In assessing the prospects of large-scale energy release by means of fission, three basic questions needed to be answered. The first, the easiest, and the least important was: How much energy is released in a fission event? This is a question that Lise Meitner could answer in December 1938, even as she and her nephew Otto Frisch were postulating the existence of fission. She knew the overall pattern of nuclear binding energies from the lightest to the heaviest elements, and reckoned that the fission of a uranium nucleus should release about 1 MeV per nucleon, or about 200 MeV per fission event—a prediction soon verified by experiment. The reason the precise magnitude of the energy release is not very important is that this energy, quickly transformed to heat, is irrelevant to the nuclear course of events. Although it is obviously of great practical significance, fission energy contributes nothing to the chain-reaction process.
The second question was: Which nuclei are most fissionable? and the third: How many neutrons are emitted in a fission event? Once these questions were answered, it could be decided whether a fission chain reaction could be self-sustaining, and if so, with what isotopes.
Nuclear fission would have been a dramatic discovery at any time. Coming as it did on the eve of a major war and in the midst of a period of persecution of Jews in Germany, its drama was heightened. The persecution drove many of Europe’s leading scientists to the United States, and the war drove fission work into secrecy. Half a dozen years after the discovery of fission, the United States emerged as the scientific leader of the world, atomic energy was a household word, and ties forged between science and government set a pattern for the support of large-scale research that has lasted to the present day.
One small act in the total drama of fission was the theoretical work of Bohr and Wheeler in 1939. Using the liquid-droplet model of the nucleus, they provided an explanation of the fission process and predicted which nuclei should be most fissionable (the second question above). Apprised of the ideas of Frisch and Meitner just before leaving Copenhagen early in January, 1939, Bohr pondered the idea of nuclear fission all the way across the Atlantic. By the time he greeted his young colleague John Wheeler on the pier in New York, he had a half-completed theory of fission in his mind (and therefore no inclination to question the validity of the evidence for fission). After five months of effort in Princeton, Bohr and Wheeler submitted for publication a paper that provided both a mathematical theory and a pictorial model of fission. (It was published on September 1, 1939, the day that Hitler’s troops marched into Poland, launching World War II.)
In essence, the Bohr-Wheeler theory is simple. Two forces are at work in a heavy nucleus: the nuclear force, holding the nucleus together, and the Coulomb electrical force, tending to blow the nucleus apart (see the figure below). For all the nuclei we know, the nuclear force is in control, but for the heaviest known nuclei, it is only barely in control. The problem of the fissionability of a nucleus can be posed this way: If a nucleus is stretched into an elongated shape, what is greater—the repulsive electric force tending to push it into an even more elongated shape, or the attractive nuclear force tending to restore it to a spherical or near spherical shape? If a nucleus like uranium is slightly distorted from its normal shape, the nuclear force wins out, tending to restore it to its original shape. If it is distorted much further, the electric force wins out and it splits in two (or occasionally three). Between these two regions is an energy barrier. In slow spontaneous fission (a rarity in naturally occurring elements), this barrier can be penetrated, just as a barrier is penetrated in alpha decay. For the rapid fission that occurs in reactors or bombs, the barrier must be surmounted. The magnitude of the energy barrier to be overcome depends sensitively on the relative magnitude of two energies of opposite sign: the Coulomb (electric) energy, arising from the mutual repulsion of the protons; and the surface-tension energy, arising from the nuclear forces. From approximate considerations of these energies, one can extract a significant parameter, which measures nuclear fissionability.
The mechanism of fission. Attractive nuclear forces create an effective surface tension tending to keep the nucleus in near-spherical form. Electrical repulsion between protons tends to blow the nucleus apart. Below a critical deformation, the surface tension wins. Beyond the critical deformation, the electrical repulsion wins.
Consider first the electric energy. Associated with each pair of protons is an energy ke2/r, where r is an average distance between the two protons. The larger the nucleus, the larger this average separation, in direct proportion to the nuclear radius R. Therefore the Coulomb energy of a pair of protons is proportional to 1/R. The number of distinct proton pairs is equal to Z(Z – 1), since each of the Z protons can pair off with Z – 1 other protons. Therefore, the total Coulomb energy is proportional to Z(Z – 1)/R, or approximately Z2/R if Z is large:
The other relevant energy is the nuclear surface-tension energy. It is proportional to the surface area of the nucleus, or to the square of the nuclear radius:
Esurface ~ R2 .
The ratio Ecoulomb/Esurface is then proportional to Z2/R3. Since the cube of the nuclear radius is proportional to the nuclear volume, which in turn is proportional to the number of nucleons in the nucleus1 (the mass number A), one can write
Bohr and Wheeler recognized that this ratio, Z2/A, is an important parameter of fission. The greater its magnitude, the more nearly does the repulsive electric force win out over the attractive nuclear force, and the more easily fissionable is the nucleus. Consider nuclei of the two principal isotopes of uranium, to each of which a neutron is added:
U235 + n = U236 : Z2/A = 35.9
U238 + n = U239 : Z2/A = 35.4 .
Because of this small difference, the fission energy barrier is slightly lower for U236 than for U239. Another and even more important contributor to the distinction between the fissionability of these two isotopes is the somewhat greater excitation energy provided when U235 absorbs a neutron than when U238 absorbs a neutron. This is attributable, ultimately, to the exclusion principle, which favors an even number of protons and of neutrons. In U235, the absorption of a slow neutron provides enough energy to surmount the fission barrier. In U238 it does not. The difference is all-important. Most of the neutrons emitted during fission are of relatively low energy, capable of inducing further fission in U235 but not in U238.
1 Experiments over the years to measure the radii of nuclei have revealed that the density of nuclear matter (nucleons per unit volume) is nearly constant from one end of the periodic table to the other.