EXPERIMENTAL VERIFICATION OF THE THEORY OF $\beta$-DECAY

As is well known, the most characteristic feature of radioactive $\beta$-decay is the continuous energy spectrum of the emitted electrons. In order not to come into contradiction with the laws of conservation of energy and momentum, it was necessary to assume the existence of a hypothetical uncharged particle of very small mass—the so-called neutrino—which, in each separate act of $\beta$-decay, is emitted from the nucleus together with the electron; in this case the sum of the energies of the neutrino and the electron remains constant for any act of decay. Obviously, to calculate this constant quantity, knowledge of which is required for computing the energy balance of certain nuclear reactions, it is necessary to determine from experimental data the upper boundary of the $\beta$-spectrum, i.e. to find the energy of those electrons which carry away all the energy released in the given act of $\beta$-decay. However, in finding the upper boundary of the $\beta$-spectrum we encounter the difficulty that the experimentally obtained curves of the distribution of electrons emitted by radioactive elements over energy near the upper boundary fall smoothly to zero, and therefore, without knowing the law according to which the curves decrease, it is difficult to establish exactly where the true upper boundary of the $\beta$-spectrum of the given active element lies. Thus, for the correct determination of the upper boundary it is necessary to know the law of the distribution of electrons over energy, by extrapolating which the desired quantity can be obtained. Such a distribution law must obviously be supplied by theory.

In 1934 Fermi[^1] created the first quantitative theory of $\beta$-decay, assuming the existence of the neutrino, which is emitted together with the electron in each act of $\beta$-decay; in this theory $\beta$-decay itself is regarded as the spontaneous transition of a neutron into a proton inside the nucleus, accompanied by the liberation of an electron and a neutrino, in the same way as, for example, a quantum of light is emitted in the transition of an excited atom to the normal state. In addition, Fermi made a certain assumption about the interaction of heavy particles with the field of the emitted electron–neutrino. Subsequently Konopinski and Uhlenbeck[^2] showed that this assumption about the interaction of the particles was not unique. Having made another assumption about the interaction, Konopinski and Uhlenbeck obtained a modified theory of $\beta$-decay, while retaining in it, however, all the remaining assumptions of Fermi’s theory.

Both Fermi’s theory and the modified theory of Konopinski and Uhlenbeck make it possible to calculate the distribution of electrons over energy or over momentum. These theories give different distribution functions. If by $N$ we denote the number of electrons with momentum between $mc\eta$ and $mc(\eta+\Delta\eta)$, where $\eta$ is the momentum of the particle measured in $mc$, then for the dependence of $N$ on momentum or energy the following expression is obtained:

$$ k\left(\frac{N}{f}\right)^{\frac{1}{\alpha}} = C - (E + 1), \tag{1} $$

where \(E\) is the electron energy, measured in \(mc^2\), \(k\) is a constant, and \(f\) is a certain function of the electron momentum and of the atomic number of the decaying element, which, for given values of \(\eta\) and \(Z\), can be calculated. \(\alpha\) in Fermi’s theory is equal to 2, and in the modified theory it is equal to 4.

It follows from this that, if Fermi’s theory is correct, then, plotting \(\left(\dfrac{N}{f}\right)^{1/2}\) as a function of the electron energy, we should obtain a straight line; if, however, the Konopinski and Uhlenbeck theory is correct, then a straight line should be obtained for the dependence of \(\left(\dfrac{N}{f}\right)^{1/4}\) on \(E\).

Recently, several experimental investigations of this question have been carried out. Let us try briefly to analyze the material obtained in them.

Kurie et al.\(^3\) recently investigated the energy distribution for electrons emitted by artificially radioactive elements arising when a substance is irradiated with deuterons of energy 5.3 MeV. In their experiments the electron energies were measured from the curvature of tracks in a Wilson chamber with a magnetic field. The \(\beta\)-spectra were studied for the active elements \({}^{13}\mathrm{N}\), \({}^{17}\mathrm{F}\), \({}^{24}\mathrm{Na}\), \({}^{32}\mathrm{P}\), Cl, \({}^{41}\mathrm{A}\), \({}^{42}\mathrm{K}\), of which the first two possess positron decay, and all the rest—electron decay. Counting the number of electron tracks for different energy intervals, the authors constructed the dependence expressed by formula (1). It was found that, for all artificially radioactive elements, a linear dependence is obtained for \(\alpha = 4\), whereas \(\alpha = 2\) gives a curved line. For Cl, \({}^{41}\mathrm{A}\), and \({}^{42}\mathrm{K}\) the dependence of \(\left(\dfrac{N}{f}\right)^{1/4}\) on \(E\) is represented by a broken line consisting of two rectilinear segments, which corresponds to the superposition of two continuous spectra with different upper boundaries.

Exactly the same results were obtained by Lauritsen et al.\(^4\) Unlike the preceding authors, they activated the substance with slower deuterons (about 1000 keV). In their work the \(\beta\)-spectra of the following active elements were investigated: \({}^{12}\mathrm{B}\), \({}^{8}\mathrm{Li}\), \({}^{11}\mathrm{C}\), \({}^{13}\mathrm{N}\), \({}^{16}\mathrm{N}\), \({}^{15}\mathrm{O}\), \({}^{20}\mathrm{F}\), of which \({}^{13}\mathrm{N}\) and \({}^{15}\mathrm{O}\) emit positron radiation, and the rest—electron radiation.

Let us note that recently analogous results were obtained for the \(\beta\)-spectra of naturally radioactive elements, namely for RaE and ThC\(_5\).

Thus all the data obtained indicate that the modified theory of \(\beta\)-decay of Konopinski and Uhlenbeck correctly conveys the shape of the \(\beta\)-spectrum curves of radioactive elements.

Since the theory establishes a definite regularity in the shape of the distribution curves, this regularity can be used for the accurate determination of the upper boundary of \(\beta\)-spectra. For this it is sufficient to find the point of intersection of the straight line expressing the dependence of \(\left(\dfrac{N}{f}\right)^{1/4}\) on \(E\) with the energy axis. The following table gives data on the upper boundaries of \(\beta\)-spectra investigated in the works mentioned. In the second column are given the values of the upper boundary for \(\beta\)-spectra calculated from the electrons of greatest energy observed in the Wilson chamber for the given element. The third column gives the values of the upper boundary obtained by extrapolating the linear dependence of \(\left(\dfrac{N}{f}\right)^{1/4}\) on \(E\) to the energy axis. Upon consideration of the table, the circumstance that immediately catches the eye is that the data of the second and third columns differ noticeably from one another. Therefore one might doubt the correctness of the extrapolation of the linear dependence given by the Konopinski and Uhlenbeck theory.

However, there is one case (\({}^{13}_{7}\mathrm{N}\)) for which it is possible to calculate

the energy emitted in $\beta$-decay by another route, namely from a comparison of the energy balance of the following three reactions:

\[ {}^{12}_{6}\mathrm{C}+{}^{2}_{1}\mathrm{H} = {}^{13}_{7}\mathrm{N}+{}^{1}_{0}\mathrm{n}+Q_{1}, \]

\[ {}^{13}_{7}\mathrm{N} = {}^{13}_{6}\mathrm{C}+e^{+}+Q_{2}, \]

\[ {}^{12}_{6}\mathrm{C}+{}^{2}_{1}\mathrm{H} = {}^{13}_{6}\mathrm{C}+{}^{1}_{1}\mathrm{H}+Q_{3} \]

for the value of $Q_{2}$ one obtains $1.45$ MeV, in good agreement with the data of the third column.

Thus it may be considered quite probable that the modified theory of $\beta$-decay of Konopinski and Uhlenbeck correctly describes the phenomenon of $\beta$-decay both with respect to the form of the electron energy distribution curve and with respect to the upper limit of the $\beta$-spectra.

L. Groshev, Moscow

LITERATURE

1. Fermi, Z. Physik, 88, 161, 1934.
2. Konopinski and Uhlenbeck, Phys. Rev., 48, 7, 1935.
3. Kurie, Richardson and Paxton, Phys. Rev., 48, 167, 1935; Kurie, Richardson and Paxton, Phys. Rev., 49, 368, 1936.
4. Fowler, Delsasso and Lauritsen, Phys. Rev., 49, 561, 1936.
5. Champion and Alexander. Nature, 137, 744, 1936.