Accuracy of the Superconductivity Approximation for Pairing Forces in Nuclei

The accuracy of the superconductivity approximation for pairing forces in nuclei is studied. To this end we have exactly diagonalized the pairing force for certain nuclei and compared the results with those obtained from the approximate calculation of Kisslinger and Sorensen. When the energy is computed by use of the approximate-wave function, which is not an eigenfunction of the number operator, it is found that the excitation energies of the low-lying states with seniority one and two are correct to within 200 kev, whereas the ground-state energies are usually not given to better than 500 kev. The wave function obtained by projecting out and normalizing that part of the variational trial function that corresponds to the correct number of particles is found to agree closely with the exact energy eigenfunction. Overlap integrals greater than 98% are found in all cases considered. The expectation values of the pairing Hamiltonian with respect to these projected wave functions are therefore in excellent agreement with the exact energy eigenvalues. The variational aspects of the superconductivity approximation are also discussed briefly.