Alexander G. Magner

The nuclear surface as a collective variable

In heavy nuclei where the thickness of the diffused edge is relatively small, a certain sharp effective surface can be defined which characterizes the shape of the nucleus, and it can be considered as a collective dynamic variable. It is shown that the problem of fluid dynamics can be simplified by reducing it to simple linearized equations for the dynamics in the nuclear interior and boundary conditions set at the effective dynamic sharp surface of the density distribution. These conditions are derived from the fluid dynamical equations. Transitional densities obtained from this simple model are compared with the numerical solution of fluid dynamical equations.

Particle-Liquid Dynamics of Nuclear Deformations. 1

Equations of a combined, liquid plus particle model are derived which accounts consistently for the short-range liquid properties of nuclei as well as for the long-range quasi-particle behavior. Macroscopic quantities are defined as statistical averages and their dynamics is determined by an Euler-type equation, which contains a term due to the interaction with the gas component, and also by the continuity equation. For the quasi-particle component the full quantal treatment is kept in the form of the Liouville equation with a time-dependent external potential related to the bulk density. It may include the residual quasi-particle interaction. It is suggested that the solution of the coupled dynamics is facilitated by the explicit use of an effective sharp nuclear surface as a dynamical variable.

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.

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.

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 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 Orbits And Deformed Shell Structure

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 three‐dimensional periodic orbits, which are born out of bifurcation of planar orbits at large prolate deformations, generate the superdeformed shell structure.

Liquid-Drop Type Expansion of Nuclear Binding and Deformation Energies with Skyrme Forces

Start i n g from the energy density formalism with e f f e c t i v e Skyrme forces,we show how a l i q u i d drop model l i k e expansion of the nuclear binding energy can be systematically obtained. For a model Skyrme force with constant e f f e c t i v e nucleon mass, a l l l i q u i d drop parameters of symmetric nuclei can be given a n a l y t i c a l l y i n terms of the Skyrme parametersNumerical tests of the leptodermous expansion are presented.