ar X iv : n uc l - th / 0 20 80 77 v 1 3 0 A ug 2 00 2 Periodic Orbits and Deformed Shell Structure 1

Relationship between quantum shell structure and classical periodic orbits is briefly reviewed on the basis of semi-classical trace formula. Using the spheroidal cavity model, it is shown that threedimensional periodic orbits, which are born out of bifurcation of planar orbits at large prolate deformations, generate the superdeformed shell structure. Introduction Existence of superdeformed (SD) nuclei is often explained in terms of the SD magic numbers for the harmonicoscillator (HO) potential with axis ratio 2:1. It appears, however, that we need a more general explanation not restricted to the HO potential, since, up to now, more than 200 SD bands have been found in various regions of nuclear chart and their shapes in general deviates from the 2:1 shape to some extent. In this talk, we shall discuss the mechanism how and the reason why the SD shell structure emerges. The major tool for this purpose is the trace formula, which is the central formula in the semiclassical periodic-orbit (PO) theory and provides a link between quantum shell structure and classical periodic orbits in the mean field. Here, shell structure is defined as regular oscillation in the single-particle level density coarse-grained to a certain energy resolution. An example of coarse-graining for the well-known axially symmetric HO model is displayed in Fig. 1. In this talk, we discuss the spheroidal cavity model, since, in contrast to the HO model, this model is very rich in periodic orbits; it is an ideal model for exhibiting the presence of various kinds of periodic orbit and their bifurcations. We present both Fourier transforms of quantum spectra and semiclassical calculations based on the PO theory, and identify classical periodic orbits responsible for emergence of the SD shell structure. The result clearly shows that three-dimensional (3D) periodic orbits, that are absent in spherical and normal deformed systems and are born out of bifurcations of planar orbits, generate a new shell structure at large prolate deformations, which may be called “the SD shell structure.” They continue to exist for a wide range of deformation, once they are born. The PO theory provides a basic tool to get a deeper understanding of microscopic origin of symmetry breaking in the mean field. It sheds light, in addition to the stability of the SD nuclei, on the reason of prolate dominance in normal deformed nuclei, on the origin of left-right asymmetric shapes, etc. It is useful for finite many-Fermion systems covering such different areas as nuclei, metallic clusters, quantum dots, etc. In this talk, we shall also touch upon such applications of the PO theory. Level Bunching and Trace Formula For the axially symmetric HO potential, the following two conditions coincide: 1) Talk presented by K.M. at the Conference on Frontiers of Nuclear Structure, July 29th August 2nd, 2002, UC Berkeley.