BETA DECAY OF LIGHT NUCLEI*)

Ya. Smorodinskii

§ 1. INTRODUCTION

The study of allowed beta decays has made it possible to draw a number of important conclusions about the form of the interaction between nucleons and light particles. A detailed analysis of experiments in this field is given in the article by Zel’dovich published above, and here we shall only briefly summarize the results.

The half-life of an allowed beta decay \(t\) is customarily written in the form

\[ ft=\frac{A}{|(1)^2+R(\sigma)^2|}. \tag{1} \]

In this formula \(f\) is the well-known Fermi function, which depends on the maximum energy of the electron and on the charge of the nucleus. This function has been tabulated (with account taken of the nucleus) in several works\(^{2,34}\). The constant \(A\) is related to the usual beta-decay constant \(g_F\) by the formula

\[ A=\frac{1}{g_F^{\,2}}\frac{2\pi^3\hbar^7}{mc^3}\ln 2 \tag{2} \]

(where \(g_F\) is expressed in units \(mc^2\left(\frac{\hbar}{mc}\right)^3\)), or numerically:

\[ A=\frac{5.57\cdot 10^{-20}}{g_F^{\,2}}. \tag{3} \]

The quantity \(A\) has been determined experimentally\(^{3}\) from the decay \(O^{14}\to C^{14*}\) (see below, § 3) and is equal to

\[ A=6550\pm150\ \text{sec}. \tag{4} \]

*) Expanded presentation of a report at the Fifth Conference on Nuclear Spectroscopy (February 1955).

This gives for the constant of $\beta$-decay (Fermi) $g_F$ the value

\[ g_F=2.92\cdot 10^{-12}\left[\text{in units }(mc^2)\left(\frac{\hbar}{mc}\right)^3=4.73\cdot 10^{-38}\ \text{erg}\cdot \text{cm}^3\right] \tag{5} \]

or, in ordinary units,

\[ g_F=1.38\cdot 10^{-49}\ \text{erg}\cdot \text{cm}^3 . \tag{6} \]

The denominator of formula (1) is the nuclear part of the squared modulus of the matrix element of the interaction operator,

\[ M=\langle 1\rangle^2+R\langle\sigma\rangle^2 . \tag{7} \]

This matrix element consists of two terms: one corresponds to an interaction independent of spin (denoted by $\langle 1\rangle$), and the other corresponds to an interaction dependent on spin (denoted by $\langle\sigma\rangle$).

As is known, scalar and vector interactions lead to the element $\langle 1\rangle$, while tensor and pseudovector interactions lead to the element $\langle\sigma\rangle$. Their relative contribution is measured by the constant $R$. Let us note that the existence of four types of interaction in the nonrelativistic case (for nucleons) is readily discovered if one proceeds from the requirement that the interaction describe both the decay of a proton and the decay of a neutron with the same constants. Then, reasoning in the usual way for quantum mechanics, we conclude that each of the matrix elements will correspond to an interaction that does not change sign under replacement of a proton by a neutron, and to an interaction that does change sign under such a replacement. The interactions $V$ and $T$ (similar, respectively, to the electromagnetic interaction of currents and of dipole moments) are odd, and consequently the remaining two, $S$ and $A$, are even with respect to the indicated replacement.*

Classification of nonrelativistic interactions

It has been established experimentally that:

1) Of the two interactions $A$ and $T$, responsible for the matrix element $\langle\sigma\rangle$, the interaction $T$ is realized in nature. (Experiments on the correlation in the decay $\mathrm{He}^6\to \mathrm{Li}^1$, Rustad and Ruby$^4$, Allen and Jentschke$^5$.)

* The fifth type of interaction—the pseudoscalar—is relativistic and must be considered simultaneously with the second approximation for the other four types.

** The pseudoscalar interaction is also even.

2) Of the two interactions \(S\) and \(V\) responsible for the matrix element \(\langle 1\rangle\), the interaction \(S\) is realized in nature. (Experiments on the correlation \(F^{19}\to Ne^{19}\), Alford and Hamilton\({}^{6}\), Maxson, Allen, and Jentschke\({}^{7}\).)

If one assumes, as seems natural, that the decays of all nuclei are described by Hamiltonians of the same structure, then, while for the time being neglecting possible small admixtures of the interactions \(S\) and \(A\) (which must also be clarified experimentally in the future), two further questions remain to be answered.

1) The relative sign of the interactions \(S\) and \(T\): whether the scheme \(S+T\) or \(S-T\) is realized\({}^{*}\).

2) The relative share of the interactions \(S\) and \(T\)—the value of the coefficient \(R\).

Because the terms of the Hamiltonian corresponding to the interactions \(S\) and \(T\) do not interfere with one another, the relative sign of \(S\) and \(T\) cannot be determined from spectra or lifetimes. For this, investigations of the polarizations of the electron and neutrino (or of the nucleus—the reaction product) are required—a very difficult task. This question is discussed in more detail in Zel’dovich’s article\({}^{1}\), and we shall not dwell on it.

We shall now be interested in the second question. The answer to it should be sought in the region of light nuclei, since for such nuclei the matrix elements can be determined with appreciable accuracy.

For this purpose we shall analyze the data on the \(\beta\)-decay of light nuclei and, on the basis of this analysis, estimate the value of \(R\).

§ 2. FORMULAS FOR MATRIX ELEMENTS

The calculation of the lifetimes of allowed \(\beta\)-decays leads, as we have seen, to the calculation of the (nuclear) matrix elements of the operators \(1\) or \(\sigma\). We shall present here the formulas necessary for calculating these matrix elements for nuclei in which the protons and neutrons are in identical states (in the shell model)—both in the initial and in the final nucleus. It is clear that in the case of nuclei in which, in the transition, a nucleon passes from one state into another (changes the angular momentum \(j\)), the matrix element depends on the radial part of the wave function, which cannot be calculated without detailed knowledge of the nuclear forces. Conversely, if the nucleon does not change its state, the radial functions drop out of the calculations (the integral over them, by virtue of normalization, is equal to unity), and the value of the matrix element is determined entirely by the configuration of the nucleons.

\({}^{*}\) Different interactions can enter the Hamiltonian only with real coefficients—this is required by the general charge invariance of quantum mechanics, i.e. invariance with respect to changes of the signs of all charges (Groot and Tolhoek\({}^{8,9,10}\)).

Let us begin with the case when there is only one nucleon in the nucleus located outside closed shells.

Obviously, in this case

\[ \langle 1\rangle=1 \quad \text{(one nucleon, any } j\text{)}. \]

To compute the matrix element \(\langle\sigma\rangle\), we proceed analogously to how the matrix element is computed for the Zeeman effect. Namely, we find the projection of the vector \(\boldsymbol{\sigma}\) onto the vector of the total angular momentum \(\mathbf{j}\). This will be the diagonal transition element (i.e., the element determining the transition \(j\to j\)).

Thus,

\[ \langle\sigma\rangle=\frac{(\boldsymbol{\sigma}\mathbf{j})\mathbf{j}}{j(j+1)}. \]

We compute the scalar product \((\boldsymbol{\sigma}\mathbf{j})\), putting \(\mathbf{j}-\frac{1}{2}\boldsymbol{\sigma}=\mathbf{l}\), whence\(^*\)

\[ (\boldsymbol{\sigma}\mathbf{j})=j(j+1)-l(l+1)+\frac{3}{4}. \tag{8} \]

From this, for the two cases \(j=l\pm\frac{1}{2}\), we obtain:

\[ (\boldsymbol{\sigma}\mathbf{j})= \begin{cases} -j, & j=l-\frac{1}{2},\\ j+1, & j=l+\frac{1}{2}. \end{cases} \tag{9} \]

Thus,

\[ \langle\sigma\rangle= \begin{cases} -\dfrac{j}{j+1}, & j=l-\dfrac{1}{2},\\[6pt] \dfrac{j}{j}, & j=l+\dfrac{1}{2}, \end{cases} \tag{10} \]

and correspondingly we obtain\(^ {**}\):

\[ \langle\sigma\rangle^{2}= \begin{cases} \dfrac{j}{j+1}, & j=l-\dfrac{1}{2},\\[6pt] \dfrac{j+1}{j}, & j=l+\dfrac{1}{2} \end{cases} \quad \text{(one particle).} \tag{11} \]

\[ \text{\(^*\) Recall that } \mathbf{j}^{2}=j(j+1),\quad \mathbf{l}^{2}=l(l+1), \]

\[ \left(\frac{1}{2}\boldsymbol{\sigma}\right)^{2} = \frac{1}{2}\left(\frac{1}{2}+1\right) = \frac{3}{4}. \]

\(^ {**}\) It is interesting to note that, knowing the diagonal matrix elements \((j\to j)\), one can easily find (not needed by us now) also the diagonal elements \((j\to j-1)\). Since there is only one such element and since, on the other hand, by the rules of matrix multiplication
\(\langle\sigma\rangle_{jj}\langle\sigma\rangle_{jj}+\langle\sigma\rangle_{j,j-1}\langle\sigma\rangle_{j-1,j}=\langle\sigma^{2}\rangle=3\),
we have

\[ |\langle\sigma\rangle|^{2}_{j,j-1} = 3-\langle\sigma\rangle^{2}_{jj} = \begin{cases} \dfrac{2j+3}{j+1}, & j=l-\dfrac{1}{2},\\[6pt] \dfrac{2j-1}{j}, & j=l+\dfrac{1}{2}. \end{cases} \]

Let us now pass to the case where more than one nucleon participates in the transition, i.e., when in the nucleus, outside closed shells, there is not one nucleon but several.

In this case, as the theory shows, the matrix element for one particle is multiplied by a certain factor. The results of theoretical calculations can be reduced to a number of rules.

1) Matrix element \(\langle 1\rangle\). The calculations are carried out according to the formula

\[ \langle 1\rangle^2=T(T+1)-T_\zeta T'_\zeta, \tag{12} \]

where \(T\) is the isotopic spin (unchanged in the transition), and \(T_\zeta\) and \(T'_\zeta\) are its projections for the initial and final nuclei. In particular, for transitions

\[ \left. \begin{array}{ll} \left(T=\dfrac{1}{2}\right) & T=\dfrac{1}{2}\to T=\dfrac{1}{2}\quad \langle 1\rangle^2=1,\\[6pt] (T=1) & T_\zeta=\pm 1\to T'_\zeta=0\quad \langle 1\rangle^2=2. \end{array} \right\} \tag{13} \]

2) Matrix element \(\langle \sigma\rangle^2\) (Talmi\(^{11}\)). The matrix element for one particle is multiplied by the factor:

\[ \begin{array}{ccl} \multicolumn{3}{c}{\text{Two particles}}\\[4pt] \dfrac{2}{3} & \text{transition} & I=1\to I=0,\\[6pt] 2 & \text{transition} & I=0\to I=1. \end{array} \left\} \tag{14} \right. \]

The spin of the nucleus changes by 1, which corresponds to transitions between ground states of even-even and odd-odd nuclei. The difference in the two coefficients is connected with summation over the spin projections in the final state, which gives a factor of 3 for final spin 1 and a factor of 1 for final spin 0.

\[ \begin{array}{c} \text{Three particles}\\[6pt] \left(\dfrac{2j-1}{3j+3}\right)^2\quad \text{transition}\quad T=\dfrac{3}{2}\to T=\dfrac{1}{2},\\[10pt] \left(\dfrac{1}{3}\dfrac{j+4}{j+1}\right)^2 = \left(\dfrac{1}{3}+\dfrac{1}{j+1}\right)^2 \quad \text{transition}\quad T=\dfrac{1}{2}\to T=\dfrac{1}{2}. \end{array} \left\} \tag{15} \right. \]

The spin of the nucleus does not change, which corresponds to transitions between ground states of odd-even nuclei.

§ 3. DECAY OF LIGHT NUCLEI

In analyzing the experimental data we shall proceed from the value of the coefficient \(A\), or, what is the same thing, the Fermi constant obtained from the decay \(O^{14}\to C^{14*}\) and equal to \(6550\pm150\) sec. (see below).

We shall restrict the analysis to light nuclei for which the configuration of nucleons in the shell scheme is known. Naturally, in doing so, we must confine ourselves only to the so-called super-

permitted transitions, i.e., transitions (for which \(\lg ft \simeq 3\)), in which the radial functions of the nucleons do not change (i.e., in which all the nucleons outside closed shells have one and the same number \(j\)). In this case we may expect to obtain a comparatively accurate estimate of the matrix elements.

We shall consider only nuclei up to neon, for which the analysis of spins and magnetic moments has made it possible to establish the shell scheme*). We shall not consider heavier nuclei, since the calculation of matrix elements for them is already connected with greater uncertainty. However, some of them will be mentioned at the end of the article.

It is interesting to note that the analysis of decay data proves to be very sensitive to the accuracy of the shell scheme, and its results may serve as an indication of the level scheme. Thus, for example, the transition \(C^{11}\to B^{11}\) has \(ft\sim 4000\), which corresponds to the complex configuration \(B^{11}\), well manifested also in the magnitude of the magnetic moment \(^{24}\).

The case \(F^{18}\to O^{18}\), with approximately the same value of \(ft\), is also of interest. This case reflects the crossing of the \(d_{5/2}\) and \(s_{1/2}\) levels in the region of nuclei between \(O^{16}\) and \(Ne^{20}\). The case \(C^{14}\) is well known, the forbidden character of whose decay was for a long time considered puzzling. Recently this case has apparently found its explanation in an accidental compensation of matrix elements. This will be discussed in more detail below.

Thus, let us proceed to the analysis of individual cases of decay.

\[ \mathrm{n}\to \mathrm{p} \]

For it the value \(ft\) is equal to \(1260 \pm 200^{12}\). Neutron decay is the only case when the nuclear matrix elements are known exactly. Namely:

\[ M(n)=\langle 1\rangle^{2}+R\langle \sigma\rangle^{2}=1+3R. \]

Thus, we have:

\[ \frac{6550\pm 150}{1+3R}=1260\;(\pm 20\%), \]

whence we obtain the first estimate for \(R\)

\[ 1\lesssim R\lesssim 1.8. \]

In principle, a careful study of neutron decay (in particular the correlation between the electron and the proton**) would make it possible

*) A good summary of the data on \(\beta\)-decay was made by King \(^{23}\).

**) The most sensitive method is the simultaneous measurement, together with the correlation, of the energies of the particles as well—for example, the study of the spectrum of protons and electrons flying in opposite directions.

determine both decay constants (\(A\) and \(R\)). As we have already noted, such experiments do not allow one to choose the sign of \(S \pm T\). It can be determined only from a simultaneous measurement of the polarization of both charged particles.

\[ \mathrm{H}^3 \to \mathrm{He}^3 \quad (ft = 1014 \pm 20)^{13} \]

If one assumes that the configuration of the nucleons is described as a “hole” in the \(s_{1/2}\) shell, then \(\langle 1\rangle^2 = 1\) (mirror nuclei) and \(\langle \sigma\rangle^2 = -\frac{1}{2}\cdot\frac{1}{2} = 3\), and we obtain:

\[ R = 1.7 \]

apparently with great accuracy.

However, it should be borne in mind that, whereas the calculation of the matrix element \(\langle 1\rangle^2\) for mirror nuclei is sufficiently accurate and the accuracy of the result is determined by the accuracy of charge invariance (the error is no more than a few percent), the calculation of \(\langle \sigma\rangle^2\) is based on considerably less rigorous ideas about shells. Therefore, despite the high accuracy of \(ft\), the accuracy in determining \(R\) is not very great. It is noteworthy that \(ft\) for the decay under consideration is smaller than the corresponding value for the neutron. Since deviations from the shell scheme can apparently (cf. Blatt \(^{34}\)) make the value of \(\langle \sigma\rangle^2\) less than 3, the published lifetime of the neutron is clearly overestimated. The actual half-life of the neutron should not exceed 10 min. (which, however, does not go beyond the experimental errors). In this connection an accurate measurement of the neutron lifetime becomes of primary interest. If it actually proves to be smaller than the value now accepted, then it will be possible to assert that the decays of light nuclei are in agreement with one another.

Considering the remaining nuclei, we shall see to what extent the value \(R = 1.7\) can be reconciled with the known values of \(ft\) for the decays. The values of \(ft\) calculated under these assumptions will be denoted conventionally by \((ft)_{\mathrm{theor}}\).

\[ \mathrm{He}^6 \to \mathrm{Li}^6 \quad (ft = 815 \pm 70)^{14} \]

The transition takes place with a change of spin \((0 \to 1)\). The structure of \(\mathrm{Li}^6\) has not been uniquely established. If it is assumed that the two nucleons are in the state \(2s_{1/2}\), then for the matrix element \(\langle \sigma\rangle^2\) we obtain the value*) \(2\cdot 3 = 6\); if the nucleons are in the state \(2p_{1/2}\), then this value will be \(2\cdot \frac{1}{3}=\frac{2}{3}\). Finally, if, as some authors assume, the \(LS\)-coupling scheme is used, then we obtain the value 6.

*) The coefficient 2 is connected with taking account of two nucleons.

Taking, on the other hand, \(A=6550\) and \(R=1.7\), we obtain:

\[ \langle \sigma\rangle^{2}=\frac{A}{R\cdot ft}=\frac{6550}{1.7\cdot 815}\simeq 4.7 . \]

We see that the agreement in this (not very clean) case is not important. Nevertheless, it may still be asserted that the configuration \(2s_{1/2}^{2}\) is not too contradictory to experiment. The assumption that agrees better with experiment is that the nucleus is in a state representing a mixture of the states \(p_{1/2}^{2}\) and \(p_{1/2}p_{3/2}\) (intermediate coupling). However, in view of the ambiguity of such a conclusion we shall not dwell on it in more detail (see \(^{2}\)).

In any case, already from this example it is clear that the value of \(\langle \sigma\rangle^{2}\) in reality lies somewhat below the value calculated under the assumption of the strict validity of the shell scheme.

\[ \mathrm{Be}^{7}\to \mathrm{Li}^{7}\qquad (ft=2140)^{15} \]

For transitions with capture of \(K\)-electrons, the value of \(ft\) is determined by the formula (see \(^{13}\))

\[ f=\frac{\pi}{2}G_{K}^{2}(W_{0}+E_{K})^{2}, \]

where \(W_{0}\) is the decay energy, \(E_{K}\) is the binding energy of the \(K\)-electron, and \(G_{K}\) is the large component of the relativistic wave function of the \(K\)-electron.

The transition occurs between mirror nuclei with \(T=1/2\). Therefore \(\langle 1\rangle^{2}\) for this transition is equal to unity; \(\langle \sigma\rangle^{2}\) for the configuration \(p_{3/2}\) is equal to

\[ \frac{5}{3}\cdot \frac{121}{225}\simeq 1, \]

whence for \(ft\) we obtain:

\[ (ft)_{\mathrm{theor}}=\frac{6550}{1+1.7\cdot 1}\simeq 2400 . \]

This value is in satisfactory agreement with experiment, especially if one takes into account the inaccuracy of the theoretical calculation of \(ft\) associated with screening of the nucleus by the inner electrons.

\[ \mathrm{C}^{10}\to \mathrm{Be}^{10*}\qquad (ft\simeq 1600)^{16} \]

A transition with a spin change \((0\to 1)\) between nuclei with configuration \(2p_{3/2}^{-2}\) (two nucleons lacking to the closed shell \(\mathrm{C}^{12}\)). For such a transition \(1.7\langle\sigma\rangle^{2}=1.7\cdot 2\cdot {}^{5}/_{3}\simeq 5.7\), whence we obtain \((ft)_{\mathrm{theo}}=1150\). In order to obtain the experimental value of \(ft\), \(R\langle\sigma\rangle^{2}\) would have to be equal to \(\sim 4\). Thus we again see that the inaccuracy of the shell model leads to a reduction of \(\langle\sigma\rangle^{2}\).

The transition \(\mathrm{C}^{10}\to \mathrm{Be}^{10*}\) is accompanied by a less probable transition to the second excited state \(\mathrm{B}^{10**}\) \((E=2.1\ \mathrm{MeV})\) attributed

established as a transition without change of spin (a \(0 \leftrightarrow 0\) transition). For this transition \(\langle 1\rangle^2=2\) and \(\langle \sigma\rangle^2=0\). The experimental ratio \(ft\) for the two transitions is equal to \(\sim 2.5\), which gives, in agreement with the estimate just made, \(R\langle \sigma\rangle^2 \simeq 5\). This estimate obviously does not depend on the adopted value of the constant \(A\). This constant can be calculated independently on the basis of \(ft\) for the transition \(\mathrm{C}^{10}\to \mathrm{Be}^{10**}\) \((ft=5900\pm2400)\). It is obtained in agreement with its value derived from considerably more accurate data on the decay \(\mathrm{O}^{14}\to \mathrm{N}^{14*}\).

\[ \mathrm{N}^{13}\to \mathrm{C}^{13}\qquad (ft=4600)^{17,23} \]

The transition is between mirror nuclei; \(\langle 1\rangle^2=1\) and \(\langle \sigma\rangle^2=\frac{1}{3}\). Hence \((ft)_{\text{theor}}=4000\), which is in satisfactory agreement with experiment.

\[ \mathrm{O}^{14}\to \mathrm{N}^{14*}\qquad (ft=3275\pm75)^3 \]

The classical case of a \(0\leftrightarrow 0\) transition (without change of spin and parity). For this transition \(\langle 1\rangle^2=2\), whence we obtain the adopted value

\[ A=6550\pm150\ \text{sec}. \]

In addition to decay to the excited level \(\mathrm{N}^{14*}\), decay to the ground state should also occur. Observation of this decay is of fundamental interest in connection with the “\(\mathrm{C}^{14}\) problem,” since the decay \(\mathrm{O}^{14}\to \mathrm{N}^{14}\) (ground state) is a transition mirror (charge-symmetric) to the decay \(\mathrm{C}^{14}\to \mathrm{N}^{14}\) (see the scheme).

Known is that the decay \(\mathrm{C}^{14}\to \mathrm{N}^{14}\) is accompanied by a change of spin by only 1 (\(0+\to1+\) transition) and nevertheless is strongly forbidden. The value of \(ft\) for this transition is \(\sim 3\cdot10^9\) sec. (we recall that superallowed transitions correspond to \(ft\) of order \(\sim 10^3\)). Such a small probability of the decay of \(\mathrm{C}^{14}\) is all the more unexpected because the nucleon configuration both for \(\mathrm{C}^{14}\) and for \(\mathrm{N}^{14}\) has been established (two nucleons are lacking in the \(p_{1/2}\) shell relative to the closed shell \(\mathrm{O}^{14}\)). The most natural explanation was recently given by Jancovici and Talmi \(^{26}\). The explanation itself is very simple and consists in the fact that, in decays whose probability is determined by the matrix element \(\langle\sigma\rangle\), the probability may turn out to be anomalously small because of an accidental compensation of the various terms in the expression for

matrix elements that arise because of an admixture of other states (in the calculation of the cited authors, the states \(p_{1/2}\)) to the ground state of \(C^{14}\) and \(O^{14}\)*). The possibility of such an effect is excluded for the matrix element \(\langle 1\rangle^2\), whose calculation is based only on the assumption of isotopic invariance of nuclei and in no way depends on further details of their structure. The interest of such an explanation is connected with the unexpected possibility of testing it. If we assume that the small matrix element for the transition \(C^{14}\to N^{14}\) is the difference of two large terms (approximately \(\sim 10^3\) times larger than the difference), then for the mirror transition \(O^{14}\to N^{14}\) (ground state) these terms should change only slightly because of the Coulomb energy. This change should be small, as we know from the matrix elements of other mirror transitions. However, if this change is small in relation to the large quantities, it should produce a large effect on the difference. With the estimates given above, a change of \(1\%\) associated with the Coulomb energy will give a tenfold increase of the matrix element for the transition \(O^{14}\to N^{14}\), and of the \(ft\) value by a factor of 100, compared with the matrix element of the transition \(C^{14}\to N^{14}\). Therefore, despite charge invariance, for the transition \(O^{14}\to N^{14}\) (ground state) we should expect an \(ft\) value approximately equal to \(10^7\). There are contradictory indications concerning the existence of this transition. Penning and Schmidt \(^{27}\) give the value \(\lg ft=6.7\) (fraction of transitions to the ground state of \(N^{14}\), \(3\%\)). At the same time, Gerhart \(^{3}\) believes that in any case \(\lg ft>7.3\) (fraction of transitions \(<0.3\%\)). Clarification of this question is very important for understanding the structure of light nuclei. If it indeed turns out that, contrary to charge symmetry, the \(ft\) values for the two mirror transitions are very different, then it may be considered proven that the \(C^{14}\) anomaly is accidental in character and is not connected with fundamental defects of the shell theory in light nuclei.

\[ O^{15}\to N^{15}\qquad (ft=3910)^{18} \]

Transition \(p_{1/2}\to p_{1/2}\) (one “hole” in the closed shell \(O^{16}\)). The values \(\langle \sigma\rangle^2=\dfrac{1}{3}\), and \(\langle 1\rangle^2=1\), whence the theoretical value of \(ft\) is 4150, which is in good agreement with the experimental value.

\[ F^{17}\to O^{17}\qquad (ft=2420)^{19} \]

According to data on the magnetic moment, the nucleus \(F^{17}\) has the structure \(O^{16}+f_{5/2}\). The same structure should also be adopted for \(O^{17}\) (mirror—

*) The objection that such a compensation is unlikely is removed by the fact that it actually occurs only for one nucleus among the many possible cases of decay.

calcium nucleus). The matrix elements \(\langle\sigma\rangle^2\) and \(\langle 1\rangle^2\) are respectively \(\frac{7}{5}\) and 1. Then we obtain \(ft=2600\)—a value close to the experimental one.

\[ \mathrm{F}^{18}\to \mathrm{O}^{18}\qquad (ft=4000)^{20} \]

The configuration of the nucleons in these nuclei is unknown. The difficulty in determining this configuration is connected with the fact that after \(\mathrm{O}^{16}\) the states \(f_{7/2}\) and the states \(s_{1/2}\) are filled simultaneously. The nucleus \(\mathrm{F}^{17}\) has the structure \(\mathrm{O}^{16}+f_{5/2}\), whereas the nucleus \(\mathrm{Ne}^{20}\) apparently has a closed shell \(s_{1/2}^4\). The transition \(\mathrm{F}^{18}\to\mathrm{O}^{18}\) occurs with a change of spin \((I=1\to I=0)\). Therefore the matrix element \(\langle\sigma\rangle^2\) is equal to \(2/3\) of the matrix element for one nucleon (see § 2). If we assume that in \(\mathrm{F}^{18}\) two nucleons are in the state \(s_{1/2}\), then

\[ \langle\sigma\rangle^2=\frac{2}{3}\cdot 3=2 \]

and \(ft=3300\); if, however, we assume that the nucleons are in the state \(f_{5/2}^2\), then

\[ \langle\sigma\rangle^2=\frac{2}{3}\cdot\frac{7}{5}\simeq 1 \]

and \(ft=6550\). The experimental value of \(ft\) lies between these two values, which confirms the mixed character of the nucleon configuration in these nuclei.

\[ \mathrm{Ne}^{18}\to \mathrm{F}^{18}\qquad (ft>1000)^{21} \]

The structure of these nuclei is also unknown. In the nuclei \(\mathrm{F}^{18}\) and \(\mathrm{O}^{18}\) we may again expect an overlap of the states \(f_{5/2}\) and \(s_{1/2}\). The transition is accompanied by a change of spin \((J=0\to J=1)\). The assumption of the configuration \(s_{1/2}^2\) leads to the value

\[ \langle\sigma\rangle^2=2\cdot 3=6 \]

and \(ft=1100\); the assumption of the configuration \(f_{5/2}^2\) gives

\[ \langle\sigma\rangle^2\,2\cdot\frac{7}{5}\simeq 3 \]

and \(ft=2200\). The actual value apparently lies between these values.

\[ \mathrm{Ne}^{19}\to \mathrm{F}^{19}\qquad (ft=1970\pm 100)^{22} \]

This decay case is interesting because, as we have already noted, the correlation between the neutrino and the electron has been investigated for it\(^7\).

The structure of these nuclei is mirror-like, so that \(\langle 1\rangle^2=1\). The quantity \(\langle\sigma\rangle^2\), under the assumption that the nucleon configuration corresponds to one hole in \(\mathrm{Ne}^{20}\) \((s_{1/2}^3)\), is equal to 3. This gives for \(ft\) the same value as for the neutron and tritium (\(\sim 1000\)). It follows from this that in this case as well the configuration \(s_{1/2}^3\) does not correspond exactly to the state of these nuclei. On the other hand, the spin of \(\mathrm{F}^{19}\) is equal to \(1/2\), which is difficult to reconcile, within the framework of the existing regularities, with any other structure (for example, with the configuration \(f_{5/2}^3\), with total spin \(1/2\)*).

\[ \text{*) Arguments in favor of such a configuration are given by Pizli}^{28}. \]

If, starting from the experimental value of $ft$, one finds $R\langle\sigma\rangle^2$, then one obtains the value 2.3. The deviation from the limiting value $1.7\cdot 3$ (the pure state $s_{1/2}^3$) should be attributed to an admixture of other states.

The correlation of the neutrino with the electron gives an independent method for determining the constant $R\langle\sigma\rangle^2$.

It is well known that the neutrino correlation function has the form (see, for example, the paper by Zel’dovich$^{1}$)

\[ f(\vartheta)=1+\lambda\frac{v}{c}\cos\vartheta, \]

where

\[ \lambda=1\quad (S);\quad -1\quad (V);\quad +\frac{1}{3}\quad (T)\ \text{and}\ -\frac{1}{3}\quad (A) \]

for the four variants, respectively. The experimental value is

\[ \lambda=-0.21\pm0.08. \]

Since variant $A$ is excluded, the only combination that can give $\lambda<0$ and is in agreement with the remaining data on $\beta$-decay is $(S,T)$. It is not difficult to show that for such a mixture

\[ \lambda=\frac{\frac{1}{3}R\langle\sigma\rangle^2-1}{1+R\langle\sigma\rangle^2} \]

or

\[ R\langle\sigma\rangle^2=\frac{\lambda+1}{\frac{1}{3}-\lambda}, \]

whence

\[ R\langle\sigma\rangle^2=1.4\pm0.4, \]

which differs from the theoretical value calculated from the shell model for $R=1.7$. Moreover, this quantity is not even consistent with the value $R\langle\sigma\rangle^2=2.3$, obtained from the experimental value $ft=1970\pm100$.

If the latter value is adopted, then one should expect for $\lambda$ the value $-0.07$ (which, however, does not fall very far outside the stated errors).

In any case, the results undoubtedly prove the variant $(S,T)$ and the limited accuracy of the assumption of the configuration $s_{1/2}^3$ for these nuclei.

§ 4. OTHER NUCLEI. CONCLUSION

Heavier nuclei, generally speaking, are unsuitable for an analysis whose purpose is to determine $R$, since the calculation of the matrix element $\langle\sigma\rangle^2$ for them is quite unreliable.

It makes sense, however, to point to the case of the decay \(Sc^{41}\to Ca^{41}\) (\(ft=2180^{29}\)), for whose nucleon configuration one may adopt the scheme \(Ca^{40}+f_{7/2}\). In this case \(\langle 1\rangle^2=1\) and \(\langle\sigma\rangle^2=9/7\), which gives \(ft=2000\), in unexpectedly good agreement with experiment.

Also of interest is the situation near the filled \(d_{5/2}\) shell. Here three superallowed decays of nuclei are known, whose configurations correspond respectively to one, two, and three holes in this shell. These are the decays (cf. \(^{23}\)):

\[ d_{5/2}^{-1}\qquad Si^{27}\to Al^{27}\qquad (ft=2260,\quad 5/2^+\to 5/2^+), \]

\[ d_{5/2}^{-2}\qquad Al^{26*}\to Mg^{26}\qquad (ft=2160,\quad 0+\to 0+), \]

\[ d_{5/2}^{-3}\qquad Al^{25}\to Mg^{25}\qquad (ft=2220,\quad 5/2^+\to 5/2^+) \]

Without giving the details of the calculation (using the formulas of § 2), let us say only that the theoretical values for these three cases will be respectively: \(\sim 1900\), 3300, and 3200. These values show an increase of the discrepancy as one moves away from the closed shell.

In conclusion we may draw the following conclusions:

1. The quantity \(R\), characterizing the admixture of the tensor and scalar variants in the Hamiltonian of \(\beta\)-decay, is equal (with a possible error \(\sim 10\%\)) to

\[ R=1.7\quad \text{(or somewhat (\(\sim 10\%\)) larger).} \]

In the literature the following values of this constant have been proposed, some of which are in clear disagreement with the data on light nuclei:

Values of \(R\) according to different authors

Accordingly, the decay constants \(g\) are equal (see § 1):

\[ g_F=2.9\cdot 10^{-12}(mc^2)\left(\frac{\hbar}{mc}\right)^3 =1.4\cdot 10^{-49}\ \text{erg}\cdot\text{cm}^3, \]

\[ g_{GT}=3.8\cdot 10^{-12}(mc^2)\left(\frac{\hbar}{mc}\right)^3 =1.8\cdot 10^{-49}\ \text{erg}\cdot\text{cm}^3. \]

1. The half-life of the neutron, for the adopted value of \(R\), should be \((10.2 \pm 0.5)\) min.

2. The \(ft\) data for light nuclei provide excellent material for testing the shell structure of light nuclei and confirm the accepted concepts.

In conclusion I express my gratitude to L. D. Landau, Ya. B. Zel’dovich, and D. A. Frank-Kamenetsky for discussing this work.

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