Periodic-orbit bifurcations and superdeformed shell structure

Abstract

We have derived a semiclassical trace formula for the level density of the three-dimensional spheroidal cavity. To overcome the divergences occurring at bifurcations and in the spherical limit, the trace integrals over the action-angle variables were performed using an improved stationary phase method. The resulting semiclassical level density oscillations and shell-correction energies are in good agreement with quantum-mechanical results. We find that the bifurcations of some dominant short periodic orbits lead to an enhancement of the shell structure for "superdeformed" shapes related to those known from atomic nuclei. Keywords: Single-particle level density, periodic orbit theory, Gutzwiller's trace formula, bifurcations, superdeformations.PACS numbers: 03.65.Ge, 03.65.Sq, 05.45.Mt Introduction -The periodic orbit theory (POT) [1][2][3][4][5][6][7] is a nice tool for studying the correspondence between classical and quantum mechanics and, in particular, the interplay of deterministic chaos and quantum-mechanical behavior. But also for systems with integrable or mixed classical dynamics, the POT leads to a deeper understanding of the origin of shell structure in finite fermion systems from such different areas as nuclear [5,[8][9][10], metallic cluster [11,12], or mesoscopic semiconductor physics [13,14]. Bifurcations of periodic orbits may have significant effects, e.g., in connection with the so-called "superdeformations" of atomic nuclei [5,6,9,15], and were recently shown to affect the quantum oscillations observed in the magneto-conductance of a mesoscopic device [14].In the semiclassical trace formulae that connect the quantum-mechanical density of states with a sum over the periodic orbits of the classical system [1-3], divergences arise at critical points where bifurcations of periodic orbits occur or where symmetry breaking (or restoring) transitions take place. At these points the stationary-phase approximation, used in the semiclassical evaluation of the trace integrals, breaks down. Various ways of avoiding these divergences have been studied [2,4,16], some of them employing uniform approximations [17][18][19][20]. Here we employ an improved stationary-phase method (ISPM) for the evaluation of the trace integrals in the phase-space representation, based on the studies in Refs. [4,18], which we have derived for the elliptic billiard [21]. It yields a semiclassical level density that is regular at all bifurcation points of the short diameter or-

Shell structure and orbit bifurcations in finite fermion systems

“…In order to avoid such singularities, we observe that the bifurcation problem is similar to the caustic singularity considered by Fedoryuk within the catastrophe theory [22,24], adopted to its specific position at the edge of the phase-space volume accessible for the classical motion (see also Appendix A in [21]). Therefore, we employ what we call the "improved SPM", in short: ISPM [18,19,20,21]. Hereby the integration over ξ in (3) is restricted to the finite limits defined by the classically allowed phase space region through the energy-conserving delta function in the integrand of (3).…”

“…6). Presently, we shall discuss the evaluation of trace integrals in the phase-space representation, following mainly the presentations in [18,19,20,21].…”

“…As one of the main subjects in this report, we apply the semiclassical theory to a 3D spheroidal (prolate) cavity [5,19,20], which may be taken as a simple model for a large deformed nucleus or a (highly idealized) deformed metallic cluster [4]. We shall investigate the role of orbit bifurcations in the shell structure in connection with the superdeformation (corresponding to the fission isomer of heavy nuclei).…”

“…2 are the predictions of the POT for the loci of the ground-state minima, using the shortest POs in a spheroidal cavity. Bifurcations of POs under the variation of a deformation parameter or the (Fermi) energy can have noticeable effects for the shell structure [5,6,7,18,19,20,21].…”

“…3.3, we use the spheroidal cavity [5,19,20,27] as a simple integrable model that allows to study semiclassically the shell structure related to the 'super-deformed' energy minimum which in realistic actinide nuclei corresponds to the fission isomers. Strutinsky's prediction concerning the importance of the enhancement of the shell structure owing to period-doubling bifurcations of three-dimensional POs from simple equatorial (EQ) orbits in the spheroidal cavity model [5] will be discussed.…”