Shell structure and orbit bifurcations in finite fermion systems

Abstract

We first give an overview of the shell-correction method which was developed by V. M.Strutinsky as a practicable and efficient approximation to the general selfconsistent theory of finite fermion systems suggested by A. B. Migdal and collaborators. Then we present in more detail a semiclassical theory of shell effects, also developed by Strutinsky following original ideas of M. Gutzwiller. We emphasize, in particular, the influence of orbit bifurcations on shell structure. We first give a short overview of semiclassical trace formulae, which connect the shell oscillations of a quantum system with a sum over periodic orbits of the corresponding classical system, in what is usually called the "periodic orbit theory". We then present a case study in which the gross features of a typical double-humped nuclear fission barrier, including the effects of mass asymmetry, can be obtained in terms of the shortest periodic orbits of a cavity model with realistic deformations relevant for nuclear fission. Next we investigate shell structures in a spheroidal cavity model which is integrable and allows for fargoing analytical computation. We show, in particular, how period-doubling bifurcations are closely connected to the existence of the so-called "superdeformed" energy minimum which corresponds to the fission isomer of actinide nuclei. Finally, we present a general class of radial power-law potentials which approximate well the shape of a Woods-Saxon potential in the bound region, give analytical trace formulae for it and discuss various limits (including the harmonic oscillator and the spherical box potentials).

Semiclassical catastrophe theory of simple bifurcations

“…This will enable us to understand more properly the shape dynamics of the finite fermion systems. In particular, the improved stationary phase method can be applied to describe the deformed shell structures where bifurcations play an essential role in formations of the superdeformed minima along a potential energy valley [10,11]. One of the remarkable tasks might be to clarify, in terms of the symmetry-breaking (restoration) and bifurcation phenomena, the reasons of the exotic deformations such as the octupole and tetrahedral ones within the suggested ISPM.…”

“…For definiteness, we will consider first a simple bifurcation defined as a caustic point of the first order [Eqs. (10) and (11)] where the degeneracy parameter K is locally increased by one. In the SPM, after performing exact 2 We call the nth-order ISPM the ISPMn, in which we use the expansion of the phase integral Φ up to the nth-order terms and the amplitude up to the (n−2)th order near the stationary point.…”

“…MP is defined as the sum of this asymptotic part (55) and the argument of the complex density amplitude (51) [10][11][12][13][14]. The total index µ (tot) MP behaves as a smooth function of the energy E and the power parameter α through the bifurcation point.…”

Chaoticity and shell effects in the nearest-neighbor distributions for an axially-symmetric potential

“…All considered potentials are axially symmetric but they are the same nonintegrable ones in the plane of the symmetry axis. The degree of symmetry which determines the classical degeneracy [22,27,28] K (see introduction) for the case of the full spectra (Figs. 1, 3, 5) is higher than for the fixed angular momentum m (K = 0).…”

Periodic-orbit approach to nuclear shell structures with power-law potential models: Bridge orbits and prolate-oblate asymmetry

“…Some preliminary results for the spherical power-law potential using the improved stationary phase method have been reported in Ref. [38]. Application of a uniform approximation to this problem is also in progress.…”