Absence of criticality in the reference hypernetted chain equation for short ranged potentials

Abstract

In this note the reference hypernetted chain equation is solved for a finite ranged potential. For the model considered no solution to the integral equation is found in a certain region of the phase diagram. To determine the presence of criticality in the no-solution line, the behaviour of the total correlation function h(r) at large distances is investigated. For finite ranged potentials, the decay of the function rh(r) at large distances can be either exponential (at low and intermediate densities) or exponentially damped oscillatory (at high densities). For the model considered, the no-solution line falls within the region where the function rh(r) shows exponential decay. The correlation length in the region where the decay of rh(r) is exponential is given by the inverse of the purely imaginary pole of the structure factor. The purely imaginary pole is analysed in the proximities of the no-solution line. It is shown that the correlation length remains finite when approaching the no-solution line. Therefore for the finite ranged potential considered in this work, the reference hypernetted chain equation does not exhibit criticality.