Nuclear Rotational Spectra, the Elliott Model, and theP2Force

The essential features of the Elliott model with the momentum-dependent quadrupole-quadrupole operator which leads to $L(L+1\mathrm{})\mathrm{}$ rotational spectra are reviewed. The operator is reduced to a momentum-independent residual interaction which differs somewhat from the $P_2$ interaction. The model Hamiltonian is separated into a rotational Hamiltonian, a deformed "intrinsic" Hamiltonian, and a perturbation term. The eigenfunctions and eigenvalues of the "intrinsic" Hamiltonian are found and used in Inglis' cranking model formula to calculate the moment of inertia. The model is modified, in a simple configuration, by taking a mixture of the long-range "$P_2$" interaction with the short-range $\delta$-function force. For an intermediate mixture the spectrum obtained resembles the spectrum predicted by the collective vibrational Model. Finally, the implications of a $P_2$ residual interaction for direct-interaction inelastic scattering processes are considered. The question is discussed whether one can actually see the $P_2$ residual interaction in rotational nuclei, and, if so, whether the strength of the $P_2$ interaction determined from such scattering experiments is consistent with the strength determined from the observed rotational spectra. Within the rough approximations made, the few experimental results available are not inconsistent with the calculation.