Connection between the Nuclear Shell Model and the Unified Model

Abstract

The main point of this paper is to discuss the connection between the unified model and a modified form of the nuclear shell model. In the latter, the effective interactions between nucleons, i.e., those not included in the central average potential, are assumed to be factorable. It can be argued that inside nuclear matter the Pauli principle greatly suppresses the off-diagonal effects of the interactions, i.e. it prohibits most inelastic collisions; so that the nucleons move nearly freely within the nucleus, at least as far as low-energy phenomena are concerned. Short-range correlations due to the interactions are not suppressed, but these are expected to manifest themselves at rather high energies.According to this view, the effective interactions between nucleons occur mainly at the nuclear surface and give rise to surface oscillations. While quadrupole oscillations predominate, higher modes also arise in a natural way. Under certain conditions, the problem of particles subject to mutual interactions may be solved by introduction of additional collective variables, which is the method of the unified model. Physically, the nuclear motion separates, at least approximately, into intrinsic and collective motions. The resulting wave functions are very similar to the ones used by Bohr and Mottelson, except that they are integrated over the collective variables. Also, the energy spectra are of the form obtained by Bohr and Mottelson, but the collective excitation spectrum is cut off, i.e., only a finite number of states occur.This method is applied to a simplified two-dimensional nuclear model. The particles are assumed to move in an isotropic harmonic oscillator potential, and to be in addition subject to (a) one-body spin-orbit forces, and (b) mutual quadrupole-quadrupole interactions. In the absence of spin-orbit coupling, the spectrum separates into a series of rotational bands, while in the absence of mutual interactions we have an independent-particle spectrum. The intermediate coupling problem is also treated in the hope that it may provide some insight into the competition between independent-particle and collective motions in nuclei. In the present example, the transition between the two limiting schemes occurs quite suddenly, in agreement with the experimental evidence.Another case of interest is the situation at the beginning of the nuclear $1p$ shell. In the limit of pure $\mathrm{LS}$ coupling, the states of maximum spatial symmetry form a rotational band, though only very few members can appear, and collective effects are not pronounced, because of the small number of particles involved. The effect of spin-orbit coupling can also be described in the language of the unified model.