Analytical Approximation of the Combinatorial Calculation of Nuclear Level Densities

An exact numerical solution of the spin-dependent level-density problem for a spherical shell-model nucleus with realistic single-particle level spacings is used to investigate the validity of analytical expressions based on approximate treatments of the model. With parameters chosen by reference to the numerical values, at least one of the analytical expressions proposed in the literature results in adequate fits to these values. Fitted parameters for the region $Z=25\mathrm{} \mathrm{to} 59$ and $N=25\mathrm{} \mathrm{to} 91$ are presented, and their physical interpretation is discussed. The dependence of the parameters on the number of nucleons of either kind is strongly affected by the shell structure of the nucleus. The level-density parameter $a$ is not simply proportional to the mass number $A$, and the spin distribution parameter can not be represented by the rigid-body moment of inertia, nor by any other monotonic function of $A$. Both parameters vary smoothly across the major proton and neutron shells, and the contributions of the two types of nucleons are additive.