Shell-structure and asymmetry effects in level densities

Level density [Formula: see text] is derived for a nuclear system with a given energy [Formula: see text], neutron [Formula: see text], and proton [Formula: see text] particle numbers, within the semiclassical extended Thomas–Fermi and periodic-orbit theory beyond the Fermi-gas saddle-point method. We obtain [Formula: see text], where [Formula: see text] is the modified Bessel function of the entropy [Formula: see text], and [Formula: see text] is related to the number of integrals of motion, except for the energy [Formula: see text]. For small shell structure contribution one obtains within the micro–macroscopic approximation (MMA) the value of [Formula: see text] for [Formula: see text]. In the opposite case of much larger shell structure contributions one finds a larger value of [Formula: see text]. The MMA level density [Formula: see text] reaches the well-known Fermi gas asymptote for large excitation energies, and the finite micro-canonical limit for low excitation energies. Fitting the MMA [Formula: see text] to experimental data on a long isotope chain for low excitation energies, due mainly to the shell effects, one obtains results for the inverse level density parameter [Formula: see text], which differs significantly from that of neutron resonances.

Semiclassical and quantum shell-structure calculations of the moment of inertia

Shell corrections to the moment of inertia (MI) are calculated for a Woods–Saxon potential of spheroidal shape and at different deformations. This model potential is chosen to have a large depth and a small surface diffuseness which makes it resemble the analytically solved spheroidal cavity in the semiclassical approximation. For the consistent statistical-equilibrium collective rotations under consideration here, the MI is obtained within the cranking model in an approach which goes beyond the quantum perturbation approximation based on the nonperturbative energy spectrum, and is therefore applicable to much higher angular momenta. For the calculation of the MI shell corrections [Formula: see text], the Strutinsky smoothing procedure is used to obtain the average occupation numbers of the particle density generated by the resolution of the Woods–Saxon eigenvalue problem. One finds that the major-shell structure of [Formula: see text], as determined in the adiabatic approximation, is rooted, for large as well as for small surface deformations, in the same inhomogenuity of the distribution of single-particle states near the Fermi surface as the energy shell corrections [Formula: see text]. This fundamental property is in agreement with the semiclassical results [Formula: see text] obtained analytically within the non perturbative periodic orbit theory for any potential well, in particular for the spheroidal cavity, and for any deformation, even for large deformations where bifurcations of the equatorial orbits play a substantial role. Since the adiabatic approximation, [Formula: see text], with [Formula: see text] the distance between major nuclear shells, is easily obeyed even for large angular momenta typical for high-spin physics at large particle numbers, our model approach seems to represent a tool that could, indeed, be very useful for the description of such nuclear systems.

Analysis of pairing phase transition in Sn-isotopes within semiclassical approach

We demonstrate that pairing phase transition (superfluid to normal) can be described quite generally in terms of the thermodynamical properties after verifying the obtained level densities with the available experimental data for [Formula: see text]- isotopes. Periodic-orbit theory conveniently connects the oscillatory part of level density to the underlying classical periodic orbits and hence, leads to the shell effects in the single-particle level density. Such methods incorporated with pairing effects can be used effectively to study the phase transitions in [Formula: see text]-isotopes. In addition to this, an interplay between pairing correlations and the shell effects has been understood by analyzing the results obtained for the critical temperatures and shell structure energies for [Formula: see text] isotopes. A relation between variation in critical temperatures caused by shell effects and the shell structure energies determined with and without pairing effects has been established. Furthermore, the systematics of the heat capacity (giving a clear signature of pairing phase transition) as function of temperature for these nuclei are investigated as well.

Quantum chaos and quantum graphs

This article examines the origins of the universality of the spectral statistics of quantum chaotic systems in the context of periodic orbit theory. It also considers interesting analogies between periodic orbit theory and the sigma model, along with related work on quantum graphs. The article first reviews some facts and definitions for classically chaotic systems in order to elucidate their quantum behaviour, focusing on systems with two degrees of freedom: one characterized by ergodicity and another by hyperbolicity. It then describes two semiclassical approximation techniques — Gutzwiller’s periodic orbit theory and a refined approach incorporating the unitarity of the quantum evolution — and highlights their importance in understanding universal spectral statistics, and how they are related to the sigma model. This is followed by an analysis of parallel developments for quantum graphs, which are relevant to quantum chaos.

Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow

We consider long-time simulations of two-dimensional turbulence body forced by $\sin 4y\hat {\boldsymbol{x}} $ on the torus $(x, y)\in \mathop{[0, 2\mathrm{\pi} ] }\nolimits ^{2} $ with the purpose of extracting simple invariant sets or ‘exact recurrent flows’ embedded in this turbulence. Each recurrent flow represents a sustained closed cycle of dynamical processes which underpins the turbulence. These are used to reconstruct the turbulence statistics using periodic orbit theory. The approach is found to be reasonably successful at a low value of the forcing where the flow is close to but not fully in its asymptotic (strongly) turbulent regime. Here, a total of 50 recurrent flows are found with the majority buried in the part of phase space most populated by the turbulence giving rise to a good reproduction of the energy and dissipation p.d.f. However, at higher forcing amplitudes now in the asymptotic turbulent regime, the generated turbulence data set proves insufficiently long to yield enough recurrent flows to make viable predictions. Despite this, the general approach seems promising providing enough simulation data is available since it is open to extensive automation and naturally generates dynamically important exact solutions for the flow.