Beta Spectra of the Mirror Nuclei

The positron spectra and half lives of all the mirror nuclei ($2Z=A\pm 1$) with $19<~A<~39$ have been systematically measured with a 180°-deflection uniform-magnetic-field spectrometer. The ground-state transition energies were used to compute Coulomb-energy differences between mirror pairs. Deviations of these Coulomb-energy differences from a smooth variation with $A$ are explained by a nuclear shell model using the potential well of an isotropic harmonic oscillator. The data support a symmetry for the proton wave functions characteristic of the state of lowest seniority, with magic-number effects at $A=14\mathrm{} \mathrm{and} 16$ as well as $A=8\mathrm{} \mathrm{and} 20$. Comparison of the ft values obtained with experimental nuclear magnetic moments gives the following values for the partial coupling constants for the Fermi and Gamow-Teller $\beta$ interactions: ${g_{\mathrm{F}}}^2=(1.5\mathrm{}\pm 0.1)\times {10}^{-4\mathrm{}}$ ${\sec }^{-1\mathrm{}}$, ${g_{\mathrm{GT}}}^2=(2.2\mathrm{}\pm 0.1)\times {10}^{-4\mathrm{}}$ ${\sec }^{-1\mathrm{}}$. Nuclear radii from $\mu$-mesic atoms, when properly interpreted, are shown to be in agreement with radii deduced from Coulomb energy differences.