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

V. Problems of β-Decay

G. A. Gamow

§ 1. In contrast to nuclear processes involving the motions of heavy particles (neutrons, protons, α-particles), the phenomenon of the emission of nuclear electrons presents, at the present time, exceedingly great difficulties for theoretical understanding.

One of the principal difficulties is the existence of a continuous spectrum of β-rays, i.e. a continuous distribution of energy among the β-particles emitted by the various nuclei of a given disintegrating element. Thanks to the work of Gurney, who studied the magnetic spectra of β-rays, we now have energy-distribution curves for β-particles for a whole series of radioactive elements. From Fig. 1, which gives a summary of the results of the investigations of Gurney and other authors, we see that the β-spectra of various elements have a fairly characteristic form, and that the mean energy of the β-particles is approximately equal to one third of the maximum value. The question of the existence of an upper limit of the continuous β-spectrum remained for a very long time in an uncertain state, and only recently was it settled in the affirmative by the investigations of Sargent, who measured the upper limits of the β-spectrum for a whole series of elements.

Fig. 1.

* See Uspekhi Fizicheskikh Nauk 10, 531, 1930; 12, 31, 1932; 12, 389, 1932; 13, 46, 1933.

What explains the continuity of the energy of the $\beta$-spectrum, which so sharply distinguishes it from the energy spectrum of heavy particles emitted by nuclei? The hypothesis that in $\beta$-decay we have the simultaneous emission of two electrons, sharing between them the energy liberated in the transformation, must immediately be rejected, for from the position of radioactive elements in Mendeleev’s periodic system it follows that in $\beta$-decay the atomic number changes by one unit. Moreover, direct measurements by Geiger showed that the number of $\beta$-particles is exactly equal to the number of disintegrated atoms. The original supposition that the observed continuity of the $\beta$-spectrum is due to secondary absorption of $\beta$-particles in the very layer of the disintegrating substance was refuted by Ellis’s experiments, in which he measured calorimetrically the total amount of energy released in the $\beta$-decay of RaE and showed that this energy coincides with the mean energy of the observed $\beta$-spectrum, and not with its upper limit, as should have been the case for secondary absorption.

Fig. 2.

In exactly the same way we must exclude the supposition that, before $\beta$-decay, different nuclei were in different energy states, or that after $\beta$-decay nuclei with different stores of energy are obtained. The point is that experiment shows that different nuclear processes (emission of $\alpha$-particles, $\gamma$-rays, etc.) proceed in exactly the same way in all nuclei both before and after the emission of the $\beta$-particle, which would be impossible if these nuclei were in different energy states.

Thus only two possibilities remain for us: to suppose that in $\beta$-decay the law of conservation of energy does not hold, or to accept that the excess energy is carried away by some mysterious particles which successfully elude observation and, in particular, do not at all get stuck in Ellis’s calorimeter.

The first hypothesis was put forward by Bohr, who pointed out that, since processes connected with nuclear electrons cannot be described by the modern wave theory and require, for their explanation, the construction of a new relativistic theory of quanta, there is no logical necessity for the conservation of the laws of conservation of energy and momentum in these processes. However, as Landau showed, rejection of the law of conservation of energy leads to a serious conflict with the general theory of gravitation. Indeed, let us imagine, for example, an atom of RaE (Fig. 2) surrounded by some closed surface of very large size. From the general theory of gravitation it is known that the mass enclosed within such a closed surface is completely determined by the value of the gravitational field on the surface itself (Gauss’s formula). It follows that if, in the $\beta$-decay of our

nucleus its mass changed by some amount, then during a time equal to the radius of our surface divided by the speed of light, somewhere in another part of our region a compensating mass must appear. A change in the total mass enclosed in the region bounded by us must necessarily be connected with the passage of the corresponding mass (or an equivalent quantity of energy) through our surface. Thus the rejection of the law of conservation of energy must necessarily lead to a change in the general equations of gravitation for empty space. This, of course, is possible, but very inconvenient.

The second difficulty connected with the rejection of the law of conservation of energy consists in the existence of an upper limit of the continuous β-spectrum, since in that case it would be natural to suppose that the energy distribution curve, like Maxwell’s curve, decreases exponentially up to arbitrarily large values of the energy. In reality, however, the β-spectrum has a quite definite upper limit, and even, as Ellis and Mott have shown, this maximum value of the energy is subject to the law of conservation. For example, Fig. 3 shows a branching point in the thorium family. If we add the energy of the α-particle and the maximum energy of the β-particles for two different paths leading from ThC to ThD (lead), then we obtain for (ThC, ThC′, ThD): \(2.20 + 8.95 = 11.15 \cdot 10^6\ \mathrm{V}\), and for (ThC, ThC″, ThD): \(6.20 + 1.82 = 8.02 \cdot 10^6\ \mathrm{V}\), i.e., different values. But in taking account of the energy balance here it is also necessary to take into consideration that ThC″ emits two extremely intense γ-lines, \(0.58 \cdot 10^6\ \mathrm{V}\) and \(2.62 \cdot 10^6\ \mathrm{V}\). Ellis’s measurements showed that for each disintegrated ThC″ nucleus one quantum of each of the two above-mentioned frequencies is emitted. This leads us to the conclusion that the observed β-spectrum of ThC″ corresponds to the formation of an excited state of the ThD nucleus, corresponding to an excess energy \(0.58 + 2.62 = 3.20 \cdot 10^6\ \mathrm{V}\)* and subsequently passing into the normal state by the emission of two γ-quanta. Accordingly, when accounting for the energy along the path (ThC, ThC″, ThD), we must add to the energy of the α- and β-particles also the energy of the γ-rays, which gives us \(8.02 + 3.20 = 11.22 \cdot 10^6\ \mathrm{V}\), a value agreeing within the errors of measurement with the value,

Fig. 3.

* Let us note that the energy balance agrees better if we replace the line 0.58 by the other known γ-line 0.51. In that case the energy of the excited level must be \(2.62 + 0.51 = 3.13\), which is exactly equal to \(11.15 - 8.02\).

obtained for the other path. A calculation of the same kind is applicable to taking into account the energy in the branches of the uranium–radium and actinium families. Both of the above-mentioned facts make the admissibility of rejecting the law of conservation of energy in $\beta$-decay highly doubtful.

The second possible hypothesis for explaining the existence of a continuous spectrum of $\beta$-rays was proposed by Pauli and consists in the assumption of the existence in nature of neutral particles, possessing the mass of the electron or an even smaller one, which have received the name neutrino. Particles of this kind, which play an extremely important role in carrying away the excess energy obtained in $\beta$-decay, would be extremely difficult to observe, for, having no charge and possessing an extremely small mass, they should hardly interact with matter and can pass through a many-kilometer layer of substance without producing any noticeable effect. It is self-evident that the introduction of such hypothetical particles, almost not subject to experimental verification, can be justified only in the case that with their aid it proves possible to construct a coherent theory of the processes of emission and absorption of nuclear electrons, which at the present time is still far from being done.

Here it is necessary to note that, besides the difficulties with the conservation of energy in the process of $\beta$-decay, difficulties also arise with the law of conservation of angular momentum. The point is that investigation of the phenomenon of the hyperfine structure of spectral lines makes it possible to determine, for a whole series of isotopes, the magnitude of the angular momentum of the nucleus. This angular momentum (or spin) turns out, as is to be expected from the quantum theory of the rotator, to be an integral multiple of one half of the Bohr angular momentum:

\[ \sigma = \frac{i}{2}\,\frac{h}{2\pi}\quad (i\text{—an integer}), \tag{1} \]

where for isotopes having an even atomic weight, $i$ always turns out to be an even number, while for odd isotopes $i$ is always odd. This is precisely what is to be expected if we assume that nuclei are built of neutrons and protons, and the angular momenta of both these elementary particles are each equal to $\frac{1}{2}\frac{h}{2\pi}$.

In the process of $\beta$-decay the atomic weight of the nucleus does not change, and, consequently, the difference between the spins of the decaying nucleus and the product nucleus must be equal to an even multiple of $\frac{1}{2}\frac{h}{2\pi}$. If the law of conservation of angular momentum were valid, then this spin difference would have to be imparted to the emitted electron. But for an electron leaving the nucleus, the angular momentum is composed of the orbital moment, always equal to an integral multiple of $\frac{h}{2\pi}$, and the electron’s own spin, equal to

as is known, \(\frac{1}{2}\frac{h}{2\pi}\), which in all gives an odd number of halves of \(\frac{h}{2\pi}\).

This discrepancy, according to Bohr’s hypothesis, must also be attributed to a violation of the law of conservation of angular momentum, and, according to Pauli’s hypothesis, must be compensated by the emission of a neutrino, to which it is therefore necessary to ascribe a spin \(\frac{1}{2}\frac{h}{2\pi}\).

§ 2. We shall now turn to the question of the stability of various nuclei with respect to the emission of electrons. The general conditions of nuclear stability were developed by Heisenberg on the basis of a model of the nucleus constructed from neutrons and protons. In calculating the energy of such a model it is first of all necessary to make certain assumptions about the character of the forces acting between the elementary particles. In Heisenberg’s model we have three kinds of interaction: neutron—neutron, proton—proton, and neutron—proton. Heisenberg takes the forces between two neutrons to be extremely weak and not to play an essential role in the structure of the nucleus. Forces of the same weak kind would have to be assumed also between protons, if the latter did not carry an electric charge; in view of the charge, however, Coulomb repulsive forces must be introduced. In the case of two different particles (neutron—proton), however, one should expect very considerable attractive forces, connected with the possibility of exchange of electric charge between these two particles (the transformation of a neutron into a proton and conversely), and decreasing extremely rapidly with distance. Using an analogy with quantum chemistry, we may say that the forces between a neutron and a proton are analogous to the forces between H and H\(^+\), leading to the formation of the very stable ion H\(_2^+\), whereas the forces between like nuclear particles are analogous to the forces binding H and H into the molecule H\(_2\). In the theory of the nucleus, as in the theory of molecular structure, it is assumed that the latter forces are far less significant. Denoting the potential of the first kind of forces by \(I(r)\), and of the second by \(K(r)\), we may formulate our assumptions as follows:

\[ \begin{aligned} \text{neutron—proton} &\quad \ldots\quad -I(r)\;(= b\cdot e^{-r/a}),\\ \text{neutron—neutron} &\quad \ldots\quad -K_n(r),\\ \text{proton—proton} &\quad \ldots\quad -K_p(r)+\frac{e^2}{r}. \end{aligned} \tag{2} \]

In view of the assumption that \(K(r)\ll I(r)\), we may in the first approximation consider only the forces due to the potential \(-I(r)\) and \(+\frac{e^2}{r}\).

Let us now consider a nucleus composed of \(n_1\) neutrons and \(n_2\) protons, and let us seek its most stable state, i.e. the state corresponding to the maximum binding energy. In

Thus we shall assume that the total number of particles \(n=n_1+n_2\) remains unchanged, but that a neutron may transform into a proton and back, emitting respectively a negative or positive electron. Considering first only the forces due to the potential \(f(r)\), we easily arrive at the conclusion that the greatest number of neutron—proton bonds will be attained if the numbers of particles of both kinds are the same \(\left(n_1=n_2=\dfrac{n}{2}\right)\).

On the other hand, the Coulomb repulsion between protons will reduce the binding energy, and from this point of view the most favorable case would be the complete absence of protons \((n_1=n;\ n_2=0)\). The resulting optimum state will obviously be some compromise between the two requirements just mentioned and will correspond to a large relative number of nuclear neutrons \(\left(n_1>\dfrac{1}{2}n;\ n_2<\dfrac{1}{2}n\right)\).

For light elements, for which Coulomb forces play a comparatively small role, the optimum state will still correspond to equal numbers of neutrons and protons \(\left(\dfrac{n_1}{n_2}=1\right)\), but as the atomic number increases the relative number of neutrons \(\left(\dfrac{n_1}{n_2}\right)\) will increase strongly. Taking into account that

\[ \frac{A}{z}=\frac{n_1+n_2}{n_2}=\frac{n_1}{n_2}+1, \]

we should expect that for light elements \(\dfrac{A}{z}=2\), while for heavier ones \(\dfrac{A}{z}>2\), in full agreement with reality.

Applying to the indicated model of the nucleus the statistical method, Heisenberg was able to calculate the binding energy of the nucleus \(E\) as a function of \(n_1\) and \(n_2\), containing, of course, the unknown coefficients \(a\) and \(b\) from the force law (2). Taking \(a=8\cdot10^{-13}\ \text{cm}\) and \(b=4.05\cdot10^{-5}\ \text{erg}\), he obtains for the binding energy (expressed in units of the proton mass) the expression:

\[ E=\left\{ \begin{aligned} &0.00347\,n_2-0.0364\,n_1+0.01211\,\frac{n_1^2}{n_2}+{}\\ &\quad + n_2^{2/3}\left(3.19-0.715\,\frac{n_1}{n_2}\right)\cdot10^{-4}+0.049 \end{aligned} \right\}. \tag{3} \]

This agrees well with the experimental mass-defect curve, constructed on the basis of Aston’s data. Starting from this formula, we can immediately obtain the relative number of nuclear neutrons \(\left(\dfrac{n_1}{n_2}\right)\) for the most stable state; for this it is necessary to find the minimum of expression (3) for

under the additional condition \(n_1+n_2=\mathrm{const}\), which is evidently written in the form:

\[ \frac{\partial E}{\partial n_1}-\frac{\partial E}{\partial n_2}=0, \tag{4} \]

or

\[ -0.0399+0.02422\,\frac{n_1}{n_2} +0.01211\left(\frac{n_1}{n_2}\right)^2 -\left(6.04-0.477\,\frac{n_1}{n_2}\right)n_2^{2/3}\cdot 10^{-4}=0. \tag{4′} \]

It is just as easy to write the condition for the boundary of \(\alpha\)-stability, which consists in the fact that the work of removing from the nucleus two neutrons and two protons must be greater than the binding energy of these particles in the \(\alpha\)-particle:

\[ -2\left(\frac{\partial E}{\partial n_1}+\frac{\partial E}{\partial n_2}\right)>\Delta M\alpha \tag{5} \]

or

\[ 0.0658-0.04814\,\frac{n_1}{n_2} +0.02422\left(\frac{n_1}{n_2}\right)^2 +\left(9.2-0.954\,\frac{n_1}{n_2}\right)^2 n_2^{2/3}\cdot 10^{-4}>0.03 \tag{5′} \]

In both formulas it has been assumed, for simplicity, that the neutron mass defect is equal to zero (i.e., that \(M_n=M_p+m_e\)). In Fig. 4 both curves (4) and (5) are plotted together with the experimental points corresponding to the known isotopes; along the abscissa axis is laid off the number of protons, and along the ordinate axis the relative number of neutrons. We see that the theoretical curves give a correct representation of the stability boundaries, but that curve (4′) lies too low, and curve (5′) too far to the right; this may be explained, first, by the inaccuracy of the statistical method used to calculate expression (3), and, second, by the arbitrary assumption that the neutron mass defect is zero. [According to Chadwick, the mass of the neutron is \(1.0065\), which gives \(\Delta M_n=0.0012\), while according to Joliot this mass is \(1.011\), which gives \(\Delta M_n=-0.003\).] It must be hoped, however, that a more exact calculation will shift the theoretical stability curves to the proper place.

Fig. 4.

The question of the width of the band of stability, whose middle is determined by curve (4,4′), cannot be decided on the basis

the above simple considerations, since the latter do not take into account the fact that even nuclear protons are bound more strongly than odd ones, owing to the formation of $\alpha$-particles in the nucleus. Taking this fact into consideration, we should expect that the energy curve as a function of the number of protons, for a given total mass of the nucleus, must split into two curves running more or less parallel to one another, as indicated in Fig. 5. This entails the consequence that certain nuclei lying on a rather high part of the energy curve will nevertheless be stable, since in order to pass downward they would first have to pass to another curve, the corresponding point on which lies somewhat higher (for example, in Fig. 5 nucleus $a$ can pass into nucleus $c$ only through $b$, which is energetically impossible). Of course, one may conceive of a direct transition with the simultaneous emission of two electrons (in our example directly from $a$ to $c$), but theoretical considerations show that the probability of such a double transition is vanishingly small. It is not excluded, however, that the very weak $\beta$-activity of potassium and rubidium is connected precisely with a process of this kind.

Fig. 5.

The fact that isotopes of elements with an even atomic number have predominantly even atomic weight, and conversely, indicates to us that, when $n_1+n_2$ is even, the lower energy curve corresponds to an even number of protons, while when $n_1+n_2$ is odd, the curve for odd $n_2$ lies lower. This is also in agreement with the fact that in the thorium and uranium-radium families (even atomic weight) elements with even atomic number emit faster $\beta$-particles than elements with odd atomic number, whereas in the actinium family (odd atomic weight) the situation apparently is the reverse. The considerations set forth above lead us to the conclusion that the region of stability with respect to the emission of electrons has a certain rather considerable width, depending on the difference in the binding energy of even and odd protons in the nucleus. The theoretically estimated width of this band of stability is in agreement with the experimental facts.

Let us consider the position of the unstable elements in the diagram of Fig. 4. In the region of large atomic weights, where the curves of $\alpha$- and $\beta$-stability approach one another and even intersect, there are located ra-

radioactive elements belonging to the three known radioactive families. In this region a single $\alpha$- or $\beta$-decay can transfer the point representing our nucleus from one unstable region to another, and we may expect the existence of series of successive transformations, stopping only when the atomic weight decreases so much that the width of the stable band becomes too large.

In the region of the lighter elements only isolated cases of unstable nuclei are known. Hevesy and Pahl discovered the radioactivity of the rare element samarium (Sm), which emits $\alpha$-particles with a range of about $1.5\ \mathrm{cm}$, has a decay period of $10^{12}$ years, and lies, as is evident from Fig. 4, very close to the boundary of $\alpha$-stability. Let us note that the observed decay period agrees very well with the value calculated by Gamow’s formula for $\alpha$-decay on the basis of the observed range of the $\alpha$-particles.

Much less clear is the case of the observed $\beta$-activity of the alkali elements potassium (K) and rubidium (Rb), belonging apparently, as Hevesy showed, to the heavier isotope

$\left({}_{19}^{41}\mathrm{K};\ {}_{37}^{87}\mathrm{Rb}\right)$.

The extraordinarily long decay period of these elements ($10^{13}$ and $10^{12}$ years) is not connected with the comparatively large energy of the $\beta$-particles emitted by them (of the order of a million volts), which leads us to suppose that here we have the above-mentioned case of the simultaneous emission of two nuclear electrons. However, investigations by Ambrosen and Vokhmintseva, who used a system of two Geiger counters to find paired $\beta$-emission, apparently give a negative result. A second possibility is that the observed $\beta$-activity belongs to comparatively short-lived products obtained from potassium and rubidium by $\alpha$-decay.

\[ {}_{19}^{40}\mathrm{K}\to{}_{17}^{36}\mathrm{Cl}+\alpha;\qquad {}_{17}^{36}\mathrm{Cl}\to{}_{18}^{36}\mathrm{A}+\bar{\beta} \tag{6} \]

and, respectively,

\[ {}_{37}^{86}\mathrm{Rb}\to{}_{35}^{82}\mathrm{Br}+\alpha;\qquad {}_{35}^{82}\mathrm{Br}\to{}_{36}^{82}\mathrm{Kr}+\bar{\beta}. \tag{6'} \]

In this case $\alpha$-decay is responsible for the long lifetime.

The energy of the $\alpha$-particles, calculated from this period according to Gamow’s formula, proves to be very small and corresponds to a range in air of the order of a millimeter. This may explain the fact that these $\alpha$-particles have not yet been detected by anyone.

Nuclei emitting positive electrons spontaneously are unknown in nature and were first produced artificially only at the beginning of the present year, 1934. Joliot showed that when certain light elements are bombarded with $\alpha$-particles, in addition to the emis-

emission of ordinary protons, the emission of neutrons and positive electrons is also observed. These observations were interpreted by him as the presence of two possible transformations leading to one and the same final product. For example, under α-bombardment of boron the following two reactions are possible

\[ {}^{10}_{5}\mathrm{B}+{}^{4}_{2}\mathrm{He}\to{}^{13}_{6}\mathrm{C}+\mathrm{p} \tag{7} \]

or

\[ {}^{10}_{5}\mathrm{B}+{}^{4}_{2}\mathrm{He}\to{}^{13}_{6}\mathrm{C}+\mathrm{n}+\beta^{+}. \tag{7'} \]

Further investigations by Joliot showed that, upon removal of the source of α-particles, the emission of positive electrons continues for several minutes more, its intensity decreasing exponentially. This leads us to the conclusion that transformation (7) in reality consists in knocking out from the boron nucleus at first only one neutron, leading to the formation of the β⁺-unstable element \({}^{13}_{7}\mathrm{N}\), which received the name radio-nitrogen. By emission of a positive electron, radio-nitrogen passes into the known isotope of carbon

\[ {}^{10}_{5}\mathrm{B}+{}^{4}_{2}\mathrm{He}\to{}^{13}_{7}\mathrm{N}+\mathrm{n}, \tag{7''} \]

\[ {}^{13}_{7}\mathrm{N}\to{}^{13}_{6}\mathrm{C}+\beta^{+}. \tag{7'''} \]

The energy of the β⁺-particles emitted by radio-nitrogen is equal to \(0.7\cdot10^{6}\ \mathrm{V}\), and the decay period is 14 min. Joliot also succeeded, by acting chemically on a piece of boron immediately after bombardment, in converting the resulting atoms of radio-nitrogen into ammonia and observing an increase of the β⁺-activity with the gaseous fraction.

In a similar way, by bombarding magnesium and aluminum with α-particles, Joliot obtained new β⁺-active elements: radio-silicon \(\left({}^{27}_{14}\mathrm{Si}\right)\), with β⁺-particle energy \(0.7\cdot10^{6}\ \mathrm{V}\) and period 2 min. 30 sec., and radio-phosphorus \(\left({}^{30}_{15}\mathrm{P}\right)\), with β⁺-particle energy \(2.2\cdot10^{6}\ \mathrm{V}\) and period 3 min. 15 sec.

Let us note that these same elements can also be obtained, as Cockcroft showed, by bombardment with protons of the corresponding stable isotopes, for example:

\[ {}^{12}_{6}\mathrm{C}+{}^{1}_{1}\mathrm{H}\to{}^{13}_{7}\mathrm{N}. \]

Using a very powerful proton beam, Cockcroft succeeded in obtaining a very strongly activated layer of carbon, which made it possible to investigate the energy distribution in the spectrum of β⁺-particles.

According to preliminary results, this energy has (besides the scattering spectrum) a quite definite value, which is an incomprehensible contrast to the continuity of the spectra of the β-active bodies hitherto known.

In Fig. 4 the points corresponding to three known β-active nuclei are plotted, and we see that their position is in excellent agreement with the theoretical boundary of β-stability. Let us note further that from the arrangement of the theoretical curves of nuclear stability it follows that we may expect β-activity only for elements with small atomic weights.

§ 3. We shall now turn to the consideration of the β-decay process itself and to the calculation of the probability of emission of an electron by an atomic nucleus. As was pointed out by Bohr, we encounter very great fundamental difficulties when trying to imagine an electron localized in so small a region as the atomic nucleus. The point is that the limits of applicability of Dirac’s relativistic quantum mechanics to a particle of mass \(m\) are confined to a region of size

\[ \Delta = \frac{h}{2\pi mc} = 3 \cdot 10^{-11}\ \text{cm}, \tag{9} \]

i.e., approximately thirty times larger than the radius of the atomic nucleus, so that in attempting to place an electron inside the nucleus we can no longer use this theory. Let us note that the difficulty arising here is quite analogous to the difficulty we would encounter in trying to place a light quantum emitted by some atom inside the latter, for the wavelength of our quantum considerably exceeds the dimensions of the atom itself. The natural consequence of this comparison is the attempt, first made by Beck, to regard the electrons emitted in β-decay as arising directly in the above-mentioned region around the nucleus as a result of the transformation of the nuclear neutron into a proton, or conversely. This point of view was successfully developed by Fermi and led him to a consistent theory of β-decay, having excellent agreement with experimental data. Following Heisenberg, we shall formally regard the neutron and the proton as two different quantum states of one and the same heavy particle, characterized by the additional coordinate \(\rho\), which can take only two values, \(\pm 1\). We shall assume that the state \(\rho=+1\) corresponds to the neutron, and the state \(\rho=-1\) to the proton. Then each particle from the neighborhood of a neutron into the state of a proton, or conversely, must be accompanied by the appearance in the surrounding space of one (negative or positive) electron and one neutrino.*

* As Bohr pointed out, on the basis of general symmetry one should also expect the existence in nature of negative protons, differing from positive ones only by the sign of the charge. Such an assumption does not alter ...

If the difference in binding energies between two nuclei, of which one contains a neutron in the state \(Un\), and the other, instead, a proton in the state \(Vn\), is \(\Delta E\), then the condition for the possibility of \(\beta\)-decay will be:

\[ \Delta E > (m+\mu)c^2, \tag{10} \]

where \(m\) and \(\mu\) are the masses of the electron and the neutrino. The excess energy obtained in this transformation will be distributed between the emitted electron and the neutrino. A condition of the same kind may be written for the inverse transition, corresponding to the emission of a positive electron. Since the transition of a particle from the neutron state to the proton state is characterized by a change of \(\rho\) from \(+1\) to \(-1\) and conversely, we can describe such transitions by means of the operators:

\[ Q=\begin{vmatrix} 0&1\\ 0&0 \end{vmatrix} \quad\text{and}\quad Q^*=\begin{vmatrix} 0&0\\ 1&0 \end{vmatrix}, \tag{11} \]

applied to the Hamiltonian function of the heavy particle containing \(\rho\).

The expression for the Hamiltonian function of our problem will be written as the sum of three terms, corresponding to the energy of the heavy particles, the energy of the light particles, and the energy of the interaction between the heavy and light particles which gives rise to the transformation under consideration. For the heavy particles we have:

\[ H_1=\frac{1+\rho}{2}N+\frac{1-\rho}{2}P, \tag{12} \]

where \(N\) and \(P\) are the energy operators for the neutron and proton. For \(\rho=+1\) our expression becomes \(N\), and for \(\rho=-1\), \(P\). The transition from one value to the other is evidently achieved by applying the operators (11). For the Hamiltonian function of the light particles one should write

\[ H_2=\sum_s A_s l_s+\sum_\sigma B_\sigma n_\sigma \tag{13} \]

where \(l_s\) and \(n_\sigma\) are the corresponding energy operators, while \(A_s\) and \(B_\sigma\) are the numbers of electrons and neutrinos in the corresponding quantum states. According to the Pauli principle we must assume that \(A_s\) and \(B_\sigma\) can take only two values: zero or unity. As the various quantum states for the light particles we must take, in the case of electrons, the stationary states \(\psi_s\) in the central Coulomb field of the nucleus (with allowance for the “shielding”1

other electrons), and in the case of the neutrino simply plane waves \(S_s\), since the ordinary forces acting on the neutrino may be neglected.

With regard to the forces acting between the heavy and light particles and leading to the transformations under consideration, we make the very plausible hypothesis that these forces act only in the immediate vicinity of the heavy particles. Under this assumption the simplest expression for the interaction-energy operator will be:

\[ H_3 = g[Q\psi(x,y,\ldots)\varphi(x,y,\ldots) + Q^*\psi^*(x,y,\ldots)\varphi^*(x,y,\ldots)], \tag{14} \]

where the values of the wave functions of the electron and the neutrino (composed of \(\psi_s\) and \(\varphi_s\)) are taken for the coordinate values \(x, y, \ldots\) of the heavy particles. The coefficient \(g\) in expression (14) for the interaction energy has the dimension \(L^5MT\) and is a new universal constant characterizing the transformation*.

On the basis of these assumptions about the interaction of nuclear particles, Fermi arrives, as a result of very long and tedious calculations, at the final formula giving the energy distribution of the continuous \(\beta\)-spectrum and the decay probability as a function of the maximum energy of the \(\beta\)-particles.

The energy distribution obtained theoretically is in general agreement with the observed shape of the \(\beta\)-spectrum; moreover, the form of the distribution curve near the upper limit of the spectrum enables Fermi to draw certain conclusions about the mass \(\mu\) of the neutrino. The point is that the energy distribution between the electron and the neutrino must depend strongly on the ratio of the masses of the neutrino and the electron \(\frac{\mu}{m}\), and \(\frac{\mu}{m} \gg 1\) leads to a rapid (vertical) drop in the number of particles near the upper limit, whereas \(\frac{\mu}{m} \ll 1\) gives a smooth gradual descent. The fact that the experimental curves of \(\beta\)-spectra have precisely the second type of distribution leads Fermi to the conclusion that the mass \(\mu\) of the neutrino is very small compared with the mass of the electron, or is even equal to zero.

For the \(\beta\)-decay constant \(\lambda\), Fermi finds (for atomic number \(Z - 82\)) the expression:

\[ \lambda = 1.75 \cdot 10^{95} g^2 F(\eta_0)\left|\int u_n v_m^{*} V^{*} d\omega\right|^2, \tag{15} \]

where the function \(F\) of the maximum value of the momentum of the emitted electrons \(\eta_0 = \frac{P_{\max}}{mc}\) is given by the formula:

\[ F(\eta_0) = \frac{2}{3}\sqrt{1+\eta_0^2} -\frac{2}{3} +\frac{\eta_0^4}{12} -\frac{\eta_0^2}{3} +0.355\left\{ -\frac{\eta_0}{4} -\frac{\eta_0^3}{12} +\frac{\eta_0^5}{30} +\frac{\sqrt{1+\eta_0^2}}{4} \lg(\eta_0+\sqrt{1+\eta_0^2}) \right\}. \tag{15'} \]

* In the further development of the theory this constant \(g\) must be expressed through other universal constants, such as: \(M, m\) (the masses of the neutron and electron), \(\kappa\) (the gravitational constant), \(e, c\), and \(h\).

G. A. GAMOW

Let us note that in formula (15) the matrix element \(\left(\int u v^{*}\,d\omega\right)^2\) is obtained in the form of a separate factor because the length of the de Broglie wave corresponding to the wave functions \(\psi\) and \(\varphi\) for the electron and neutron may be taken to be very large in comparison with the dimensions of the nucleus (i.e., with the wavelength of the neutron and proton).

Formula (15) leads us at once to very interesting conclusions concerning the selection principle in \(\beta\)-transformations. Indeed, it is easy to see that the matrix element \(\left(\int u v^{*}\,d\omega\right)^2\) will differ from zero (and will be approximately equal to unity) in those and only those cases when the initial neutron and the proton obtained after \(\beta\)-emission have identical angular momenta of rotation, i.e. when in \(\beta\)-decay the total spin of our nucleus is not changed \((i=i')\).

In the case of a change of spin \((i\ne i')\), formula (15) gives \(\lambda=0\), and the given transition is forbidden.

Fig. 6.

However, since formula (15) is only approximate, this prohibition will not in reality be fulfilled quite strictly. The fact that the wavelength of the emitted electron is not infinitely large in comparison with the dimensions of the nucleus (as was assumed in the calculations) will lead to the existence of a certain finite probability for forbidden transitions. It is easy to see, by analogy with the corresponding phenomena in the theory of the emission of light by atoms, that the probability of such a transition will, however, be approximately

\[ \left(\frac{\Lambda}{r_0}\right)^2 \sim 100 \]

times smaller than the value given by formula (15) (here \(\Lambda=\dfrac{h}{2\pi p}\) is the wavelength of the emitted electron). Another factor that violates the strict fulfillment of the above-mentioned selection principle is the presence of relativistic terms in the expressions for the energy of the heavy particles in the nucleus [also omitted in deriving formula (15)]; this second factor also lowers the probability of decay by about a factor of 100.

For comparison of the theory set forth above with experimental facts, we shall make use of the recent measurements of Sargent, who determined the upper limits of the \(\beta\)-spectrum for a whole series of radioactive elements. In Fig. 6 there is presented the graph obtained by Sargent and indicating the existence of a definite relation between the decay constant and the energy of the emitted \(\beta\)-particles. We see that the \(\beta\)-decaying bodies are sharply divided into two classes, the bodies of the second class having a decay probability approximately 100 times smaller than the bodies of the first class.

Recalling the above-stated consequences of Fermi’s theory, we must regard the β-transformations lying on the upper curve as allowed \((i=i')\), and those lying on the lower one as forbidden \((i\ne i')\).

For comparison of Sargent’s experimental curves with Fermi’s theory it is necessary to assign a definite value to the constant \(g\). In Fig. 6 the theoretical curves are drawn for

\[ g=4\cdot 10^{-50}\ \text{erg}/\text{cm}^{3}, \tag{16} \]

and we see that the general course of the curves represents excellently the distribution of the experimental points.

§ 4. As Gamow has shown, we can continue our analysis if we accept, following Ellis and Mott, that in cases where β-disintegration is accompanied by strong γ-radiation, the observed curves of energy distribution in the continuous β-spectrum are the sum of several components corresponding to the excitation of different energy levels of the nucleus.

In several cases in which the scheme of levels of the nucleus produced in the β-disintegration is known to us, we can carry out such a decomposition of the observed β-spectrum, assuming that the upper limit of the various components is less than the maximum value of the energy of the given β-disintegration by the amount of the excitation energy of the corresponding level, and that the relative intensities of the different components are determined by the excitation of the corresponding levels. Thus, for example, for the nucleus RaC′, which is the product of the β-disintegration of RaC, the energies and percentage of excitation of the principal quantum levels are given in Table 1, constructed on the basis of data concerning

TABLE 1

γ-rays and long-range α-particles of this element. In the last two columns are given the corresponding upper limits and disintegration probabilities for the various β-components that cause excitation of the RaC′ nucleus. From Fig. 6, where the various components of the β-spectrum of RaC are represented by small circles, we see that \(\beta_4\) and \(\beta_5\) apparently correspond to allowed transformations \((i=i')\),

and \(\beta_1, \beta_2\), and \(\beta_3\), lying on the lower curve, to forbidden transitions (probably \(i = i' \pm 1\)). The point \(\beta_0\), corresponding to an unexciting decay, lies very low, which indicates that the spin difference between the normal states RaC and RaC′ is apparently very large (\(i = i' \pm 2\) or even more). Conclusions of the same kind can be drawn with respect to a number of other elements.

We shall present here Gamow’s results relating to the analysis of the thorium family from the point of view of the selection rule given above. Fig. 7 gives the scheme of transformations of this family, beginning with ThA and ending with the stable final product ThD, or, simply, lead (\({}^{208}\mathrm{P}\)). In addition to the above-mentioned Fermi selection rule for \(\beta\)-transformations* we shall also have to use the explanation of the fine structure of \(\alpha\)-rays proposed by Gamow, according to which the presence of intense components in the structure of \(\alpha\)-radiation indicates that, in the given \(\alpha\)-decay, the spins of the initial and final nuclei have different values. Turning to our case, we must conclude that, since the whole series of \(\alpha\)-transformations leading from RaTh to ThB (four \(\alpha\)-decays) is devoid of fine structure, all the corresponding nuclei (RaTh, ThX, ThEm, ThA, ThB) have one and the same spin. Since all nuclei with even atomic number and even atomic weight investigated up to now have spin equal to zero, we may almost with certainty assume that in the present case as well

\[ i_0(\mathrm{RaTh}) = i_0(\mathrm{Thx}) = \cdots = i_0(\mathrm{ThB}) = 0. \]

Turning to the \(\beta\)-transformation leading from ThB to ThC, we note from Fig. 6 that the observed \(\beta\)-spectrum belongs to the allowed class. However, this \(\beta\)-transformation, as experiment shows, always leads not to the normal, but to an excited state of the product nucleus. Indeed, it is well known that ThC possesses an extremely strong \(\gamma\)-line \(0.237 \cdot 10^6\ \mathrm{V}\), corresponding in the secondary \(\beta\)-spectrum to an electron group with absolute intensity 0.25 (number of electrons per one disintegration). The internal conversion coefficient calculated for this line by the formula of Mott and Taylor (Mott a. Taylor), is equal to 0.026 under the assumption of a dipole transition and 0.205 under the assumption of a quadrupole transition**, which gives, for the two corresponding hypotheses, absolute intensities of the \(\gamma\)-line (number of \(\gamma\)-quanta per one disintegration) 9.6 and 1.2. Since the absolute intensity of a \(\gamma\)-line cannot be greater than unity, we must exclude the hypothesis of a dipole transition and assign to the \(\gamma\)-line \(0.237 \cdot 10^6\ \mathrm{V}\) a quadrupole character and inten-

* It should be noted that the selection rule for transformations used here (\(i = i'\)) is much more general than Fermi’s theory and will follow practically from any theory that regards \(\beta\)-decay as the transformation of a nuclear neutron into a proton.

** As is known, Mott and Taylor showed that the internal-conversion coefficient (number of secondary electrons : number of primary \(\gamma\)-quanta) is considerably greater for quadrupole radiation than for dipole radiation.

intensity 1 (with 0.2 attributed to errors of measurement and calculation). Thus we arrive at the conclusion that the ThC nucleus is almost always formed in an excited state (with an excess energy of \(0.237\cdot 10^6\) V) and, since the \(\beta\)-emission leading to this state belongs to the first class \(i_n(\mathrm{ThC})=i_0(\mathrm{ThB})=0\), as indicated in Fig. 7, the transition from the excited state to the normal one is accompanied by the emission of a quadrupole \(\gamma\)-line, which gives two possible values of the spin for the normal state: 0 or 2. The value \(i_0(\mathrm{ThC})=0\) is excluded, for then \(\beta\)-decay between the normal states of ThB and ThC would be allowed, leaving only one possibility, \(i_0(\mathrm{ThC})=2\). Comparing ThC and ThC′, we must take into account that, in view of the absence of fine structure of the \(\alpha\)-rays from ThC′, it is necessary to take
\[ i_0(\mathrm{ThC}')=i_0(\mathrm{ThD})=0 \]
(the absence of spin and \(\mathrm{ThD}={}^{208}\mathrm{Pb}\) has been shown experimentally). Hence it follows that the \(\beta\)-transition ThC—ThC′ (not accompanied by intense \(\gamma\)-radiation) belongs to the forbidden class in accordance with Fig. 6. As we indicated above, the \(\beta\)-decay leading from Th″ to ThD is accompanied by two \(\gamma\)-lines, \(2.62\cdot 10^6\) V and \(0.58\cdot 10^6\) V, with absolute intensities 1. Of these lines the first, apparently, is a quadrupole transition, and the second a dipole one. This compels us to assign to the level \(2.62+0.58=3.20\cdot 10^6\) V the spin value 1 or 3. Let us note that, as has already been pointed out, the energy balance along the two paths leading from ThC to ThD agrees better if we replace the line \(0.58\cdot 10^6\) by another strong \(\gamma\)-line \(0.51\cdot 10^6\), corresponding to a quadrupole transition and having absolute intensity 0.3*. In this case the excited state of the ThD nucleus will correspond to an excess energy \(3.13\cdot 10^6\), and simple considerations lead us to the conclusion that the rotational moment of this excited nucleus is equal to four. Since the \(\beta\)-transition leading from the normal state of the ThC nucleus to the excited state of ThD belongs to the allowed class, we must assume that
\[ i_0(\mathrm{ThC}'')=i_{II}(\mathrm{ThD}), \]
i.e. 1 or 3 in the case of the first possibility and

Fig. 7.

* According to the general rule, dipole radiation corresponds to transitions \(i\to i\pm 1\), and quadrupole radiation to transitions \(i\to i\pm 2\) or \(i\to i\) (for \(i\ne 0\)).

** The remaining intensity must be accounted for by other \(\gamma\)-lines of ThC″.

4 in the second case. All three possibilities lead to the result that \(i_0(\mathrm{ThC}^{\prime\prime})\) is different from \(i_0(\mathrm{ThC})(=2)\), which is in good agreement with the presence of fine structure in the \(\alpha\)-rays of ThC. Let us note that the observed distribution of intensity among the components of the fine structure of the \(\alpha\)-rays from ThC is more easily explained on the assumption that the difference of the spins of the corresponding normal states of the nuclei is equal to two \([\,\text{i.e., that } i_0(\mathrm{ThC}^{\prime})=4\,]\), which speaks in favor of the second possibility. An analysis of the same kind may also be carried out for other radioactive families and gives us a number of interesting data on the angular momenta of radioactive nuclei.

§ 5. We see that Fermi’s theory gives us the possibility of successfully describing a whole series of features characteristic of \(\beta\)-decay. However, the very essence of the process of transformation of a neutron into a proton remains obscure, and the mystery is hidden in formula (14), or more precisely, in the new constant \(g\), expressing the intensity of the interaction between heavy and light particles. The exceedingly small value of the constant \(g\) \((4\cdot 10^{-50}\ \mathrm{erg}/\mathrm{cm}^3)\) accounts for the slowness of \(\beta\)-transformations, but it should not be forgotten that this numerical value of the new constant has by no means been determined theoretically, but has been chosen so as to explain the experimental values of the decay constants.

In order to determine the order of magnitude of the forces responsible for the transition of a neutron into a proton and conversely, we must first decide at what distances these forces act, for the quantity \(g\) is the product of a certain energy (erg) by a certain volume \((\mathrm{cm}^3)\). Assuming that the forces act at nuclear distances \((10^{-12}\ \mathrm{cm})\), we obtain for the corresponding mean energy the value

\[ \frac{5\cdot 10^{-50}}{(10^{-12})^3}=5\cdot 10^{-14}\ \mathrm{erg}, \]

i.e., many millions of times smaller than the ordinary Coulomb forces

\[ \left( \frac{e^2}{r_0} = \frac{(5\cdot 10^{-10})^2}{10^{-12}} = 2\cdot 10^{-7}\ \mathrm{erg} \right). \]

This appears very strange, and therefore it seems necessary to reduce considerably the radius of action of the forces under consideration. Assuming that the forces act only at the direct approach of heavy particles, i.e., at distances of the order

\[ \frac{\hbar}{Mc^2} = \frac{(5\cdot 10^{-10})^2}{10^{-24}(3\cdot 10^{10})^2} = 2\cdot 10^{-16}\ \mathrm{cm}, \]

we obtain for the energy of interaction

\[ \frac{5\cdot 10^{-50}}{(2\cdot 10^{-16})^3} = 5\cdot 10^{-3}\ \mathrm{erg}, \]

which coincides with the magnitude of the potential energy of two charges at the same distances,

\[ \frac{e^2}{r} = \frac{5\cdot 10^{-10}}{2\cdot 10^{-16}} = 10^{-3}\ \mathrm{erg}. \]

Thus we arrive at the possibility that \(\beta\)-transformations are caused by forces of electromagnetic character, acting when charged heavy particles directly interpenetrate one another. The question of the nature and exact laws governing these interactions is, of course, as yet completely unclear and is evidently closely connected with the problem of elementary charge.

1. of the formalism set forth here. It will be necessary to assume that the number \(\rho\) may take three values \(+1; 0; -1\) (proton, neutron, negative proton), correspondingly changing the form of the formulas. The emission of an electron of a given sign will be possible by two paths; which path takes place in one case or another will depend on the energy conditions of the nuclei. ↩