Electric Monopole Transitions inC12andO16

Abstract

The matrix elements for the electric monopole (${0}^{+}$\ensuremath{\rightarrow}${0}^{+}$) transitions between the ground and 7.68-Mev state in ${\mathrm{C}}^{12}$ and between the ground and 6.06-Mev state in ${\mathrm{O}}^{16}$ may be estimated from inelastic electron scattering and from the pair emission lifetime, respectively. The two are equal to each other within the rather large error of the electron scattering determination, and are given by ${(\ensuremath{\Sigma}{P}^{}{{r}_{P}}^{2})}_{f0}\ensuremath{\cong}3.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}$ ${\mathrm{cm}}^{2}$, where 0 and $f$ represent initial and final states of the nucleus, and ${r}_{P}$ is the radial distance of a proton from the center of the nucleus. Calculations based on the alpha-particle model and on an elastic-fluid model yield three to five times this experimental value. Therefore, a calculation was made in the case of ${\mathrm{C}}^{12}$, based on the $\mathrm{jj}$-coupling independent-particle model, according to which two nucleons undergo transitions between the ${p}_{\frac{3}{2}}$ shell and the ${p}_{\frac{1}{2}}$ shell. The matrix element vanishes if there are no internucleon forces. Pair forces are included to first order, and the sum over configurations is performed exactly by means of a Green's function. For simplicity it is assumed that the independent-particle potential is an infinitely deep square well, and that the pair interaction has zero range. Even assuming that the pair interaction has its free-space triplet magnitude, the calculated matrix element is only about one-sixth the experimental value. It is concluded, therefore, that a model that is more collective than the independent-particle model with pair interactions, and less collective than the alpha-particle or elastic-fluid models, is required to account for the experimental results.