Diverging correlation lengths in electrolytes: Exact results at low densities

The restricted primitive model of an electrolyte (equisized hard spheres ca rrying charges +/-q(0)) is studied using Meeron's expressions [J. Chem. Phy s. 28, 630 (1958)] for the multicomponent radial distribution functions g(s igma tau)(r;T,rho), that are correct through terms of relative order rho, t he overall density. The exact second and fourth moment density-density corr elation lengths xi(N,1)(T, rho) and xi(N,2)(T,rho), respectively, are there by derived for low densities: in contrast to the Debye length xi(D) = (k(B) T/44 pi q(0)(2)rho)(1/2), these diverge when rho-->0 as (T rho)(-1/4) and ( T/rho(3))(1/8), respectively, with universal amplitudes. The asymptotic exp ressions agree precisely with those obtained by Lee and Fisher [Phys. Rev. Lett. 76, 2906 (1996)] from a generalization of Debye-Huckel (GDH) theory t o nonuniform ion densities. Other aspects of this GDH theory are checked an d found to be exact at low densities. Specifically, with the further aid of the hypernetted-chain resummation, the corresponding charge-charge correla tion lengths xi(Z,1) and xi(Z,2) and the Lebowitz length, xi(L) (which rest ricts charge fluctuations in large domains), are calculated up to nonuniver sal terms of orders rho In rho and rho. In accord with the Stillinger-Lovet t condition, one finds xi(Z,1)=xi(D) although the ratios xi(Z,2)/xi(D) and xi(L)/xi(D) deviate from unity at nonzero rho.