Asymptotic decay of pair correlations in a Yukawa fluid

We analyze the $r\to \mathrm{\infty }$ asymptotic decay of the total correlation function $h\left(r\right)$ for a fluid composed of particles interacting via a (point) Yukawa pair potential. Such a potential provides a simple model for dusty plasmas. The asymptotic decay is determined by the poles of the liquid structure factor in the complex plane. We use the hypernetted-chain closure to the Ornstein-Zernike equation to determine the line in the phase diagram, well removed from the freezing transition line, where crossover occurs in the ultimate decay of $h\left(r\right)$, from monotonic to damped oscillatory. We show that (i) crossover takes place via the same mechanism (coalescence of imaginary poles) as in the classical one-component plasma and in other models of Coulomb fluids and (ii) leading-order pole contributions provide an accurate description of $h\left(r\right)$ at intermediate distances $r$ as well as at long range.

Figure 1

Crossover line (diamonds joined by a solid line) separating the region of the $(\kappa ,\mathrm{\Gamma })$ plane where the asymptotic decay of $h\left(r\right)$ is damped oscillatory from that where it is monotonic. In the OCP, corresponding to $\kappa =0$, crossover occurs at ${\mathrm{\Gamma }}_K=1.21$ [22]. We also display the fluid-solid (+) and solid-solid (×) phase boundaries given in Ref. 16. In the inset we display the crossover line in the $(\rho ,T)$ plane. Note that the freezing transition present at low $T^{*}\equiv k_{B}T∕ϵ$ is not visible on this scale.

Figure 2

Pair correlation functions for a reduced temperature $T^{*}=1$. (a) $g\left(r\right)$ calculated for a range of densities; the lines denote HNC results and the symbols MC results (see legend). Crossover occurs at $\rho {\lambda }^{-3}\simeq 0.47$ for this temperature. For densities larger than this the decay of $g\left(r\right)$ is damped oscillatory. (b) Comparison between the full HNC result for $h\left(r\right)$ (solid line) and the contribution from the leading-order single imaginary pole (dashed line) for $\rho {\lambda }^{-3}=0.1$. (c) The full HNC result for $h\left(r\right)$ (solid line) and the contributions from the two leading-order purely imaginary poles (dotted and dashed lines) and their sum (dot-dashed line), for $\rho {\lambda }^{-3}=0.45$. (d) The full HNC result for $h\left(r\right)$ (solid line) and the contribution from the leading-order conjugate pair of complex poles (dotted line), giving damped oscillatory decay, for $\rho {\lambda }^{-3}=40$. The insets in (c) and (d) show ln $\left[r\mid h\left(r\right)\mid \right]$ versus $\lambda r$.

Figure 3

The leading-order poles (smallest imaginary part, ${\alpha }_0$) along the isotherm $T^{*}=1$ for increasing $\rho {\lambda }^{-3}$. Only poles with positive ${\alpha }_1$ are shown; complex poles occur in conjugate pairs. At low densities the leading order pole (×) is purely imaginary. As the density is increased a second purely imaginary pole (+) descends and, at a density $\rho {\lambda }^{-3}\simeq 0.47$, the two purely imaginary poles coalesce to form a pair of complex poles (◻). At this (Kirkwood) point the asymptotic decay of $h\left(r\right)$ crosses over from monotonic to damped oscillatory.