Radial distribution function: Difference between revisions

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In [[computational mechanics]] and [[statistical mechanics]], a '''radial distribution function''' (RDF), ''g''(''r''), describes how the density of surrounding matter varies as a function of the distance from a particular point.

Suppose, for example, that we choose a molecule at some point O in the volume. What is then the average density at some point P at a distance r away from O? If <math>\rho=N/V</math> is the average density, then the mean density at P ''given'' that there is a molecule at O would differ from &rho;ρ by some factor g(r). One could say that the radial distribution function takes into account the correlations in the distribution of molecules arising from the forces they exert on each other:

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in which additional functions <math>g_{1}(r), \, g_{2}(r)</math> appear which may depend on temperature <math>T</math> and distance <math>r</math> but not on density, <math>\rho</math>.

Given a [[potential energy]] function, the radial distribution function can be found via computer simulation methods like the [[Monte Carlo method]]. It could also be calculated numerically using rigorous methods obtained from [[statistical mechanics]] like the [[Perckus-Yevick approximation]], or the [[Hypernetted-chain_equation|chain equation|Hypernetted Chain Theory]].

==Importance of g(r)==

g(r) is of fundamental importance in thermodynamics because macroscopic thermodynamic quantities can be calculated using g(r). A few examples:

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==Experimental==

It is possible to measure g(r) experimentalyexperimentally with [[neutron scattering]] or [[x-ray scattering]] diffraction data. In such an experiment, a sample is bombarded with neutrons or x-rays which then diffract in all directions. The average molecular density at each distance can be extracted according to [[Snells law]]: r=wavelength/sin(scattered angle), where r is the distance the neutron traveled during diffraction.<br />

For an example of an RDF experiment see [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000125000001014508000001&idtype=cvips&gifs=yes Eigen vs. Zundel structures in HCl solution, 2006]

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#D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976

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[[it:Funzione di distribuzione radiale]]