Radial Distribution Function for Hard Spheres

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In statistical mechanics, the distribution of interparticle separations determines the radial distribution function [1].


This Demonstration shows the radial distribution function of a three-dimensional liquid composed of identical hard spheres of diameter , making use of the exact solution of the Percus–Yevick integral equation [2, 3]. You can vary the packing fraction of the liquid, that is, the fraction of the total volume occupied by the spheres themselves. For a packing fraction greater than about 0.49, the corresponding can be regarded as that of a metastable liquid since the true stable phase is a crystal.

The Demonstration also includes three functions directly related to the radial distribution function : the structure factor , the direct correlation function [6], and the bridge function [7].


Contributed by: Andrés Santos (May 2013)
Open content licensed under CC BY-NC-SA



The exact solution of the Percus–Yevick integral equation for hard spheres was first obtained in [2] and [3]. The entirely analytic forms for the static structure factor were first presented in [4].


[1] Wikipedia. "Radial Distribution Function." (Mar 8, 2013) en.wikipedia.org/wiki/Radial_distribution_function.

[2] M. S. Wertheim, "Exact Solution of the Percus–Yevick Integral Equation for Hard Spheres," Physical Review Letters, 10(8), 1963 pp. 321–323. doi:10.1103/PhysRevLett.10.321.

[3] E. Thiele, "Equation of State for Hard Spheres," The Journal of Chemical Physics, 39(2), 1963 pp. 474–479. doi:10.1063/1.1734272.

[4] N. W. Ashcroft and J. Lekner, "Structure and Resistivity of Liquid Metals," Physical Review, 145(1), 1966 pp. 83–90.

[5] Wikipedia. "Structure factor." (May 9, 2013) en.wikipedia.org/wiki/Structure_factor.

[6] Wikipedia. "Ornstein–Zernike equation." (April 20, 2013) en.wikipedia.org/wiki/Ornstein% E2 %80 %93 Zernike_equation.

[7] SklogWiki. "Bridge function." (November 7, 2012) www.sklogwiki.org/SklogWiki/index.php/Bridge_function.

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