NUCLEAR PHOTOEFFECT¹)

N. Bohr

Bothe and Gentner observed the emission of neutrons from heavy nuclei under the action of gamma rays that had an energy of about \(17 \cdot 10^{6}\ \mathrm{eV}\) and were produced in collisions of protons with lithium. In these remarkable experiments there was revealed a sharply selective character of this nuclear photoeffect. Thus, for a few light elements (in whose distribution no regularity is detected) the cross sections for such an effect proved to be of the order of \(10^{-26}\ \mathrm{cm}^{2}\), whereas for the vast majority of the elements studied no appreciable effect was observed. As various authors have pointed out, such a selective character of the phenomenon is difficult, at first sight, to reconcile with our views on the mechanism of nuclear reactions—views to which the study of phenomena produced by collisions with neutrons leads. Indeed, if one considers the distribution of the energy levels of the compound nucleus formed in such collisions, the following turns out to be the case. For all heavier elements and for excitation energies exceeding \(10 \cdot 10^{6}\ \mathrm{eV}\), the distribution of these levels would seem to have to be practically continuous; meanwhile, the nuclear photoeffect evidently requires, even for much stronger degrees of excitation, the presence of sharply bounded regions of energy and an especially sensitive “tuning” within each of them.

This apparent contradiction, however, disappears if we examine certain features of the distribution of the energy levels of compound nuclei, which, as is known, form an intermediate stage in nuclear transformations produced by collisions. The distribution of the energy levels of these compound nuclei represents a set of stationary states corresponding to more or less coupled proper oscillations. The photoeffect, however, is determined primarily by interaction with certain special oscillatory motions possessing special radiative properties. Thus, in nuclear transformations produced by high-frequency radiation, we are not dealing with any completely definite intermediate state for which there is a competition between the probabilities of decay and radiation. In our case

¹) Nature, 141, 326, 1938. Translation by V. A. Fock.

we must consider the balance between the processes of radiation and those processes which occur as a result of the coupling between the given particular vibrational motion of the nucleus and other possible vibrational states. This coupling will promote the rapid damping of the features of the initial type of excitation and its replacement by a more stable state of the excited nucleus, in which the energy is distributed among all the normal vibrations, just as occurs for the thermal vibrations of a solid at low temperatures. As soon as such a kind of state of excitation of the nucleus has been established, the character of the photoeffect is practically determined. Indeed, in this state the radiative properties of the nucleus will be similar to the properties of an absolutely black body with a temperature of several million volts; therefore the probability that all the excitation energy will be emitted in the form of a single quantum of \(17 \cdot 10^6\) eV will be negligibly small. Moreover, for those high degrees of excitation which are in question here, the total probability of all radiative processes will be much smaller than the probability of decay of the nucleus (the latter also increases with temperature according to an exponential law, as for ordinary evaporation processes).

In these considerations it is assumed that, in the energy region under consideration, the cross section for the nuclear photoeffect is expressed by a formula of the same form as the well-known formula of optics for selective absorption, namely:

\[ \sigma = \frac{\lambda^2}{4\pi}\sum_i \frac{\Gamma_R \Gamma_C} {(\nu - \nu_i)^2 + \frac{1}{4}(\Gamma_R + \Gamma_C)^2}, \]

where \(\lambda\) and \(\nu\) are, respectively, the wavelength and frequency of the \(\gamma\)-rays, while \(\nu_i\) is one of those frequencies which correspond to the greatest resonance. Further, \(\Gamma_R\) is the probability of emission of a secondary quantum \(h\nu\) from the initial special state of excitation of the nucleus, and \(\Gamma_C\) is the probability of transformation of this special state into an ordinary state of excitation with the same energy. As can be seen, this latter process presents a close analogy with that observed in the absorption of light in gases at high pressures, namely, it corresponds to the influence of collisions of gas molecules on the reduction of the sharpness of the resonance.

The available experimental data do not make it possible directly to detect, for any element, a change in the cross section of the selective photoeffect with the frequency of the \(\gamma\)-rays.

The character of the change of this cross section in passing from one element to another for one and the same frequency of \(\gamma\)-rays \((h\nu = 17 \cdot 10^6\ \mathrm{eV})\), however, makes it possible to draw certain conclusions concerning the distances between the resonance maxima and concerning their sharpness. Namely, for the energy region under consideration the distance between the maxima is probably several million volts, while the width of each of them is comparable with that change, multiplied by \(h\), of the frequency \(\nu\) of the incident \(\gamma\)-rays, which

Nuclear Photoeffect

The former is due to the natural width of the line and the Doppler effect in collisions of protons with lithium. This amounts, for the width of the maxima, to about 50,000 V. If, further, one assumes that the largest observed cross sections correspond to the resonance maximum, then the formula derived above gives for \(\Gamma_D\) and \(\Gamma_C\) values of the order, respectively, of \(10^{15}\ \mathrm{sec}^{-1}\) and \(10^{19}\ \mathrm{sec}^{-1}\). Both values appear quite reasonable. Indeed, owing to the high frequency of the \(\gamma\)-rays under consideration, we should expect \(\Gamma_R\) to be several times greater than the probability of emission of ordinary \(\gamma\)-rays in nuclear transformations; the latter probability, judging from neutron-capture processes, will be of the order of \(10^{14}\ \mathrm{sec}^{-1}\). Further, the value of \(\Gamma_C\) must be much smaller than the frequency \(10^{21}\ \mathrm{sec}^{-1}\) of the primary \(\gamma\)-rays and at the same time much larger than the probability of decay of the excited nucleus; the latter, if judged by analogy with the evaporation process, will, for energies of about \(17\cdot 10^6\ \mathrm{eV}\), be of the order of \(10^{16}\ \mathrm{sec}^{-1}\).

Thus the ratio between the duration of the initial transient state and the total lifetime in the excited state is, for \(17\cdot 10^6\ \mathrm{eV}\), about \(10^{-3}\); for lower energies this ratio will be still smaller, since the coupling between different normal modes of vibration decreases, one must suppose, much more slowly than the probability of neutron emission. At the same time, the ratio between the probabilities of emitting a quantum \(h\nu\) from the initial state of excitation of the nucleus and from the subsequent more stable state will rapidly decrease. For \(17\cdot 10^6\ \mathrm{eV}\) this ratio is extremely large, and despite the rapidity of its decrease it will hardly reach the order of unity before we find ourselves in the very middle of the region of discrete nuclear levels. Even in the upper part of this region we must therefore expect a selective character of the nuclear photoeffect, similar to that found in the region of the continuous spectrum; only each resonance maximum breaks up (in experiments with sufficiently monochromatic \(\gamma\)-rays) into a narrow band of sharp absorption lines corresponding to individual levels. When the initial state of excitation ceases to be predominant with respect to radiation, this kind of selectivity will soon disappear and be replaced by a line absorption spectrum of the ordinary type, which, of course, will be accompanied by the nuclear photoeffect only so long as \(h\nu\) is large enough to cause the decay of the nucleus.

A more detailed account of these problems will be given in a subsequent article in the Proceedings of the Copenhagen Academy of Sciences.

For valuable assistance in discussing these problems I here express my gratitude to my colleagues at the Institute of Theoretical Physics, and in particular to Fritz Kalckar, whose sudden death a few weeks ago is for all of us a most grievous loss.