Ornstein-Zernike equation for a two-Yukawac(r) with core condition

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Abstract
The work begun in an earlier paper [Høye, J. S., Stell, G., and Waisman, E., 1976, Molec. Phys., 32, 209] is continued. The nature of the singularity that defines the critical point of the fluid described by the Ornstein-Zernike equation with a direct correlation function c(r) of two-Yukawa form is used to simplify the algebra relating the parameters appearing in its solution. In addition, solutions under the core conditions h(r) = -1 and h(r) = 0 for r < 1, where h(r) is the pair correlation function, are related to one another through the study of the more general core condition h(r) = -ε for r < 1. This relation elucidates a remarkable functional linearity in the solution of the equation as well as defining a generalization of the ‘permeable-sphere model’ used in previous work by one of the authors. The above results represent an important ingredient in the analytic simplification of a two-Yukawa generalized mean spherical approximation and is used in that context in a companion paper. They also permit simplified treatment of the conditions of continuity on h(r) and dh/dr at r = 1, which appear in the generalized permeable-sphere model and various other applications. Finally they facilitate the evaluation of critical exponents for the mean spherical approximation with a c(r) of two-Yukawa form for r > 1. In particular, we find δ = 5 and γ = 2 via the compressibility equation, as expected.