COMPOUND NUCLEI*

F. Friedman and V. Weisskopf

1. Bohr’s address, delivered at the Copenhagen Academy in 1936,\(^1\) exerted a very strong influence on our views. During the eighteen years since its publication, the ideas set forth in it have proved decisive for the analysis of nuclear reactions.

What was the situation in nuclear physics at the time this address appeared? At that time only a very limited amount of qualitative data concerning certain nuclear reactions was known. From earlier work it was known that most reaction cross sections were of the order of nuclear dimensions. (For charged particles the nuclear effects were even smaller because of the presence of the Coulomb field.) Somewhat later Fermi and his collaborators\(^2\) discovered much larger cross sections for reactions of slow neutrons on certain elements.

Earlier theoretical attempts\(^3\) to explain such differences in the magnitudes of cross sections had all been based on an extremely primitive model of the nucleus—the potential-well model. According to this model, the action of the target nucleus on the incident particle can be described, at least in first approximation, by means of a potential corresponding to attraction. The quantum-mechanical state of the system is specified in the form

\[ \Psi=\varphi(\mathbf r)\chi(\mathbf r_1,\mathbf r_2,\ldots,\mathbf r_A), \tag{1} \]

where \(\chi\) is the wave function of the target nucleus, and \(\varphi(\mathbf r)\) is the wave function of the incident particle. It is assumed that the function \(\chi\) is not distorted by the interaction taking place in the system, whereas \(\varphi(\mathbf r)\) is taken to be the solution of the problem of the motion of a single particle in a field with potential \(V(r)\). In the simplest potential-well model the potential \(V(r)\) is a rectangular well:

\[ \begin{aligned} V&=-V_0 &&\text{for } r<R,\\ V&=0 &&\text{for } r>R, \end{aligned} \tag{2} \]

where \(R=r_0 A^{1/3}\). As usual, \(r_0\) is a constant of the order of \(10^{-13}\) cm, and \(A\) is the mass number of the target nucleus. \(V_0\) is a quantity of the order of nuclear energies, say several tens of MeV.

The results obtained on the basis of the potential-well model are quite simple. Two types of reactions are possible: elastic scattering and various emission phenomena. All phenomena are characterized by their cross sections. Scattering cross sections predominate. These cross sections are, generally speaking, of the order of nuclear dimensions, but they attain considerable values in resonances contained within a broad interval.

* The Compound Nucleus. From the volume dedicated to the seventieth anniversary of the birth of Niels Bohr: Niels Bohr and the Development of Physics. Pergamon Press, London, 1955.

The resonance region amounts, for a given angular momentum \(l\), to from 10 to 20 MeV; as for the scattering cross section, in the resonance region it is approximately equal to \((2l+1)4\pi\lambda^2\). If the resonance occurs at low neutron energies, then the scattering cross section becomes very large.

At first sight, the large magnitude of the resonance cross section might serve as an explanation of the large value of the cross section for thermal neutrons in certain nuclei, found by Fermi and his collaborators. However, further consequences following from consideration of the potential-well model point to the inadequacy of this explanation. First, the capture cross section turns out in general to be very small, and even at resonance scattering predominates over capture. According to this model, neutrons remain inside the potential well for a very short time, whence it follows that the probability of their radiative transition into a bound state is extremely small. Secondly, the broad resonance region reflects the most characteristic feature of the behavior of both cross sections: both cross sections depend only weakly on energy. Divided by \(\lambda^2\), these cross sections vary only slightly within an interval of 1 MeV or less. Such a weak dependence on energy and the predominance of scattering even in the resonance region are typical features of the behavior of a single particle.

Nature on two counts expresses disagreement with the potential-well model. Quite shortly before Bohr’s famous address, experiments carried out by Bjerge and Westcott, Moon and Tillman, Szilard, Fermi, and others\(^4\) revealed that neutron cross sections vary appreciably within a few electron-volts: the resonances are extremely narrow and closely adjacent to one another. At the same time it turned out that the cross section at resonance is due mainly to capture.

Thus, in 1936 it was already clear that the potential-well model had to be replaced by something else. Narrow resonances, situated close to one another, and large capture probabilities at resonance required changes in the description of nuclear reactions. And indeed, a complete change in views on nuclear reactions followed; it was precisely to these changes that Bohr’s address was devoted.

The basic concept introduced by Bohr was the concept of compound nuclei\(^*\). Immediately after the incident particle enters the nucleus, a state arises in which many particles participate; this concept is the antithesis of the concept in which a single particle was considered. In justification of such a view the following was stated: the observed, closely adjacent resonances in heavy nuclei indicate that the states formed during the reaction are states characteristic of a many-body system. Expression (1) cannot describe reactions in which closely adjacent resonances participate. Level spacings of only a few electron-volts in systems of nuclear scale can appear only if a considerable number of particles participate in the excitation process. It is therefore incorrect to think that the passage of the incident particle through the nucleus can occur without appreciable disturbance of the whole target nucleus. On the contrary, in order to explain the excitation of many particles, it is natural to take the other extreme point of view, which consists in assuming that all nucleons making up the target nucleus and the incident particle interact strongly with one another.

It was supposed that, as a result of such strong interaction, the incident particle and the nucleus into which it enters unite, forming a compound nucleus in which all the particles, or most of them, are already—

\(^*\) In the Russian literature the terms “compound nucleus” and “intermediate nucleus” are used; both terms have been used in the translation. (Translator’s note.)

locally; resonances are energy levels corresponding to the quantum levels of the compound system. These states, strictly speaking, are not stationary, since they have a finite lifetime: the compound nucleus may decay by re-emission of the primary particle, by emission of a \(\gamma\)-quantum, or in some other way. The width of the resonances, moreover, indicates that the lifetime of the compound state formed by low-energy particles is relatively large compared with the time of direct passage through the nucleus by the incident particle. Thus these states are almost stationary and, in their general properties, should not differ essentially from true stationary states of compound systems at lower energies.

This new idea proved extremely fruitful for describing experiments with low-energy neutrons. With its help it was possible not only to explain the appearance of narrow resonances closely adjacent to one another, but also rather easily to interpret the predominance of capture over scattering in the resonance region at low energies. The quite long lifetime of the intermediate state allows electromagnetic radiation to compete with the other possible modes of decay. A large part of the resonance cross section from scattering, as it had been in the potential-well model, passed in Bohr’s model to capture.

At higher energies the resonances broaden and begin to overlap. The observed phenomena are readily explained by the following two circumstances: first, by the fact that the probability of particle emission increases with increasing energy, and second, by the fact that with increasing energy the number of possible reaction channels increases, each of which makes its contribution to the total width.

Modern experimental data, as well as a certain reasonable extrapolation, indicate that sharp, sharply bounded resonances exist only for incident particles with energies of the order of \(1\)—\(2\) Mev, apart from the case of very light nuclei. At higher energies the width becomes comparable with, or even greater than, the spacing between levels, and the resonance structure disappears.

1. At the same time that Bohr put forward his general ideas concerning nuclear reactions, Breit and Wigner\(^5\) began a more quantitative consideration of resonance phenomena. Subsequently this work was generalized in various directions; among the most important generalizations are the works of Bethe and Placzek\(^6\) and, later, those of Wigner with collaborators\(^7\). Careful measurements carried out during the last two decades have shown that all resonance phenomena in nuclear reactions fit well within the framework of the Breit—Wigner formula. The agreement of this description with experiment may serve as evidence for the existence of well-defined compound states; the characteristics of these states—their lifetimes, the probabilities of decay through various channels, etc.—can be measured and systematized.

In the region where the levels of the compound nucleus overlap, the Breit—Wigner formula must be generalized to the case of many levels, but this description is practically useless, since it depends to the strongest degree on the unknown phase relations between the resonances. In order to obtain conclusions of practical significance concerning nuclear reactions in this region, it is necessary to simplify the whole picture greatly.

Such a simplification is usually made by following the suggestion put forward by Bohr in the speech mentioned above. We divide the nuclear reaction into two stages: the formation of the compound nucleus and its subsequent decay. It is then assumed that the decay does not depend on the mode of formation of the compound nucleus. According to this “independence hypothesis,” it is of no importance what constituted

by the incident particle and what the target nucleus was; what is important is only which compound nucleus was formed.

In the resonance region the Breit—Wigner formula automatically splits into two factors corresponding to the two stages of Bohr’s description—the cross section for formation of the compound nucleus and the probability of its decay into a definite final state. In the region where resonances overlap, the separation into factors is already a special assumption. If this assumption is adopted, then the cross section of a reaction of type \((a,b)\) may be written in the form

\[ \sigma(a,b)=\sigma_c(a)\left(\frac{\Gamma_b}{\Gamma}\right)_c, \tag{3} \]

where \(\sigma_c(a)\) is the cross section for formation of the compound nucleus by particle \(a\), and the ratio \((\Gamma_b/\Gamma)_c\) is the probability that particle \(b\) will be emitted from the compound state.

This formula can be used to calculate the cross sections of various reactions. The cross section \(\sigma_c(a)\) can be roughly determined on the basis of the assumption that the compound nucleus is formed directly at the moment when the incident particle reaches the surface of the nucleus. All that is required here is to calculate the probability that the incident particle reaches the surface of the nucleus, a problem that can be solved by simple quantum-mechanical calculations.^8 The probability of decay of the compound nucleus can be determined by considering the inverse process. Since decay through a definite channel is a process inverse to the formation of a compound nucleus through the same channel, any method that permits one to calculate \(\sigma_c(a)\) can also be used to calculate the factor \((\Gamma_b/\Gamma)_c\) entering into (3).

The method just described for calculating reaction cross sections is often called the “statistical method.” Some consequences of this method can be set forth very conveniently with the aid of thermodynamic concepts based on Bohr’s idea of the distribution of energy among the constituent parts of the intermediate nucleus. Excited compound nuclei are regarded as heated systems, and the subsequent decay of such nuclei as the evaporation of particles.^9

The statistical method for determining the yield of nuclear reactions to a considerable extent embraces their most characteristic features. For example, one may conclude that reactions induced by protons will be weaker than reactions induced by neutrons, differing by a certain factor corresponding to the energy of penetration through the barrier. An analogous factor should also appear when one considers the ratio between the yields of two reactions induced by identical particles but ending with emission, in the first case, of a proton and in the second, of a neutron. Further, the mean energy of the emitted particles must be small in comparison with the total energy of the compound nucleus; the excess energy is transferred to the residual nucleus in the form of excitation.

These statements are qualitatively quite correct. The dependence of reaction cross sections on the energy of the incident particles, in particular the energy-yield curves in reactions induced by protons or alpha particles, can be explained quite satisfactorily.^10 The relative yields of \((x,n)\) or \((x,2n)\) reactions (where the symbol \(x\) is to be understood as a neutron, proton, or \(\alpha\)-particle) can in essential features be predicted. The energy distribution of the reaction products can also be indicated with sufficient accuracy. The spectrum of neutrons and protons emitted by a nucleus bombarded by neutrons of energy \(14\ \text{MeV}\) or by protons of the same energy approximately corresponds to the predicted Maxwellian distribution of the evaporating compound nucleus, whose “temperature” does not differ too greatly from the expected value.^11

However, the achievements of statistical theory are limited to a qualitative description of the most characteristic features. The accumulation of quantitative data in recent years has revealed an ever increasing number of quantitative deviations, and some phenomena display features that plainly do not conform to statistical concepts. For example, discrepancies have been found in studying the dependence of the total neutron cross section on energy and the yield of those reactions in which charged particles are emitted. If neutron cross sections are observed by means of instruments having low energy resolution, then their energy dependence appears somewhat similar to the energy dependence of scattering by a potential well[^12]; such a dependence is, from the standpoint of statistical theory, unexpected. As for charged particles, they are often emitted with energies considerably greater than could be expected in evaporation, and with an angular distribution having a maximum in the forward direction[^13].

These discrepancies, as well as certain others that will be discussed below, are serious enough to make it necessary to subject to a thorough analysis the two basic assumptions on which the statistical treatment of nuclear reactions is based: the independence of the decay of the compound nucleus from the manner in which it was formed, and the immediate formation of a compound nucleus as soon as the incident particle has reached the nuclear surface.

1. We shall begin the investigation of the assumptions underlying the reaction scheme under consideration with an analysis of the independence hypothesis. We shall begin with a review of the experimental data that bear on this hypothesis, and only afterward turn to its logical justification.

There is very little material at our disposal that would make it possible to test this assumption directly. It is rather difficult to obtain the same energy values of compound nuclei in different reactions. Nevertheless, some considerations can be put forward.

So long as the energy of the compound nuclei lies in the resonance region, the independence hypothesis always proves to be valid[^14]. At higher energies the hypothesis becomes more doubtful. Ghoshal[^15] found that the independence hypothesis can be accepted for compound nuclei with excitation energy from 15 to 40 MeV, formed by protons on Cu\(^ {63}\) nuclei or by \(\alpha\)-particles on Ni\(^ {60}\) nuclei. In many other cases, however, it proved not to be entirely satisfactory. Cohen and his collaborators[^15] pointed out that the probability of decay of a compound nucleus in some cases depends very sharply on the mode of formation of this compound nucleus. For example, Cohen and Newman[^16] compared the relative probabilities of emission of protons and neutrons from a compound nucleus formed either by protons or by neutrons. Nuclei with mass numbers from 48 to 71 were bombarded with protons of energy 21 MeV and neutrons of energy 14 MeV. If the reaction was induced by protons, it turned out that proton emission was more probable than neutron emission.

On the other hand, other data indirectly indicate the untenability of the independence hypothesis. According to Bohr’s ideas, independence follows from the fact that the acquired energy is distributed among all the particles and therefore the direction along which the particle exciting the reaction entered the nucleus, or its location there, is of no significance for the compound nucleus that has been formed. In fact, as has already been mentioned, it often happens that the reaction products are emitted with an energy considerably greater than would be expected if the energy of the incident particle were distributed among all the constituents of the compound nucleus; moreover, the reaction products often exhibit a nonuniform distribution ...

…in directions (predominantly forward) relative to the direction of motion of the incident particle. In these cases the intermediate state “remembers” the initial process to a considerably greater extent than it should according to the independence hypothesis.

What are the logical arguments in favor of the independence hypothesis, and where should one look for the causes of its inadequacy? If the energy of the incident particle lies in the region of sharp and strongly pronounced resonances, and if this energy coincides with the resonant energy or is close to it, the assumption under consideration can evidently be justified. In this case the nuclear reaction leads only to a single quantum state of the compound nucleus. The properties of this quantum state obviously cannot depend on the manner in which it was obtained*). The validity of this conclusion is limited only by the circumstance that resonances are not stationary states in the strict sense of the word, since they possess a finite width. In fact, because of the overlap of the “tails” of neighboring resonances, we never deal with the realization of a single quantum state. Nevertheless, deviations from the picture corresponding to a single state are small and are of the order of the ratio of the level width to the spacing between levels.

In the region of appreciable overlap of resonances, the validity of the independence hypothesis is by no means obvious**). In this case an incident particle of definite energy excites several states of the compound nucleus, and the relative phases of these states depend on the mode of excitation. Consequently, if one and the same compound nucleus is excited to a certain definite energy by different methods, one should expect different phase relations between the states of the intermediate nucleus, and also different modes of decay, since the probabilities of emission of particles for a linear combination of states depend on the relations between their phases. Thus, in the region of overlapping resonances of compound nuclei the independence hypothesis is by no means trivial; on the contrary, it requires justification (and a justification different from that required for a single state). In other words, this assumption loses its force as a hypothesis.

But when the density of overlapping states is very large and many states of the intermediate nucleus are excited simultaneously, a new possibility opens up. Since in this case very many states participate in the reaction and, consequently, very many different phases, then, although the relative phases are determined in the process of excitation, with respect to the act of decay these phases may behave as if they had a random distribution. It is therefore not excluded that the second stage of nuclear reactions will behave, in the region where there is strong overlap of states, independently of the first stage.

The fact that such possible independence can indeed take place seemed plausible long ago; this was indicated within the framework of the semiclassical consideration to which we have already referred. The reasoning is as follows: the energies at which resonance overlap occurs are sufficiently large to allow the possibility of a classical treatment. Using a purely classical treatment, we first of all point out that the incident particle and the particles composing the target nucleus inter—

*) Such states may exhibit “memory” with respect to certain directions, but not an asymmetry with respect to the “forward” direction. These states possess a plane of symmetry located perpendicular to the direction of motion of the initial particle. (The initial beam and the target nuclei are assumed to be unpolarized.)

**) The arguments that follow were advanced mainly in 1939 by Bohr, Peierls, and Placzek \(^{17}\).

interact with one another so strongly that the energy is distributed among a large number of participating particles in a time interval small compared with the time of free passage of the particle through the nucleus. Then, since the particles continuously and rapidly exchange energy, the state of statistical equilibrium is reached even before the intermediate nucleus begins to decay. And, finally, the decay proceeds from a state of statistical equilibrium, and therefore the probabilities of decays of various kinds do not depend on the manner in which the energy was acquired at the beginning of the reaction.

The quasiclassical treatment also explains the origin of deviations from the picture predicted by the independence hypothesis. This picture is valid only for such energies of the incident particles as are sufficiently large to allow a classical treatment. On the other hand, at high energies of the incident particles, relatively short-lived intermediate states arise, and there is no certainty that there exist at all such energy intervals to which classical reasoning is applicable and in which the lifetime is sufficiently long for statistical equilibrium to be established before the onset of decay.

But not only does the time remaining for the attainment of equilibrium decrease with increasing energy of the incident particle; the interaction cross section between the incident particle and the nucleons also becomes smaller, hindering the establishment of thermal equilibrium. The mean free path within the nucleus for nucleons of energy 100 MeV and higher becomes comparable with the nuclear radius, and some nuclear reactions at such energies can be successfully described under the assumption that an energetic nucleon, passing through the nucleus, interacts with individual nucleons and not with a system consisting of many particles[^18]. In such reactions all the features of the formation of a compound nucleus are significant.

At intermediate energies, when the interaction is still strong, the limiting case of describing nuclear reactions as individual collisions of a nucleon with a nucleon is no longer suitable. However, there is also no need to return to the opposite case of states in which all nucleons are treated on an equal footing. A certain intermediate point of view is reasonable; one can easily imagine a mechanism leading to nuclear reactions in which the decay of the intermediate nucleus depends on the manner of its formation.

For this intermediate energy region Bethe[^19] pointed out the possibility of “local heating.” When the incident particle strikes the nuclear surface, its energy is transferred primarily to the nucleons located near the point of collision*). Thus, in a small region of the nucleus a relatively high temperature arises, and from it a nucleon may be ejected instantaneously with an energy considerably greater than would be expected if the total energy were distributed throughout the whole nucleus. Something different, though close to local heating, occurs when energy is transferred directly to an individual nucleon as the incident particle just grazes the surface. In both cases the energy of the emitted particle will be considerable, but local heating corresponds, in angular distribution, roughly speaking, to emission of the particle in the direction opposite to that of the incident particle, whereas a grazing collision corresponds to the “forward” direction. Another possible mechanism of this type is that the initial impulse leading to a deformation of the nuclear surface propagates through the nucleus, is focused on the opposite side of the nucleus, and may cause the emission of a particle. Reactions occurring by such mechanisms serve as examples in which the independence hypothesis is already invalid.

*) The accuracy of determining all positions in such quasiclassical reasoning is of the order of the wavelength, which for energies of the order of 10 MeV is small in comparison with nuclear dimensions.

In his original paper, Bohr emphasized that the notion of compound nuclei and the two-stage analysis of nuclear reactions are not suitable for considering reactions occurring in very light nuclei. Deuteron stripping reactions and “pick-up” reactions are examples of such reactions, although a representation involving a compound nucleus that includes one of the neutrons participating in the reaction is very fruitful for a detailed theory of these reactions. Since in the reactions under consideration a true compound nucleus is often not formed, this representation plays a role intermediate between the picture of local heating and that of contact, because the statistical distribution of the energy obtained, used in these representations, is untenable.

It is of interest once again to analyze the independence hypothesis from the following point of view. The quasiclassical and quantum descriptions of intermediate systems are connected with one another by the correspondence principle. In particular, the mean interval between energy states \(D\), for systems possessing simple periodicity, corresponds to the period of motion \(\tau \simeq \dfrac{2\pi\hbar}{D}\). In complex systems it is difficult to determine the period of motion; however, the time \(\tau = \dfrac{2\pi\hbar}{D}\) nevertheless plays the role of the time interval during which the classical system passes through all configurations compatible with the initial conditions: after a time of order \(\tau\), all configurations will be similar to those that existed before this \(^{20}\) (of course, within the limits allowed by the uncertainty relation).

In the region of strongly overlapping states of the intermediate nucleus, the widths of the states \(\Gamma\) are greater than the spacing \(D\), which means that the lifetime \(\tau_c = h/\Gamma\) is considerably less than the characteristic recurrence time \(\tau\). In other words, the system does not have enough time to pass through all configurations compatible with its initial energies, angular momentum, etc. The system could pass through all such configurations if it were possible to arrange that all decay channels were closed. If the reaction proceeds through these channels so slowly that all possible configurations of the system are traversed several times, then the decay of the intermediate nucleus will no longer depend on the true initial state. Then both the decay time and the decay probability will not depend on the method by which the compound nucleus was formed. If, however, all reaction channels are open, the lifetime becomes so short that only some configurations have time to be realized; and when the system passes through only a small number of configurations, the question of which configurations in particular are traversed depends critically on the initial conditions. Hence the decay of the intermediate nucleus may depend on the way in which it was formed.

We again arrive at the conclusion that, if the independence hypothesis gives correct results in the region of strong overlap, then it must be valid despite the fact that different configurations are realized through different modes of formation of the intermediate nucleus. Since all decay probabilities have a number of common features, it is possible that they do not depend on individual configurations; they may depend on certain averaged configurations in which some details of individual configurations are lost. Nevertheless, above the energy region of sharp and sharply separated resonances the independence hypothesis becomes unreliable.

In general, one should conclude that the independence hypothesis is a far-reaching assumption. Its significance is different in the region of sharp resonances and in the region of overlapping widths. In the resonance region the independence hypothesis is evidently valid; its validity follows from the circumstance that well-defined quantum states arise, as follows from the very existence of sharp resonances. At high energies its valid-

...becomes doubtful. There is a certain intermediate energy region where the width of the resonances becomes, in order of magnitude, equal to the distance between them, and the hypothesis of independence becomes entirely inapplicable. In this case several states are excited simultaneously, the phase relations between which depend strongly on the manner in which the nucleus is excited. Finally, in the region of strong overlap the independence hypothesis again becomes applicable, at least in individual cases. In these cases its applicability represents the limiting case of statistical disorder attained in the intermediate state before its decay; but since the decay time rapidly decreases with increasing energy, it is not altogether clear whether such a chaotic state is ever reached.

It is therefore not surprising that reactions have been found for which the idea of a two-stage course of the reaction and of the independence of the decay from the mode of formation of the compound nucleus proved inapplicable. We should have expected such nuclear reactions, in which the various properties and probabilities pertaining to the decay products are most closely connected with the initial configuration. Although the independence hypothesis is a very fruitful idea, at the same time it is too strong an oversimplification of reality.

1. We now return to the second hypothesis used in the statistical method of calculating nuclear reactions—the assumption that the compound nucleus is formed at the moment when the incident particle reaches the surface of the nucleus. There is no doubt that intermediate states are sometimes formed by the incident particle; the best proof is furnished by the observed close groups of resonances. There remains, however, the question of how these states, into which a large number of particles enter, arise “immediately” after the particle has penetrated into the nucleus.

The assumption of the instantaneous formation of the intermediate nucleus is based on the classical picture of a particle incident on a system whose individual components are strongly bound to one another. The strong interaction between nucleons would, it would seem, have to lead to a rapid exchange of energy between the particles. The success of the shell model in describing the properties of the lower-energy states of the nucleus, achieved in recent years, casts doubt on the validity of this conclusion. Some data, clarified in connection with the shell model, indicate that nucleons move comparatively freely within the nuclear volume. Such nucleons apparently have their own angular momentum and move in a well-defined individual orbit.

At present we do not have a satisfactory explanation of this circumstance. The obvious contradiction between the clearly expressed orbit of an individual particle and the strong interaction of particles, revealed in experiments on the scattering of nucleons by nucleons, can be explained in two ways: either the forces acting between nucleons are considerably weakened when the nucleons are inside nuclear matter, or the configuration of particles in the lower states of the nucleus is such that it can be described with the aid of independent orbits of individual particles, despite the strong interaction between the particles.

At present there are not very many data in favor of the first explanation. Perhaps our present difficulties in understanding the saturation of nuclear forces on the basis of the interaction of free nucleons indirectly suggest that certain changes occur in the potential of the internucleon interaction when the nucleons are collected in a small volume. It appears, however, that even small changes will make it possible to explain saturation; for example, the introduction of repulsive forces acting among three particles, or the introduction of a broad repulsive core in the potential.

interaction of two particles²¹. These changes by themselves are still insufficient to ensure the behavior of nucleons as free particles in nuclear matter, since they do not exclude a strong interaction between two neighboring nucleons. More decisive changes make the free motion of particles obvious. An example of such a scheme is the nonlinear behavior of the nuclear potential. The saturation value for the potential is caused, it is assumed, by the high density of particles²². If this saturation value is reached within nuclear matter of normal density, the interaction of nucleons ceases, even if they still come closer together.

The second explanation has not yet been formulated in a satisfactory form. The motion of nucleons as free particles in lower excited states, perhaps, can be understood even in the presence of strong interaction between particles, by using the exclusion principle, according to which the transfer of momentum or energy from one particle to another is forbidden because all the states into which the particle could be scattered are already occupied²³. Be that as it may, until it has been proved that the states of free particles are indeed the lower energy states, these considerations remain no more than plausible conjectures.

Despite the absence of a satisfactory explanation, there is no doubt that the lower energy states of the nucleus can be described surprisingly well if one considers a system in which the nucleons move in a common potential well, interacting weakly with one another; thus, a certain group of facts is selected that requires a re-evaluation of ideas about the formation of a compound nucleus by an incident particle falling on a target nucleus.

In undertaking a reconsideration of the process of formation of a compound nucleus, we may be guided by certain discrepancies between the predictions of the statistical model and the experimental data. We have already mentioned, for example, that the dependence of the total neutron cross section on energy over a wide interval of energies does not agree with statistical theory. In considering the neutron cross section, the statistical model proceeds from the assumption that the compound nucleus is formed immediately, as soon as the incident neutron reaches the surface of the nucleus; however, the cross section describing the reaching of the surface of the nucleus turns out to be a monotonically decreasing function of energy, of the form \(E^{-1/2}\) at low energies and reaching the asymptotic value \(2\pi R^2\) at high energies⁸. This does not agree with the experimental data; the observed cross sections show a more complex behavior which apparently indicates the existence of some combination of states of an individual particle and of the compound nucleus. Looking with a microscopic view at experiments of high resolving power in energy, we find narrow peaks, in agreement with Bohr’s theory. Looking more coarsely, Barshall and his collaborators¹² represented the observed cross sections on a three-dimensional diagram as functions of the energy \(E\) and atomic number \(A\). This surface displays a systematic regularity with maxima and minima at those values of \(E\) and \(A\) at which the older theories, proposed even before Bohr and based on the idea of a potential well, indicated the presence of maxima and minima.

Although these widely scattered maxima are not as clearly expressed as would follow from the potential-well model, they are quite unexpected for the statistical model. They provide a weighty argument in favor of a partial return to the old potential-well model. Evidently, to the extent that experiments with slow neutrons indicate the presence of compound nuclei in which the energy is distributed among all the particles comprising it, the old description with the notion of a single particle cannot be literally correct; but a certain averaged ...

picture between the pure description of an individual particle and the instantaneous formation of a compound nucleus. In this intermediate picture, under certain circumstances, at least, the motion of the incident particle inside the nucleus should be approximately the same as the motion of an individual particle in a potential well.

1. One such attempt to combine the concept of an individual particle and of a compound nucleus is embodied in the optical model of the nucleus²⁴. This model describes the influence of the nucleus on the incident particle by means of a potential well—\(V_0(r)\), but allows for the possibility of formation of a compound nucleus by introducing into the potential an imaginary part with a negative coefficient, \(-iV_1(r)\). This part of the potential leads to absorption of the incident wave inside the nucleus; it is assumed that this absorption represents the formation of a compound nucleus. Since \(V_1\) rather determines what is removed from the description corresponding to an individual particle than what is introduced in some other particular method of description, in the present case “formation of a compound nucleus” must be understood more broadly. It includes not only processes in which the energy of the incident particle is distributed over all nucleons, and all the particles form an intermediate state in the orthodox sense; it also includes processes of the type described in item 3, in which the particle interacts only with some part of the particles constituting the target nucleus. What is essential here is that these processes include any process that removes the incident particle from the entrance channel.

According to the optical model, formation of a compound nucleus occurs neither immediately nor with complete certainty. Even if the incident particle has penetrated into the nucleus, it leaves the state corresponding to the state of a free particle with some delay and with a definite probability. If \(V_0(r)\) and \(V_1(r)\) inside the nucleus are taken equal to some reasonable constants, one can determine the coalescence coefficient; that is, the probability of formation of the intermediate system per unit path length of the incident particle in nuclear matter. For an incident particle with energy \(E\), the coalescence coefficient \(K\) is determined in the form

\[ K=\left[-\frac{m}{2(E+V_0)}\right]^{1/2} 2\left(\frac{V_1}{\hbar}\right). \tag{4} \]

The quantity \((K)^{-1}\) determines the distance that the particle must travel inside the nucleus for formation of a compound nucleus to occur with appreciable probability, while the ratio \((\hbar/V_1)\) is the mean time up to the moment at which coalescence occurs. According to this representation, when the incident particle enters the nucleus, it is reflected back and forth exactly as in the old potential-well model until coalescence or emission. Where in the old model there were virtual states, they appear here as well, but as precursors of the final compound nucleus. A nuclear reaction may be conceived as consisting of two stages—the rapid formation of states corresponding to states of an individual particle, followed by the emission of this particle or by coalescence.

In the discussion that unfolded several years ago, when certain data concerning shell structure had accumulated and certain inconsistencies in the concept of intermediate nuclei had become clearly apparent, Bohr noted that the new information reveals to us new details of nuclear reactions and, in particular, makes it possible to investigate the possibility of introducing earlier stages of the reaction before the formation of the final compound state. The intermediate stage introduced by the optical model into the picture of a nuclear reaction—the stage in which an individual particle is inside the nucleus—is perhaps the simplest realization of Bohr’s proposal. During this intermediate stage, the optical model automatically combines reflections

waves from the edges of the potential well with the coupling that occurs inside the nucleus, and indicates how often the compound nucleus is formed.

If one chooses a suitable expression for the functions \(V_0\) and \(V_1\), the action of a potential of the form

\[ V=-[V_0(r)+iV_1(r)] \tag{4'} \]

on an incident beam of particles can be calculated. One can obtain the scattering and absorption cross sections \(\sigma_{el}^{op}\) and \(\sigma_a^{op}\) as functions of the energy. The absorption cross section \(\sigma_a^{op}\) in this model corresponds to the cross section for formation of the compound nucleus in an actual nuclear reaction. The scattering cross section \(\sigma_{el}^{op}\), obtained on the basis of this model, is associated with elastic scattering. By comparing the cross sections calculated in this model with the experimental data, one may hope to determine the function \(V(r)\).

When the total neutron cross section observed experimentally is averaged over a sufficiently large interval of energy (large enough to smooth out the narrow resonances), an unexpectedly good agreement with the predictions of the optical model is obtained. Even the simplest assumption of a rectangular well for \(V_0\) and for \(V_1\) makes it possible to reproduce the characteristic maxima and minima of the dependence of the averaged cross section on energy and mass number, covering all values of the mass number and the energy interval from zero to several Mev. The most suitable values of the potential turn out to be \(^{25}\):

\[ \left. \begin{array}{ll} V_0=40\ \text{Mev}, & \\ 1\ \text{Mev}<V_1<2\ \text{Mev} \end{array} \right\} \quad \text{for } r<R, \qquad \left. \begin{array}{l} V_0=V_1=0 \quad \text{for } r>R, \end{array} \right\} \tag{5} \]

where

\[ R=1.45\cdot 10^{-13}\cdot A^{1/3}. \]

We therefore conclude that the mean free path of slow neutrons penetrating into the nucleus for the formation of a compound nucleus is approximately \(1\text{--}2\cdot 10^{-12}\ \text{cm}\). A neutron does not form a compound nucleus immediately after it has penetrated into the target nucleus; in fact, the probability of formation of a compound nucleus is about \(0.3\) for a path equal to the dimensions of an average nucleus.

1. The fact that such a simple model as the model with a complex potential gives such good agreement with the experimental data is rather surprising. In order to find out whether the reason for this agreement can be understood, it is necessary to examine more deeply than before the relation between the state of affairs in a real nucleus and the optical model. In particular, the question of what is meant by the formation of a compound nucleus requires especially careful consideration.

In the optical model, formation of a compound nucleus and absorption are synonymous and correspond to the removal of the particle from the entrance channel of the reaction. Reality is undoubtedly more complex. Among the complications that we encounter is the fact that the intermediate nucleus may emit the incident particle back through the same entrance channel.

Such re-emission causes no concern in the region of high energies. If the incident particle enters the continuum region rather than the resonance region, then so large a number of possible decay channels of the intermediate nucleus is open to it that the probability of re-emission becomes negligibly small. Formation of a compound nucleus and absorption may be identified. In this sense “reality” and the optical model coincide.

On the other hand, if the energy of the incident particles is small, resonances that are closely adjacent to one another in reality by no means correspond to the smooth energy dependence adopted in the optical model; the presence of resonances can be expressed by means of boundary conditions imposed on the incident wave, which differ substantially from the boundary conditions adopted in the optical model. (Thus, for example,\(^{26}\) the conditions for the formation of resonances require that the derivative of the external wave function vanish at the nuclear surface.) At the same time, a smaller number of possible channels has already been opened; we no longer have confidence that re-emission into the entrance channel can be neglected. Consequently, the formation of the compound nucleus can no longer be identified with absorption.

In the region of resonances there are definite difficulties in establishing the relation between the optical model and reality. To overcome these difficulties, it is necessary to carry out a kind of averaging over all resonances. Such averaging is necessary in order to smooth out the individual peaks. It may also be used to overcome the difficulties in establishing the connection between absorption and the formation of the compound nucleus in the case of re-emission of the particle into the entrance channel. Finally, it can give us the resolution of the paradox consisting in the fact that the boundary conditions for resonances change rapidly with energy, whereas in the optical model the dependence on energy is very small.

How averaging over resonances can lead to correspondence between “reality” and the optical model can be explained in two ways, one of which is somewhat mathematized, the other relatively visual. Since the more mathematized method makes it possible to give a considerably more concise account of the relations of interest to us, we shall begin with an exposition of this method, and then consider the results obtained from a more intuitive point of view.

For simplicity we shall restrict ourselves to considering neutron reactions with partial waves \(l = 0\). The wave function

\[ \Psi = \frac{A}{r}\left(e^{-ikr} - \eta e^{ikr}\right) \tag{6} \]

outside the region of interaction with the nucleus \(R\) has a sufficiently general form if \(\eta\) is a function of the energy \(E\) of the incident particle, a function that changes sharply as \(E\) passes through a resonance. The quantity \(\eta\) is related to the cross sections in the following way\(^{26}\):

\[ \left. \begin{aligned} \sigma_{el} &= \frac{\pi}{k^2}|1-\eta|^2,\\ \sigma_{r} &= \frac{\pi}{k^2}(1-|\eta|^2),\\ \sigma_{tot} &= \frac{\pi}{k^2}\cdot 2(1-\operatorname{Re}\eta), \end{aligned} \right\} \tag{7} \]

where \(\operatorname{Re}\eta\) denotes the real part of \(\eta\). The cross sections given are the true cross sections of elastic scattering, absorption, and, finally, the total cross section for the given energy \(E\) of the incident neutrons, to which the wave number \(k\) corresponds, i.e.,

\[ E = \frac{(\hbar k)^2}{2m}. \]

We shall now average these cross sections over some energy interval \(I\), containing many resonances. The mean is defined by the condition that basically the quantity \(k^2\sigma\) is averaged, and not \(\sigma\) itself; however, when the mean energy is made large in comparison with the interval \(I\), the factor \(k^2\)

will be discarded; only at very small energies will some specialization be required. With these reservations one may define

\[ \langle f(e)\rangle=\frac{1}{I}\int_I f(\varepsilon)\,d\varepsilon, \]

and also

\[ \bar{\sigma}=\frac{1}{k^2(E)}\langle k^2\sigma\rangle . \]

From this definition it follows that

\[ \bar{\sigma}_r=\frac{\pi}{k^2}\left(1-\langle|\eta|^2\rangle\right) =\frac{\pi}{k^2}\left(1-|\langle\eta\rangle|^2\right)-\sigma_{fl}, \tag{8} \]

where

\[ \sigma_{fl}=\frac{\pi}{k^2}\langle|\Delta\eta|^2\rangle, \]

and

\[ \langle|\Delta\eta|^2\rangle=\langle|\eta|^2\rangle-|\langle\eta\rangle|^2 \]

is the mean square fluctuation of the coefficient of the outgoing wave in the interval \(I\). Similarly,

\[ \bar{\sigma}_{tot}=\frac{\pi}{k^2}\cdot 2(1-\operatorname{Re}\langle\eta\rangle), \]

where it depends only on \(\langle\eta\rangle\), as it should, since \(\sigma_{tot}\) is linear with respect to \(\eta\). (\(\bar{\sigma}_r\), of course, depends on \(\langle|\eta|^2\rangle\).) \(\sigma_{fl}\) is called the fluctuation cross section.

In establishing the correspondence between reality and the optical model, we must identify the total cross section calculated on the basis of the model, \(\sigma^{op}_{tot}\), with the averaged “real” total cross section taken at a definite energy. Therefore one of the relations establishing such a correspondence will be

\[ \sigma^{op}_{tot}=\bar{\sigma}_{tot} =\frac{\pi}{k^2}\cdot 2(1-\operatorname{Re}\langle\eta\rangle). \tag{9} \]

To establish the correspondence completely, we wish to identify the absorption cross section calculated in the model with the cross section for formation of the compound nucleus. Absorption in the model corresponds to the sum of all reactions and that part of the elastic scattering which is effected by decay of the compound nucleus through the entrance channel. The analogous cross section in resonance theory has the form

\[ \sigma_c=\sigma_r+\sigma_{ce}, \]

where \(\sigma_r\) is the cross section for formation of the compound nucleus, which is composed of the reaction cross section \(\sigma_r\) and the cross section of elastic scattering of the intermediate nucleus \(\sigma_{ce}\). Averaging, we obtain:

\[ \sigma^{op}_a=\bar{\sigma}_c=\bar{\sigma}_r+\bar{\sigma}_{ce}. \tag{10} \]

Let us proceed to the determination of \(\bar{\sigma}_c\). Using (8) and (10), we find

\[ \bar{\sigma}_c=\frac{\pi}{k^2}\left(1-|\langle\eta\rangle|^2\right)-\sigma_{fl}+\bar{\sigma}_{ce}. \tag{11} \]

At this stage we do not have a special expression for the quantity \(\bar{\sigma}_{ce}\), but we know the following about it. At high energies \(\bar{\sigma}_{ce}\) tends to

to zero because of the competition of a large number of other possible modes of decay of the compound nucleus. At these energies one may expect that the function \(\eta\) will be smooth, whence its fluctuations \(\sigma_{fl}\) may be regarded as negligibly small. Therefore in this energy region one may take

\[ \bar{\sigma}_c \approx \frac{\pi}{k^2}\left(1-|\langle\eta\rangle|^2\right). \tag{12} \]

At small values of the energy the cross sections \(\sigma_{fl}\) and \(\bar{\sigma}_{ce}\) increase. We shall verify that expression (12) remains valid and that the cross sections \(\sigma_{fl}\) and \(\bar{\sigma}_{ce}\) are equal. To establish the validity of formula (12) in the region of small energies, we turn to the limiting case of well-pronounced resonances \((\Gamma \ll D)\). We shall use the necessary data, borrowed from resonance theory, in the form of a certain approximation for \(\eta\), which describes well an isolated resonance arising in a compound system if the energy of the incident neutron is equal to \(E_s\):

\[ \eta_{BW}=e^{i2\delta}\left(1-\frac{i\Gamma_n}{E-E_s+i\frac{\Gamma}{2}}\right), \tag{13} \]

where \(\delta\) is a slowly varying phase, depending only on the energy states of the particle and therefore of no significance if we restrict ourselves throughout to definite energy intervals. For this special case, as is seen by substituting \(\eta_{BW}\) in (7), the reaction cross section takes the form

\[ \sigma_r=\frac{\pi}{k^2}\, \frac{\Gamma_n(\Gamma-\Gamma_n)} {(E-E_s)^2+\left(\frac{\Gamma}{2}\right)^2}, \]

in accordance with the Breit—Wigner formula, and

\[ \sigma_c=\frac{\pi}{k^2}\, \frac{\Gamma_n\Gamma} {(E-E_s)^2+\left(\frac{\Gamma}{2}\right)^2}, \qquad \sigma_r=\sigma_c\,\frac{\Gamma-\Gamma_n}{\Gamma} \quad \text{and} \quad \sigma_{ce}=\sigma_c\,\frac{\Gamma_n}{\Gamma}. \]

Averaging, we obtain:

\[ \bar{\sigma}_c=\frac{\pi}{k^2}\,\frac{2\pi\bar{\Gamma}_n}{D}, \]

where

\[ \frac{\bar{\Gamma}_n}{D}=\frac{1}{I}\sum_s \Gamma_n^{(s)} \]

is the average neutron width divided by the mean distance between levels;

\[ \bar{\sigma}_{ce}=\frac{\pi}{k^2}\,\frac{2\pi}{I} \sum_s \frac{\left(\Gamma_n^{(s)}\right)^2}{\Gamma^{(s)}}. \tag{14} \]

Let us compare the result obtained for \(\bar{\sigma}_c\) with the value \(\frac{\pi}{k^2}(1-|\langle\eta\rangle|^2)\), replacing \(\eta\) by \(\eta_{BW}\). We have:

\[ \langle\eta_{BW}\rangle = e^{i2\delta}\left(1-\pi\frac{\bar{\Gamma}_n}{D}\right), \]

and therefore

\[ \frac{\pi}{k^2}\left(1-|\langle\eta_{BW}\rangle|^2\right) = \frac{\pi}{k^2}\,\frac{2\pi\bar{\Gamma}_n}{D} \left(1-\frac{\pi}{2}\frac{\bar{\Gamma}_n}{D}\right). \]

By virtue of the fact that \(\Gamma_n \ll \Gamma \ll D\), the last term in the brackets may be neglected. Consequently, formula (12) is valid even in this limiting case. (The neglect which we made in proving the equivalence of the last formula with (12) is, in order of magnitude, equal to the accuracy of the approximate resonance theory used by us.) The result obtained is equivalent, according to (11), to the assertion

\[ \sigma_{fl}=\bar{\sigma}_{ce}, \]

the validity of which can also be confirmed by a direct estimate of the quantity \(\langle|\Delta\eta|^2\rangle\) for the determination of \(\sigma_{fl}\) and by subsequent comparison with the quantity \(\sigma_{ce}\), determined from (14).

Thus, we have obtained the complete system of relations defining the correspondence:

\[ \sigma_{to}^{op}=\frac{\pi}{k^2}\,2(1-\operatorname{Re}\langle\eta\rangle),\qquad \sigma_a^{op}=\frac{\pi}{k^2}(1-|\langle\eta\rangle|^2) \]

and, as a consequence,

\[ \sigma_{el}^{op}=\frac{\pi}{k^2}|1-\langle\eta\rangle|^2. \]

All these relations may be replaced by a single one:

\[ \eta^{op}=\langle\eta\rangle. \]

Some additional light on the last relation may be shed by the averaging of elastic scattering:

\[ \bar{\sigma}_{el}=\frac{\pi}{k^2}\langle|1-\eta|^2\rangle =\frac{\pi}{k^2}|1-\langle\eta\rangle|^2+\frac{\pi}{k^2}\langle|\Delta\eta|^2\rangle, \]

whence

\[ \sigma_{se}=\bar{\sigma}_{el}-\sigma_{fl} =\frac{\pi}{k^2}|1-\langle\eta\rangle|^2. \]

This is the smooth, nonfluctuating part of the scattering, sometimes called scattering of the elastic type (shape elastic). Since \(\sigma_{se}=\sigma_{fl}\) and \(\sigma_{el}^{op}=\bar{\sigma}_{el}-\bar{\sigma}_{ce}\), we again find that such elastic-type scattering corresponds to scattering in the model (that is, \(\sigma_{el}^{op}=\frac{\pi}{k^2}(1-\langle\eta\rangle)^2\)). Consequently, the optical model is closely connected with an averaging in which all fluctuations are completely excluded; these fluctuations are transferred from scattering to absorption:

\[ \sigma_{el}^{op}=\bar{\sigma}_{el}-\bar{\sigma}_{ce},\qquad \sigma_a^{op}=\bar{\sigma}_r+\bar{\sigma}_{ce}. \]

The validity of this correspondence depends on whether the fluctuations with elastic scattering of the intermediate nucleus are equivalent.

The decisive point in our argument was the establishment of the relation \(\bar{\sigma}_{fl}=\bar{\sigma}_{ce}\). One can penetrate more deeply into the essence of this connection if one follows the change of the neutron wave packet in time. Fluctuation scattering of the wave packet appears later in comparison with the rest of the packet; this makes it possible to identify this scattering with that part of the wave function which is delayed, forming the intermediate state. In the main, the different behavior in time for the scattering associated with \(\sigma_{se}\) and \(\sigma_{fl}\) can be seen directly from the definition. \(\sigma_{se}\) changes slowly, following the slow changes of \(\langle\eta\rangle\); \(\sigma_{fl}\) perceives the rapid changes of \(\eta\), omitted in \(\sigma_{se}\). Elastic-type scattering must behave with respect to time more or less similarly to the wave packet in

old model, where a separate particle was considered. Fluctuational elastic scattering, on the other hand, depends on the magnitude \(\Delta\eta=\eta-\langle\eta\rangle\), and the behavior of \(\Delta\eta\) near the resonance of an intermediate state introduces the width \(\Gamma^{(s)}\) into the characteristic of the time behavior of the reflected packet.

More explicitly, we write (6) in the form:

\[ \Psi=\frac{A}{r}\left(e^{-ikr}-\langle\eta\rangle e^{ikr}\right) -\frac{A}{r}(\eta-\langle\eta\rangle)e^{ikr} \quad \text{for } r>R . \tag{15} \]

In this way we introduce the interaction, described by the averaged phase, into the first term, while the fluctuations of the quantity \(\eta\) are isolated in the second. If the second term is omitted, we obtain a model in which the scattering is \(\sigma_{se}\) and the absorption is \(\sigma_c\). It will represent that part which we identified with the optical model.

In accordance with the averaging we have adopted over an energy interval \(I\), large compared with the spacings between levels \(D\), we now form a wave packet from waves of the form (15), the incoming parts of which pass through the given point at the time \(T\sim\hbar/I\). Then we can study the time behavior of the outgoing parts of the wave packet at the moment when they are at a definite distance from the center; we can consider separately the term corresponding to scattering of the elastic type,

\[ \frac{A}{r}\langle\eta\rangle e^{ikr} \]

and the term

\[ \frac{A}{r}(\eta-\langle\eta\rangle)e^{ikr}. \]

Since the mean value \(\langle\eta\rangle\) is constant over the entire energy interval in the pulse, the scattered pulse corresponding to the first of these two terms has exactly the same form as the primary pulse and appears immediately after the primary pulse has passed through the nucleus.

The second term, corresponding to fluctuations, is of greater interest. The principal features of its time dependence*) can be clarified from the expression

\[ f(t)=\int_I(\eta-\langle\eta\rangle)e^{-i\frac{Et}{\hbar}}\,dE. \]

Substituting for \(\eta\) the Breit--Wigner approximation \(\eta_{BW}\), defined by formula (13), we can determine the dependence of \(f(t)\) on time. For \(t>T\) we obtain

\[ |f(t)|^2\sim \sum_{s,s'}\Gamma_n^{(s)}\Gamma_m^{(s')} e^{\,i\frac{(E_s-E_{s'})t}{\hbar}} e^{-\frac{(\Gamma^{(s)}+\Gamma^{(s')})t}{2\hbar}} . \tag{16} \]

In relation (16) all periods

\[ \frac{2\hbar}{\Gamma^{(s)}+\Gamma^{(s')}} \]

are of order \(\hbar/\Gamma^{(s)}=\tau_c^s\). Consequently, \(|f(t)|^2\) spreads out over a time interval of order \(\tau_c^s\).

We may recall the inequality \(\hbar/\Gamma^{(s)}\gg\hbar/D\gg\hbar/I\). (This means that the lifetime \(\tau_c^s\) of the intermediate system is considerably greater than the time of recurrence of motion within the system \(\tau\), which, in turn, is much greater than the duration \(T\) of the wave packet in time.) Consequently, \(\tau_c^s\gg T\). From this we conclude that the emission of the fluctuation wave packet occurs almost entirely after the time \(T\), i.e., almost without any interference with scattering of the elastic type. As was to be expected, the fluctuations give rise to elastic scattering which is somewhat delayed, which proceeds according to (16) with the decay period of the intermediate syste-

*) The motion of the wave packet back and forth is observed in the immediate vicinity of the nucleus, so that during this time the wave packet does not have time to spread.

...s. Our interpretation of $\sigma_{fl}$ as $\bar{\sigma}_{ce}$ is thus supported by the time behavior*).

It is now possible to sketch the development of the nuclear reaction in the following way. When the incident wave packet reaches the nucleus, most of the pulse is scattered by the nuclear potential well; this scattered pulse, to a rough approximation, preserves the form of the initial pulse and leaves the nucleus immediately. At the same time (within the time of coupling $\hbar/V_1$, according to the optical model) part of the incident pulse forms a compound nucleus. Since this part of the pulse, forming the compound nucleus, returns to the entrance channel only after a time interval $\tau$, it has no effect on the initially scattered pulse.

Here we have used a circumstance already discussed in point 3, namely that the quantity

\[ \tau=\frac{2\pi\hbar}{D} \]

represents the “revolution time” of the intermediate state; this time is of the order corresponding to the time required for the pulse to reappear at the entrance channel after the formation of the intermediate state. Since $\tau \gg T$, the reappearance of the pulse at the entrance channel occurs considerably later than the initial scattering occurred. Consequently, the initial scattering of the pulse cannot be changed by imposing any special boundary conditions, or, in other words, the internal pulse forming the intermediate system cannot interfere with the initial pulse and alter the immediate instantaneous reflection. Consequently, the re-emission of a particle by the intermediate nucleus through the initial channel also occurs considerably later, after a time $\hbar/\Gamma$, and may therefore be separated from the scattering that occurs without delay. Hence, for an initial pulsed neutron beam, the formation of the compound nucleus can be clearly separated and appears as the absorption of part of the initial pulse, despite the fact that re-emission may occur later. Finally, reversing our reasoning, one may say that, since

*) In (16) the sum can be divided into

\[ \sum_s \left(\Gamma_n^{(s)}\right)^2 e^{-\Gamma^{(s)}t/\hbar}+\sum_{s'<s}^{\prime}, \]

where $\sum'$ contains all terms mixed between different resonances. In addition to the exponential decay, each mixed term contains the oscillatory time factor $\cos(E_s-E_{s'})t/\hbar$. Under some special conditions these terms can interfere at a definite instant of time and give a significant contribution to $|f(t)|^2$. However, these terms are small in comparison with the diagonal $(s=s')$ sum if one averages over time, since $\Gamma \ll D$. Roughly speaking, because of this,

\[ |f(t)|^2 \sim \sum_s \left(\Gamma_n^{(s)}\right)^2 e^{-\Gamma^{(s)}t/\hbar}, \]

that is, $|f(t)|^2$ exhibits the decay of separate intermediate states.

It is interesting to note that the intermediate elastic wave packet, as a function of energy, extends over the entire energy interval $I$, but it is concentrated, as should have been expected, in narrow energy bands around the resonance energies of the compound system. This concentration is connected with the considerable intermediate time $\tau_c^s$, during which the intermediate elastic scattering appears.

We also note (under the condition $\Gamma/D\ll1$) that in

\[ \frac{1}{T}\int |f(t)|^2\,dt = C\left( \sum_s \frac{\left(\Gamma_n^{(s)}\right)^2}{\Gamma^{(s)}}+\int \sum^{\prime} dt \right) \]

the second term may be neglected, so that

\[ \frac{1}{T}\int |f(t)|^2\,dt \sim \frac{1}{T}\sum_s \frac{\left(\Gamma_n^{(s)}\right)^2}{\Gamma^{(s)}} \]

in full agreement with expression (14) for $\bar{\sigma}_{ce}$.

COMPOUND NUCLEI

the pulse time \(T\) must be small compared with the recurrence time \(\tau = 2\pi\hbar/D\), and the energy spread in the incident beam must be considerably larger than the quantity \(D\). The cross section determined in this way must be averaged over the resonances of the compound nucleus.

1. The predictions obtained on the basis of the model with a complex potential agree rather well with experimental data. We shall divide the experimental data into three groups: total cross sections, data on elastic scattering, and cross sections for the formation of a compound nucleus.

Total cross sections. As has already been noted, the magnitude of the calculated total neutron cross section shows astonishing agreement with experiment, even when a simple rectangular well is taken as the potential. This agreement is especially noteworthy in the region of low energies, between 0 and 2 MeV, in which the total cross section, averaged over resonances, exhibits distinct maxima and minima. These characteristic features are excellently reproduced also in the calculated data if, for the computations, the constants given in (5) are used. The calculations turn out to be very sensitive to changes in \(V_0\), \(V_1\), and \(R\), so that comparison with the experimental data determines the latter quantities quite accurately[^25].

At somewhat higher energies the total cross sections become less sensitive to the values of these constants. They are roughly approximated[^8] by the formula \(\sigma_{tot} = 2\pi(R+\lambda)^2\) and depend only very weakly on the imaginary part of the potential. Nevertheless, the observed deviations from this formula may give some information concerning the constants of the potential well. At present the chief obstacle to determining the value of \(V_1\) above an energy of 4 MeV is a purely mathematical circumstance: it is very difficult to carry out the calculation of scattering by a potential well whose dimensions are large compared with the wavelength, but whose depth is too great for the Born approximation to be used.

For scattering at considerably higher energies, the first application of the optical model was made by Fernbach, Serber, and Taylor[^24]. This important work was essentially the first attempt to reproduce phenomena in the nucleus by means of a complex potential. Recently Taylor[^27] generalized and refined the method, applying it to scattering above 40 MeV. He found that the experimental results above 40 MeV can be reproduced by a potential well somewhat less deep than (5); consequently, the depth

Coalescence coefficient \(K\) in nuclear matter as a function of the energy of the incident particle. The data are taken from the following sources:

At 1 MeV: Feshbach, Porter and Weisskopf, Phys. Rev. 96, 448 (1954).

At 4 MeV: Yost and Beyster, in press.

At 10 MeV: Prus and Hossein, private communication.

(The data in[^31] correspond to the value of \(K\) marked with a cross. Unfortunately, we do not know the accuracy of the measurements. We indicate an error which undoubtedly has a basis, relying on similar cases.)

At 14 MeV: Gittings, Barshall and Everhart, Phys. Rev. 75, 610 (1949); Phillips, Davis and Graves, Phys. Rev. 88, 600 (1952).

(Only a lower limit for \(K\) has been established.)

At 20 MeV: Saxon and Wood, Phys. Rev. 95, 577 (1954).

Above 40 MeV, the curve has been borrowed from T. B. Taylor, Phys. Rev. 92, 831 (1953).

the potential well must decrease with increasing energy. The imaginary part of the potential turns out to be considerably larger than it was at low energies, but it also decreases as the energy is increased further (see the figure).

Elastic scattering. The scattering cross section \(\sigma_{el}^{op}\) and the absorption cross section \(\sigma_{a}^{op}\), calculated on the basis of a model with a complex potential, are more difficult to compare with experimental data. The point is that scattering according to the optical model does not include elastic scattering on the intermediate nucleus, which plays an essential role at energies below 1 or 2 Mev. Similarly, the calculated absorption cross section, which includes formation of the compound nucleus, also cannot be measured directly, since it contains all possible reactions, including elastic scattering on the intermediate nucleus.

If one estimates the elastic scattering on the intermediate nucleus, then certain conclusions can already be drawn on the basis of the experiments available at present.

If reasonable values are adopted for the ratio \(\Gamma_n/\Gamma\) (that is, \(\sigma_{el}/\sigma_a\)), then the very same complex potential which reproduces the total cross section between 0 and 3 Mev also gives the observed angular distribution of scattering at \(1\) Mev \(^{28,25}\).

At higher energies, when elastic scattering on the intermediate nucleus is suppressed by competing processes, the observed scattering can be compared directly with the model. Recent measurements of the angular distribution of elastic scattering of neutrons with energy \(4.1\) Mev (Walt and Barschall \(^{29}\)) agree rather well with the calculated distribution obtained by means of a complex potential well, using the previously adopted values for \(V_0\) and \(R\). As for the imaginary part \(V_1\), it has to be increased by at least a factor of three, which indicates an increase in the probability of formation of the compound nucleus with increasing energy.

If the neutron energy is raised still further, the general scheme of angular scattering is little sensitive to all details of the potential well, with the exception of its radius. The dependence on the radius corresponds to the well-known diffraction scheme for a circular disk. Only very delicate experiments can in this case give us new information about the potential well.

Up to now such experiments have been carried out only with protons; for several elements the scattering of protons of energy \(10\) Mev \(^{30}\) was measured, and the result for oxygen was analyzed with the aid of the optical model. These results were well reproduced by the optical model if one takes \(V_0 = 30\) Mev and \(V_1 = 5\) Mev \(^{31}\). Measurements were also made of the scattering of protons with energy about \(20\) Mev \(^{32}\); Saxon and Woods \(^{33}\) attempted to explain these results on the basis of the optical model. They successfully reproduced the main features of the experiment by increasing the imaginary part to \(10\) Mev and smoothing the corners of the rectangular potential well. The smoothing region, in which the potential rises from the value \(-V_0\) to zero, must have a width of about \(1 \times 10^{-13}\) cm. Such smoothing is quite natural, and one should expect that it will be important at high energies and in scattering through large angles.

Formation of the compound nucleus. The difficulties in comparing theoretical results for the cross section \(\sigma_c\) for formation of the compound nucleus with experimental data can be overcome in the following way. The calculated value of \(\sigma_c\) at low energies is assumed to be averaged over resonances; consequently, one can use the fact that the resonances are rather well determined by the Breit–Wigner formula. Thus, we obtain the following relation—

...solution for \(\sigma_c\) in the case of very small energies (only for neutrons with \(l=0\)):

\[ \sigma_c = 2\pi^2 \lambda^2 \left( \frac{\overline{\Gamma_n}}{D} \right) = \frac{C(A)}{v}, \]

where \(\left( \frac{\overline{\Gamma_n}}{D} \right)\) is the mean of the neutron width divided by the distance between levels, taken for neighboring levels, and \(C(A)\) is a function of the atomic number. This expression permits a direct check of the calculated \(\sigma_c\) at low energies.

It is easy to see that the model indicates the appearance of maxima for the expression \(v\sigma_c = C(A)\) at those values for which the formation of a standing wave inside the nucleus is possible. These will be the cases in which \(\sqrt{2mV_0}\,R = \pi\hbar(n + 1/2)\), where \(n\) is an integer. Consequently, within the limits of admissible nuclear radii we may expect a maximum of the expression \(v\sigma_c\), and also of \(\left( \frac{\Gamma_n}{D} \right)\), near the mass numbers \(A \simeq 11.55\) and 155. The maximum at \(A \sim 155\) was established by Carter, Harvey, Hughes, and Pilcher \(^{34}\). This maximum is not as sharp as was predicted by the optical model with constants borrowed from scattering experiments. However, fluctuations in the dependence of the extremal value of \(A\) on the radius and strong deviations of nuclei from spherical shape may lead to a flattening of the expected maximum. At \(A \sim 55\) and 11 the spacings between levels are large, and it is much more difficult to measure the constants of several levels so as to obtain a reliable mean value. Nevertheless, quite recently the maximum at \(A \sim 55\) was detected by Cote and Bollinger \(^{35}\).

The prediction of the optical model concerning \(\sigma_c\) can also be checked by comparison with the observed cross sections of neutron reactions \(\sigma_r\). The cross section \(\sigma_c\) should be greater than \(\sigma_r\), and the difference should be attributed to elastic scattering on the compound nucleus. The quantitative agreement of experiment with the results following from the model of a rectangular potential well is considerably less satisfactory than the agreement for the total neutron cross sections.

At an energy of \(1\) MeV the experiment gives distinct minima and maxima as a function of \(A\) \(^{28}\). The theory also indicates the presence of maxima and minima, but if one adopts the value of \(R\) used above, then for heavy elements \(^{25}\) these extrema occur at the wrong values of \(A\). If a smaller nuclear radius is adopted for these elements, this discrepancy can be removed, at least within the limits of present experimental accuracy*). (The necessary change in \(R\) shifts the maximum of \(v\sigma_c\) only slightly and even improves its agreement with \(\sigma_{\mathrm{tot}}\).) At high energies the maxima and minima are expressed less sharply; this is equally true in theory and in experiment. However, the experimental cross sections are, in general, somewhat smaller than they should be according to the rectangular-potential-well model \(^{29}\). Smoothing the edges of the potential well leads to an increase in the theoretical values.

This application of a complex potential to elastic scattering and to the formation of a compound nucleus is, in some respects, a stronger test than the agreement of the measured total cross sections

*) It should be noted that, strictly speaking, comparison of theory with experiment for reaction cross sections in this energy interval is a test for both of them. The reaction cross section is measured as the difference between the total cross section and the integrated differential elastic cross section. This difference amounts to only about \(\frac{1}{6}\) of the total cross section.

with the calculated ones. In this case the agreement depends on the details of the transfer of fluctuations from one cross section to another, whereas the total cross section depends directly on \(\langle \eta \rangle\) and is therefore completely independent of this division into absorption and scattering. Since the test here is more sensitive than in determining the total cross section, it is not at all surprising that the agreement with experiment is somewhat worse.

Despite these shortcomings, the values of the reaction cross sections also indicate a considerable change in absorption with increasing energy. At \(1\) MeV the reaction cross sections decrease within the limits predicted theoretically, if the absorption is derived from the total cross section. At \(14\) MeV all cross sections are close\(^{37}\) to the maximum geometrical value \(\pi(R+\lambda)^2\). These data indicate weak absorption at \(1\) MeV and strong absorption at \(14\) MeV.

In the figure (p. 443) are collected the values of the sticking coefficient, determined by formula (4), which must be adopted in various energy intervals in order to reproduce the formation of the compound nucleus. The curve connects the few known values of \(K\) and is intended only as a qualitative guide. At present a more accurate curve has no meaning, because the data known from experiment are determined, at best, to within a factor of two and, moreover, they depend on assumptions concerning the shape of the potential well. The most characteristic feature of the magnitude \(K\) is its rapid rise within the first ten MeV. This rise very probably reflects the rapid increase in the number of possible ways in which the incident particle can excite the nucleus. The fall at higher energies may be ascribed to a decrease in the cross section of the elementary nucleon–nucleon interaction with increasing relative energy.

1. In the present review we have tried to show that a more careful use of Bohr’s original idea and the analysis of the rapidly growing experimental material lead to a modification of the primitive picture of nuclear reactions that was used earlier. According to the former view, the incident particle, striking the target nucleus, forms a compound nucleus, in which its energy is distributed among all the constituents of the intermediate nucleus. The intermediate system decays into some reaction products by a path independent of the process of formation of the compound nucleus.

This point of view must now be changed in several respects: if the incident particle penetrates into the nucleus, the formation of an intermediate system is not at all obligatory. The action of the target nucleus on the particle can be described fairly satisfactorily (if an averaging over resonances is carried out) by means of a complex potential. Part of this action is simply scattering, in which the target nucleus acts exclusively as a potential well. Another part is the formation of a compound nucleus, which occurs with a considerably smaller probability than was previously assumed. Here by the formation of a compound nucleus we mean any process in which the incident particle is removed from the initial channel. This includes not only processes in which the incident nucleon distributes its energy over the whole nucleus, but also those cases in which this energy is transferred to one or several constituent particles of the target nucleus.

The decay of the intermediate system into reaction products depends on the details of the mechanism of energy transfer. Only in certain limiting cases is the decay independent of the mechanism of formation of the intermediate nucleus. In general, the reaction products themselves and their distributions in energy and angle depend on the characteristic conditions that existed at the moment of formation

of the compound nucleus. In order to understand, classify, and calculate the various mechanisms that come into play in nuclear reactions, a detailed study is required of the interaction between the incident particle and the individual and collective motion of the particles composing the nucleus.

The concept of the compound nucleus proposed by Bohr made it possible to look more deeply into many phenomena; indeed, it provided “exceptional possibilities for a broad interpretation of the specific properties of nuclei, admitting the division of nuclear reactions into two clearly separated stages to a degree that has no parallel in the mechanical motion of atoms.”¹ Now, twenty years later, our knowledge has already surpassed the limits of this definition.

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