Intermediate Coupling as Encountered in Some of thep-Shell Nuclei

The manner in which a transition into intermediate coupling, between the extremes of ($\mathrm{jj}$) coupling and ($\mathrm{LS}$) coupling, accounts for much of the failure of the ($\mathrm{jj}$) coupling shell model in the light nuclei of the $p$-shell is illustrated by some typical examples. The set of secular equations for the configurations $p^2$ and $p^{-2\mathrm{}}$ applying to ${\mathrm{Li}}^6$+${\mathrm{He}}^6$ and to ${\mathrm{N}}^{14}$+${\mathrm{C}}^{14}$ is sufficiently simple to be worked out in detail. In the configuration $p^8$ which applies to ${\mathrm{C}}^{12}$+${\mathrm{B}}^{12}$ the states are much more numerous and the equations are too complex to be solved in detail, but the solutions for the low states of interest may be treated by approximate methods, relying on some knowledge of the asymptotic behavior in the extremes. The general features of observed energy spectra for $p^{-2\mathrm{}}$ and for $p^8$ are both compatible with the intermediate coupling scheme with the same ratio $\frac{a}{K}\approx 5$ of the spin-orbit coupling parameter $a$ to the "exchange integral" $K$, which separates multiplets. The criterion for whether the spectrum of the low states slightly resembles ($\mathrm{LS}$) coupling, as it does in ${\mathrm{C}}^{12}$+${\mathrm{B}}^{12}$, rather than ($\mathrm{jj}$) coupling, as in ${\mathrm{N}}^{14}$+${\mathrm{C}}^{14}$, is not just the value of $a$ relative to $K$ but the magnitude of $a$ relative to the multiplet separations provided by $K$ in ($\mathrm{LS}$) coupling, which are exceptionally large among the low states of ${\mathrm{C}}^{12}$.