Nuclear shell model: Hartree-Fock approximation with Gammel-Thaler two-nucleon potential

Abstract

To investigate how good or bad the Hartree-Fock approximation is for the atomic nuclei, a calculation has been carried out with the phenomenological two-nucleon potential V( r i − r j ) of Gammel-Thaler containing a repulsive core, a central, a tensor and an L· S interaction. Harmonic oscillator wave functions are used for the one-nucleon wave functions. The parameters in the wave functions and a multiplicative parameter in V( r i − r j ) are determined from O 16 by the variational principle and the empirical total energy. These (now fixed) parameters are then used to calculate the energy of the configuration (ls) ν 2 (lp) ν 5 (ls) π 2 (lp) π 6 of O 15. It is found that the L· S term in V( r i − r j ) gives rise to a doublet {ggggggggg} MeV. The total binding energy is E( J=½)=−114.8 MeV compared with the empirical value −111.97 MeV. The sign of this doublet corresponds, on the picture of the individual nucleon moving in an effective central field and possessing an l i · s i interaction, to a sign of l i · s i opposite to that of the electron, and is in agreement with the fundamental hypothesis of the nuclear shell model of Mayer and of Jensen et al. The magnitude of the doublet, however, is too large, by a factor of about 5. These results are discussed.