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.

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

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 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:

Suppose, for example, that we choose aan moleculeatom 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 aan moleculeatom at O would differ from ρ by some factor g(r). One could say that the radial distribution function takes into account the correlations in the distribution of moleculesatoms 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 moleculesatoms that g(r) takes into account are due to the potential <math>\phi</math>(r) that aan moleculeatom at P feels owing to the presence of aan moleculeatom 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 moleculesatoms 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 aan moleculeatom at O would not influence the presence or absence of any other moleculeatom 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 moleculesatoms.

[[File:Lennard-Jones Radial Distribution Function.svg|thumb|600px300px|Radial distribution function for the [[Lennard–Jones potential|Lennard-Jones model fluid]] at <math>T^* = 0.71, \; n^* = 0.844</math>.]]

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

[[File:Lennard-Jones Radial Distribution Function.svg|thumb|600px|Radial distribution function for [[Lennard–Jones potential|Lennard-Jones fluid]] at <math>T^* = 0.71, \; n^* = 0.844</math>.]]

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) resemble the [[virial equation]], which is an expansion in the density:

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==Experimental==

It is possible to measure g(r) experimentally 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 molecularatomic 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]

==Formal derivation==

Consider a system of ''N'' particles in a volume ''V'' and at a temperature ''T''. The probability of finding moleculeatom 1 in <math>d \rm{r}_{1}</math>, moleculeatom 2 in <math>d \rm{r}_{2}</math>, etc., is given by

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where <math> \beta = \frac{1}{kT}</math> and <math>Z_{N}</math> is the configurational integral. To obtain the probability of finding moleculeatom 1 in <math>d \rm{r}_{1}</math> and moleculeatom ''n'' in <math>d \rm{r}_{n}</math>, irrespective of the remaining ''N-n'' moleculesatoms, one has to integrate (7) over the coordinates of moleculeatom ''n'' + 1 through ''N'':

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Now the probability that ''any'' moleculeatom is in <math>d \rm{r}_{1}</math> and ''any'' moleculeatom in <math>d \rm{r}_{n}</math>, irrespective of the rest of the moleculesatom, is

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<math>g^{(n)}</math> is called a correlation function, since if the moleculesatoms 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 moleculesatoms.

From (9) it can be shown that

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In the theory of liquids <math>g^{(2)}(\rm{r}_{1},\rm{r}_{2})</math> is of special importance for it can be determined experimentally using [[X-ray diffraction]]. If the liquid contains spherically symmetric moleculesatoms <math>g^{(2)}(\rm{r}_{1},\rm{r}_{2})</math> depends only on the relative distance between moleculesatoms, <math>\rm{r}_{12}</math>. People usually drop the subscripts: <math> g(r)=g^{(2)}(r_{12})</math>. Now <math> \rho g(r) d\rm{r}</math> is the probability of finding aan atom molecule at '''r''' given that there is aan moleculeatom at the origin of '''r'''. Note that this probability is not normalized:

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

Currently, 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 available, and scientists rely on approximations based upon [[statistical mechanics]].