Shell-structure and asymmetry effects in level densities

Level density \(\rho(E,N,Z)\) is derived for a nuclear system with a given energy \(E\), neutron \(N\), and proton \(Z\) particle numbers, within the semiclassical extended Thomas-Fermi and 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 \(S\), and \(\nu\) is related to the number of integrals of motion, except for the energy \(E\). For small shell structure contribution one obtains within the micro-macroscopic approximation (MMA) the value of \(\nu=2\) for \(\rho(E,N,Z)\). In the opposite case of much larger shell structure contributions one finds a larger value of \(\nu=3\). The MMA level density \(\rho\) reaches the well-known Fermi gas asymptote for large excitation energies, and the finite micro-canonical limit for low excitation energies. Fitting the MMA \(\rho(E,N,Z)\) 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 \(K\), which differs significantly from that of neutron resonances.