  The quantum statistical treatment of the Rutherford model, considering matter
as a system of point charges (electrons and nuclei) is analyzed. First, in the
historical context, the solutions of different fundamental problems, such as
the divergence of the partition function, elaborated by Herzfeld, Planck,
Brillouin and Rompe - most of the relevant papers published in the Annalen der
Physik, are discussed. Beyond this, the modern state of art is presented and
new results are given which explain, why bound states according to a discrete
part of the spectra occur only in a valley in the temperature-density plane.
Based on the actual state of the quantum statistics of Coulomb systems, virial
expansions within the canonical ensemble and the grand ensemble and
combinations are derived. The following transitions along isotherms are
studied: (i) the formation of bound states occurring by increasing the density
from low to moderate values, (ii) the disappearance of bound state effects at
higher densities due to medium effects. Within the physical picture we
calculate isotherms of pressure for Hydrogen in a broad density region and show
that in the region between $20\, 000$ K and $100\, 000$ K and particle
densities below $10^{22}$ cm$^{-3}$ the cross-over from full to partial
ionization may be well described by the contributions of extended ring diagrams
and ladder diagrams.
