Exact nuclear level density formula for periodic spectra

The total nuclear level density is obtained for periodic single particle spectra. First, we express the canonical partition function in terms of Jacobi theta functions for degenerated fermion systems. Secondly, we use a relation for the number of representations of an integer number as sums of smaller numbers to find an exact formula for the nuclear level density. The shell structure is incorporated in the result in a simple way. An asymptotic relation can be easily obtained from the exact result and shown to be the well known back-shifted Bethe-formula.