Evaluation of the Quantum-Mechanical Ring Sum with Boltzmann Statistics

A procedure is given for the evaluation of the quantum-mechanical ring sum at finite temperature. The method is used for the evaluation of the quantum corrections to the classical Debye-Hückel free energy for an electron gas obeying Boltzmann statistics. The ring sum is shown to be of the form (βe2/πλD)P(γ), where γ=ƛ/λD,ƛ=ℏ/(2mkT)1/2, and λD is the Debye screening length. The quantum effects for finite γ are due only to the operation of the uncertainty principle. The function P(γ) decreases monotonically from the classical value π/3, and the form is shown to be P(γ)=(π/3)1+ ∑ n=2anγ2(n−1)3/2− ∑ n=2bnγ2n−3. The coefficients an and bn are evaluated exactly for small n and asymptotically for large n. The two series converge for γ2 γc2 = 2.042 …. For γ ≫ γc the function P(γ) is also evaluated as an asymptotic expansion in inverse powers of γ1/2. Thus the low-temperature correlation energy of distinguishable electrons is obtained in random phase approximation. At zero temperature, the result is the same as the correlation energy obtained by Foldy for charged bosons. The correlation pressure in this approximation is negative and diverges as ρ1/4 at high density.