Ken-ichiro Arita

Symmetry Breaking and Bifurcations in the Periodic Orbit Theory. II Spheroidal Cavity

We derive a semiclassical trace formula for the level density of a three-dimensional spheroidal cavity. To overcome the divergences and discontinuities occurring at bifurcation points and in the spherical limit, the traceintegrals over the action-angle variables are performed using an improved stationary phase method. The resulting semiclassical level density oscillations and shell energies are in good agreement with quantum-mechanical results. We find that the births of three-dimensional orbits through the bifurcations of planar orbits in the equatorial plane lead to considerable enhancement of the shell effect for superdeformed shapes.

Symmetry Breaking and Bifurcations in the Periodic Orbit Theory. I: Elliptic Billiard

We derive an analytical trace formula for the level density of the two-dimensional elliptic billiard using an improved stationary phase method. The result is a continuous function of the deformation parameter (eccentricity) through all bifurcation points of the short diameter orbit and its repetitions, and possesses the correct limit of the circular billiard at zero eccentricity. Away from the circular limit and the bifurcations, it reduces to the usual (extended) Gutzwiller trace formula which for the leading-order families of periodic orbits is identical to the result of Berry and Tabor. We show that the circular disk limit of the diameter-orbit contribution is also reached through contributions from closed (periodic and non-periodic) orbits of hyperbolic type with an even number of reflections from the boundary. We obtain the Maslov indices depending on deformation and energy in terms of the phases of the complex error and Airy functions. We find enhancement of the amplitudes near the common bifurcation points of both short-diameter and hyperbolic orbits. The calculated semiclassical level densities and shell energies are in good agreement with the quantum mechanical ones.We derive an analytical trace formula for the level density of the two-dimensional elliptic billiard using an improved stationary phase method. The result is a continuous function of the deformation parameter (eccentricity) through all bifurcation points of the short diameter orbit and its repetitions, and possesses the correct limit of the circular billiard at zero eccentricity. Away from the circular limit and the bifurcations, it reduces to the usual (extended) Gutzwiller trace formula which for the leading-order families of periodic orbits is identical to the result of Berry and Tabor. We show that the circular disk limit of the diameter-orbit contribution is also reached through contributions from closed (periodic and non-periodic) orbits of hyperbolic type with an even number of reflections from the boundary. We obtain the Maslov indices depending on deformation and energy in terms of the phases of the complex error and Airy functions. We find enhancement of the amplitudes near the common bifurcation points of both short-diameter and hyperbolic orbits. The calculated semiclassical level densities and shell energies are in good agreement with the quantum mechanical ones.

Shell structure and orbit bifurcations in finite fermion systems

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 self-consistent 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.C. 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 far-going 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 Approach for Bifurcations in a Smooth Finite-Depth Potential

The analytical trace formula for a dense cascade of bifurcations was derived using the improved stationary phase method based on extensions of the semiclassical Gutzwiller path integral approach. For the integrable version of the famous Henon-Heiles Hamiltonian, our analytical trace formula solves all bifurcation problems, in particular, in the harmonic oscil- lator limit and the potential barrier limit. We obtain nice agreement with quantum results for gross to finer shell structures in level densities and for the shell structure energies, even near the potential barrier where there is a rather dense sequence of bifurcations.

Semiclassical trace formula for the two-dimensional radial power-law potentials.

The trace formula for the density of single-particle levels in the two-dimensional radial power-law potentials, which nicely approximate up to a constant shift the radial dependence of the Woods-Saxon potential and its quantum spectra in a bound region, was derived by the improved stationary phase method. The specific analytical results are obtained for the powers α=4 and 6. The enhancement of periodic-orbit contribution to the level density near the bifurcations are found to be significant for the description of the fine shell structure. The semiclassical trace formulas for the shell corrections to the level density and the energy of many-fermion systems reproduce the quantum results with good accuracy through all the bifurcation (symmetry breaking) catastrophe points, where the standard stationary-phase method breaks down. Various limits (including the harmonic oscillator and the spherical billiard) are obtained from the same analytical trace formula.

Periodic-orbit bifurcation and shell structure at exotic deformation

We have investigated the semiclassical origin of superdeformed shell structure and also of reflection-asymmetric deformed shapes by means of the periodic orbit theory and the deformed cavity model. Systematic analysis of the quantum-classical correspondence reveals that bifurcation of equatorial orbits into three-dimensional ones play predominant role in the formation of these shell structures.

Shells, orbit bifurcations, and symmetry restorations in Fermi systems

The periodic-orbit theory based on the improved stationary-phase method within the phase-space path integral approach is presented for the semiclassical description of the nuclear shell structure, concerning themain topics of the fruitful activity ofV.G. Soloviev. We apply this theory to study bifurcations and symmetry breaking phenomena in a radial power-law potential which is close to the realistic Woods–Saxon one up to about the Fermi energy. Using the realistic parametrization of nuclear shapes we explain the origin of the double-humped fission barrier and the asymmetry in the fission isomer shapes by the bifurcations of periodic orbits. The semiclassical origin of the oblate–prolate shape asymmetry and tetrahedral shapes is also suggested within the improved periodic-orbit approach. The enhancement of shell structures at some surface diffuseness and deformation parameters of such shapes are explained by existence of the simple local bifurcations and new non-local bridge-orbit bifurcations in integrable and partially integrable Fermi-systems. We obtained good agreement between the semiclassical and quantum shell-structure components of the level density and energy for several surface diffuseness and deformation parameters of the potentials, including their symmetry breaking and bifurcation values.

Semiclassical origin of the anomalous shell effect for tetrahedral deformation in the radial power-law potential model

Shell structures in single-particle energy spectra are investigated against regular tetrahedral type deformation using a radial power-law potential model. Employing a natural way of shape parametrization which interpolates sphere and regular tetrahedron, we find prominent shell effects for rather large tetrahedral deformations, which bring about shell energies much larger than the cases of spherical and quadrupole type shapes. We discuss the semiclassical origin of these anomalous shell structures using periodic orbit theory.

Semiclassical catastrophe theory of simple bifurcations

The Fedoriuk-Maslov catastrophe theory of caustics and turning points is extended to solve the bifurcation problems by the improved stationary phase method (ISPM). The trace formulas for the radial power-law (RPL) potentials are presented by the ISPM based on the second- and third-order expansion of the classical action near the stationary point. A considerable enhancement of contributions of the two orbits (pair consisting of the parent and newborn orbits) at their bifurcation is shown. The ISPM trace formula is proposed for a simple bifurcation scenario of Hamiltonian systems with continuous symmetries, where the contributions of the bifurcating parent orbits vanish upon approaching the bifurcation point due to the reduction of the end-point manifold. This occurs since the contribution of the parent orbits is included in the term corresponding to the family of the newborn daughter orbits. Taking this feature into account, the ISPM level densities calculated for the RPL potential model are shown to be in good agreement with the quantum results at the bifurcations and asymptotically far from the bifurcation points.

Periodic orbit bifurcations and local symmetry restorations in exotic-shape nuclear mean fields

The semiclassical origins of the enhancement of shell effects in exotic-shape mean-field potentials are investigated by focusing attention on the roles of the local symmetries associated with the periodic-orbit bifurcations. The deformed shell structures for four types of pure octupole shapes in the nuclear mean-field model having a realistic radial dependence are analyzed. Remarkable shell effects are shown for a large Y 32 deformation having tetrahedral symmetry. Much stronger shell effects found in the shape parametrization smoothly connecting the sphere and the tetrahedron are investigated from the view-point of the classical–quantum correspondence. The local dynamical symmetries associated with the bridge orbit bifurcations are shown to have significant roles in the emergence of exotic deformed shell structures for certain combinations of the surface diffuseness and the tetrahedral deformation parameters.