Octupole Instability of the Closed-Shell Configurations in the Superdeformed Oscillator Potential

Abstract

We show that the closed-shell configurations in the axially-symmetric harmonic-oscillator poten­ tial with the frequency ratio wJ../wz=2 are unstable against octupole deformation when their upper­ most shell quantum numbers N:h are even, whereas they are stable when N:h are odd, in agreement with the recent suggestion by Nazarewicz and Dobaczewski. We also suggest a possible relationship between the octupole instability of the shape and the supershell effect in refiection­ asymmetric potentials. In recent years, octupole instability of some nuclei has been suggested in shell-structure energy calculations by means of the Strutinsky me­ thod. 1l 6 ) Concerning the physical condition for the occurrence of octupole instability, N azarewicz et aFH) have discussed dynamical symmetry of the harmonic-oscillator potential with frequencies in rational ratio, and suggested that the octupole instability might occur for closed-shell configurations in the axially-symmetric poten­ tial with frequency ratio (J).l./(J)z=2 (which we call superdeformed oscillator for brevity)when the single-particle levels are filled up to the major shells with NSh=even, NSh being the shell quantum number defined by NSh = 2n.l. + nz. 10 ) The single-particle levels in the potential can be· divided into two classes according to whether NSh are even or odd. Each class corresponds to the single­ particle levels in a spherical potential having frequency 2(J)z.8) This dynam­ ical symmetry was previously discussed by Bengtsson et al.l!) For the closed-shell configurations whose uppermost shell-quantum number N[h are even, the particle numbers belonging to two spherical potentials are unequal so that one can expect a tendency toward a reflection-asymmetric shape under the assumption that each spherical corresponds to a spatial cluster. Evaluating octupole sus­ ceptibility of shell-structure energy by the second-order perturbation, they have shown 7 ),9) that the shell-energy octupole-stiffness coefficient is negative (positive) when N[h is even (odd). The purpose of this paper is twofold: Firstly, in order to examine the N~ dependence of the octupole instability, we evaluate the shell-structure energies of the closed-shell configurations in the potential as functions of octupole-deformation parameters by means of the Strutinsky method. The calcula­ tion is done such that the volume-conservation condition is rigorously fulfilled. Secondly, we show that the supershell effect,10) which is intimately connected with the dynamical symmetry of the potential, becomes more pro. nounced when the reflection symmetry is broken by the octupole deformation term. This result indicates that the octupole instability is related with the super-shell