NUCLEAR STRUCTURE AND INTERPRETATION OF FISSION PHENOMENA

D. Hill and J. Wheeler*)

The authors have made an attempt to bring together into a single picture the features emphasized by experience in the structure of the nucleus, generalized both as the model of a liquid drop and as the model of independent particles. As the starting point for the investigation of this generalized representation—the collective model of the nucleus—the assumption is adopted of a strong coupling, according to which the potential of a nucleon inside the nucleus depends almost not at all on the positions of the other nucleons and falls to a small value at a distance from the nuclear surface. In this model a distinction is made between the nucleonic state of the system, determined by the states of the individual nucleons, and the states of oscillation and rotation of the nucleus as a whole. On the basis of quantum-mechanical considerations it is shown how the kinetic energy of these motions can be calculated in terms of the degrees of freedom of the individual particles. In the case of the nucleus, just as in the case of the electronically vibrational-rotational description of molecules, it makes sense to regard the sum of the energies of the states of the individual particles and of the interaction energies as the potential energy of deformation, depending on the form of the system. Different states of the set of individual particles lead to different many-sheeted potential-energy surfaces. A given sheet of the potential-energy surface touches another, adjacent to it, upper or lower, only in certain isolated “funnels,” as in the case of polyatomic molecules. In order for the collective model to be useful, it is necessary that the frequency of nonadiabatic transitions from one surface to another be small in comparison with the frequency of rotations and surface vibrations, so that these collective motions have a quite definite meaning. The mathematical treatment of the collective model is still not sufficiently developed to make it possible to give a complete, or even approximate, answer to the question of whether such a condition of self-consistency is fulfilled sufficiently well, or only approximately, or, finally, is not fulfilled for any excitation energy interval at all. The frequencies of oscillations in general correspond to the frequencies given by the simple model of the liquid drop, though with certain characteristic quantum-mechanical differences. The basic provisions of the Franck–Condon principle are applied by analogy with polyatomic molecules. Some conclusions following from the collective model, or from its simplified liquid-drop model, are discussed with respect to energy levels,

*) D. L. Hill and J. A. Wheeler, Phys. Rev. 83, 1102 (1953).

compatibility of strong neutron capture with effects in the binding energy of individual particles, quadrupole moments, $\alpha$-decay, fission thresholds, photofission, spontaneous fission, asymmetry of nuclear fission, hydrodynamics of the fission process, $\alpha$-particles of fission, and excitation of fragments.

I. THE LIQUID DROP AND INDEPENDENT PARTICLES

Nuclear fission is a nuclear process of an unusual type. If a system consisting of many nucleons divides into two approximately equal parts, then this phenomenon cannot be explained on the basis of the motion of one nucleon or of a small number of nucleons. Here we encounter the collective behavior of the nucleus as a whole, which may ideally be represented by the liquid-drop model. The behavior of nuclear matter is analogous to that of a drop of an almost incompressible liquid, possessing a more or less uniform volume density of electric charge and approximately the same binding energy per particle. To this must be added the influence of the energy of electrostatic interaction of the separate parts of the liquid and the decrease in the binding energy of particles located near the surface, which is proportional to the size of the surface of the drop and leads to the appearance of a surface tension analogous to the surface tension in ordinary liquids[^1],[^2]. Such a system may undergo deformations (Fig. 1*).

The stabilizing action of surface tension exceeds the destructive action of the electrostatic repulsive forces even in heavy nuclei having a normal, approximately spherical, configuration. Consequently, small perturbations cause oscillations about the equilibrium shape. However, a significant “dumbbell-like” change in the shape of the nucleus leads to the predominance of the electrostatic repulsion of the two parts of the system over the forces of surface tension drawing the drop together[^3]. Thus, a significant deformation of a heavy nucleus brings the nucleus to instability. An even greater stretching arises, in which the decrease of electrostatic energy proceeds faster than the increase of surface-tension energy; the motion begins to accelerate and, in the end, the nucleus divides into two (or more) parts. The act of fission passes through several stages[^4]: 1) an increase of the excitation energy of the nucleus to a certain level; 2) concentration of a considerable part of this energy in the energy of surface oscillations, sufficient to obtain the critical deformation (Figs. 2 and 3); 3) spontaneous increase of this deformation (Fig. 4); 4) separation (scission) into fragments of irregular shape; 5) separation of the fragments from one another; 6) “calming” of the new nuclei.

* The figures and their explanations are placed in the “Appendix,” at the end of the article.

Although the fission process indicates the capacity of the nucleus for collective forms of motion, in recent times a large amount of data has accumulated that corresponds to the behavior of nucleons as particles possessing individual and almost independent states \(^{5-8}\). These data are drawn from measurements of the spins and magnetic moments of nuclei and from regularities in nuclear binding energies, expressed in the shell structure. Thus, we are compelled to conclude that two such different points of view, the liquid-drop model and the model of independent particles, are different aspects of some more general picture. Consequently, it is necessary to consider and understand the behavior of the nucleus as a collective, proceeding from the properties of the individual nucleons. Having set out on this path, we must calculate, from hydrostatic ideas, the general character of the dependence of the critical fission energy on the atomic number and mass number, and explain the possible deviations from these average quantities in different nuclei by the individual character of nuclear states.

The magnitudes of the quadrupole moments of nuclei clearly show that neither the liquid-drop model nor the model of individual nucleons moving in a spherically symmetric field is, each by itself, complete (Figs. 5 and 29). Both models cannot explain so great an asymmetry in the distribution of the electric charge of the nucleus, which is observed in a large number of cases \(^{9,10}\). It is necessary, however, to take into account that the pressure exerted by several nucleons on the surface of the nucleus will lead to a change in the shape of the entire ensemble of nuclear charges (Fig. 6). Allowance for this phenomenon leads to the correct order of magnitude of the quadrupole moments, as was first indicated by Rainwater \(^{11}\).

The quadrupole moments show that the interaction of nucleons with one another through the surface of the nucleus proves to be stronger than the direct interaction of nucleons. This conclusion is confirmed by experimental data concerning nuclear binding energies. The energy of a nucleon surrounded on all sides by neighboring nucleons at a distance of the order of \(10^{-13}\) cm depends little on whether there are still other nucleons at a greater distance or not. Moreover, in both cases there is no noticeable influence on the average spatial distribution of the nearest neighbors. Why nuclear forces become saturated, and what the origin of these forces is, we do not know \(^{12,13}\). It is also unclear to us why the values of the spins and magnetic moments and the fine structure of nuclear binding-energy levels agree with the model of individual nucleons moving almost independently in an average potential field. Nevertheless, the experimental data force us to accept an idealized picture in which each particle moves in a potential well of approximately constant depth inside the whole nucleus and suddenly vanishing at a distance of order \(r_0\) from the surface of the nucleus.

Finer details of the structure of nuclear shells have led to a hypothesis in which the individual nucleons are subject to the action not only of the nuclear potential field, but also of spin-orbit coupling.³ Without in any way casting doubt on this fruitful hypothesis, we are nevertheless entitled at first to neglect the influence of spin-orbit coupling when considering the relation between the independent-particle model and other models of the nucleus that assume a closer interaction of the particles.

II. THE COLLECTIVE MODEL OF THE NUCLEUS

Let us consider the collective model of the nucleus, based on the following assumptions:

1. Assumptions valid for any model

1) An approximately constant density—one particle per volume
\(\frac{4}{3}\pi r_0^3\), where \(r_0 \sim 1.4 \cdot 10^{-13}\ \mathrm{cm} = \frac{e^2}{2mc^2}\).

2) The maximum kinetic energy \(F\) per nucleon is almost independent of the mass number and is equal to \(\sim 24\) MeV.

3) The charge distribution over the volume is uniform to an accuracy of 25% (or better).

4) The binding energy of nucleons lies within the limits \(5 \div 10\) MeV; the average potential energy is of the order of \(-30\) MeV.

5) Saturation of nuclear forces.

2. Special assumptions

1) Strong saturation; the potential field for a typical nucleon inside the nucleus depends hardly at all on the positions of the other nucleons; the potential falls off rapidly at a small distance from the surface of the nucleus. This idealization is opposite to the model of an impenetrable liquid drop, in which the forces depend sharply on the positions of neighboring nucleons regardless of whether the particle under consideration is inside the nucleus or near its surface. In the liquid-drop model the direct interaction between particles is regarded as so strong that the individual states of nucleons have no meaning. (The observations that correspond to the concept of almost independent particle orbits apply for the most part to the ground and low-lying excited levels.) We have no serious arguments either for or against such a conception. Nevertheless, for the collective model, the neglect of this direct interaction of particles in comparison with the indirect interaction occurring through the intermediary of a mobile

of the boundary of the potential well (the surface of the nucleus) is an essential initial idealization. From this point of view the collective model is a first approximation to the truth, whose validity can be established only by a complete examination of the consequences following from it. The theory of the collective model is still at too early a stage of development to permit a full comparison with experiment.

2) The state of the entire system in the first approximation is determined by the states of motion of the individual nucleons or, as will be seen below, by one of many potential curves and by the corresponding quantum numbers of rotation of the system and of collective oscillations along this potential curve. In reality, owing to the coupling of both forms of motion through the surface of the nucleus, an exchange of energy will occur with a certain probability between the vibrations of the nucleus and the excitation of individual particles. This exchange is not an accidental aspect of the phenomenon in the collective model; it is an essential aspect of the kinetics of the process. But if the frequency of exchange becomes comparable with the frequency of the vibrations, then the separation of the energy into a part belonging to the collective and a part belonging to the individual particles cannot be carried out rigorously, and the collective model will lose its meaning. We must therefore determine whether the collective model is self-consistent. Up to now we are not in a position to carry out a detailed comparison of the frequency of exchange with the frequency of the vibrations. The rough estimates given below indicate the possibility of realizing an intermediate case, in which the two frequencies are comparable in magnitude and in which the separation into vibrations and motions of individual nucleons will prove to be partially blurred. It is still too early to exclude either the possibility that the relation between the two frequencies will prove more favorable, or the possibility that the numerical values will turn out such that the usefulness of the collective model will be severely limited. In that case the general conclusions concerning the influence of quadrupole moments on $\alpha$-decay, concerning fluctuations of the fission-barrier height from element to element, etc., will remain unaffected, but most of the quantitative applications of the model will prove almost useless. We, however, hope for the consistency of assumption 2); for this reason we investigate in the present work certain mathematical details of the collective model and its applications.

§ 3. Comparison of nuclei with atoms and molecules

The idealization that we shall adopt for describing the behavior of nucleons in a nucleus is in many respects derived from the picture of the motion of electrons in an atomic field. There are analogous grounds for speaking of individual quantum states

and in transition probabilities. The same considerations may be used in calculating the self-consistent nuclear potential. However, there is an important difference in the foundations. In the case of electrons the field of forces is determined by the nucleus. The action of the field on one electron in a many-electron atom changes relatively little if another electron crosses the orbit of the first. The potential may be regarded as static.

In the case of the nucleus, the magnitude of the potential inside the nucleus ideally does not depend on the orbit of any individual nucleon, but the boundaries of this potential, from the point of view of some particle, depend very strongly on the motions of a small number of other particles constituting the nucleus. In this sense the potential field fluctuates or oscillates.

We could neglect fluctuations in the position of the nuclear surface if our discussion were restricted to the ground states of nuclei, for which the amplitude of the oscillations4 is of the order of \(r_0\). However, it is also necessary to consider excited states and the fission process, when the amplitude of the oscillations is comparable with the dimensions of the nucleus. How can we describe, in a quantum-mechanical way, a system of many independent particles with characteristic new features of collective forms of motion?

A useful comparison is that of the collective model of the nucleus with a typical molecule. In the case of a molecule, the electrons move with high velocity in a field of forces which itself changes owing to the change in the distance between the nuclei with a frequency 10–100 times smaller. Thus each electron has time to follow almost adiabatically the instantaneous values of the potential. On the other hand, the total energy of the electron system at any given instant possesses a certain store of potential energy. From this source the molecule can draw kinetic energy for internuclear motion, which it returns when the amplitude of the oscillations reaches its limits. Analogously to this picture, the characteristic time of the radial motion of a nucleon possessing an average kinetic energy \(T = 15\) MeV is, for \(U^{236}\),

\[ t_{\mathrm{nucl}} = \oint \left[\frac{2T}{M} - \frac{l(l+1)\hbar^2}{M^2 r^2}\right]^{-1/2} dr = \]

\[ = \frac{2R}{[2T/M]^{1/2}} \left[1 - \frac{l(l+1)\hbar^2}{2MTR^2}\right]^{1/2} < \frac{2R}{[2T/M]^{1/2}} \equiv \frac{2R}{v} = \frac{2A^{1/3} r_0}{0.18c} = 0.3 \cdot 10^{-21}\ \mathrm{sec}. \tag{1} \]

i.e., fifteen times smaller than the period of the lowest-order surface oscillations for the same nucleus, equal to

\[ t_2 = \frac{2\pi\hbar}{\hbar\omega_2} = 2\pi \cdot 0.658 \cdot 10^{-21}\, \frac{\mathrm{MeV}\ \mathrm{sec}}{0.8\ \mathrm{MeV}} = 5 \cdot 10^{-21}\ \mathrm{sec}. \tag{2} \]

Thus, in the case of nuclei one may likewise expect that the states of the particles will follow the changes of the nuclear boundary; in this case the probability of a nonadiabatic jump from one state to another will be small, provided only that the deformations considered are simple.

4. Interaction of nucleons with the surface of the nucleus

In the case of complex deformations of the surface, a particle requires a longer time in order to “probe” the whole surface, and the adiabatic conditions will not be satisfied so simply. It seems reasonable to require that, for surface deformations of order \(n\), the quantity \(\dfrac{n}{2} t_{\text{nucl}}\) be less than the period \(t_n\) of the deformation—a condition which, evidently, cannot be satisfied for deformations of very high order (for example, for \(n > \sim 6\)). Such deformations have no definite meaning in the collective model.

The natural minimal limit of the area of deformations capable of producing a response of the nucleus is, evidently, the square of the de Broglie wavelength of the particle

\[ \chi = \frac{\lambda}{2\pi}. \]

The time necessary for this response must be of the order of the product of the number of collisions (needed for contact with all parts of the surface \(S\)) by the time interval between two collisions, i.e.

\[ \sim \frac{S}{\chi^2}\cdot\frac{R}{v} \sim \frac{V}{\chi^3}\frac{\hbar}{E} \sim \frac{\hbar}{\Delta E}, \]

where \(V\) is the volume accessible to the particle, \(E\) is its kinetic energy, and \(\Delta E\) is the typical splitting (Fig. 11) between individual levels in the given region. This splitting is so small, and the corresponding time interval so large in comparison with the period of oscillations, that there can be no question of adiabatic conditions for small cell-like perturbations. If two particles participate in the motion and they interact strongly with one another, then the time required for the response of this two-particle system to small cell-like perturbations of the surface increases by a factor of \(\dfrac{V}{\chi^3}\) (the number of different positions of the second particle within the volume \(V\)). If the number of strongly interacting particles increases, then this time increases by a factor of \(\dfrac{V}{\chi^3}\) for each particle, for different particles, and somewhat less for particles,

satisfying the Pauli principle; however, the increase is always such that the time interval considered is equal to \(\hbar\), divided by the splitting of the levels of the entire system. Obviously, such times will be too large for an adiabatic reaction to surface oscillations. This means that the collective model can be justified only when the interaction of individual particles with one another, according to the law of strong saturation, does not depend on the position of any other particle inside the nucleus.

If we adhere to the analogy with molecular potential-energy curves as functions of internuclear distances, then we arrive at the notion of a nuclear potential-energy curve as a function of the deformation of the nuclear surface. Just as in a polyatomic molecule there are many independent degrees of freedom, in the case of the nucleus it is necessary to introduce into consideration a certain number of parameters to determine the shape of the nuclear surface (Fig. 1). Thus, we shall be dealing with a potential-energy surface. As is known, in molecules the minima of different potential surfaces do not coincide with one another; still less does this occur in the case of nuclei. The equilibrium quadrupole moment changes from state to state in accordance with the features of the interaction of the odd (with respect to the closed shell) nucleon with the surface. The energy of the system will consist of the energy of the nucleons, the energy of vibrations, and the energy of rotations.

5. Kinetic energy of collective motion

In the case of molecules, the vibrational kinetic energy is explicitly connected with the motion of atoms. In the case of nuclei, can one indicate a mass whose motion is responsible for the vibrational kinetic energy?

Is not the energy of motion of the nucleons already included in the energy curve of the vibrational potential? This vibrational potential is defined as the sum of the potential and kinetic energies of all individual states of the nucleon, calculated for a static configuration of the deformed nucleus. However, such a calculation, by its nature, does not take into account slow changes in the shape of the nucleus. These changes necessarily cause the displacement of considerable masses from one place to another. In such a state of affairs, there inevitably arises a kinetic energy of the entire nucleon system in addition to the energy attributed to the individual nucleons located in the static potential well. This additional kinetic energy may be interpreted as the vibrational energy of motion of the system.

In the quantum-mechanical description (Figs. 7 and 8), the motion of the nuclear liquid is manifested in the displacement of the nodes of the wave function of a bound nucleon from place to place in accordance with the motions of the surface. In the simplest case the nodal surfaces move like ink lines carried along bodily in the irrotational motion of an imaginary liquid. The wave function of a nucleon \(\psi\) at any instant of time during a slow deformation of the surface differs from the value \(u\) which it would have for a stationary surface of the same form by a factor equal (in this approximate description of the motion of the nodes) to

\[ e^{-\frac{iM\varphi}{\hbar}}, \]

where \(\varphi\) is the velocity potential of the motion of the liquid under consideration. It follows from this that the kinetic energy of the nucleon is greater than the value it would have in the absence of motion of the nuclear surface by an amount proportional to the square of the velocity of motion of this surface. The coefficient of proportionality is equal to the coefficient of proportionality calculated for the classical irrotational motion of a liquid enclosed in the same bounding surface.

Of special interest among the motions of the surface is the motion which leaves its shape unchanged, i.e. pure rotation of the surface.

In this case the amount of matter moving from place to place is determined not by the total mass of the body, but by the magnitude of the deviation of the surface from a spherical shape. The effective moment of inertia of the system proves to be considerably smaller than the moment corresponding to rotation of the system as a whole. The corresponding rotational levels lie considerably higher. All these phenomena, following from the quantum picture of rotations of the nucleus, were pointed out by A. Bohr\(^{14,15}\), who also indicated their importance for the analysis of nuclear spins and magnetic moments.

The energy of the lowest rotational levels, although much greater than the energy of the levels corresponding to rotation of a rigid body, amounting to only a few tens of keV, nevertheless remains smaller than most quanta of vibrational energy. It therefore seems possible to follow the manner of presentation adopted in molecular physics and, in those cases where the angular momentum of the rotational motion vanishes or is negligibly small, to use the term “potential energy surface.”

Rotational moments of larger magnitude produce changes in the potential surface of the same type as those known in molecular spectroscopy\(^{16}\). These changes of the potential surface caused by rotation lead to a number of complex and interesting effects. In this article, however, we shall neglect them in comparison with the phenomena of surface vibration and nucleon excitation.

Under certain circumstances, deformation of the surface leads to such motion of the nodal surfaces that it can no longer be described even approximately as irrotational motion of a liquid. It is then necessary to consider vortical motion of the carrier liquid. Its effect on the wave function of the nucleons (Fig. 10) can be represented by the quantum analogue of vortices arising in a classical liquid. Since these effects appear only in one of the many states of the nucleons, the kinetic energy of the whole system in the presence of such deformations will not differ greatly from the value corresponding to the simple model of a liquid drop.

6. Quantum description of collective motion

Assuming that the kinetic energy of the droplet model is manifested in the collective motion of the nucleus for a prescribed motion of its surface, let us consider how the motion of the surface itself can be represented as part of the quantum-mechanical description of a system of \(N\) particles, and not as something specified from outside.

Can one not explain everything by a complete set of degrees of freedom, without using the concept of surface motion? How is one to describe oscillations of the surface in terms of degrees of freedom, if the surface has none?

Physically this means that fluctuations in the position of the surface and the collective character of their action on a system of \(N\) particles are an inevitable consequence of the strong binding of the particles near the surface of the nucleus. Mathematically this can be described as follows. Let the position of the surface be determined by external parameters \(\alpha\); then the wave function of the system—disregarding inessential details—will have the form of a determinant

\[ \Psi_{\text{stationary}} = \left| \begin{array}{cccc} u(1,x_1;\alpha) & \cdot & \cdots & u(1,x_N;\alpha)\\ \cdot & & & \cdot\\ \cdot & & & \cdot\\ \cdot & & & \cdot\\ u(N,x_1;\alpha) & \cdot & \cdots & u(N,x_N;\alpha) \end{array} \right|, \tag{3} \]

where \(u(n,x_j;\alpha)\) is the wave function of an individual particle in a potential well of prescribed shape. For a variable deformation (caused by external causes) \(\alpha\) becomes a function of time, and in order to obtain an approximate value of the wave function of the system of nucleons, the determinant on the right must be multiplied by

\[ e^{\frac{iM}{\hbar}\,[\varphi(x_1)+\cdots+\varphi(x_N)]}. \tag{4} \]

Here \(\varphi\) (expressed in \(\mathrm{cm}^2/\mathrm{sec}\)) denotes the velocity potential of the irrotational motion of the imaginary fluid. Conversely, if one assumes that the nucleons determine the potential energy of deformation corresponding to the coordinate \(\alpha\), then the vibrational state of the system—if it can at all be regarded as existing independently—will be characterized by a quantity \(h_n(\alpha)\) corresponding to a quasi-harmonic oscillator. It would be incorrect to write the wave function of the entire system as the product of the determinant by the velocity potential and the function of a harmonic oscillator, since in this product there would be too many independent variables for a system of \(N\) particles. However, integration of this product over \(\alpha\) leads to a wave function depending only on the coordinates of the particles\(^*\)

\[ \Phi(x_1,\ldots,x_N)=\int \Psi(x_1,\ldots,x_N;\alpha)\, e^{-\frac{iM}{\hbar}\sum_j \varphi(x_j)}\,h_n(\alpha)\,d\alpha . \tag{5} \]

Nevertheless, this function gives a description of the collective motion under consideration that has physical meaning: a) It is antisymmetric. b) It is large in the neighborhood of the classical turning points of the oscillations; the meaning of this statement is that if one nucleon is located at some small distance from the mean position of the surface, then (with a certain probability) the other nucleons will also be at the same distance from it, and conversely, if for some particle there is a probability of not being at the given distance from the mean position of the surface, then this is accompanied by an increase in the probability that the other nucleons likewise are not there; these correlation probabilities

\(^*\) The factor \(\dot{\alpha}\) in the velocity potential entering the exponent (Figs. 7 and 8) must, of course, before integration over \(\alpha\), be replaced by the operator

\[ \frac{\hbar}{iM_\alpha}\frac{\partial}{\partial \alpha}. \]

Conversely, if for \(h_n(\alpha)\) we use its semiclassical approximation (the Jeffreys–Wentzel–Kramers–Brillouin approximation), then the exponential operator acting on this oscillatory function gives two additional terms. In one of these terms the factor \(\dot{\alpha}\) is given by the value \(+\{2[E-V(\alpha)]/M_\alpha\}^{1/2}\), and the exponential function is multiplied by that part of \(h_n(\alpha)\) which represents a wave traveling to the right; the same is true in the other term, where \(\dot{\alpha}\) is given with the opposite sign. Although the wave function (5) is formulated on the basis of physical considerations, one may, of course, conversely regard \(h_n(\alpha)\) as an almost undetermined function which must be chosen so as to transform \(\Phi\) into the “best possible wave function” in the sense of the Ritz variational principle. The question of how to apply this approximation method for deriving the wave function for \(h_n(\alpha)\) is, in principle, identical with the question of formulating the method of “resonating group structures” \(^{56}\). In particular, it is not at all necessary that the potential \(V(\alpha)\) be quasi-elastic or that \(h_n(\alpha)\) be the wave function of a harmonic oscillator.

are largest when the distances under consideration are compared with the amplitudes of the corresponding classical surface oscillations. c) Two determinants from among the number combined with one another in the integration are almost orthogonal (Fig. 9) if the displacements $\alpha_1$ and $\alpha_2$ to which they correspond differ, in the direction perpendicular to the surface, by an amount of order

\[ \frac{r_0}{N}, \]

where $N$ is the number of identical particles, and

\[ 4 \cdot \frac{4}{3}\pi r_0^3 \]

is the volume assigned to each of these particles. Physically this means a very high degree of correlation between the probability of the particle distribution and the magnitude of the “hidden” deformation variable $\alpha$. This approximate orthogonality of the functions permits one, with some justification, to regard the variable $\alpha$ as an almost independent coordinate.

The deformation coordinate can indeed be regarded as a variable quantity characterizing the particle; the collective model by no means requires an unlimited increase in the total number of degrees of freedom. How can we, in general, justify counting the number of states of the system to which we assign not only the indices of the individual states of the particles, but also the quantum states of the surface oscillators? A system of functions is complete when the quantum numbers of the oscillators have definite values. If we also wished to sum over the vibrational quantum numbers, we would obtain the very same series several more times.

The solution of this paradox is that these extra summations are not implied mathematically and have no physical meaning. Only oscillations of low orders have a quite definite physical meaning. Moreover, the rate of energy exchange between vibrational and nucleonic motion depends on the choice of the potential surface and becomes the greater, the higher the excitation of the nucleons in the state under consideration. Thus there is a limiting value of the energy above which oscillations of any kind lose their meaning. The problem of the collective model is not that we have too many states, but that we have too few of them. The inapplicability of the model at energies above several tens of MeV can be demonstrated from physical considerations. A strongly excited nucleus may break up into several parts, as is evident from observations of stars in cosmic rays. The collective model is suitable only for not too large excitations.

The usual expansion of the nuclear wave function into individual states of particles in a spherical potential field turns out to be very poorly adapted for the description of collective vibrations and rotations. The number of types of multiple excitations of nucleons required to describe combinations of vibrational and nucleonic excitations would in this case be enormous.

III. DEFORMATION POTENTIAL

1. Definition and general formula

The key to a collective description is the concept of the potential energy of deformation, by which is meant the sum of the kinetic and potential energies of the individual nucleons moving inside a nuclear surface of fixed shape. In an initial consideration of the energy surfaces it is meaningful to make the following simplifications: a) neglect spin-orbit coupling. b) Assume that the intranuclear forces have such strong saturation that the potential energy of a particle is constant inside the nuclear matter, while at the surface it undergoes a sharp jump. In reality, the wave function penetrates into the region of negative kinetic energies over distances of order \(r_0\). When operating with the wave functions of bound nucleons it is often convenient to regard the potential jump at the boundary surface as infinitely large. Then the calculated wave function of the nucleons does not penetrate at all beyond the boundary surface. This effect, and its influence on the displacement of the nodal surfaces inside the potential well and on the change of the eigenvalues of the energy, can be corrected with sufficient approximation by a corresponding change in the quantities adopted for nuclear dimensions. Then the contribution of each particle to the total energy is equal to the corresponding eigenvalue

\[ \nabla^2 \psi_n + \frac{2 M E_n}{\hbar^2}\,\psi = 0, \]

satisfying the boundary conditions, diminished by the constant quantity \(B_0\), of order \(14\) MeV, which represents the saturation binding energy per nucleon. c) We shall represent the decrease of the binding energy of particles at the nuclear surface by means of a term (in the total nucleon energy of the system) proportional to the surface area \(S\). We shall denote the coefficient of proportionality by \(O_p\) (potential) and shall assume that it is equal not to the full surface tension[^17] of nuclear matter

\[ O = O_p + O_k \sim \frac{14}{4\pi r_0^2}\ \text{MeV}, \]

but only to that part of it which is connected with the specific nuclear forces. The remaining part, \(O_k\) (kinetic), of the usual surface tension is connected with that part of the total kinetic energy of the particles which depends on the surface of the potential well, and not on its volume. This kinetic part has already been included in b). Whatever the difference in the dependence of the kinetic energy on the magnitude of the deformation in a statistical analysis (for which the constants \(O_k\) are typical) or in a detailed analysis (by summing the eigenvalues \(E_n\)), the latter should be regarded as more definite. As for the dependence of the specific nucleon potential energy on deformation, it would be

somewhat more accurate, if this is practically possible, to estimate the expected value of the interaction energy from a wave function having the form of a determinant constructed from the eigenfunctions \(\psi_n\), than to use the expression \(-AB_0 + O_pS\) for \(A\) nucleons as a statistical method for estimating this nucleon expression. The difference between the two methods of estimation is the smaller, the closer the actual forces of interaction between nucleons are to that highest degree of saturation which is meant in the idealized collective model. г) The electric energy of interaction of the protons can likewise be estimated by means of a wave function constructed in the form of a determinant, but again it makes sense to represent the Coulomb interaction approximately through the electric energy of a uniformly charged liquid enclosed within the prescribed boundary:

\[ V_e=\rho_e^2\iint \frac{d(\nu_0\rho)_1\,d(\nu_0\rho)_2}{2r_{12}} . \]

One could consider—although we shall not do so here—a refinement of this analysis in which: 1) for protons and neutrons slightly different boundaries of the volumes in which they move are adopted\(^{18}\); 2) the motion of the two particles is considered in two different potential wells; 3) both potential wells have nonconstant depth; 4) the gradient of the potential is such that the number of protons in the outer part of the nucleus and the number of neutrons in the inner part are somewhat greater than the numbers corresponding to a homogeneous proton–neutron ratio\(^{19}\); and 5) oscillations of the neutrons as a whole relative to the protons are possible, as was indicated by Goldhaber and Teller and by Jensen and Steinwedel\(^{20*}\), in particular in connection with the observed maximum of the nuclear photoabsorption cross section in the region \(10 \div 20\) MeV.

Thus, in the collective model one considers the deformation potential function \(V(a_2,a_3,\ldots;\,n_1,\ldots)\), depending on the coordinates \(a_2, a_3\), etc., which determine the shape of the potential surface, and on the quantum numbers \(n_l\), etc., of the occupied nucleon states. This potential function has the following form:

\[ V(a,n)=-AB_0+O_pS(a)+\Sigma E_n(a)+V_e(a), \tag{6} \]

2. Density of Levels

It is of interest to consider the dependence of the deformation potential on the deformation at a fixed state \(n=(n_1,\ldots)\) of the whole system of nucleons and on changes of the quantum state for a prescribed configuration of the nuclear wall (the bounding surface

* The presence of such oscillations was pointed out as early as 1940 by A. B. Migdal\(^{57}\). (Ed.)

nucleus). The change of the potential when the configuration of the walls changes determines a certain surface in the space \((V,\alpha)\). One may imagine that the point representing the system moves over the potential surface like a ball. The potential surface usually has at least one minimum—the equilibrium point of the collective oscillations of the nucleon system. The position of the minimum determines the normal equilibrium deformation (shape) of the nucleus. A first, crude approximation for the curvature of the potential surface near the minimum is obtained by equating the expression \(V(a,n)\), up to an additive constant and a shift of the origin of the coordinates of \(\alpha\)-space, to the expression

\[ V_{\text{liquid drop}}(\alpha)=(O_p+O_h)S(\alpha)+V_e(\alpha). \]

For a fixed value of the deformation coordinate \(\alpha\), \(V\) may take different values depending on the individual states of the nucleons. The spatial arrangement of the corresponding potential surfaces qualitatively follows from the distribution of levels of a spherical nucleus consisting of \(A\) particles and having the same statistical relation between the density of eigenvalues of the total energy and the density of eigenvalues of the individual states of the nucleons \(^{21}\): a) the lowest level of the whole system is found by filling the individual particle levels (starting from below) until all \(A\) particles have been used (the kinetic energy for the highest state of one particle is \(F \sim 24\) Mev and for all particles together \(\sum E_n \sim \dfrac{3}{5}AF\)). b) The first several excited levels of the whole system are separated from one another by an amount of the same order as the mean spacing between the levels \(\Delta E_F\) of a single particle with excitation \(F\). c) For excitations \(E_N=\sum E_N\), exceeding \(\Delta E_F\) by several times but still much smaller than \(AF\) (where the collective model has long since ceased to be valid), the density of the system of levels grows roughly exponentially (neglecting possible power-law factors) with exponent

\[ \pi \sqrt{\frac{8E_N}{3\Delta E_F}}. \]

Consequently, the number of potential surfaces of the nucleus, just like the corresponding number of potential surfaces of a polyatomic molecule, grows ever faster with excitation of the nucleons in one case and of the electrons in the other.

The density of states \(\dfrac{dZ}{dE}\) in the collective nuclear model at the corresponding energies increases with energy even faster than the density of potential surfaces \(\dfrac{dZ_N}{dE}\), since there are many ways in which the total energy \(E=E_N+E_{\text{osc}}\) may

be distributed between the excitation of individual particles and collective oscillations:

\[ \frac{dZ}{dE} = \int_0^E \left(\frac{dZ}{dE}\right)_{E_N} \left(\frac{dZ_{\text{osc}}}{dE}\right)_{E-E_N} \,dE_N . \tag{7} \]

As Bergeland has shown \(^{22}\), the density of levels of surface oscillations increases with energy at a rate also determined by an exponential factor, which depends on the energy in approximately the same way as the factor for the density of nucleon levels. However, instead of comparing the two expressions for the level density, we combine them by means of the indicated integration. As a result, the exponent in the expression for the level density increases by approximately a factor of \(\sqrt{2}\) for those excitations which are not yet so large as to exceed the limits of applicability of the collective model. We shall not consider the level splitting associated with angular momentum, nor shall we make a comparison with the experimental data collected in the review by Blatt and Weisskopf \(^{21}\).

3. “Crossing” and “sliding”

An interesting and important property of the collective model is the connectedness of the many-sheeted potential surface. There are two possibilities:
a) The potential surfaces never intersect and do not even touch, provided only that the configurations of the boundary surface under consideration possess no symmetry—neither rotation, nor reflection, nor inversion. Let the point representing the system lie far from those sharply defined regions of \(\alpha\)-space which correspond to symmetric forms. Further, let jumps from one surface to another be excluded. Then the connectedness of the sheets is such that there are no paths in \(\alpha\)-space, even very winding ones, which could move the point representing the system from one sheet to another. The surfaces can be numbered in a unique way, and this canonical classification \(V_1(\alpha)\), \(V_2(\alpha), \ldots\) remains unchanged for any \(\alpha\), provided only that \(\alpha\) does not lie in the forbidden region of “symmetric forms.”

b) If the deformations possess one or several symmetries, then different energy surfaces intersect when one or another deformation parameter is varied. Consequently, there exist such \(\alpha\) for which two successive surfaces \(V_k(\alpha)\) and \(V_{k+1}(\alpha)\) touch one another. With the corresponding symmetries, an encounter of more than two surfaces is possible, and the point of contact has a higher order.

If the point representing the system is free in its choice of the values of \(\alpha\), then, with an appropriate choice of path, it can ascend

upward from surface to surface, sliding smoothly from \(V_k\) to \(V_{k+1}\) at one point of \(\alpha\)-space and from \(V_{k+1}\) to \(V_{k+2}\) at another point. This fundamental process, which we shall call “sliding,” has an analogue in the theory of polyatomic molecules, known as the “nonradiative transition.”\({}^{23}\) To describe the form of the potential surface near the crossing point, we shall also use the term “funnel.”

For a quantitative description of a many-sheeted potential surface it is necessary: a) to assign to each point of intersection the corresponding set of indices, indicating the number of surfaces that meet and the smallest value of the deviation \(\delta\alpha\) from the value corresponding to the point of intersection, at which the expression vanishes; b) to determine the magnitude of the coefficients in the first terms of the expansion of the energy around this point; c) to estimate the frequency distribution of transition points in \(\alpha\)-space for a typical potential surface; d) to determine the order of the height or depth of the vertex of the funnel (on the energy scale) relative to the surrounding potential surface; e) to determine the order of magnitude of the curvature of the given surface (as a whole), disregarding the funnels; this quantity is described by the curvature coefficients along the various axes of \(\alpha\)-space.

In order to discuss these points in detail, it is necessary to do somewhat more in this direction than has been done up to now. Nevertheless, considering this subject from various sides (Figs. 10–25) can give some idea of the state of affairs. It should be mentioned that in most of the idealized cases that we consider, it is necessary to pay attention to the behavior of the energy of an individual particle \(E_n(\alpha)\), and not to the behavior of the total potential vibrational energy \(V_k(\alpha)\) [equation (6)]. Of course, every crossing of surfaces causes a crossing of the surfaces \(E_n(\alpha)\), so that the connection between these two types of surfaces is not remote.

4. Review of the Behavior of Energy Levels

In Fig. 10 two energy levels are shown for a particle located in a rectangular potential well. The potential well has a high degree of symmetry, which is broken only by a small irregularity of the walls of the well. This irregularity is sufficient to prevent the crossing of energy surfaces, which otherwise would have occurred.

In the case under consideration the function has a definite number of nodal surfaces normal to each coordinate surface—a number that does not change throughout the entire deformation. The energy of the state decreases or increases according as the direction of propagation of the main

parts of the wave toward the direction of stretching, as should be expected from the analogous classical problem. It is interesting to make a comparison with the one-dimensional case, in which stretching in one direction lowers all energy levels; two levels never intersect. Such is the character of the behavior of the levels also in the three-dimensional problem in the absence of symmetry.

Figs. 11 and 12 relate the one-nodal curves \(E_n(a)\) to the general potential curves \(V_k(a)\). It is necessary, however, to point out that Fig. 12 presents a cross section of the energy surfaces along a specially chosen line in the space of deformations; namely, a cut has been chosen that intersects all the surfaces so as to show their intersections. If some irregularity were superposed on the deformations under consideration, then each intersection would be replaced by a curve analogous to the curve in Fig. 10.

Noteworthy is the qualitative connection between the “jagged” potential curve and the smooth curve following from the statistical conception of kinetic surface tension. In this simple example the potential energy and electrostatic energy have not been included in the energy of surface tension. One can see that the statistical conception of a constant surface tension is limited in accuracy.

It is easy to estimate the frequency of intersections of the energy levels of individual particles. Stretching the system in one direction by a relative amount \(a\) raises or lowers the energy levels \(F\) by an amount of order \(Fa\), according as the propagation vector is initially perpendicular or parallel to the direction of stretching. The mean distance between the levels of a spinless particle in the vicinity of the \(\dfrac{A}{4}\)-th level—this level has energy \(F\)—is of order \(\dfrac{8F}{3A}\) (Fig. 11). Consequently, increasing \(a\) by an amount of order \(\dfrac{1}{A}\) will on the average be sufficient for a given level to intersect once more a level initially situated above or below it, provided only that the symmetry of the surface permits an intersection.

Much can be said regarding the energy levels of a single particle located inside an ellipsoidal well. Inequality of the three semiaxes (Fig. 13) removes rotational symmetry and leaves only reflection and inversion symmetry. Figs. 14–18 illustrate the splitting of levels caused by a small ellipsoidal deviation from sphericity. Large deviations, which bring the sphere to fission, are considered in Figs. 19–22; here it is assumed that the wall of the well is axially symmetric, and we are dealing with a non-typical case of many intersections.

The relation between the levels of an individual particle and the levels of a system of particles is further illustrated in Fig. 23. This diagram shows how the quadrupole forces in nearly spherical nuclei increase to a maximum when the number of nucleons is such that almost half the shell is filled. The diagram presented becomes unsuitable when there are substantial deviations from symmetry, since then the levels change their character, as is seen from Figs. 10 and 19–22. Further information concerning the splitting of the levels of a single particle and the behavior, as a whole, of the levels of a system of particles is given in Figs. 24–25. Irregularities in the curve of the total potential energy of a many-particle system will hardly, in order of magnitude, be greater than the protrusions in the potential curve corresponding to the higher states of a single particle. The rise of the lower filled level in the form of an “inverted funnel” and the fall of the next higher level in the form of a “straight funnel” are perturbations which, when added together, to a considerable extent cancel one another.

IV. QUADRUPOLE MOMENTS OF GROUND STATES

In Fig. 26 the expected qualitative dependence of quadrupole moments on the degree of filling of shells is considered; it is assumed here that the deformations are not so large as to lead the system to a “funnel.”

In Fig. 27 the possible consequences are clarified of the existence of two well-separated minima in the potential surface of the ground state of the system under the same limiting conditions with respect to the magnitude of the deformations. Since the passage of a heavy nucleus through a high barrier of this kind apparently requires a considerably longer interval of time in comparison with characteristic nuclear times, bypassing the barrier in the $\alpha$–$\gamma$ plane (Fig. 28) through a sequence of triaxial ellipsoids may more readily occur. Our knowledge concerning potential surfaces corresponding to nuclear deformations is still too meager to permit a sufficiently well-founded discussion of the energies and lifetimes of such deformation-isomeric states.

In Fig. 29 experimental data are collected concerning the periodicity of nuclear quadrupole moments. It is still too early to discuss in detail the connection of the experimental data with the model under consideration, without having more detailed information on potential surfaces (see Ford’s article$^{58}$ on this subject).

Accurate measurements of the magnitude of quadrupole moments are of substantial importance for the interpretation of $\alpha$-decay phenomena, as indicated in Figs. 30 and 31. Quadrupole moments should also cause a broadening of the minima in nuclear diffraction scattering.

(effect which, evidently, can be observed if the incident beam is sufficiently limited in its spread of energy and direction and has a sufficiently small wavelength. Asymmetry in the shape of the nucleus is not, as is evident, an isolated phenomenon, but, on the contrary, is closely connected with important fundamental questions of nuclear physics. Apparently, for the development of nuclear physics it is important to study the quadrupole moments of a considerably larger number of nuclei, especially heavy nuclei and $\alpha$-emitters. In this connection, diffraction scattering, anomalies in the periods of $\alpha$-decay (Fig. 31) and in isotope shifts$^{24}$, and the splitting of the levels of mu-mesonic $K$-radiation$^{25}$ constitute additional means for increasing the experimental data.

(To be concluded in the next issue)

ADDENDUM.

FIGURES AND COMMENTS

Figure 1

The first three orders of small oscillations of a liquid moving under the influence of surface-tension forces are shown. The addition of a uniform volume electric charge, characteristic of the liquid-drop model,

Fig. 1. Independent types of small oscillations of a liquid drop.

drop, does not affect the form of the orthogonal oscillations, but changes their frequency somewhat, since the decrease of Coulomb energy partly compensates the increase of surface energy when the figure of the drop deviates from an equilibrium spherical shape.

To estimate the frequencies of these oscillations, for example, for a model of the uranium nucleus, let us note that these oscillations are represented (with the exception of the term

with index \(i=1\), corresponding only to a displacement of the center of mass) in the expression

\[ R(\mu)=a_0\left[1+\sum_{i=1}^{N} a_i P_i(\mu)\right] \]

for the distance \(R\) from the center of mass to the surface of an axially symmetric nucleus in terms proportional to Legendre polynomials in \(\mu=\cos\theta\) (\(a_0\) is chosen for normalization of the volume). Having found the coefficients in the quadratic form of the kinetic and potential energies of small oscillations, we find the frequency of oscillation of order \(n\) in the form:

\[ \nu_n=A^{-1/2}\left[ \frac{O}{3M_p r}\, n(n-1)\, \frac{(2n+1)(n+2)-20x}{2n+1} \right]^{1/2}, \]

where \(M_p\) is the proton mass, and the remaining notation is given in the caption to Fig. 2. Quantization of the surface oscillations gives for the zero-point energy

\[ \left\langle E_n\right\rangle_{\text{zero}} = \frac{1}{2}h\nu \]

and for the mean square value of the zero-point amplitude

\[ \left\langle a_n^2\right\rangle_{\text{zero}} = A^{-7/6} \left\{ \frac{\hbar^2}{12M_p r_0^2(4\pi r_0^2 O)} \times \frac{n(2n+1)^3}{(n-1)\left[(2n+1)(n+2)-20x\right]} \right\}^{1/2}. \]

Calculating these quantities for oscillations of the first three orders, we obtain:

\(n\)	\(\nu_n\) in sec\(^{-1}\)	\(\left\langle E_n\right\rangle_{\text{zero}}\) in MeV	\(\left\langle a_n^2\right\rangle_{\text{zero}}^{1/2}\)
2	\(2.15\cdot 10^{20}\)	0.45	0.064
3	\(6.21\cdot 10^{20}\)	1.29	0.054
4	\(10.78\cdot 10^{20}\)	2.23	0.053

It is necessary to note that the mean square quantities \(a_n\) given in the table are approximately 5–6 times smaller than the quantities corresponding to the orthogonal oscillations shown in the figure. The latter were calculated under the assumption \(a_n=0.3\), with all the other \(a_i'=0\).

The infinite sequence of possible oscillations of a liquid drop must be limited, for an atomic nucleus, by a number \(n\) lying between 6 and 10, in view of the finite number of nucleons making up the nucleus. Defining the smearing of the nuclear surface in the collective model of the nucleus in the ground state by the formula

\[ \alpha_{\text{eff}} = \left[ \sum_{i=2}^{N} \left\langle a_n^2\right\rangle \right]^{1/2}, \]

we obtain an overall uncertainty in the position of the surface of the order of 15% of the nuclear radius.

Figure 2

The critical figures of unstable equilibrium presented depend (in the approximation of the simple liquid-drop model) only on the ratio of the square of the charge to the first power of the mass number, or, in a more convenient form, on the dimensionless parameter

\[ x=\frac{(\text{charge})^2}{10\times \text{volume}\times \text{surface tension}} = \]

\[ =\frac{Z^2e^2}{10A\cdot \frac{4}{3}\pi r_0^3 O} = \frac{(Z^2/A)}{(Z^2/A)_{\text{lim}}}, \]

where

\[ (Z^2/A)_{\text{lim}} = 2\,\frac{4\pi r_0^2O}{3\,\frac{e^2}{5\,r_0}} \simeq 47.8 . \]

For the imaginary nucleus “cosmium,” lying far beyond the known limit of stability, \(x=1\), and the nucleus will be unstable with respect to fission even for a spherical figure. For \(x\) close to unity, the critical figures are described by the equation\(^4\)

\[ R=R_0\sum a_n P_n(\cos\theta), \]

where \(R_0\) is the initial radius, \(a_0=1\), \(a_2=\dfrac{7}{3}(1-x)\), and the remaining coefficients \(a\) are negligibly small.

For four values of \(x\) the critical figures were calculated by Frankel and Metropolis\(^27\). The sequence of figures shown in the drawing was found by interpolation and extrapolation of their data.

Fig. 2. Critical figures of unstable equilibrium.

Each figure is an equilibrium figure for the corresponding nucleus. It is also possible to regard this sequence of figures as a sequence of changes of the figure of a specified nucleus, i.e., a nucleus with fixed \(x\), in the course of possible deformations. In this sense the figures shown should not be regarded as equilibrium figures. From this point of view it is convenient to characterize each individual figure by the magnitude of the parameter

\(y = 1 - x\). Then \(y\) describes the shape of the figure, and \(x\) is the ratio of the square of the charge to the mass for that drop which has the prescribed critical figure; we may, consequently, speak of the deformation \(y\) for systems with different \(x\).

The curves were computed from interpolation and extrapolation formulas:

\[ a_0 = 1 - y^2 \left[ 1.06 + \frac{9.76 \cdot 10^{-4}}{(0.49 - y)^4} \right], \]

\[ a_2 = y \left[ 2.3 + \frac{5.42 \cdot 10^{-4}}{(0.49 - y)^4} \right], \]

\[ a_4 = y^2 \left\{ 1.6 + y \left[ 3.0 + \frac{2.84 \cdot 10^{-3}}{(0.49 - y)^4} \right] \right\}, \]

\[ a_6 = - \frac{2.36 \cdot 10^{-5}}{(0.49 - y)^4}, \]

\[ a_8 = - \frac{4.72 \cdot 10^{-5}}{(0.49 - y)^4}, \]

the remaining \(a_i\) are equal to zero. It is possible that, for values of \(x\) close to 0.65, the symmetric equilibrium figure does not correspond to the lowest minimum in the multidimensional dependence of the energy on the shape. It may be that two asymmetric figures, which are mirror images of one another, correspond to lower values. It is also possible that the compressibility of the nucleus and the redistribution of neutrons and protons between the peripheral and internal parts of the nucleus may change both the figures themselves and the values of \(x\) corresponding to the predominant formation of asymmetric figures. These effects have the smaller influence the closer \(y\) is to zero, if only one assumes \(y = 1 - x^*\). Here \(x^* = x + z\), where \(x\) has the definite value given above and \(z\) is a measure of the compressibility and redistribution effect. The approximate value of Finberg, Stewart, and Swiatecki \(^7\) equal, for nuclei in the uranium region, to a value of about 0.04 (with an accuracy to within a factor of about two). The introduction of the quantity \(z\) requires changing the value \((Z^2/A)_{\mathrm{lim}} = 47.8\) so that the height of the fission barrier remains unchanged.

Figure 3

Each curve of this family corresponds to the ratio of the deformation energy to the total surface energy of the sphere for the deformation shapes of Fig. 2. The different curves correspond to nuclei with different \(x\). It should be noted that in the general formula

\[ \xi(x,y) = 2.178(1-x)y^2 - 4.09(1-0.645x)y^3 + \]

\[ +\,18.64(1-0.894x)y^4 - 13.33y^5 \]

the value of \(y\) corresponding to the maximum value of \(\xi\) for a given \(x\) corresponds to the critical figure of unstable equilibrium of the given nucleus. Consequently, substituting \(y = 1 - x\), we obtain an expression for the fission-threshold energy as a function of \(x\)

\[ \xi_{\max} = 0.728(1-x)^3 - 0.661(1-x)^4 + 3.330(1-x)^5. \]

Consideration of the surfaces of the deformation potential (see Fig. 4) shows that, for the sequence of shapes defined by the values \(y\),

Fig. 3. Dependence of the energy on deformation.

the curves \(\xi(x,y)\) are an approximation to the shortest path toward fission along a “valley” on the potential surface corresponding to the given \(x\).

Figure 4

The sequence of shapes leading the uranium nucleus to fission is shown in connection with the potential surfaces corresponding to deformations in the \(a_2\)—\(a_4\) plane (see Fig. 1) (borrowed from the article by Frankel and Metropolis\(^{27}\)). The sequence of events leading to fission is described as follows. When a neutron is captured, the uranium nucleus is transformed into an excited intermediate nucleus, in which the excitation energy is distributed between the internal motion of the individual nucleons and the collective motion of the entire nucleus. Typical components of the second type of motion are represented in Fig. 1. The collective motion may be represented as the motion of the point representing the system over the energy hypersurface. The surface shown in the figure is a projection of the hypersurface onto the \(a_2\)—\(a_4\) plane. The point representing the system describes Lissajous figures with an amplitude that changes as the energy of the nucleus passes from one form of excitation (internal) to another (collective).

Consider the case in which the energy of excitation only slightly exceeds the height of the barrier on the potential surface corresponding to the fission threshold. Such a case occurs when uranium captures a thermal neutron. Then fission cannot occur until 1) almost all the excitation is concentrated in the collective motion, 2) the phases and amplitudes of the various types of collective motion combine in such a way as to carry the point representing the system along the “valley”

potential surface toward the saddle. It is clear that, owing to the large number of degrees of freedom of the internal and collective motions, the system will pass through many “nuclear epochs” before the capture of the slow neutron leads, on the average, to fission. The same considerations also apply to the competing process of neutron or \(\gamma\)-quantum emission.

If, however, the point representing the system passes over the saddle, then the increase of surface energy upon further elongation will become

In the figure: dependence of the total energy on the parameters \(\alpha_2,\alpha_4\), for \(x=0.74\) (in relative units).

Fig. 4. Sequence of events in fission.

greater than the compensating decrease of the Coulomb energy, and the motion will “self-accelerate” until fission occurs. The resulting fragments will be strongly excited by both internal and collective forms of excitation and will emit neutrons and \(\gamma\)-quanta.

Figure 5

Neither the liquid-drop model nor the model of independent particles can explain the magnitudes of the quadrupole moments or the asymmetry in the distribution of electric charge over typical nuclei. A negative quadrupole moment arises in the liquid-drop model if it is assumed that the uniformly charged nucleus as a whole rotates with angular momentum \(l\). For a nucleus of the size of uranium, minimizing the sum of the deformation energy (Fig. 3) and the rotational energy, we find \(n\):

\[ -\frac{Q}{eR^2} = \frac{6\alpha Z}{5} = -\frac{l^2\,2.8\,ZA^{-7/3}}{1-x}. \]

Since the nuclear spin rarely exceeds 4, the magnitude of the quadrupole moment in this model would have to depend sharply on the magnitude of the orbital angular momentum of the particle, which, according to experimental data, varies within the range from 1 to 10 units. Moreover, in the liquid-drop model it is difficult to imagine a mechanism leading to an elongation of the drop, i.e., to a positive quadrupole moment.

The independent-particle model also cannot explain the magnitudes of the observed quadrupole moments. The shaded annular region is the region of large amplitude of the wave function of the single proton that is in excess relative to the configuration of closed

Simple spherical liquid drop

Absence of a quadrupole moment

Independent-particle model. The quadrupole moment is caused by an asymmetric distribution of the charge of several particles. The effect is too small

The wave function of the excess particles is large

Collective model. The pressure of the excess particles distorts the shape of the drop. Quadrupole moments increase by an order of magnitude

Fig. 5. Quadrupole moments and nuclear models.

shells (spherical symmetry). The quadrupole moment corresponding to it in typical cases is 5–15 times smaller than the observed one. From the wave function \(\Psi_n(r)\) of a particle in a spherical potential well, the quadrupole moment can be computed by assuming that the charge density, depending on the distance of the proton, is proportional to \(|\Psi_n(r)|^2\). Taking the total magnitude of the charge thus determined to be equal to unity, we find that the largest value of the relative quadrupole moment \(\left(\dfrac{Q}{eR^2}\right)\), even for large angular momenta and any quantum numbers \(n\), is 0.5, while the expected average value is 0.2. Thus, if even 4 or 5 nucleons simply added their contributions, then even in this case one could not explain the observed magnitudes.

However, Rainwater\(^{11}\) pointed out that the pressure of a nucleon on the surface of the nucleus acts against the forces of surface tension and can produce a noticeable deformation. The resulting large displacement of charge leads to an increase of the quadrupole moment by an order of magnitude compared with the value due directly to the extra particle. Thus, the study of quadrupole moments gives us the first convincing evidence for the possibility of interaction of nucleons with the surface, which is the basic assumption of the collective model of the nucleus.

Figure 6

The upper drawing presents the dependence of the energy of an individual particle on static deformation. If the angular momentum is parallel to the axis, then the waves propagate azimuthally and the wavelength increases for a flattened configuration.

The lower drawing presents the dependence of the sum of the energies of a peripheral particle and the energy of the remaining filled shells of the nucleus on deformation; the parabolas shown are calculated on the basis of the following idealization:
\((\text{energy}) = 2/5 \times (\text{radius})^2 \times (\text{surface tension}) \times (\text{deformation}, \alpha)^2\). The sign of the resulting quadrupole moment changes depending on the magnitude of the projection of the angular momentum on the symmetry axis.

Text in the diagram:

Kinetic energy of a single particle moving freely inside a spheroidal pit (inversely proportional to the square of the mean wavelength)

Oblate; large \(\alpha^2\).
Circular orbit.
The pressure is great in the equatorial plane.

Short waves

Long waves

Energy

\(|m|=l\)

\(m=0\)

Waves move parallel to the axis.
Pressure at the poles.

Total energy: kinetic energy of a single particle in an unfilled shell plus the surface-tension energy corresponding to the deformation of a liquid drop.

Energy

\(|m|=l\)

\(m=0\)

Negative quadrupole

Positive quadrupole

Deformation

Fig. 6. Deformations caused by an excess neutron.

Figure 7

In the case of a fixed boundary the wave function is equal to

\[ \left(\frac{2}{L}\right)^{1/2}\sin\frac{n\pi x}{L}\exp\left(-i\int^t E\frac{dt}{\hbar}\right), \]

where

\[ E=\frac{n^2\pi^2\hbar^2}{2ML^2}=\text{const.} \]

In the case of a slowly moving wall, the expression is multiplied by a factor of the known dependence of \(L\) on \(t\), and by the factor \(\exp\left(-\dfrac{M i \varphi(x,t)}{h}\right)\). Here \(\varphi(x,t)\) is the classical velocity potential; its derivative \(v(x,t)=\dfrac{\partial\varphi}{\partial x}\) is equal to the velocity with which the classical gas (with the speed of sound infinitely large) responds to the motion of the boundary. Owing to the term containing the velocity potential, the wave function satisfies the continuity equation

\[ \frac{\partial}{\partial x}\left\{\frac{\hbar}{2Mi}\left(\psi^*\frac{\partial\psi}{\partial x}-\psi\frac{\partial\psi^*}{\partial x}\right)\right\} +\frac{\partial}{\partial t}(\psi^*\psi)=0. \]

It describes the motion of matter to the right when the boundary moves in the same direction. The resulting wave function satisfies the wave equation, free of the velocity potential, in the space between the limiting walls, if the distance \(L\) changes linearly with time. In the case of a more general dependence of \(L\) on time, the wave function satisfies the equation

\[ i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2M}\frac{\partial^2\psi}{\partial x^2} -\frac{Mx^2\ddot L}{2L}\psi . \]

Here the additional term on the right is negligibly small for small accelerations of the confining wall. In this approximation the effect of the motion of the wall on the state of the particle is completely described by a factor containing the classical velocity potential.

Fig. 7. Influence of the motion of the boundary on the state of a particle in the one-dimensional case.

Figure 8

We shall prove here that the energy of several identical particles enclosed within a slowly varying surface that bounds a constant volume is, at each given instant of time, equal to the energy of the particles for a fixed surface plus the kinetic energy of an incompressible fluid of the same mass \(M\), brought into irrotational motion by the same bounding surface. Let \(x\) denote the three spatial coordinates of one particle, and let \(\alpha\) determine the configuration of the boundary,

Deformation shifts the nodes.

Fig. 8. Influence of the motion of the boundary on the state of a particle in the three-dimensional case.

\(f(x,a)\) and \(E(a)\) are the wave function and energy for a fixed boundary, and \(\varphi(x,a,\dot a)\) is the velocity potential of the classical motion of the fluid;

\[ \frac{\partial \varphi}{\partial t}-\frac{1}{2}(\nabla\varphi)^2-\frac{p}{\rho}=0;\qquad \nabla^2\varphi=0; \]

\[ -\frac{\partial\varphi}{\partial n} \]

denotes the normal velocity of the boundary. Let us represent the wave function of the particles in the approximate form:

\[ \psi(x,t)=f(x,a(t))\exp\left[-i\int^t \frac{E(a(t))\,dt}{\hbar-\dfrac{iM\varphi}{\hbar}}\right] =f(x,a(t))\exp[*]. \]

Substituting this function into the Schrödinger equation, we obtain:

\[ i\hbar\frac{\partial\psi}{\partial t} =\exp[*]\left[i\hbar\dot a\,\frac{\partial}{\partial a}+E+M\frac{\partial\varphi}{\partial t}\right]f . \]

Let us now assume that, in the first approximation, the nodes and the quantities \(f\) are transported with the classical velocity of the fluid in such a way that

\[ \dot a\,\frac{\partial f}{\partial a}=(-1)^2\nabla\varphi\cdot\nabla f . \]

Then

\[ i\hbar\frac{\partial\psi}{\partial t} =\exp[*]\left[ i\hbar\nabla\varphi\cdot\nabla f -\frac{\hbar^2}{2M}(\nabla^2 f) +\frac{Mf}{2}(\nabla\varphi)^2 +\frac{Mp}{\rho}f \right] = -\frac{\hbar^2}{2M}\nabla^2\psi+\frac{Mp}{\rho}\psi . \]

We see that the wave function under consideration satisfies approximately the Schrödinger equation in a cell dependent on velocity; in it one may neglect only terms containing the acceleration in the form \((\text{pressure}/\text{density})\). The kinetic energy of the state is equal to:

\[ \frac{\hbar^2}{2M}\int(\nabla\psi^*)(\nabla\psi)\,d\Omega = \]

\[ =\frac{\hbar^2}{2M}\int(\nabla f^*)(\nabla f)\,d\Omega +\frac{i\hbar}{2}\int(f^*\nabla f-f\nabla f^*)\,\nabla\varphi\,d\Omega +\frac{M}{2}\int(\nabla\varphi)^2 f^*f\,d\Omega, \]

where \(\Omega\) is the volume.

It may be shown that the last term in this expression (summed over all occupied states), for large quantum numbers, passes into the kinetic energy of the classical fluid. The first term represents the energy of the particles in the static potential well. The second term represents the interaction of unpaired particles with the rotation of the boundary.

Figure 9

The general principles are illustrated by the example of a one-dimensional potential well extending from \(x=0\) to \(x=a=R_0(1+\alpha)\) or to \(x=b=R_0(1+\alpha)(1+\varepsilon)\), according as we are dealing with one or another value of \(\alpha\). The determinant wave function \(\Phi\) of a system of \(N\) particles is composed of the wave functions of the individual particles,

\[ u(n,x)=\left(\frac{2}{a}\right)^{1/2}\sin\frac{n\pi x}{a}. \]

The matrix element of the wave functions of one

and of the same nucleon state, but for slightly different \(a\), is equal to:

\[ \sum_{\substack{\text{permutations}\\ 1,2,\ldots,N\\ \alpha,\beta,\ldots,\gamma}} (-1)^P \int u(n_1,x,a)u(n_\alpha,x,b)\,dx \times \cdots \times \int u(n_N,x,a)u(n_\gamma,x,b)\,dx, \]

where summation over all permutations \(P\) of the indices \(n_1,\ldots,n_N\) of the occupied states is understood. If the indices in one of the integrals are the same, then the approximate value of the integral is

\[ 1-\varepsilon^2\left[\frac{1}{8}+\frac{n^2\pi^2}{6}\right], \]

and if the indices are different, then this value is equal to

\[ \varepsilon(-1)^{n+m}\frac{2mn}{m^2-n^2}. \]

Representing the factors \((1-\varepsilon_1)(1-\varepsilon_2)\ldots(1-\varepsilon_N)\) in the form of the exponent \(\exp(-\varepsilon_1-\varepsilon_2\ldots-\varepsilon_N)\), we obtain for the value of the scalar product \(\int\ldots\int \Psi_a\Psi_b\,d\Omega\) the approximate formula

\[ \exp\left[-\varepsilon^2\left\{\frac{N}{4}+\frac{41}{288}+\frac{(6N^2+6N+1)\ln 3.526N}{12}\right\}\right], \]

from which follows the result cited in the text. Thus, the wave functions of the one-dimensional system are approximately orthogonal for

Fig. 9. Approximate orthogonality of the nucleon wave functions of equation (3) in the case of two slightly different values of the deformation parameter \(a\).

relative stretching \(\varepsilon \sim N^{-1}(\ln 3.562N)^{-1/2}\), or relative change of the volume \((\ln 3.562N)^{-1/2}\) of the cell occupied by a typical particle. In generalizing to three dimensions we assume that the wave functions of a system of nucleons become practically orthogonal as soon as the volume, not common to both configurations, exceeds some small fraction of one cell.

Figure 10

Let us consider the mixing of two states of a particle in a potential well having the form of a rectangular box with a small irregularity, under the condition that two dimensions of the box are varied in such a way that its volume remains unchanged. The figure shows the nodal lines of the common wave function for successive stages of deformation. In the absence of an irregularity of the boundary, the two levels cross without interaction in the process of deformation. Owing to the existing irregularity, the wave functions are mixed in a comparable proportion only near the crossing point. The nodes were found by solving the equation
\(R\sin 4x\cdot\sin z+\sin x\times\sin 5z=0\), where \(x\) and \(z\) are distances from the lower corner of the rectangle, normalized so that \(x=\pi\) and \(z=\pi\) for the diagonally opposite corner.

Fig. 10. Influence of a small asymmetry on the mixing of two states that are orthogonal in the absence of asymmetry.

Figure 11

We shall “derive” here the asymptotic formula for the number of solutions of the equation \(\nabla^2\psi+k^2\psi=0\), \(\psi_{\mathrm{surf}}=0\), with wave numbers in the interval from \(k\) to \(k+dk\):

\[ dN=\frac{Vk^2}{2\pi^2}\,dk-\frac{Sk}{8\pi}\,dk+\frac{\int \gamma\,dS}{8\pi^2}\,dk . \]

Here \(V\) is the volume of the region under consideration, \(S\) is its surface, and \(\gamma\) is the total local curvature of the surface. Consider the case of a rectangular parallelepiped with dimensions \(a,b,c\). To each eigen-solution
\(\sin k_xx\cdot\sin k_yy\cdot\sin k_zz\) there corresponds a point in \(k\)-space with coordinates

\[ k_x=\frac{l\pi}{a},\qquad k_y=\frac{m\pi}{b},\qquad k_z=\frac{n\pi}{c}, \]

with which is associated the volume of an elementary cell

\[ \frac{\pi}{a}\cdot\frac{\pi}{b}\cdot\frac{\pi}{c}. \]

When the states are filled up to a prescribed wave number \(k_{\max}\), the elementary cells fill the octant of a sphere (left figure), except for small regions near the coordinate surfaces. The excluded regions are shown in more detail in the right figure (in the case where the number of states between \(k\) and \(k+dk\) is considered). Their

the volume is equal to the volume of the shell minus a correction for the volume of annular strips, plus a correction for the volume of corners, subtracted twice in calculating the volume of the rings:

\[ \frac{4\pi}{8}k^2\,dk-\frac{\pi}{2}k\,dk\left[\frac{\pi}{2a}+\frac{\pi}{2b}+\frac{\pi}{2c}\right]+ \]

\[ +\,dk\left[\frac{\pi}{2a}\cdot\frac{\pi}{2b}+\frac{\pi}{2a}\cdot\frac{\pi}{2c}+\frac{\pi}{2b}\cdot\frac{\pi}{2c}\right]. \]

Dividing by the volume of an elementary cell, we obtain the number of states:

\[ dN \simeq abc\,k^2\,\frac{dk}{2\pi^2}-(2ab+2bc+2ac)\frac{k\,dk}{8\pi}+(4a+4b+4c)\frac{dk}{16\pi}. \]

Generalization of this expression to a cavity of not too irregular shape gives the formula quoted above. Then the number of states with wave number less than \(k\) is equal to:

\[ N \simeq \frac{Vk^3}{6\pi^2}-\frac{Sk^2}{16\pi}+\frac{Lk}{8\pi^2}, \]

and their total kinetic energy \(E\) is given by the expression

\[ \frac{2ME}{\hbar^2} = \frac{Vk^5}{10\pi^2} - \frac{Sk^4}{32\pi} + \frac{Lk^3}{24\pi^2} = \frac{V}{10\pi^2}\left(\frac{6\pi^2N}{V}\right)^{5/3} + \frac{S}{32\pi}\left(\frac{6\pi^2N}{V}\right)^{4/3} + \]

\[ + \frac{6\pi^2N}{V}\left(\frac{S^2}{128V}-\frac{L}{12\pi^2}\right). \]

The proportionality of part of this energy to the surface area gives a contribution to the surface tension of kinetic origin\(^{73,61}\). For nuclei having approximately equal numbers of neutrons and protons and volume per particle \(\frac{4}{3}\pi r_0^3\), the corresponding term in the surface tension is equal to

Fig. 11. Asymptotic frequency of eigenvalues.

\[ O_{\mathrm{kin}}=\frac{4\hbar^2}{64\pi M}\left(\frac{9\pi}{8r_0^3}\right)^{4/3}, \]

i.e.

\[ 4\pi r_0^2 O_{\mathrm{kin}}=28\ \text{MeV}. \]

This calculated value is twice as large as the experimental value \(14\ \text{MeV}\), equal to the sum of the kinetic and potential terms

in the surface tension. However, the calculated value is reduced to a more reasonable magnitude if the depth of the potential well is taken not as infinite, but finite[^74].

After the article had been written by us, Prof. Feinberg kindly acquainted us with Hammaq’s dissertation[^75], in which the asymptotic density of levels of independent particles in a potential well of infinite and finite depth is considered, and a three-term asymptotic formula, given above, is proposed.

Figure 12

The deformation energy depends on the nucleon state of the system, as well as on the deformation amplitude. Each potential curve in the diagram is obtained by summing the energies of 60 particles occupying 60 different states.

Labels visible in the figure:

• Potential curves for rapid deformations
• \(n\)
• Number of occupied states for a given \(n\)
• Surface-tension formula of second order with correction for curvature
• Minimum deformation potential
• Deformation energy for a rectangular box

\[ E_{lmn}=\frac{\hbar^{2}}{2M}\left\{(l^{2}+m^{2})e^{\alpha}+\frac{n^{2}}{e^{2\alpha}}\right\} \]

\[ \frac{2M}{\hbar^{2}}\left\{V(\alpha)=\sum E_{lmn}\right\} \]

• Dependence on surface tension
• 60 particles, fermions without spin
• \(\pi e^{\alpha}\)
• \(\pi e^{-\alpha/2}\)
• \(\pi e^{-\alpha/2}\)
• \(\alpha\)—deformation amplitude

Fig. 12. Potential curves of a deformed rectangular box.

different states. In order for the system of particles to retain a minimum value of the energy, it is necessary that the distribution of particles over states change during the deformation process. The jagged curve gives a qualitative representation of the type of potential curve of real nuclei. If the deformation occurs so rapidly that the probability of transition of the system from one potential curve to another is small, or such transitions are altogether forbidden owing to the high symmetry of the system, as in the case

box with smooth rectangular walls, then the distribution of particles over states will not change in the course of deformation. Then the energy needed for the perturbation will be much greater than the energy determined from surface tension. The dashed curve is obtained by applying statistical considerations and the formula of Fig. 11 to the present problem for \(N=60\), volume \(\pi^3\), surface \(2\pi^2 e^{-\alpha}+4\pi^2 e^{\alpha/2}\), and mean curvature \(2\pi^2 e^\alpha+4\pi^2 e^{-\alpha/2}\)

\[ \sum \frac{2ME}{\hbar^2} = \frac{\pi}{10}\left(\frac{360}{\pi}\right)^{5/3} + \frac{\pi}{16}\left(\frac{360}{\pi}\right)^{4/3} \left(e^{-\alpha}+2e^{\alpha/2}\right) + \frac{45}{4}e^{-2\alpha} + \left(45-\frac{120}{\pi}\right)e^{-\alpha/2} + \left(45-\frac{60}{\pi}\right)e^\alpha . \]

Figure 13

Ellipsoidal deformation with preservation of volume can be represented in polar coordinates on the plane: \(\alpha\) is the deformation amplitude,

Fig. 13. Coordinates \((\alpha,\gamma)\) of an ellipsoidal deformation preserving volume.

\(\gamma\) is the shape parameter. These coordinates are related to the principal semiaxes of the ellipsoid

\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \]

by the equations:

\[ a=R_0\exp\left[\alpha\cos\left(\gamma-\frac{2\pi}{3}\right)\right], \qquad b=R_0\exp\left[\alpha\cos\left(\gamma+\frac{2\pi}{3}\right)\right], \]

\[ c=R_0\exp[\alpha\cos\gamma], \]

which satisfy the condition of constancy of volume \(abc=R_0^3\). Thus, for example, a deformation consisting in changing the axes \(x,y,z\) by \(-1\%\), \(-2\%\), \(+3\%\), or by some small amount proportional to these quantities, is described by a positive value of the deformation amplitude \(\alpha\) and the shape parameter \(\gamma=10.895^\circ\).

Another example: let \(\alpha\) be a given positive number, for example \(0,02\), and let \(\gamma\) increase, starting from \(\gamma=0^\circ\). Then the ellipsoid changes in the following way:

\(\gamma\)	\(\dfrac{a}{R_0}\)	\(\dfrac{b}{R_0}\)	\(\dfrac{c}{R_0}\)	Shape	Axes of symmetry
\(0^\circ\)	0,990	0,990	1,020	prolate spheroid	\(x\)
\(30^\circ\)	1,000	0,983	1,017	ellipsoid	none
\(60^\circ\)	1,010	0,980	1,010	oblate spheroid	\(y\)
\(90^\circ\)	1,017	0,983	1,000	ellipsoid	none
\(120^\circ\)	1,020	0,990	0,990	prolate spheroid	\(z\)

With a further increase of \(\gamma\), the cycle is repeated under the condition of a cyclic permutation of the indices \(x, y, z; a, b, c\).

Figure 14

In considering this effect we neglect the spin-orbit coupling. This diagram and several of the following figures should be regarded as illustrative.

Fig. 14. Qualitative picture of the influence of small ellipsoidal deviations from sphericity on several of the first levels of a single nucleon.

They may be interpreted as illustrative. The deformation of the ellipsoid is expressed in polar coordinates \(\alpha\) and \(\gamma\) (Fig. 13). The simplest case is the deformation

into an elongated spheroid, axially symmetric about the \(z\)-axis (\(\gamma = 0\)). Then the quantum number \(m\) has a definite meaning. A level with orbital angular momentum \(l\hbar\) is split into sublevels separated from the original level by the amount

\[ \delta E = 2\alpha E \frac{3m^2 - l(l+1)}{(2l-1)(2l+3)}, \]

which depends on the quantum number \(m\). The levels with \(m = +|m|\) and \(m = -|m|\) coincide. The diagram does not show the following effects:

1) The effect of displacements on the energy surfaces is so large that \(\delta E\) is no longer proportional to \(\alpha\). Then terms of order higher than the first must be taken into account. The resulting curvature of the energy surfaces corresponds to the repulsion of the surfaces from one another everywhere, except in the neighborhoods of special points. At these points two surfaces usually form a double funnel, as shown in Fig. 35. On an enlarged diagram many such funnels would be visible.

2) The dependence of the energy on deformations of order higher than the second is not shown here because of lack of space. In studying the asymmetry of nuclear fission it is necessary to consider deformations of third order (\(n = 3\)), as well as the ellipsoidal deformations (\(n = 2\)) presented in this figure.

Figure 15

The diagram shows the displacement of the level \(\delta E\), corresponding to a small deformation \(\alpha\), as a function of the parameter \(\gamma\). \(E\) is the kinetic energy.

Fig. 15. Splitting of the levels of a \(p\)-particle (orbital angular momentum \(l = 1\)) in a potential well of constant depth in the case when its boundaries have received a small ellipsoidal deformation with conservation of volume.

of the nucleon in the original spherical potential well, and \(\alpha\) and \(\gamma\) are defined in Fig. 13. Note that the \(p\)-particle has the minimum possible energy when it is a constituent part of a nucleus having the form of an elongated spheroid.

Figure 16

Five sublevels merge into three sublevels under an axially symmetric deformation \((\gamma = 0^\circ,\ \pm 60^\circ,\ \pm 120^\circ\), etc.). The energy of a single \(d\)-nucleon corresponding to the lowest sublevel, in the first approximation, does not depend on the shape parameter \(\gamma\). The case under consideration \((l = 2)\)

Fig. 16. Splitting of the levels of a \(d\)-particle (orbital angular momentum quantum number \(l = 2\)) as a function of the parameter \(\gamma\) of ellipsoidal deformation.

is, in a known sense, neutral. For \(l = 1\), an elongated deformation gives the greatest lowering of the energy; for \(l = 3\) or more, a flattened deformation is energetically more favorable.

Figure 17

A single \(g\)-particle assumes the minimum possible energy in a flattened ellipsoid \((\gamma = 0,\ \pm 60^\circ,\ \pm 180^\circ)\). The calculation of the magnitude of the energy shifts

Fig. 17. Removal of the ninefold degeneracy of a \(g\)-nucleon \((l = 4)\) under a small ellipsoidal deformation.

in the first approximation is carried out as follows: let \(u_1\) be a wave function that vanishes on the surface of the sphere and satisfies

the equation \(\nabla^2 u_1 + k_1^2 u_1 = 0\) inside the sphere; let a slightly different function \(u_2\) vanish on the surface of the ellipsoid and satisfy the equation \(\nabla^2 u_2 + k_2^2 u_2 = 0\) inside the ellipsoid. Then, according to Green’s equation,

\[ k_2^2 - k_1^2 = \frac{\displaystyle \int \left( u_2^{*}\frac{\partial u_1}{\partial n} - u_1 \frac{\partial u_2^{*}}{\partial n}\right)\, dS} {\displaystyle \int u_1 u_2^{*}\, d\Omega}, \]

where the integral in the denominator is taken over the volume common to both bodies, and the integral in the numerator is taken over the surface enclosing this common volume. Let \(S_1\) be the portion of the surface of the sphere lying inside the ellipsoid. Then on \(S_1\), \(u_1=0\), and, introducing the normal displacement \(\delta n\) of the surface of the ellipsoid \(S_2\) relative to the surface \(S_1\), we find:

\[ u_2 = -\delta n \frac{\partial u_2}{\partial n} = -\delta n \frac{\partial u_1}{\partial n}. \]

The same result will hold if \(S_1\) lies outside the surface \(S\). Neglecting terms of order higher than the first, we obtain:

\[ \delta(k^2) = -\frac{\displaystyle \int \left| \frac{\partial u_1}{\partial n}\right|^2 \delta n\, dS} {\displaystyle \int |u_1|^2\, d\Omega}, \]

provided only that the difference between \(u_2\) and \(u_1\) is small, i.e. if we choose for \(u_1\), as is done below, an appropriate linear combination of the wave functions of the original, degenerate levels. We denote the corresponding linear combination of spherical harmonics by

\[ Y(\theta,\varphi)=\sum c_m Y_l^{(m)}. \]

Let, further, \(F_l(\rho)\) denote the regular solution of the radial equation

\[ \frac{d^2 F_l}{d\rho^2} + \left[ 1-\frac{l(l+1)}{\rho^2} \right]F_l=0, \]

so that

\[ \int_0^\rho F_l^2\, d\rho = \frac{1}{2}\rho\left[(F_l)^2+(F_l')^2\right] -\frac{1}{2}F_l F_l' -\frac{l(l+1)}{2\rho}F_l^2 = \frac{1}{2}\rho(F_l')^2, \]

if \(\rho\) is a zero. Then the unperturbed wave function is

\[ u_1=\frac{1}{r}F_l(kr)Y(\theta,\varphi), \]

and the perturbation of the energy of first order is equal to

\[ \frac{\delta E}{\alpha E} = \frac{\delta(k^2)}{\alpha k^2} = -\frac{\displaystyle \frac{2}{\alpha R_0} \int Y^{*}(\delta R)Y\, d\Omega} {\displaystyle \int Y^{*}Y\, d\Omega}. \]

Let \(I\) be an operator which, acting on any surface harmonic

\[ U=\sum_{l,m} b_{l,m}Y_{l,m}(\theta,\varphi), \]

annuls all terms except those corresponding to the value of \(l\) that interests us. These latter it leaves unchanged. Then

\[ \int Y^* \Lambda Y\,d\Omega=\int Y^* I\Lambda Y\,d\Omega, \]

where \(\Omega\) is the solid angle and \(\Lambda\) is an abbreviated notation for the perturbation

\[ \Lambda=-\frac{2\delta R(\theta,\varphi)}{aR_0}. \]

A suitable linear combination \(Y\) satisfies the characteristic equation

\[ I\Lambda(\theta,\varphi)Y(\theta,\varphi)=\lambda Y(\theta,\varphi), \]

where \(\lambda\) is a numerical constant. This is the secular equation for the coefficients \(c_m\) and the energy displacement

\[ \delta E=\lambda aE. \]

Let us write the equations of the ellipsoid in the form

\[ x=(R_0+\delta R)\sin\theta\cos\varphi,\quad y=(R_0+\delta R)\sin\theta\sin\varphi,\quad z=(R_0+\delta R)\cos\theta \]

and substitute these equations into the formulas for the semiaxes \(a,b,c\) as functions of \(\alpha\) and \(\gamma\) (Fig. 13). The change \(\delta R\) in the length of the radius vector from the origin to a point on the surface is, in the first approximation, given by the equation

\[ \Lambda(\theta,\varphi)=-\frac{2\delta R(\theta,\varphi)}{aR_0} =3^{1/2}\sin^2\theta\cos2\varphi\sin\gamma-2\bar P_2(\cos\theta)\cos\gamma . \]

From this expression it is clear that the matrix element of the spherical harmonics with one and the same value of \(l\) (the one that interests us) and different \(m\) vanishes, with the exception of the following cases:

1) equal \(m\) (diagonal elements of the matrix)

\[ \Lambda_{m,m}=\frac{2\cos\gamma\,[3m^2-l(l+1)]}{(2l-1)(2l+3)}, \]

2) the quantities \(m\) differ by 2 units

\[ \Lambda_{m\pm1,m\mp1}= \frac{3^{1/2}\sin\gamma\,[l^2-m^2]^{1/2}[(l+1)^2-m^2]^{1/2}} {(2l-1)(2l+3)} . \]

The solution of the secular equation:

\[ \text{determinant}\ \left|\Lambda_{jk}-\lambda\right|=0 \]

gives the magnitude of the energy displacement in this and the preceding cases.

Here I have calculated the influence of deformation on the displacement of energy levels in the case of an infinite potential well; the values obtained, as indicated by Finberg and Hammack,⁸ are reduced in the case of a finite potential well.

Figure 18

In this case the secular equation (Fig. 17) has so many roots that their distribution can be represented statistically. Along the horizontal axis is plotted the value of the root of the secular equation

\[ \lambda=\frac{\delta E}{aE}. \]

Along the vertical axis is plotted the relative number of roots per unit interval of \(\lambda\),

\[ \int \frac{df}{d\lambda}\,d\lambda=1. \]

Whatever the value of the shape parameter \(\gamma\), the distribution function has a singularity at one value of the dimensionless parameter of the magnitude

Fig. 18. Splitting of levels in the case of small ellipsoidal deformations, but very large orbital angular momenta.

of the level displacement \(\lambda\). In other words, in the neighborhood of the special value \(\lambda_{\mathrm{особ}}\) there are many levels:

\[ \lambda_{\mathrm{особ}}=\cos\left(\gamma-\frac{2\pi}{3}\right), \]

if \(\gamma\) lies in the interval \(0\)—\(60^\circ\). The extreme values of the level displacement in the case under consideration of large \(l\) are:

\[ \lambda_{\min}=\cos\left(\gamma+\frac{2\pi}{3}\right) \quad \text{and} \quad \lambda_{\max}=\cos\gamma. \]

On the right, for comparison, the distribution of levels found from the secular equation in the case \(l=4\) is presented; moreover, each level has been broadened so as to approximate a continuous distribution. There is qualitative agreement between the level distribution for \(l=4\) and \(l=\infty\). As the conclusions of quantum mechanics approach those of classical mechanics for large quantum numbers, the method of calculation may be replaced by the following:

\[ \lambda=\frac{\delta E}{aE}= \left( \text{the average value of the perturbation, } \Lambda(\theta,\varphi)= \right. \]

\[ \left. = -\,\frac{2\delta R(\theta,\varphi)}{aR_0} \text{ over an unperturbed classical motion} \right). \]

In the case of unperturbed classical motion the particle moves in a plane passing through the center of the sphere. Its motion is a series of straight line segments, suddenly changing their direction when the particle strikes the surface. For all orbits, with the exception of a set of measure zero, the period of the motion is incommensurable with \(2\pi\), and the particle will with the passage of time pass arbitrarily close to any point of some great circle. Let the normal to this great circle have polar angles \(\theta^*\) and \(\varphi^*\). Then the average value \(\Lambda\) on this great circle is equal to:

\[ \Lambda_{\mathrm{av}}=\lambda=\cos\gamma P_2(\cos\theta^*)+ \frac{\sqrt{3}}{2}\sin\gamma \sin^2\theta^*\cos 2\varphi^*. \]

If we require that this value lie within the interval \(\lambda\) and \(\lambda+d\lambda\), then it is necessary to choose on the surface of the sphere a strip of points, each of which determines the direction of the angular momentum of the corresponding orbit. The solid angle subtended by this strip gives us the fraction of eigenvalues lying between \(\lambda\) and \(\lambda+d\lambda\)

\[ \frac{df}{d\lambda} = \frac{1}{4\pi}\frac{d\Omega}{d\lambda} = \pi^{-1}3^{-1/4}(\sin\gamma)^{-1/2}(\cos\gamma-\lambda)^{-1/2}K(\chi), \]

where \(K\) is the complete elliptic integral, and

\[ \chi= \left\{ \frac{(\lambda-\cos120^\circ+\gamma)\sin120^\circ+\gamma} {(\cos\gamma-\lambda)\sin\gamma} \right\}^{1/2}. \]

This expression has meaning if \(\gamma\) lies in the interval \(0\text{–}60^\circ\) and, in addition, \(\lambda\) lies in the interval \(\lambda_{\min}\text{–}\lambda_{\mathrm{sp}}\). If \(\lambda\) lies in the interval \(\lambda_{\mathrm{sp}}\text{–}\lambda_{\max}\), then the corresponding formula takes the form

\[ \pi^{-1}3^{-1/4}(\lambda-\cos120^\circ+\gamma)^{-1/2} (\sin120^\circ+\gamma)^{-1/2}K(\chi^{-1}). \]

From these formulas the distribution curves \(\dfrac{df}{d\lambda}\) were constructed.

Figure 19

For definiteness, the sequence of Franck–Metropolis figures is considered. Along the vertical is plotted the dimensionless energy quantity

\[ \rho^2=\frac{2MR_0^2E}{\hbar^2}, \]

where \(R_0\) is the radius of the initial sphere. The magnitude of the deformation is determined by the parameter

\[ \alpha_2=\frac{7}{3}y=\frac{7}{3}(1-x), \]

where \(x\) and \(y\) have the meaning assigned to them in Fig. 2. This definition of \(\alpha_2\) agrees with the definitions given in Fig. 1 and Fig. 13 for small perturbations, but all three definitions differ for large perturbations.

Fig. 19. Effect of the deformation leading to fission on the even “\(+\)” levels of a single particle.

On the right-hand side of the diagram are plotted the energy levels in the case when the nucleus is deformed into two spheres of half volume, joined to one another by a small neck. Whatever the magnitude of the single deformation parameter considered here, the boundary is invariant with respect to inversion \((x,y,z\to -x,-y,-z)\) and reflection in the plane passing through the origin and perpendicular to the axis of symmetry \((x,y,z\to x,y,-z)\).

Consequently, the levels split into 4 classes:

Ratio of the values of the wave functions at two inverse points \(= (-1)^l\)	Parity	Ratio of the values of the wave functions at two mirror points \(= (-1)^{l+m}\): “+”	Ratio of the values of the wave functions at two mirror points \(= (-1)^{l+m}\): “−”
Ratio of the values of the wave functions at two inverse points \(= (-1)^l\)	even	\(l\) even. \(m\) even. Fig. 19	\(l\) even. \(m\) odd. Fig. 20
Ratio of the values of the wave functions at two inverse points \(= (-1)^l\)	odd	\(l\) odd. \(m\) odd. Fig. 22	\(l\) odd. \(m\) even. Fig. 21

In this figure only the even “+” levels are represented. In Figs. 19, 20, 21, and 22, only those levels are shown in detail which correspond to the quantum numbers of the projection of the angular momentum on the symmetry axis \(m = 0\) or \(m = 3\). The curves drawn as continuous lines are calculated as indicated in the caption to Fig. 21. The curves drawn with a dotted line are schematic. The quantitative basis for them consisted only of the initial and final ordinates, as well as the slopes and inflection points, chosen so as to ensure the proper jumps of the levels, preserving the necessary relation between the number \(n\) of nodal surfaces in the initial and final wave functions.

Figure 20

The energy-level diagram satisfies the well-known correlation principle: two levels do not intersect if and only if (taking into account an arbitrary deformation amplitude) they belong simultaneously to one and the same symmetry class out of four possible ones (in the given case, to the even “−” class) and are characterized by identical values of \(m\). This principle is applicable insofar as the system has axial symmetry. If, on the contrary, the surface is a triaxial ellipsoid, then \(m\) is not a quantum number and no two levels from among those belonging to one and the same symmetry class intersect one another. For configurations close to the initial sphere, the energy levels

\[ E = \frac{\hbar^2 \rho^2}{2MR_0^2} \]

are calculated from the perturbation formula (Fig. 17)

\[ \rho^2 = \rho_{n,l}^{\,2}\left[1 + 2\alpha_2 \frac{3m^2 - l(l+1)}{(2l-1)(2l+3)}\right]. \]

Here \(\rho_{n,l}\) is the \((n-l)\)-th root of the regular solution \(F(\rho)\) of the diffe-

differential equation

\[ -\frac{d^{2}F}{d\rho^{2}}+\left[1-\frac{l(l+1)}{\rho^{2}}\right]F=0 \]

for the radial part of the wave function:

\[ \psi(r,\theta,\varphi)=\frac{1}{r}F(kr)P_l^{(m)}\cos\theta e^{im\varphi}. \]

All values of \(\rho_n^{2}\) less than 200 [see the table of spherical Bessel functions] are given on p. 127.

Fig. 20. Even “—” levels of the Franck–Metropolis figure.

The number \(n\) represents, in the case of a spherical or nearly spherical configuration, the total number of nodal surfaces of the wave function, counted as follows: \(m\) in the direction of the coordinate \(\varphi\), \(l-m\)

\(l\)	\(n-l=1\)	\(n-l=2\)	\(n-l=3\)	\(n-l=4\)
0	9,870	39,479	88,897	157,914
1	20,191	59,679	118,899	197,858
2	33,218	82,719	151,854
3	48,831	108,516	187,635
4	66,955	137,005
5	87,531	168,130
6	110,519	201,850
7	135,886
8	163,605
9	193,649

in the direction of the coordinate \(\theta\) and \(n-l\) in the radial direction (counting the boundary surface as a nodal one). As the magnitude of the deformation increases, \(l\) ceases to be a quantum number, and even the total number \(n\) of nodal surfaces becomes nonconstant, since surfaces shift toward the outer boundary or merge within it. The only unchanged principle of ordering is the requirement that levels with one and the same \(m\) must not intersect.

Figure 21

Only the levels corresponding to \(m=0\) are presented in detail. The wave functions satisfy the equation \(\nabla^{2}\psi + k^{2}\psi = 0\) inside the surface and the condition \(\psi=0\) on the surface of an infinite potential well. Elsasser \(^{76}\) and Mottelson \(^{77}\) obtained, in this approximation, the splitting of nucleon levels in nuclei having a spherical shape. Neglect of the penetration of the wave function into the region of negative kinetic energies (i.e., beyond the boundary of the finite potential well) may be approximately compensated by a corresponding small change in the dimensions of the figure: by shifting the surface perpendicularly by a distance \(\hbar[2M(W-E)]^{1/2}\), where \(W\) is the height of the potential well and \(E\) is the kinetic energy of the state under consideration. In the exact calculations (smooth curves) the wave function of the odd “\(-\)” levels with quantum number \(m\) was represented as the product of the factor \(e^{im\varphi}\) and the sum

\[ \sum C_l i^l \frac{pr}{R_0} P_l^{(m)}(\cos\theta), \]

where \(R_0\) is the initial radius of the sphere and \(\rho^2=\dfrac{2MR_0^2E}{\hbar^2}\) is the dimensionless unit of energy. The summation is carried out over even values of \(l\) in the case of levels of positive parity and over odd values of \(l\) in the case of levels of negative parity. This wave function automatically satisfies the differential equation. The vanishing of the solution on the boundary surface

\[ \frac{r}{R_0}=f(\cos\theta), \]

defined in Fig. 2, leads to the imposition of a continuous continuum of conditions on the infinite set of coefficients \(C_l\). As an approximation, in the calculation only five terms of the sum were used, and the requirement that the sum vanish only at 5 points of the surface for which \(\mu=\cos\theta=\mu_s\),

where \(\mu_s=0.19;\ 0.38;\ 0.57;\ 0.76;\ 0.95\) in the “\(-\)” levels and \(\mu_s=0.105;\ 0.315;\ 0.525;\ 0.735;\ 0.945\) in the “\(+\)” levels. A system of five equations with five unknown coefficients \(C_l\) has a solution if and only if \(\rho\) makes the determinant vanish

Fig. 21. Odd “\(-\)” levels in the sequence of Franck–Metropolis figures.

\[ \left|\, j_l(\rho f_s)\,P_l^{(m)}(\mu_s)\,\right|, \]

where \(s=1,2,\ldots,5\) are the indices of the columns, and \(l=0,2,4,6,8\) are the indices of the rows in the case \(m=0\) and positive parity, and \(l=1,3,5,7,9\) in the case \(m=0\) and negative parity. The functions \(j\) and \(P\) were calculated on an electronic computer; the Legendre functions (divided by \((1-\mu^2)^{m/2}\)) from finite power expressions; the spherical Bessel functions \(j(x)\), satisfying the equation

\[ \frac{1}{x^2}\frac{d}{dx}\left(x^2\frac{dj}{dx}\right)+\left[1-\frac{l(l+1)}{x^2}\right]j=0, \]

—from the ordinary expansion in a power series, if the ratio \(\dfrac{x}{\sqrt{l+\dfrac{3}{2}}}\) is less than 5, and for large values of the argument—from the known expression

\[ \left(\text{finite polynomial in } \frac{1}{x}\right)\times \sin x + \left(\text{finite polynomial in } \frac{1}{x}\right)\times \cos x . \]

The roots of the \(P_l\) determinant were found by trial.

Figure 22

In the right-hand part of this figure, as in the three preceding ones, the levels of two spheres with radius

\[ R_f=\frac{R_0}{\sqrt[3]{2}}, \]

connected by an aperture of radius \(a\), small in comparison with \(R_0\), are marked. When \(a=0\), the levels coincide with the levels of the sphere and the corresponding values of the dimensionless energy parameter

Fig. 22. Odd “+” levels in the sequence of Frankl–Metropolis figures leading to fission.

\(p^2\) is \(2^{1/2}\) times greater than the values given in the caption to Fig. 20. When the opening is slightly enlarged, the wave functions in both spheres are joined, and it is necessary to distinguish two cases: 1) the “\(+\)” case—the wave functions of both spheres are joined in such a way that no node is formed at the point of junction (mirror symmetry with respect to the plane of the opening); 2) the “\(-\)” case—at the point of junction a node of the common wave function is formed (the wave function is antisymmetric with respect to reflection in the plane of the opening). The half-volume sphere contains, up to a specified energy (the boundary of the Fermi distribution), approximately half the number of levels as a sphere of full volume. Thus, doubling the number of states owing to the two possibilities for joining the wave functions through the opening gives the correct number of states of the individual particles of the whole system. The energies of the “\(-\)” states in the first approximation do not depend on the size of the opening (the wave function vanishes at the point of perturbation). The energies of the “\(+\)” states, on the contrary, decrease as the opening is widened, and the relative magnitude of this decrease is

\[ \frac{1}{\rho^2}\,\delta(\rho^2) \simeq -\,\frac{2}{3\pi}(2l+1)\frac{a^3}{R_f^3} \]

for states with \(m=0\), and is considerably larger (of higher degree in \(\frac{a}{K_0}\)) for large values of \(m\). This result may be obtained by comparing the “\(+\)” wave function \(\psi_0\), corresponding to a completely closed opening, and the function \(\psi_1\), corresponding to a small opening. Both functions coincide everywhere, except for points at distances of order \(a\) from the center of the opening.

The wave function in the left sphere is

\[ \psi_0=\frac{1}{r}F_l(kr)P_l^{(m)}(\cos\theta)e^{im\varphi} \]

(in the right sphere the wave function is the mirror image of the wave function of the left sphere) and behaves, for positive \(m\), approximately as \(\psi_0=Qz\rho^m e^{im\varphi}\), where \(z\) is the distance along the perpendicular from the opening to the center of the left sphere, \(\rho\) is the corresponding cylindrical polar coordinate, and the constant \(Q\) has the value

\[ Q=-R_f^{-1-m}kF_l(kR_f)\frac{(l+m)!}{2^m m!(l-m)!}. \]

In the case \(m=0\), the approximate value of \(\psi_0\) increases linearly with increasing distance from the surface. We are dealing with a wave-mechanical analogy of a constant electric field. According to electrical terminology, an enlargement of the neck opening leads to a leakage of lines of force. Superposed on the linearly varying electric potential is another term—a local perturbation—which also approximately satisfies Laplace’s equation. The second derivatives of this term in the directions perpendicular and parallel to the plane of the opening \(\left(\frac{\partial^2\psi}{\partial\rho^2}\ \text{and}\ \frac{\partial^2\psi}{\partial z^2}\right)\) are opposite in sign and both are large in comparison with \(k_1^2\psi\).

We neglected the quantity \(k_1^2\psi\) when considering the correction \(\delta\psi=\psi_1-\psi_0\) in the wave function. Consequently, we found this difference from Laplace’s equation satisfying the boundary conditions: 1) \(\delta\psi\) decreases relative to \(\psi_0\) at large distances from the opening,

2) \(\dfrac{\partial(\delta\psi)}{\partial z}\) vanishes on all portions of the surface (considered here as plane), except for the aperture; 3) at the aperture the correction term has such a normal derivative that it makes the derivative of \(\psi_1\) itself vanish (the condition of mirror symmetry), i.e.
\[ \frac{\partial(\delta\psi)}{\partial z}=-Q\rho^m e^{im\varphi}. \]
In what follows the equations are understood for positive \(m\), since the case \(m=0\) is of no interest. Passing to the new coordinates: \(z=auv\), \(\rho=a[(1-u^2)(1+v^2)]^{1/2}\), we reduce Laplace’s equation to an equation with separable variables of the type \(P_n^{(m)}(u)f(v)e^{im\varphi}\). Here the function \(f\) satisfies the equation
\[ \frac{d}{dv}(1+v^2)\frac{df}{dv} +\left[\frac{m^2}{1+v^2}-n(n+1)\right]f=0. \]

The initial wave function \(\psi_0\) at distances not too far from the midplane has exactly the same form, with \(n=m+1\) and \(f(v)\) a multiple of \(P_{m+1}^{(m)}(iv)\):
\[ \psi_0=Qa^{m+1}u(1-u^2)^{m/2}v(1+v^2)^{m/2}e^{im\varphi}. \]

In the corresponding expression for \(\delta\psi\) everything remains the same, with, however, the exception that the function \(f_{m+1}^{(m)}(v)\) decreases for large positive \(v\):
\[ \delta\psi =Qa^{m+1}u(1-u^2)^{m/2}v(1+v^2)^{m/2}e^{im\varphi} \frac{\displaystyle\int_v^\infty v^{-2}(1+v^2)^{-m-1}\,dv} {\displaystyle\int_0^\infty v^{-2}\left[1-(1+v^2)^{-m-1}\right]\,dv}. \]

Here the definite integral in the denominator normalizes \(\delta\psi\) so as to satisfy the boundary conditions. The change in the wave number caused by the perturbation is found, in the first approximation, by substituting into the numerator of Green’s exact equation
\[ k_1^2-k_0^2= \frac{\displaystyle\int \psi_1^*\left(\frac{\partial\psi_0}{\partial n}\right)dS} {\displaystyle\int \psi_1^*\psi_0\,d\Omega} \]
the expression \(\psi_1=\psi_0+\delta\psi\) in ellipsoidal coordinates,
\[ \frac{\partial\psi_0}{\partial n} =-Qa^m(1-u^2)^{m/2}e^{im\varphi}, \]
\[ \psi_1^*=\delta\psi^* =Qa^{m+1}u(1-u^2)^{m/2}e^{-im\varphi} \frac{2\cdot2\cdot4\cdot6\cdots(2m)} {\pi\cdot3\cdot5\cdot7\cdots(2m+1)} \]
and, by replacing in the denominator \(\psi_1^*\) by \(\psi_0^*\),
\[ \int \psi_1^*\psi_0\,d\Omega = \frac{R_f}{2}\,[F_l(kR_f)]^2\, \frac{4\pi}{2l+1}\cdot\frac{(l+m)!}{(l-m)!}. \]

Thus, the displacement of the “$+$” levels caused by a small hole is equal to

\[ \frac{k_1^2-k_0^2}{k_0^2} = -\frac{2(2l+1)}{\pi}\cdot \frac{(l+m)!}{(l-m)!}\cdot \frac{1}{1^2\cdot 3^2\cdot 5^2\cdots (2m+1)^2\cdot (2m+3)} \left(\frac{a}{R_f}\right)^{2m+3}. \]

This formula was used to construct the curves in the right-hand part of the diagrams of the “$+$” levels.

Figure 23

The total energy of nucleons filling a closed shell and a partially filled $g$-shell up to the indicated number of particles, in the case of small ellipsoidal deformations $\alpha$, is equal to $E=E_0+aE_g f(\gamma)+c\alpha^2$ in an approximation corresponding to an idealized collective model of the nucleus. Here $f(\gamma)$ is represented graphically as a function of the shape parameter $\gamma$ (Fig. 13), while $E_0$, $E_g$, and $c$ are constants, with $E_g$ the kinetic energy of one nucleon with $l=4$. For simplicity, the diagram is drawn as if the nucleons have no spin and as if only 9 particles are required to fill the shell. The necessary correction for twice the number of nucleons can easily be made. The quantity $f(\gamma)$ was obtained by summing the corresponding coefficients for the individual particles in Fig. 17 according to the number of particles in the distribution of nucleons, taking in each case the smallest energy (or the most negative coefficient) allowed by the Pauli principle. From the diagram presented it is seen that the quadrupole forces, measured by the quantity $f(\gamma)$, increase to a maximum for a half-filled shell and then decrease. Oblate deformations are preferable at the beginning of shell filling, elongated ones at the end. Of course, these considerations apply only to the intrinsic quadrupole moments, and not to the moments averaged owing to precession about the axis of the nuclear spin.

Fig. 23. Dependence of the total nucleon energy on $\gamma$ in the case when from one to nine nucleons are in the state $l=4$.

Figure 24

Levels which cross without interaction under deformations of high symmetry experience mutual repulsion under deformations of low symmetry. In the case of axially symmetric deformations, the intersec-

ing levels are usually themselves doubly degenerate, except for the case \(m=0\), so that, when an additional small ellipticity is introduced, 4 levels arise. The deformation coordinate \(\eta=a\sin\gamma\) (Fig. 13) is perpendicular to the plane of the upper diagram. The cylinder, shown in perspective, intersects the four-sheeted surface of the energy level along the four curves represented as functions of the angle in the lower diagram. The figures show an example of the intersection of the doubly degenerate levels \(l=3,\ m=\pm1\) and \(l=5,\ m=\pm3\).

Fig. 24. Splitting of two double levels near the crossing point.

Let \(\xi=a\cos\gamma-a_0\) denote the distance from the crossing point visible in the upper diagram. Then, near the crossing point, the energy matrix of the four levels (omitting inessential details) has the form

\[ H= \left\| \begin{array}{cccc} s\xi+t\eta & 0 & f\eta & -g\eta\\ 0 & s\xi-t\eta & g\eta & -f\eta\\ f\eta & g\eta & -s\xi+u\eta & 0\\ -g\eta & -f\eta & 0 & -s\xi-u\eta \end{array} \right\|, \]

where \(s,t,u,f,g\) are constants, while the diagonal elements represent the positions of the levels in the absence of coupling between one pair of levels and the other. Setting \(\xi=r\cos\theta\), \(\eta=r\sin\theta\), and \(y=\dfrac{\text{energy}}{r}\), we obtain the secular equation for the four roots, represented graphically in the lower diagram:

\[ y^4-\left[2s^2\cos^2\theta+(2f^2+2g^2+t^2+u^2)\sin^2\theta\right]y^2+ \]

\[ +\left[2s(u^2-t^2)\cos\theta\sin^2\theta\right]y+s^4\cos^4\theta+ \]

\[ +s^2(2f^2+2g^2-t^2-u^2)\sin^2\theta\cos^2\theta+(g^2-f^2+ut)^2\sin^4\theta=0. \]

Figure 25

Figure 25 schematically depicts the lower sheet of the many-sheeted potential surface as a function of the ellipsoidal deformation parameters \(a, \gamma\) (Fig. 13). This sheet touches the surface lying above it only at the vertices of inverted “funnels,” i.e. only for spheroidal deformations of flattening and elongation, for which \(\gamma = 0^\circ, 60^\circ, 120^\circ\), etc. The upper sheet, not shown in the figure, in turn has both straight and inverted “funnels.” The probability of slipping from one surface to another, located higher or lower, depends substantially on the total curvature of the surface and, in particular, on the size of the region of the surface accessible to the point representing the system at a given value of the total (potential \(+\) vibrational kinetic) energy.

Fig. 25. Qualitative picture of inverted “funnels” on the lower potential surface.

Figure 26

The figure presents a qualitative picture of the influence of the degree of shell filling on the intrinsic quadrupole moment, i.e. on the shape of the nucleus. Here the shape of the nucleus is meant from the point of view of particles moving inside the nucleus, and not from the point of view of atomic electrons, since

Fig. 26. Expected dependence of deformation potentials and quadrupole moments on the degree of shell filling.

in the latter case an averaging of the nuclear precession is added—an averaging which reduces to zero the quadrupole moment in atomic spectra when the nuclear spin is equal to \(1/2\) or 0 (even-even nuclei). Several nucleons in a partially filled shell lead to deformations in the same way as a single nucleon (Fig. 6), with the following corrections:

a) The particles combine. Coupling with the surface brings the orbit of the second particle into the plane of the first, which proves energetically more favorable. In the first approximation the deformation caused by two particles is twice as large, and the lowering of the energy (relative to the spherical configuration) is four times as large as in the case of one particle.

b) As the number of particles increases, the quadrupole forces (see, for example, Fig. 23) pass through a maximum, showing some symmetry between the beginning of the filling of a shell (oblate figures are energetically more favorable) and the end of the filling of the shell (elongated figures are energetically more favorable).

c) When shells are filled, roughly speaking within the limits from \(1/3\) to \(2/3\), both configurations, oblate and elongated, correspond to relative minima of the energy curves, which means the possibility of configuration-isomeric figures of one and the same nucleus.

d) All these arguments have been carried out under the assumption that the contribution of all the remaining closed shells to the deformation potential is a quadratic function of the deformation parameter \(\alpha\). In reality, a greater possible cause leads to a rearrangement of the order of the levels, and even the term “closed shell” no longer has a simple meaning. Owing to this circumstance one should not expect regular changes of quadrupole moments with increasing degree of filling of the outer shell.

Figure 27

Mutations of the figure of the nucleus from oblate to elongated, or conversely, depending on the sign of the released energy, according to the data of

Labels in the figure: mutation threshold; unstable isomer (a); mutation energy sufficient for neutron evaporation; \(r(\theta)=R[1+\alpha P_2(\theta)]\); surface-oscillation levels; energy; deformation \(\alpha\); stable isomer (b); \(R\); \(\theta\); axis of symmetry.

Fig. 27. Mutations of the figure of the nucleus.

Fig. 26, are possible in nuclei in which the shell is filled from \(1/3\) to \(2/3\). It seems improbable that, for such large deformations, deforma-

the shell energy of closed shells is still proportional to $\alpha^2$. Consequently, this diagram should be regarded only as an idealized schematic of the true potential surface and only for spheroidal nuclear shapes (axial symmetry). The spherical configuration is a potential barrier against transitions from stable to unstable isomeric shapes.

Figure 28

The energy is idealized as the sum of the surface energy and the energy of individual nucleons occupying as low a position as possible in the potential well of the given shape. Minima of the total energy occur in cases where the angular momenta of the nucleons are directed as nearly parallel or antiparallel as possible to the axis of symmetry (an oblate figure), or when they are directed as nearly perpendicular as possible to the axis of symmetry (an elongated figure). For the transition from an elongated figure to an oblate one through a spherical one, a large energy is required. As Teller pointed out, the most accessible path is the transition along the ridge of the potential surface. This ridge corresponds to the shape of a triaxial ellipsoid.

Fig. 28. Contour diagram of the dependence of the energy on the deformation variables $\alpha$ and $\gamma$ of Fig. 13 for the ground state of heavy nuclei with a shell filled slightly more than half.

Figure 29

Circles denote the moments of nuclei having an odd number of protons or odd numbers of protons and neutrons (except for Li$^6$ and Cl$^{34}$), as a function of the number of protons. Crosses denote the moments of nuclei having an odd number of neutrons, as a function of the number of neutrons.

Arrows indicate the filling of the main nucleon shells. The solid line marks regions of accurately established quadrupole moments; the dashed line marks doubtful regions. The figure is taken from the paper by Townes et al.\(^{10}\). The relation between \(Q\) and \(a\) is given in Fig. 5.

Fig. 29. Effective nuclear quadrupole moments of hyperfine structure, divided by the square of the nuclear radius \(\left(1.5\cdot 10^{-13}A^{1/3}\right)^2\).

Axis labels in the figure: ordinate \((Q/R^{2}e)\); abscissa “Number of odd nucleons.”

Figure 30

\(\alpha\)-active nuclei usually have a quadrupole moment and therefore emit \(\alpha\)-particles predominantly from protruding parts of the surface, where the height of the potential barrier is lowered. The wavelength

\[ \bar{\lambda}=\frac{\lambda}{2\pi} \]

of an \(\alpha\)-particle with energy \(5\) MeV, equal to \(1\cdot 10^{-13}\) cm, is small in comparison with the diameter of a typical heavy nucleus, of order \(18\cdot 10^{-13}\) cm. Consequently, the radiation will be directed with respect to the nuclear axis—a fact that is established better than the correlation of the nuclear axes themselves, in relation to the direction selected in space. In particular, a nuclear state with angular momentum equal to zero is characterized by isotropic \(\alpha\)-radiation in the laboratory coordinate system, although in the nuclear coordinate system the radiation is anisotropic with respect to the nuclear surface. The maximum directionality will be observed when the following conditions are simultaneously fulfilled: the nuclear spin \(I\) is large, its projection \(I_z\) on the selected axis (strong magnetic field,

low temperature) is equal to \(\pm I\), and the nucleus under consideration is, on the average, elongated in time along the axis of the nuclear spin \(I\). In this case the radiation will have a preferred direction parallel and antiparallel to the magnetic field. If, under the same conditions, a maximum is observed in the direction perpendicular to the field, this means that the ellipsoid is flattened.

Directed \(\alpha\)-radiation

[Diagram labels: Energy; relative change of the old curve at the new distance; barrier for an elongated nucleus; \(\alpha\)-radiation; distance; \(0\), \(R_0\), \(R_0(1+\alpha P_2(\cos\theta))\); \(\frac{3}{2}\alpha P_2\), \(-\alpha P_2\).]

Fig. 30. Directed \(\alpha\)-decay.

Figure 31

The Geiger–Nuttall relation between the energy of \(\alpha\)-particles and the lifetime of a nucleus with respect to \(\alpha\)-decay is not single-valued, as was shown in a recent review of even-even nuclei by Perlman, Ghiorso, and Seaborg \(^{79-81}\) (see also \(^{78}\)). Perlman \(^{78}\) and others gave analo—

[Graph labels: Half-life periods (seconds); Energy of \(\alpha\)-particles (MeV). Curve labels include: \(U^{238}\), \(U^{232}\), \(Th^{230}\), \(Th_0^{230}\), \(Ra^{226}\), \(Rn^{226}\), \(Po^{208}\), \(Po^{210}\), \(U^{234}\), \(U^{232}\), \(Th^{229}\), \(Pu^{238}\), \(Th^{228}\), \(Pu^{236}\), \(U^{230}\), \(Cm^{242}\), \(Cm^{240}\), \(Em^{212}\), \(Th^{226}\), \(Em^{220}\), \(Ra^{222}\), \(Em^{222}\), \(Ra^{224}\), \(Po^{218}\), \(Po^{216}\), \(Em^{218}\), \(Po^{214}\), \(Po^{212}\).]

Fig. 31. Half-life periods as a function of the energy of \(\alpha\)-particles.

ical curves for nuclei of other classes. This unambiguity should not have been expected. Treatment of the passage through the barrier leads to the formula (see also \(^{82}\))

\[ \begin{pmatrix} \text{Probability}\\ \alpha\text{-decay}\\ \text{per second} \end{pmatrix} = \begin{pmatrix} \text{Internal}\\ \text{probability of}\\ \alpha\text{-emission} \end{pmatrix} \cdot e^{-2\frac{2Z}{137}\sqrt{\frac{2M_{\alpha}c^2}{E_{\alpha}}}\, f(u)}, \]

into which enter not only the energy of the \(\alpha\)-particle (we disregard here the well-known small correction for the finiteness of the recoil nucleus mass and energy), but also the charge and radius of the nucleus,

\[ u=\frac{E_{\alpha}}{\text{barrier height}} =\frac{E_{\alpha}R_0}{2Ze^2} =\frac{E_{\alpha}A^{1/3}}{4mc^2Z}, \]

\[ f(u)=\arccos u^{1/2}-u^{1/2}(1-u)^{1/2}. \]

Nevertheless, the deviations of the points in the diagram from the smooth curve corresponding to a given \(Z\) appear too large and too irregular to be connected exclusively with the resulting dependence on \(Z\) and \(A\). Anomalies in the binding energy near filled shells lead to irregular variations of \(E_{\alpha}\) from element to element, but these variations in the energy by themselves cannot explain the deviations from the formula for the barrier penetrability. However, the presence of quadrupole moments leads to deviations from this formula that are very sensitive to the filling of nucleon shells—or, more precisely, sensitive to deformations of the nuclear shape caused by the asymmetric pressure on the surface from nucleons of unfilled shells. For a deformation

\[ R=R_0[1+aP_2(\cos\theta)] \]

the potential energy of an \(\alpha\)-particle outside the nucleus, in first approximation, is equal to

\[ V=\frac{2Ze^2}{r}\left[1+\frac{3R_0^2}{5r^2}aP_2(\cos\theta)\right]. \]

Re-evaluating the integral

\[ \frac{2}{\hbar}\int_{R(\theta)}^{\text{turning point}} [2M_{\alpha}(V-E_{\alpha})]\,dr, \]

in the exponent of the formula for the barrier penetrability, we find that the factor \(f(u)\) changes by the amount

\[ \delta f(u)=-\frac{2}{5}aP_2(\cos\theta)u^{1/2}(1-u)^{1/2}(2-u). \]

We have neglected here the fact that the area of the surface from which appreciable emission occurs is in reality somewhat smaller than \(4\pi R_0^2\). This circumstance is more than compensated by the increase in the barrier penetrability due to the decrease of its thickness. Accordingly, we substitute the values of the angle \(\theta\) corresponding to maximum emission:

\[ -aP_2(\cos\theta)=-a \quad \text{for elongated ellipsoids,} \]

\[ -aP_2(\cos\theta)=a/2 \quad \text{for flattened ellipsoids,} \]

\[ \delta f(u)\ \text{is a negative number in both cases.} \]

The increase in the rate of \(\alpha\)-decay under ellipsoidal deformation is characterized by the factor

\[ \exp \left[-2\,\frac{2Z}{137}\sqrt{\frac{2M_\alpha c^2}{E}}\,\delta f(u)\right], \]

which, for \(Z=90\), \(A=234\), a barrier height before deformation of \(30\) MeV, an \(\alpha\)-decay energy \(E=5\) MeV, a total energy of the \(\alpha\)-particle \(Mc^2=3700\) MeV, and the dimensionless ratio

\[ u=\frac{5\ \text{MeV}}{30\ \text{MeV}}, \]

is equal to

\[ \exp(-101\,\delta f)=\exp[-101(-0.273\alpha P_2)] = \begin{cases} \exp 28\alpha & \text{(elongation)},\\ \exp(-14\alpha) & \text{(flattening)}. \end{cases} \]

Thus, a deformation corresponding to elongation of the axis with \(\alpha=0.1\)—a value lying within the range corresponding to the observed quadrupole moments—can lead to an increase in the decay rate by approximately \(e^{2.8}=16\) times. Of course, from data on decay rates associated with a given process of \(\alpha\)-decay, one cannot simultaneously determine the nuclear radius and the quadrupole moment. Only a certain combination of both quantities can be determined, which, in the absence of other phenomena, is usually denoted the effective nuclear radius. Between the effective radius, the radius of the sphere of equal volume, and the quadrupole moment there exists the relation

\[ R_{\text{eff}} = \frac{2Ze^2}{E_\alpha}u_{\text{eff}} = \frac{2Ze^2}{E_\alpha} \left[ u+\frac{\delta f}{\left(\dfrac{df}{du}\right)} \right] = R_0 \left[ 1+\frac{2}{5}(2-u)\alpha P_2 \right], \]

or, with an accuracy sufficient for most nuclei,

\[ \frac{R_{\text{eff}}-R_0}{R_0} = 0.533\alpha P_2 = \begin{cases} 0.533\alpha & \text{(elongation)},\\ -0.267\alpha & \text{(flattening)}. \end{cases} \]

In connection with these conclusions it is of interest to consider the \(\alpha\)-activity of samarium. Measurements of the energy of the \(\alpha\)-particles, carried out by Jessop and Sadauskis \(^{83}\), gave the value \(2.18\) MeV, and the activity is attributed to the isotope \(\mathrm{Sm}^{147}\) (see \(^{84,85}\)). We owe the discussion of this example to Perlman, who took the nuclear radius to be \(7.81\cdot 10^{-13}\) cm and calculated a half-life of \(2.9\cdot 10^{12}\) years \(^{86}\), which is 10–20 times greater than the experimental value. Perlman also found that increasing the \(\alpha\)-decay energy by \(80\) MeV or increasing the radius of the nucleus by approximately \(10\%\) would remove the discrepancy. On the other hand, we may note that an elongation by

\[ \frac{10\%}{0.533}=19\% \]

or a flattening by

\[ \frac{10\%}{0.267}=37\% \]

would lead to the same result, provided only that the adopted radius \(7.81\cdot 10^{-13}\) cm is indeed a suitable approximation for the radius of the sphere of the same volume. Quadrupole deformations of this order have been found experimentally for nuclei in the immediate vicinity of samarium. Briggs and Kauffman \(^{87}\) expressed a number of considerations concerning the half-life of samarium. Of course, the decrease of lifetimes occurring in \(\mathrm{Sm}^{147}\) to a large extent also occurs to a greater or lesser extent in most other nuclei. Consequently, nuclear radii found from an analysis of \(\alpha\)-decay data should on average be somewhat larger than the corresponding radii of spheres of equal volume. From this point of view, the radii

STRUCTURE OF THE NUCLEUS AND THE INTERPRETATION OF FISSION PHENOMENA

the polonium isotopes \(^{89}\), previously regarded as anomalously small, should now be considered as “normal” quantities. Obviously, direct measurements of quadrupole moments of “active” nuclei, where they are possible, would be very desirable. It is also necessary to mention anomalies in the isotopic shifts of atomic spectra \(^{21}\), which are readily explained if one assumes that, owing to ellipsoidal deformation, the electron-effective radius of the nucleus increases (a circumstance noted by Wilets). It is possible that it will be possible to develop a new experimental method for studying quadrupole moments by investigating the so-called Chang radiation, associated with the transition of the \(\mu\)-meson from the Bohr orbit \(2p\) to \(1s\). This method should be close to the method of studying atomic spectra, with the difference, however, that the shifts and splittings of terms in percentage terms are here many times larger than in atomic spectra, owing to the much greater amount of time spent by the \(\mu\)-meson near the nucleus. The half-life of \(\alpha\)-radiation depends not only on the penetrability of the barrier (the Gurney–Condon formula), but also on the Franck–Condon principle, i.e. not only on the equilibrium deformation of the initial nucleus, but also on the difference between the initial and final states of the nucleons. If this difference is great, then it is unlikely that the residual nucleus will be formed in the state of zero vibrations. On this account there will be a decrease in the rate of \(\alpha\)-decay not taken into account in the arguments presented above. Transitions to the lower vibrational state will be forbidden relative to transitions to vibrational states preferable from the point of view of the Franck–Condon principle. This circumstance should be taken into account in interpreting the fine structure of \(\alpha\)-spectra.

Figure 32

Can one experimentally distinguish diffraction or shadow scattering on ellipsoidal nuclei from scattering on spherical nuclei? A qualitative analysis carried out on this figure shows that the ratio of the minimum to the maximum in diffraction by randomly oriented nuclei of a given shape is indeed a sensitive indicator of nuclear deformation. This ratio is equal to zero (in the idealized picture of diffraction scattering) only in the case of spherical scattering centers. A quantitative analysis can be carried out as follows: let \(f\) and \(g\) be the principal axes of the projection of the ellipsoid onto the plane of the receiver, and \(X\) and \(Y\) axes oriented parallel to \(f\) and \(g\). The contribution of the element \(dS=d\xi\,d\eta\) of the area of the projection of the ellipsoid to the amplitude of the scattered wave at a distance \(r_{12}\), at a not very large angle \(\theta\) to the initial direction, is equal to

\[ -\frac{ik}{2\pi r_{12}}\exp(ikr_{12})\,dS, \]

where \(k\) is the wave number. The differential scattering cross section is equal to:

\[ \frac{d\sigma}{d\Omega} = \left| \frac{k}{2\pi} \int \exp\left[ \frac{-ik(x\xi+y\eta)}{r} \right] \,d\xi\,d\eta \right|^2 . \]

Here the scattering direction is determined by the ratios \(\frac{x}{r}\) and \(\frac{y}{r}\). Denoting \(\xi=fu\cos\theta\), \(\eta=gu\sin\theta\) and integrating first over \(\theta\), then over \(u\) from 0 to 1, we obtain:

\[ \frac{d\sigma}{d\Omega} = \left\{ \frac{ kfgJ_1\left[ k^2 f^2 \frac{x^2}{r^2} + k^2 g^2 \frac{y^2}{r^2} \right]^{1/2} }{ \left( k^2 f^2 \frac{x^2}{r^2} + k^2 g^2 \frac{y^2}{r^2} \right)^{1/2} } \right\}^2 . \]

for scattering from a single orientation of the ellipsoid. Superposition of such curves gives the final distribution. Richardson, Ball, Leith, and Møller\(^{90}\) were the first to mention this effect, which, however, at the existing accuracy of the experiments, remains unnoticed.

Fig. 32. Diffraction by ellipsoidal nuclei.

(To be continued in the next issue)