the year was out, measurable isotope enrichment had actually been demonstrated, both by the barrier-diffusion method and with an electro-magnetic technique being developed at the Berkeley campus of the University of California—the two technologies that ultimately produced the U(235) used in the Hiroshima bomb. Continuing their systematic scale-up toward a self-sustaining chain reaction, Fermi’s group in September completed a 40-ton graphite-moderated reactor based on the lattice design, whose behavior proved to be approximately as predicted.

Using the powerful Berkeley cyclotron, plutonium 239 was created and isolated; its fissibility was measured and found to exceed that of U(235). Newly isolated pure samples of U(235) permitted direct measurement of its fission cross section. On the basis of the new measurements the Frisch-Peierls critical mass was recalculated in March 1941: Eight kilograms would be sufficient for a bomb, reducible to 4 kilograms with the addition of a thick neutron-reflecting outer layer. In a meeting with the President of the United States in October, Vannevar Bush cited the round figure of 25 pounds (11 kilograms) for the critical mass. In light of this cumulative progress an atomic bomb was now seen by the Allies as likely, not merely possible.

A more careful recalculation would later supersede the Frisch-Peierls estimate, and with new experimental values for the fission parameters the critical mass would eventually be revised upward to 60 kilograms; but by then the Manhattan Project had been set massively and irreversibly in motion toward its goal. On December 6th, 1941, the S-1 section of the U.S. Office of Scientific Research and Development was formally reorganized and charged with overseeing a full-scale effort to develop an atomic bomb.

In German laboratories none of these things had been accomplished. The Clusius-Dickel isotope-separation method, pursued as most promising, had been defeated by the disintegrating corrosivity of uranium hexafluoride gas. The fast fission properties of U(235) had not been directly measured. No experiments had been done with plutonium because not one atom of it had been produced. Even had they possessed an adequate understanding of a fast-fission bomb and a realistic estimate of the critical mass, the Germans would still have lacked the confidence-building results that might have emboldened them to attempt an atomic bomb. This presumably is what Heisenberg had in mind at Farm Hall when he made the comment that neither he nor his colleagues would have had the “moral courage” to recommend to the Nazi government the enormous expenditure required for a full-scale bomb project. The day before the S-1 section was reorganized in Washington, as it happens, German Army Weapons Research had ordered a review of the uranium project. Declaring that he could no longer support projects not expected to yield results in the foreseeable future, the head of the Heereswaffenant was considering cancellation of all army support for fission research in Germany.

It is sometimes suggested that technical comparisons of the German and Allied fission programs are inherently unfair because so much more money and material was available on the Allied side. Wealthy and secure across the ocean, it is said, America could afford a billion-dollar gamble on an atomic weapon whereas Germany could not. This may be true for the period beginning in late 1942, when large sums of money first began to be committed to the Allied program and the war had started to turn against Germany; but Allied fission research was by no means lavishly supported in the earlier period we are considering here. In its first two years, as an index, the Advisory Committee on Uranium created late in 1939 by President Roosevelt expended a total of $50,000 on fission research. A further year later, in December 1942, the total cost of Fermi’s CP1, the first self-sustaining reactor, was about $1 million, a sum available in Germany for the asking. The German program comprised some 200 people, heavily weighted toward reactor work; Fermi’s group totaled 43. By the spring of 1942, nonetheless, a large scientific disparity had developed between the two sides, and a sharply contrasting technical outlook.

Heisenberg on the Critical Mass, 1945

At the moment Werner Heisenberg heard the news of Hiroshima in August 1945, he was evidently still under the impression that tons of uranium—”ten tons of pure U(235)”—would be needed for a bomb. If he had a more accurate fix on the critical mass at some point during the war, there is apparently no record of it.

Soon after they recovered from the initial shock of the Hiroshima news on August 6, Heisenberg and Otto Hahn had an extended discussion about how a fission bomb might work. Although he was to receive a Nobel Prize for his role in the discovery of nuclear fission, Hahn was a chemist by training rather than a physicist, and he asked for an explanation. By this point, 16 pages into the transcripts of August 6 and probably late into the night, Heisenberg has been persuaded that the Hiroshima bomb really is atomic.

Heisenberg first responds by explaining that only a fast-neutron chain reaction can be used to make a bomb. With neutrons slowed by a moderator as in a nuclear reactor, he observes, “the reaction is so slow that the thing explodes sooner, before the reaction is complete.” (Here he goes too far; as pointed out above, even a fast fission explosion does not get anywhere near completion.) “How does The bomb explode?” Hahn asks directly. Heisenberg’s answer indicates that although his broad conception of a bomb had advanced since 1940, his estimate of the amount of uranium required remained wildly inaccurate.

If I have pure 235 each neutron will immediately beget two children and then there must be a chain reaction which goes very quickly. Then you can reckon as follows. One neutron always makes two others in pure 235. That is to say that in order to make 10(24) neutrons I need 80 reactions one after the other. Therefore I need 80 collisions and the mean free path is about 6 centimetres. In order to make 80 collisions, I must have a lump of a radius of about 54 centimetres and that would be about a ton.

This attractively clear description of the fission process, which has the ring of a practiced explanation (Heisenberg seems to have given it without preparation), conceals a fundamental misunderstanding. To see what is wrong here, it is necessary to consider in some detail the model Heisenberg is implicitly invoking and to compare it with a well-reasoned solution to the problem. Heisenberg’s calculation of the critical mass involves the fission mean free path, l, the average distance a neutron travels in U(235) before fissioning a nucleus; the neutron number, v, which is the number of neutrons emitted, on average, in each fission; and the density of uranium. The shorter the mean free path and the higher the neutron number, the easier it will be to create a chain reaction.

Heisenberg defines the problem by requiring that 10(24) fissions take place. The figure 10(24) corresponds to roughly one mole, or about he number of uranium nuclei in a quarter-kilogram of uranium metal, and he presumably chose it as a round figure for the amount of uranium that must have been fissioned to produce the Hiroshima explosion. This follows from the fact that fission energy is about 10(7) times larger than chemical energy. (Thus, as was pointed out on the front page of the May 5, 1940 Sunday New York Times, the fission energy in a pound of U(235) is the equivalent of 15 kilotons of TNT.)

To estimate the critical mass, Heisenberg assumes that a neutron diffusing from the center of a U(235) sphere must produce 10(24) fissions before reaching the surface and being lost to the process. As a simple model of neutron diffusion and multiplication in a chain reaction, Heisenberg assumes what is called a random walk, in which, at each collision of a neutron with a uranium nucleus, a fission occurs and two neutrons are created. A random walk is an elementary representation of a diffusion process, adequate for the purpose at hand. Heisenberg’s assumption is equivalent to taking the value v = 2 for the neutron number, which is not far from the true value, and ignoring all neutron collisions that do not result in a fission. Including the effect of these unproductive collisions would alter the result, but not by a large amount.

Heisenberg then says that since the number of neutrons doubles at each successive collision, a neutron starting from the center must undergo an average number of collisions n determined by the formula 2(n) = 10(24), if 10(24) fissions are to be produced. This gives n = 80, approximately. In the random-walk model of diffusion, the mean distance traveled by a particle that undergoes n collisions is d = l -sq. root of- n, where l is the average distance traveled between collisions, which Heisenberg equates with the fission mean free path. Thus the critical radius is given approximately by the simple formula R(c)= 9l.

Taking 6 centimeters for the mean free path, Heisenberg figures the critical radius to be 54 centimeters for the hypothetical sphere of U(235), which corresponds to a critical mass of about 13 tons if the density of uranium, 19 grams per cubic centimeter, is taken into account. (According to the transcript Heisenberg, never reliable with numbers, stated the result as “about a ton,” although Hahn echoes the number back to him as “two tons”.)

Although Heisenberg’s reasoning sounds plausible, it is incorrect. There is no need for every neutron, or even for the average neutron, to give rise to so many fissions before it leaves the mass of uranium. New neutrons are constantly being created in fission events, and these cause further fissions and the production of still further neutrons. So long as the number being created exceeds the number escaping from the surface, the chain reaction will continue to grow. Heisenberg’s criterion does not account for this, and so is much too demanding. A chain reaction will indeed occur in a 13-ton sphere of U(235) as Heisenberg predicts, but it will also occur in a sphere 200 times smaller.

The intuitive appeal of Heisenberg’s argument helps to explain how he could have made such a gross mistake and not realized it. It is not unreasonable to imagine that the critical radius has to be substantially larger than the mean free path; if it is not, one might think, the neutrons released by most of the fissioning uranium nuclei will escape from the mass of uranium before they have had a chance to encounter another nucleus, and the chain reaction will be bound to fail. It is actually a little surprising to discover from the calculations performed at Los Alamos that the critical radius is actually smaller than the fission mean free path; for U(235), R(c) = 0.7l.

This result becomes easier to appreciate when one remembers that the mean free path only represents the average distance neutrons travel before producing a fission; a substantial percentage of them will cause a fission before traveling half that distance. Slightly more than two neutrons are released in each fission, and for the chain reaction to continue it is only necessary that one of these, on average, produce a further fission. Thus the critical radius need not be even as large as the mean free path. Heisenberg evidently had not thought this through, and had gone on believing what was commonly assumed in 1939, that the critical radius must be a good deal larger than the mean free path. That would explain his puzzled exchange with Otto Hahn immediately following the news of Hiroshima. Hahn insists that he remembers hearing that the critical mass was only 50 kilograms; but Heisenberg will not agree that his is possible:

Hahn: I thought that one needed only very little 235.

Heisenberg: If they only enrich it slightly, they can build an engine which will go but with that they can’t make an explosive which will

Hahn: But if they have, let us say, 30 kilograms of pure 235, couldn’t they make a bomb with it?

Heisenberg: But it still wouldn’t go off, as the mean free path is still too big.

Hahn: But tell me why you used to tell me that one needed 50 kilograms of 235 in order to do anything. Now you say one needs two tons.

Heisenberg: I wouldn’t like to commit myself for the moment, but it is certainly a fact that the mean free paths are pretty big.

(The radius of a 50-kilogram sphere of uranium is about 8.6 centimeters; Heisenberg’s working value for the fission mean free path, at this point, is 6 centimeters.)

Since all of Heisenberg’s reports and lectures consistently describe uranium masses of tons, the figure Hahn remembers may have been a reference to the amount of uranium that must undergo fission to produce a city-leveling blast, which, on account of the inefficiency of uranium bombs, is much less Ulan the minimum amount of uranium needed to get the explosion to occur. Likewise, The comment Heisenberg is said to have made during the question period at a June 1942 Berlin presentation to the effect that London could be destroyed by exploding a bomb no larger than a pineapple—a pineapple’s worth would be more than enough. Or there may have been a misunderstanding; the only account of the June remark is not from a scientist but the postwar recollection of one of the Nazi officials in the audience.

Recent Commentary

One of the more fantastic claims to have been made about the Farm Hall reports since their release is that they demonstrate a “sophisticated understanding of bomb physics” on Heisenberg’s part and by extension on the part of the German project as a whole. It is difficult to imagine how even a nonscientist could read this view into the fumbling confusion and chaotic guesswork transcribed at Farm Hall. The record shows the Germans trying out every idea they can think of to explain how the ALlied bomb could have been made. Protoactinium? Neptunium from a temperature-stabilized reactor? Plutonium? An admixture of moderator to slow down the neutrons? Ionium? Still, three recent books go to considerable lengths to argue that no misreckoning of the difficulty or lack of technical progress, but other reasons entirely, kept the Germans from building a bomb.

Two books by Mark Walker—German National Socialism and the Quest for Nuclear Power (published before the release of the transcripts) and Nazi Science—Myth, Truth, and the German Atomic Bomb (published afterward) directly address the question of why the Germans did not attempt a full-scale bomb project. As their titles suggest, the books are put forward as corretives to a received history the author sees as biased and misleading, and so have they been accepted, the first of them having quickly become the standard scholarly reference on the topic. In Walker’s revisionary view, the decisive actors are no longer scientists but administrators of the German Army Weapons Department. These administrators, he argues, fully appreciated from the reports of their scientists that atomic bombs could be built, but after considering Germany’s wartime situation and weighing the costs involved they came to the conclusion that a nuclear power project would be a wiser use of resources. That the German scientists’ fission work had failed to set the groundwork, or that a mistaken understanding of bomb physics figured in the decision, are two of several “lingering myths” Walker seeks to dispel.

The true state of affairs, according to Walker, is that through early 1942 German scientists “had performed the same sort of experiments, had made the same type of calculations, and had come to similar conclusions as the Allies—for example, the estimate of explosive critical mass…” How completely this misrepresents the situation should be evident by this point, but Walker goes further, in both books, to write that “German researchers in February 1942 were working with an estimate of 10-100 kilograms, comparable to the Allied estimates at this time.” This assumption about the relative standing of the two projects is the starting point for Walker’s “demythologizing” of the German fission project.

Although extensive archival references are a prominent feature of both books, the source for this statement about the critical mass turns out to be a single document, an anonymous February 1942 report to Army Weapons summarizing progress in German fission research up to that time. Turning to page 13 of the report, one does find (in passing, in parentheses) an isolated sentence stating that “10-100kg” of material might be sufficient for a bomb. No supporting calculation or discussion appears, however, and no reference is supplied. (The authorship of this report is unknown. Walker attributes the estimate to Heisenberg who, he says, “most likely” would have been asked.)

The context is not in fact a possible uranium bomb but a speculation about plutonium. Elsewhere in the report, however, the subject of an explosive chain reaction in uranium is discussed—on pages 42 and 43, and recapitulated in the summary on page and what appears on those pages 48—is the misguided description of a huge mass of enriched uranium escaping from equilibrium, as in Heisenberg’s original 1939-1940 report. None of this, strangely, earns a mention in Walker’s writings. As to Farm Hall, Walker simply dismisses in advance the critical-mass calculation there as “irrelevant” because, he says, it is certain that Heisenberg and the other German physicists knew the correct critical mass in 1942, and it is “unlikely that they would have forgotten.”

In another recent volume, a New York Times “best book of the year” titled Heisenberg’s War, journalist Thomas Powers intricately plots a 600-page theory that Heisenberg “kept the bomb from Hitler” by deceiving the Nazi authorities into believing a bomb was too difficult to achieve—in particular, according to Powers, by suppressing the knowledge that plutonium would provide an “open road” to building one. The complete lack of evidence for such a conspiracy (Heisenberg, although he disapproved of the Nazis, was a committed patriot who explicitly hoped for a German victory) is explained by the author as the natural result of conspiratorial secrecy.

Lacking direct evidence, the book advances primarily by other means of persuasion, including dramatized, overtly fictional episodes and an imagined “shadow history” of the scientists’ unspoken thoughts. But the time does come when the author has to deal with the question of what Heisenberg actually understood about fission bombs. If he and his fellow scientists saw a bomb as not even remotely feasible, of course, hen The notion of a risky conspiracy to block its construction is nonsense. Putting aside The fact that in his major Berlin presentation to Nazi officials in February 1942 Heisenberg emphasized the plutonium possibility (as a reason for supporting his reactor experiments, presumably), and a host of other stark factual contradictions, it is possible to test the book’s reasoning against what we have learned about the critical mass.

According to Powers (who is joined in this by Stanley Goldberg, Daniel Kevles and several other writers), Heisenberg’s initial calculation of the critical mass at Farm Hall on August 6 yielded the wrong answer only because accurate experimental values for the fission parameters were not at hand. Had the correct fission parameters been available to Heisenberg at Farm Hall, it is argued, he would have quickly reached the correct conclusions. This is something we can check.

In calculating the critical mass on August 6, Heisenberg used a value of the fission mean free path less than half the actual value, 6 centimeters instead of the 13 centimeters he would have computed from the best available figure for the U(235) fast-fission cross section. As we saw above, in his model the critical radius is proportional to the fission mean free path, so substituting the correct value increases the critical radius by a correspondiing amount. Since the critical mass is proportional to the cube of the critical radius, inserting the true value results in a tenfold increase in the critical mass.

Heisenberg’s assumption that the neutron number is equal to 2 alters the result slightly in the opposite direction, but he also made a large arithmetical mistake. If this is corrected and the best available values for both of the nuclear constants are substituted (see Figure 11), Heisenberg’s method of calculation would have yielded M(c) = 96,00 kilograms, a roughly 2,000-fold overestimate of the actual critical mass, and 160,000 times larger than the Frisch-Peierls estimate. So much for the correct fission parameters. (Figure 11 omitted)

In attempting to get his mistaken formula to agree with the values in the newspaper reports, Heisenberg desperately inserted increasingly optimistic values for the nuclear constants (doubling the value for the crucial neutron number v, for example, beyond the value suggested by his own reactor experiments); even with these wishful assumptions, the critical mass still came out much too high.

Over the week following Hiroshima, as might be expected, Heisenberg intensely reconsidered the question of the Allied bomb and the problem of the critical mass to see where he had gone wrong, a question, he says to Harteck, that “has worried me considerably.” Newspaper reports gave further information about the bomb, including its approximate mass and that of the plutonium bomb exploded over Nagasaki three days later. With this information in hand, Heisenberg did on August 14 at last produce a calculation that gave approximately the correct result for the critical mass.

The week-later calculation, a primitive adaptation of Heisenberg’s reactor-theory formalism to the problem of a bomb, does not tell us anything about what his working assumptions had been during the war, of course, although that is what numerous commentators have tried to make of it. Walker cites the “surprisingly accurate” lecture as proof of Heisenberg’s solid mastery of bomb physics. Thomas Powers and Stanley Goldberg transport it through time to the same evening as the Hiroshima announcement, transforming eight intense days of thought and calculation into a few dazzling minutes, and present the lecture as triumphant proof of the knowledge Heisenberg had “hidden” throughout the war. What is really demonstrated is that there never had been any such knowledge to conceal.

In this article I have focused on two reasons the German fission project did not advance further along the road to an atomic bomb: limited laboratory progress in its first two years, and a mistaken estimate of the critical mass. Other factors were surely no less important. A certainty of their own superiority led the Germans to be complacent, and rivalries between the two research groups hindered the sharing of scarce resources. The campaign against the Jews cost Germany many of the talented minds that were to shine so brightly on the Allied side, and a few of the gifted scientists who remained, such as Max von Laue, would do no work for the Nazi regime. Heisenberg at Farm Hall says he was relieved when he found that a bomb was beyond Germany’s reach. And there is also the caprice of history, placing one individual rather than another at a consequential juncture.

In one of the conversations at Farm Hall there is some grumbling that Heisenberg, an abstract theoretician, had been a poor choice to lead the German program. As if to prove the point, a few pages after Heisenberg’s confused critical mass calculation in the transcripts, physical chemist Paul Harteck roughly sketches a better estimate. As Rudolf Peierls recently recalled,

… Heisenberg, though a brilliant theoretician, was always very casual about numbers. When I was his student in the late 1920s the first assignment he gave me was to check whether a recent observation in a spectroscopic experiment could be explained as an example of his uncertainty principle. A simple back-of-an-envelope estimate would have shown that the effect was 100 or even 1,000 times greater than could be explained by his hypothesis.

In this Heisenberg was just the opposite of his counterpart on the Manhattan project, Enrico Fermi. Twice gifted as experimenter and theorist, Fermi had a cultivated talent for quick but reliable estimates of relative magnitudes. At Alamogordo, while banks of spectrographs and ionization chambers triggered into action to assimilate the complex signals of the first atomic explosion, Fermi released a handful of shredded paper into the still New Mexico air. As he had calculated in advance, the one-second pulse of overpressure from the blast displaced the paper falling from his hand. From the distance it was carried by this first surge of nuclear force, about 2.5 meters, Fermi instantly, and accurately, estimated the yield as approximately 10 kilotons TNT equivalent. It is hard to imagine the eminent theorist in Berlin, for all his insight into the abstract world of the quantum, ever improvising this kind of brilliantly simple experiment.