Shell structure of theA=6ground states from three-body dynamics

Three-body ($\alpha \mathrm{NN}$) models of the ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ ground states are used to investigate their shell structure. Three models for each nucleus are considered: simple, full ($\mathrm{nn}$), and full ($\mathrm{np}$) for ${}_{}{}^6{}_{}{}^{}\mathrm{He}$, and simple, full (0%), and full (4%) for ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$. The full models in both cases are obtained by including the $S_{\frac{1}{2}}$, $P_{\frac{1}{2}}$, and $P_{\frac{3}{2}}$ partial waves of the $\alpha N$ interaction, whereas the simple model truncates to only the strongly resonant $P_{\frac{3}{2}}$ wave. The ${}_{}{}^6{}_{}{}^{}\mathrm{He}$ full models distinguish between use of the $\mathrm{nn}$ or $\mathrm{np}$ parameters for the ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ $\mathrm{NN}$ interaction, while the ${}_{}{}^6{}_{}{}^{}\mathrm{Li}$ full models have either a pure ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ $\mathrm{NN}$ interaction (0%) or a ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}-{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$ interaction that leads to a 4% $d$-wave component in the deuteron (4%). These models are used to calculate the probabilities of the orbital components of the wave functions, the configuration-space single-particle orbital densities, and the configuration-space two-particle wave function amplitudes in $j-j$ coupling with the nucleon coordinates referred to the alpha particle as the "core" or "center of force." The results are then compared with those from phenomenological and realistic-interaction shell models. Major findings of the comparison are the following: None of the shell models considered have a distribution of orbital probabilities across shells like that predicted by three-body models; the orbital rms radii from three-body models indicate an ordering of the orbits within shells, i.e., $p_{\frac{1}{2}}$ outside