STRUCTURE OF THE NUCLEUS AND INTERPRETATION OF FISSION PHENOMENA

D. Hill and J. Wheeler

(Conclusion)*)

V. RATE OF ENERGY EXCHANGE BETWEEN VIBRATIONAL AND NUCLEONIC EXCITATION

1. Cross section of “sliding”

The distinction between deformations devoid of symmetry and with a regular arrangement of energy levels, on the one hand, and deformations of high symmetry with crossing energy levels, on the other hand, is not very great. The most suitable illustration of this point is the probability of a jump from the lower potential surface to the upper one when the system undergoes an axially symmetric elongation on which a small ellipsoidal deformation is superposed, transforming any circular cross section of the body into an elliptical one. The two potential surfaces under consideration intersect if the elliptical deformation is removed. The coupling that actually exists splits the levels, as shown in Fig. 33. It follows from this that if the elongation were carried out very slowly, the system would remain on the lower potential curve, as was already indicated above. However, if the time of passage through the critical region is comparable with, or less than, the minimum interval of time

\[ \left[\frac{\hbar}{E_{\text{upper}}-E_{\text{lower}}}\right]_{\min}, \]

associated quantum-mechanically with the splitting, then a jump from the lower state to the upper one can occur with appreciable probability (Fig. 34)\(^{26}\).

The possibility of nonradiative transitions is seen more clearly from Fig. 35, in which two potential surfaces are represented

* See UFN, vol. LII, no. 1, p. 20.

near the conical point of contact. In Fig. 36 one of the limiting cases of leakage from the upper “funnel” to the lower one, or the reverse process, is shown. Consideration of Fig. 35 leads to the formula

\[ \sigma = \sqrt{\frac{\hbar a}{s}}, \]

which represents the cross section for a collision with the transition cone from the lower surface to the upper one (“sliding”). Here \(a\) is the classical velocity of motion of the point representing the system, \(s\) is the angle of inclination of the cone (energy per unit deformation); the cone, for simplicity, is taken to be straight and circular. The point representing the system continues its motion in the space of deformations \((a,\gamma)\) after the transition to the upper potential surface. However, part of the energy, which previously existed as vibrational energy, now passes into the energy of excitation of the nucleons, since the system is moving on a higher potential surface. Such nonradiative slidings near the points of contact of two potential surfaces make possible a transition of nucleonic excitation into vibrations and, conversely, damping of vibrations and an increase in the energy of nucleonic excitation; in this exchange of energy proceeds comparatively smoothly, without jump-like changes of the velocity or of the potential energy of deformation.

Obviously, we may regard the collective model as self-consistent for those values of the energy and states of excitation, if such exist, for which the probability of sliding from a given potential surface during one oscillation is substantially less than unity. Even if the contrary is true and the characteristic sliding time is small compared with the period of the oscillations, the model remains self-consistent if the change of vibrational energy during one period of oscillation is small compared with the total energy. In other words, the general requirement for the consistency of the model is as follows: the coefficient of “damping” of the oscillations during one period must be small compared with unity.

“Sliding,” obviously, is an elementary act in viscous phenomena. This primary process is, of course, reversible, as in all frictional processes. Any irreversibility arises from asymmetry in time in the initial conditions. Consider, for example, the case in which the point representing the system oscillates rapidly and with large amplitude on the lower potential surface. The distribution of energy between nucleonic excitation and oscillations may be any of those possible. Then the statistical result of slidings, sometimes upward, sometimes downward, will on the average lead to a transition of vibrational energy into the energy of nuc-

of excitation. If the collective model is self-consistent, then this “damping” must be sufficiently small to allow a statistical leakage with a kind of macroscopic coefficient of friction. In contrast to most ordinary physical systems, this system has a quite limited number of degrees of freedom. Consequently, the degradation of energy cannot continue indefinitely. Statistical fluctuations in the distribution of energy over the degrees of freedom will continue until one of the forms of excitation has accumulated all, or the greater part, of the available energy. If the excitation energy of the nucleus is sufficiently large, then the accumulation of energy may occur in surface oscillations of lower order and lead to fission; or the energy may accumulate in the excitation of a single nucleon and lead to neutron emission. In both cases the process of concentration of energy will have its own history, the reverse history of the corresponding dissipative process. If the circumstances are such that dissipation is described by a kind of coefficient of friction, then the reverse concentration of energy will also be described by a coefficient of the same magnitude, but of opposite sign.

2. Estimate of the magnitude of the damping

At present it is not yet possible to say anything definite about the magnitude of the dissipation in one period and, in this way, to decide the question of the self-consistency of the collective model. However, one cannot fail to make use of a first rough estimate in order to point out several factors, among the many, that must be taken into account in an exact calculation. We shall restrict ourselves to considering collective oscillations in the plane $\alpha-\gamma$, i.e. deformations of order $n=2$ (Fig. 1). In reality, the point representing the system moves in a multidimensional space. It may turn out that the probability of its “slipping” from one potential surface to another depends on the number of dimensions taken into consideration. However, let us consider the influence of deformation coordinates of higher order $\alpha_n$. The characteristic energy quantum associated with a deformation of this type is large. Consequently, it is reasonable to suppose that higher-order deformations are always in the lower quantum states. The amplitude of zero oscillations will be small. Moreover, per unit amplitude of the coordinate $\alpha_n$, the displacement of the levels of the individual particles will also be small, since the corresponding wave functions do not “feel” very well a deformation with a small wavelength. Consequently, the influence of higher-order displacements may, in a corresponding approximation, be neglected. In this sense the theory of the sliding velocity is invariant with respect to the number of dimensions of the deformation space taken into account,

if only this number is sufficiently large. One may, however, express doubt that restricting ourselves to deformations of the 2nd order, as we do, is sufficient. In this case the deformation space is two-dimensional in the dimensionless parameter \(\alpha\); the probability of slipping in the neighborhood of a “funnel” is measured by a cross section \(\sigma\), one-dimensional in the parameter \(\alpha\), and the “damping” coefficient has the form:

\[ (\text{damping coefficient})= \frac{ (\text{probability of slipping per second})(\text{energy exchange in slipping}) }{ (\text{angular frequency})(\text{energy of oscillation}) } < \]

\[ <\sim \frac{\alpha\sigma}{\omega}\, \frac{(\text{number of funnels per oscillator})} {(\text{area of \(\alpha\)-space per oscillator})}. \]

To estimate this expression, let us assume that the potential surface under consideration has, on the whole, the same curvature as follows from the model of a simple liquid drop. Then the order of magnitude of the amplitude of the changes \(\alpha\) (Fig. 1) will be equal to

\[ \delta\alpha \sim \frac{\left(\nu+\frac{1}{2}\right)^{1/2}}{A^{7/12}}, \]

where \(\nu\) is the vibrational quantum number, \(A\) the mass number, and the angular frequency

\[ \omega \sim \frac{24}{\hbar A^{1/2}}\,\text{MeV}. \]

Let us assume that the slope of the funnel (energy per unit \(\alpha\)) is of the order of the Fermi energy

\[ F \sim 25\,\text{MeV}. \]

The number of “funnels” per oscillator is equal to \(\sim A\cdot\delta\alpha\), provided that we confine ourselves to the consideration of low-lying potential surfaces. With regard to the area of \(\alpha\)-space per oscillator, we have the least information. It is simplest to suppose that this area is, in order of magnitude, equal to \((\delta\alpha)^2\), although it is easy to imagine potential surfaces of such a form for which this quantity is too large or, conversely, too small (see Fig. 28). As a very rough estimate of the damping magnitude we obtain:

\[ <\sim \frac{\omega\cdot\delta\alpha\cdot\sigma}{\omega} \left[\frac{A\cdot\delta\alpha}{(\delta\alpha)^2}\right] = A\sigma = A\frac{\hbar\omega\alpha}{(\text{slope of the funnel})^{1/2}} \sim \]

\[ \sim A\left(\frac{\alpha}{\sqrt{A}}\right)^{1/2} \sim \left(\nu+\frac{1}{2}\right)^{1/4} A^{11/24}. \]

If we suppose that the area of \(\alpha\)-space per oscillator is, in order of magnitude, equal to \(\mathrm{const}\,\delta\alpha\) (normal quadrupole moment; the region in \(\alpha\)-space has the form of a circle), then the order of magnitude of the damping coefficient will be equal to

\[ \sim \left(\nu+\frac{1}{2}\right)^{3/4} A^{-1/8}. \]

Not knowing the numerical coefficients, we cannot say whether these quantities will be greater or less than unity.

“Damping” is one of the ways of interpreting the exchange of energies between vibrational and nucleonic excitation; another way consists in finding the proper energy values of the entire system, taking into account the coupling between these two types of excitation. The latter path is hardly likely to be fruitful in the study of nuclei. The number of particles in the system is so large, and the number of ways of distributing the energy among them so enormous, that, apparently, it is impossible to reveal all the couplings and their consequences. In Fig. 37 a one-dimensional case is presented, illustrating the influence of a small and a large sliding effect on the distribution of energy levels. In the presence of a large number of degrees of freedom, the interpretation of the case of strong sliding would be exceptionally complicated. The case of weak sliding is, obviously, much simpler. It corresponds to the concepts of the idealized collective model. The distribution of energy between nucleonic excitation and vibrations is, in a first approximation, well defined.

We shall confine ourselves to the study of the collective model, since we do not yet have another mathematically analyzable picture that would carry out the necessary combination of a model of independent particles with the liquid-drop model. If it turns out that the rate of sliding is too great for the collective model to be considered self-consistent, then, apparently, we shall have to turn to a less clear conception of the nucleus, but inevitably to one that combines the characteristics of individual particles and of the liquid drop.

3. The Franck–Condon Principle

The molecule-like distribution that we have ascribed to vibrational and nucleonic energy is based on the Franck–Condon principle and its many applications in molecular physics. It may be expected that an analogous consequence of this principle in the case of nuclei is the excitation of vibrations either by nuclear photoabsorption, or by $\mu$-meson charge-exchange reactions, or by the impact of a fast particle that directly excites only one nucleon. The apparent anomalies in the arrangement of nuclear levels are also, apparently, consequences of the Franck–Condon principle.

4. Coupling with the Surface as a Doppler Effect

The collective model regards the wall of the nucleus as a means of coupling between the energy of individual particles and the energy of vibrations. From this point of view it is interesting to consider this coupling as a function of time. The mixing of a particle’s energy levels $\delta E_n$, caused by a displacement of the wall $\delta\alpha$, has hitherto been considered without relation

with the velocity with which it occurs:

\[ \delta E_n=\frac{\partial E_n}{\partial a}\,\delta a . \]

On the other hand, one may suppose that the change of the level is caused by the Doppler effect from the moving wall. Namely, the change of energy upon one reflection from the wall is, in order of magnitude,

\[ \delta_1 E_n \sim E_n \cdot \frac{\text{(velocity of the surface)}}{\text{(velocity of the nucleon)}} \sim R_0 \dot a E_n / v_n; \]

the number of reflections per unit time is \(\sim v_n/R_0\), and the time is equal to \(a/\dot a\); then the total change is \(\delta E_n \sim E_n a\). In estimating the order of magnitude we have omitted a numerical coefficient of order unity, negative if the direction of propagation of the wave and the motion of the surface coincide, and positive in the opposite case. We are speaking here of a neutron situated inside the nucleus and reflected from within by the bounding surface; however, the same reasoning is applicable to a neutron coming from outside and undergoing a change of wavelength upon entering the nucleus (Fig. 39). Then the particle will have a smaller energy than for a stationary surface, provided only that it enters the nucleus in the region of expansion of the nuclear surface. The opposite picture occurs in the region of compression. If the energy of the incident neutron is less than the mean depth of the potential inside the nucleus, then the change of energy corresponding to the motion of the wall of the nucleus is still, in order of magnitude,

\[ \delta_1 E_n \sim R_0 \dot a E_n / v_n, \]

where \(E_n\) and \(v_n\) are the energy and velocity inside the barrier. The mechanism by which the energy of a particle incident on the nucleus is transferred to collective oscillations for subsequent redistribution among other nucleons is of interest in connection with the process of neutron capture (Fig. 40).

5. The nucleus as a quantum liquid

In our discussion we have encountered certain properties of an idealized quantum liquid. It appears to us completely transparent with respect to the internal motions of its constituent particles and is perturbed only through deformation of the surface. Its almost complete incompressibility arises not from collisions of the particles with one another, as in an ordinary liquid, but as a consequence of more subtle phenomena. The liquid can undergo collective oscillations, which, however, are organized by the wall of the nucleus, and not by the interaction of the nucleons. The oscillations experience “damping,” but the mechanism of damping is unlike that which we encounter in ordinary liquids. The liquid can evaporate a particle, but by a path quite different from evaporation in ordinary liquids. The wave function of the particle is spread

over the whole nucleus and its energy is “drawn in” by means of the Doppler effect; it is not concentrated near the region of the surface before the act of particle emission. In general, we are dealing with a very interesting new form of matter.

VI. THE FISSION PROCESS

1. Barriers and Thresholds

In the picture of fission developed earlier^4 it was assumed that the energy imparted to the nucleus by irradiation or by impact of a material particle is distributed over the entire system and later, owing to an accidental fluctuation, may become concentrated either on a neutron (the evaporation process), or in such a form of deformation as leads to fission. In order for fission to occur with appreciable probability, it is necessary that the energy of the nucleus exceed the fission threshold (Fig. 3), i.e., the energy required to obtain the critical deformation (Fig. 2). This energy was at first estimated roughly, and then calculated more accurately by Frankel and Metropolis^27 on the basis of a simple liquid-drop model. According to this model, deformations of the nucleus possessing lower energy oscillate quasiperiodically with a characteristic quantum energy (see Fig. 41), but fission can nevertheless occur with very small probability by passage through the barrier. Their conclusions concerning the height of the barrier, the frequency of oscillations, and the probability of the tunnel effect are apparently qualitatively correct, but there are characteristic deviations in the details. Peculiarities in the potential surfaces which cause quadrupole deformations of order \(\alpha \sim 0.03\) in the basic nuclear states lead to fluctuations of the fission threshold of order \(\sim 1\) MeV relative to the monotonic dependence

\[ \frac{Z^2}{A} \]

(Fig. 42). The experimental data^28,29, collected in Table I (see at the end of the article), demonstrate analogous fluctuations relative to the quantities calculated on the basis of the liquid-drop model.

2. Cross Sections

The dependence of the neutron fission cross section on energy (Fig. 43) is characterized first of all by a threshold, below which the cross section falls, with decreasing energy, mainly exponentially. Above the threshold, the cross section increases, forming a plateau. Writing the fission cross section in the form

\[ \sigma_f=\sigma_{\mathrm{geom}}\frac{\Gamma_f}{\Gamma_f+\Gamma_n}, \]

where \(\Gamma_f\) and \(\Gamma_n\) are the probabilities (per second) for the compound nucleus to undergo fission or to emit a proton, we may conclude from

of the plateau, that the ratio \(\Gamma_f\) to \(\Gamma_n\) does not change very greatly, although both constants separately, of course, increase strongly with energy. At neutron energies of the order of \(8\) MeV the fission curve shows a tendency to rise to a new plateau, associated with the possibility of undergoing fission after emission of the first neutron[^30]. The experimental data collected in Fig. 43 demonstrate, in addition to these general tendencies, interesting features of the fission cross section in the low-energy region. Since the distribution of the shapes and densities of the potential surfaces reveals a number of features, and these features apparently affect the fission cross section, it is necessary to try to identify this connection in greater detail.

The features of the potential surfaces should also influence the lifetime with respect to spontaneous fission (Figs. 44 and 45). The most important are fluctuations in the height of the fission barrier from one nucleus to another, on which the probabilities of the tunnel effect depend substantially; however, differences in the shape of the barriers of two nuclei having almost the same barrier height are also important. From this point of view one can understand how it proves possible that the lifetime of \(U^{235}\) with respect to spontaneous fission is longer than the lifetime of \(U^{238}\), although the monotonic dependence on \(Z^2/A\) following from the liquid-drop model leads to the opposite picture[^31].

3. Fission asymmetry

The release of energy does not explain fission asymmetry. A characteristic feature of the fission process is the difference in the sizes of the two fragments (Fig. 46). The probability of fission by thermal neutrons into two equal parts is 600 times smaller than the greatest probability of fission into parts related to each other in mass as \(2:3\). Nevertheless, the amount of energy released in the two cases is almost the same, as was shown by the experiments of Brunton and Hanna[^32]. Moreover, calculation shows that the critical diameter of the uranium nucleus at the moment of passing through the fission barrier is still equal to \(11 \cdot 10^{-13}\) cm, i.e., the same as for a copper nucleus. It is difficult to imagine how the behavior of a system so far from the actual act of fission can be “controlled” by the energy or by the nature of fragments that may potentially arise.

The shell structure does not explain fission asymmetry. The so-called “magic” numbers and coupling effects in shells have been considered as possible causes of fission asymmetry[^33]. Although we still know very little about shell structure, from the consideration of a spherical potential well carried out in Section II we saw that the order of levels in a deformed

in the nucleus differs greatly from the ordinary one. It is difficult to imagine in what way a nucleus in a transition state can “feel” some potential shell structures in products that have not yet been formed. No systematic differences have been found in the abundance of fission fragments having even or odd charge, nor in the energies of even-even and odd-even nuclei. True, recently certain features have been found in the mass-yield curve that are connected with shell structure (Fig. 47), but precisely this circumstance indicates the implausibility of the assumption that the two broad maxima on the mass curve are caused by the same reason. Attempts have been made to show statistically \(^{34}\) that shell structure determines the most probable fission of masses. But any calculation of fragment abundances that does not take into account the nature of the transition state seems to us untenable, however carefully the statistical weight of various final configurations may have been analyzed. A simple accounting of statistical factors, whether connected or not connected with consideration of the magnitude of the energy released, would lead, for example, to the probability of fission into 3 parts being much greater than the probability of fission into 2 parts, contrary to the experimental data (see Table III).

Penetration through the barrier does not explain the asymmetry of fission. Frenkel \(^{35}\) gave an entirely different explanation of the asymmetry of fission, based on passage through the barrier (tunnel effect), and not on passage over the barrier. Starting from the expression for the probability of passage through the barrier \(^{4}\) in spontaneous fission:

\[ e^{-\frac{2}{\hbar}\int \left\{ 2 [V(\alpha)-E]\sum_i M_i \left(\frac{dr_i}{d\alpha}\right)^2 \right\}^{1/2} d\alpha} = \]

\[ = e^{-\frac{2\sqrt{2}}{\hbar}\int \left\{(\text{potential minus total energy})\cdot(\text{effective mass})\right\}^{1/2} d(\text{distance})} \]

he ascribes the greater probability of fission into unequal parts to the smaller value of the reduced mass of the system in this case. However, this picture evidently does not apply to induced fission, in which the excitation energy exceeds the critical energy and passage over the barrier is much more probable than passage through the barrier. If, at low excitations, the tunnel effect were of serious importance, then the dependence of the cross section on energy would have a form different from the observed one. Also, the absolute value of the fission probability would be too small. Thus, for example, the difference in the reduced mass of the fragments in fission into \(140—94\) and \(117—117\) is approximately one twenty-fifth (\(56.2\) as against \(58.5\)), which leads to a difference in the exponent according to

considered hypothesis, to one fiftieth. This difference would have to be explained by a factor of order \(10^{-2}\) in the relative probability of symmetric and asymmetric fission. Then the absolute probability of passage through the barrier would have to be \((10^{-2})^{50}\), and fission would not occur, in contradiction with experiment.

Indications of the symmetry of the critical saddle-shaped configuration of the nucleus. Suggestions were made that asymmetric fission is more probable because the critical figure of the nucleus itself in unstable equilibrium is asymmetric.^36 The potential barrier through which the nucleus must pass is determined, as was shown in Fig. 4, by specifying the deformation energy through the quantities \(\alpha_2,\ \alpha_3,\ldots\), which determine the shape of the nucleus. In a nucleus possessing just enough energy to pass through the barrier, the unfolding of the motion, in accordance with classical ideas, should occur in a unique manner and lead to fragments of definite mass. If there are several minima in the potential threshold capable of leading to fission, then one may expect several possible paths for the fission process to pass. Each of these paths would have to lead to a specific and definite division of the nuclear mass. This classical picture of the development of the process after passage through the barrier is, of course, quite untenable. At best one may allow for some correlation between the masses of the fragments and one or several critical nonequilibrium shapes of the nucleus, corresponding to one or several possible transitions through the potential threshold. The passage of the fission process along different paths would have to occur with a relative probability depending on the various critical energies \(E_f\) and on the effective temperature \(T\), in accordance with the well-known Boltzmann formula.

Using these considerations, one could try to explain the small probability of symmetric fission at low excitations and its increase with increasing energy of the bombarding particles.

At present there are no theoretical prerequisites indicating the above-supposed asymmetry of the critical figure of the nucleus. Frankel and Metropolis,^27 using the electronic calculating machine ENIAC, investigated the critical figure of the nucleus and found that, before the nucleus divides into two parts, a distinct symmetric saddle-shaped figure is formed. They also found that the energy of the potential threshold under asymmetric deviations from this symmetric figure increases. No evidence has been found for the formation of other saddle-shaped figures of the nucleus, symmetric or asymmetric. Swiatecki^17 showed that the polarizability and compressibility of the nucleus only slightly change the dependence of the deformation energy in the simple liquid-drop model, and suggested that

this effect acts in the direction of splitting a symmetric saddle into two asymmetric ones. Even in the absence of this Wigner–Finberg–Swiatecki effect, which consists in a redistribution of the electric charge over the volume of the nucleus, such a splitting of the saddle should be expected at small values of the fission parameter \(x\) (proportional to \(Z^{2}/A\)), as indicated in Fig. 48. The redistribution effect shifts the point of first appearance of an asymmetric saddle toward larger \(x\). However, the magnitude of the redistribution effect is certainly insufficient for the formation of an asymmetric critical figure of such a nucleus as uranium, while for heavier nuclei the calculated critical figure approaches ever closer to a sphere (Fig. 49).

Viscosity depending on the shape promotes asymmetric fission. It may be assumed that in the collective model of the nucleus certain symmetric processes also promote asymmetric deformation of the nucleus. In a completely symmetric deformation, the individual states of the particles can be divided into two classes according to whether the wave function changes its sign under reflection (the “\(-\)” function) or does not change it (the “\(+\)” function*). For small deviations from symmetry, both series of states are filled approximately up to one and the same energy \(F\). As the deformation increases, the energy of the “\(+\)” states rises relative to the energy of the “\(-\)” states, since this class of wave functions is more subject to the action of the compression arising when a constriction is formed in the nucleus. Consequently, the energy of the deformed system can be lowered and the fission process facilitated if the possibility appears for particles to pass from the higher “\(+\)” states to the “\(-\)” states. Such sliding is impossible for perfectly symmetric deformations, but it will occur in the presence of appreciable asymmetry. Consequently, there is a quite tangible mechanism promoting the occurrence of asymmetric critical configurations.

Whatever the shape of the nucleus at the moment of passage through the barrier, there is sufficient possibility for variation of the ratio of fragment masses from one fission event to another, for the same constituents of the nucleus and the same excitation energy only slightly exceeding the barrier. There exists a quantum-mechanical scatter of possible paths from the moment of formation of the critical figure to the moment of actual fission. The simplest way to obtain a qualitative idea of this effect is to imagine that the trajectory of the point representing the system in con-

* The authors use, to denote these classes of states, the terms “gerade” and “ungerade,” which in translation are not distinguished from the terms “even” and “odd” used to denote even and odd functions. We shall use the designations “\(+\)” and “\(-\),” by analogy with the designation of the terms \(\Sigma^{+}\) and \(\Sigma^{-}\) in diatomic molecules. (Ed.)

in configuration space is fully determined classically, but the direction of this point at the instant of passage over the barrier is determined with an accuracy corresponding to the amplitude of the inevitable zero-point oscillations of the surface.

Hydrodynamic instability promotes asymmetric fission. The inevitable small asymmetry of the nucleus at the instant of passage over the barrier can lead to a sharp asymmetry in the fragments if there exists some internal hydrodynamic instability that increases the amplitude of the disturbances. Qualitative arguments in favor of the existence of such a phenomenon are readily seen from Fig. 50. They are apparently confirmed by preliminary calculations carried out up to the present time (Figs. 51 and 52). It may perhaps be said that there is nothing paradoxical in the phenomenon of asymmetric fission. On the contrary, it is necessary to decide which of two effects acting in the same direction is more important quantitatively: differences in the “+” and “−” states of the individual particles, or hydrodynamic instability.

The more the nucleus is disturbed at the instant of passage through the fission barrier as a result of excess energy, the smaller is the influence of the comparatively subtle factors that promote asymmetric fission, and the greater is the yield of fragments of equal mass. The experimental data concerning the change in the fragment-yield curve with increasing excitation of the compound nucleus are consistent with this point of view (Fig. 53). The ratio of symmetric fission to asymmetric fission changes qualitatively in accordance with the usual formula of statistical mechanics for the ratio of the rates of competing processes \(\left(e^{-(\text{difference of activation energies})/(\text{temperature})}\right)\), where in the case of a nucleus the temperature is approximately proportional to the square root of the excitation. The difference in activation energies is interpreted as an additional contribution (expressed in units of energy) of the perturbation of the nuclear surface that compresses the nucleus (which is in the critical state of unstable equilibrium) along its equatorial plane of symmetry. Any attempt to estimate the critical magnitude of this energy difference from experimental data encounters great difficulties, since at high energies we are dealing with a superposition of fissions of primary compound nuclei and fission of secondary nuclei successively formed upon emission of one or more neutrons. A preliminary estimate of the difference in activation energies, neglecting this complication in the identification of fission, gives a reasonable value of the order of several \(M_{\mathrm{eV}}^{37,38}\).

The distribution of fragment sizes in spontaneous fission is also only weakly connected with the release of energy, as in induced fission. It may be assumed that energy factors are related only to the process of passing through the critical state.

... phenomena. The sequence of events that play out is apparently the same as in induced fission. The final result thus depends on the same phenomena of mirror symmetry and effects of hydrodynamic instability. Therefore we should not expect a large difference in the distribution of fragments and the emission of neutrons in spontaneous fission and in induced fission with moderate excitation, in agreement with observations^39 (see Fig. 46).

4. Angular Distribution of Fragments

It is customary to assume that, in a simple liquid-drop model, from the moment of excitation of the nucleus by irradiation or by the impact of a particle up to the moment of concentration of energy in the form leading to fission, there occurs a complex multistage process of redistribution of energy in the nucleus. From this picture there follows the absence of a correlation between the direction of incidence of the exciting particle and the direction of flight of the fragments. Meanwhile Halpern and Winhold^40,41 found a correlation between these two directions in the photofission of thorium, in the form \(1 + b \sin^2 \theta\), where \(\theta\) is the angle between the directions; preliminary measurements of the value of \(b\) gave \(0.3 \pm 0.1\) at photon energy \(16\) MeV and approximately 4 times larger at \(8\) MeV. What follows from the collective model? In accordance with the Franck–Condon principle (Fig. 38), absorption of energy by a nucleus in its ground state transfers it to an excited potential-energy surface, while the nuclear wall retains its former shape, which, generally speaking, is no longer an equilibrium one. Consequently, rather appreciable oscillations are established about the new equilibrium surface of the nucleus. If the deviations from sphericity during this motion are sufficiently large, then the contracting forces weaken, the potential curve bends, and we shall be dealing not with a periodic phenomenon but with passage through a potential barrier leading to fission. From this point of view the correlation between the act of energy absorption and the act of fission appears more direct than in the model of a liquid impermeable drop. We are, of course, now considering an idealized version of the collective model, in which we have neglected the frictional forces responsible for sliding phenomena. It is still not clear from the theory how greatly these frictional effects complicate the picture; one must hope that this can be learned from experiments.

The qualitative aspect of the direct correlation in photofission may be represented by the following schematic picture: 1) Photons are absorbed by means of the photoelectric effect on a single proton. 2) Those protons absorb the energy most strongly whose orbits have the greatest angular momentum and lie in a plane,

perpendicular to the direction of the incident ray. 3) The most probable absorption is that in which the angular momentum increases by one, while the plane of the orbit remains unchanged. Points 1), 2), 3) are simple consequences of the theory of light absorption by an individual charged particle. 4) Large centrifugal forces, caused by the rapid motion of the excited particle, exert pressure on part of the wall of the nucleus in a direction perpendicular to the direction of motion of the photons. 5) Oscillations of the surface, excited as a consequence of the Franck–Condon principle, lead to preferential fission in the observed direction.

If this picture of the asymmetry of directions in photofission is correct, then an analogous effect should also be observed in fission by neutrons with an energy of 1 MeV and more. However, in this case the preferential direction of fragment emission is parallel to the direction of incidence of the exciting particle. The pressure exerted by the neutron on the wall of the nucleus in the process of capture (Figs. 39 and 40) leads to preferential elongation of the nucleus in the direction of the incident beam. The Franck–Condon principle is applicable to this process just as it is to photofission*).

Of interest are the questions of how the probabilities of fission or neutron emission depend on the excitation energy and angular momenta, how the fission cross section changes in passing from the region of the tunnel effect to the region of free passage through the fission barrier, and how one can explain the irregularities in the dependence of the fission cross section on energy; however, at present we cannot yet give an answer to these questions from the standpoint of the collective model.

) Addition in proof.* After this article had been written, the angular distribution of fragments in fission by neutrons was studied by Dickinson and Brolley. They measured the relative number of fission fragments in directions parallel and perpendicular to the direction of the incident neutrons (thermal and with energy 14 MeV). Their results are in agreement with the considerations given above:

| Target nucleus | \multicolumn{2}{c}{\(I(0^\circ)/I(90^\circ)\)} |
|---|---:|---:|
| Target nucleus | thermal neutrons | 14 MeV neutrons |
| \( \mathrm{U}^{233} \) | \(1.00 \pm 0.08\) | \(1.32 \pm 0.11\)* |
| \( \mathrm{U}^{235} \) | \(0.99 \pm 0.09\) | \(1.27 \pm 0.17\) |
| \( \mathrm{Th}^{232} \) | — | \(1.53 \pm 0.21\) |
| \( \mathrm{U}^{238} \) | — | \(1.53 \pm 0.17\) |
| \( \mathrm{Np}^{237} \) | — | \(1.20 \pm 0.13\) |

*) Statistical errors correspond to the limits of 95% confidence.

5. Charge distribution

Another characteristic feature of the act of fission is the variation in the number of protons belonging to a fragment with a given mass number. This question obviously cannot be considered by treating the fissioning nucleus as a liquid medium with a precisely defined charge-to-mass ratio. But it would be still more incorrect to consider the behavior of the two kinds of nucleons as the independent motion of gas molecules obeying statistical laws. On the contrary, the binding energy of the neutron and the proton is sufficiently large, and the separation of these two components at the moment of nuclear fission must be regarded as the result of quantum zero-point oscillations.

An exact analysis of charge fluctuations on the basis of these zero-point oscillations evidently presents great difficulties. It is possible, however, to obtain an approximate estimate if one considers only one form of motion, called by Teller and Goldhaber^21 the “dipole oscillation” of the nucleus, i.e. the motion of the protons of the nucleus as a whole relative to the neutrons. They expressed the idea that the maxima in the fission cross section and, presumably, in the photoneutron-process cross section, observed for U and Th at about 17 MeV, are due to this oscillation. The oscillation frequency and the elastic force associated with it are large, but not so large as to exclude some variation in the number of protons entering one of the two fragments. Replacing the actual, rather complicated shape of the fissioning nucleus by a sphere and denoting the displacement of the neutrons relative to the protons by \(x\), we obtain for the excess of protons in the left fragment

\[ \delta Z=\frac{Z}{\frac{4}{3}\pi R^3}\,\pi R^2 x, \]

and for the relative probability of a displacement \(x\) the usual quantum-mechanical expression

\[ e^{-\frac{M_N M_Z}{M_N+M_Z}\cdot\frac{\omega x^2}{2\hbar}}. \]

Thus, the charge variation corresponding to a probability equal to one half of the maximum, in this approximation, is

\[ \delta Z_{1/2}=\frac{3}{4}Z\sqrt{\frac{0.693\hbar^2}{M_{\mathrm{red}}R^2\hbar\omega}} \sim 69\left(0.693\cdot 0.010\,\frac{M\!eV}{17\,M\!eV}\right)^{1/2} \sim 1.38 . \]

Two corrections of opposite sign must be introduced into this result. First, it is necessary to take into account higher orders of oscillations, up to \(n\sim A^{1/3}=6\), which introduce under the radical a factor approximately of the following form:

\[ 1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{6}. \]

Secondly, the energy quantum \(17\ \mathrm{MeV}\) in the denominator must be substantially increased, since in the deformed nucleus a given electric moment corresponds to a greater energy than in the spherical one. An analysis of the elongated shape of the uranium nucleus near fission (Fig. 2) shows that the second correction exceeds the first and reduces the magnitude of \(\delta Z\) to unity or less. Finally, it is necessary to consider still another source of fluctuations responsible for the emission of secondary neutrons. Fission chains with mass number \(A\) begin partly with fragments of mass \(A+n\), which emit \(n\) neutrons, and partly with fragments of mass \(A+n+1\), which emit \(n+1\) neutrons. A difference in mass of \(\pm 0.5\) unit in the corresponding region of the table of nuclei corresponds to a charge change of about \(\pm 0.2\) unit.

Taking into account the quadratic law of addition of independent fluctuations, we see that the influence of variations in the emission of secondary neutrons on the fluctuations of the electric charge may be neglected. The order of magnitude of the fluctuations in the length of a given fission chain that we have obtained is apparently correct and agrees with the experimental value

\[ \delta Z_{1/2}=1.0, \]

observed by Glendenin, Coryell, and Edwards\(^{42}\). Thus it appears plausible to conclude that the fluctuations in the charge distribution are indeed directly connected with the inevitable quantum “zero-point” uncertainty in the positions of the nucleons.

6. Fission neutrons

The emission of neutrons accompanying fission requires a certain amount of energy, which is drawn from three sources: hydrodynamic disturbances at the moment of fission, excitation of the fragments after fission, and excitation of the fragments after \(\beta\)-decay. The last effect gives the well-known delayed neutrons\(^{43,44}\) (see Table II), which constitute approximately 1% of the total number of secondary neutrons in uranium fission. The remaining 99% are emitted within less than \(10^{-12}\) sec from the moment of fission and must be attributed to the first two excitation processes. It is difficult to make any simple comparison of the influence of hydrodynamic disturbances immediately before fission and immediately after fission on the excitation of a neutron up to the emission threshold. A rapid change in the shape of the bounding surface of the nucleus in the course of fission, through the Doppler effect, increases the energy of neutrons having a correspondingly oriented propagation vector. It may happen, however, that final attainment by the neutron of the emission threshold occurs only when the newly formed fragments of неправильной

forms will return back to the spherical form. From this point of view one may expect that the majority of prompt neutrons are emitted after that stage of fission which we have called separation. The limiting surfaces of both fragments are at this moment in a state of strong contraction along the fission axis, and the Doppler effect in the internal reflections of nucleons gives the maximum energy to those neutrons whose propagation vector is parallel to the fission axis. Thus, one should expect a maximum in the emission of neutrons in a direction parallel to the fission axis.

The experimentally observed distribution of fission neutrons (Fig. 54) qualitatively corresponds to the picture of isotropic emission of neutrons from moving fission fragments and to an energy distribution in the moving coordinate system that is, in general, characteristic of the evaporation picture

$$ \frac{dN}{dE}\sim Ee^{-E/T}. $$

This representation was introduced at first for reasons of simplicity, but there is no doubt that the preferential emission of neutrons parallel and antiparallel to the direction of motion of the fragments is also consistent with the observed energy distribution.

Observations of the angular distribution fall into two classes. De-Benedetti and co-workers \(^{45}\) measured the angular correlation of prompt neutrons in the fission of \(U^{235}\) by neutrons. They found that the number of coincidences is approximately constant in the range from \(30^\circ\) to \(90^\circ\) and increases approximately twofold from \(90^\circ\) to \(180^\circ\). Assuming that the principal direction of motion of the neutrons coincides with the direction of motion of the fragments that emitted the neutrons, they came to the conclusion that fission neutrons are emitted mainly by different fragments; at least twice as many neutron pairs are emitted by different fragments as by one and the same fragment.

Fraser \(^{46}\) measured the correlation of prompt neutrons with collimated fission fragments and found that, in the direction parallel to the fission axis, \(4.35\pm0.15\) times more neutrons are emitted than in the perpendicular direction; he also found that light fragments emit on the average \(\sim 30\%\) more neutrons than heavy ones. It is premature to consider that these data indicate preferential emission of neutrons in the moving coordinate system, parallel to the fission axis.

7. Fission into Three Parts

Another interesting phenomenon consists in the formation of \(\alpha\)-particles (Fig. 55) and other light nuclei in certain fission events (Table III) \(^{47-55}\). This effect can easily be explained on the basis of the liquid-drop model of fission. From classical hydrodynamics it is well known that the breakup of a liquid drop is accompanied by the formation of small droplets between the large ones.

It is therefore not surprising that in the fission of a nucleus, besides large fission fragments, smaller particles of nuclear matter are sometimes found. In this connection it is necessary to distinguish between $\alpha$-particles, protons, and neutrons. Of these, only $\alpha$-particles are almost saturated nuclear formations, and only they (from the energy point of view) can be emitted from the original nuclei even in an unexcited state. But near the surface of the original nucleus the $\alpha$-particle is situated appreciably below the Coulomb barrier, owing to its binding with the surrounding matter, whereas in the region of separation the $\alpha$-particle is near the maximum of the Coulomb potential and is surrounded by a considerably smaller amount of nuclear matter binding it.

Thanks to the change in the shape of the nucleus at the moment of fission, this $\alpha$-particle has been raised to a level only slightly below the height of the barrier. In this position it has a considerable probability of passing through the barrier. Thus, it is expedient to relate the energy of the emitted $\alpha$-particles to the magnitude of the electrostatic potential in the small interval between the fission fragments just formed. From this point of view the $\alpha$-particle should be ejected in a direction approximately perpendicular to the line of separation, with an energy of about $20$ MeV. The unequal repulsion from the lighter and the heavier fragment leads to some deviation from perpendicularity, which is in fact observed.

Similar effects should also be expected for other light nuclear fragments, with the only difference that here the potential barrier will be higher and the probability of emission smaller.

The emission of protons proves to be practically forbidden in comparison with the emission of $\alpha$-particles, since the binding of the proton with the nuclear matter even near the fission neck leads to the proton energy lying considerably below the Coulomb barrier. The observed protons apparently arise in the process of collision of the fission fragments with one another and with the surrounding matter. The energy distribution of the protons agrees with this picture and is directly opposite to what might have been expected if the protons were emitted directly by the fission system or by the fission fragments.

VII. CONCLUSION

The present incomplete outline of the collective model of the nucleus and of its connection with the fission process indicates ways of uniting the liquid-drop model with the model of independent particles into a somewhat more general conception consistent with experiment; it also emphasizes the importance of the nuclear wall in processes of energy exchange within the nucleus.

APPENDIX. FIGURES AND COMMENTS

Figure 33

Levels close to an intersection, in the absence of interaction, are determined by the equations

\[ E_a = E_0 + a(a-a_0);\qquad E_b = E_0 + b(a-a_0). \]

The interaction of the equations establishes a connection \(H_{ab}=H_{ba}\); we neglect its influence on \(a\) in the immediate vicinity of \(a_0\). The actual wave function of the stationary state of the system is equal to

\[ \psi = A\psi_a + B\psi_b. \]

The system of equations for the coefficients \(A\) and \(B\) and the energy \(E\), in the case of any fixed value of the deformation parameter \(a\), has the form:

\[ i\hbar\dot{\psi}=E\psi=H\psi, \]

\[ EA=E_aA+H_{ab}B, \]

\[ EB=H_{ba}A+E_bB. \]

Fig. 33. Behavior of two levels near the crossing point \(a=a_0\).

In the diagram two eigenvalues of the energy are represented,

\[ E_{\text{upper}}=\frac{1}{2}E_a+\frac{1}{2}E_b+ \left\{\left(\frac{1}{2}E_a-\frac{1}{2}E_b\right)^2+|H_{ab}|^2\right\}^{1/2}, \]

\[ E_{\text{lower}}=\frac{1}{2}E_a+\frac{1}{2}E_b- \left\{\left(\frac{1}{2}E_a-\frac{1}{2}E_b\right)^2+|H_{ab}|^2\right\}^{1/2}. \]

Figure 34

The probability of a jump is very small if the interaction \(H_{ab}\) between the levels is very weak or the rate of deformation \(\dot a\) is very large, or if both factors exist simultaneously. A system in the state \(\psi_a\) and with energy \(E_a\) tends to remain in the same state after passing through the crossing region during the deformation process. In other words, the probability of a jump from the lower level \(E_{\text{lower}}\) to the upper \(E_{\text{upper}}\) is large, as is always the case in nonadiabatic excitation. The situation will be different in the case of strong interaction or slow deformation, or, more generally, in the case of large values of the dimensionless “interaction parameter” \(G\), equal to

\[ G=\frac{|H_{ab}|^2}{\dfrac{\hbar\dot a\, d(E_a-E_b)}{2\,da}}. \]

The dependence of the transition probability on \(G\) can be found in the following way. Consider the equation

\[ i\hbar\dot{\psi}=H(t)\psi \]

or the system

\[ i\hbar\dot A=E_a(t)A+H_{ab}B, \]

\[ i\hbar\dot B=H_{ba}A+E_b(t)B. \]

To abstract from the unimportant general slope of the cone, let us pass to dimensionless variables

\[ A(t)=f(t)\exp\left[-i(E_a+E_b)\frac{t}{2\hbar}\right], \]

\[ B(t)=g(t)\exp\left[-i(E_a+E_b)\frac{t}{2\hbar}\right] \]

and

\[ t^2=\frac{x^2\cdot 2\hbar}{(a-b)\dot{\alpha}}, \]

where \((a-b)\) is the difference of the slopes of the two unperturbed potential curves,

\[ \frac{d(E_a-E_b)}{d\alpha}. \]

Fig. 34. Probability of transition from the state \(\psi_a\) to the state \(\psi_b\) in the case when the deformation parameter passes near the intersection point \(\alpha_0\) (Fig. 33) with constant velocity \(\dot{\alpha}\) [the graph is plotted versus \(G=\dfrac{H_{ab}}{\sqrt{\hbar\dot{\alpha}\,d(E_a-E_b)/2\,d\alpha}}\); vertical axis: jump probability].

Next, let us assume that \(H_{ab}\) is real and positive, since any imaginary part of \(H_{ab}\) can always be included in the probability amplitudes \(A\) and \(B\). The origin of time is chosen at the moment when the deformation passes through the crossing point \(\alpha=\alpha_0\). The problem is to find the probabilities \(|f|^2\) and \(|g|^2\) of being in the states \(a\) and \(b\) after a long interval of time, from the differential equations

\[ i\frac{df}{dx}=xf+Gg, \]

\[ i\frac{dg}{dx}=Gf-xg, \]

or from the equivalent second-order equation

\[ \frac{d^2 f}{dx^2}+(G^2+x^2+i)f=0 \]

with the following initial conditions:

\[ \text{for } t=-\infty \quad \text{or} \quad x=-\infty, \]

\[ f\sim \exp\left[-\frac{i x^2}{2}-\frac{iG^2}{4}\ln 2x^2\right], \]

\[ g\sim 0. \]

The solution of the equation has the form

\[ f=\frac{1}{\sqrt{2\pi}}\int_0^\infty \exp\left[\frac{i x^2}{2}-\frac{y^2}{2}-\frac{iG^2}{2}\ln y\right]\,dy \left\{ \exp\left[\frac{\pi G^2}{8}-(1-i)xy\right]+ \right. \]

\[ \left. +\exp\left[-\frac{3\pi G^2}{8}+(1-i)xy\right] \right\}. \]

For large positive times \((\chi = +\infty)\) its asymptotic value is equal to:

\[ \exp\left[-\frac{\pi G^2}{2}-\frac{i\chi^2}{2}-\frac{iG^2}{4}\ln 2\chi^2\right]. \]

The probability of a jump is, obviously,

\[ I(G)=\frac{|f_{\text{final}}|^2}{|f_{\text{initial}}|^2}=\exp(-\pi G^2). \]

This function is represented graphically in the diagram. It should be noted that the analysis set forth for the transition probability treats the motion of the nucleons quantum-mechanically, since the wave function \(\psi\), or the probability amplitudes \(A\) and \(B\) of being in the states \(\psi_a\) and \(\psi_b\), has been introduced into the consideration. The process of deformation, however, is treated as purely classical motion \(a = at\). This interpretation is legitimate, since the mass of an individual nucleon is small in comparison with the mass performing the surface oscillations.

Figure 35

The figure illustrates the concept of the effective cross section for a jump from one potential surface to another, first introduced by Teller\(^{23,60}\) in connection with the consideration of vibrations of polyatomic molecules. Energy is plotted vertically. In the two horizontal directions are plotted two coordinates (from among many) necessary for specifying the deformation. The two potential surfaces meet in a double cone, which in the present case, for simplicity, has been chosen to be straight and circular. In the figure only a small part of both potential surfaces is shown in enlarged form (for the rest, see Fig. 25). It is assumed that the kinetic energy of the collective vibrations is large in comparison with the energy considered here. This means that the deformation proceeds in a direction and with a velocity that do not depend on the small irregularity in the potential surface (appearance in the plane). When the “vibrational coordinates” pass near the apex of the cone, i.e., when the point representing the system has a small deviation parameter \(a_{\min}\), the probability of a jump of the nucleon system from the lower surface to the upper one is rather large (case \(a\)). If the “vibrational coordinates” are far from the critical values, the probability of a jump is negligibly small (case \(b\)). If the system performs complicated oscillations, then the point representing the system describes Lissajous figures in the plane of the plane (not shown in the figure).

Labels in Fig. 35: direction of motion; apex of cone; plane; appreciable probability of a jump to a higher state; negligible probability of a jump.

Fig. 35. Cross section of the transition in the case of conical contact of energy surfaces.

In this case it is better to treat the probability of a jump statistically, in terms of the “cross section” of the jump. In the case under consideration the cross section \(\sigma\) has the dimension of the deformation parameter \(a\) to the first power, although the space of motions is two-dimensional.

From dimensional considerations Teller showed that

\[ \sigma=\mathrm{const}\sqrt{\frac{\hbar a}{s}}, \]

where \(s\) is the slope of the circular cone. The magnitude of the cross section follows directly from the probability of a jump \(I(G)\) (Fig. 34)

\[ \sigma=\int_{-\infty}^{+\infty} I(G)\, da_{\perp}, \]

where \(a_{\perp}\) is the deviation parameter, or the magnitude of the closest distance from the vertex of the cone. Let the deformation \(a\) be measured from the center of the circular cone as the origin of coordinates, so that (Fig. 34)

\[ E_a-E_b=2sa_x,\qquad H_{ab}=sa_y, \]

\[ E_{\text{upper}}=-E_{\text{lower}}=s\left(a_x^2+a_y^2\right)^{1/2}=sa, \]

\[ G^2=\frac{a_y^2s}{\hbar \dot a}. \]

Integration gives, for Teller’s constant in the expression for the cross section, the value

\[ \mathrm{const}=\int_{-\infty}^{+\infty} I(G)\, dG=1. \]

In the more general case the two potential surfaces meet in an inclined, elliptic cone. Let the axes \(x\) and \(y\) be oriented along the principal axes of the ellipse; then

\[ E_{\text{upper}}=c_1a_x+c_2a_y+\left(s_x^2a_x^2+s_y^2a_y^2\right)^{1/2}, \]

\[ E_{\text{lower}}=c_1a_x+c_2a_y-\left(s_x^2a_x^2+s_y^2a_y^2\right)^{1/2}. \]

The directions of motion of the representative point of the system are now inequivalent and must be considered separately:

\[ a_x=\dot a t\cos\theta-a_{\perp}\sin\theta, \]

\[ a_y=\dot a t\sin\theta+a_{\perp}\cos\theta. \]

Then

\[ G^2=\frac{\left(s_x^2\sin^2\theta+s_y^2\cos^2\theta\right)a_{\perp}^2} {\hbar\dot a\left(s_x^2\cos^2\theta+s_y^2\sin^2\theta\right)^{1/2}}, \]

and the cross section for motion in the direction \(\theta\) is equal to

\[ \sigma_\theta=(\hbar\dot a)^{1/2} \left(s_x^2\cos^2\theta+s_y^2\sin^2\theta\right)^{1/4} \left(s_x^2\sin^2\theta+s_y^2\cos^2\theta\right)^{-1/2}, \]

while the effective cross section is found by averaging this expression over all angles.

Figure 36

We shall consider here a special case of small energy of the collective oscillations and such a form of the potential surface that the system oscillates near the opening of the cone with small quantum numbers of the oscillation energy. In this case both the nucleonic and the collective oscillations must be treated quantum-mechanically. In the first approximation the system behaves as a system having characteristic energy levels corresponding to a conical potential well. In the next approximation it is necessary to consider the probability of a jump to an unstable potential surface without a change in the total energy, but with a large increase in the kinetic energy. From quantum-mechanical considerations

![Figure 36 diagram with labels: bound state; leakage; incident wave; effective potential including the centrifugal forces; straight potential funnel; inverted potential funnel; energy; deformation.]

Fig. 36. Probability of a nonradiative jump in a system connected by a potential of conical form.

one can derive, in the case of large quantum numbers, the same formula for the cross section as was derived and discussed in connection with Fig. 35. Let us consider in greater detail the case of a right circular cone with slope

\[ \frac{dE}{da}=s. \]

Let \(\psi_a(a_x,a_y)\) represent the probability amplitude for the values of \(a_x\) and \(a_y\) for collective coordinates and the state \(a\) of the system; the analogous meaning is possessed by \(\psi_b\). Then the wave equation of the stationary state of the system will be

\[ E\psi_a=-\frac{\hbar^2}{2M_a}\left(\frac{\partial^2\psi_a}{\partial a_x^2}+\frac{\partial^2\psi_a}{\partial a_y^2}\right)+sa_x\psi_a+sa_y\psi_b, \]

\[ E\psi_b=-\frac{\hbar^2}{2M_a}\left(\frac{\partial^2\psi_b}{\partial a_x^2}+\frac{\partial^2\psi_b}{\partial a_y^2}\right)+sa_y\psi_a-sa_x\psi_b, \]

where \(M_a\) is the coefficient in the classical formula \(\frac{1}{2}M_a\dot a^2\) for the kinetic energy of the oscillator. Write:

\[ a_x=\left(\frac{\hbar^2}{2M_aE}\right)^{1/2}u\cos\theta,\qquad a_y=\left(\frac{\hbar^2}{2M_aE}\right)^{1/2}u\sin\theta,\qquad \beta=\frac{\hbar s}{\sqrt{2M_aE^3}}, \]

and, using the axial symmetry of the system for separation of variables,

\[ \psi_a=u^{-1/2}\left[F(u)\cos\frac{\theta}{2}+G(u)\sin\frac{\theta}{2}\right]e^{im\theta}, \]

\[ \psi_b=u^{-1/2}\left[F(u)\sin\frac{\theta}{2}-G(u)\cos\frac{\theta}{2}\right]e^{im\theta}; \]

we find the radial wave equations

\[ \frac{d^2F}{du^2}+\left[1-\beta u-\frac{m^2}{u^2}\right]F=-\frac{im}{u^2}G, \]

\[ \frac{d^2G}{du^2}+\left[1+\beta u-\frac{m^2}{u^2}\right]G=\frac{im}{u^2}F. \]

In the absence of the “coupling” term on the right-hand side, the first equation describes the stable motion of the system with a given angular momentum in a funnel with its flare upward. The second equation describes the “diverging” motion of the system with the same energy and angular momentum under the action of a potential having the form of a funnel with its flare downward. The coupling term makes possible transitions from stable states to unstable ones. The coupling term vanishes for \(m=0\), in contrast to what one would expect from the classical analysis of Fig. 35, where the probability of the beginning is greater the smaller the deflection parameter. However, this supposition, found from classical considerations, must be confirmed by detailed calculations of the jump probability for \(m>0\) in the presence of coupling terms in the equations. Despite the proportionality of these terms to the magnitude \(m\), the probabilities of “divergence” rapidly decrease with increasing \(m\). These probabilities can in principle be calculated by the perturbation method. We calculate \(F\) from the first equation, neglecting the term \(G\) on the right-hand side. Then we substitute the eigenfunction \(F\) into the right-hand side of the second equation and determine \(G\):

\[ G(u)=\left\{\int_0^u G_1(u)G_2(v)+\int_u^\infty G_2(u)G_1(v)\right\}\frac{im}{v^2}F(v)\,dv, \]

where \(G_1\) and \(G_2\) are two independent solutions of the homogeneous equation for \(G\); the first solution is irregular at the origin and represents an “outgoing” plane wave at infinity; the second solution is regular at the origin. The normalization is chosen so that

\[ G_2\frac{dG_1}{du}-G_1\frac{dG_2}{du}=1. \]

For large \(u\) these functions have the form:

\[ G_1\div -\left(1+\beta u-\frac{m^2}{u^2}\right)^{-1/4} \exp\left[i\left\{\frac{\pi}{4}+\int_{u_{\min}}^u \left(1+\beta u-\frac{m^2}{u^2}\right)^{1/2}du+\delta\right\}\right], \]

\[ G_2\div \left(1+\beta u-\frac{m^2}{u^2}\right)^{-1/4} \sin\left\{\frac{\pi}{4}+\int_{u_{\min}}^u \left(1+\beta u-\frac{m^2}{u^2}\right)^{1/2}du+\delta\right\}, \]

where \(\delta\) is the correction for the phase shift in the quasiclassical approximation. The probability \(A\) of transition per second from the upper potential surface to the lower one can be found by comparing the flux of asymptotically “escaping” particles at large \(u\) with the number of particles bound in the state \(F\),

\[ A= \frac{ 2\pi \left[ G(u^2)\dfrac{du}{dt}\right]_{u=\infty} }{ 2\pi \displaystyle\int_0^\infty F^2(u)\,du } = \frac{ \dfrac{2E}{\hbar} \left[ \displaystyle\int_0^\infty G_2(v)\,\frac{m}{v^2}\,F(v)\,dv \right]^2 }{ \displaystyle\int_0^\infty F^2(u)\,du }. \]

The dependence of the transition probability on the quantum number of rotational oscillations is derived from semiclassical considerations in the case of small \(m\) and large energies. The transition occurs only when (if it occurs at all) the point representing the system in \(\alpha\)-space is near the point of “closest approach” \(\alpha=\alpha_{\min}\). In this case the point representing the system describes an almost rectilinear path. Consequently, one may apply the formula derived above for the probability of a jump during one passage (Fig. 35)

\[ \exp(-\pi G^2)= \exp\left[-\frac{\pi s^2\alpha_{\min}^2}{\hbar \alpha \dot{s}}\right]. \]

The number of passages per unit time in the periodic motion of the representative point follows from the classical mechanics of a system with energy \(E\), mass \(M_\alpha\), and angular momentum \(m\hbar\), moving in the conical potential well \(V=s\alpha_r\). The period is

\[ \oint \frac{d\alpha_r}{\dot{\alpha}_r} = \oint \frac{d\alpha_r}{ (2/M_\alpha)^{1/2} \left(E-s\alpha_r-\dfrac{m^2\hbar^2}{2M_\alpha\alpha_r^2}\right)^{1/2} } \doteq (8M_\alpha E/s^2)^{1/2}. \]

Substituting into the expression for the transition probability the quantities

\[ \alpha_{\min}= \frac{m\hbar}{\sqrt{2M_\alpha E}} \qquad \text{and} \qquad \dot{\alpha}= \sqrt{\frac{2E}{M_\alpha}} \]

and dividing by the period, we obtain for the probability of a jump per second the expression

\[ A= \frac{E}{\hbar}\, \frac{\hbar s}{2^{3/2}E^{3/2}M_\alpha^{1/2}} \exp\left( -\frac{\pi m^2\hbar s}{2^{3/2}E^{3/2}M_\alpha^{1/2}} \right), \]

\[ A \cong \frac{E}{\hbar}\,\frac{\beta}{2}\, \exp\left(-\pi\beta\,\frac{m^2}{2}\right). \]

As an example, let us consider a cone with slope \(10\) Mev per unit \(\alpha\) of ellipsoidal deformation. The kinetic energy is

\[ \frac{1}{2}M_\alpha \dot{\alpha}^{\,2} = \frac{1}{2}\rho\,\frac{4\pi R_0^3}{3}\cdot \frac{3R_0^2\dot{\alpha}^{\,2}}{10}, \]

so that in the case of \(\mathrm{U}^{236}\) the effective “mass” is

\[ M_\alpha = 236M\,\frac{3r_0^2}{10}\cdot 236^{2/3}. \]

Finally, suppose that the energy of oscillations about the apex of the cone is \(E=4\) MeV. Then the dimensionless quantity \(\beta\), the reciprocal of which is related to the number of states in the conical potential well with energy less than \(E\), is equal to

\[ \beta = 10\,\text{MeV}\, \frac{ \left(\dfrac{\hbar^2}{0.6Mr_0^2}\right)^{1/2} }{ 236^{5/6}(4\,\text{MeV})^{3/2} } = 0.0781. \]

The probability of a nonradiative jump from the state associated with the conical potential surface to the lower state is

\[ A = \frac{4\,\text{MeV}} {0.66\cdot 10^{-21}\,\text{MeV}\cdot \text{sec}} \cdot 0.039\exp(-0.123) = 2.1\cdot 10^{20}\ \text{sec}^{-1} \]

and corresponds to a level width \(\hbar A=0.14\) MeV.

Figure 37

We shall describe the phenomenon of “sliding” in terms of the displacement of stationary energy levels, and not in terms of transition probabilities. The latter path is more natural in the case of an idealized collective model, in which the transition probabilities are relatively small. A description in terms of levels becomes invalid if the number of ways of distributing the energy between oscillations and excitation of nucleons is very large. In contrast to this, the figure shows an idealized case, not realized in real nuclei, but useful for illustrating the basic principles, when there are only two nucleonic states \(a\) and \(b\). The probability amplitude has two components \(\psi_a(a)\) and \(\psi_b(a)\), the squares of whose absolute values give the probabilities of finding the system in one or the other state for a given deformation of the surface \(a\). We write the Schrödinger equation in the form

\[ i\hbar\frac{\partial \psi_a}{\partial t} = -\frac{\hbar^2}{2M_a}\frac{\partial^2\psi_a}{\partial a^2} + V_a(a)\psi_a + H_{ab}\psi_b, \]

\[ i\hbar\frac{\partial \psi_b}{\partial t} = -\frac{\hbar^2}{2M_a}\frac{\partial^2\psi_b}{\partial a^2} + V_b(a)\psi_b + H_{ba}\psi_a. \]

Fig. 37. Influence of “sliding” on the distribution of levels in the one-dimensional case.

In the limiting case of very weak coupling \(H_{ab}\), the system decomposes into two characteristic equations having the energy eigenvalues shown in the upper diagram. In the opposite limiting case of very strong coupling, the description is simplified if the distance between the levels is small in comparison with the coupling constant \(H_{ab}\), in other words, if the mass \(M_a\) is very

large. It is then more convenient to consider the characteristic equations

\[ E\psi_a=V_a\psi_a+H_{ab}\psi_b, \]

\[ E\psi_b=V_b\psi_b+H_{ba}\psi_b. \]

The two energy values \(E_1(a)\) and \(E_2(a)\), found from these equations, constitute two new potential curves. The oscillations now occur with respect to these new potential curves, unless the latter can exert some influence on the particle of large mass. In the case of intermediate coupling, the order of arrangement of the levels is very complicated. However, in all three cases the total number of levels with energy less than a specified large energy \(E\) increases with increasing \(E\) in the same way.

Figure 38

In the idealized collective model of the nucleus, just as in the molecule, one may consider excitation processes (collision with a particle of large energy, photodisintegration of one particle—nucleon or electron) in the first approximation as if the remaining heavy part of the system (collective oscillator, atomic nuclei) retains its position and velocity unchanged. One may assume that the individual particle has made a jump from one potential surface to another at an unchanged value of the deformation potential. The Franck–Condon principle

Fig. 38. Franck–Condon principle in nuclear transitions.

indicates that the energy absorbed in the primary act is not connected by a simple relation either with the difference of the energies corresponding to the minima of the potential curves, or with the distance between these curves at zero deformation, or with the magnitude of that part of the absorbed energy which passes into the energy of vibrations (figure on the right). It is obvious that in the case of a nucleus, sliding from one potential surface to another will occur after manyfold oscillations, and the distribution of energy over the entire system will eventually become statistically random even before radiation carries away this energy. It is seen from the diagram that the distance between levels may depend on the ellipticity of the configuration in which it is measured. It is possible that this circumstance explains a significant part of the anomalies in the arrangement of the levels of Pb and Bi\(^{91}\).

Figure 39

The Doppler change in the wavelength of the nucleon wave function, caused by the motion of the wall of the nucleus, is an elementary mechanism for the exchange of energy between nucleon excitation and collective vibrations. However, an analysis of this process, illustrated in the figure,

is illegal. In order to observe the phase modulation of the incident wave, it is necessary to get rid of multiple reflections. Only then will the dimensions of the system and the time allow a sufficiently accurate analysis of the frequency spectrum; in order to detect the sidebands that arise when one or several vibrational quanta exchange energy with the surface, the time must exceed several periods of collective vibrations. But in reality the time of intersection of the nucleus by a nucleon is much less than the period of vibrations. Frequency analysis is opposed by a more correct method, in our case, for describing the exchange of energy between the particles and the surface, consisting in the adiabatic “following” of the wave function of the particle after the configuration of the surface. As the surface of the nucleus changes its shape, the wave function and the energy of the particle also change, and this change in energy may be attributed to the Doppler effect at the moving wall. An important point in this argument is the circumstance that the particle does not “know” that the motion of the surface is periodic. Therefore the frequency-phase treatment of the reflection process is unacceptable.

Outside         Inside
Wave function at a stationary wall

Outside         Inside
Wave function at an oscillating wall
Fourier analysis of the wave function inside gives the values of the energy indicated by the horizontal lines.

Fig. 39. Phase modulation of the wave function of a nucleon near the nuclear surface.

Figure 40

The drawn potential well is needed only in order to show the relative position of the levels of zero kinetic energy inside and outside the potential well, and is not connected with the main part of the diagram—

—we use a plot of the dependence of the potential surfaces of the intermediate nuclei on a typical deformation parameter. The neutron enters the nucleus and forms a virtual level of the intermediate nucleus near \(A\). If the vibrational coordinate reaches \(B\) before the neutron is emitted from the virtual level, then it will be captured, at least temporarily. The point representing the system may, in principle, perform oscillations

Fig. 40. Schematic representation of neutron capture in the collective model.

along the potential curve \(ABCFG\), etc. However, the excitation is so large that there can be no exactly defined distribution of energy between vibrational and nucleonic excitation. In other words, the system jumps from one potential curve to another with a frequency greater, in comparison with the frequency of those natural oscillations which would occur in the absence of crossings.

Consequently, the neutron very rapidly gives up its energy to the entire system. According to our conception, the transfer of energy takes place only through the wall of the nucleus. It may be described by means of the Doppler effect, as was shown in the preceding figure. The neutron is captured because the nodes of its wave function are oriented with respect to the direction of motion of the surface in such a way that it loses its energy upon reflection from the “receding” surface.

Figure 41

The figure shows the frequencies of oscillation of the nuclear surface \(\omega_n = 2\pi\nu_n\) as functions of the mass number, and also the corresponding magnitudes of the energies of zero oscillations. The indicated quantities are obtained from a classical analysis (Fig. 1). For small mass numbers the excitation energies increase rapidly, but the conception

Fig. 41. Dependence of the frequency of oscillations of the nuclear surface on the mass number.

of the collective model becomes, in this case, unsuitable. As indicated in Fig. 1, the larger the mass number of the nucleus, the higher the maximum order of oscillations that it is still reasonable to consider, but even for \(A=240\) it is less than 10.

Figure 42

Here the differences in the course of fission of the liquid-drop model and of the collective model are represented schematically. The same features of the potential surfaces that are responsible for quadrupole moments also lead to anomalies in the height of the fission barrier. The order of magnitude of these anomalies can be estimated by comparing the potential curve of the liquid drop \(V=V(a)\), which near the minimum varies approximately according to the law \(50a^2\) MeV and

Fig. 42. Displacement of the fission barrier associated with the quadrupole moment of the ground state.

has a maximum \(\sim 5\) MeV at \(a \sim 0.7\), with another potential curve \(V^* = V(a) - Ca\), where the constant \(C\) is a rough measure of the forces leading to the quadrupole moment. The minimum of \(V^*\) lies near \(a_0 = \dfrac{C}{100}\) MeV, and the lowering of the barrier height is approximately
\[ \delta E_f \sim C a_{\max} = 100 a_0 a_{\max}\ \text{MeV} \sim 2\ \text{MeV}, \]
for typical quadrupole moments \(a_0 \sim 0.03\).

Of course, the preceding argument is highly schematic; in reality the potential surface is not such a smooth function of the deformation.

Figure 43

The general form of all five curves corresponds to the results of the discussion carried out in Article 4. At low energies the probability of fission is negligible until the total excitation of the intermediate nucleus (the neutron binding energy plus the kinetic energy of the neutron outside the nucleus) approaches the height of the potential fission barrier (see Figs. 3 and 4). For excitations somewhat smaller than this value, the cross sections increase sharply, which is characteristic of penetration through the barrier (Fig. 44). Then the cross section grows more slowly, being connected above all with the competing process of neutron emission as an alternative form of decay of the intermediate nucleus. At excitations exceeding the barrier by several MeV (but less than the binding energy of two neutrons), one should expect a smooth dependence of the fission cross section on energy. The fission cross section again increases if the excitation is so great that fission can occur after the intermediate nucleus has emitted one or more neutrons.\(^{30}\)

An illustrative example of the possible influence of the internal state of a nucleus on the probability of fission was described by Stritom, Gioco, and Thompson[^92]. They observed the formation of two isomers of \( \mathrm{Am}^{242} \), when a neutron is captured by \( \mathrm{Am}^{241} \), one of which transforms into the other with a half-life of about 16 hours. The fission cross section of \( \mathrm{Am}^{242} \) in the ground state by thermal neutrons turned out to be

Fig. 43. Dependence of the neutron fission cross section on energy for five fissioning elements[^88].

equal to \(6000 \cdot 10^{-24}\ \mathrm{cm}^2\), and in the excited state—\(2000 \cdot 10^{-24}\ \mathrm{cm}^2\). Whether this difference in fission cross sections is the result of a different probability of neutron capture and emission, or of the removal of excitation energy by radiation, or of the effect of the different internal states of the isomers (Figs. 6 and 42), is still unclear.

In studying the photofission of thorium and uranium, Baldwin and Klaiber[^93] found that the cross section has a maximum at \(17\ \mathrm{MeV}\) with a half-width of \(3\ \mathrm{MeV}\). As Steinwedel and Jensen[^94] pointed out, there is a large amount of analogous data.

Figure 44

The probability of overcoming the fission barrier as a function of energy is given by the expression

\[ P = \frac{1}{1 + e^{2\pi b}} . \]

Here \(b\) is the energy deficit relative to the height of the barrier, divided by

characteristic quantum of energy \(E_{\text{curv}}\), depending on the curvature of the top of the barrier and on the effective mass associated with the deformation coordinate. In order to clarify the meaning of \(E_{\text{curv}}\), let us imagine that the sign of the potential energy is changed, so that the barrier is transformed into a trough. Then the system will behave as a harmonic oscillator oscillating about the critical point with circular frequency \(\omega_{\text{imag}}\) and characteristic quantum of energy \(\hbar \omega_{\text{imag}}\). The latter quantity is, by definition, equal to \(E_{\text{curv}}\). In

Energy relative to the top of the barrier at \(E_{\text{curv}}=0.8\) MeV

Vertical axis: Probability of overcoming the barrier
Labels in the plot: Above the barrier; Below the barrier
Horizontal axis: \(b\)—energy deficit/\(E_{\text{curv}}\); \(b\)—energy excess/\(E_{\text{curv}}\)

Fig. 44. Probability of overcoming the fission barrier as a function of energy.

the first approximation it is reasonable to take \(\hbar \omega_{\text{imag}}\) equal to the characteristic quantum of energy \(\hbar \omega_2\) of the surface oscillations of the system of lowest order relative to its normal (nearly spherical) equilibrium figure. This follows also from such considerations: the potential energy, represented in the form of a power function of the deformation, has the leading terms of the expansion

\[ V(a)=Aa^2-Ba^3. \]

But at its maximum this function has the same second derivative (apart from sign) as at its minimum, and consequently the same frequency \(\omega_{\text{imag}}=\omega_2\). For uranium, \(\hbar\omega_2\) is of the order of \(0.8\) MeV and the characteristic quantum \(E_{\text{curv}}\) is also equal to \(\sim 1\) MeV. Thus we come to the scale of energies indicated in the figure. Note that the probability of overcoming the barrier is equal to only \(0.5\) if the excitation energy reaches the critical value, and approaches unity only when the barrier is considerably exceeded. This result contradicts the conclusions of the usual formula for passage through a barrier, which gives:

\[ P=\exp\left[-\frac{2}{\hbar}\int 2M_a\left(E_{\text{upper}}-\frac{1}{2}V''a^2-E_{\text{exc}}\right)^{1/2}\,da\right] \]

\[ =\exp\left[-2\pi\left(E_{\text{upper}}-E_{\text{exc}}\right)\frac{1}{\hbar\omega_{\text{imag}}}\right] =\exp[-2\pi b]. \]

A more complete formula for passage through a barrier can be derived if one writes the wave equation in dimensionless variables:

\[ \frac{d^2\psi}{dx^2} + (x^2 - 2b)\psi = 0. \]

The required solution is, on the right-hand side of the barrier, a wave traveling to the right, and on the left-hand side, the incident and reflected waves. In terms of parabolic-cylinder functions it has the form:

\[ \psi = D_{-1/2-ib}[(1-i)x] = \left[\Gamma\left(\frac{1}{2}+ib\right)^{-1}\right]\times \]

\[ \times \int_0^\infty \exp\left[ \frac{i x^2}{2}-(1-i)xt-\frac{t^2}{2} -\left(\frac{1}{2}-ib\right)\ln t \right]dt \]

with the asymptotic value for large positive \(x\):

\[ \approx 2^{-1/4}x^{-1/2} \exp\left[ \frac{i x^2}{2} -\frac{ib}{2}\ln 2x^2 +\frac{i\pi}{8} -\frac{\pi b}{4} \right], \]

and for large negative \(x\):

\[ \approx 2^{-1/4}|x|^{-1/2} \exp\left[ \frac{i x^2}{2} -\frac{ib}{2}\ln 2x^2 -\frac{3i\pi}{8} +\frac{3\pi b}{4} \right] \]

(reflected wave),

\[ +(2\pi)^{1/2}\Gamma\left(\frac{1}{2}+ib\right)^{-1} 2^{-1/4}|x|^{-1/2}\times \]

\[ \times \exp\left[ -\frac{i x^2}{2} +\frac{ib}{2}\ln 2x^2 +\frac{i\pi}{8} +\frac{\pi b}{4} \right] \]

(incident wave).

Comparison of the intensities of the incident and transmitted waves gives the above formula for transmission. It retains its meaning only so long as the portion of the potential curve, represented as a function of the dynamical deformation coordinate, is close to the inverted curve of a harmonic oscillator.

Figure 45

The graph of the dependence of the half-period of spontaneous fission on \(Z^2/A\) demonstrates the close connection that exists between the rate of spontaneous fission and the fissility parameter of the liquid drop \({}^{95}\) (see Fig. 2 and Table 1). The indicated linear dependence extends not only within the range of the designated nuclei, but even to the nucleus “cosmium” (Figs. 2, 51), for which \(Z^2/A \approx 47\) and the spontaneous-fission time is approximately equal to the period of oscillations of the lowest

of the order for such a nucleus as, for example, \(U^{238}\) (Fig. 1). Small deviations from linearity, in accordance with the considerations developed above (Figs. 6, 26, 42), indicate variations in the quadrupole moments of the ground states and their influence on the height of the fission barrier (Fig. 3). The analogy with the \(\alpha\)-decay (Fig. 30) is obvious. Not included in the graph is \(U^{235}\), for which the spontaneous-fission rate is less than that of \(U^{238}\), although \(Z^2/A\) is greater.\(^{31}\)

Fig. 45. Dependence of the periods of spontaneous fission on \(\dfrac{Z^2}{A}\).

Figure 46

The figure shows the curve of the mass distribution of fragments in the fission of \(U^{235}\) by thermal neutrons\(^{96}\) and by 14-MeV neutrons\(^{97}\) (solid circles). The clearly visible tendency toward symmetric fission with increasing excitation is also confirmed by the mass-distribution curve for fragments of heavy elements under bombardment by particles of very high energy.\(^{98,99}\) A simple hydrodynamical explanation of the asymmetry of the curve and of its changes with excitation energy is given in Figs. 50 and 51. Of interest is the observation\(^{39}\) that, in the case of \(Cm^{242}\) with half-

Fig. 46. Mass distribution of fragments of \(U^{235}\) in fission by slow neutrons and by 14-MeV neutrons.

...with a period of spontaneous fission of \(7.2 \cdot 10^6\) years, fission events so frequent that they make it possible to study the distribution of the kinetic energy of the fragments. The resulting picture also indicates the distribution of the masses of the fragments, as in induced fission by thermal neutrons.

Figure 47

A curve of the distribution of fragment masses, drawn in such a way that the light and heavy fractions are combined into a single curve \(^{100}\), demonstrates a sharp deviation, in a narrow mass interval, from the smooth curve known from still earlier works \(^{96}\). The mass values corresponding to the anomalous portion of the curve agree with the hypothesis of the influence of nuclear shells on fission. This effect is superimposed on the dominant process of asymmetric fission.

Fig. 47. Fine structure of the curve of the yield of fragment masses.

Figure 48

Five independent, distinct forms of oscillations of a sphere of order \(n = 2\) can be expressed through radial elongations proportional to the five harmonics \(P_2(\cos \theta)\), \(P_2^{(1)}(\cos \theta)[\cos \varphi\ \text{or}\ \sin \varphi]\), \(P_2^{(2)}(\cos \theta)[\cos 2\varphi\ \text{or}\ \sin 2\varphi]\). But these same oscillations can also be expressed in terms of the rotation of a “hump” on the surface of the sphere (three degrees of freedom) and two types of pure oscillations. In this it is assumed that the system as a whole has no angular momentum; otherwise the presence of rotation would lead to a certain number of complex and interesting branchings of the curves shown in the figure. Both types of pure oscillations are clearly manifested in the critical figure of unstable equilibrium. The first type of oscillation consists in deviations from the saddle-shaped figure in the direction of decreasing potential energy, i.e., in stretching leading to fission, or in compression in the direction of the normal deforma-

...of the figure. The second type consists in ellipsoidal transverse oscillations of neighboring figures (perpendicular to the axis of symmetry) without a change in the length of the dumbbell-shaped figure of the nucleus.

Let us consider one more type of oscillation of the figure of unstable equilibrium of third order, namely: the axially symmetric oscillation of lowest frequency, which is described by the harmonic \(P_3(\cos\theta)\) the better the closer \(x\) is to unity. This oscillation represents a displacement of the liquid through the neck of the “dumbbell” back and forth, i.e., an oscillation of asymmetry. Its frequency decreases as \(x\) decreases and becomes zero at some \(x = x_{\text{crit}}\), which is unknown to us but apparently close to \(x = 0.65\). At smaller values of \(x\) the symmetric saddle-shaped configuration of the nucleus still exists, but two new asymmetric saddle-shaped figures also arise, corresponding to a smaller value of the energy. Since \(x\) for uranium is considerably larger than \(x_{\text{crit}}\), the asymmetric saddle-shaped figure of the nucleus in this case does not arise and therefore has no relation to the asymmetry of uranium fission. Svyatetskii \(^{101}\), in a short note, indicated that the compressibility and polarizability of nuclear matter promote the formation of an asymmetric saddle point, i.e., suppress the lower component of the oscillations \(\omega_3\). It is not clear, however, whether this effect is large enough to lead to asymmetric saddle points in the case of uranium.

Fig. 48. Characteristic angular frequencies \(\omega\) of oscillations of the lower orders of an incompressible, uniformly charged liquid drop with mass \(AM\) and surface tension \(O\), as a function of the fissility parameter \(x\). (In the case of the spherical figure—solid lines; in the case of the symmetric critical figure of unstable equilibrium—dashed lines.)

Labels in the figure: \(\omega^2/(32\pi O/3MA)\); \(\omega_4\); \(\omega_3\); \(\omega_2\); “ellipsoidal transverse displacement”; “asymmetric figure”; “symmetric figure”; “region of instability”; \(x\).

Figure 49

In this case a small deformation brings the system to a saddle-shaped figure preceding the overcoming of the fission barrier. In the diagram only ellipsoidal deformations are considered, since deformations described by spherical harmonics of higher order do not lead to instability. Ellipsoidal deformations are described in polar coordinates, defined in Fig. 13, by the “deformation amplitude” \(\alpha\) (radius) and the “shape parameter” \(\gamma\) (angle). In the diagram there are three points corresponding to the saddle-shaped figure, since elongation along any one of the three principal axes of the ellipsoid leads to fission. The basis for this diagram was the idealized liquid-drop model.

Here neither the internal quadrupole moment nor the angular momentum of the nucleus is taken into account. The deformation energy in this first approximation is equal to:

\[ V(a,\gamma)=4\pi R_0^2 O\left\{\left[2(1-x-z)\frac{1}{5}\right]a^2-\left[(12x+20z)\frac{1}{105}\right]a^3\cos 3\gamma\right\}= \]

\[ =B\left[\frac{u^2}{2}-\frac{u^3}{18}\cos 3\gamma\right]=Bf(u,\gamma). \tag{*} \]

Here \(4\pi R_0^2 O=14A^{2/3}\) MeV is the energy of ordinary surface tension, and \(x\) and \(z\) are parameters defined earlier; their sum \(x^*\) is the critical parameter under consideration. In the alternative form of writing the deformation energy (the second line of equation (*)), \(B\) is a constant expressed in such units of energy that the height of the barrier is equal to 6.

Fig. 49. Contour diagram of the fission barrier in the case when the critical fission parameter \(x^*\) is close to unity.

The contour diagram gives the dimensionless quantity

\[ f=\frac{V}{B} \]

as a function of \(\gamma\) and of the quantity \(u\), a multiple of \(a\),

\[ u=\frac{18x+30z}{7(1-x-z)}\,a. \]

Figure 50

The critical figure of unstable equilibrium of the uranium nucleus is close to an elongated cylinder (Fig. 2). In view of this, let us consider the stability of a long jet of incompressible, uniformly charged liquid possessing surface tension. If the electric charge density were equal to zero, then, as is known, the jet would be unstable with respect to ...

small perturbations with a wavelength exceeding the circumference of the jet. Such perturbations are the first step in the redistribution of the liquid into individual spherical drops, which together have the same mass but a smaller surface area. The dimensions of such spheres are the greater, the greater the wavelength. The presence of even a small volume electric charge leads to an energetic disadvantage in the separation of the liquid into very large spheres. The charge would, as it were, stabilize the jet with respect to long-wave excitations, whereas surface tension prevents short-wave deformations. The intermediate zone of wave numbers \(k\), corresponding to unstable configurations, becomes the narrower the greater the charge density. Complete stability occurs if

\[ y=\frac{2(\text{charge per unit length})^2}{\pi(\text{radius})(\text{surface tension})}>y_{\text{critical}}\approx 1.125. \]

This condition is satisfied by \(U^{235}\) for the critical Franckel–Metropolis figure, corresponding to the value of the fission parameter

\[ x=\frac{4\pi R_0^3 \rho_e^2}{30\cdot O}=0.74. \]

The radius \(R\) of the cylindrical part of the figure is equal to \(0.64R_0\), and

\[ y=\frac{2\pi R^3\rho_e^2}{O}= \]

\[ =2\pi\cdot0.74\,\frac{30}{4\pi}\left(\frac{R}{R_0}\right)^3=2.92. \]

How can the forces of repulsion lead to such stability? Only because the ends of the cylinder prove to be removed to infinity.

Consider now a cylinder of finite length. Capillary forces round off its ends, but they cannot resist the repulsive forces of the electric charge concentrated in the given cylinder. As a result, a large drop of liquid will separate from the end of the cylinder, and the act of fission will be highly asymmetric. What is the measure of the asymmetry? Let \(L\) be the distance from the place where the constriction of the cylinder begins to form and let \(\varepsilon\) be the decrease of the radius at this place. Through the neck into the incipient drop there is forced a volume of liquid of order \(\varepsilon 2\pi LR\), which separates the drop and decreases the electrical energy of the system by the amount

Fig. 50. Qualitative picture of the classical hydrodynamic interpretation of the asymmetry of nuclear fission.

\[ \sim 2\pi^2\varepsilon L^2R^2\rho_e^2, \]

which is transformed into kinetic energy

\[ \sim \rho_m\frac{\varepsilon^2L^2}{R^2}\pi R^2L. \]

Consequently, the relative magnitude of the constriction \(\dfrac{\varepsilon}{R}\) initially grows as

\[ \left(\frac{t}{t_{\text{critical}}}\right)^2, \quad \text{where } t_{\text{critical}}\sim\left(\frac{L\rho_m}{R\rho_e^2}\right)^{1/2}. \]

The most rapid formation of a drop occurs when the neck length \(L\) is of the same order as the radius of the jet, since the character of the expression for \(t_{\text{critical}}\)

changes at smaller \(L\). We see that successively separating fragments in such “jet-like” fission will be of approximately the same size.

In the lower part of the figure the pattern of fission of a nucleus of the type \( \mathrm{U}^{236} \) is presented for the case in which the excitation only slightly exceeds the fission threshold. On reaching the threshold, the system has such a small kinetic energy of motion that a considerable interval of time passes before the shape of the nucleus is far from the elongated figure of unstable equilibrium. This figure resembles a cylinder of electrically charged liquid. In a rough approximation one may suppose that the behavior of the cylinder at one end is as if the other end extended without limit. At each end one may expect the onset of fission of the “jet-like” type. At which end the constriction of the cylinder will begin is a matter of chance. But once it has begun at one end, this type of deformation will grow so rapidly that it will soon outstrip all processes that can occur at the other end. As a result there will appear a fragment whose length is comparable with its diameter (a light fragment).

The probability of fission from both ends is in fact small. There is a strong interaction between the ends. The beginning of a constriction near one end increases the curvature and, consequently, the surface tension at the other end. This effect prevents the formation of constrictions at the other end, which otherwise would arise there somewhat later. Thus fission into a large and a small fragment occurs. The proposed mechanism of asymmetric fission is the result of the direct application of elementary concepts of the surface tension of electrostatics and hydrodynamics.

Figure 51

The calculations were carried out on electronic computing machines. The calculation began with a configuration close to the position of unstable equilibrium. The subsequent motion was calculated on the basis of the classical hydrodynamic equations for an incompressible uniformly charged liquid possessing surface tension. The preceding figure suggests the working hypothesis that the inevitably arising small asymmetry in the figure of a nucleus in a transitional state is amplified in the course of the subsequent motion and, in the overwhelming majority of cases, leads to the formation of a constriction of the elongated figure near one end or the other. It is assumed that the hydrodynamic phenomena proceed classically. However, the magnitude and origin of the small initial asymmetry are associated with quantum zero-point oscillations of the surface of various orders.

As the excitation of the intermediate nucleus increases, the irregularities of the nuclear surface at the moment of passage through the fission barrier increase, being superposed on the effect of zero-point oscillations. These irregularities substantially influence the position of the point at which the constriction begins. A displacement of this point will lead to the predominance of symmetric fission. Thus, with increasing excitation energy the probability of symmetric fission increases.

Two nuclei were considered: 1) “cosmium”—a nonexistent nucleus for which the critical fission parameter

\[ x=\frac{(Z^2/A)}{(Z^2/A)_{\text{critical}}}=1. \]

2) A drop with \(x=0.74\), closely imitating the nucleus \( \mathrm{U}^{235} \). In both cases only axially symmetric figures are considered. The first calculation was carried out on the assumption that the deformation of “cosmium” begins with a small deformation of second order, remaining symmetric as it grows (\(t=20\)

Cosmium

Spherical figure of unstable equilibrium

\(t=0\)
\(t=20\)
\(t=30\)

\(e\)
\(z\)

Symmetric motion

a)

Cosmium

\(t=0\)
\(t=20\)
\(t=40\)
\(t=60\)
\(t=80\)
\(t=90\)

Spherical figure of unstable equilibrium

\(e\)
\(z\)

Initial energy: 2.5 MeV of lower-order symmetric deformation
1.3 MeV of lower-order asymmetric deformation

b)

Uranium \(^{235}\)

\(t=0\)
\(t=30\)
\(t=50\)

\(e\)
\(z\)

Figure at the fission threshold

Initial energy: 2.5 MeV of lower-order symmetric deformation
1.3 MeV of lower-order asymmetric deformation

c)

Fig. 51. Results of a dynamic analysis of nuclear fission, carried out on the basis of the liquid-drop model \(^{102}\).

means a time of \(20\cdot 0.66\cdot 10^{-23}\) sec). In the second calculation it was assumed that a third-order deformation with an amplitude corresponding to the energy of zero-point vibrations was superposed on the symmetric deformation of the “cosmium.” In this case the small initial asymmetry increases with time and leads to fission into fragments of different masses. The growth of the asymmetry follows from the dynamics of the fission process. Here there can be no question of asymmetry of the initial saddle-shaped configuration of the nucleus, since it is the sphere itself. In the third case (\(x=0.74\)) a small initial asymmetry is superposed on the initial symmetric saddle-shaped figure of Frankel–Metropolis. The asymmetry increases and leads to predominant fission into two unequal parts. The calculations were stopped in all cases considerably earlier than the formation of a neck, since the mesh with which the calculation was performed was not sufficiently fine for calculating the narrow part of the neck. Although these results cannot be regarded as a proof, nevertheless they at least agree with the view of fission asymmetry as a classical hydrodynamic effect.

Figure 52

Irrotational motion of an incompressible fluid is expressed through the velocity potential \(\mathbf{u}=-\operatorname{grad}\varphi\), where \(\nabla^2\varphi=0\) \(\left(u_n=-\dfrac{\partial\varphi}{\partial n}\right.\) is the normal component of the velocity at the surface of the liquid\(\left.\right)\); the dependence of the velocity potential on time is given by the equation

\[ \frac{\partial\varphi}{\partial t} = -\frac{1}{2}(\operatorname{grad}\varphi)^2+V+P=H. \]

The quantity \(H-\dfrac{1}{2}(\operatorname{grad}\varphi)^2\) is the acceleration potential; \(P\) is the pressure divided by the density; on the surface \(P\) assumes a value depending only on the local curvature \(\chi\)

\[ P_{\mathrm{surf}} = \frac{O\chi}{\rho_m} = \frac{O}{\rho_m} \left\{ \rho^{-1} \left[ 1+\left(\frac{d\rho}{dz}\right)^2 \right]^{-\frac{1}{2}} - \frac{d^2\rho}{dz^2} \left[ 1+\left(\frac{d\rho}{dz}\right)^2 \right]^{-\frac{3}{2}} \right\}, \]

where \(O\) is the specific surface energy and \(\rho_m\) is the density; \(V\) is the electrical potential energy per unit mass, expressed in cylindrical coordinates \(z\) and \(\rho\) as follows:

\[ V_1 = \frac{\rho_e^2}{\rho_m} \int \frac{d\tau_2}{r_{12}} = 2\frac{\rho_e^2}{\rho_m} \int \rho_2 \frac{ \left[ \rho_2+\rho_1+(z_1-z_2)\frac{d\rho_2}{dz} \right]K-2\rho_1D }{ \left[(\rho_1+\rho_2)^2+(z_1-z_2)^2\right]^{1/2} } \,dz_2, \]

where \(K(k)\) and

\[ D(k)=\frac{K(k)-E(k)}{k^2} \]

are complete elliptic integrals of argument \(k\)

\[ k^2=\frac{4\rho_1\rho_2}{(\rho_1+\rho_2)^2+(z_1-z_2)^2}, \]

and \(\rho_e\) is the density of electric charge.

The real system, with its infinite number of degrees of freedom, is replaced by a system with a limited number of degrees of freedom, namely: the surface

defined by the positions of eleven marked points. The two points located at the poles of the figure are denoted by the coordinates \(z_0\) and \(z_{10}\). The coordinates of the other nine points \(z_i\) are distributed uniformly between \(z_0\) and \(z_{10}\). Eleven independent spatial coordinates \(z_0, \rho_1, \rho_2, \ldots, \rho_9, z_{10}\) and eleven corresponding velocities describe the state of the system at a specified instant of time. The new positions (at the next instant of time)

Fig. 52. Scheme for carrying out the hydrodynamic computations of Fig. 51.

Flowchart labels:

Left branch:

• Initial state; determined by the coordinates and velocities of 11 surface points \(z_j, \rho_j\)
• Computation of the curvature of the surface and of the liquid pressure on it \(P\)
• Computation of the electric potential energy \(V\), in the form of a flat integral at each point of the surface
• Summation of \(V\) and \(P\), multiplication by weights \(B_j\), and integration over the surface \(j=1,2\ldots 8\)
• Computation of the 64 elements of the matrix \(M_{nj}=\int B_j\cdot B_n\,dS\)

Right branch:

• Solution of the equation \(\sum M_{jn} b_n=f_j\) for the eight \(b_n\) and finding the potential acceleration \(H=\sum b_n B_n\)
• Computation of the normal derivative \(H\) on the surface; computation of the acceleration of the surface
• Obtaining new velocities from the old accelerations and velocities
• Obtaining new coordinates from the old coordinates and velocities
• New state

are computed kinematically from the old positions and velocities, and the new velocities from the old accelerations obtained from the acceleration potential:

\[ \frac{d\mathbf{u}}{dt}=-\operatorname{grad}\left(H-\frac{u^2}{2}\right). \]

The Laplacian \(H\) is equal to zero inside the surface, and on the surface it assumes a value known from the old position of the system. In the computations we neglect the term \(\dfrac{u^2}{2}\) both in the expression for the boundary conditions for \(H\) and when subtracting it from \(H\), since the kinetic energy, 170 MeV, is small in comparison with the probable value of the surface energy, 540 MeV, and the electrostatic energy, 780 MeV. In the computations presented—

represent \(H\) in the form of the sum

\[ H(z,\rho,t)=\sum_{n=1}^{8} b_n(t) B_n(z,\rho) \]

of eight spherical harmonics, where \(b_n\) are chosen in the time interval under consideration so as to reduce to a minimum the deviation of \(H\) from its boundary value

\[ \int (H-V-P)^2_{\text{surface}}\, ds \to \text{minimum}. \]

This requirement leads to a system of eight linear equations for the coefficients \(b_n(t)\)

\[ \sum M_{jn} b_n = f_j, \]

where

\[ M_{jn}=\int B_j B_n\, dS \quad \text{and} \quad f_j=\int B_j(V+P)\, dS \]

are found from the preceding state of the system. Thus, the cycle of calculations is completed and a new state of the system is found.

Figure 53

The ratio of the yield of Ag\(^{111}\) fragments to the yield of Ba\(^{140}\) fragments is a measure of the asymmetry of the yield curve for fragments of different masses, since it has been established experimentally (Fig. 46) that the complete curve of fragment yield in general retains its form with increasing excitation energy. The drawing presents the dependence of the measure of asymmetry on the excitation energy of the initial nucleus \(^{37,38}\). This dependence was obtained in the bombardment of various fissile nuclei by various particles. It has been noted that at high excitations the intermediate nucleus usually emits several neutrons before fission. Nevertheless, we note the general tendency toward a symmetric mass distribution with increasing excitation energy of the initial nucleus.

Fig. 53. Dependence of the fission asymmetry on the initial excitation.

Figure 54

Recently, works \(^{103—105}\) have been published concerning slow, intermediate, and fast neutrons from the fission of U\(^{235}\). These works make it possible to draw several qualitative conclusions regarding the manner in which neutrons are liberated. Measurements of the neutron spectrum in all three works agree with the hypothesis of neutron emission from excited, rapidly moving fragments, with an energy distribution generally characteristic of nuclear evaporation \(^{81}\). The broad excitation spectrum of fission fragments leads

Axis labels: vertical — “Intensity (in arbitrary units)”; horizontal — “Energy in MeV.”

Fig. 54. Fission neutron spectrum.

Text in figure: “Energy distribution of \(\alpha\)-particles from \(U^{235}\) fission.”

Legend:
— — — \(\displaystyle N(E)=22.8\exp\left[-\left(\frac{E-15.2}{6.1}\right)^2\right]\)
— averaged distribution

Axis labels: vertical — \(k\times N(E)\); horizontal — “Energy in MeV.”

Fig. 55. Spectrum of fission \(\alpha\)-particles.

to a broad spectrum of the number and energy of neutrons emitted in individual fission events. Finally, according to this hypothesis neutrons are emitted over a wide angle relative to the direction of motion of the parent fragments. The kinetic energy of the latter, according to the data of Brunton, Hanna, and Thompson\(^{106}\), has a spread of the order of 20 MeV in a typical mass fission. The energy spectrum of neutrons in the laboratory coordinate system is the result of repeated averaging over various factors of the energy spread.

The curve presented is taken from Hill’s paper\(^{104}\). It was obtained by registering recoil protons with a large number of proportional counters. The maximum near 0.8 MeV was confirmed by Bonner, Ferrell, and Rinehart\(^{103}\). The spectrum falls toward higher energies exponentially with a coefficient of recession of 1.6 MeV, in agreement with Watt’s data\(^{105}\). The number of fission neutrons is 2.5 neutrons per fission event of \(U^{235}\) by a thermal neutron. This value coincides with that announced earlier (data of the U.S. Atomic Energy Commission).

Figure 55

Of the various types of ternary fission listed in Table III, fission with the simultaneous emission of long-range \(\alpha\)-particles has been studied most intensively. Figure 55 reproduces the energy distribution of these particles. The dotted curve is drawn in order to show the closeness of the distribution to a Gaussian.

Figure 56

The figure presents the curve of the energy distribution of protons (see Table III), which are believed to be emitted in rare cases of ternary fission. The number of protons is too large to ascribe their appearance to \((n,p)\) or (fragment, \(p\)) reactions. At the same time, the energy distribution indicates that they are not connected with the fission process in any way analogous to the way in which \(\alpha\)-particles are connected. For particles penetrating through a potential barrier, one should expect a maximum in the right-hand part of the curve, not a monotonic fall. It is difficult to imagine that these “fission” protons really have a direct relation to the fission event.

Fig. 56. Energy distribution of fission protons.

Table I

Comparison of the experimental fission thresholds by neutrons and photons with values calculated on the basis of the simple liquid-drop model, neglecting the correction for polarizability and compressibility.

The irregular deviations of the observed quantities from the calculated ones correspond approximately to the expected influence of typical quadrupole moments (Fig. 29).

Fission threshold by neutrons	Fission threshold by neutrons	Fission threshold by neutrons	Fission threshold by neutrons	Fission threshold by neutrons	Fission threshold by neutrons
Target nucleus	Compound nucleus	Threshold energy for neutrons \(E_n\) (in MeV)*)	Neutron binding energy \(B_n\) (in MeV)***)	Observed barrier \(E_n+B_n=F_n\) (in MeV)	Calculated barrier ****) \(F_n\) (in MeV)
\({}_{90}\mathrm{Th}^{232}\)	\({}_{90}\mathrm{Th}^{233}\)	1.05	\(4.9 \pm 0.2\)	\(6.0 \pm 0.2\)	6.5
\({}_{91}\mathrm{Pa}^{231}\)	\({}_{91}\mathrm{Pa}^{232}\)	0.45	\(4.9 \pm 0.4\)	\(5.4 \pm 0.4\)	5.0
\({}_{92}\mathrm{U}^{238}\)	\({}_{92}\mathrm{U}^{239}\)	0.92	\(4.6 \pm 0.2\)	\(5.5 \pm 0.2\)	5.5
\({}_{92}\mathrm{U}^{234}\)	\({}_{92}\mathrm{U}^{235}\)	0.28	\(4.9 \pm 0.4\)	\(5.2 \pm 0.4\)	4.5
\({}_{93}\mathrm{Np}^{237}\)	\({}_{93}\mathrm{Np}^{238}\)	0.25	\(5.0 \pm 0.4\)	\(5.3 \pm 0.4\)	4.2

Photofission threshold	Photofission threshold	Photofission threshold
Target nucleus (compound nucleus)	Observed fission threshold **)	Calculated fission threshold
\({}_{90}\mathrm{Th}^{232}\)	\(5.40 \pm 0.22\)	6.21
\({}_{92}\mathrm{U}^{233}\)	\(5.18 \pm 0.27\)	4.19
\({}_{92}\mathrm{U}^{235}\)	\(5.31 \pm 0.27\)	4.53
\({}_{92}\mathrm{U}^{238}\)	\(5.08 \pm 0.15\)	5.24
\({}_{94}\mathrm{Pu}^{239}\)	\(5.31 \pm 0.25\)	3.40

*) See \(^{28}\).

**) See \(^{29}\).

***) The tabulated neutron binding energies were obtained by interpolation and extrapolation of binding energies calculated from \((\gamma,n)\)- and \((d,p)\)-reactions, systematized in the work of Harvey \(^{59}\), with allowance for systematic deviations caused by the shell at 126 neutrons in the values of the true binding energies relative to the values corresponding to the semiempirical Weizsäcker–Fermi mass formula \(^{60,61}\). The fission barriers were calculated according to the formula of Fig. 3.

****) The formula for the fission barrier in Fig. 3 gives the tabulated values \(F_n\), for \(y=1-x\). Here the “fissility parameter” \(x\) is equal, as in Fig. 2, to one half the ratio of the Coulomb energy to the surface energy.

Table II

Known periods, energies, and yields of delayed neutrons*)

Half-life period	Energy in keV	Yield (in % relative to the total neutron yield)	Literature
0.05 sec.	—	0.025	43
0.43 »	420	0.085	43
1.52 »	620	0.241	43
4.51 »	430	0.213	43
22.0 »	560	0.166	43
55.6 »	260	0.025	43
3 min.	—	\(8\cdot 10^{-7}\)	44
12 »	—	\(3\cdot 10^{-9}\)	44
120 »	—	\(1.3\cdot 10^{-10}\)	44

of a spherical nucleus

\[ x=\frac{1}{2}\frac{E_{\mathrm{k}}}{E_{\mathrm{n}}} =\frac{1}{2}\, \frac{\dfrac{3}{5}\dfrac{e^2}{r_0}} {4\pi r_0^2 O}\cdot \frac{Z^2}{A} =\frac{(Z^2/A)}{(Z^2/A)_{\mathrm{lim}}}. \]

If we substitute \(r_0=\dfrac{e^2}{2mc^2}=\dfrac{e^2}{1.022}\) MeV and \(4\pi r_0^2 O=14\) MeV, we obtain
\((Z^2/A)_{\mathrm{lim}}=45.7^{62,61}\). Owing to the inaccuracy of the constants, we determine \((Z^2/A)_{\mathrm{lim}}\) by another method: using \(\xi_{\max}\), given in Fig. 3, we obtain:

\[ E_{\mathrm{n}}=4\pi r_0^2 O A^{2/3}\xi_{\max}. \]

We choose \(x\) so as to obtain the experimental value \(5.5\) MeV for the fission barrier of \(U^{238}\), and substitute two values \(4\pi r_0^2O=13.0\) MeV and \(14.0\) MeV, following from the formulas of Weizsäcker–Fermi and Finberg, respectively. The resulting values \((Z^2/A)_{\mathrm{lim}}\) are 46.78 and 46.45. We do not introduce here the small correction (\(-0.38\)), which follows from the fact that the zero-point excitation decreases from 0.45 MeV for the spherical form to zero for the form corresponding to the maximum of the barrier curves in Fig. 3. Noting the insensitivity of the result to the magnitude of the surface energy, we arbitrarily choose the value \(4\pi r_0^2O=13\) MeV, for which the barrier quantities given in the table were calculated.

We see that the calculated and observed quantities differ more than the experimental errors, and the calculated quantities change more sharply with variation of \(Z^2/A\) than the experimental ones.

*) The origin of the delayed neutrons accompanying fission in about 1% of cases is explained, according to \(^{5}\), in the following way: 1) fission occurs; 2) the excitation of the fragments is removed by radiation; 3) \(\beta\)-decay of the fragments occurs; 4) in some definite fission products, the energy released in \(\beta\)-decay is greater than the binding energy of a neutron in the daughter nucleus; 5) in these cases the daughter nucleus sometimes proves to be excited and instantaneously emits a neutron. The average kinetic energy of the delayed neutrons, measured by some groups, agrees with the above explanation. Some \(\beta\)-active sources have been identified radiochemically. In these cases, odd–odd transitions permit \(\beta\)-decay processes with the release of energy exceeding the neutron binding energy.

Table III

Fission into three particles*)

Compound nucleus	Mass of the third particles, in atomic mass units	Range, in centimeters of air equivalent	Energy in Mev	Number of double fissions per one triple fission	Literature
\( \mathrm{U}^{235} +\) slow neutron	from 40 to 90	—	\(>40\)	\(7 \cdot 10^{6}\)	47
\( \mathrm{U}^{233} +\) slow neutron	\(13 \pm 3\)	from 0 to 0.8		75	48
\( \mathrm{U}^{235} +\) slow neutron	\(13 \pm 3\)	from 0 to 0.8		75	48
\( \mathrm{Th}^{232} +\) neutron 2.5 Mev	8	—	20	\(10^{3}\)	49
\( \mathrm{U}^{235} +\) photon 23 Mev	8	—	20	\(10^{4}\)	50
\( \mathrm{U}^{233} +\) slow neutron	4	from 10 to 50	from 5 to 25; maximum at 15	400	51
\( \mathrm{U}^{235} +\) slow neutron	4	from 10 to 50	from 5 to 25; maximum at 15	500	51
\( \mathrm{Pu}^{239} +\) slow neutron	4	from 10 to 50	from 5 to 25; maximum at 15	450	51
\( \mathrm{Th}^{232} +\) neutron 2.5 Mev	4	—	from 5 to 23	\(\sim 400\)	52
\( \mathrm{U}^{238} +\) neutron 2.5 Mev	4	—	from 5 to 21	\(\sim 400\)	52
\( \mathrm{Th}^{232} +\) photon 23 Mev	4	—	—	\(\sim 400\)	53
\( \mathrm{U}^{238} +\) photon 23 Mev	4	—	17	—	54
\( \mathrm{U}^{235} +\) slow neutron	1	—	up to 2.1	5000	55

Various kinds of fission into more than two charged particles were identified following the work of Alvarez \(^{54}\). The earliest published investigations concerned \(\alpha\)-particles coincident with fission \(^{63—65}\). This work was refined and extended \(^{66—67}\). The most recent and complete works are indicated for each type of fission in the table. The angular distribution of fission \(\alpha\)-particles was also investigated \(^{68,69}\), and proved to be approximately Gaussian relative to the direction \(82^\circ\) (according to

*) The range is expressed in centimeters of air equivalent. One centimeter of air equivalent is an equivalent unit of stopping power corresponding to a decrease in the range of an \(\alpha\)-particle or proton by one centimeter of air at \(15^\circ\)C and a pressure of 760 mm Hg.

with respect to the direction of the light fragment) with a half-width of \(25^\circ\) [52]. The small probability of fission into 3 equal parts again emphasizes the principle (already obvious in fission into two parts) that mass division is determined above all by the dynamics of the transition forms (Fig. 50), and not by the overall energy balance of the final nuclei, which is considerably lower in the case of symmetric fission into three parts than in fission into two parts.

In the emission of \(C^{12}\) or \(H^4\), the distribution of masses among the heavier fragments is close to the mass division in fission into two parts [50, 69].

Rows 4 and 5 of the table refer to \(Be^8\), which in fact is observed as two \(\alpha\)-particles flying in close directions.

Data have also been published [66] concerning the possibility of fission into four fragments with masses greater than 20. However, Titterton [70] raised serious objections to these considerations.

References

1. N. Bohr, Nature, 137, 344, 351 (1936).
2. N. Bohr, F. Kalckar, Kgl. Danske Videnskab. Selskab. Mat-fys. Medd. 14, No. 10 (1937).
3. L. Meitner, O. R. Frisch, Nature 143, 239 (1939).
4. N. Bohr, J. A. Wheeler, Phys. Rev. 56, 426 (1939).
5. M. G. Mayer, Phys. Rev. 78, 16, 22 (1950).
6. Haxel, Jensen, Suess, Zeits. f. Physik 128, 295 (1950).
7. L. W. Nordheim, Phys. Rev. 75, 1894 (1949).
8. E. Feenberg, K. C. Hammack, Phys. Rev. 75, 1877 (1949).
9. W. Gordy, Phys. Rev. 76, 139 (1949).
10. Townes, Foley, Low, Phys. Rev. 76, 1415 (1949).
11. J. Rainwater, Phys. Rev. 79, 432 (1950).
12. L. Rosenfeld, Nuclear Forces, New York (1948).
13. V. F. Weisskopf, Helv. Phys. Acta 23, 187 (1950); Science 113, 101 (1951).
14. A. Bohr, Phys. Rev. 81, 134 (1950).
15. A. Bohr, Kgl. Danske Videnskab. Selskab. Mat-fys. Medd. 26, No. 14 (1952); A. Bohr, B. R. Mottelson, Phys. Rev. 89, 316 (1953).
16. G. Herzberg, Molecular Spectra and Molecular Structure, New York (1950).
17. W. J. Swiatecki, Phys. Rev. 83, 178 (1951); C. F. Weizsäcker, Zeits. f. Physik 96, 431 (1935); E. Feenberg, Phys. Rev. 60, 204 (1941).
18. J. H. D. Jensen, P. Jensen, Zeits. Naturforsch. 5a, 343 (1950).
19. E. P. Wigner, Nuclear Masses and Binding Energies, p. 27, pt. IV; Nuclear Physics, Philadelphia (1941); E. Feenberg, Rev. Modern Phys. 19, 239 (1947); W. J. Swiatecki, Proc. Phys. Soc. (London) A64, 226 (1951).
20. M. Goldhaber, E. Teller, Phys. Rev. 74, 1046 (1948); H. Steinwedel, J. H. D. Jensen, Phys. Rev. 79, 1019 (1950); Zeits. Naturforsch. 5a, 413 (1950).
21. N. Bohr, F. Kalckar, Kgl. Danske Videnskab. Selskab. Mat-fys. Medd. 14, 10 (1937); V. F. Weisskopf, Phys. Rev. 52, 295 (1937); Л. Ландау, Physik. Zeits. Sowjet. Un. 11, 556 (1937); H. A. Bethe, Phys. Rev. 50, 332 (1936); J. Bardeen, Phys. Rev. 51, 799 (1937); J. Bardeen, E. Feenberg, Phys. Rev. 54, 809 (1938); C. Van Lier, G. E. Uhlenbeck, Physika 4, 531 (1937); L. Motz, E. Feenberg, Phys. Rev. 54, 1055 (1938); V. F. Weisskopf, D. H. Ewing, Phys. Rev. 57, 472, 935 (1940); I. N. Sneddon, B. F. Touchek, Proc. Cambridge Phil. Soc. 44, 391 (1948); J. M. Blatt, V. F. Weisskopf, Theoretical Nuclear Physics, New York (1952), ch. VIII, section 6.

1. H. Wergeland, Skrifter Norske Videnskaps-Akad., Oslo, No. 1 (1941); H. Wergeland, Fysik. Verden 3, 223 (1945); see also Bethe, Rev. Modern Phys. 9, 86 (1937).

2. E. Teller, J. Phys. Chem. 41, 109 (1937); J. Neumann, E. P. Wigner, Physik. Zeits. 30, 467 (1929).

3. L. Wilets, Phys. Rev. 87, 1018 (1952).

4. W. Y. Chang, Rev. Modern Phys. 21, 166 (1949).

5. C. Zenner, Proc. Roy. Soc. (London) A137, 696 (1932).

6. S. Frankel, N. Metropolis, Phys. Rev. 72, 914 (1947); R. D. Present, J. K. Knipp, Phys. Rev. 57, 751, 1188 (1940); Present, Reines, Knipp, Phys. Rev. 70, 557 (1946).

7. U. S. Atomic Energy Commission, AECU-2040 (1952), unpublished.

8. Koch, McElhinney, Gasteiger, Phys. Rev. 77, 329 (1950).

9. N. Bohr, Phys. Rev. 58, 864 (1940).

10. E. Segrè, Phys. Rev. 86, 21 (1952).

11. D. C. Brunton, G. C. Hanna, Phys. Rev. 75, 990 (1949).

12. M. G. Mayer, Phys. Rev. 74, 235 (1948); G. C. Wick, Phys. Rev. 76, 181 (1949); K. H. Kingdon, Phys. Rev. 76, 136 (1949).

13. T. D. Newton, Phys. Rev. 87, 187 (1952); P. Fong, Phys. Rev. 89, 332 (1953).

14. Ya. Frenkel, J. Phys. USSR 10, 533 (1946); E. Bagge, Zeits. Naturforsch. 2a, 565 (1947).

15. R. D. Present, J. K. Knipp, Phys. Rev. 57, 751 (1940).

16. Jones, Fowler, Paehler, Phys. Rev. 87, 174 (1952).

17. J. L. Fowler et al., Phys. Rev. 88, 71 (1952).

18. Hama, Harvey, Moss, Tunnicliffe, Phys. Rev. 81, 466 (1951).

19. Winhold, Demos, Halpern, Phys. Rev. 87, 1139 (1952).

20. I. Halpern, E. J. Winhold, Progress Report of the Laboratory of Nuclear Science and Engineering (1952), unpublished.

21. Glendenin, Coryell, Edwards, Radiochemical Studies, The Fission Products, New York (1951); Plutonium Project Record, vol. 9, part IV.

22. Hughes, Dabbs, Cahn, Hall, Phys. Rev. 73, 111 (1948).

23. Kunstadter, Floyd, Borst, Weremchuk, Phys. Rev. 83, 235 (1951).

24. de Benedetti, Francis, Preston, Bonner, Phys. Rev. 74, 1645 (1948).

25. J. S. Fraser, Phys. Rev. 85, 726 (1952).

26. L. Rosen, A. M. Hudson, Phys. Rev. 78, 533 (1950).

27. K. W. Allen, J. T. Dewan, Phys. Rev. 82, 527 (1951).

28. E. W. Titterton, Phys. Rev. 83, 1076 (1951).

29. Goward, Titterton, Wilkins, Nature 164, 661 (1949).

30. K. W. Allen, J. T. Dewan, Phys. Rev. 80, 181 (1950).

31. E. W. Titterton, Phys. Rev. 83, 673 (1951).

32. E. W. Titterton, private communication (1950).

33. E. W. Titterton, F. K. Goward, Phys. Rev. 76, 142 (1949).

34. D. L. Hill, Phys. Rev. 87, 1049 (1952).

35. J. A. Wheeler, Phys. Rev. 52, 1107 (1937).

36. A. B. Migdal, ZhETF (1940).

37. K. W. Ford, Phys. Rev. 90, 29 (1953).

38. J. A. Harvey, Phys. Rev. 81, 353 (1950).

39. E. Fermi, Nuclear Physics, IL, Moscow, 1951.

40. E. Feenberg, Rev. Modern Phys. 19, 239 (1947).

41. E. Feenberg, Phys. Rev. 55, 504 (1939).

42. L. Green, D. Livesey, Conference on Physics of Fundamental Particles (1946).

1. Tsien, Chastel, Ho, Vigneron, Comptes Rendus 223, 986, 1119 (1946); 224, 272 (1947).
2. Farwell, Segrè, Wiegand, Phys. Rev. 71, 327 (1947).
3. Tsien, Ho, Chastel, Vigneron, J. phys. et rad. 8, 165 (1947); 8, 200 (1947).
4. L. Green, D. Livesey, Phil. Mag. A241, 323 (1948).
5. Wollan, Moak, Sawyer, Phys. Rev. 72, 447 (1947).
6. L. Marschall, Phys. Rev. 75, 1939 (1949).
7. E. W. Titterton, Nature 170, 794 (1952).
8. W. E. Swiatecki, Proc. Phys. Soc. A63, 1208 (1950).
9. K. Way, Phys. Rev. 55, 964 (1939).
10. C. F. Weizsäcker, Die Atomkerne, Leipzig (1937).
11. E. Feenberg, private communication.
12. K. C. Hammack, Topics in Nuclear Structure, Washington (1951).
13. W. Elsasser, J. phys. et rad. 5, 625 (1934).
14. H. Margenau, Phys. Rev. 46, 613 (1934).
15. Perlman, Ghiorso, Seaborg, Phys. Rev. 77, 26 (1950).
16. A. Bertelot, J. phys. et rad. 3, 17 (1942).
17. I. Kaplan, Phys. Rev. 81, 962 (1951).
18. J. M. Blatt, V. F. Weisskopf, Theoretical Nuclear Physics, New York (1952), ch. XI.
19. Rassmussen, Thompson, Ghiorso, Phys. Rev. 89, 33 (1953).
20. W. P. Jesse, J. Sadauskis, Phys. Rev. 78, 1 (1950).
21. A. J. Dempster, Report ANL—4355 (1949), unpublished.
22. Rasmussen, Reynolds, Thompson, Ghiorso, Phys. Rev. 80, 475 (1950).
23. I. Perlman, T. J. Ypsilantis, Phys. Rev. 79, 30 (1950).
24. P. Brix, H. Kopfermann, Zeits. f. Physik 126, 344 (1949).
25. Phys. Rev. 77, 26 (1950).
26. Richardson, Ball, Leith, Moyer, Phys. Rev. 86, 29 (1952).
27. F. O. Rice, E. Teller, J. Chem. Phys. 6, 489 (1938).
28. J. A. Harvey, Phys. Rev. 79, 241 (1950).
29. Street, Ghiorso, Thompson, Phys. Rev. 84, 135 (1952).
30. G. C. Baldwin, G. S. Klaiber, Phys. Rev. 71, 3 (1947).
31. H. Steinwedel, J. H. D. Jensen, Zeits. Naturforsch. 5a, 413 (1950).
32. G. T. Seaborg, Phys. Rev. 85, 157 (1952).
33. Plutonium Project, Rev. Modern Phys. 18, 539 (1946).
34. R. W. Spence, Report BNL-C-9 (1949), unpublished.
35. P. R. O’Connor, G. T. Seaborg, Phys. Rev. 74, 1259 (1948).
36. R. H. Goeckermann, I. Perlman, Phys. Rev. 73, 1127 (1948).
37. Glendin, Steinberg, Inghram, Hess, Phys. Rev. 84, 860 (1951).
38. W. J. Swiatecki, Phys. Rev. 83, 178 (1951).
39. D. L. Hill, Phys. Rev. 78, 330 (1950); 79, 197 (1950); doctoral dissertation (unpublished).
40. Bonner, Ferrell, Rinehart, Phys. Rev. 87, 1032 (1952).
41. D. L. Hill, Phys. Rev. 87, 1034 (1952).
42. B. E. Watt, Phys. Rev. 87, 1037 (1952).
43. Brunton, Hanna, Thompson, Can. J. Research 28A, 190 (1950); 28A, 498 (1950).