Statistical Mechanics of High-Temperature Quantum Plasmas Beyond the Ring Approximation

A quantum mechanical perturbation expansion of the partition function is used to evaluate the free energy of the electron gas and multicomponent plasmas to logarithmic accuracy in the particle number density, thus including the next important contribution beyond the ring approximation. The quantum generalization of Abe's work on the classical electron gas is given for the ladder interactions with the dynamic screened Coulomb potential, and each ladder is shown to be separately finite because of the finite size of the wave packets describing point electrons of the order of the thermal de Broglie wavelength ƛ = h̄(β/2m)1/2. The results show that quantum effects due to the uncertainty principle persist at high temperature, and that when kT Ryd plasmas are quantum systems, rather than classical, because ƛ is greater than the average distance of closest approach, βe2. Results are also obtained for the Wigner-Kirkwood wave mechanical diffraction corrections to the classical electron-gas free energy which are valid for low temperature (kT Ryd). The connection between the high- and low-temperature formula is discussed, and it is shown how the logarithmic divergence in the free energy that is cut off at βe2 in the low-temperature electron gas in the Abe theory is cut off at ƛ in the high-temperature case. Also it is shown that the quantum diffraction effects contained in the Montroll-Ward ring sum formula are valid only for kT Ryd, even though the quantum ring sum formula contains the classical Debye-Hückel result.