An Outline of the Development of the Theory of the Atomic Nucleus*

G. A. Gamow (Leningrad)

II. Excited States of the Nucleus and γ-Rays

§ 1. A very important method for obtaining information about the structure of the atomic nucleus is the study of the spectra of γ-rays emitted by nuclei excited in the process of radioactive decay. In view of the very short wavelength of γ-rays, the ordinary diffraction methods for investigating spectra are almost inapplicable in this region, and almost all the data at our disposal have been obtained by analyzing the secondary electron groups ejected by γ-rays from the various electron shells of the decaying atom. The study of the distribution of velocities of the secondary electron groups not only gives us data on the spectral composition and relative intensity of the γ-radiation lines, but also, as we shall see below, makes it possible to draw a number of conclusions about the quantum characteristics of the various energy levels of the nucleus. To a given γ-quantum there corresponds, in the magnetic spectrum of secondary electrons, a number of groups ejected from the various electron shells of the atom \((K, L_1, L_2, L_3, M_1, \ldots)\) and separated from one another (on the energy scale) by distances equal to the differences in the binding energies of the indicated shells. We see that this method of investigation can also directly answer the question of the atom of which element emits

* See the first article of this review, Uspekhi fiz. nauk, vol. X, p. 531 (1930).

a given γ-line (for different elements the binding energies of the electron shells are different), which is important in those cases when, with the modern experimental technique, it is impossible to obtain the given radioactive element in pure form (for example, in the case of the C-, C′- and C″-products of radioactive families).

It must not be forgotten that, in view of the fact that the time of γ-radiation is much shorter than ordinary periods of decay, the γ-rays belonging to the nucleus of some element will be observed simultaneously with the decay of the preceding element (the decay that was the cause of the excitation). Thus the γ-rays observed in the decay, say, of ThB, in reality belong to the excited nucleus of the following element (ThC). In calculating the energy of γ-quanta from the energy of secondary electron groups, we must take the binding energy of the electron shells of the product atom.

Turning to the process of the ejection from the atom of secondary electron groups, one might think that here we are dealing with the ordinary photoelectric effect of the γ-radiation coming from the nucleus in the electron shells of the atom. A detailed experimental and theoretical analysis of the question shows, however, that this is far from the case, that the chief role here is played by factors directly connected with the atomic nucleus itself, and that only a very small part of the secondary electrons is due to the ordinary photoelectric effect. We arrive at this conclusion by comparing the experimental data on the relative number of secondary electrons and the corresponding γ-quanta (the coefficient of internal conversion) with the theoretically calculated values of the photoelectric coefficient for the γ-rays under investigation. We take the experimental data from the investigations of Ellis,* who succeeded in developing a method for the simultaneous quantitative determination of the intensities of γ-rays and of secondary electron groups. The intensities of the strongest groups in the spectrum of RaC, according to Ellis’s measurements, are indicated in the second and third columns of Table I, where the numbers of γ-quanta are given directly.

* C. D. Ellis and G. H. Aston, Proc. Roy. Soc. A., Vol. CXXIX, p. 1930.

TABLE 1

Internal conversion of γ-rays from RaC (levels of the RaC′ nucleus).

quanta and the number of electrons torn from the \(K\)-shell, calculated per one decay event.

The number of secondary electrons corresponding to one γ-quantum is given in the fourth column, while the fifth column gives the photoeffect coefficient calculated from the formula:

\[ {}_k K = \frac{1}{Zp}\,\alpha^2(2\alpha)^{2\sqrt{1-\alpha^2}}\, \left| \frac{\Gamma\left\{\sqrt{1-\alpha^2}-i\alpha\right\}} {\Gamma\left\{1+2\sqrt{1-\alpha^2}\right\}} \right|^2 e^{-2\alpha \operatorname{arc}\cos \alpha}, \tag{1} \]

where

\[ \alpha=\frac{Ze^2}{hc} \quad \text{and} \quad p=\frac{\left(\frac{v}{c}\right)} {\sqrt{1-\left(\frac{v}{c}\right)^2}}, \tag{1′} \]

derived specially for the case of very hard electromagnetic radiation in atoms of heavy elements*.

Here \(Z\) is the atomic number of the emitting atom, and \(v\) is the velocity of the ejected electron. We see that the numbers in the fifth column are ten to twenty times smaller than the observed effect, which indicates that, for the γ-rays listed in the table, the main role is played by some other interaction factor.

* H. Casimir, Nature, December, 20, 1930.

In view of the fact that at large distances from the nucleus we can be quite certain of the applicability of our calculations, the cause of the effect must be sought in the interaction of the atomic electron with the nucleus at distances comparable with the dimensions of the nucleus. We may speak of an effect of collision of the electron with the nucleus and of the transfer of energy by direct interaction, without the mediation of radiation forces (analogously to collisions of the second kind). In view of our ignorance of the forces acting on the electron at such small distances from the nucleus, we cannot at present estimate theoretically the probability of such a process. Nevertheless, we can put forward one important conclusion, justified by experiment. Since the process under consideration is caused by the collision of an atomic electron with the nucleus, it must be very weak for those electron orbits of the atom for which the probability density at the center is zero, and for the other orbits it must be proportional to the density at the center. For hard $\gamma$-rays this expectation is brilliantly justified by experiment: conversion of $\gamma$-rays is observed only for the $K$, $L_1$, $M_1$, $N_1$, ... orbits, for which, as is known, the probability density at the center differs from zero; likewise, the relative intensities of the secondary electron groups corresponding to one and the same $\gamma$-line agree rather accurately with the probability density in the region of the nucleus calculated for the atomic electron. In the region of soft $\gamma$-rays the probability of the photoeffect is considerably greater, and here it begins to predominate over the effect of direct collision. This is evident from the fact that here conversion is observed on all, without exception, orbits of the atomic electrons.

Here we must pay special attention to the $\gamma$-line $2.267 \cdot 10^{-6}$ erg from the spectrum of RaC. As is seen from Table 1, this line is completely absent as $\gamma$-radiation, whereas very intense groups of secondary electrons corresponding to it are observed. The explanation of this fact must be sought in the supposition that the corresponding radiative transition in the excited RaC nucleus is for some reason forbidden—this may, for example, occur, as a detailed investigation shows, if the initial and final

state of the nucleus both have azimuthal quantum number equal to zero. Since the “selection principle for radiative transitions” need not be applicable to the process of direct transfer of energy by an excited nucleus to an atomic electron, we can understand how, and in what way, secondary electron groups can exist without the presence of the $\gamma$-line itself. Up to the present time only one forbidden $\gamma$-line is known to us, but it may be hoped that, with a more detailed investigation of $\gamma$-spectra, a number of others will be discovered, which will help us in solving the problem of constructing a classification of the levels of the atomic nucleus.

We can estimate the probability of $\gamma$-radiation on the assumption that we are dealing here with the radiation of an oscillating charged particle (dipole); this may be an $\alpha$-particle or a proton*. For this case we may write for the transition probability:

\[ \chi = \frac{8\pi}{3}\cdot \frac{Z^2 e^2 \nu_{n,m}^{\,2}}{h^2 c^3}\, r_{n,m}^{\,2}, \tag{2} \]

where $Ze$ is the charge of the radiating particle, $h\nu_{n,m}$ is the energy of the emitted $\gamma$-quantum, and $r_{n,m}$ is the matrix element corresponding to the transition. From the well-known relation of wave mechanics:

\[ \sum_{m=0}^{n-1} \frac{4\pi m \nu_{n,m}}{h}\, r_{n,m}^{\,2} = 1 \tag{3} \]

we have:

\[ r_{n,m}^{\,2} < \frac{h}{4\pi m \nu_{n,m}}. \tag{3'} \]

And formula (2) is reduced to the form:

\[ \chi < \frac{8\pi^2}{3}\cdot \frac{Z^2 e^2}{c^3 h^2 m}\,(h\nu_{n,m})^2. \tag{2'} \]

Of course, expression (2′) gives us only an upper limit, and the actual value of the probability may be several times smaller.

* The assumption that $\gamma$-rays are emitted by nuclear electrons does not withstand criticism; in that case the $\gamma$-lines could not be as sharp as they are in reality.

We have already said that, in view of our ignorance of the forces, we cannot estimate the probability \(\mu\) of the ejection of a secondary electron upon collision with a nucleus. One may, however, attempt to estimate this probability on the assumption of a Coulomb interaction between the electron approaching the nucleus and the \(\alpha\)-particles moving inside the nucleus. Such a very approximate estimate was made by Fowler* and gives, for the line \(2.267 \cdot 10^{-6}\) erg, the value \(10^{11}\ \mathrm{sec}^{-1}\). In view of the fact that the actual interaction forces must be supposed to be considerably greater than the Coulomb forces, this quantity can give us only a lower limit for the probability sought.

§ 2. We now turn to a very important phenomenon closely connected with quantum transitions of the excited nucleus: \(\alpha\)-particles of anomalously high velocity (long-range). If an \(\alpha\)-particle is in the nucleus at an excited level, it is possible that, instead of falling to a lower level with the emission of a \(\gamma\)-quantum, the \(\alpha\)-particle will cross the potential barrier surrounding the nucleus and will fly out with the full energy of the excited level. Since the probability of transition through the barrier for radioactive nuclei is many times smaller than the probability of \(\gamma\)-emission, we may hope to observe such \(\alpha\)-particles only for the shortest-lived nuclei, and even then only in very small numbers. Indeed, up to the present it has been possible to observe \(\alpha\)-particles of high velocity only in the case of ThC′ \((\lambda = 10^{9}\ \mathrm{sec}^{-1})\) and RaC′ \((\lambda = 10^{5}\ \mathrm{sec}^{-1})\). The energy excesses of the various high-velocity groups relative to the normal group, as well as their intensities, are indicated in Tables IIa and IIb.

TABLE IIa

\(\alpha\)-groups of high velocity from ThC′

* R. H. Fowler, Proc. Roy. Soc. A., Vol. 129, p. 1 (1930).

TABLE IIb

High-velocity α-groups from RaC′

As is easily seen, the energy differences indicated in the third column of Tables IIa and IIb give us directly the positions of the various quantum levels of the disintegrating nucleus. The problem of arranging the known \(\gamma\)-lines into the serial scheme outlined by the study of high-velocity \(\alpha\)-particles has still not been solved. In the case of ThC′ the hard \(\gamma\)-rays belonging to the decay of this element have not been detected at all—evidently their intensity is too small (less than one percent).

In the case of RaC′ we have a spectrum very rich in strong lines (up to 0.6 \(\gamma\)-quanta per decay process); but although some lines can be definitely localized in the level scheme given by Table IIb*, the inaccuracy in determining the latter has so far not made it possible to solve the problem completely.

From the relative value of the intensity of the various high-velocity \(\alpha\)-groups we can draw important conclusions concerning the probability of various transitions in the excited nucleus. Indeed, if \(\chi^{(n)}\) is the probab-

* For example, the \(\gamma\)-line \(h\nu = 0.933 \cdot 10^{-6}\) erg with intensity 6.0 almost certainly corresponds to the transition from level \(\alpha^{II}\) to the normal level \(\alpha_0\).

G. A. GAMOV

the probability of the radiative transition, and \(\lambda^{(n)}\) is the decay constant for the excited level, then the total number of \(\alpha\)-particles of velocity greater than that of our level will be:

\[ N^{(n)}=p^{(n)}\frac{\lambda^{(n)}}{\chi^{(n)}};\qquad \chi^{(n)}=\frac{p^{(n)}\lambda^{(n)}}{N^{(n)}} , \tag{4} \]

where \(p^{(n)}\) is the excitation percentage (\(p<0.01\) for ThC′ and \(p\simeq 1\) for RaC′). The quantity \(\lambda^{(n)}\) can be calculated by the usual formula for radioactive decay:

\[ \lambda^{(n)}=\frac{h}{4mr^2}\, e^{-\frac{8\pi^2 e^2 (Z-2)}{h}\frac{1}{v_n} +\frac{16\pi e\sqrt{m}}{h}\sqrt{(Z-2)r^{(n)}}}, \tag{5} \]

where \(m\) and \(v\) are the mass and velocity of the emitted \(\alpha\)-particle, and \(r^{(n)}\) is the radius of the nucleus for the excited particle. For a preliminary estimate we may assume that the critical radius \(r^{(n)}\) for the \(n\)-th excited state of the \(\alpha\)-particle is the same as in the unexcited state; however, it must be remembered that, owing to the expansion of the potential crater of the nucleus upward, the radius for particles with large energy may have a somewhat larger value.

For the \(\alpha\)-particles of the two known groups of ThC′ and for the main group of RaC′ we can calculate, by formula (5), the values of the decay constants and obtain:

\[ \lambda=3\cdot 10^{15},\ 5\cdot 10^{12}\ \text{and}\ 7\cdot 10^{9}\ \text{sec}^{-1}. \]

Assuming \(p<0.01\) for ThC′ and \(p\simeq 1\) for RaC′ and taking the value \(N\) from Tables 2a, b, we have from formula (4):

\[ \chi\simeq 5\cdot 10^{13},\ 3\cdot 10^{14}\ \text{and}\ 6\cdot 10^{14}\ \text{sec}^{-1}, \]

whereas formula (2′) gives:

\[ \chi\simeq 6\cdot 10^{15},\ 3\cdot 10^{16}\ \text{and}\ 2\cdot 10^{16}\ \text{sec}^{-1}. \]

Thus we see that the values we have obtained for the probability of emission are considerably smaller than would be expected from formula (4′). To explain this fact we may make two different hypotheses. The first hypothesis: the probability of emission is indeed much smaller than corresponds to dipole radiation. This would mean that

Outline of the Development of the Theory of the Atomic Nucleus

excited nuclei, owing to the special character of the symmetry, are deprived of a dipole moment, and we are dealing with radiation of a higher order (quadrupole). The second hypothesis for calculating the decay constants for highly excited $\alpha$-particles in the nucleus is incorrect. This would mean that, for excited states, the radius is somewhat larger (by 20–30% for the groups we considered), which in general agrees with our model of the nucleus not as a “rectangular box,” but as one that expands somewhat upward. Which of the two hypotheses indicated above actually holds cannot yet be decided because of the insufficiency of the experimental material.

§ 3. Up to now we have spoken of processes occurring in the excited nucleus; now we shall turn to the consideration of the excitation process itself.

We shall consider only excitation of the nucleus that is a consequence of $\alpha$-decay, for the process of $\beta$-decay and all phenomena connected with it are at the present time very unclear.

If, after the emission of an $\alpha$-particle, the product nucleus remains in an excited state, then it is clear that the emitted $\alpha$-particle will possess less energy than in the case when the decay process leads to the normal state of the product nucleus.

The difference in energy between the various groups of slow $\alpha$-particles must correspond to the spacings of the levels of the excited product nucleus. After the decay, the nucleus that remains excited will pass into the normal state with emission of the excess energy in the form of $\gamma$-rays. If $E_{\alpha}$ is the energy of the normal group of $\alpha$-particles of the decaying nucleus, and $E_0, E_1', E_2' \ldots$ are the energies of the excited levels of the product nucleus, then the various slow $\alpha$-groups will have energies:

\[ E_1 = E_{\alpha} - [E_1' - E_0'],\quad E_2 = E_{\alpha} - [E_2' - E_0'] \ldots \]

The intensity of the various groups will decrease very rapidly with decreasing energy, owing to the increasing difficulty of penetration through the potential surrounding the nucleus—

... barrier. The existence of such groups of low velocity has in fact recently been discovered for a whole series of $\alpha$-decaying elements by the method of deflection in a strong magnetic field (Rosenblum) and by the differential method of analyzing the ranges of $\alpha$-particles (Rutherford)*. Since the separation of the groups lies almost at the limit of the old measurement methods, these groups have received the name “fine structure of $\alpha$-rays.” Rosenblum was able to show that the $\alpha$-radiation of ThC consists of five closely spaced groups, as indicated in Table III.

TABLE III

$\alpha$-groups of low velocity from ThC

From what has been said, one should expect to find in the spectrum of $\gamma$-rays emitted in the decay of ThC lines corresponding to the various transitions between the levels indicated in Table 3. Such $\gamma$-lines can indeed be found (0.0649; 0.231; 0.259; 0.444; 0.698; $0.762\cdot 10^{-6}$ erg). By selecting the energies of the levels marked by the low-velocity $\alpha$-groups so as to obtain the smallest mean error in comparison with the $\gamma$-rays, we arrive at the system of levels indicated in Table IVa.

Table 4b shows the agreement between the differences of the level energies and the measured energies of the $\gamma$-quanta—as can be seen, the agreement is more than satisfactory.

* S. Rosenblum, C. R., Vol. CXC, p. 1124 (1930).
** E. Rutherford, Proc. Roy. Soc. A. Vol. CXXIX, p. 211 (1930).

Essay on the Development of the Theory of the Atomic Nucleus

TABLE IVa

Energy levels of the nucleus ThC″

TABLE IVb

Levels and \(\gamma\)-rays of ThC″

In the spectrum of \(\gamma\)-rays three lines are absent (transitions \(C-A\); \(D-A\); \(D-B\)), which should be expected according to our serial scheme; this points to the existence of certain forbidden transitions in the nucleus.

The relative intensity of the various low-velocity groups can be calculated as the ratio of the decay constants. Using the general formula (5), we find:

\[ \frac{I_{\alpha_n}}{I_{\alpha_0}} = e^{ \frac{[[unclear: numerator]]}{h}(Z-2) \left(\frac{1}{v_n}-\frac{1}{v_0}\right) + \frac{16\pi e[[unclear]]}{h}\sqrt{Z-2} \left(\sqrt{r_n}-\sqrt{r_0}\right) }, \tag{6} \]

where \(r_n\) is the radius of the nucleus in the \(n\)-th excited state. Setting \(r_n=r_0\), we obtain from formula (6) a decrease of intensity considerably more rapid than that observed. This indicates that we must ascribe to the excited

the nucleus a somewhat larger radius than in the normal state. The radii of the excited states of the ThC′ nucleus, calculated by formula (6), are given in the last column of Table 3; as we see, the radii increase quite regularly with increasing excitation of the nucleus. We must note that the possibility, mentioned in the preceding paragraph, of assigning to excited $\alpha$-particles leaving the nucleus a larger value of the critical radius of the nucleus is closely connected with the result obtained here.

The fine structure of RaC and AcC was investigated by Rutherford; he found in each of these elements the presence of one strong low-velocity $\alpha$-group. For RaC the slow group is separated by $0.27 \cdot 10^{-6}$ erg, and in the case of AcC by $0.54 \cdot 10^{-6}$ erg, from the main one. In view of the small percentage of RaC atoms giving $\alpha$-decay, the $\gamma$-radiation is very weak and has not been observed. In the $\gamma$-spectrum of AcC, however, there is a strong line $0.562 \cdot 10^{-6}$ erg, which evidently coincides with the measured fine structure of the $\alpha$-rays of this element. In general we must expect the existence of fine structure of $\alpha$-radiation and the presence of a $\gamma$-spectrum in all cases of $\alpha$-decay where the first excited levels of the nucleus do not lie too high. Conversely, the absence of low-velocity $\alpha$-groups and $\gamma$-rays should indicate to us that a very large energy is necessary for excitation of the nucleus.

§ 4. Up to the present time we have at our disposal too few known systems of levels to infer any regularities of the latter. It is possible, however, to attempt, on the basis of general assumptions about the structure of nuclei, to obtain theoretically a system of levels for comparison with the available experimental material. We know that the nuclei of radioactive elements consist of a large number of $\alpha$-particles, several protons, and free electrons; on the basis of energetic considerations we may, however, conclude that the latter do not play an essential role in obtaining systems of discrete quantum levels of the nucleus. We must, however, distinguish between the levels of an excited $\alpha$-particle and of an excited nuclear proton. In view of the relatively smaller mass of the proton, one should expect

…to expect that the proton levels will lie at greater distances from one another than in the case of $\alpha$-particles. It is also necessary to take into account the possibility of the existence of combination levels (two or more excited particles), whose energy, owing to the strong interaction between the constituent parts of the nucleus, will differ from the sum of the energies of the simple levels.

For calculating the various quantum levels for an $\alpha$-particle or a proton in the nucleus, the most rational model is that of a potential crater, which, for lack of more accurate data, must be taken as rectangular. Under this hypothesis, the calculation of the energy levels reduces to finding the eigenvalues of the equation:

\[ \frac{d^2 \Psi}{dr^2} + \frac{8\pi^2 m}{h^2} \left[ E - \frac{h^2}{8\pi^2 m}\frac{j(j+1)}{r^2} \right]\Psi = 0 \tag{7} \]

with the boundary conditions $\Psi(r_0)=0$.

Here, $m$ is the mass of the particle, $j$ is the angular momentum, and $r_0$ is the radius of the nucleus. Finding the roots of equation (7) can be readily carried out with the aid of tables of Bessel functions. Calculating from equation (7) the energy levels of the $\alpha$-particle and the proton and constructing all possible combination levels, we obtain a very complex system, comparison of which with the observed levels has so far not led to a positive result. It must be hoped, however, that further experimental investigations of $\alpha$-particles of high and low velocity will enable us to obtain more data for comparison with our theoretical conceptions and will lead to a complete solution of the question of the serial schemes of the nucleus.