TRANSFORMATIONS OF ATOMIC NUCLEI1

Niels Bohr, Copenhagen

The author has already had occasion earlier[^2] to point out that, in order to understand the typical features of nuclear transformations caused by impacts of material particles, it is necessary to assume that the first stage of every collision process consists in the formation of an intermediate, semistable system from the original nucleus and the particle that has collided with it. It must also be assumed that, in this state, the excess energy is temporarily concentrated in the complex motions of all the particles of the composite system that has been formed. The possible subsequent loss of this excess by the nucleus, with the liberation of some elementary or complex nuclear particle, may from this point of view be regarded as a separate event not directly connected with the first stage of the collision process. Therefore one may say that the final result of the collision depends on the competition of all the processes of disintegration and radiation occurring in the composite system in accordance with the laws of conservation of energy and mass.

Fig. 1.

Figure 1 shows a simple mechanical model illustrating these features of nuclear collisions. In a shallow bowl there are—

some number of billiard balls. If the hollow of the bowl were empty, then a ball released into it would roll down one slope and jump out on the other side with its former energy. But if there are other balls in the bowl, then the ball released toward them will not be able to pass freely through the bowl. First it will give up part of its energy to one of the balls, then both will give up part of their energy to two other balls, and so on until the initial kinetic energy has been distributed among all the balls. If the hollow and the balls were ideally smooth and elastic, the collisions would continue until a sufficiently large part of the kinetic energy again became concentrated in a ball close to the rim. If the energy of the released ball is not very great, then the remaining balls will not retain enough total energy to allow any one of them to climb the slope. If, however, there is even the slightest friction between the balls and the bowl, or if the balls are not perfectly elastic, it may well turn out that none of the balls will have a chance to jump out before, owing to friction, so much energy has been lost in the form of heat that the remaining energy is already insufficient for any one of them to jump out.

Such a comparison very aptly illustrates what happens when a fast neutron strikes a heavy nucleus. In view of the large number of particles of which the system consists in this case, and in view of their strong interaction with one another, we may expect, on the basis of this simple mechanical analogy, that the lifetime of the compound nucleus will be very large in comparison with the time required by the fast neutron to pass through the nucleus. This model explains, first of all, the fact that although the probability of emission of electromagnetic radiation by the nucleus during such an interval of time is extremely small, nevertheless, in view of the long lifetime of the compound nucleus, there exists a not too insignificant probability that the system, instead of liberating the neutron, will give up its excess energy in the form of electromagnetic radiation. Another experimental fact which is now easy to understand from such a picture is the surprisingly large value of the probability of an inelastic collision leading to the emission of a neutron with much less energy than that of the neutron which collided with the nucleus. Indeed, from the considerations set forth above it is clear that the decay process of the compound system, which requires the concentration of a smaller amount of energy on a single particle, will take place more readily than the process of disintegration in which the entire excess of energy proves to be concentrated on the particle that has flown out.

At first sight one might think that such a simple mechanical interpretation contradicts a fact so well established by studies of the spectra of radioactive γ-rays, namely that nuclei, like atoms, possess a discrete distribution of energy levels. For in the preceding discussion what was essential was that the compound system would have had to possess

used at practically any kinetic energy of a neutron colliding with the nucleus. However, we must clearly realize that, in collisions of fast neutrons, we are dealing with such an excitation of the compound system as is far greater than the usual excitation levels of γ-rays. Whereas the latter for the most part reach a few million volts, in the former case the excitation will considerably exceed the energy necessary for the complete removal of the neutron from the nucleus in its normal state. This energy may be estimated, approximately, at 8,000,000 eV, from measurements of the mass defect.

Fig. 2 schematically shows the general character of the distribution of energy levels for a heavy nucleus. The lower levels, which are separated from one another on the average by several hundred thousand volts, correspond to the γ-ray levels found in radioactive nuclei. As the excitation is increased, the levels rapidly approach one another, and at an excitation of about 15 million volts, corresponding to the collision of the nucleus with a fast neutron, they are apparently distributed continuously. The nature of the structure of the upper part of the level scheme is shown by means of two high-magnification lenses placed on the diagram—one in the above-mentioned region of continuous energy distribution, and the other in the region corresponding to the excitation obtained in the compound system when a very slow neutron is attached to the original nucleus. The dotted line in the middle of the field of the lower magnifying glass represents the excitation energy of the compound nucleus in the case when the kinetic energy of the incident neutron is exactly equal to zero. The distance from this line to the ground state is therefore just equal to the binding energy of the neutron in the compound system.

Fig. 2.

How the energy levels are situated in the region lying near this line may be judged from the results of experiments on the capture of very slow neutrons with energies of the order of fractions of a

volts. Thus, for example, if the kinetic energy of an incident neutron exactly corresponds to the energy of one of the stationary states of the composite system, then the effect of quantum-mechanical resonance may give such an effective cross-section for the capture of neutrons that exceeds the ordinary cross-sections of nuclei by several thousand times. Such a selective effect was indeed found in the case of several elements, and subsequently it was established that the width of the resonance region in all these cases does not exceed a certain small fraction of a volt. On the basis of the relative distribution of the effect of selective neutron capture among the heavy elements and of the sharpness of the resonance, one can roughly estimate the order of magnitude of the mean distance between energy levels in this region, \(10—100\ \mathrm{eV}\). In the field of view of the lower magnifying glass in Fig. 2 several such levels are shown, and the fact that one of these levels lies very close to the dashed line corresponds to the possibility of selective capture of very slow neutrons in this particular case \(^{1}\).

The distribution of the energy levels of the nucleus shown in Fig. 2 is, in its character, entirely different from that with which we are familiar in the case of ordinary atoms, where the excitation of an atom can usually be ascribed to a perturbed quantum state of an individual particle, since the coupling between the individual electronic jumps in the field around the nucleus is insignificant. Nevertheless, the distribution of nuclear levels is precisely of the kind that we have a right to expect from an elastic body, where the energy is concentrated in the vibrations of the body as a whole. Indeed, since with an increase in the total energy of the system the probability of combinations of simple frequencies of such motions increases extraordinarily, the distance between neighboring levels will very rapidly decrease on passing to higher excitations. It is well known that such an interpretation was used in the discussion of the heat capacity of solids at low temperatures.

In discussing the question of the disintegration of a composite system with the liberation of material particles, the application of thermodynamic analogies proves very fruitful. The case of neutron emission, in which no forces act beyond ordinary nuclear distances, presents a particularly successful analogy with the evaporation of a liquid or solid body at low temperatures \(^{2}\). Indeed, it has proved possible, from the approximately known nuclear levels at small excitations, to obtain an estimate

\(^{1}\) The phenomenon of selective capture of slow neutrons, which reveals an interesting formal analogy with optical resonance, was specially studied in the work of Bohr and Wheeler \(^{2}\). The first estimates, from these data, of the widths of levels were made by Frisch and Placzek \(^{3}\) and were discussed in detail in a recent paper by Bethe and Placzek \(^{4}\).

\(^{2}\) Ya. Frenkel \(^{5}\) was the first to propose applying to the probability of emission of a neutron from a composite nucleus the usual formula for evaporation. A more detailed study, based on general statistical mechanics, was given in the paper by Weisskopf \(^{6}\).

“temperature” of the compound nucleus. From this value one obtains such a probability of evaporation for the neutron as agrees with the lifetime of the compound nucleus, calculated from experimental data, formed in collisions with fast neutrons.

Fig. 3 shows the process of collision between a fast neutron and a heavy nucleus. To simplify the reasoning, an imaginary thermometer has been inserted into the nucleus. The scale of the thermometer shown in the figure is given in billions of degrees Celsius, and, for comparison, another, more commonly used measure is also given, namely a scale with divisions in millions of electron-volts. The figure shows the various stages of the collision process. At first the original nucleus is in the normal state, and the temperature is equal to zero. After the impact on the nucleus of a neutron with kinetic energy of approximately ten million volts, a compound nucleus is formed with an energy of 18 million volts, and the temperature rises from zero to approximately one million volts. The irregular outline of the nucleus symbolizes oscillatory deformations reaching an amplitude corresponding to the various oscillations excited at the given temperature. The following figure shows how the neutron flies out of the excited system, and the temperature correspondingly falls somewhat. In the last stage of the process the remaining energy is lost in the form of electromagnetic radiation, and the temperature falls to zero.

Fig. 3.

This collision process described above is most probable

then, when the energy of the incident neutron is large. But for smaller values of the neutron energy, the probabilities of neutron emission and radiation become of the same order of magnitude, which causes a considerable increase in the probability of neutron capture by the nucleus. Finally, if we descend to the region of very slow neutrons, then, as is known from experiment, the probability of radiation even becomes much greater than the probability of neutron emission. It is clear, however, that in this case the analogy between neutron emission and evaporation will be completely inadequate, since the mechanism of emission, like the formation of the compound nucleus, here requires a specific quantum-mechanical explanation, which does not admit so simple an interpretation.

Indeed, a quantitative comparison between ordinary evaporation and neutron emission can be carried out only in those cases where the excitation energies of the compound system are very large in comparison with the energy required to remove a separate neutron from the nucleus, because only in these cases is the excitation remaining in the nucleus after neutron emission almost equal to the excitation of the compound nucleus. This same condition is also regarded as fulfilled in the phenomena of ordinary evaporation, where the change in the heat content of the body under consideration when an individual molecule of gas escapes is very insignificant.

In the experiments carried out so far with impacts of fast neutrons, the conditions for applicability of the analogy with evaporation are, generally speaking, not strictly fulfilled; nevertheless there is a large number of rather qualitative consequences from this analogy which may be of great use in discussing such collision processes. For example, the above-mentioned large probability of energy loss in collisions between fast neutrons and nuclei corresponds exactly to the fact that molecules escaping in ordinary evaporation do not have the full energy of the heated body, but in general leave with a much smaller store of energy per degree of freedom than that corresponding to the temperature of the evaporating body. Further, proceeding from the thermodynamic analogy, we may expect that the energy of the emitted particles will be distributed about this mean value according to the Maxwell distribution law. Moreover, if the energy of the incident neutron is several times greater than the binding energy falling to one particle, then one may predict that not one individual particle, but several particles, each with an energy smaller than that of the incident particle, will leave the compound nucleus gradually, in a series of successive, partial processes of decomposition. Indeed, it has been found that nuclear reactions of this type take place in a number of cases.

These considerations may also be applied to the emission from a compound nucleus of charged particles, such as protons and α-particles. It should only be borne in mind that in this case the latent heat of evaporation is not simply the binding energy of the charged particle, and that to the latter must be added the electrostatic

...the mutual repulsion energy of the emitted particle and the remainder of the nucleus. In addition, this repulsion will act in an accelerating manner on the particles after their emission from the nucleus, and the average kinetic energy of a charged particle will therefore be greater than that of a neutron by an amount corresponding to this repulsion. We are therefore entitled to expect that the energy of the emitted particle will most probably be equal to the sum of the thermal energy and the energy of electrostatic repulsion, and that the probability of emission of a charged particle will, as in the case of neutrons, decrease with increasing energy according to Maxwell’s exponential law. Such a preference for decay processes in which the emitted particle carries away not all the energy available to it is indeed one of the most remarkable features of the large number of nuclear reactions proceeding with the emission of protons or \(\alpha\)-particles.

Up to now we have dealt chiefly with nuclear processes caused by neutron impacts. Nevertheless, similar considerations concerning the formation of an intermediate state will also be applicable to collisions between charged particles and nuclei. But in this case it must be taken into account that the electric repulsive forces acting between positively charged nuclei can, at small values of the kinetic energy of the incident particles, completely prevent, or make less probable, the contact necessary for the formation of the compound nucleus. The action of this electrostatic repulsion of nuclear particles at large distances, in combination with their strong attraction at small distances, can be described simply by means of the concept of the so-called “potential barrier” surrounding the nucleus. An incident charged particle has to overcome this barrier in order to come into contact with the nucleus. After the law governing the spontaneous \(\alpha\)-decay of radioactive nuclei had been explained, it became known that according to quantum mechanics a charged particle has a certain probability of passing through such a potential barrier, even if, from the point of view of classical mechanics, this particle ought to have stopped on the slope of the barrier because of insufficient energy. The same quantum-mechanical effect also explains the experimental fact that slow protons, striking not very heavy nuclei, have a rather large probability of causing nuclear decay processes even at such energy values for which, from the classical point of view, contact of the particle with the bombarded nucleus would have been prevented by electrostatic repulsion.

Another interesting feature of collisions between charged particles and light nuclei is the remarkable resonance effects experimentally established in cases of decay induced by impacts of protons and \(\alpha\)-particles. As in the selective capture of slow neutrons, such resonance may be ascribed to the coincidence of the value of the sum of the energies of the incident particle

and of the original nucleus, with an energy of a stationary state of the compound system corresponding to some quantized collective type of motion of all the particles making up the system.^1 Especially much information on the distribution of high excitation levels in light nuclei has been obtained in studying this resonance effect in the case of impacts by α-particles. In contrast to the close spacing of levels found in heavy nuclei, the levels in this case are separated from one another by several hundred thousand volts of excitation, considerably exceeding ten million volts. This result is easy to understand if one clearly imagines that the lower excitation levels in the case of light nuclei are farther removed from one another than in heavy nuclei, and therefore the number of possible combinations of these levels in the given energy region is much smaller in the first case than in the second.

Not only the distance between resonance levels, but also their half-width is much greater in light nuclei than in heavy ones, which indicates that the lifetime of the compound system in the first case is much shorter than in the second. Above all this follows from the circumstance that resonance in heavy nuclei occurs, as it has turned out, only in the case of very slow particles, for which the probability of escape is extremely small, so that the lifetime of the compound system is determined only by the probability of emission of electromagnetic radiation. In light nuclei, however, the lifetime is, generally speaking, determined entirely by the probability of release of comparatively fast particles. Incidentally, quite independently of this, we have the right to expect that the lifetime of a heavy nucleus—even if that nucleus were so strongly excited that it could emit a fast particle—will be much greater than in light nuclei, owing to the fact that heavy nuclei must be assigned a lower temperature than light nuclei for a given excitation energy.

The simple considerations set forth here apparently can indeed convey, in general outline, the characteristic features of nuclear reactions caused by collisions. It also seems quite possible to us to explain the characteristic differences in the radiative properties of nuclei and atoms by means of a similar treatment. These differences likewise arise essentially from the extreme ease of energy exchange between the closely packed particles of the nucleus, as compared with the approximate independence of each bound electron in the atom. However, a more detailed discussion of these problems lies outside the scope of this brief survey.^2

^1 Besides the total energy of the compound system, as has often been indicated, its spin and other symmetry properties may also be important for the study of resonance phenomena. The question of how such a treatment can be connected with the general picture of nuclear reactions presented in the present article is discussed in an article by Kalckar, Oppenheimer, and Serber.^7

^2 A more complete report on the development of the ideas set forth here will soon be published by Kalckar and the author in Proc. Copenhagen. Acad.

References

1. N. Bohr, Nature, 137, 344, 1936.
2. G. Breit and E. Wigner, Phys. Rev., 49, 642, 1936.
3. O. R. Frisch and G. Placzek, Nature, 137, 357, 1936.
4. H. Bethe and G. Placzek, Phys. Rev., 51, 450, 1937.
5. J. Frenkel, Sow. Phys., 9, 533, 1936.
6. V. Weisskopf, Phys. Rev. (in press).
7. F. Kalckar, J. R. Oppenheimer, and R. Serber, Phys. Rev. (in press).

1. A shortened account of a lecture delivered in Moscow in June 1937. The illustrations are reproduced from three slides shown at this lecture. Translated from the manuscript by V. Vasiliev. ↩