Semiclassical shell-structure micro-macroscopic approach for the level density

Level density $\rho(E,A)$ is derived for a one-component nucleon system with given energy $E$ and particle number $A$ within the mean-field semiclassical periodic-orbit theory beyond saddle-point method of Fermi gas model. We obtain $~~\rho \propto I_\nu(S)/S^\nu$, $I_\nu(S)$ being modified Bessel function entropy $S$. Within micro-macro-canonical approximation (MMA), small thermal excitation energy, $U$, respect to rotational excitations, $E_{\rm rot}$, one obtains $\nu=3/2$ $\rho(E,A)$. In case $U$ larger than rot}$ but smaller neutron separation finds value $\nu=5/2$. A role fixed spin variables rotating nuclei discussed. The MMA level $\rho$ reaches well-known grand-canonical ensemble limit (Fermi asymptotic) large $S$ related energies, also finite micro-canonical combinatorial at low energies (the constant "temperature" model). Fitting experimental data taking into account shell and, qualitatively, pairing effects, inverse parameter $K$ which differs essentially from that on resonances.