New Shell Structure Originated from the Combination of Quadrupole and Octupole Deformations†

Semiclassical analysis of the shell structure for a reflection-asymmetric deformed oscillator potential with irrational frequency ratio ω⊥/ωz = √ 3 is presented. Strong shell effects associated with bifurcations of short periodic orbits are found, which occur for a combination of quadrupole and octupole deformations. Submitted to Physis Letters B. E-mail: [email protected] 1 Study of reflection-asymmetric deformation is one of the exciting current subjects in both nuclear structure and micro-cluster physics [1–4]. As is well known, nuclear deformation is intimately related with the single-particle level structure. The system favors such shapes that exhibit strong shell effects and make the level density at the Fermi surface lower. We have analyzed in Refs. [5–7] shell structures in a superdeformed (SD) oscillator potential with the octupole deformation added. The model Hamiltonian adopted is H = p 2M + ∑ i Mω i x 2 i 2 − λ30Mω 0[rY30(Ω)], (1) where the double primes denote that the variables in the square bracket are defined in terms of the doubly-stretched coordinate x i ≡ (ωi/ω0)xi. The frequency ratio ω⊥/ωz is taken to be 2 with ωx = ωy = ω⊥. At λ30 = 0, the shell structure for this Hamiltonian is mainly determined by the figure-8 type family of periodic orbits. However, the planar orbit family in the x-y plane having shorter period give rise to an interference effect on the shell oscillations and result in modulation pattern in the smoothed level density, called supershell. We found that the supershell effect significantly develops at octupole deformation parameter λ30 ≃ 0.4 [5]. This is explained as due to the changes in stabilities of the two classes of periodic orbits against the octupole deformation [6, 7]. As a continuation of the above work, we have carried out semiclassical analysis of the shell structure for the Hamiltonian (1) varying both λ30 and the axis ratio ω⊥/ωz. The details will be reported elsewhere [8]. As an illustrative example, we discuss here the case of an irrational axis ratio ω⊥/ωz = √ 3. Figure 1 shows the single-particle spectrum calculated as a function of λ30. There is no prominent shell structure at λ30 = 0 because of irrationality of the frequency ratio. However, a significant shell structure appears at λ30 ≃ 0.3. The oscillating level density smoothed by the Strutinsky method is shown in Fig. 2. One clearly sees a prominent shell oscillation. Evidently, this shell structure arises as a consequence of the combination of quadrupole and octupole deformations. One may also notice a supershell structure, which is found to be associated with the interference between classical periodic orbits with the periods T ≃ 2π/ω⊥ and 2π/ωz. In classical mechanics, there are only two types of periodic orbits at λ30 = 0; one is the Isolated Linear orbit along the z-axis (IL) and the other is the continuous family of elliptic orbits in the x-y plane. Among the latter family, only the Planar-A type (PA) orbits (in the plane containing the symmetry axis) and an isolated circular orbit survive when the 2 octupole deformation is added. In Fig. 3, we show some short periodic orbits for λ30 = 0.3. It is noteworthy that orbit PB appears at λ30 ≃ 0.29 due to an isochronous bifurcation of orbit IL. Likewise, orbits PC and PD are born by a saddle-node bifurcation at λ30 ≃ 0.28. Orbits PE and PF arise from a period-tripling bifurcation of orbit PA at λ30 ≃ 0.2. To show how these bifurcations occur, we plot in Fig. 4 traces of the monodromy matrices [9] for these orbits as functions of λ30. The monodromy matrix M describing linearized dynamics about the periodic orbit is defined by Mr(Zr0) = ∂Z r (t = Tr) ∂Z r (t = 0) ∣