Zero-Zero Transition in Carbon-12

The electric-monopole transition density between the ground state and the 7.68-Mev level of carbon-12 is examined on the basis of the nuclear shell model. It is found to vanish for all stages of intermediate coupling if only the ${\mathrm{}\left(1p\right)\mathrm{}}^8$ configuration is involved. A nonzero value is obtained by including states of the ${\mathrm{}\left(1p\right)\mathrm{}}^7\mathrm{}\left(2p\right)\mathrm{}$ configuration, brought in by a residual central internucleon interaction. For computational simplicity, both this and the spin-orbit interaction are first treated as perturbations about the $\mathrm{LS}$ limit. Since the results show that the residual interaction is probably too large for this to be reliable, an attempt is made to diagonalize the Hamiltonian exactly, but the severe restriction on the number of states considered makes the result rather unsatisfactory. It is concluded that if all of the possible states of the low-lying configurations were to be included in the diagonalization, agreement with experiment might result, but that in this case some semicollective model might better be applied to the problem.