  We consider a partially ionized gas at thermal equilibrium, in the Saha
regime. The system is described in terms of a quantum plasma of nuclei and
electrons. In this framework, the Coulomb interaction is the source of a large
variety of phenomena occuring at different scales: recombination, screening,
diffraction, etc. In this paper, we derive a cluster expansion adequate for a
coherent treatment of those phenomena. The expansion is obtained by combining
the path integral representation of the quantum gas with familiar Mayer
diagrammatics. In this formalism, graphs have a clear physical interpretation:
vertices are associated with recombined chemical species, while bonds describe
their mutual interactions. The diagrammatical rules account exactly for all
effects in the medium. Applications to thermodynamics, van der Waals forces and
dielectric versus conductive behaviour will be presented in forthcoming papers.
