Spin-Orbit Doublet Separation inO17Using Hard-Core Harmonic-Oscillator Wave Functions

Abstract

Calculations of the spin-orbit doublet separation in ${\mathrm{O}}^{17}$ are performed. It is assumed that, to a first approximation, the individual-particle potential experienced by each nucleon in the nucleus is given by the harmonic-oscillator potential. In this approximation the two-nucleon wave function for nucleons in the nucleus is separable in their relative and center-of-mass coordinates so that, taking into account only two-body interactions which depend on the relative coordinate, the $K$ matrix elements are essentially functions of quantum numbers of relative motion only and of the relative space coordinate. The nucleus ${\mathrm{O}}^{17}$ is considered as consisting of the nucleus ${\mathrm{O}}^{16}$ as a core plus a neutron outside. The spin-orbit doublet separation is the difference in energy of ${\mathrm{O}}^{17}$ with the outside neutron in the states $J=\frac{5}{2} \mathrm{and} \frac{3}{2}$, and is evaluated in the approximation of taking interactions of the outside neutron with each of the sixteen core nucleons and neglecting interactions between nucleons in the core. Numerical calculations are done using only the spin-orbit part of the Gammel-Thaler potential, but treating it as a perturbation using hard-core harmonic-oscillator wave functions as the unperturbed wave functions. A value of 5.95 MeV is obtained for the spin-orbit splitting.