MODERN STUDIES OF THE SHAPE OF BETA SPECTRA

Chien-Shiung Wu *)

INTRODUCTION

The year 1949 may be called a year of major advances in beta spectroscopy. Many laboratories joined in the active study of beta spectra, and many important and interesting discoveries were made. Most essential of all is that agreement between theory and experiment was achieved on certain points that had long troubled physicists. Theorists working in the field of beta decay, perhaps for the first time, experienced a certain satisfaction from results concerning the behavior of both allowed and forbidden spectra.

FERMI’S THEORY OF BETA DECAY¹

Pauli’s neutrino hypothesis

The modern theory of beta decay was proposed by Fermi² in 1934. A central place in the understanding of this process is occupied by Pauli’s neutrino hypothesis. The apparent nonconservation of energy in the continuous beta spectrum is explained by the neutrino hypothesis through the existence of a new, hypothetical particle, produced in beta decay and carrying away the missing energy. To explain the difficulty of detecting this particle, it is postulated that it is electrically neutral and has a small mass. The basis for introducing this new particle was initially rather negative, but the success of Fermi’s theory of beta decay and the results of experiments on the detection of recoil nuclei give undoubted support to the neutrino hypothesis.

*) Chien-Shiung Wu, Rev. Mod. Phys. 22, 386 (1950). Translated from the English.

Transformation of Nucleons

At the present time we consider that the nucleus consists only of neutrons and protons. Electrons do not exist in the nucleus. Beta particles emitted by radioactive nuclei arise at the moment of decay. Therefore beta transformation may be regarded as the decay of one neutron into one proton, one electron, and one antineutrino, according to the scheme:

\[ \mathrm{n}^{1}=\mathrm{H}^{1}+e^{-}+\nu^{*}. \]

In an analogous way one may write the emission of a positron:

\[ \mathrm{H}^{1}=\mathrm{n}^{1}+e^{+}+\nu. \]

Five Types of Interaction*)

In order to calculate the probabilities of these processes, one must introduce a new type of interaction between the nucleon and the light particles (the electron and neutrino), by analogy with the interaction between charge and the electromagnetic field in quantum electrodynamics. If the type of interaction has been chosen, then the problem of determining the probability of a beta transition is solved by the methods of ordinary time-dependent perturbation theory. The difficulty of the problem consists in choosing the type of interaction. Fermi proposed the simplest type of interaction satisfying the basic requirements. In fact, there are five invariant expressions suitable for the same purpose:

1) scalar

\[ S=(U_f^{*}\beta^{0}U_i)(\psi_e^{*}\beta^{0}\varphi_\nu); \]

) L. D. Landau drew attention to the circumstance that the five variants of the theory of beta decay considered by Wu and usually accepted are not the only possible ones (see the footnote to the article by Berestetskii and Pomeranchuk, ZhETF 19, 756 (1949)). In the usual set it is assumed that the relative parity of the wave functions of the neutron and proton, on the one hand, and of the electron and neutrino, on the other hand, with respect to mirror reflection at the origin of coordinates is the same. For example, if the parity of the wave functions of the neutron and proton is the same, then the parity of the wave functions of the electron and neutrino is also the same; or if the parity of the neutron and proton is different, then the parity of the electron and neutrino is also different. Landau observed that five other variants are possible, each of which is alternative to the ordinary one. Namely, one may admit that the relative parity of the light and heavy particles is pairwise different. Then, in order to obtain invariant expressions, in one of the pairs one must take such a combination of the operators \(\beta,\gamma\) which is transformed under mirror reflection oppositely to the combination entering the other pair. For example, in the pseudoscalar variant one must multiply by \(\gamma_5\). It has not been clarified how far Landau’s variant changes the shape of forbidden spectra and the angular correlation of the electron and neutrino. (Translator’s note.*)

2) four-dimensional polar vector

\[ V=(U_f^* O U_i)(\psi_e^* \varphi_\nu)-(U_f^* \alpha O U_i)(\psi_e^* \alpha \varphi_\nu); \]

3) tensor

\[ T=(U_f^* \beta \sigma U_i)(\psi_e^* \beta \sigma \varphi_\nu) -(U_f^* \beta \alpha O U_i)(\psi_e^* \beta \alpha \varphi_\nu); \]

4) axial vector

\[ A=(U_f^* \sigma O U_i)(\psi_e^* \sigma \varphi_\nu) -(U_f^* \gamma_5 O U_i)(\psi_e^* \gamma_5 \varphi_\nu); \]

5) pseudoscalar

\[ P=(U_f^* \beta \gamma_5 O U_i)(\psi_e^* \beta \gamma_5 \varphi_\nu). \]

Here \(U_i\), \(U_f\), \(\psi_e\) and \(\varphi_\nu\) are the wave functions of the initial nucleon, the final nucleon, the electron, and the neutrino; \(O\) is the operator causing the nucleon to make a transition; \(\beta\), \(\alpha\), and \(\gamma_5\) are Dirac operators, and \(\sigma\) is the ordinary spin operator.

Two approximations for allowed transitions

I

It is known that the eigenvalues of the operators \(\alpha\) and \(\gamma_5\) have an order of magnitude \(\frac{v}{c}\). The velocities of nucleons in the nucleus are of order \(c/10\) (where \(c\) is the speed of light). Therefore the second term in the expressions \(V\), \(T\), and \(A\) has order of magnitude \(1/10\) in comparison with the first (when squared, \(1/100\)), and in calculating allowed transitions it may be neglected.

II

Further, the neutrino practically does not interact with matter. For simplicity, let us also temporarily disregard the action of the Coulomb field of the nucleus on the electron. Then the wave functions of the electron and the neutrino may be taken in the form of plane waves:

\[ \psi_e=A\exp\left[i\,\frac{p}{\hbar}\,r\right], \tag{1} \]

\[ \psi_\nu=B\exp\left[-i\,\frac{q}{\hbar}\,r\right]. \tag{2} \]

Let us write them in expanded form and substitute into the following functions:

\[ (\psi_e^* \beta \varphi_\nu) =(A^* \beta B)\left[ 1-i\,\frac{p+q}{\hbar}\,r -\frac{1}{2}\left(\frac{p+q}{\hbar}\,r\right)^2\ldots \right] \tag{3} \]

and

\[ (\psi_e^* \alpha\gamma_5 \varphi_\nu) =(A^* \alpha\gamma_5 B)\left[ 1-i\,\frac{p+q}{\hbar}\,r -\frac{1}{2}\left(\frac{p+q}{\hbar}\,r\right)^2\ldots \right]. \tag{4} \]

where \(p\) and \(q\) are the momenta of the electron and the neutrino. The energy released in beta decay limits \(p+q\) to several units of \(mc\); \(r\) has nuclear dimensions, and therefore is at most of order \(\frac{1}{40}\frac{\hbar}{mc}\); the successive terms in the expansion

\[ (\psi_e^* \beta \sigma \psi_n) \]

decrease at least in the ratio \(1/10\) (or \(1/100\) when squared). Therefore all terms except the first in expansion (3) may be neglected. In other words, for allowed transitions one need take only the first term in this expansion. The term \((\psi_e^* \alpha \varphi_\nu)\) or \((\psi_e^* \gamma_5 \varphi_\nu)\) is called relativistic and, in allowed transitions, may be discarded for the reasons explained in approximation (I). In this approximation the probability of beta transformation is calculated directly with the aid of the ordinary nonstationary perturbation theory. Then the most important distinguishing feature of the results is the independence of the energy distribution of the beta particles from the form of the interaction*).

Selection rules

Nevertheless, the different types of interaction determine selection rules with respect to spin and parity. For example, for an interaction of the polar-vector type, originally se-

) I. M. Shmushkevich (ZhETF 21* (1951)) has shown that the generally accepted view that the shape of the allowed spectra is identical in all five variants of the theory with respect to the pseudoscalar variant is applicable only to a very limited extent. In any case, the beta spectrum of the free neutron, if only pseudoscalar forces acted, would have to have a quite different shape. Shmushkevich noted that, if one passes to the nonrelativistic limiting case for the pseudoscalar variant, the combination of the wave functions of the free nucleons

\[ \psi_p^* i\beta\gamma_5\psi_N \]

goes over into

\[ -\frac{\hbar}{2Mc}\operatorname{div}(\psi_p^*\sigma\psi_N). \]

It is clear that now one cannot simply take the wave functions of the light particles outside the integral, since the integral would then vanish. One must first integrate by parts, so that the operation \(\operatorname{div}\) is referred to the wave functions of the light particles. Taking them in the form of plane waves, Shmushkevich obtained an extra factor \(p+q\), as compared with the other four variants, in the matrix element and, correspondingly, an extra factor (\(E\) is the energy in units of \(mc^2\))

\[ E^2-1+(E_0-E)^2-\frac{2}{3E}(E^2-1)(E_0-E) \]

in the beta-spectrum curve. This factor appears in the pseudoscalar variant in the lowest order for which the selection rules for the nucleon spin are \(0,\ \pm 1\).

For a nucleon bound in the nucleus, the nonrelativistic limit for the wave functions may fail to give the exact divergence. Then an allowed spectrum of the ordinary type will be obtained. (Translator’s note.)

for the selected Fermi interaction, for allowed transitions one obtains the matrix element \(\left|\int (U_f^* 1 U_i)\,d\tau\right|^2\). The unit operator \(1\), of course, does not change the symmetry properties of \(U_f\) and \(U_i\); therefore no difference in angular momenta or parity between \(U_f\) and \(U_i\) is allowed. We write this as \(\Delta I=0\), no, where “no” means that a change of parity is forbidden. Since the operator \(\beta\) is a scalar, just as \(1\) is, the same selection rules for allowed transitions are obtained for it as well: \(\Delta I=0\), no.

The pseudoscalar \(\gamma_5\) changes sign under mirror reflection; therefore the parity of \(\gamma_5 U_i\) is opposite to the parity of \(U_i\). Consequently, it leads to the selection rules \(\Delta I=0\), yes, where “yes” means that the parity changes.

For tensor and axial-vector interactions one obtains selection rules entirely different from those for the original Fermi interaction.

The Pauli spin operator \(\sigma\) brings the initial state \(U_i\) into coincidence with the final \(U_f\) according to the following selection rules (Teller):

\[ \Delta I=0,\ \pm 1,\ \text{no, and }0\to 0\text{ is forbidden.} \]

Spectrum of Allowed Transitions

The spectrum of an allowed beta transformation can be written as follows:

\[ P(E)\,dE = \]

\[ = \frac{g^2 m_0^5 c^4}{2\pi^3 \hbar^7} \left|\int U_f^* O U_i\,d\tau\right|^2 F(Z,E)(E_0-E)^2(E^2-1)^{1/2}E\,dE. \tag{5a} \]

Since in a magnetic beta spectrometer the number of counts at a given value of the magnetic field, divided by the momentum \((\eta)\), is proportional to the number of beta particles per unit momentum, it is more convenient to write the expression for the spectrum in the following form:

\[ P(\eta)\,d\eta \sim \left|\int U_f^* O U_i\,d\tau\right|^2 \eta^2(E_0-E)^2F(Z,\eta)\,d\eta. \tag{5b} \]

Here \(g\) is a universal constant, known as the Fermi constant and determining the strength of the interaction leading to the transition. \(\int U_f^* O U_i\,d\tau\) is called the nuclear matrix element and is assumed not to depend on \(E\) and \(Z\). \(\left|\int U_f^* O U_i\,d\tau\right|^2\) measures the overlap of the initial and final states. \(E_0\) is the maximum electron energy, including the rest energy. The term \(\eta^2(E_0-E)^2\) is the statistical distribution of momentum between the electron and the neutrino and must arise in any acceptable theory that takes into account the distribution of the energy \(E_0\) between the two particles. \(F(Z,\eta)\) is the Coulomb correction factor, account-

…expressing the action of the Coulomb field of the nucleus on the wave function of the electron. In Fermi’s original paper this factor was written in the following form:

\[ F(Z,\eta)=\gamma^{2S} e^{\pi\delta}\left|\Gamma(1+S+i\delta)\right|^2, \tag{6} \]

where

\[ S=(1-\gamma^2)^{1/2}-1;\quad \gamma=\frac{Z}{137};\quad \delta=\gamma\frac{E}{\eta}. \]

There are, however, no suitable tables of the gamma function of a complex argument.* For very light elements, where the relativistic correction is small, one may take the nonrelativistic Coulomb factor\(^3\)

\[ F_N(Z,\eta)=\frac{2\pi\delta}{1-e^{-2\pi\delta}}. \tag{7} \]

\(\delta\) is positive for electrons and negative for positrons; therefore the Coulomb effect is very different for electrons and positrons. At one and the same energy in the region of low energies, one should expect very few positrons, but rather many electrons, in comparison with the curve constructed without taking account of the Coulomb factor. A better approximation, especially for large \(Z\), was given by Bethe and Bacher\(^4\):

\[ F(Z,\eta)\sim F_N(Z,\eta)\eta^{2S}\left(\delta^2+\frac{1}{4}\right)^S = \]

\[ =F_N(Z,\eta)\left\{E^2(1+4\gamma^2)-1\right\}^S . \tag{8} \]

The accuracy of this approximation up to atomic numbers less than 84 is \(1\%\). In reality, atomic electrons also influence the form of the spectrum at low energies. This screening effect was calculated by Rose and Longmire, Brown, and Reit\(^5\), and was found to be positive both for electrons and for positrons. It is especially important for positrons with energy less than 300 kev at \(Z>25\).

Fermi and Curie Plots

If one constructs the curve

\[ \left[\frac{P(\eta)}{\eta^2 F(Z,\eta)}\right]^{\frac{1}{2}}\sim E_0-E \tag{9} \]

*) Feister recently published the results of a comparison of the distribution of beta electrons according to Fermi, found by three different approximate methods. [Phys. Rev. 78, 375 (1950)].

depending on the energy, then for an allowed beta spectrum it must be a straight line intersecting the abscissa axis at \(E = E_0\). This curve is called the “Curie plot”\(^6\), or the “Fermi plot,” and is often used in studies of the shape of beta spectra. Great success was achieved by Lawson and Cork\(^7\) in 1939 in their work with \(\mathrm{In}^{114}\), and by Tyler\(^8\) in work with \(\mathrm{Cu}^{64}\). Using comparatively thin sources (several \(\mathrm{mg/cm^2}\)), they were able to show good agreement between experiment and the original version of Fermi’s theory.

STUDY OF ELECTRONS OF VERY LOW ENERGIES

The investigation of low-energy electrons is very difficult. The principal difficulty is the effect of scattering and absorption of electrons in the source, of finite and moreover nonuniform thickness, and in the material of the backing. Great difficulties at very low energies are also introduced by absorption in the counter window. Whereas the main part of the spectrum closely follows the Fermi distribution, there are always some deviations in the region below 200 kev. If these differences were genuine, Fermi’s theory would have to be revised.

Spectra of \(\mathrm{S}^{35}\) and \(\mathrm{Cu}^{64}\)

Many laboratories concentrated their efforts on clarifying the situation in the low-energy region. Cook and Langer\(^9\) used a large magnetic spectrometer of high resolving power to investigate the electron and positron spectra of \(\mathrm{Cu}^{64}\). This is an ideal case for testing the theory of beta decay, since the electrons and positrons have approximately the same energy and comparable intensity. In particular, the ratio of the numbers of electrons and positrons is free from errors connected with elastic scattering. The results show that the Curie plot is linear for the electron spectrum up to an energy of 190 kev and for the positron spectrum up to an energy of 270 kev. However, deviations below this energy were regarded as genuine and as not caused by the apparatus, since it was difficult to imagine how a source with a thickness of only about \(10\,\mathrm{\mu g/cm^2}\) could affect the spectrum up to such high energies as 200 kev. But since it is now known that a source prepared from a solution by crystallization can have thickness variations reaching 100:1, the average thickness of an inhomogeneous source of \(100\,\mathrm{\mu g/cm^2}\) simply has no meaning. The Columbia group investigated the spectra of \(\mathrm{S}^{35}\) and \(\mathrm{Cu}^{64}\) with the aid of a solenoidal spectrometer with large transmission\(^ {10,15}\) (Fig. 1) and paid special attention to the homogeneity of the sources. A gradual and consistent decrease of the deviations in the low-energy region was found.

with decreasing source thickness. In the case of \(S^{35}\), when it was possible

Fig. 1. Schematic of a solenoidal magnetic spectrometer.
Visible labels in the diagram: correcting coils; scintillator-filling system; main coils; shutter; to the pump; seal; source holder; diaphragm system; G–M counter; counter shield; counter leads; shutter control; \(10\ \mathrm{cm}\).

to use a source without a carrier, the Kurie plot was obtained as a straight line up to an energy of \(20\ \mathrm{keV}\), where absorption in the counter window begins to have an effect (Fig. 2). In the case of \(Cu^{64}\), a thin colloidal suspension of it was used in order to avoid the effects that arise during crystallization from solution. The spectra obtained from the thinnest and most carefully prepared sources \((100\ \mu\mathrm{g}/\mathrm{cm}^{2})\) showed much smaller deviations in the low-energy region (Fig. 3) than those observed in earlier measurements (in the ratio \(1:4\)). In particular, the experimentally found ratio of the number of electrons to the number of positrons is in excellent agreement with the prediction of the Fermi theory of beta decay, in contrast to the large deviations,

Fig. 2. Fermi plot for the beta spectrum of \(S^{35}\).
Visible labels in the graph: \(\left(\frac{N}{f}\right)^{1/2}\); Energy in \(\mathrm{keV}\); Curve \(A\)—\(1\ \mu\mathrm{g}/\mathrm{cm}^{2}\); Curve \(B\)—\(2\ \mu\mathrm{g}/\mathrm{cm}^{2}\); Curve \(C\)—\(5\ \mu\mathrm{g}/\mathrm{cm}^{2}\).

Fig. 3. Curie plots for the electron and positron spectra of \( \mathrm{Cu}^{64} \).

Text and labels in the figure:

• Vertical axis, left: \(\left(\dfrac{N}{f}\right)^{1/2}_{e^-}\)
• Vertical axis, right: \(\left(\dfrac{N}{f}\right)^{1/2}_{e^+}\)
• Horizontal axis: energy in keV
• Curve labels: \(e^-\), \(e^+\)
• Endpoint labels: 571 keV; 657 keV
• Legend:
• \(\triangle\) without corrections for screening and relativistic corrections
• \(\circ\) with corrections
• Inset:
• “% corrections for screening and relativistic corrections for \( \mathrm{Cu}^{64} \)”
• Curves: \(e^+\), \(e^-\)
• Horizontal axis: energy in keV

Fig. 4. Ratio of the number of positrons to the number of electrons as a function of energy.

Text and labels in the figure:

• Vertical axis:

\[ \lg \frac{N^+}{N^-} \]

• Horizontal axis:

\[ x=\frac{2\pi Z}{137}\,\frac{(1+\eta^2)^{1/2}}{\eta} \]

• Energy scale: 500, 200, 100, 50, 30, 18 keV
• Legend:
• \(\triangle\) without corrections for screening and relativistic corrections
• \(\circ\) with corrections
• \(\bullet\) Bakus, Kuhn, and Langer
• Label on curve: theoretical curve

found by Bakos^[12] and Cook and Langer^[9] (Fig. 4). On the basis of the good agreement between the theoretical and experimental values of this ratio, Wu and Albert^[11] concluded that Fermi’s theory of beta decay, in all probability, correctly predicts the distribution of electrons and positrons at low energies. This conclusion was recently confirmed by independent work by Langer, Moffat, and Price^[13] and by Owen and Cook.^[14] In order to obtain a microscopically homogeneous source, they improved the method of evaporating the active metal Cu^64 onto a thin film in vacuum. An autoradiogram of this source showed its complete homogeneity. Curie plots constructed from data obtained with such thin and homogeneous sources, with thicknesses from a few to 75 μg/cm², showed no deviations from Fermi theory for all energies above 50 keV (Fig. 5). This conclusion is very favorable for Fermi’s theory.

Fig. 5. Fermi plot for the electron spectrum of Cu^64. The shaded circles denote data obtained with a 75 μg/cm² source on a 0.18 mg/cm² aluminum sheet. The open circles refer to data obtained with a source of less than 5 μg/cm² on 15 μg/cm² cellophane.

Further experimental confirmation in the low-energy region

Further experimental confirmation of the correctness of the Fermi allowed spectra was obtained by Price, Motz, and Langer.^[15] They found that Curie plots constructed for Pm^147 and S^35 were linear down to 8 keV. Gross and Hamilton^[16] reported that, using a new type of electrostatic beta spectrometer for studying the low-energy region of the S^35 spectrum, they likewise found that the Curie plot is a straight line in the region from 30 to 7 keV. Deviations below 7 keV are associated with backscattering and absorption in the source and backing.

Method of proportional counters and the spectrum of H^3

In view of range straggling and reflection effects in sources of finite thickness and backings, as well as absorption and scattering in the windows of counters, the ordinary method of investigation

electrons of low energies cannot be extended into the energy region noticeably below 10 keV. The most suitable method of measurement in the lower-energy region apparently consists

Fig. 6. Fermi plot for H³.

in the use of proportional counters into which a radioactive gas is introduced. Curran, Angus, and Cockcroft^17 and Hanna and Pontecorvo^18 investigated tritium by this method and found that the Curie plot is a straight line from the maximum energy 18.6 down to 0.5 keV (Fig. 6). Since H³ is a simple nucleus, this case may be a good test of the Fermi theory of beta decay. Furthermore, since the upper energy limit is exceptionally low, the exact form of the spectrum near the upper—

Fig. 7. Comparison of the theoretical and experimental Fermi curves for tritium near the upper end.

the boundary is very sensitive to the magnitude of the neutrino mass, if it is not equal to zero. A careful study[^19] of the tritium beta-ray spectrum limits the neutrino mass from above to a value of \(1\ \mathrm{kev}\) (Fig. 7). Therefore the neutrino mass may be taken to be zero in most applications of beta-decay theory.

The case of \( \mathrm{Cu}^{61} \)

Among the beta spectra of allowed transitions there was one spectrum that was difficult to understand. Namely, this was the spectrum of \(\mathrm{Cu}^{61}\). Cook and Langer[^20] used a \(\mathrm{Cu}^{61}\) source many times thinner than \(0.1\ \mathrm{mg}/\mathrm{cm}^{2}\) on a collodion film \(0.02\ \mathrm{mg}/\mathrm{cm}^{2}\) thick. The deviation found proved to be much larger than for \(\mathrm{Cu}^{64}\), beginning at about \(500\ \mathrm{kev}\). Owen and Cook[^21] repeated the investigation of the \(\mathrm{Cu}^{61}\) spectrum, evaporating pure \(\mathrm{Cu}^{61}\) onto an aluminum backing, and likewise found a clearly revealed deviation at about \(500\ \mathrm{kev}\). They also repeated[^14] the experiments with \(\mathrm{Cu}^{64}\), using this improved technique for preparing the source, and were able to eliminate a large part of the deviations at low energy found in their previous work[^9]. This was a weighty indication of the possibility of a complex character of the positron spectrum of \(\mathrm{Cu}^{61}\). If this is so, then one or several gamma quanta emitted with very low intensity must be associated with the beta transition. Böhm’s group[^22] and Owen and Cook[^23] searched for these gamma rays and found them. There are three gamma transitions, corresponding to energies of \(0.652\), \(0.279\), and \(0.070\ \mathrm{Mev}\). The complexity of the spectrum probably arises from a transition to the \(\mathrm{Ni}^{61}\) level with excitation energy \(0.652\ \mathrm{Mev}\). The end point of this spectrum should lie at about \(1.205 - 0.652 = 0.553\ \mathrm{Mev}\), which agrees with the deviations obtained earlier at energies of about \(511\ \mathrm{kev}\).

BETA SPECTRUM OF \( \mathrm{He}^{6} \)

The beta radiation of \(\mathrm{He}^{6}\) \(\bigl(\mathrm{He}^{6} \to \mathrm{Li}^{6} + \beta^{-} + \nu^{+}\bigr)\) was one of the first historical cases to indicate Teller’s selection rules. \(\mathrm{He}^{6}\) emits beta rays with energy above \(3\ \mathrm{Mev}\) and has a half-life of less than one second. The product \(ft\), which in this case is of the order \(10^{2}\)—\(10^{3}\) sec., shows that a superallowed type of transition takes place. On the other hand, one may consider \(\mathrm{He}^{6}\) as consisting of one \(\mathrm{He}^{4}\), i.e. an alpha particle, and two neutrons, which in the ground state must have spin equal to zero. Therefore one should expect that the spin of \(\mathrm{He}^{6}\) is zero, as in all known cases when the nucleus consists of an even number of neutrons and protons. \(\mathrm{Li}^{6}\) may be regarded as consisting of an alpha particle and a deuteron. The spin of the deuteron is 1; therefore \(\mathrm{Li}^{6}\) must have spin \(I = 1\).

Therefore the change of spin in this transition is \(\Delta I = 1\). The selection rules for all allowed transitions obtained for the originally chosen type of interaction (scalar and polar-vector), known as the Fermi selection rules, forbid a change of spin. The experimentally determined allowedness of the transition of \(\mathrm{He}^6\) undoubtedly contradicts selection rules of this type. However, the tensor and axial-vector types of interaction allow the spin to change by one unit also in allowed transitions, owing to the properties of the spin operator \(\sigma\) used in these types of interaction (see the section on selection rules).

Fig. 8. Fermi plot for the beta spectrum of \(\mathrm{He}^6\).

Much attention has always been paid to \(\mathrm{He}^6\). But since it decays rapidly and is in the gaseous state, for a long time it was impossible to obtain information either about the upper limit or about the shape of the beta spectrum. Recently Brown and Perez-Mendez\(^{23a}\) improved the method of circulating the radioactive gas between the activating chamber of the cyclotron and the source chamber of the beta spectrometer, as a result of which they were able to determine the upper energy limit of \(\mathrm{He}^6\), equal to \(3.230 \pm 0.15\) MeV. Further, the Fermi plot gives a straight line, as it should for allowed transitions, from the upper limit of the spectrum down to 150 keV, i.e. over 95% of the extent of the entire energy interval\(^{23a}\) (Fig. 8).

SIMPLEST RADIOACTIVE NUCLEI: n, \(\mathrm{H}^3\) and \(\mathrm{He}^6\)

The magnitude of the nuclear matrix element remains, in the theory of beta decay, generally speaking, indeterminate. Indeed, little can be said about the degree of overlap of the initial and final wave functions, since the wave functions themselves ...

not much is known. But for the simplest mirror nuclei, such as n and H\(^3\), and for the nucleus He\(^6\), these matrix elements have been calculated by Wigner\(^ {34}\) both for the Fermi and for the Teller selection rules. Since the upper limit of the spectrum and the half-life times for H\(^3\) and He\(^6\) are now much better known, it is of interest to determine to what extent the values of \(M^2 ft\),

Table I

Data on n, H\(^3\), and He\(^6\)

\(*\) The neutron decay energy 0.790 MeV was calculated from the new value of the deuteron binding energy, from which the neutron mass was found, and may contain an error of several percent.

\(**\) The limiting energy of tritium was taken to be 18.6 keV and may have an error of up to 0.5 keV (2.4%, which may give an 8% error in \(f\)). The half-life time of 12.46 years may also have an inaccuracy of several percent.

\(***\) The limiting energy and half-life time of He\(^6\) have recently been measured with accuracies of 0.3 and 0.5%, respectively, so that the error in \(ft\) does not exceed 2%. However, the value of \(M^2\) for He\(^6\) is probably somewhat smaller than that calculated on the assumption that the wave function of the two nucleons changes from singlet to triplet.

calculated for H³ and He⁶. Although exact agreement between them is not to be expected because the spin changes in He⁶, it is possible to determine the order of magnitude that should be assigned to the half-life of the neutron, calculated from data relating to H³ and He⁶. Table 1 gives the principal data relating to n, H³, and He⁶, and the corresponding values of \(M^2ft\), calculated in accordance with both Fermi’s and Teller’s selection rules. The acceptable half-life of the neutron should be of the order of 10–12 min.; according to preliminary experimental results it lies between 9 and 25 min.

FORBIDDEN SPECTRA

The significance of forbidden spectra

As was indicated at the beginning, the energy distribution in an allowed spectrum is determined essentially by the statistical distribution of momentum between the electron and the neutrino. Good agreement between theory and experimental results for allowed spectra gives no indication as to the choice of one of the five possible types of interaction. On the other hand, it may be expected that there exist certain forbidden spectra whose form will differ substantially from the form of allowed spectra, and that the precise form of the momentum distribution will help to determine the character of the interaction between the nucleon and the electron-neutrino field.

Brief remarks on the theory of forbidden transitions

Transitions that violate the selection rules for allowed transitions are called forbidden. In such cases the nuclear matrix element obtained from the first term of (3), which gives allowed transitions, becomes zero. Only such beta radiation can be emitted that carries away the difference of angular momenta, analogously to electromagnetic radiation of high multipolarity. We must therefore take the second term of (3) and the first term of (4) together. In the case of first-order forbiddenness for the polar-vector interaction \(V\), these two terms give the matrix elements

\[ \int U_f^* \frac{p+q}{\hbar}\cdot r U_i\,d\tau \quad \text{and} \quad \int U_f^* \alpha U_i\,d\tau . \]

The basic property of these matrix elements is that they lead to an intensity diminished by a factor of

\[ \left[\frac{p+q}{\hbar}\cdot r\right]\sim 1/100; \quad \text{or} \quad \alpha^2\sim 1/100; \]

moreover, the first of them explicitly depends on the energy of the emitted electron. Further, since

\(\int U_f^* r U_i\,d\tau\) and \(\int U_f^* \alpha U_i\,d\tau\) are polar vectors, and the selection rules are as follows:

\[ \Delta I = 0,\ \pm 1 \quad (\text{no } 0 \to 0), \quad d\alpha . \]

Teller’s selection rules for the first forbidden transition are determined by quantities composed of \(\sigma\), \(\gamma\), and \(\alpha\). These rules are as follows:

\[ \Delta I = 0,\ \pm 1,\ \pm 2 \quad \left( \text{no } 0 \to 0,\ 1 \to 0,\ \frac{1}{2} \to \frac{1}{2} \right), \quad d\alpha . \]

Of course, in the case where the selection rules are not fulfilled either for allowed transitions or for first-order forbidden transitions, one must go further, introducing higher powers from (3) and (4) and thus obtaining selection rules for second-, third-, etc.-order forbidden transitions. The first theoretical calculation of a forbidden spectrum belongs to Konopinski and Uhlenbeck\(^{25}\). The final results are presented in the form of a “correction factor” \(C\), by which the allowed distribution must be multiplied in order to obtain the given forbidden spectrum. These correction factors depend not only on the energy but also, to one degree or another, on the type of interaction. It should be expected that there must be certain forbidden transitions whose spectra differ markedly from allowed ones, and a careful investigation of the exact form of the distribution may provide certain information about the type of interaction.

For first-order forbidden transitions there are, however, known difficulties, since the relativistic terms \(\int U_f^* \alpha U_i\,d\tau\) or \(\int U_f^* \gamma_5 U_i\,d\tau\), entering into the interactions \(V\), \(T\), and \(A\), do not depend on the energy. If the contribution of these relativistic terms is much greater than the contribution of the term \(\int U_f^* r U_i\,d\tau\), which depends on the energy, then the shape of the first forbidden spectrum will coincide with the shape of an allowed one. In cases where the Coulomb potential energy \(\dfrac{Ze^2}{r}\) is large, it is also possible that the electron will receive a small kinetic energy; then the term \(\left[\dfrac{p+q}{\hbar}\cdot r\right]\) must be replaced by the term \((\alpha Z)\), also independent of the energy. Therefore it is theoretically possible that a spectrum forbidden in first order will have the shape of an allowed one.

Beta spectrum of RaE

About a year ago the beta spectrum of RaE (Fig. 9) was the only one for which it was considered generally accepted that its shape differs from the spectrum of an allowed transition\(^{26}\). Although the shape of the RaE spectrum has been found quite satisfactorily\(^{25}\), the interpre-

calculation includes much arbitrariness because of the computation of many matrix elements entering into the forbidden multiplier; therefore, on the basis of the data on RaE alone it was not possible to arrive at any definite conclusions about the form of the interaction.

Fig. 9. Fermi plot for the beta spectrum of RaE.

Unambiguously determined spectra of forbidden transitions for which \(\Delta I=2, da\)

Langer and Price\(^ {27}\) found that \(\mathrm{Y}^{91}\) (a fission product) has a spectrum of undoubtedly forbidden type. Moreover, judging by its comparative half-life \((f_3 \sim 5\cdot 10^8\ \mathrm{sec.})\), it should be classified as twice forbidden. But from an analysis of the structure of the nuclear shells\(^ {38}\) it is seen that this transition must be accompanied by a change of the total angular momentum by two units, a change of parity, \({}_{39}\mathrm{Y}^{91}\,(p_{1/2}\ \text{odd}) \to {}_{40}\mathrm{Zr}^{91}\,(d_{5/2}\ \text{even})\), and, according to the rule-

According to Teller’s selection rules, such a transition is classified as first forbidden.^29 According to the theory of forbidden spectra, developed by Konopinski, Uhlenbeck, and Greuling,^30 if in decay the change of the total angular momentum exceeds the degree of forbiddenness by one, then the transition probability contains a matrix element of only one type. And this means that the dependence on energy is determined uniquely, differing from the shape of the allowed spectrum by the factor

\[ \alpha = p^2 + q^2 = (E^2 - 1) + (E_0 - E)^2 . \]

Here \(E\) is the electron energy and \(E_0\) the total released energy, expressed in units of \(mc^2\). The factor \(\alpha\) increases the fraction of energetic particles and may also increase the fraction of particles of low energy if \(E_0 > 2\). Therefore the uncorrected Kurie plot has a tendency to bend upward at high energies, but may also rise at low energies, having a point of inflection at \(E = \frac{1}{2}E_0\) (Fig. 10).

Fig. 10. Fermi plot for the beta spectrum of \(Y^{91}\).

Immediately after the discovery of the spectrum of \(Y^{91}\), reports appeared^31–38 that \(Y^{90}\), \(Sr^{89}\), \(Sr^{90}\), \(Sr^{91}\), \(Cs^{137}\), \(Rb^{86}\), \(K^{42}\), \(Cl^{38}\), \(Sb^{124}\), \(Sn^{123}\), and \(Sn^{125}\) all have a forbidden spectrum of the “\(\alpha\)-type.” The existence of forbidden spectra of this type is strong support for Teller’s selection rules, as well as for the theory of nuclear shells. It is therefore clear that at least part of the true interaction must be of tensor or axial-vector type.

Shull and Feenberg^39 also pointed out that the values of \(ft\) for this class of transitions, multiplied by the corresponding values \((E_0^2 - 1)\), in order thereby to take account of the factor \(\alpha\), have one and the same order of magnitude, namely \(10^{10}\). Table II gives, for reference, the values of \(ft\) and \((E_0^2 - 1)\) for all such nuclei. Further, according to the spin–orbit coupling scheme in the nuclear shell model, odd nuclei undergoing transitions of this kind change the orbital angular momentum by one unit, \(\Delta L = 1\), and the total angular momentum^40 by two: \(\Delta I = 2\). For example:

\[ \begin{array}{ll} {}_{38}Sr^{89}\ (d_{5/2}\ \text{even}) & {}_{39}Y^{89}\ (p_{1/2}\ \text{odd}) \\ {}_{38}Sr^{91}\ (d_{5/2}\ \text{even}) & {}_{39}Y^{91}\ (p_{1/2}\ \text{odd}) \\ {}_{39}Y^{91}\ (p_{1/2}\ \text{even}) & {}_{40}Zr^{91}\ (d_{5/2}\ \text{odd}) \\ {}_{18}A^{41}\ (f_{7/2}\ \text{even}) & {}_{19}K^{41}\ (d_{3/2}\ \text{odd}) \end{array} \]

Table II

Values of \((E_0^2 - 1)\,ft\) for beta transitions with \(\Delta I = 2,\ d\alpha\)

Beta spectrum of \(\mathrm{Cl}^{36}\)

At approximately the same time, Wu and Feldman obtained a certain amount of radioactive \(\mathrm{Cl}^{36}\) with a specific activity of \(0.05\ \mu\mathrm{C}/\mathrm{mg}\). At that time a quadrupole-focusing device for a solenoidal spectrometer, designed according to the theoretical calculations of Frankel \(^{41}\) and Persico \(^{42}\), had already been completed and tested. It turned out that it increases the transmission by a factor of four in comparison with the previous device (Fig. 11). A source with a surface density of \(0.1\ \mathrm{mg}/\mathrm{cm}^2\) was prepared, and its spectrum was studied with a new diaphragm system \(^{43}\). At a glance at the pulse-distribution curve (Fig. 12) one clearly sees a pronounced shift toward higher energies, indi-

Fig. 11. Increase in transmission with ring focusing.

...indicating a high degree of forbiddenness. The Curie plot has a large convexity away from the abscissa axis, extending all the way to the region of the smallest energies. Such a form of the forbidden spectrum is observed for the first time.

Fig. 12. Momentum distribution on the curve for \( \mathrm{Cl}^{36} \).

The spin of \( \mathrm{Cl}^{36} \) was not known when its spectrum was taken. By means of the usual method of interpreting forbidden spectra, the best agreement with experiment was obtained by introducing the so-called \(D_2\)-factor (in Marshak’s notation \(^{44}\)):

\[ D_2 \sim \frac{1}{30}(E_0-E)^4+ \frac{1}{9}(E^2-1)(E_0-E)^2+ \frac{1}{30}(E^2-1)^2 \]

(see Fig. 13), which is a unique correction factor for \( \mathrm{Be}^{10} \), for which the spin changes by three. Further, \( \mathrm{Cl}^{36} \) has approximately the same half-life and upper energy limit as \( \mathrm{Be}^{10} \) (see Table III). It could theoretically have been considered an analogue of \( \mathrm{Be}^{10} \), if the change of spin had remained unknown. But when Townes \(^{45}\) determined the spin of \( \mathrm{Cl}^{36} \) from the radio-frequency spectrum of \( \mathrm{Cl}^{36}\mathrm{CN} \), comparing it with various

Fig. 13. Fermi plot for the beta spectrum of \( \mathrm{Cl}^{36} \).

theoretically possible spectra for different spin values of Cl\(^{36}\), it turned out that the spin of Cl\(^{36}\) is equal to two. A\(^{36}\) contains an even number of protons and neutrons. Its spin is probably equal to zero. But with

Table III

Data on \({}_{4}\mathrm{Be}^{10}\) and \({}_{17}\mathrm{Cl}^{36}\)

a change of spin by two, with changed or unchanged parity, many different beta transitions can be chosen in accordance with the selection rules. Table IV gives the matrix elements

Table IV

Matrix elements allowing the transition
\(2\to 0\), and the parity change associated with them \(*\)

\(*\) Here the notation of the paper by Konopinski and Uhlenbeck\(^{30}\) is adopted.

which give the transition \(2\to 0\) with the corresponding change of parity. Correction factors for these matrix elements were given by Konopinski and Uhlenbeck\(^{30}\) and by Greiling\(^{30}\). None of these factors, taken separately, agrees with experiment. In other words, no single type of interaction, taken separately, fits the experimental data (Fig. 14). Longmire tried linear combinations of interactions\(^{47}\). These combinations give cross terms with other forms of spectra. The results are shown in Fig. 15. The combinations \((2S,\,2V)\) and \((2T,\,2A)\) are excluded according to Fierz’s theoretical result, which showed that these combinations seriously change the form of allowed spectra by a term \((1\pm CE)\), where \(E\) is the electron energy. The combinations \((2S,\,2T)\)

Fig. 14. Correction factors for electrons of C$^{13}$. The experimental data are enclosed between the two dashed curves. Curve (a) represents $2T(T_{ij})$ and (approximately) $\frac{1}{4}2V(R_{ij})$; curve (b) is on the same scale as (a), for $2A(T_{ij})$ and (approximately) $\frac{1}{4}2S(R_{ij})$; curve (c) is for $2T(A_{ij})$ and $2V(A_{ij})$; curve (d) is the best fit for comparison of the experiment with $2T$, or $2V$ (approximately). The correction factor has for $3V$ approximately the same form as (b). The ordinate scale is arbitrary (see$^{47}$).

Fig. 15. Correction factors for electrons of C$^{13}$. The experimental data are enclosed between the dashed curves. Curve (a) is a combination $(2S,\ 2T)$ or, approximately, $(2A,\ 2V)$. Curve (b) is the best fit with the combination $(2S,\ 2V)$ (see$^{47}$).

and \((2A, 2V)\) are almost identical and agree well with the experimental data. Furthermore, these combinations of interactions are in good agreement with experiment in those cases in which a correction multiplier \(a\) is introduced, such as \(Y^{91}\), \(Y^{90}\), \(Sr^{90}\), \(Sr^{89}\), \(Cs^{137}\), \(Rb^{86}\), and \(K^{42}\), since the cross multiplier is reduced in all cases where the change of spin is one unit greater than the order of forbiddenness. In any case it is still too early to draw final conclusions concerning \(Cl^{36}\). It may be that the determination of the spectrum or of the spin is erroneous. It may turn out, as Marshak has noted, that this case will help in selecting the correct linear combination of the five types of interactions. Recently, Bouché and Nataf \(^{48}\) indicated that this case can be explained with a pure invariant, if one takes into account a modification of the usual selection rules, which may be necessary for light elements.

Beta Spectrum of \(Be^{10}\)

Let us now consider two interesting cases, \(Be^{10}\) and \(K^{40}\). \(Be^{10}\) has a relative half-life of \(4.4 \cdot 10^{13}\) sec. and a change of spin by three units \((0 \to 3)\). For \(K^{40}\) the relative half-life is \(10^{18}\) sec. and the change of spin is by four units. The spectra

Fig. 16. Fermi plot for the beta spectrum of \(Be^{10}\).

of these were theoretically investigated and predicted by Marshak \(^{44,49}\). In the case of \(Be^{10}\) the observed half-life requires the rejection of all matrix elements except four, connected with the energy spectrum, which is determined by the multiplier \(D_2\). Of these four, three \((2T, 2_{1}^{5}A, 3T)\) belong to the Teller type of interactions and one \((3V)\)—to the Fermi type. The activity of \(Be^{11}\), obtained in the usual way, is very small because of the large time

half-life and small activation cross section. Nevertheless, Bell and Cassidy\(^ {50}\) used a source several \(mg/cm^2\) thick and, with their scintillation spectrometer, found a deviation of the Curie plot from the allowed form in the region of high energies. Further, having found the \(D_2\) spectrum in Cl\(^ {36}\), Wu and Feldman carried out an extensive comparative study\(^ {51}\) of various identical sources: Y\(^ {91}\), RaE, P\(^ {32}\), Cl\(^ {36}\), and Cu\(^ {64}\), and came to the conclusion\(^ {52}\) that the true distribution in the beta spectrum may well be of the \(D_2\) type, as had been predicted theoretically. Immediately thereafter Wright and Milton\(^ {53}\) reported that their work with a high-pressure proportional counter gives results consistent with the assumed form of the Be\(^ {10}\) spectrum—\(D_2\). Studying spectra of BeO sources less than \(0.5\ mg/cm^2\) thick in a magnetic spectrograph, Alburger, Hodges, and Engelk\(^ {54}\) and Feldman and Wu\(^ {55}\) found good agreement between the experimentally measured and theoretical spectrum (Fig. 16). Bell and Cassidy\(^ {56}\) obtained the same result with a scintillation spectrometer. This is undoubtedly a great success of the theory of beta decay.

Beta Spectrum of K\(^ {40}\)

In the case of K\(^ {40}\) the spin changes by four units. Theoretical calculations were performed independently by Marshak and Greuling\(^ {49,30}\). If the parities of K\(^ {40}\) and Ca\(^ {40}\) are different, then for the beta transition a third-order forbiddenness is required for tensor or axial-vector interaction. According to a special property of the theory of forbidden spectra, when the change of spin exceeds the order of forbiddenness by one, the spectrum is determined uniquely. If the parities of K\(^ {40}\) and Ca\(^ {40}\) are the same, the same transition must be fourth-order forbidden. The corresponding factors \(C_{4S}\), \(C_{4P}\), and \(C_{4A}\) give a uniquely determined energy spectrum; \(C_{4V}\) and \(C_{2T}\) possess greater flexibility. Unfortunately, the half-life of K\(^ {40}\) is \(2.7 \cdot 10^9\) years and its abundance is only \(0.016\%\).

To obtain the true spectrum of K\(^ {40}\), a highly enriched sample must be used. Feldman and Wu\(^ {51}\) at Columbia University studied, with thick sources, the K\(^ {40}\) spectrum on the high-energy side and concluded that the true spectrum should most likely be concave in this region on the Curie plot. Recently Bell, Weaver, and Cassidy\(^ {57}\) used an enriched KCl source \(2.5\ mg/cm^2\) thick and found in their scintillation spectrometer that the Curie plot is concave toward the energy axis at energies above \(700\ kev\) (Fig. 17). The end point is located at \(1.36 \pm 0.05\ Mev\). If a correction factor is introduced for a tensor or axial-vector interaction forbidden in third order, a straight line is obtained up to an energy of \(0.7\ Mev\) (one-half of the energy interval). The strong bend at

at lower energies, characteristic of a scintillation spectrometer, inevitably arises because of scattering in the crystal. Recently Alburger^58 investigated the beta spectrum of K^40 (enriched to 7%) from a source of thickness 2.4 mg/cm^2 in a spectrometer “with a thin lens” with a resolution of 17%. The Curie plot, corrected for the third-order forbiddenness in the case of tensor or axial-vector interaction, remains straight down to 500 keV. The deviation below 500 keV is interpreted as being due to the influence of the source thickness,

Fig. 17. Fermi plot for the beta spectrum of K^40.

which was confirmed by an auxiliary experiment with P^33. The most recent investigations by Feldman and Wu), carried out with an enriched K^40 source (2.5 mg/cm^2) on a solenoidal spectrometer with a resolution of about 10%, also confirm tensor or axial-vector interaction with a third-order forbiddenness and exclude four-times-forbidden scalar axial-vector or pseudoscalar interaction. Further theoretical study of the correction factors $C_{AV}$ and $C_{AT}$ by a method analogous to that applied for Be^10, by estimating the magnitude of the matrix elements according to Greuling, showed*), that these two correction factors give the same single spectrum as for the triply forbidden axial-vector or tensor interaction. Therefore a four-times-forbidden tensor or vector interaction can explain the observed form of the spectrum just as well. Nevertheless, the theory of nuclear shells predicts a change of parity in this transition. If this is so, the fourth-order forbiddenness

) In press.
*) Private communication.

naturally excluded because of the change in parity, both by the Fermi and by the Teller selection rules. The maximum energy in the spectrum, extrapolated from the corrected Curie plot, is \(1325 \pm 15\) keV, somewhat less than that given in earlier reports.

CONCLUSION

Some time ago it caused perplexity that evidently forbidden transitions have spectra of an allowed type. Theoretically it is quite possible that a first forbidden transition will lead to a spectrum of allowed shape. However, spectra forbidden twice will have the shape of allowed spectra only under very special circumstances. On the other hand, strongly forbidden forms will appear only for prohibitions of high order: \(\Delta J = 2, 3, 4\), as is obtained theoretically. Therefore, upon a careful review of the material on beta spectra and their interpretation that has accumulated in recent times, it becomes clear that determining the order of forbiddenness from a single value of \(ft\) leads to values of the order that are too high. The smallest \(\lg ft \simeq 3\) or 4 are allowed transitions between nuclei with similar wave functions. These transitions are called superallowed, whereas transitions between nuclei with not very similar wave functions have \(\lg ft\) lying between four and six. If the transition probability for each successive order of forbiddenness decreases by a factor of \(1/100\), then for first-order forbiddenness one should expect \(\lg ft\) from six to eight or more. Transitions forbidden in the second order or higher will, generally speaking, have \(\lg ft > 0\). Therefore it is not surprising that the majority of spectra have an allowed shape\(^{40}\).

If this is so, one can try to understand why so few cases of beta–gamma correlation have been observed\(^{59}\). According to theory, angular correlation should exist not only for allowed but also for forbidden beta transitions, if they have a spectrum of allowed form. Only three definitely anisotropic distributions are known\(^{60–62}\)—these are Rb\(^{86}\), Tm\(^{170}\), and Sb\(^{134}\). The shape of the beta spectrum of Sb\(^{134}\) has recently been studied, and it was established\(^{37}\) that it indeed has a spectrum of the \(\alpha\)-forbidden type. It is therefore highly desirable to study also the corresponding spectra of Rb\(^{86}\) and Tm\(^{170}\). Any additional information on the angular beta–gamma correlation will undoubtedly be useful for the theory of beta decay.

The three experimentally studied, unambiguously determined spectra Y\(^{91}\), Be\(^{10}\), and K\(^{40}\) agree with their theoretical form, which is a triumph of the theory of forbidden spectra. The so-called \(\alpha\)-type spectrum of the Y\(^{91}\) group is typical for a first-order forbidden transition

order with a change of spin \(\Delta I = 2\), \(da\). It is strong confirmation of the fact that, at least, part of the true interaction is tensor or axial-vector. The spectrum of \(\mathrm{Be}^{10}\) (\(\Delta I = 3\)) and \(\mathrm{K}^{40}\) (\(\Delta I = 4\)) can be interpreted by means of a tensor or axial-vector interaction, if the degree of forbiddenness (second or third) is one less than \(\Delta I\), and by means of a tensor or polar-vector interaction, if the degree of forbiddenness (third or fourth) is equal to the change of moment. But if one takes into account the change of parity occurring in the transitions \(\mathrm{Be}^{10}\) (\(нет\)) and \(\mathrm{K}^{40}\) (\(da\)), as predicted by the nuclear-shell model, then the spectra of \(\mathrm{Be}^{10}\) and \(\mathrm{K}^{40}\) must be classified as \(\alpha\)-type, for which the change of spin is greater by one unit than the degree of forbiddenness. They are allowed only by the Teller selection rules, i.e. by a tensor or axial-vector interaction. But in order to choose between these two types of interaction, additional information is needed. The spectrum of \(\mathrm{Cl}^{36}\) cannot be explained by a simple interaction; for it a linear combination \((2S, 2T)\), \((2V, 2A)\) and possibly \((2V, 2T)\) is needed. These combinations of interactions are consistent also with the unambiguously determined spectra of \(\mathrm{Y}^{91}\), \(\mathrm{Be}^{10}\), and \(\mathrm{K}^{40}\). In order to learn more about the true linear combinations, it is necessary to find and study more cases of beta decay in which the change \(\Delta I\) is equal to the degree of forbiddenness. The only known case of this class is \(\mathrm{Rb}^{87}\). In it the spin changes by three, and it is most suitable for the third order of forbiddenness. But its exceptionally long half-life and the low end point of the spectrum make the study of the beta spectrum very difficult.

Such is the brief summary of the present-day study of beta spectra.

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