Single particle spectrum of a nucleon in the harmonic oscillatormean field with spin-orbit coupling - a semiclassical view

摘要

We have presented the single, particle spectrum for a particle in a mean-field of an isotropic harmonic oscillator with~l·~scoupling based onour semiclassical approach. It has been seen that this spectrum, without~l·~scoupling, exactly matches with the quantum mechanical one(without nuclear constraints). In this case, periodicity conditions give only pendulating orbits coinciding withl=0 axis, which fully supportsthe observations reported by Bohr and Mottelson [28]. The orbits withl≥0 are generated by reflecting the particle from the nuclear surface,R0, instead of infinity, which is the usual nuclear constraint. The mean-field strength is fixed by virial theorem. The resulting spectrumcompares reasonably with the quantum spectrum for a particle enclosed in a perfectly reflecting walls. The variation of particle number withenergy helps us to identify the significant quantum numbers ‘n’ and ‘l’ in this semiclassical method. Finally, the~l·~scoupling splits eachlevel and the splitting width of these levels compares well with that of nuclear splitting. Thus the complete nuclear shell model (with magicnumbers) is reproduced without any fitting parameter.