  We characterize in a novel manner the physical properties of the low
temperature Fermi gas in the degenerate domain as a function of temperature and
chemical potential. For the first time we obtain low temperature $T$ results in
the domain where several fermions are found within a de Broglie spatial cell.
In this regime, the usual high degeneracy Sommerfeld expansion fails. The other
known semi-classical Boltzmann domain applies when fewer than one particle is
found in the de Broglie cell. We also improve on the understanding of the
Sommerfeld expansion in the regime where the chemical potential is close to the
mass and also in the high temperature regime. In these calculcations we use a
novel characterization of the Fermi distribution allowing the separation of the
finite and zero temperature phenomena. The relative errors of the three
approximate methods (Boltzmann limit, Sommerfeld expansion, and the new domain
of several particles in the de Broglie cell) are quantified.
