ABSENCE OF CRITICALITY IN THE REFERENCE HYPERNETTED-CHAIN EQUATION FOR SHORT RANGED POTENTIALS

In this note the reference hypernetted chain equation is solved for a finite ranged potential. For the model considered no solution to the i ntegral equation is found in a certain region of the phase diagram. To determine the presence of criticality in the no-solution line, the be haviour of the total correlation function h(r) at large distances is i nvestigated. For finite ranged potentials, the decay of the function r h(r) at large distances can be either exponential (at low and intermed iate densities) or exponentially damped oscillatory (at high densities ). For the model considered, the no-solution line falls within the reg ion 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. T he purely imaginary pole is analysed in the proximities of the no-solu tion line. It is shown that the correlation length remains finite when approaching the no-solution line. Therefore for the finite ranged pot ential considered in this work, the reference hypernetted chain equati on does not exhibit criticality.