Application of Elliott's SU(3) model to the triaxially deformed harmonic oscillator

The eigenstate of many-body Hamiltonian is usually obtained by projecting angular momentum, nucleon number, parity, etc from the Slater determinant which is composed of the products of the intrinsic single-particle wave functions. We adopt the triaxially deformed harmonic oscillator model to describe such intrinsic wave functions. We have extended Elliott's SU(3) model [1] to an axially symmetric deformed state [2]. In this paper we extend this model further to the triaxially deformed case in order to see the degeneracy in single-particle energies as a function of γ deformation. Suppose that the three frequencies of harmonic oscillator have a rational ratio, i.e., 3 : 2 : 1, which corresponds to γ = 30o, then the eigenvalue and eigenfunction of H is given by following expression. As the least common multiplier (L.C.M.) is 6 for numbers 3, 2 and 1, we construct new boson d from a biproduct of the cx bosons, f boson from a triproduct of cy bosons and s boson from sixfold products of cz bosons. Then, 8 biproducts of d, f and s bosons form a new realization of the group SU(3), and commute with H. Subsequently, there is the sixfold degeneracy in the vacuum. Here, we remark that Ngh is not a simple sum of nd + nf + ns. Applying the analogy of Elliott's group operators, we obtain new set of group operators from d, f and s bosons, i.e. Qq for q = 0, ±1 and ±2, and lk for k = d, f and s. Then, the commutation relations among these 8 operators are closed, and they commute with H. We find the Casimir operator for single-particle states. The diagonal number operators for the new bosons are given. Together with these two operators, Casimir operator classifies Nsh, which shows sixfold degeneracy. The ratio of frequencies is not unique, but there are other possibilities for the case of γ = 30o, for example 4 : 3 : 2 or 5 : 4 : 3. If more complex ratios are chosen, the L.C.M. becomes larger and the deformation 5 becomes smaller. Similarly, when γ ∼ 19o, we find two simple ratios, i.e., 4 : 3 : 1 which corresponds to the prolate shape, and 1 : 2 : 4 which corresponds to oblate shape. The latter case is fourfold degeneracy as we construct g boson as a tetraproduct of cx bosons, and d boson as a biproduct of cy bosons. In this case the vacuum has fourfold degeneracy, and the resulting SU(3) has tetrahedral nature.(author)