Level density of the H\'enon-Heiles system above the critical barrier Energy

We discuss the coarse-grained level density of the H\'enon-Heiles system above the barrier energy, where the system is nearly chaotic. We use periodic orbit theory to approximate its oscillating part semiclassically via Gutzwiller's semiclassical trace formula (extended by uniform approximations for the contributions of bifurcating orbits). Including only a few stable and unstable orbits, we reproduce the quantum-mechanical density of states very accurately. We also present a perturbative calculation of the stabilities of two infinite series of orbits (R$_n$ and L$_m$), emanating from the shortest librating straight-line orbit (A) in a bifurcation cascade just below the barrier, which at the barrier have two common asymptotic Lyapunov exponents $\chi_{\rm R}$ and $\chi_{\rm L}$.Comment: LaTeX, style FBS (Few-Body Systems), 6pp. 2 Figures; invited talk at "Critical stability of few-body quantum systems", MPI-PKS Dresden, Oct. 17-21, 2005; corrected version: passages around eq. (6) and eqs. (12),(13) improve

Paring correlations within the micro-macroscopic approach for the level density

Level density $\rho(E,N,Z)$ is calculated for the two-component close- and open-shell nuclei with a given energy $E$, and neutron $N$ and proton $Z$ numbers, taking into account pairing effects within the microscopic-macroscopic approach (MMA). These analytical calculations have been carried out by using the semiclassical statistical mean-field approximations beyond the saddle-point method of the Fermi gas model in a low excitation-energies range. The level density $\rho$, obtained as function of the system entropy $S$, depends essentially on the condensation energy $E_{\rm cond}$ through the excitation energy $U$ in super-fluid nuclei. The simplest super-fluid approach, based on the BCS theory, accounts for a smooth temperature dependence of the pairing gap $\Delta$ due to particle number fluctuations. Taking into account the pairing effects in magic or semi-magic nuclei, excited below neutron resonances, one finds a notable pairing phase transition.Pairing correlations sometimes improve significantly the comparison with experimental data.Comment: 8 pages, 2 figures, 2 table

Microscopic-macroscopic level densities for low excitation energies

Level density $\rho(E,{\bf Q})$ is derived within the micro-macroscopic approximation (MMA) for a system of strongly interacting Fermi particles with the energy $E$ and additional integrals of motion ${\bf Q}$, in line with several topics of the universal and fruitful activity of A.S. Davydov. Within the extended Thomas Fermi and semiclassical periodic orbit theory beyond the Fermi-gas saddle-point method we obtain $\rho\propto I_\nu(S)/S^\nu$, where $I_\nu(S)$ is the modified Bessel function of the entropy $S$. For small shell-structure contribution one finds $\nu=\kappa/2+1$, where $\kappa$ is the number of additional integrals of motion. This integer number is a dimension of ${\bf Q}$, ${\bf Q}=\{N, Z, ...\}$ for the case of two-component atomic nuclei, where $N$ and $Z$ are the numbers of neutron and protons, respectively. For much larger shell structure contributions, one obtains, $\nu=\kappa/2+2$. The MMA level density $\rho$ reaches the well-known Fermi gas asymptote for large excitation energies, and the finite micro-canonical combinatoric limit for low excitation energies. The additional integrals of motion can be also the projection of the angular momentum of a nuclear system for nuclear rotations of deformed nuclei, number of excitons for collective dynamics, and so on. Fitting the MMA total level density, $\rho(E,{\bf Q})$, for a set of the integrals of motion ${\bf Q}=\{N, Z\}$, to experimental data on a long nuclear isotope chain for low excitation energies, one obtains the results for the inverse level-density parameter $K$, which differs significantly from those of neutron resonances, due to shell, isotopic asymmetry, and pairing effects.Comment: 24 pages, 4 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2109.0183

Nuclear level density in the statistical semiclassical micro-macroscopic approach

Level density $\rho$ is derived for a finite system with strongly interacting nucleons at a given energy E, neutron N and proton Z particle numbers, projection of the angular momentum M, and other integrals of motion, within the semiclassical periodic-orbit theory (POT) beyond the standard Fermi-gas saddle-point method. For large particle numbers, one obtains an analytical expression for the level density which is extended to low excitation energies U in the statistical micro-macroscopic approach (MMA).The interparticle interaction averaged over particle numbers is taken into account in terms of the extended Thomas-Fermi component of the POT. The shell structure of spherical and deformed nuclei is taken into account in the level density. The MMA expressions for the level density $\rho$ reaches the well-known macroscopic Fermi-gas asymptote for large excitation energies U and the finite combinatoric power-expansion limit for low energies U. We compare our MMA results for the averaged level density with the experimental data obtained from the known excitation energy spectra by using the sample method under statistical and plateau conditions. Fitting the MMA $\rho$ to these experimental data on the averaged level density by using only one free physical parameter - inverse level density parameter K - for several nuclei and their long isotope chain at low excitation energies U, one obtains the results for K. These values of K might be much larger than those deduced from neutron resonances. The shell, isotopic asymmetry, and pairing effects are significant for low excitation energies.Comment: 31 pages, 7 figures, 1 tabl

Semiclassical approach to the low-lying collective excitations in nuclei

For low-lying collective excitations we derived the inertia within the semiclassical Gutzwiller approach to the onebody Green’s function at lowest orders in h. The excitation energies, reduced probabilities and energy-weighted sum rules are in agreement with main features of the experimental data

Analytical perturbative approach to periodic orbits in the homogeneous quartic oscillator potential

We present an analytical calculation of periodic orbits in the homogeneous quartic oscillator potential. Exploiting the properties of the periodic Lam{\'e} functions that describe the orbits bifurcated from the fundamental linear orbit in the vicinity of the bifurcation points, we use perturbation theory to obtain their evolution away from the bifurcation points. As an application, we derive an analytical semiclassical trace formula for the density of states in the separable case, using a uniform approximation for the pitchfork bifurcations occurring there, which allows for full semiclassical quantization. For the non-integrable situations, we show that the uniform contribution of the bifurcating period-one orbits to the coarse-grained density of states competes with that of the shortest isolated orbits, but decreases with increasing chaoticity parameter $\alpha$.Comment: 15 pages, LaTeX, 7 figures; revised and extended version, to appear in J. Phys. A final version 3; error in eq. (33) corrected and note added in prin

Analytic approach to bifurcation cascades in a class of generalized H\'enon-Heiles potentials

We derive stability traces of bifurcating orbits in H\'enon-Heiles potentials near their saddlesComment: LaTeX revtex4, 38 pages, 7 PostScript figures, 2 table