ON THE TRANSFORMATIONS OF ATOMIC NUCLEI INDUCED BY COLLISIONS WITH MATERIAL PARTICLES1

I. GENERAL THEORETICAL CONSIDERATIONS

N. Bohr and F. Kalckar

Preface

As is evident from the title, the present article was intended to constitute the first part of a work consisting of three parts, which were to appear immediately one after another. The second part was conceived as a more detailed exposition of the theory of nuclear collisions on the basis of the general considerations set forth below; the third part was to contain an analysis of the available experimental data on the transformations of atomic nuclei, based on the same ideas. However, the publication of this article, which was in press in January 1937, was delayed, and the completion of the remaining parts was postponed; the reason for this was the authors’ trip to several American universities in order to take part in a whole series of conferences at which problems of the nucleus were discussed. In the meantime, the theory of the nucleus developed rapidly owing to the publication of a number of important papers that appeared in recent months. Moreover, an excellent and complete survey of the present state of nuclear dynamics was published by Bethe2. This survey also included a detailed discussion of some of the considerations developed by us below, based on oral reports by the authors delivered at the conference in Washington in February 1937. In view of these circumstances, we have temporarily abandoned our plan to publish a more detailed work. In order to make our article more appropriate to the present moment, we have appended to it an addendum, written in October 1937; it contains references to the most important of the recent works on our subject, as well as brief comments on them.

§ 1. Basic Ideas

In a recently published article1 it was pointed out that the extraordinary ease with which energy is exchanged between the closely packed particles in atomic nuclei plays a decisive role in the course of their transformations caused by collisions of nuclei with material particles. In considering such collisions it was usually assumed that the transformation of the atomic nucleus consists essentially in the direct transfer of energy from the incident particle to some particle of the original nucleus, which entails the subsequent ejection of the latter. Such an assumption, however, must be abandoned. On the contrary, we must clearly envisage that every transformation of an atomic nucleus passes through an intermediate stage in which the energy is temporarily distributed among all the particles of the compound system formed by the nucleus and the particle that has collided with it. At the small distances in question, large forces arise between any two of the nuclear particles. Owing to this, the connection between the particles of the compound system proves to be extremely close. Any possible decay of this system—whether the emission of an “elementary particle,” such as a neutron or proton, or the emission of a “complex” nuclear particle, such as a deuteron or an \(\alpha\)-ray—must therefore be regarded as a separate event, independent of the first stage of the collision process. Thus one may say that the final result of the collision depends on free competition among all the various decay or radiation processes of the compound system that are compatible with the usual conservation laws.

From this point of view, the study of transformations of atomic nuclei caused by collisions must consist first of all in considering the balance between those separate processes of which the formation and decay of the intermediate semistable system are composed.

Although simple mechanical analogies are very illustrative, the development of this question is evidently impossible without the corresponding quantum considerations. Indeed, above all, the laws of quantum mechanics impose general restrictions on the possible energy states of the compound system; moreover, the very formation or decay of this system is often connected with characteristic quantum effects, well known from the successful explanation of the laws of radioactive decay given by Condon, Gurney, and Gamow. The proposed here

the close connection between the motions of the particles of the nucleus, however, forces one to introduce considerable changes into the usual method of treating such problems, based on the assumption that, in the first approximation, a particle inside the nucleus moves in a constant force field. But we shall see that the extreme interdependence of the particles in the nucleus leads to well-known simplifications, making it possible to draw a number of simple conclusions of a general character concerning nuclear reactions.

The representation of atomic nuclei as quantum-mechanical systems consisting exclusively of neutrons and protons, as is known, has led to extremely interesting results concerning the structure of such nuclei. This representation gives, first of all, an explanation of one fact revealed in the study of band spectra and the hyperfine structure of spectral lines; namely, it explains why the intrinsic spin of the nucleus of any isotope is equal to an even or odd multiple of the unit \(\frac{h}{4\pi}\), depending on whether its atomic weight is even or odd; moreover, this representation explains in general terms how the stability of the nucleus (and hence the existence of isotopes and the magnitude of their mass defect) changes with atomic weight and number. In this connection it should be especially noted that the important data obtained from this by Heisenberg and his collaborators concerning the forces acting between particles in the nucleus at small distances are, in essence, based on an estimate of the mean kinetic energy of these particles in the normal state of the nucleus. In view of the fact that protons, like neutrons, obey the Pauli principle, this kinetic energy will indeed be almost independent of the conditions of motion of the particles in the nucleus; as for its order of magnitude, it is always comparable with that value which is obtained for the energy if one assumes that each particle moves in a separate cell inside the nucleus.

In considering the structure of atomic nuclei it is usually assumed that, in the first approximation, nuclear particles move independently of one another in a conservative force field, like the extranuclear electrons in atoms. However, because of the much closer connection between the particles of the nucleus, one cannot expect that an investigation of atomic nuclei based on this usual method of consideration would yield results comparable with the actual properties of the nucleus. Despite many promising attempts at a more exact calculation of the structure of the lightest nuclei, we must at present be satisfied with regarding atomic nuclei as a state of matter of extraordinarily high density and electrization; the properties of this state can be studied only by analyzing experimental data concerning nuclear reactions. In this, the task is facilitated by the circumstance that in ordinary experiments on the transformation of atomic nuclei the excitation energy of the compound nucleus is very small in comparison with the total energy required for the complete separation of all the particles composing the nucleus; this permits, as we shall see

further on, to liken many properties of nuclear matter to the properties of ordinary solid or liquid substances.

§ 2. Distribution of Nuclear Levels

As was shown,² the distribution of the energy levels of excited nuclei differs sharply from what one might have expected if these excited states were caused, as was usually supposed, by an anomalously large energy of some one particle in the nucleus. Thus, experimental data concerning the capture of fast and slow neutrons by heavy nuclei, accompanied by radiation, show that the distances between the energy levels of such nuclei rapidly decrease as the excitation increases; as a result, the distribution of energy levels becomes practically continuous. This will occur even for such excitation energies as, although sufficient for the emission of a neutron with large kinetic energy, are far from sufficient to change the quasi-stable character of the composite system. Even within the region of a continuous distribution, the mean lifetime of the composite system is probably more than a hundred thousand times greater than the time interval during which a fast neutron would pass through a region of the size of the nucleus. However, the typical features of the distribution of nuclear levels can easily be explained if we imagine that the stationary states of the nucleus must correspond to some quantized collective type of motion of all the particles composing it. Indeed, the rapid convergence of neighboring nuclear levels with increasing energy resembles² in its character the multitude of linear combinations that can be formed from a certain number of independent quantities (see Appendix I). The distribution of levels in the nucleus therefore bears a great resemblance to the distribution of quantum states of a solid body, well known from the theory of heat capacity at low temperatures (see Appendix II).

This analogy gives grounds for making a more direct comparison between the excitation of a nucleus and the vibrations of elastic bodies; this comparison is simplified by the fact that, with the exception of the lightest nuclei, the density of matter and of energy is practically the same in all nuclei. Indeed, if we denote by \(N\) the total number of protons and neutrons in such a nucleus, then the volume is expressed approximately as

\[ V = N\delta^3, \tag{1} \]

where \(\delta\) is approximately equal to \(3\cdot 10^{-13}\) and may be taken as the diameter of the cell occupied by each individual particle in the nucleus. Further, the mean kinetic energy of each particle in such nuclei is approximately expressed by the simple formula

\[ K = \frac{h^2}{8\delta^2\mu}, \tag{2} \]

where \(h\) is Planck’s constant, and \(\mu\) is the mass of the proton or neutron (since their masses are almost equal to one another). This gives for \(K\) approximately \(20\) MeV, and since measurements of the mass defect give for the average binding energy of a neutron or a proton approximately \(10\) MeV, the average loss of potential energy per nuclear particle turns out to be about \(30\) MeV. If the quantity \(\delta\) may be regarded as a unit of length characteristic of nuclear problems, then the unit of time appropriate for such problems will be the interval of time \(\tau\) required for an elementary particle with kinetic energy \(K\) to traverse the distance \(\delta\). The order of magnitude of this interval of time, approximately expressed by the formula

\[ \tau = 2 \frac{\mu \delta^2}{h}, \tag{3} \]

is equal to \(10^{-22}\) sec.

Further, the excitation energy of heavy nuclei is very small in comparison with the total kinetic energy \(NK\) of the normal state of the nucleus; this fact suggests to us an analogy between nuclear excitations and oscillations of the volume and shape of a certain sphere, arising under the action of forces of elasticity \(\varepsilon\) or surface tension \(\omega\), given by expressions of the type

\[ \varepsilon = C_\varepsilon K\delta^{-3}; \quad \omega = C_\omega K\delta^{-2}, \tag{4} \]

where the dimensionless factors \(C_\varepsilon\) and \(C\) must be approximately constant for all nuclei except the very lightest. Thus, \(\nu_\varepsilon\) and \(\nu_\omega\), the frequencies of oscillations of the simplest kind for a sphere of volume \(V\) and density \(\sigma\), are expressed by the usual formulas

\[ \nu_\varepsilon \sim \varepsilon^{\frac12} V^{-\frac13} \sigma^{-\frac12}; \quad \nu_\omega \sim \omega^{\frac12} V^{-\frac12} \sigma^{-\frac12}, \tag{5} \]

which can easily be checked from dimensional considerations. Taking \(\sigma = \mu\delta^{-3}\) and using (1), (2), and (4), we obtain from (5), for the differences of energies between consecutive quantum states of the nucleus corresponding to such oscillations, the following expressions:

\[ \Delta_\varepsilon E = h\nu_\varepsilon \sim \sqrt{8C_\varepsilon}\, N^{-\frac13} K; \]

\[ \Delta_\omega E = h\nu_\omega \sim \sqrt{8C_\omega}\, N^{-\frac12} K. \tag{6} \]

Since the numerical values of the constants \(C_\varepsilon\) and \(C_\omega\) are difficult to determine, the chief interest of these formulas lies in the fact that they give the change of the energy differences as a function of \(N\). Thus, the fact that the average differences of energy between the lowest excited states of nuclei vary definitely faster than \(N^{-\frac13}\), and even somewhat faster than \(N^{-\frac12}\), shows,

that, at least in the case of heavier nuclei, the weakest excited states cannot be ascribed to oscillations corresponding to \(\Delta_{\omega}E\); the presence of such oscillations can be expected only at stronger excitations. Nevertheless, the fact that the expression for \(\Delta_{\omega}E\) better corresponds to the way in which the mean spacing between the lowest levels decreases with \(N\) suggests a more direct comparison between surface oscillations and the fundamental frequencies (degrees, modes) of nuclear excitation that determine the distribution of levels. However, the proper surface energy of nuclei, calculated on the basis of the mass-defect curves\(^4\) and substituted in (4) and (6), gives for \(\Delta_{\omega}E\) values exceeding a million volts even for heavy nuclei, for which the mean spacing between levels is probably not more than several hundred thousand volts. This indicates that such comparisons encounter great difficulties (see Addendum III).

It is evident that all such simple considerations can, at best, serve for a first orientation in the question of the possible origin of nuclear excitation. For a more exact discussion of this question, more detailed considerations are required concerning the special character of the interaction between the individual particles of the nucleus, and also concerning the stability of nuclei and the mechanism of their excitation. The insufficiency of simple considerations alone is clear from the well-known periodicity of the mass-defect curves and from the noticeable difference in the distances from the ground level to the excited levels observed for nuclei with even and with odd atomic weight and number. These effects must evidently be ascribed to the different degree of saturation of the bonds between pairs of nuclear particles; we have in mind bonds that can be obtained for such nuclei under a more rigorous quantum-mechanical treatment of the corresponding many-body system on the basis of the restrictions prescribed by the Pauli principle. In view of the presence of a close connection between the motions of the particles of the nucleus, it is still rather difficult to say how reliable are conclusions concerning the exchange character or spin dependence of the specific nuclear forces, if these conclusions are based on the study of nuclear models with weak coupling between the particles.

In particular, any attempt to explain the values of spin by ascribing orbital angular momenta to individual particles in the nucleus appears to us to be completely unjustified. In fact, we must assume that every orbital angular momentum is distributed among all the particles composing the nucleus, like the angular momentum of a rotating rigid body. Denoting the moment of inertia by \(J\), we obtain the quantity

\[ \Delta_r E = \frac{h^2}{8\pi^2 J} \sim N^{-\frac{5}{3}}K \tag{7} \]

as an approximate estimate of the magnitude of the energy differences

between the very lowest quantum rotational states. For heavy nuclei formula (7) gives values small in comparison with the mean distance between γ levels; therefore it is possible that formula (7) gives an explanation of the fine structure of many energy levels observed in heavy nuclei. However, part of this fine structure, and perhaps many other characteristic features of the structure of the distribution of low levels, can probably be attributed1 to the mutual orientation of the spins belonging to the particles of the nucleus, which gives rise to an angular momentum of the nucleus equal to that obtained from it (see Addendum IV).

§ 3. Radiative properties of nuclei

The study of the so-called internal conversion of γ-rays shows that the polarity properties of the radiation emitted by excited nuclei are often substantially different from the polarity of the radiation of an excited atom, in which only one electron is in an anomalously high quantum state. In the case of the atom the most intense radiation is always of the dipole type; in the case of a radiating nucleus, however, radiations corresponding to poles of higher order turn out to be comparatively intense. True, this is precisely what might have been expected if we believed that nuclei consist entirely of constituent parts similar to α-particles, possessing one and the same mass and identical charges; for in that case the electric center would always coincide with the center of mass, which excludes the appearance of a dipole moment. In general, however, one must assume that nuclei are built from protons and neutrons. In this case one should evidently expect the appearance of dipole moments. This will be quite independent of the character of the forces acting between the particles, provided only that we suppose the coupling between the particles to be so weak that the state of the nucleus can be described by assigning to each individual particle quite definite quantum states.

If, on the contrary, the coupling between the motions of the individual particles is assumed to be so close that we are dealing with collectively quantized states of the whole nucleus, then the situation will obviously be entirely different. Indeed, if the excitation is not so great as to substantially affect the relative position of neighboring particles, then one should expect that the radiative properties of the nucleus will strongly resemble the radiation of a rotating body possessing practically uniform electrification; owing to the approximate coincidence of the centers of mass and of charge, dipole moments will under these conditions be absent, or at least strongly suppressed. Such a comparison also makes it possible to estimate quantitatively the probability of radiation processes associated with neutron capture. In the very

indeed, when nuclear matter oscillates with frequency and amplitude \(\alpha\), the quadrupole radiation emitted per unit time will be approximately

\[ R \sim (2\pi\nu)^6 \frac{E^2}{c^5}\,\alpha^2 d^4, \tag{8} \]

where \(E = ze\) is the total electric charge, and \(d = \delta \cdot N^{\frac{1}{3}}\) is the diameter of the nucleus. Further, for a low quantum state we have

\[ h\nu \sim (2\pi\nu)^2 \alpha^2 d^2 M, \tag{9} \]

where \(M = N\mu\) is the total mass of the nucleus. Eliminating \(\alpha\) from (8) and (9), we obtain, for the probability of a transition accompanied by radiation, per unit time, the expression

\[ \Gamma_r = \frac{R}{h\nu} \sim \tau^{-1} (2\pi\nu)^4 \frac{e^2}{hc}\,\frac{z^2 \delta^4}{N^{\frac{1}{3}} c^4}. \tag{10} \]

But the lifetime of the excited states of a nucleus formed in the collision of a slow neutron with a heavy nucleus corresponds to a value of \(\Gamma_r\) approximately equal to \(\tau^{-1}\cdot 10^{-7}\). This agrees with (10) if \(h\nu\) is of the order of a million volts for the most probable transition accompanied by radiation; such a value of \(h\nu\) is, in general, consistent with the experimental data.

Of course, formula (10) is valid only for the case of a transition actually accompanied by quadrupole radiation. For those excited states which correspond to radial pulsations or simple rotations, the quadrupole moment also vanishes, and transitions accompanied by radiation become still more improbable1. As for the question of radiation-accompanied transitions between any two levels of an excited nucleus, it must be noted that the various possible types of oscillations will not, generally speaking, be independent of one another. Indeed, a calculation of the amplitude of these oscillations made with the aid of formula (9) shows that even for heavy atoms these amplitudes will be small in comparison with the dimensions of the nucleus only for the very lowest quantum states. Therefore, in general, there probably exists a close coupling between elastic oscillations of different types, which may explain the often observed—

...the observed appearance of comparatively hard radiation from excited nuclei, corresponding to transitions between individual nuclear levels.⁶ In this connection one may hope that further experiments on the radiation emitted by excited nuclei, and on the disintegration of nuclei caused by γ-rays, will help to clarify the question of the mechanism of nuclear excitation (see Addendum V).

§ 4. Emission of neutrons from excited nuclei

As was already indicated in § 1, the disintegration of the compound system formed in the process of nuclear transformation should be regarded as an event depending exclusively on the state of this system, and by no means on the path by which it was formed. In fact, for such a disintegration it is necessary that, on an individual particle (which then flies off), there should be concentrated, so to speak accidentally, a considerable part of the energy that had previously existed in the form of internal motions of nuclear matter. These characteristic features of nuclear dynamics appear especially clearly in the case of such a disintegration of a compound system as a result of which neutrons are emitted. Indeed, in the case of the emission of charged particles, the electric repulsion extending beyond the range of action of the nuclear forces proper may, under certain circumstances, have a significant influence on the probability of disintegration; as we shall see below in § 6, this essentially quantum effect cannot always be separated with complete precision from the kinetic conditions for the detachment of a particle from nuclear matter. Even in the case of collisions with neutrons, considerations of classical mechanics cannot be applied to the motion of the neutron outside the nucleus; this is permissible only if the de Broglie wavelength

\[ \lambda = \frac{h}{\mu v} \tag{11} \]

is smaller than the dimensions of the nucleus, or at least comparable with them. Strictly speaking, if \(\lambda\) is not comparable with \(\delta\), then there can be no question of a definitely established interaction between a free neutron and any particle inside the nucleus. Indeed, the formation of a metastable compound system (and under these conditions such a system is obtained in almost all cases as the result of the contact of the incident neutron with the nuclear surface) is similar to the adhesion of a vapor molecule to the surface of a liquid or solid body. Conversely, the disintegration of a compound system in which a neutron is liberated presents a vivid analogy with the evaporation of liquids or solids at low temperatures.

This analogy was pointed out by Frenkel in a recently published article, in which, by comparison with the known formulae for evaporation, he derived an expression for the probability of electron emission from an excited nucleus; in our notation this formula can be...

can be written in the form

\[ \Gamma_n = N^{\frac{2}{3}} \zeta^{-1} e^{-\frac{W}{kT}}, \tag{12} \]

where \(W\) is the work required to liberate a neutron from nuclear matter, \(T\) is the effective temperature, and \(k\) is Boltzmann’s constant. Frenkel estimates the thermal energy of the nucleus by assuming that the excitation energy is distributed, according to Planck’s formula, among a multitude of vibrators, the number of which is equal to the number of degrees of freedom of the system consisting of \(N\) particles. If \(U\) is the total energy of the excited nucleus, this gives

\[ U = \sum_i \frac{h \nu_i}{e^{\frac{h \nu_i}{kT}} - 1}, \tag{13} \]

where the summation extends over all vibrators. Assuming further that the frequencies of these vibrators are all comparable with the lowest frequencies of the radiation emitted by excited nuclei, Frenkel obtains, for the composite system formed by the collision of a neutron with a heavy nucleus, values of \(kT\) equal to several hundred thousand electron-volts. Substitution into (12) then gives values for \(\Gamma_n\) considerably smaller than the probabilities of neutron emission calculated from experiments. However, since \(W\) is about \(10\ \mathrm{MeV}\), this formula is very sensitive to how we estimate the magnitude of \(T\); in fact, much better agreement with the experimental data can be obtained if one takes into account that the possible oscillations of nuclear matter have very different frequencies, ranging from the values given by formulas similar to (7) up to quantities of the order of \(\frac{k}{h}\).

In practice, all the excitation energy of the composite system is collected in a small number of oscillations of nuclear matter with the lowest frequencies, and, consequently, the temperature of the nucleus calculated from formula (13) will be several times greater than that obtained by Frenkel; this temperature proves quite sufficient to ensure approximate agreement with the observed decay probabilities in those cases where formula (12) may be expected to be sufficiently accurate. A quantitative comparison between ordinary evaporation and the emission of a neutron from the composite system is in fact limited not only by the difficulties associated with an exact calculation of the effective temperatures of this system, but also by the circumstance that the excitation of the residual nucleus left after neutron emission will usually be much smaller than the excitation of the composite system; whereas in the ordinary phenomenon of evaporation, on the contrary, during the separation of a single gas molecule the change in the thermal energy of the bodies participating in the reaction is so small that it may be neglected. Therefore, from a formula such as (12),

one may expect approximately correct results only in the case when the mean excitation of the residual nucleus, being less than the excitation of the compound system, will nevertheless be of the same order of magnitude (see Appendix VI).

In such cases the analogy between the emission of a neutron from the compound system and ordinary evaporation also gives a simple explanation for the relative probabilities of various decay processes leading to different excitation states of the residual nucleus. Indeed, formula (12) gives, above all, an estimate of the probabilities for those decay processes in which the energy of the emitted neutron is approximately the same as the energy of a gas molecule at the corresponding temperature; as for the relative probabilities of emission of neutrons with higher velocities, one should expect them to be smaller, approximately in accordance with the Maxwellian distribution of velocities of gas molecules. In fact, such a comparison gives a simple explanation of the following fact, observed in nuclear reactions leading to the separation of a neutron: the probability that this neutron will leave the nucleus taking with it all the available energy is, generally speaking, very small if this energy is large in comparison with the thermal energy (see Appendix VII).

Similar arguments are also in qualitative agreement with the observed high probability of energy transfer in collisions between nuclei and such neutrons as possess kinetic energy greater than the energy difference between the normal and the lowest excited states of the nucleus. This effect, which is in such striking contradiction with the usual ideas about nuclear collisions, is easily explained from the new point of view2. Namely, in such decays of the compound system in which the residual nucleus remains in an excited state, a smaller concentration of the energy present in nuclear matter is required for the emission of a neutron than in those decay processes in which the nucleus remains in the normal state. In very violent collisions, when the energy of the compound system is comparable with \(K\) or even greater than \(K\), we must further expect that this system will be left by several particles as a result of successive separate decay processes. If such a decay process leads to the emission of a neutron1, then its most probable energy will be of the same order of magnitude as the thermal energy of the compound system; if, however, a charged particle is liberated, then its energy will be greater owing to the additional effect of electrical repulsion beyond the surface of the nucleus, which in cases of this kind has only secondary importance for the separation process itself (see § 6).

§ 5. Collisions with Slow Neutrons

As has already been indicated, in the case of a collision between nuclei and neutrons possessing such a small kinetic energy that their de Broglie wavelength (11) is very large in comparison with the dimensions of the nucleus, we can no longer speak with any definiteness about contact between the neutron and the nucleus.

Consequently, we evidently lose any basis for applying the usual mechanical description of the processes of formation or decay of the composite system. A convincing confirmation of this may be provided by the remarkable phenomenon of the absorption of slow neutrons; for these processes effective nuclear cross sections have been found several thousand times larger than their simple cross sections. In these highly selective phenomena we are evidently dealing with a typical quantum resonance effect. Therefore, although here too one may divide the collision process into sufficiently sharply delimited stages, the probabilities of these successive stages cannot be calculated independently of one another.

In the first attempts to explain the existence of such a resonance it was assumed that the neutron moves inside the nucleus in a constant field forming the so-called potential well. Owing to the large drop of the potential, the kinetic energy of the neutron inside the well will indeed be so large that its wavelength becomes smaller than the diameter of the well, although this wavelength outside was much larger. Such a considerable change of wavelength entails almost complete reflection of the neutron wave from the inner walls of the well; for suitable values of the neutron energy, owing to this reflection, a standing wave of considerable intensity is formed. As a consequence of the existence of such metastable states of motion of the neutron inside the nucleus, for these values of the energy we shall have, first, an anomalously large scattering effect, corresponding to the secondary emission of the neutron from its metastable state, and, second, a considerable probability of neutron capture as a result of a transition, accompanied by radiation, to a lower energy level inside the potential well.

Although this picture illuminates in a very instructive way the essential features of the resonance effect, it (as soon became clear) proves insufficient to explain the details of the observed phenomena. In particular, calculation of the probabilities of the radiation effect in such simple-collision processes shows that this probability will always be greater than, or comparable with, the probability of capture, which contradicts the experimental data. Experience shows that the often observed unusually large probability of capture of slow neutrons is never accompanied by an equally large scattering effect.

To get around this difficulty, G. Breit and E. Wigner⁸ proposed a somewhat different explanation of resonance effects in colli-

collisions with slow neutrons. In their view, in a certain intermediate state the following occurs: the incident neutron enters into interaction with another nuclear particle and transfers it from its normal state to a higher quantum state; the neutron itself proves to be bound in the field of the nucleus in some stationary state with an energy too small for it to fly out immediately. Indeed, the wave of the incident neutron has only a very small ability to penetrate a potential well of nuclear dimensions; therefore, as the authors mentioned have shown, in such collisions even a comparatively small probability of energy transfer from the neutron to another intranuclear particle is sufficient to reverse the balance between the processes of scattering and capture accompanied by radiation. However, as was already indicated earlier², the observed extraordinary sharpness of resonance phenomena and their comparatively frequent occurrence require a much longer lifetime of the intermediate system and a much closer arrangement of energy levels than can be provided by any model of the nucleus with weak coupling between the individual particles. The method of considering the resonance problem proposed by Breit and Wigner, consisting in the derivation of general formulas for the variation of cross sections in the scattering and capture of neutrons in the resonance region, nevertheless represents a definite step forward, since these formulas are very valuable for the analysis of experimental data. Denoting by \(\Gamma_n\) and \(\Gamma_r\), respectively, the probability of decay of the compound system with emission of a neutron and the probability of its transition accompanied by radiation, one may write these formulas for the cross sections in the form

\[ \sigma_{sc}=\frac{\lambda^2}{4\pi}\cdot \frac{\Gamma_n^2}{\left(\frac{E-E_0}{\hbar}\right)^2+\frac14(\Gamma_n+\Gamma_r)^2}, \tag{14} \]

\[ \sigma_r=\frac{\lambda^2}{4\pi}\cdot \frac{\Gamma_n\Gamma_r}{\left(\frac{E-E_0}{\hbar}\right)^2+\frac14(\Gamma_n+\Gamma_r)^2}, \tag{15} \]

where \(\lambda\) and \(E\) are, respectively, the wavelength and the kinetic energy of the incident neutron, and \(E_0\) is the value of the energy that should be assigned to the metastable stationary state of the compound system.

The similarity between (14) and (15) and the well-known dispersion formulas of optics is a remarkable circumstance that makes it possible to draw a number of conclusions by analogy. In particular, this analogy shows how difficult it is in resonance collisions to separate the probability of formation of the compound system from the probability of the radiation and decay processes of this system. The relative number of scattered and captured neutrons (the ratio of the cross sections) is determined exclu-

...by the ratio of the probability of decay to the probability of radiation; the dependence of the absolute values of these cross sections on \(\Gamma_n\) and \(\Gamma_r\) shows in what way these probabilities affect the sharpness of the optimal resonance and, thereby, the probability of formation of the compound system.

In analyzing experimental data with the aid of formulas (14) and (15), it is especially important that measurement of the width of the resonance region

\[ \beta = h(\Gamma_n+\Gamma_r) \tag{16} \]

and of the maximum cross section for capture

\[ (\sigma_r)_{\max}=\frac{\lambda^2}{\pi}\cdot \frac{\Gamma_n\Gamma_r}{(\Gamma_n+\Gamma_r)^2} \tag{17} \]

make it possible, in principle, to determine both quantities, \(\Gamma_n\) as well as \(\Gamma_r\). A more exact analysis of the phenomena shows that, for heavier elements, \(\Gamma_r\) will be of the order of \(10^{14}\ \mathrm{sec}^{-1}\), and that the ratio of \(\Gamma_r\) to \(\Gamma_n\) for neutrons of thermal velocity is about \(10^3\). It should be expected that, as the energy changes, \(\Gamma_r\) will, over a considerable energy interval, change only slowly; as for \(\Gamma_n\), it follows from quite simple quantum considerations that in that energy region where the neutron wavelength is large in comparison with the dimensions of the nucleus, \(\Gamma_n\) must be directly proportional to the velocity of the incident neutron. Indeed, in such a case the balance between the processes will depend only on the probability of finding the neutron near the nucleus\(^{1}\). Therefore we must expect that, for neutron energies close to \(10^5\ \mathrm{V}\), \(\Gamma_n\) and \(\Gamma_r\) will be of the same order of magnitude. For still greater energies one should expect that \(\Gamma_n\) will increase still more rapidly and will soon become much larger than \(\Gamma_r\), which agrees with the experimental data on collisions with fast neutrons\(^{2}\).

In formulas (14) and (15) it is assumed that the cause of the anomalous change of the cross sections for capture and scattering

\(^{1}\) As was pointed out by Frisch and Placzek\(^{9}\) and by Wicks, Livingstone, and Bethe\(^{10}\), such simple reasoning gives a direct method for determining small neutron velocities. Indeed, for the process of nuclear disintegration caused by collisions with slow neutrons and leading to the emission of fast \(\alpha\)-rays, the cross section will, to a high degree of approximation and for a large energy region, be inversely proportional to the neutron velocity. In fact, in such a case the lifetime of the compound system will be very small, and all typical resonance phenomena will disappear; this is also evident from formula (15), if \(\beta\), as determined from (1), is very large in comparison with the energy of the incident neutrons throughout the region considered.

\(^{2}\) In a recent article Bethe and Placzek\(^{11}\) give a detailed analysis of experimental data concerning collisions with slow neutrons. The article derives formulas of a somewhat more general type than (14) and (15), in which the influence of the spin properties of the nuclei under consideration on resonance phenomena is explicitly taken into account.

is only one quasi-stationary state of the compound system. But in precisely the same way as in the case of optical dispersion, one can also take into account the combined effects of several resonant levels, provided only that the width of each level is small in comparison with the distance between neighboring levels. If, however, for the compound system, in the energy region under consideration, the levels are distributed continuously, such an analysis does not lead to a definite result. But if in this region the wavelength of the incident neutron is nevertheless large in comparison with the dimensions of the nucleus, then the cross section for scattering and capture will be expressed by the simple formula (17), provided only that by \(\Gamma_n\) and \(\Gamma_r\) one understands the slowly varying probabilities of decay and radiation of the compound system. Indeed, in contrast to the case of collision with fast neutrons, in this region the cross sections are determined by the balance between the processes of formation and decay of the compound system, which is very reminiscent of what occurs in complete resonance (Appendix VIII).

§ 6. Emission of Charged Particles by the Nucleus

As is well known from the quantum explanation of the decay of radioactive nuclei, in which \(\alpha\)-rays are emitted, a charged particle can fly out of the nucleus even if its potential energy in the region immediately adjacent to the very surface of the nucleus is greater than its kinetic energy at large distances. And indeed, we have a very instructive explanation of the characteristic dependence between the energy with which \(\alpha\)-rays fly out of radioactive nuclei and the mean lifetime of such nuclei; this explanation follows from a comparison between this kind of decay and the jumping of a particle through the constant potential barrier surrounding the nucleus. Such a barrier is formed owing to the combined action of the attractive forces between nuclear particles at small distances and their electrostatic repulsion outside the range of these forces. As is well known from Gamow’s theory, in this way, for the probability of decay per unit time, one obtains the expression

\[ \Gamma_\alpha \sim \tau^{-1}\exp\left(-\frac{4\pi}{h}\int_a^b \sqrt{2m(P(r)-E)}\,dr\right), \tag{18} \]

where \(m\) and \(E\) are the mass of the particle and the energy with which it flies out, \(P(r)\) is the potential of the particle at distance \(r\) from the center of the nucleus, \(a\) is the inner radius of this barrier, and \(b\) is the classical distance of closest approach.

Formula (18) served, in particular, as the basis for estimating the radii of radioactive nuclei from their known decay constants. However, the reliability of such calculations was called into question after the decisive influence of energy exchange between individual nuclear particles on the probability of emi-

emission of uncharged particles from the compound system formed as a result of nuclear collisions. In fact, we must bear in mind that the $\alpha$-particle cannot be regarded as having moved freely in the potential well before its emission. On the contrary, we must regard its emission from the nucleus as a process consisting of two more or less sharply separated stages. The first of them consists in the detachment of the $\alpha$-particle from the nuclear matter, and the second in its penetration, as a free particle, through the potential barrier. Comparing the first stage of this process with the emission of fast neutrons and strongly excited nuclei, Bethe, in a recently published paper,^12 came to the conclusion that the permeability of the barrier for $\alpha$-particles must be many times greater than has hitherto been assumed. In this way he obtained considerably larger values for nuclear radii than those usually adopted; such radii would require a radical change in all calculations of the influence of extranuclear electric forces on reactions of charged particles.

In evaluating this kind of argument, one should not forget that only the outer slope of the barrier is entirely determined by the electric repulsion between individual nuclear particles acting at large distances; its inner rise, however, depends chiefly on the special nuclear forces acting at small distances. Consequently, nuclear forces will not hinder the disintegration of the imaginary nucleus that would remain after complete removal of the barrier to the same extent as they hinder the emission of a neutral particle from a real nucleus. It is evident that the difference between these two processes will be the greater, the higher the crest of the potential barrier rises above the energy of the emitted particle. In the particular case of radioactive nuclei in the normal state, when the height of the barrier for $\alpha$-rays is of the same order of magnitude as $K$, the instability of the nuclear system remaining after removal of this barrier is apparently so great that the probability of nuclear disintegration is practically determined by the action of the barrier alone. Therefore, despite the inaccuracy inherent in all determinations of nuclear radii made without a more precise account of the difference between the various possible types of nuclear reactions, the radii of radioactive nuclei calculated by means of formulae of type (18) are hardly likely to change greatly if one takes into consideration the new formulation of the problem as a many-body problem (see Appendix IX).

As was already noted in § 4, in strongly excited compound nuclei formed in collisions, the direct action of the repulsive forces is often reduced simply to the subsequent acceleration of charged particles evaporating from the nuclear matter; therefore the relation between the influence of these repulsive forces and the influence of the exchange of energy between individual nuclear particles on the probability of disintegration will be exactly the reverse of that observed in the $\alpha$-decay of radioactive nuclei in their normal state. The indicated action of the repulsive forces (acceleration of the emitted—

which have flown out) is manifested especially clearly in the well-studied transformations of a nucleus caused by collisions of \(\alpha\)-rays with light nuclei, as a result of which very fast protons are emitted. It turns out that, after the emission of a proton, it is more probable that the nucleus will remain in an excited, rather than in the normal, state, provided only that the energy is sufficiently large; this resembles the circumstances accompanying the emission of a neutron. The only difference between the relative number of the various groups of protons appearing in such transformations and the same factor for the corresponding groups of neutrons consists in the fact that, owing to the repulsion, even the slowest protons possess energies considerably exceeding the temperature of the compound nucleus. As regards the calculation of absolute values of the decay probabilities by means of evaporation formulae of type (12), it should be remembered that one must not simply identify the latent heat of evaporation with the energy needed to remove a proton to infinity when the compound nucleus is in its normal state; to this energy one must add the potential of the proton at the very surface of the nucleus, on its outer side.

§ 7. Collisions between charged particles and nuclei

If the energy of charged particles colliding with a nucleus is sufficiently large (for example, comparable with the energy of fast neutrons), then in nuclear transformations caused by such collisions we may regard the formation of the compound system as a direct consequence of the contact of the incident particle with the original nucleus. In the case of charged particles the energy must, of course, be so great that, even after the particle has passed through the repulsive electrostatic field surrounding the nucleus, its wavelength remains small in comparison with the dimensions of the nucleus. For impacts of very fast \(\alpha\)-particles on lighter nuclei, the approximate fulfillment of these conditions (which are the conditions for the possibility of an elementary treatment of the mechanism of formation of the compound system) is proved by the fact that the total yield of the decay processes depends hardly at all on the velocity of the incident particles. This is seen especially clearly in those cases where, as a result of the collisions, both protons and neutrons may be emitted in comparatively large numbers; for these cases it was found that the sum of the numbers of emitted protons and neutrons remains remarkably constant over a wide range of \(\alpha\)-ray energies, even though their relative numbers change strongly within this range1. At the same time this observation shows very convincingly that the protons and neutrons emitted in such collisions have no direct individual relation to the incident \(\alpha\)-rays, but that the emission of protons and neutrons represents two competing processes of decay of the compound system1. In the case—

In the case of collisions with $\alpha$-rays possessing lower energy, we encounter a more complicated situation, partly because the energy levels of the composite system will no longer be distributed continuously, but will be separated from one another more or less sharply; and partly because the establishment of contact between the incident particle and the original nucleus is in itself a typical quantum problem. As for the latter question, it is well known to everyone that Gamow’s theory of the passage of a particle through a potential barrier makes it possible satisfactorily to explain the change in recoil with increasing energy of the $\alpha$-rays in many cases of nuclear disintegration caused by collision with $\alpha$-rays. In some cases of nuclear disintegration accompanied by the emission of very fast protons, remarkable maxima of recoil (yield) have been observed for definite $\alpha$-ray energies. It is evident, however, that these maxima cannot be explained in the usual way, which consists in the following: the incident $\alpha$-particle is ascribed a metastable state inside the barrier; from this metastable state the $\alpha$-particle may (according to this usual explanation) pass to some lower quantum state, this transition being accompanied by the transition of a proton from the normal energy level inside the nucleus to a level sufficiently high for this proton to be emitted. In such explanations of the resonance effect both the $\alpha$-particle and the proton are assumed, in the first approximation, to move in the constant field of the nucleus.

But no such explanation can in any way be reconciled with the observed high probability of proton emission as a result of collisions with faster $\alpha$-particles, which should easily penetrate into the nucleus. Indeed, already several years ago Mott^15 pointed out the following. The aforementioned experimental fact compels one to suppose that the coupling between the $\alpha$-particle and the proton must be so close that resonance cannot develop even at the lower $\alpha$-ray energies, for which penetration of the $\alpha$-particle into the nucleus would already depend essentially on the potential barrier, whereas for the proton the excess energy would still be sufficient for it to jump over the top of the barrier without hindrance.

The resonance effect under consideration should evidently be attributed to the coincidence of the sum of the energies of the free $\alpha$-particle and the original nucleus with the energy of some stationary state of the composite system, corresponding to some quantized collective type of motion of all the particles composing it. The sharpness of these states, and therefore also of the resonance effect, will depend on the lifetime of the composite system, which is determined by the sum of the probabilities of the various mutually competing—

—mechanisms of nuclear reactions, has for several years already begun to defend the point of view that nuclear transformations always begin with the formation of a composite system.

of the decay processes of the system. Except in special cases, the most probable process will be the emission of protons; as is clearly seen from the distribution, mentioned in § 6, of the velocities of the emitted protons, their emission is connected with a process similar to the evaporation of nuclear matter, and the presence outside the nucleus of repulsive forces will affect it only indirectly. This agrees with the presence of a resonance in that energy region where the proton can easily jump over the potential barrier; moreover, it explains the fact that the width of the resonance levels for not-too-fast α-rays changes only slowly with increasing α-ray energy, although the ease with which an α-particle passes through the potential barrier ought to increase very rapidly with an increase in its energy.

In a more detailed discussion of nuclear transformations caused by impacts of α-rays, it is further necessary to take into account that the wavelength of the α-particle, even in the resonance region, is usually of the same order of magnitude as the dimensions of the nucleus; therefore one must especially take into account the possibility of different values of its angular momentum, of its momentum relative to the nucleus, and their influence on the absolute values of the effective cross sections for the decay process. This influence, in particular, will show itself in estimating the relative role of the potential barrier and of intranuclear exchange of energy in creating the probability of emission of α-particles in the given energy region. In this connection it is also interesting to note that the phenomenon of the so-called anomalous scattering of α-rays in close collisions with nuclei cannot be ascribed exclusively to the deflection of the α-ray in a constant force field, as is usually done; this phenomenon may depend essentially on the possibility that the α-particle is temporarily captured and included in the compound nucleus, which then emits it as a result of an independent decay process.

In transformations of nuclei caused by artificially accelerated protons, the predominant influence on the whole phenomenon will be exerted by the repulsive forces; this will be due to the comparatively small energy of the incident particle. This is also evident from the great accuracy with which Gamow’s theory gives the relative change in the number of emitted particles as a function of the proton energy (except in cases of an especially sharp resonance). Simple calculations of the probabilities for protons to pass through the potential barrier cannot, however, explain why, when different nuclei are bombarded, such strong differences are obtained in the absolute values of the yield of transformation processes. These specific effects in fact show very convincingly how strongly the probability of formation of the compound system can (in the properly quantum region) depend on the probability of the decay processes of that same system; this latter probability, in turn, may depend to a large degree on the spin properties of the original nucleus and of the products of its decay \(^{16,1}\).

\(^{1}\) See Addendum IV.

In the particular case of the strongly selective capture of slow protons by certain light nuclei, we encounter a particularly instructive analogy with the capture of slow neutrons; this analogy concerns how the cross section for capture depends on the probability of proton emission and on the probabilities of transitions accompanied by radiation. At the same time, these two phenomena (proton capture and slow-neutron capture) differ extremely in mechanical respect. Indeed, the cross section for proton capture and the width of the resonance region can evidently be expressed by general formulas of the same type as (15) and (16). But the probability of neutron emission \(\Gamma_n\) depends only on the exchange of energies within the matter of the nucleus, whereas the corresponding probability of proton emission \(\Gamma_p\) will also depend strongly on the extranuclear repulsion. Owing to the strong excitation of the compound system, however, the situation differs essentially from that considered in § 6 for the \(\alpha\)-decay of radioactive nuclei in their normal state, and the influence on the mechanism of emission of the proton from the nucleus will here be comparable with the action of the barrier.

In transformations caused by collisions with deuterons, substantially new features appear. The recoil in these transformations is often (for fairly large energy ranges) much greater than expected on the basis of calculating the quantum probability that a material point with the same charge and the same mass as the deuteron will reach the surface of the nucleus. As Oppenheimer and Phillips\({}^{17}\) have pointed out, we must take into account here that, owing to the comparatively large dimensions and low stability of the deuteron, it may break up during the collision; as a result, the neutron is captured by the nucleus, while the proton is repelled by the external field of the nucleus. For the very smallest deuteron velocities this hypothesis does indeed seem to give a satisfactory explanation of the experimental data. For somewhat larger deuteron velocities, but when the energy is still too small for the penetration of a charged material point into the nucleus to become sufficiently probable, it is already necessary to take into account the following circumstance: if the regions in which the elementary particles constituting, respectively, the nucleus and the deuteron move overlap one another even partially, then as a result there may occur a complete fusion of the two systems into a metastable compound nucleus.

Owing to the weak binding energy of the deuteron, the excitation of the compound nucleus will now be almost twice as large as the excitation in a collision with a neutron or proton. But here, too, the excitation energy of the compound system will be so small in comparison with the total binding energy of its particles that the collision can be divided into two well-separated stages, just as this can be done in the study of other nuclear transformations. The only exception may be the case of the mutual collision of two deuterons; in this case an intermediate state of any appreciable stability cannot form because the total

energy of the system differs too little from the energy of two free protons and two free neutrons.

The high excitation of the compound system obtained in collisions with deuterons is precisely what accounts for the great variety of processes of its decay; it gives many instructive examples of competition among different possibilities, as a result of which the final product of the nuclear reaction is obtained.

Addenda

I. Under the simplifying assumption that each level represents a combination of a certain number of quantities taking almost equally spaced values, one can simply calculate the density of nuclear levels for high excitations. Let \(p(n)\) denote the number of possible ways of representing an integer as a sum of smaller positive integers. For \(p(n)\) G. Hardy and S. Ramanujan \({}^{18}\) derived an asymptotic formula, to which our attention was recently drawn. For large values of \(n\) this formula may be written approximately in the form

\[ p(n) \sim \frac{1}{4\sqrt{3}\,n}\, e^{\pi\sqrt{\frac{2}{3}n}} . \]

Let us take as the unit an energy value equal to \(2\cdot 10^{5}\ \mathrm{eV}\), approximately corresponding to the mean distance between the lowest levels of heavier nuclei. For the number of combinations by means of which one can obtain an excitation energy equal to \(8\cdot 10^{6}\ \mathrm{eV}\), we then find the value \({}^{1)}\) \(p(40) \sim 2\cdot 10^{4}\). This means that the mean distance between levels is about \(10\ \mathrm{eV}\), which roughly corresponds to the densities of the level distribution calculated from collisions with slow neutrons.

II. A more exact theoretical consideration of the characteristic features of the distribution of levels in the nucleus was given by Bethe \({}^{19}\). On the basis of general theorems of statistical mechanics, which give a connection between the entropy of a thermodynamic system and the mean energy, Bethe made an estimate of the density of nuclear energy levels that is quite possible for two different simplified models of nuclear excitation. In the first model, for the sake of simplicity, the connection between the motions of the individual particles in the nucleus is completely disregarded, and the excitation energy is compared with the energy of the so-called Fermi gas at low temperatures. In the second model the connection is assumed to be strong, but all the excitation energy is ascribed to capillary oscillations of nuclear matter (oscillations of the same type as those mentioned above in the text). Although the actual conditions in the nucleus are not reproduced correctly in either of these models, Bethe’s calculations are very interesting. They give instructive examples of the precise manner in which the typical character of the scheme of nuclear levels results from the idea that the excitation energy is distributed among the nuclear particles as it would correspond to thermal equilibrium.

Further interesting results on this problem were obtained by V. Weisskopf \({}^{20}\). Here, without any special assumptions concerning the origin of the excitation of the nucleus, the density of nuclear levels was calculated thermodynamically on the supposition that

\({}^{1)}\) The exact value of \(p(40)\) is 37 338 (Editor’s note).

the mean value of the excitation energy for a heavy nucleus is proportional to the square of its absolute temperature. This condition (which was already fulfilled in the first of the two particular cases considered by Bethe) in fact means that the fundamental motions (proper oscillations) in nuclei possess almost equally spaced energy values. It is therefore interesting to note that the formulas for the density of nuclear levels obtained from thermodynamic analogies agree in practice, at least with respect to the exponential dependence on the total excitation energy of the nucleus, with the expression for $\rho(n)$ from addition I, if by the number $n$ one understands the measure of the total energy expressed through the difference of energies between the lowest levels, taken as unity.

III. The question of the origin of nuclear excitation is connected with great difficulties, arising not only from the insufficiency of our information about specific nuclear forces, but also from the complex character of the corresponding quantum problem. Therefore the purpose of our simple remarks in the text consists, above all, in discussing certain possibilities of a simplified semi-empirical approach. In this respect the presence of quasi-elastic vibrations of the nucleus is suggested by considerations based directly on the correspondence principle; it is, however, very doubtful whether such arguments are legitimate when applied to the analogy between nuclear excitation and capillary oscillations. In fact, this analogy is connected with likening the nucleus to a nonviscous liquid, which is scarcely justified in view of the close connection between the motions of individual particles in the nucleus. Moreover, as Prof. Peierls pointed out in the recent discussion in Copenhagen, such a comparison would force us to consider also other types of nuclear motions, which, in particular, would be incompatible with the comparison mentioned in the text of rotational motion inside the nucleus with the rotational motion of a rigid body.

IV. The question of the interaction between the orbital angular momenta and the spin vectors of nuclear particles has often been discussed not only in connection with the values of nuclear spins, but also in attempts to explain the remarkable selection rules for various nuclear transformations. Usually these effects are ascribed to a weak coupling between the orbital momenta of the individual particles and their spin vectors, similar to the coupling in atoms. In a recent article by F. Kalckar, J. Oppenheimer, and R. Serber,^21 however, it is shown that these rules can apparently be explained on the basis of the assumption that the total angular momentum and the total spin of nuclear particles are coupled sufficiently weakly that one may speak of their mutual orientation^1).

V. An attempt to give such an analysis of the nuclear photoeffect, which would be in agreement with the views set forth here on the excitation of nuclei and on their radiation, was made in a recent article by F. Kalckar, J. Oppenheimer, and R. Serber.^21 In particular, it is shown there how, from the remarkable experiments of W. Bothe and W. Gentner, carried out with $\gamma$-rays of high energy,^23 one may calculate the probabilities associated with the emission of transitions from excited states of the nucleus to the normal state. For nuclei of medium atomic weight and for an excitation of 17 MeV these probabilities in some cases turn out to be of the order $\tau^{-1}10^{-9}$, i.e. about $\dfrac{1}{100}$ of the greatest radiation probability for such nuclei. This comparatively large probability of such distant transitions differs sharply from what one might at first sight expect if one were simply to compare the radiation of an excited nucleus with the radiation of a black

^1) This assumption is apparently incorrect; but it is not needed for explaining the selection rules, since these rules follow from the properties of the symmetry of the nucleus with respect to reflections (Editor’s note).

bodies at a temperature of about a million volts per degree of freedom (see § 4). Such a comparison is, however, associated with difficulties arising from the high polarity of nuclear radiation and from the close connection mentioned in the text between the various modes of excitation (intrinsic vibrations). In addition, the fact that the productivity (yield, emission) of the nuclear photoeffect varies from element to element in a rather capricious way suggests that in the transitions from these highly excited states of the nucleus to its normal state there appear certain special properties of the radiation mechanism, perhaps connected with the appearance of dipole moments.

VI. A more detailed consideration of the conditions for the applicability of the ordinary evaporation formula to problems of nuclear decay is given by V. Weisskopf in his recent article, mentioned in Appendix II. In this article, on the basis of the general methods of statistical mechanics, there is discussed both the limitedness of simple thermodynamic analogies in nuclear problems—which arises from the comparatively small number of degrees of freedom of the system under consideration—and their generalizations of the usual thermodynamic methods necessary for the correct treatment of such systems.

VII. The energy distribution of neutrons emitted from highly excited nuclei has been studied in especially great detail for the case of the ordinary neutron source—beryllium bombarded by α-rays. In this case the distribution of fast neutrons proves to be in good agreement with what is theoretically expected; as for the less fast neutrons, here there is observed a relative abundance of neutrons possessing energies much lower than the calculated temperature of the compound nucleus. However, this difficulty is only apparent. It disappears if one assumes that the slow neutrons should be attributed to a certain more complex process, as was first proposed by P. Auger²⁴. The first stage of such a process consists in the emission, by the compound system, of an α-ray, after which a beryllium nucleus remains in an excited state; the second stage consists in the subsequent decay of this nucleus into two α-particles and one slow neutron. This picture of the process has received confirmation in the latest experimental investigations of T. Bjerge²⁵.

VIII. The question of quantum resonance effects in the case of a continuous distribution of levels has recently been discussed by F. Kalckar, I. Oppenheimer, and R. Serber in the article appended in Appendix V. This article is devoted to the nuclear photoeffect, which presents a number of features making it analogous to the problem of transformations of the nucleus caused by impacts of slow particles. A more detailed quantum study of nuclear reactions will be given in the near future in an article by F. Kalckar, in which an attempt will be made to develop the general considerations concerning such an interpretation of the problems of the atomic nucleus, which is based on the principle of correspondence.

IX. The question of the proper estimate of nuclear radii by analyzing the α-decay of radioactive nuclei is considered further by Bethe in his review of nuclear dynamics¹. In this review he makes extensive use of the increased values of radii proposed by him in the article cited on p. 319. In connection with this he also expresses himself regarding the criticism of his method of calculating nuclear radii, which is given here in the text and which was reported at the Washington conference (see the preface). Meanwhile, important results on this question were obtained in the article cited in Appendix II. In this work it was possible to derive, from very general considerations, a widely applicable formula for the probability of decay of a nucleus accompanied by the emission of charged particles. This formula gives the dependence of this probability on the external repulsion, as well as on the density of distribution of the levels of the nucleus in the energy region under consideration. In the case of radioactive decay, where the levels are widely separated from one another, this formula leads to such values of nuclear radii which differ only slightly—

differ from the values derived from the usual formulas for the potential barrier, but differ substantially from the values proposed by Bethe.

References

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1. See O. Haxel[^13]. This view was recently also supported by D. Harkins[^14], who, without examining in greater detail the question of the mecha- ↩↩↩↩↩↩↩

2. Bethe. ↩↩