Four-Quasiparticle Excitations and Two-Phonon Vibrational States in Spherical Nuclei. Even-Parity States of Even Tin Isotopes

A microscopic theory of the low-lying states of even-even spherical nuclei is developed in which eigenvectors are linear combinations of two- and four-quasiparticle excitations. The quasiparticles are defined by the Bogoliubov-Valatin canonical transformation. The method is called the quasiparticle second Tamm-Dancoff (QSTD) approximation, since no ground-state correlations are included. It is found that the spurious kets due to the particle-number nonconservation must be absolutely projected out of the secular matrices before their diagonalization. Such a procedure is described and applied. Formulas are given for the electromagnetic transition probabilities. The theory is applied to the study of the $2^{+}$, $4^{+}$, and $0^{+}$ states of the even tin isotopes. The single-particle radial wave functions employed are those of a Saxon-Woods potential and of a harmonic-oscillator potential. The two-nucleon residual interaction potential is spin-dependent and of zero range. Satisfactory numerical agreement with the observed $2^{+}$ and $4^{+}$ low-lying levels is obtained with the Saxon-Woods wave functions for a reasonable strength constant of our zero-range force. Appreciable admixtures of the four-quasiparticle creation components are found even in the lowest lying levels. Poor agreement is obtained for the $0^{+}$ states, for which a more refined theory is necessary (rather unreasonable values of the strength constant of the zero-range potential are required to fit the $0^{+}$ data). Generally, markedly worse $2^{+}$ results are obtained if we replace the Saxon-Woods wave functions with harmonic-oscillator wave functions.