Regularity and chaos in the nuclear masses

Shell effects in atomic nuclei are a quantum mechanical manifestation of the single--particle motion of the nucleons. They are directly related to the structure and fluctuations of the single--particle spectrum. Our understanding of these fluctuations and of their connections with the regular or chaotic nature of the nucleonic motion has greatly increased in the last decades. In the first part of these lectures these advances, based on random matrix theories and semiclassical methods, are briefly reviewed. Their consequences on the thermodynamic properties of Fermi gases and, in particular, on the masses of atomic nuclei are then presented. The structure and importance of shell effects in the nuclear masses with regular and chaotic nucleonic motion are analyzed theoretically, and the results are compared to experimental data. We clearly display experimental evidence of both types of motionComment: 40 pages, 10 figures, Lectures delivered at the VIII Hispalensis International Summer School, Sevilla, Spain, June 2003 (to appear in Lecture Notes in Physics, Springer--Verlag, Eds. J. M. Arias and M. Lozano

Random matrices, random polynomials and Coulomb systems

It is well known that the joint probability density of the eigenvalues of Gaussian ensembles of random matrices may be interpreted as a Coulomb gas. We review these classical results for hermitian and complex random matrices, with special attention devoted to electrostatic analogies. We also discuss the joint probability density of the zeros of polynomials whose coefficients are complex Gaussian variables. This leads to a new two-dimensional solvable gas of interacting particles, with non-trivial interactions between particles.Comment: 8 pages, to appear in the Proceedings of the International Conference on Strongly Coupled Coulomb Systems, Saint-Malo, 199

Correlations in Nuclear Masses

It was recently suggested that the error with respect to experimental data in nuclear mass calculations is due to the presence of chaotic motion. The theory was tested by analyzing the typical error size. A more sensitive quantity, the correlations of the mass error between neighboring nuclei, is studied here. The results provide further support to this physical interpretation.Comment: 4 pages, 2 figure

Normal forms and complex periodic orbits in semiclassical expansions of Hamiltonian systems

Bifurcations of periodic orbits as an external parameter is varied are a characteristic feature of generic Hamiltonian systems. Meyer's classification of normal forms provides a powerful tool to understand the structure of phase space dynamics in their neighborhood. We provide a pedestrian presentation of this classical theory and extend it by including systematically the periodic orbits lying in the complex plane on each side of the bifurcation. This allows for a more coherent and unified treatment of contributions of periodic orbits in semiclassical expansions. The contribution of complex fixed points is find to be exponentially small only for a particular type of bifurcation (the extremal one). In all other cases complex orbits give rise to corrections in powers of $\hbar$ and, unlike the former one, their contribution is hidden in the ``shadow'' of a real periodic orbit.Comment: better ps figures available at http://www.phys.univ-tours.fr/~mouchet or on request to [email protected]

Semiclassical Theory of Bardeen-Cooper-Schrieffer Pairing-Gap Fluctuations

Superfluidity and superconductivity are genuine many-body manifestations of quantum coherence. For finite-size systems the associated pairing gap fluctuates as a function of size or shape. We provide a parameter free theoretical description of pairing fluctuations in mesoscopic systems characterized by order/chaos dynamics. The theory accurately describes experimental observations of nuclear superfluidity (regular system), predicts universal fluctuations of superconductivity in small chaotic metallic grains, and provides a global analysis in ultracold Fermi gases.Comment: 4 pages, 2 figure

On the distribution of the total energy of a system on non-interacting fermions: random matrix and semiclassical estimates

We consider a single particle spectrum as given by the eigenvalues of the Wigner-Dyson ensembles of random matrices, and fill consecutive single particle levels with n fermions. Assuming that the fermions are non-interacting, we show that the distribution of the total energy is Gaussian and its variance grows as n^2 log n in the large-n limit. Next to leading order corrections are computed. Some related quantities are discussed, in particular the nearest neighbor spacing autocorrelation function. Canonical and gran canonical approaches are considered and compared in detail. A semiclassical formula describing, as a function of n, a non-universal behavior of the variance of the total energy starting at a critical number of particles is also obtained. It is illustrated with the particular case of single particle energies given by the imaginary part of the zeros of the Riemann zeta function on the critical line.Comment: 28 pages in Latex format, 5 figures, submitted for publication to Physica