Extension of the Shell Model for Heavy Spherical Nuclei

The Bardeen-Bogoliubov-Belyaev treatment of the pairing correlations is applied to spherical nuclei with a general nuclear force. The interaction between quasi-particles is treated by the method of linearized equations of motion. An advantage of this treatment is that the same equations describe single-particle excitations and collective excitations, so that the former are orthogonal to the latter and the total number of states is correct. Another advantage is that the spurious states due to the fluctuations in the number of particles are automatically eliminated. The equations to be solved resemble those for a two-body shell model calculation. Simple estimates, based on delta-function or quadrupole forces, are made for the vibrational frequencies in various modes and transition matrix elements. It is concluded that the method is as powerful as other known methods for dealing with collective states by the shell model, and that the same order of magnitude for the effective nuclear force seems capable of fitting all the data.