Number-Conserving Approximation to the Shell Model

The broken-pair-approximation (BPA) formalism is presented in this paper in complete form for the description of the nuclear properties of medium and heavy spherical nuclei. Starting from an approximate ground state of even nuclei having BCS-type pair distribution in the valence shells, the model Hilbert space is constructed by replacing one, two,... pairs in the assumed approximate ground state by arbitrary two, four,... particle configurations. The BPA states for odd-mass nuclei are obtained by coupling the odd nucleon to zero-, one-,... broken-pair states. It is shown that the Hilbert spaces spanned in the projected quasiparticle theories and in the BPA are the same in a certain limit, and that in practice these two formalisms are physically equivalent. The BPA is an improvement on the quasiparticle theories and is an approximation to the seniority shell model. All the relevant expressions for calculating energy matrices, transition rates, inelastic electron-scattering form factors, and spectroscopic factors for one- and two-nucleon transfer reactions are presented in a coherent form. The corresponding expressions for the case of first BPA are further elaborated and presented in a form suitable for numerical computation.