Advances in the theory of collective motion in nuclei

Abstract

The theory of collective motion in nuclei has as its geometric origin the comparison of certain nuclear phenomena with properties of a liquid drop. Bohr and Mottelson (1952,53) [1,2] introduced the idea of a irrotational flow and explained a variety of collective phenomena by the deformations and vibrations of a nuclear fluid. The nuclear shell theory succeeded later on in the representation of collective excitation by coherent superpositions of many single-particle excitations. Elliott (1958) [3] showed that collective levels in the shell theory can be connected to a group SU(3). Independently of the shell theory, various attempts were made to develop the geometric ideas implicit in the BohrMottelson model. Weaver, Biedenharn and Cusson [4,5,6] introduced the group SL(3, ~) of volume-preserving deformations into the collective theory. With this group, they connected kinematical transformations of the system of A nucleons, the vortex spin, and a spectrum generating algebra. Inclusion of the mass quadrupole tensor leads to a natural extension of this group which was also studied by Rowe, Rosensteel and collaborators [7,8]. In the geometric models, it is the final goal to explain collective phenomena from the point of view of many-body dynamics. Therefore one has to link the collective coordinates to the singleparticle coordinates. This program was already started by Lipkin (1955) [9] and by Villars (1957) [10]. Whereas these authors tried to keep the single-particle coordinates, new viewpoints were developed later by Zickendraht [11], by Dzyublik et al. [12], and by Buck Biedenharn and Cusson [13] by use of the orthogonal intrinsic group SO(n, ~) acting on the particle indices and commuting with $0(3, ~). Rowe and Rosensteel (1980) [14] were the first to analyze this scheme through an orbit analysis in configuration space. Vanagas (1977) [15] pointed out the close relation of the group SO(n, ~) to the symmetric group of orbital permutations and proposed the group SO(n, ~) as a symmetry group of the collective hamiltonian. In the following sections, the dynamical implications of this proposal will be analyzed, based on work done with Z. Papadopolos, M. Saraceno and W. Schweizer.