Quantum plasmas with or without a uniform magnetic field. II. Exact low-density free energy

The exact analytical expression of the free energy f of a quantum Coulomb plasma in the presence of a uniform magnetic field B-0 is produced at low density rho. This regime corresponds to low degeneracy, weak Coulomb coupling but any strength of the magnetic field and fully quantum dynamics. Thus f is expanded around its value for an ideal gas in the Maxwell-Boltzmann (MB) approximation which provides a description of orbital diamagnetism and Pauli paramagnetism. The rho expansion for f is derived from an adequate Mayer diagrammatic representation of the ratio between the plasma density and the density of the ideal gas with the same chemical potentials and MB statistics. A systematic scaling analysis of the dependence of Mayer bonds upon density is devised. This provides a natural truncation of the trace of the two-body Gibbs factor as well as diffraction contributions specific to the long range of the Coulomb potential. The rho(3/2) term in f is the purely classical Debye contribution. From order rho(2) on, B-0 is involved through quantum dynamical and statistical effects which are the root of ferromagnetism. (Moreover, we retrieve the purely classical contributions at order rho(5/2) in a very compact form.) Our results are compared with semiclassical expressions in the case of the one-component plasma. [S1063-651X(98)02710-X].