Analyzing shell structure from Babylonian and modern times

We investigate ``shell structure'' from Babylonian times: periodicities and beats in computer-simulated lunar data corresponding to those observed by Babylonian scribes some 2500 years ago. We discuss the mathematical similarity between the Babylonians' recently reconstructed method of determining one of the periods of the moon with modern Fourier analysis and the interpretation of shell structure in finite fermion systems (nuclei, metal clusters, quantum dots) in terms of classical closed or periodic orbits.Comment: LaTeX2e, 13pp, 8 figs; contribution to 10th Nuclear Physics Workshop "Marie and Pierre Curie", 24 - 28 Sept. 2003, Kazimierz Dolny (Poland); final version accepted for J. Mod. Phys.

Quantum fluid-dynamics from density functional theory

A partial differential eigenvalue equation for the density displacement fields associated with electronic excitations is derived in the framework of density functional theory. Our quantum fluid-dynamical approach is based on a variational principle and the Kohn-Sham ground-state energy functional, using only the occupied Kohn-Sham orbitals. It allows for an intuitive interpretation of electronic excitations in terms of intrinsic local currents that obey a continuity equation. We demonstrate the capabilities of this non-empirical approach by calculating the photoabsorption spectra of small sodium clusters. The quantitative agreement between theoretical and experimental spectra shows that even for the smallest clusters, the resonances observed experimentally at low temperatures can be interpreted in terms of density vibrations.Comment: RevTeX file with 2 figures. Update on April 17 2001: Typos corrected, references updated, larger axes labels on Fig. 1. Accepted for publication in Phys. Rev.

Super-shell structure in harmonically trapped fermionic gases and its semi-classical interpretation

It was recently shown in self-consistent Hartree-Fock calculations that a harmonically trapped dilute gas of fermionic atoms with a repulsive two-body interaction exhibits a pronounced {\it super-shell} structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of the beating mode depends on the strength of the interaction. We give a qualitative interpretation of the beat structure in terms of a semiclassical trace formula that uniformly describes the symmetry breaking U(3) $\to$ SO(3) in a 3D harmonic oscillator potential perturbed by an anharmonic term $\propto r^4$ with arbitrary strength. We show that at low Fermi energies (or particle numbers), the beating gross-shell structure of this system is dominated solely by the two-fold degenerate circular and (diametrically) pendulating orbits.Comment: Final version of procedings for the 'Nilsson conference

Periodic orbit theory including spin degrees of freedom

We summarize recent developments of the semiclassical description of shell effects in finite fermion systems with explicit inclusion of spin degrees of freedom, in particluar in the presence of spin-orbit interactions. We present a new approach that makes use of spin coherent states and a correspondingly enlarged classical phase space. Taking suitable limits, we can recover some of the earlier approaches. Applications to some model systems are presented.Comment: LaTeX2e, 10pp, 5 figs; contribution to 10th Nuclear Physics Workshop "Marie and Pierre Curie", 24 - 28 Sept. 2003, Kazimierz Dolny (Poland

Supershell structure in trapped dilute Fermi gases

We show that a dilute harmonically trapped two-component gas of fermionic atoms with a weak repulsive interaction has a pronounced super-shell structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of this beating mode depend on the strength of the interaction. We give a simple interpretation of the beat structure in terms of a semiclassical trace formula for the symmetry breaking U(3) --> SO(3).Comment: 4 pages, 4 figures; In version 2, references added. The semiclassical explanation of super-shell structure is refined. Version 3, as appeared in Phys. Rev.

Wavefunction localization and its semiclassical description in a 3-dimensional system with mixed classical dynamics

We discuss the localization of wavefunctions along planes containing the shortest periodic orbits in a three-dimensional billiard system with axial symmetry. This model mimicks the self-consistent mean field of a heavy nucleus at deformations that occur characteristically during the fission process [1,2]. Many actinide nuclei become unstable against left-right asymmetric deformations, which results in asymmetric fragment mass distributions. Recently we have shown [3,4] that the onset of this asymmetry can be explained in the semiclassical periodic orbit theory by a few short periodic orbits lying in planes perpendicular to the symmetry axis. Presently we show that these orbits are surrounded by small islands of stability in an otherwise chaotic phase space, and that the wavefunctions of the diabatic quantum states that are most sensitive to the left-right asymmetry have their extrema in the same planes. An EBK quantization of the classical motion near these planes reproduces the exact eigenenergies of the diabatic quantum states surprisingly well.Comment: 4 pages, 5 figures, contribution to the Nobel Symposium on Quantum Chao

On the canonically invariant calculation of Maslov indices

After a short review of various ways to calculate the Maslov index appearing in semiclassical Gutzwiller type trace formulae, we discuss a coordinate-independent and canonically invariant formulation recently proposed by A Sugita (2000, 2001). We give explicit formulae for its ingredients and test them numerically for periodic orbits in several Hamiltonian systems with mixed dynamics. We demonstrate how the Maslov indices and their ingredients can be useful in the classification of periodic orbits in complicated bifurcation scenarios, for instance in a novel sequence of seven orbits born out of a tangent bifurcation in the H\'enon-Heiles system.Comment: LaTeX, 13 figures, 3 tables, submitted to J. Phys.