OUTLINE OF THE DEVELOPMENT OF THE THEORY OF THE STRUCTURE OF THE ATOMIC NUCLEUS

G. Gamow, Leningrad

IV. General Structure of the Nucleus*

§ 1. In the preceding articles of the present outline we have dwelt in detail on the consideration of a whole series of nuclear processes, such as natural and artificial transformations of nuclei and the excitation of the nucleus associated with these transformations, leading to the emission of γ-rays. We now turn to the general question of the constituent parts of the nucleus and of the forces that bind them into one whole. According to modern ideas, every nucleus is made up of two kinds of elementary particles, protons and electrons. The number of the former is given directly by the atomic weight \(M\), while the number of the latter is given by the difference between the atomic weight of the nucleus and its atomic number \(Z\). As is well known, the mass of any nucleus is not equal to the sum of the masses of the protons and electrons entering into its composition, but is smaller than the latter by a certain amount \(\Delta M\), called the total mass defect and connected with the total energy of the internal binding of the nucleus by the relativistic relation:

\[ E = \Delta M \cdot c^2 \tag{1} \]

where \(c\) is the velocity of light. Precise measurements of the atomic weights of various isotopes, for which we are indebted chiefly to Aston’s work, make it possible for us to calculate these binding energies for a whole series of nuclei. The material now available on this question is represented graphically in Fig. 1, where the internal binding energy is plotted as a function of the atomic weight. We see that, to a first approximation, we may regard the total binding energy as proportional to the number of constituent particles of the nucleus. It naturally suggests itself, however, to suppose that in complex nuclei its elementary constituent parts (protons and electrons) combine into certain stable forma-

* See Uspekhi fizicheskikh nauk 10, 531, 1930; 12, 31, 1932; 12, 389, 1932.

OUTLINE OF THE DEVELOPMENT OF THE THEORY OF THE STRUCTURE OF THE ATOMIC NUCLEUS

formations that play an independent role in complex nuclei. Such second-order units may, for example, be the recently discovered simplest nuclei—neutrons (proton + electron), the nuclei of hydrogen of mass two (two protons + electron), and, finally, the long-known, extremely stable helium nuclei, or α-particles (four protons + two electrons). By making definite hypotheses about the composition of the nucleus, we can obtain the energy binding these constituent parts together by subtracting from the total energy of the nucleus the internal energy of these formations.

Until recently the most probable hypothesis was considered to be that within the nucleus there is formed

Fig. 1.          Fig. 2.

the maximum possible number of α-particles, while in the remainder there always remain no more than three extra-alpha protons and some number of extra-alpha electrons. This hypothesis was based chiefly on the relatively enormous mass defect of the α-particle, equal, as is known, to \(42.3 \cdot 10^{-6}\) ergs. On the basis of this hypothesis we can calculate the binding energy between the α-particles and the protons and electrons entering into their composition. This energy, obtained simply as the difference between the curve in Fig. 1 and the straight line (shown by a dashed line) with an angular coefficient equal to the binding energy of one α-particle, is presented in Fig. 2. We see that approximately up to the middle of its course the curve descends downward rather smoothly, but farther on it begins to rise in the most unusual manner: the experimental points give segments of the curve that still descend from left to right, but these segments themselves are separated by enormous jumps. Such a course of the curve is very strange and arouses the suspicion that it may be the consequence of the incorrectness of the hypothesis of the formation of the maximum number of α-particles in a complex nucleus. Indeed, the curve of Fig. 2 can be smoothed if we assume that in heavy nuclei part of the α-particles

dissociation, and that the jumps present on the curve are due to this not having been taken into account. Such an assumption is also confirmed by a number of indications from other fields; for example, with no more than three protons in the nucleus at all times, it would be very difficult to explain the large rotational moments observed in a number of heavy nuclei.

Fig. 3.

An entirely new assumption concerning the constituent parts of the nucleus is the supposition that was the immediate consequence of the discovery of the neutron, according to which each nuclear electron is bound, in the first place, with one of the nuclear protons, forming a neutron. Thus we have in a nucleus \(Z\) protons and \(A-Z\) neutrons, which, in turn, combining into groups of two pairs, form \(\alpha\)-particles. In this way we obtain the following composition of the nucleus: for an even atomic number, \(Z/2\) \(\alpha\)-particles and \(A-2Z\) neutrons; for an odd atomic number,

\[ \frac{Z-1}{2} \]

\(\alpha\)-particles, \(A-2Z+1\) neutrons, and one proton. We see that, under such an assumption, the number of \(\alpha\)-particles in heavy nuclei will be somewhat smaller than under the former assumption (for example, for mercury \(Z=80\), \(A=200\), the number of \(\alpha\)-particles, according to the new hypothesis, is only 40 instead of 50). The curve of the internal binding energy of the nucleus, calculated according to this latter hypothesis concerning structure, is presented in Fig. 3, in which, as can be seen, the curve is now quite smooth, beginning to rise upward only in the region of the radioactive elements, which confirms the correctness of the assumption made.

Fig. 4.

Unfortunately, despite Aston’s truly assiduous efforts, the data on mass defects are still far from complete and are not very accurate, which does not make it possible to carry out a more detailed analysis of the experimental-

...tal curves, necessary for obtaining information on the distribution of binding energy among α-particles, neutrons, and protons.

Only in the region of the light elements, using both the results of direct measurement of the mass defect and data on the energy balance in the artificial transformation of elements (the latter gives us the difference between the internal energies of the initial nucleus and the nucleus that is the product of the transformation), can one construct the curve of energy more or less satisfactorily. Such a curve, shown in Fig. 4, may be very valuable for predicting the energy balance of one or another nuclear reaction.

Fig. 5.

§ 2. Another very essential factor for understanding the internal structure of the atomic nucleus is knowledge of its angular and magnetic moment. The moment of the atomic nucleus can be observed through its action on the energy levels of the outer atomic electrons, which under the influence of this action split into several closely spaced sublevels, the number of which is connected in a definite way with the angular momentum of the nucleus, while the magnitude of the splitting itself is determined by the magnetic moment. (Another method for determining the angular momentum of the nucleus is based on the study of the distribution of intensities in band spectra of molecules, but it appears to be less convenient.) In Fig. 5 are presented the values of the angular momentum of various nuclei, expressed in units of the rotational quantum \(h/2\pi\). It is immediately striking that nuclei with even atomic weight (with the exception of nitrogen) are usually altogether devoid of angular momentum, whereas in the case of odd weight the angular momentum is always different from unity, being for the light elements usually equal to one half, and for the heavy...

...heavy ones—sometimes assuming rather large values. What can the rotational moment tell us about the structure of the nucleus?

First of all, we must take into account that the α-particle (as experimental data show) is completely devoid of rotational moment. Since, moreover, all α-particles of a nucleus in its normal state are at the fundamental energy level, likewise deprived of moment, we come to the conclusion that the rotational moment of the nucleus is due exclusively to protons and neutrons not included in the nuclear α-particles. The moment of the proton is, as is known, \(\pm \frac{1}{2}\); the same, evidently, holds for the neutron*. Moreover, since the number of neutrons in heavy nuclei reaches fifty-four, and the Pauli principle forbids more than two neutrons to sit in one and the same orbit, the moments of different neutron orbits may play a role in the formation of the rotational moment of the nucleus.

The moment of the nucleus observed by us is, of course, only the total result of the intrinsic and orbital rotational moments of the neutrons and protons (for odd \(Z\)) in the nucleus, but knowledge of it is essential for testing one or another hypothesis concerning the distribution of neutrons over the various quantum levels inside the nucleus.

Unfortunately, the very large number of attempts now being made to explain the observed values of nuclear moments on the basis of various assumptions about the distribution of nuclear particles over different quantum levels has not yet led to an unambiguous result.

§ 3. Let us now proceed to consider the question of the stability of the atomic nucleus with respect to various transformations. For this it is necessary first of all to make certain assumptions about the character of the interaction between the various constituent parts of the nucleus. For the interaction of two protons, which we may here regard as point charges (since the radius of the proton \(r_p=\frac{e^2}{m_p c^2}=2\cdot 10^{-16}\) cm is considerably smaller than the radius of the nucleus), we may safely adopt the Coulomb repulsive forces with potential.

* The study of the rotational moments of nuclei showed long ago that an electron, when in the nucleus, loses its moment. This fact is understandable from the point of view of modern theory, and for its explanation one should await the appearance of a quantum theory of relativistic motion, as yet nonexistent, which is destined to explain all the riddles connected with nuclear electrons.

The interaction between a proton and a neutron, or between two neutrons, will obviously manifest itself only at distances comparable with the dimensions of the neutron (i.e. several \(\times 10^{-13}\) cm) and will decrease very rapidly as the particles are separated.

Using an analogy taken from the domain of the interaction of atoms and ions, we may suppose that in both cases attractive forces will occur; moreover, in the interaction of a proton with a neutron the mutual potential energy \(-I(r)\) will be considerably greater than the energy \(-K(r)\) corresponding to the interaction of two neutrons. Here it is necessary to point out that, with respect to the potentials \(-I(r)\) and \(-K(r)\), one must make one further additional assumption, namely: when the particles approach one another too closely, these potentials must begin to increase, giving rise to repulsive forces, since otherwise the model of the nucleus will not be stable, showing a tendency to contract to a point.

As for the interaction between \(\alpha\)-particles, it will obviously be composed of the Coulomb repulsion and of the mean force of the exchange interaction of the protons and neutrons entering into their composition. The latter leads, as can be shown, to attraction with a potential energy close to the interaction of neutrons (the forces associated with the potential \(I(r)\) cancel one another), so that we may write for the potential energy of two \(\alpha\)-particles:

\[ +\frac{4e^2}{r}-L(r), \]

where \(L(r)=K(r)\), and likewise decreases very rapidly with distance.

Exact expressions for the potentials \(-I(r)\), \(-K(r)\), \(-L(r)\) are at present unknown. Their theoretical derivation is impossible without the relativistic quantum theory; experimentally, however, they may be derived from data on the scattering of \(\alpha\)-particles in helium and in hydrogen, of neutrons in hydrogen, etc. However, in view of the mathematical difficulties of such a calculation, and partly owing to the lack of precise experimental data, no such calculation has as yet been carried out.

Let us now consider how an aggregate of such particles, with masses of approximately the same order, attracting one another by forces that decrease very rapidly with distance (the Coulomb repulsive forces inside the nucleus may, in the first approximation, be neglected), will behave. The state of such a system must be very analogous to what we have in a small drop of liquid, where the internal forces,

G. GAMOW

acting on any particle balance out (for the radius of action of the forces is smaller than the radius of the nucleus), while near the surface powerful forces arise that prevent the particle from leaving the drop (surface tension) more often. Although an exact solution of the problem for such an aggregate is not yet available, we can draw a number of interesting conclusions about the properties of such a model. First of all, we must assume that the volume of such a model will be approximately proportional to the number of particles, so that the radius will vary approximately as the cube root of the atomic weight*. The potential energy for a given particle inside such a model must be more or less constant and rise sharply at the boundaries, thus forming a kind of “potential well.”

Fig. 6.

From what has been said above about the nature of the forces of interaction between the various particles in the nucleus, it follows that the “bottom” of this “well” for the proton will lie considerably lower than for neutrons or $\alpha$-particles (Fig. 6). The total energy of such a model must be approximately proportional to the number of particles. We must not, however, forget the presence of Coulomb repulsive forces. These forces cannot substantially alter the distribution of the potential inside the nucleus, where the forces of attraction play the chief role. However, these forces will affect the value of the potential at greater distances and will lead to the formation around the nucleus of a potential barrier, which plays such an important role in

Fig. 7.

* That such a dependence, it is true, is rather roughly obeyed for atomic nuclei is well known.

of the theory of nuclear transformations. This raising of the potential well relative to the value of the potential at infinity will, obviously, be entirely absent for neutrons, which are devoid of charge, while for the proton it will be half as large as for the $\alpha$-particle. The distribution of the potential in the nucleus when the Coulomb forces are taken into account is indicated in Fig. 7, where the case of a heavy nucleus is taken, in which the level of the $\alpha$-particle has already risen above the zero level, thereby making possible spontaneous $\alpha$-decay.

The proton level, even for the heaviest nuclei, remains still in the negative region, because even without taking the Coulomb forces into account the proton level lies much deeper than the $\alpha$-particle level; moreover, the raising of the level by the repulsive forces for the proton is half as great. For the neutron, which has no charge, there will be no raising of the level by Coulomb forces at all.

All that has been said above explains to us both the occurrence of $\alpha$-decay in the heavy elements and the absence of the phenomena of spontaneous emission of a proton or a neutron.

Fig. 8.

§ 4. Up to now we have considered the neutrons present in the nucleus as indivisible units, and therefore could construct a model of the nucleus on the basis of ordinary mechanics. Now we shall turn to the decay of the nuclear neutron into a proton and an electron and the ejection of this latter beyond the limits of the atom, i.e. to the very mysterious phenomenon of $\beta$-decay.

As is well known, $\beta$-decay constitutes one of the most striking examples of the electron’s disobedience to all the principles of modern theory. Whereas in nuclear reactions with the participation of heavy particles we always deal with sharply expressed quantum levels and strict observance of the energy balance, in the case of $\beta$-transformations neither the one nor the other takes place. As the experimental investigations of Ellis have shown, the electrons emitted in the decay of different atoms of one and the same substance have the most varied values of energy, varying continuously between zero and arbitrarily large values, and the distribution curve has a form very similar to an error curve (Fig. 8). There is no other radiation that could compensate in this way the equality of energy between the disintegrating nuclei, and yet all the properties and the subsequent behavior of the nuclei before and after the decay are completely identical. From a purely experimental point of view,

from this point of view, the matter here appears as though we were dealing with a violation of the law of conservation of energy. In addition to this basic fact, there is also a whole series of no less fundamental arguments indicating that things are bad with the old electrons; these include, for example, inconsistencies in the statistics of nuclei and in the magnitudes of their angular momenta. The causes of all these irregularities lie in the fact that, as Bohr indicated, here we are already going beyond the limits of the domain in which the classical concept of the electron may be applied. Indeed, for the radius of the electron we have, according to the classical theory, the value

\[ r_0=\frac{e^2}{m_e c^2}=6\cdot 10^{-13}\ \mathrm{cm}, \]

i.e., a quantity comparable with the dimensions of that region in which the electron is compelled to move; and under these conditions such a crude representation of the electron as a charged sphere is, of course, inapplicable.

In this connection it is worth noting the fact that, in estimating the possible velocity of the electron in the nucleus according to the foundations of quantum theory, we arrive at a quantity so close to the speed of light \((0.9998\,c)\) that there can be no question of neglecting the theory of relativity; meanwhile, we still do not have a relativistic theory of quanta.

Until such a general theory, representing an organic synthesis of the modern nonrelativistic quantum theory (wave mechanics) and nonquantum relativistics, has been constructed, there can be no question of a true understanding of the process of \(\beta\)-decay. Nevertheless, even now we may try to construct working theories of \(\beta\)-decay, making use of the old concepts. The basic proposition of the theory of \(\beta\)-stability and \(\beta\)-decay, recently proposed by Heisenberg, consists in the following: closing our eyes to the uncertainty in the energies of \(\beta\)-particles, the necessary and sufficient condition for the possibility of decay is taken to be the positivity of the corresponding energy balance.

Let us consider a nucleus consisting exclusively of \(n\) neutrons “clinging together” with one another. Since between neutrons there exist only forces of attraction, such a nucleus will, of course, be stable with respect to neutrons, i.e., in extracting a neutron from the nucleus we shall expend a certain amount of work, which, obviously, will be of the order of \(-K(r)\), where \(r\) is the mean distance between the particles in the nucleus. Let us decompose the extracted neutron into a proton and an electron, for which work will be required that is determined by the internal binding energy of the neutron \(D\) (this quantity is very small and is equal, according to Chadwick’s measurements, to only one or two million electron-volts, whereas the energies \(K(r)\) and \(I(r)\) are measured in de-

... by tens of millions). We shall now return the obtained proton to the nucleus, thereby obtaining an energy of the order of \(+I(r)\); since \(|I(r)| \gg |K(r)|\), in such a process we shall have a positive energy balance. It is not difficult, however, to see that the reaction carried out is simply equivalent to the extraction of one electron from the nucleus and, since the energy balance is positive, we must expect the occurrence of spontaneous \(\beta\)-decay. Thus, the initially neutral nucleus will begin to emit a successive series of \(\beta\)-particles; the total number \(n_1\) of neutrons composing it will begin to decrease, giving rise to an ever larger number \(n_2\) of protons. However, this process will not go on to completion; in view of the increase of the positive charge of the nucleus, the introduction of new protons into it will be opposed by Coulomb repulsive forces, and finally “replacement of a neutron by a proton” will become energetically disadvantageous. To find the condition of equilibrium, Heisenberg has to make a certain hypothesis concerning the dependence of the work required to extract from the nucleus one neutron or one proton on the total number of neutrons and protons in the nucleus. This hypothesis is made in two steps: first, it is assumed that this work in both cases is a function only of the relative number of neutrons and protons \(\left(f\!\left(\frac{n_1}{n_2}\right)\right.\) and \(\left.g\!\left(\frac{n_1}{n_2}\right)\right)\); secondly, it is assumed that these functions are linear.* Since the work expended against the forces of Coulomb repulsion in bringing a proton into the nucleus (charge \(n_2 e\) and radius \(r_0\)) is equal to \(n_2 e^2/r_0\), we can determine the region of instability with respect to \(\beta\)-decay by the inequality:

\[ - f\!\left(\frac{n_1}{n_2}\right) - D + g\!\left(\frac{n_1}{n_2}\right) - \frac{e^2 n_2}{r_0} \geqslant 0 \tag{3} \]

or, assuming the linearity of \(f\) and \(g\) and putting \(r_0 \sim \sqrt[3]{\,n_1+n_2\,}\) (which approximately corresponds to reality):

\[ \frac{n_1}{n_2} \geqslant C_1 + C_2 \sqrt[3]{\frac{n_2}{n_1+n_2}} . \tag{3'} \]

* In fact, assuming that the interaction of a neutron with the nucleus is chiefly due to its attraction to the nuclear protons, and that of a proton to its attraction to the nuclear neutrons (as Heisenberg does), and that the range of action of these attractions is small in comparison with the dimensions of the nucleus, we may expect that the work of extracting from the nucleus a particle of one kind (a neutron or a proton) will be a monotonically increasing function of the concentration in the nucleus of particles of the other kind. Thus for these works we should write

\[ f'\!\left(\frac{n_1}{n_1+n_2}\right) \quad \text{and} \quad g'\!\left(\frac{n_1}{n_1+n_2}\right) \]

and assume that both functions increase with increasing argument. Since the form of the functions \(f'\) and \(g'\) is unknown, we may pass from here

Considering now the condition for the possibility of α-decay (as was indicated in the preceding paragraphs, the emission of an α-particle must begin much earlier than the emission of protons), Heisenberg writes for the boundary of the region of instability with respect to the emission of an α-particle:

\[ \frac{n_1}{n_2} \leq C'_1 + C'_2 \sqrt[3]{\frac{n_2}{n_1+n_2}}.^{*} \tag{4} \]

For comparison of the considerations set forth with experiment, the graph in Fig. 9 may serve, where on the abscissa is plotted the total number of particles in the nucleus (i.e. \(n=n_1+n_2\)), and on the ordinate the ratio of the number of neutrons to the number of protons for various known nuclei. Two boundary lines are drawn in accordance with equations (4′) and (5), the coefficients being chosen in such a way that the curves best encompass the experimental points (\(C_1=1.173;\ C_2=0.0245;\ C'_1=0.47;\ C'_2=0.077\)).

[[Figure: Fig. 9. Graph with ordinate \(n_1/n_2\), abscissa \(n_2\), and curves labeled “\(\beta\)-boundary” and “\(\alpha\)-boundary.”]]

Fig. 9.

In the region of radioactive elements both curves come very close together (Fig. 10), thereby explaining the fact that a nucleus located in the β-unstable region and having emitted two electrons (the fact that we always have two consecutive β-decays can likewise be obtained from the theory set forth) “jumps over” the region of general stability and falls into the α-unstable region. In

* In fact, α-decay will begin at sufficiently large values of the specific charge of the nucleus,

\[ \frac{n_2}{n_1+n_2}, \]

when the Coulomb repulsive forces acting on the α-particle exceed the forces of intranuclear attraction, or else at sufficiently small values of

\[ \frac{n_1}{n_2}. \]

This is perhaps a less successful expression of Heisenberg’s conclusion that \(f\!\left(\frac{n_1}{n_2}\right)\) will decrease and \(g\!\left(\frac{n_1}{n_2}\right)\) increase with increasing argument. As for the hypothesis of the linearity of the functions \(f\) and \(g\), this hypothesis is, of course, much more dangerous and is justified only by the fact that the actual form of these functions is completely unknown to us.

following this series of α-decays, the point representing the nucleus on the diagram gradually rises, finally again entering the β-unstable region. Such a process of alternating α- and β-decay will continue until, owing to the gradual decrease in the number \(n\), the region of universal stability becomes sufficiently broad not to allow it to be crossed in a single step. Here lie the stable products of radioactive families.

Fig. 10.

Thus Heisenberg’s arguments describe rather well the phenomenon of stability and instability with respect to β-decay, although they do not at all touch upon all the basic difficulties connected with the continuity of β-spectra. Likewise, the existence of such long periods in β-decay, and the complete definiteness of the lifetime of the nuclei of a given element, independently of the different values of the energy of the β-particles, remains entirely unexplained.