Beta Decay ofC14and Nuclear Forces

Abstract

The anomalously long lifetime of ${\mathrm{C}}^{14}$ is interpreted as being due to an accidental cancellation in the $\ensuremath{\beta}$-decay matrix element which is complete to about one part in 160. In the shell model with oscillator wave functions, general two-nucleon interactions are introduced which are linear in the velocities and which satisfy various invariance requirements. As many of the force parameters are determined from low-energy experimental nuclear data as possible. The strength of the spin-orbit force is fixed by the ${p}_{\frac{3}{2}}\ensuremath{-}{p}_{\frac{1}{2}}$ splitting in ${\mathrm{N}}^{15}$, while the tensor force is taken from meson theory. The conditions for cancellation are then shown to be satisfied, although uncertainties in the central force prevent an exact calculation of the wave functions. These uncertainties are therefore eliminated by requiring cancellation, in addition to fitting the three lowest energy levels of ${\mathrm{N}}^{14}$. The resulting ${\mathrm{N}}^{14}$ ground state wave function is mainly $D$-state, containing only 13% $P$- and 3% $S$-component, and checks satisfactorily against various experimental properties of ${\mathrm{N}}^{14}$. In addition, the $\ensuremath{\beta}$ decay of ${\mathrm{O}}^{14}$ is investigated and it is shown that the ${\mathrm{C}}^{14}$-${\mathrm{O}}^{14}$ difference in $\mathrm{ft}$-values is not adequately accounted for by the Coulomb repulsion theory of Jancovici and Talmi. The discrepancy is removed by including the slight difference in spin-orbit coupling strength for protons and neutrons, due to electromagnetic interaction with their magnetic moments. It is also pointed out that a nonlinearity of about 10% should be expected in the ${\mathrm{O}}^{14}$ Fermi-Kurie plot. In the appendices are a treatment of nucleon hole conjugation, an extension of a proof by Inglis on the impossibility of cancellation without the tensor force, a calculation of the $n\ensuremath{-}\ensuremath{\alpha}$ spin-orbit splitting, and an approximate treatment of the binding energy of the $\ensuremath{\alpha}$ particle.