An Outline of the Development of the Doctrine of the Structure of the Atomic Nucleus

The Theory of Radioactive Decay

G. A. Gamow, Cambridge

§ 1. It is known to everyone (or, at least, to everyone who takes this journal in hand) that the atom consists of a heavy nucleus carrying a positive charge and of a system of electrons revolving around it like a small planetary system. This is the so-called Rutherford–Bohr model of the atom.

Over the last two decades, the concerted collaboration of experimenters and theorists has made it possible to study in detail and to explain the laws of the electronic system of the atom, and at the present time the theory of atomic structure may be regarded as practically complete. The study of the atom has forced us to reconsider the applicability of the laws of classical mechanics, which turned out to be correct only approximately, and has led to a new, coherent system of quantum (or wave) mechanics.

In parallel with this, the doctrine of the structure of the nucleus was developing. The phenomenon of radioactivity, discovered already at the end of the last century, indicated that the atomic nucleus is not a simple unit, but has a very complex structure. The particles α and β observed in the radioactive decay of elements were interpreted by Rutherford as constituent parts of the nucleus, ejected from the unstable nuclei of heavy atoms, while the very hard radiation observed in decay, the γ-rays, was interpreted as electromagnetic disturbances caused by the rearrangement of nuclei after decay.

Further experiments by Rutherford also showed the possibility of artificially splitting the nuclei of ordinarily stable elements under the influence of external energetic actions.

The discovery of isotopes and the investigations of Aston, which showed that their atomic weights are expressed by numbers very close to integers, made it more than probable to suppose that the nuclei of all elements are built of protons and electrons, with formations consisting of four protons and two electrons (the $\alpha$-particle) and possessing very great stability playing a very large role in the structure of the nucleus.

Very precise measurement of the atomic weights of isotopes revealed small deviations from integers (mass defect), which led to the possibility of determining the total energy binding the separate structural elements of the nucleus into one whole.

Detailed investigations of $\gamma$-ray spectra, which showed their line structure—investigations for which we are indebted chiefly to Ellis and Meitner—led to the conclusion that in the atomic nucleus we are dealing with the existence of definite quantum energy levels, entirely analogous to those which we encounter in the electronic system of the atom.

Finally, quite recently, observation of the hyperfine structure of the lines of the optical spectrum gave an indication of the existence of a definite magnetic moment of the nucleus and the possibility of determining it.^1

At the present time we have an extraordinarily rich, but, truth be told, extraordinarily disorderly experimental material concerning the atomic nucleus, and it is just time for theorists, armed with the powerful instrument of modern quantum mechanics, to take up the question of the structure of the nucleus and the explanation of the observed facts and regularities.

^1 Cf. the article by S. Frisch in the present issue of Uspekhi fizicheskikh nauk.

§ 2.

In view of the extraordinary difficulty of acting upon the atomic nucleus by the means at our disposal, the material obtained in this way is exceedingly meager, and it is natural to expect that the first theoretical conclusions about the structure of the nucleus should be obtained from the study of the natural decay of heavy atoms (the phenomenon of radioactivity),—a field which at the present time has been investigated in very considerable detail.

The most surprising fact that we encounter in the theory of spontaneous nuclear decay is those, often incredibly long, intervals of time during which an unstable nucleus remains in statu quo, before emitting an $\alpha$- or $\beta$-particle. The mean lifetime of radioactive elements varies from an insignificant fraction of a second to extraordinarily long periods of many millions of years and, for each given element, is a quite definite quantity.

It seemed very difficult to find the causes delaying the emission of a particle for such long intervals of time, if the particle has sufficient energy to leave the nucleus,—and yet the $\alpha$- and $\beta$-particles ejected from the nucleus carry very, very substantial stores of energy.

It had long been known that there exists a quite definite dependence between the energy of the emitted particle and the mean period of its stay in the nucleus in the unstable state (the period of decay of the nucleus). In 1912 Geiger and Nuttall observed that if, for elements possessing $\alpha$-decay, we plot on the abscissa axis the energy of the $\alpha$-particles and on the ordinate axis the logarithm of the corresponding decay constant, then for a given radioactive family the points will lie approximately on a straight line. The three radioactive families known to us—uranium-radium, thorium, and actinium—are represented by three parallel straight lines. The Geiger–Nuttall diagram is shown in Figure 1, where we can note a number of deviations from the linear law. First, the values of $\log \lambda$ corresponding to very large or very small values of the energy of the $\alpha$-particle prove to be systematically

smaller than is required by the linear law (a fact first noticed by Jacobson and Gudden), indicating that in reality we are dealing with a certain curve turned with its concavity downward.

Secondly, we note (especially sharply for AcX) cases where the experimental point jumps out of the Geiger–Nuttall straight line, indicating some kind of sharp anomaly.

It is quite clear that before trying to explain the regularities connected with the emission of α-particles from the nucleus, we must know something about the forces acting on the α-particle near and inside the nucleus itself.

Fig. 1

The positively charged α-particles will, of course, experience Coulomb repulsion from the remainder of the nucleus. The potential energy of the Coulomb forces may be written in the form:

\[ U_c(r)=+\frac{2(Z-2)e^2}{r}, \]

where \(Z\) is the atomic number of the disintegrating nucleus, and \(e\) is the elementary charge. To explain the existence of α-particles inside the nucleus, it is necessary to assume the existence of certain additional attractive forces, acting only at very short distances from the nucleus. As to the nature of these forces, we may construct various hypotheses—they may be either polarization forces (decreasing with distance as \(1/r^5\)) or forces of quantum interaction (Austauschenergien) between the internal structures of the α-particle and

of the remaining nucleus—these forces decrease exponentially with distance. The existence of such attractive forces can be observed experimentally: in experiments on the scattering of $\alpha$-particles in various elements, the particles approach the nucleus to very small distances and may enter the region of action of these forces.

The experiments of Rutherford and Chadwick showed that, in the case of very close collisions of $\alpha$-particles with the nuclei of light elements, deviations are observed in the number of scattered particles from the formula derived under the assumption of Coulomb interaction. The observed deviations can be explained by assuming the existence of the indicated attractive forces; in this way we can form an idea of the range of action and of the laws of these forces. Unfortunately, at present there is no sufficiently detailed investigation of the anomalous scattering of $\alpha$-particles, and the theoretical conclusions amount, approximately, to the following. For light elements (Mg, Al) anomalous attractive forces begin to make themselves felt at distances of the order of $10^{-12}\ \text{cm}$, varying approximately in inverse proportion to the fourth or fifth power of the distance, and they overpower the Coulomb repulsion at a distance of about $3\cdot 10^{-13}\ \text{cm}$ from the center of the nucleus; at smaller distances the $\alpha$-particle is evidently already under the influence of the resultant attractive forces. For the nuclei of the heavy radioactive elements of interest to us, in view of their large charge, the $\alpha$-particles available to us cannot approach to such small distances and reach the region of anomalous forces. Rutherford and Chadwick, in experiments on the scattering of $\alpha$-particles in uranium, were able to reach (using the fastest $\alpha$-particles) only distances of $3\cdot 10^{-12}\ \text{cm}$, and no deviations from normal scattering were observed—the region of attractive forces here evidently lies much closer to the nucleus than $3\cdot 10^{-12}\ \text{cm}$.

It would seem that the results of these experiments with uranium can help us very little—since the region of attractive forces could not be reached; however, precisely

in these experiments lay the key to the solution of the phenomenon of α-decay.

In comparing them with the data on the decay of the uranium nuclei themselves, these experiments lead to a paradox, completely inexplicable from the point of view of classical mechanics. Indeed: the nuclei of uranium atoms are unstable and emit α-particles with an energy of about \(6.8 \cdot 10^{-6}\) erg. According to our assumption concerning the existence of attractive forces near the nucleus, an α-particle sitting in the nucleus of a radioactive element is surrounded by a kind of potential barrier, as shown in Figure 2.

Fig. 2

The fact that even at distances of \(3 \cdot 10^{-12}\) cm we have only Coulomb forces indicates that the maximum height of the barrier is in any case greater than

\[ \frac{2(Z-2)e^{2}}{3 \cdot 10^{-12}} = 14 \cdot 10^{-6}\ \text{erg} \]

(for uranium \(Z = 92\)).

How can a uranium α-particle with an energy of only \(6.8 \cdot 10^{-6}\) erg “roll over” such a barrier? In other words: if the \(\alpha\)-particles of RaC′, used in scattering experiments in uranium, while “rolling in” along the outer slope of the barrier, still could not reach its summit, how can uranium α-particles, possessing a considerably smaller

energy, roll over the barrier and fly out? From the point of view of classical mechanics, an α-particle passing through such a barrier, higher than its total energy, would have to possess inside the barrier “negative kinetic energy” and, consequently, an “imaginary velocity.”

However, the possibility of such a phenomenon, which is in sharp contradiction with classical mechanics, is a direct consequence of modern wave mechanics. Just as in wave optics light, falling on the interface between two media at an angle greater than the angle of total internal reflection, partly penetrates into the second medium—so, in precisely the same way in wave mechanics de Broglie–Schrödinger waves can partly penetrate into the region of “imaginary velocity,” making it possible for particles to “roll” through the barrier.

We shall now analyze the simplest case of a rectangular barrier and derive formulas for its “penetrating power.” We shall specify the potential distribution by the conditions:

\[ \begin{aligned} U(x)&=0 \qquad &&x<0,\\ U(x)&=U_0 \qquad &&0<x<l,\\ U(x)&=0 \qquad &&l<x . \end{aligned} \]

The Schrödinger equation is written in the form:

\[ \frac{\partial^2 \psi}{\partial x^2} -\frac{4\pi i}{h}\cdot \frac{\partial \psi}{\partial t} +\frac{8\pi^2 m}{h^2}\,U(x)=0 \]

Putting:

\[ \varphi(x,t)=\Psi(x)e^{\frac{2\pi}{h}Et}, \tag{4} \]

where \(E\) is an arbitrary constant giving the energy of the system, we have, for the determination of \(\Psi\):

\[ \frac{\partial^2 \Psi}{\partial x^2} +\frac{8\pi^2 m}{h^2}\,[E-U(x)]\,\varphi=0. \tag{5} \]

We consider the case of penetration through a barrier which is classically impenetrable, and therefore \(E<U_0\).

The solutions of equation (5) in regions I, II and III will be, respectively:

\[ \Psi_{\mathrm{I}}(x)=A_{+}e^{ikx}+A_{-}e^{-ikx} \tag{6} \]

\[ \Psi_{\mathrm{II}}(x)=B_{+}e^{k'x}+B_{-}e^{-k'x} \tag{6'} \]

\[ \Psi_{\mathrm{III}}(x)=C_{+}e^{+ikx}+C_{-}e^{-ikx}, \tag{6''} \]

where

\[ k=\frac{2\pi}{h}\sqrt{2mE} \qquad k'=\frac{2\pi}{h}\sqrt{2m(U_0-E)}. \]

These solutions must, at the boundaries of the potential discontinuity \((x=0,\ x_1=l)\), satisfy the conditions of continuity of the function itself and of its first derivative.

Fig. 3

Substituting the values (6) into (4), we see that the expressions (6) and \((6'')\) each represent two waves traveling in opposite directions, with amplitudes \(A_{+}\) and \(A_{-}\), and respectively \(C_{+}\) and \(C_{-}\). According to the physical meaning of the sought solution, we must have two waves (the incident and the reflected) in region III, but only one wave (the one that has passed through the barrier) in region I.

Accordingly, in formula \((6'')\) we must set \(A_{-}=0\) (and \(A_{+}=A\)). The continuity conditions at the boundaries give:

\[ B_{+}=\frac{1}{2}A\left(1+i\frac{k}{k_1}\right); \qquad B_{-}=\frac{1}{2}A\left(1-i\frac{k}{k_1}\right) \tag{8} \]

and

\[ C_{+}=A(\operatorname{ch}k'l+iD\,\operatorname{sh}k'l); \qquad C_{-}=iAS\,\operatorname{sh}k'l\,e^{+ikl} \tag{9} \]

where

\[ S=\frac{1}{2}\left(\frac{k}{k'}+\frac{k'}{k}\right); \qquad D=\frac{1}{2}\left(\frac{k}{k'}-\frac{k'}{k}\right) \tag{9'} \]

From (9) we obtain:

\[ |A|^2=|C_{+}|^2-|C_{-}|^2, \tag{10} \]

which gives the law of conservation of the particle flux. The coefficient of permeability of the barrier, given by the ratio of the squares of the amplitudes of the transmitted and incident waves, turns out to be equal to:

\[ \chi=\frac{|A|^2}{|B|^2}=\frac{1}{\operatorname{ch}^2 k'l+D^2\operatorname{sh}^2 k'l}. \]

In the case \(k'l\gg 1\), which always obtains for the barriers encountered in \(\alpha\)-decay, we may replace the hyperbolic functions by \(\frac{1}{2}e^{+k'l}\), and we obtain for the coefficient of transparency:

\[ \chi=\frac{4}{(1+D)^2}e^{-\frac{4\pi\sqrt{2m}}{h}\sqrt{U_0-E}\cdot l} \tag{11'} \]

From (11) we see that here the chief role is played by the exponential factor

\[ e^{-\frac{4\pi\sqrt{2m}}{h}\sqrt{U_0-E}\cdot l} \tag{12} \]

which, for sufficient height and width of the barrier, can be extremely small; for radioactive nuclei this factor proves to be of the order of \(10^{-30}\), which explains the very long periods of radioactive decay.

We have analyzed the case of a rectangular barrier; however, it can be shown that a completely analogous formula will be valid for a barrier of any shape, if the permeability of this barrier is small. In this case the factor (12) must be replaced by:

\[ e^{-\frac{4\pi}{h}\sqrt{2m}\int_{r_1}^{r_2}\sqrt{U(r)-E}\,dr} \tag{13} \]

where the integration is carried out over the whole region of imaginary velocity (i.e. where \(U(r)>E\)).

§. 3. Turning to the question of the emission of an \(\alpha\)-particle from a nucleus surrounded by a certain potential barrier (see § 3), we must first of all know the form of this barrier. We have already seen that the course of the potential of the anomalous attractive forces near and inside the nucleus (the inner slope) is not at all

known; on the other hand, it is easy to see that the exact course of the potential on the inner circle of the descent of the barrier has comparatively little effect on its penetrability. In this case it is most rational to make the simplest assumptions about its form; for the subsequent calculations we shall adopt the barrier model given by formulas (14) (see Fig. 4):

\[ U(r)=\frac{2(Z-2)e^2}{r}\quad \text{for } r>r_0 \]

\[ U(r)=U_i=\text{const}\quad \text{for } r<r_0. \]

This model is characterized by two unknown quantities: the radius of the nucleus \(r_0\) and the internal potential \(U_i\).

Fig. 4

The question of the emission of an \(\alpha\)-particle from the space surrounded by the potential barrier reduces to the solution of the wave equation which gives, outside the nucleus, a diverging spherical wave. This problem leads to a series of discrete (quantum) energies of the \(\alpha\)-particle sitting inside the barrier, and to a series of corresponding probabilities of emission.

In the present essay, however, we shall not dwell on the exact solution of the problem, and shall content ourselves with an approximate derivation, nevertheless quite sufficient for comparison with experimental data. In view of the great height of the barrier, we may, in a first approximation, regard the motion of the particle inside the nucleus as confined between infinitely high walls, disregarding the fact that after a million or two years the particle will nevertheless escape. We shall be interested only in the state of least energy (the fundamental orbit), since it may now be considered more than probable that all \(\alpha\)-particles in the nucleus have quantum number one.

In this case,¹ as is known, the kinetic energy of the particle is expressed by the following formula:

\[ K=\frac{h^{2}}{8\pi m r_{0}^{2}} . \]

Taking into account that the bottom of our well is at the level \(U_i\), we have, for the total energy with which the \(\alpha\)-particle can fly out, the value:

\[ E=U_i+\frac{h^{2}}{8\pi r_{0}^{2}} . \]

The probability of escape can be calculated approximately as the product of the “number of collisions of the \(\alpha\)-particle with the barrier” and its penetrability, i.e.

\[ \lambda= \frac{\sqrt{E-U_i}}{\sqrt{2m}\,r_0}\, e^{-\frac{4\pi}{h}\sqrt{2m}\int_{r_0}^{\frac{2(Z-2)e^2}{E}} \sqrt{\frac{2(Z-2)e^2}{r}-E}\,dr} \tag{16} \]

Taking into account that within the interval of integration

\[ \frac{2(Z-2)e^{2}}{r} \gg E \]

and integrating, we have:

\[ \lambda= \frac{\sqrt{E-U_i}}{\sqrt{2m}\,r_0}\, e^{-\frac{4\pi^{2}e^{2}\sqrt{2m}}{h}\cdot\frac{Z-2}{\sqrt{E}} +\frac{16\pi e\sqrt{m}}{h}\sqrt{(Z-2)r_0}} \tag{16'} \]

or, introducing the velocity of the \(\alpha\)-particle \(v\):

\[ \lambda= \frac{\sqrt{v^{2}-\frac{2U_i}{m}}}{2r_0}\, e^{-\frac{8\pi^{2}e^{2}}{h}\cdot\frac{Z-2}{v} +\frac{16\pi e\sqrt{m}}{h}\sqrt{(Z-2)r_0}} \tag{16''} \]

Formulas (15′) and (16′) are sufficient for calculating the energy and the decay constant for the given model of the nucleus, and also for the inverse calculation of the constants of the model \(r_0\) and \(U_i\) for known radioactive elements.

Here we must emphasize the substantial difference between the applicability of both formulas to real cases.

¹ The problem is reduced to finding the fundamental frequency of a spherical resonator (acoustics).

In the formula determining $\lambda$, the dominant role is played by the exponential factor, which, apart from known quantities, depends only on the radius of the nucleus $r_0$. The quantity $U$, which determines a certain mean potential inside the nucleus and depends essentially on the type of model, enters only into the first factor, which plays a negligibly small role.^1

In view of this, formula (16′) (and similar formulae) may serve for a very accurate calculation of $r_0$, the radius of the nucleus of radioactive elements. The formula for $E$, on the contrary, depends very strongly on the adopted model of the interior of the nucleus, and therefore the values of $U$ obtained in this way can as yet give only a very general idea of the internal potential.

Formula (16′), which gives the exponential dependence of the decay constant on the energy of the $\alpha$-particle, is a mathematical expression of the Geiger–Nuttall law. Expression (16″) shows that $\lg \lambda$ is not a linear function of $E$ and may be taken as such only for small changes of $E$; in reality the graph $(\lg \lambda;\ E)$ is a curved line concave toward the $E$-axis, which agrees well with the experimental data (see § 2). The second important conclusion following from the theory is that $\lg \lambda$ depends not only on $E$, but also on the atomic number of the element $Z$, and the graph $(\lg \lambda, E)$ is in fact not real. However, owing to the fact that, in the series of radioactive elements, the energy of the emitted $\alpha$-particles usually changes in parallel with the atomic number $Z$, the Geiger–Nuttall graph gives a more or less smooth curve. In those places where the parallel course of the energy of the $\alpha$-particles and of the atomic number of the element is disturbed (for example, for AcX), anomalies in the course of the Geiger–Nuttall curve should be expected. This explains the long-known deviations from this law; the observed angularities in the graph $(\lg \lambda, E)$ fully coincide with the prediction of the theory.

^1 This is indicated at least by the fact that the five methods so far proposed for obtaining the formula for $\lambda$ give five different expressions for this coefficient, which, however, has scarcely any effect on the numerical results.

We have already indicated that formula (16″) can serve for a very accurate determination of the radius of the nucleus. The values obtained for the radius \(r_0\) of our model for the uranium-radium family are given in Fig. 5. We see that the radius decreases quite regularly with decreasing atomic weight of the nucleus. The decrease of the radius is approximately inversely proportional to the cube root of the atomic weight (this regularity also extends into the region of light elements, for which the radius can be determined from the anomalous scattering of \(\alpha\)-particles), which leads to the conclusion that the density of the nucleus always remains constant.

§ 4. It would seem that the phenomenon of \(\beta\)-decay should be easily explained on the same general grounds as \(\alpha\)-decay.

In fact, the phenomenon of the ejection of a nuclear electron is in many respects analogous to the ejection of an \(\alpha\)-particle. We encounter here the same very long periods and quantitatively the same dependence between energy and decay period: slower \(\beta\)-particles correspond to longer lifetimes of the nucleus.

Fig. 5.

A significant difference, however, is the fact that the spectrum of \(\beta\)-particles is diffuse.

Ellis’s investigations have established quite reliably that \(\beta\)-particles leave nuclei with velocities varying within very wide limits; on the other hand, there is absolutely no process capable of compensating for this diffuseness of energies and balancing the total energy of the nucleus. According to the law of conservation of energy, the nuclei obtained after \(\beta\)-decay ought to have the most varied reserve of energy, whereas the discreteness of the velocities of \(\alpha\)-particles and the line structure of \(\gamma\)-spectra indicate a quite definite discrete energy of nuclei.

We thus arrive at the conclusion that, for electrons located inside the nucleus and emitted from it, the law of conservation of energy proves to be inapplicable.

This, together with a whole series of other difficulties connected with the question of the motion of electrons inside the nucleus, indicates that here we have encountered something entirely new, something that cannot be explained on the basis of contemporary theoretical conceptions. There is no doubt that all these difficulties in quantizing particles moving with velocities very close to the speed of light are directly connected with those fundamental contradictions that modern theoretical physics has met in its attempts to generalize wave mechanics to the case of relativistic motion. The study of the properties of electrons in the nucleus is at present the only field capable of providing experimental material for the further development of the basic principles of theoretical physics.