Radial distribution function: Difference between revisions
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In classical mechanics, and on the assumption of additivity of intermolecular forces, one has | In classical mechanics, and on the assumption of additivity of intermolecular forces, one has | ||
:<math>{\rm g}_1 (r_{12})= \int f (r_{13}) f(r_{23}) ~{\rm d} | :<math>{\rm g}_1 (r_{12})= \int f (r_{13}) f(r_{23}) ~{\rm d}{\mathbf r}_3</math> | ||
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+ 2\psi (r_{12}) + \frac{1}{2} \chi (r_{12})</math> | + 2\psi (r_{12}) + \frac{1}{2} \chi (r_{12})</math> | ||
where <math>r_{ik}</math> is the distance <math>| | where <math>r_{ik}</math> is the distance <math>|{\mathbf r}_i -{\mathbf r}_k|</math>, where <math>f(r)</math> | ||
is the [[Mayer f-function]] | is the [[Mayer f-function]] | ||
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and | and | ||
:<math>\varphi (r_{12}) = \int f (r_{13}) f (r_{24}) f (r_{34}) ~ {\rm d} | :<math>\varphi (r_{12}) = \int f (r_{13}) f (r_{24}) f (r_{34}) ~ {\rm d}{\mathbf r}_3 {\rm d}{\mathbf r}_4</math> | ||
:<math>\psi (r_{12}) = \int f (r_{13}) f (r_{23}) f (r_{24}) f (r_{34}) ~ {\rm d} | :<math>\psi (r_{12}) = \int f (r_{13}) f (r_{23}) f (r_{24}) f (r_{34}) ~ {\rm d}{\mathbf r}_3 {\rm d}{\mathbf r}_4</math> | ||
:<math>\chi (r_{12}) = \int f (r_{13}) f (r_{23}) f (r_{14}) f (r_{24}) f (r_{34}) ~ {\rm d} | :<math>\chi (r_{12}) = \int f (r_{13}) f (r_{23}) f (r_{14}) f (r_{24}) f (r_{34}) ~ {\rm d}{\mathbf r}_3 {\rm d}{\mathbf r}_4</math> | ||
==References== | ==References== | ||
#[http://dx.doi.org/10.1063/1.1723737 John G. Kirkwood and Elizabeth Monroe Boggs "The Radial Distribution Function in Liquids", Journal of Chemical Physics '''10''' pp. 394-402 (1942)] | #[http://dx.doi.org/10.1063/1.1723737 John G. Kirkwood and Elizabeth Monroe Boggs "The Radial Distribution Function in Liquids", Journal of Chemical Physics '''10''' pp. 394-402 (1942)] |
Revision as of 15:27, 10 July 2007
The radial distribution function is a special case of the pair distribution function for an isotropic system. A Fourier transform of the radial distribution function results in the structure factor, which is experimentally measure able.
Density Expansion of the radial distribution function
The radial distribution function of a compressed gas may be expanded in powers of the density (Ref. 2)
where is the number of molecules per unit volume and is the intermolecular pair potential. The function is normalized to the value 1 for large distances. As is known, , , ... can be expressed by cluster integrals in which the position of of two particles is kept fixed. In classical mechanics, and on the assumption of additivity of intermolecular forces, one has
where is the distance , where is the Mayer f-function
and
References
- John G. Kirkwood and Elizabeth Monroe Boggs "The Radial Distribution Function in Liquids", Journal of Chemical Physics 10 pp. 394-402 (1942)
- B. R. A. Nijboer and L. Van Hove "Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation", Physical Review 85 pp. 777 - 783 (1952)