Angular Momentum Coupling and the Internucleon Interaction in the Calcium Isotopes

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 ${{\left(f_{\frac{7}{2}}\right)}^2}_0{p}_{\frac{3}{2}}$ component of the lowest $J=\frac{3}{2^{-}}$ 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 ${\sigma }_1·{\sigma }_2$, 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 $H_{12}\simeq -3[3-{\sigma }_1·{\sigma }_2]\exp (-\frac{r^2}{{r_0}^2})$ Mev with $r_0\sim 2.7\times {10}^{-13\mathrm{}}$ 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.