Dielectric versus conductive behaviour in quantum gases: exact results for the hydrogen plasma

We study the electrical susceptibility of a hydrogen gas at equilibrium, partially ionized by thermal excitations. The gas is described as a quantum plasma of point protons and electrons, interacting via the Coulomb potential. Using the newly developed diagrammatical technique of screened cluster expansions, we calculate exactly the wavenumber-dependent susceptibility in the atomic limit, where most charges are bound into hydrogen atoms. A transition from conductive to dielectric behaviour occurs when the wavelength is decreased well below the Debye screening length. The standard formula for the dielectric function of an ideal gas of hydrogen atoms is recovered in an appropriate scaling limit. The derivation treats all effects arising from the Coulomb interaction (screening, binding, polarization) in a fully coherent way, without intermediate approximation nor modelization.