Angular Momentum Coupling and the Internucleon Interaction in the Calcium Isotopes

Abstract

A partial analysis is made of the results of the Massachusetts Institute of Technology Van de Graaff group on ($d, p$) reactions leading to the levels of ${\mathrm{Ca}}^{41}$, ${\mathrm{Ca}}^{43}$, and ${\mathrm{Ca}}^{45}$. The results demonstrate that the coupling scheme is rather close to the $\mathrm{jj}$ limit; more precisely they give small values for the amplitudes of certain minor components of the wave functions for low-lying odd-parity states (e.g., the ${{({f}_{\frac{7}{2}})}^{2}}_{0}{p}_{\frac{3}{2}}$ component of the lowest $J=\frac{3}{{2}^{\ensuremath{-}}}$ level in ${\mathrm{Ca}}^{43}$). The amplitudes determined in this way are used to deduce some features of the effective internucleon interaction. It turns out that the ($d, p$) amplitudes for the low-lying "multiparticle" states are essentially proportional to a matrix element which vanishes identically for any spin-dependent nucleon-nucleon interaction (of either ${\ensuremath{\sigma}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\sigma}}_{2}$, tensor or vector type) and thus we obtain directly an approximate range vs depth relationship for the spin-independent interaction. Finally, an approximate spin-dependent central interaction is determined by considering the level structures. A fairly satisfactory interaction between identical nucleons is found to be ${H}_{12}\ensuremath{\simeq}\ensuremath{-}3[3\ensuremath{-}{\ensuremath{\sigma}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\sigma}}_{2}]\mathrm{exp}(\ensuremath{-}\frac{{r}^{2}}{{{r}_{0}}^{2}})$ Mev with ${r}_{0}\ensuremath{\sim}2.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm. The general features of the results are similar to those of Levinson and Ford, who consider level structures and magnetic moments, but the exchange nature of the interaction is different.