Diverging correlation lengths in electrolytes: Exact results at low densities

The restricted primitive model of an electrolyte (equisized hard spheres carrying charges +/-q(0)) is studied using Meeron's expressions [J. Chem. Phys. 28, 630 (1958)] for the multicomponent radial distribution functions g(sigma tau)(r;T,rho), that are correct through terms of relative order rho, the overall density. The exact second and fourth moment density-density correlation lengths xi(N,1)(T, rho) and xi(N,2)(T,rho), respectively, are thereby 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 expressions agree precisely with those obtained by Lee and Fisher [Phys. Rev. Lett. 76, 2906 (1996)] from a generalization of Debye-Huckel (GDH) theory to nonuniform ion densities. Other aspects of this GDH theory are checked and found to be exact at low densities. Specifically, with the further aid of the hypernetted-chain resummation, the corresponding charge-charge correlation lengths xi(Z,1) and xi(Z,2) and the Lebowitz length, xi(L) (which restricts charge fluctuations in large domains), are calculated up to nonuniversal terms of orders rho In rho and rho. In accord with the Stillinger-Lovett 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.