Radial distribution function: Difference between revisions

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Revision as of 16:26, 2 December 2008

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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 distinguishedparticular 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; 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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As long as the gas is '''dilute''', the correlations in the positions of the molecules that g(r) takes into account are due to the potential <math>\phi</math>(r) that a molecule at P feels owing to the presence of a molecule at O. Using the Boltzmann distribution law:

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If <math>\phi(r)</math> was zero for all r - i.e., if the molecules did not exert any influence on each other, then g(r) = 1 for all r. Then from (1) the mean local density would be equal to the mean density <math>\rho</math>: the presence of a molecule at O would not influence the presence or absence of any other molecule and the gas would be ideal. As long as there is a <math>\phi(r)</math> the mean local density will always be different from the mean density <math>\rho</math> due to the interactions between molecules.

When the density of the gas gets higher, the so called low-density limit (2) is not applicable anymore because the molecules attracted to and repelled by the molecule at O also repel and attract each other. The correction terms needed to correctly describe g(r) resemblesresemble the [[virial equation]], itwhich is an expansion in the density:

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==Importance of g(r)==

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

''The virial equation for the pressure:''

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==Experimental==

It is possible to measure g(r) experimentaly with [[neutron scattering]] or [[x-ray scattering]] diffraction data. In such an experiment, a sample is bombarded with neutrons or x-rayrays which then diffract toin all directions. The average molecular density at each distance can be extracted in 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]

==Formal derivation==

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For ''n'' = 1 the one -particle distribution function is obtained which, for a crystal, is a periodic function with sharp maxima at the lattice sites. For a (homogeneous) liquid:

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<math>g^{(n)}</math> is called a correlation function, since if the molecules are independent from each other <math>\rho^{(n)}</math> would simply equal <math>\rho^{n}</math> and therefore <math>g^{(n)}</math> corrects for the correlation between molecules.

From (9) it can be shown that

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In fact, equation 13 gives us the number of molecules between r and r + d r about a central molecule.

As of currentCurrently, information on how to obtain the higher order distribution functions (<math>g^{(3)}(\rm{r}_{1},\rm{r}_{2},\rm{r}_{3})</math>, etc.) is not knownavailable, and scientists rely on approximations based upon [[statistical mechanics]].

==References==