Radial distribution function: Difference between revisions

Revision as of 08:13, 24 November 2009

Revision as of 11:50, 24 November 2009

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(mean local density at distance r from O) = <math>\rho</math>g(r) (1)

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

<center>

<math>g(r) = e^{-\phi(r)/kT} \,</math> (2)

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If <math>\phi(r)</math> was zero for all r - i.e., if the atoms 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 an atom at O would not influence the presence or absence of any other atom 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 atoms.

[[File:Lennard-Jones Radial Distribution Function.svg|thumb|300px|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 atoms attracted to and repelled by the atom 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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<math>g(r)=e^{-\phi(r)/kT}+\rho g_{1}(r)+\rho^{2}g_{2}(r)+\ldots</math> (3)

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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|Hypernetted Chain Theory]].

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''The virial equation for the pressure:''

:<math>p=\rho kT-\frac{2\pi}{3}\rho^{2}\int d r r^{3} u^{\prime}(r) g(r, \rho, T) </math> (4)

''The energy equation:''

:<math>\frac{E}{NkT}=\frac{3}{2}+\frac{\rho}{2kT}\int d r \,4\pi r^{2} u(r)g(r, \rho, T) </math> (5)

''[[Compressibility equation|The compressibility equation]]:''

:<math>kT\left(\frac{\partial \rho}{\partial p}\right)=1+\rho \int d r [g(r)-1] </math> (6)

==Gases==

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

<center>

<math>g(r) = e^{-\phi(r)/kT} \,</math> (2)

</center>

If <math>\phi(r)</math> was zero for all r - i.e., if the atoms 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 an atom at O would not influence the presence or absence of any other atom 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 atoms.

When the density of the gas gets higher, the so called low-density limit (2) is not applicable anymore because the atoms attracted to and repelled by the atom 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:

<center>

<math>g(r)=e^{-\phi(r)/kT}+\rho g_{1}(r)+\rho^{2}g_{2}(r)+\ldots</math> (3)

</center>

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>.

==Experimental==

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<math> P^{(N)}(\rm{r}_{1},\ldots,\rm{r}_{N}) d \rm{r}_{1}\cdots d \rm{r}_{N}=\frac{e^{-\beta U_{N}}d\rm{r}_{1} \cdots d \rm{r}_{N}}{Z_{N}} \, </math> (7)

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<math> P^{(n)}(\rm{r}_{1},\ldots,\rm{r}_{n}) =\frac{\int \cdots \int e^{-\beta U_{N}}d\rm{r}_{n+1} \cdots d \rm{r}_{N}}{Z_{N}} \, </math> (8)

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<math>\rho^{(n)}(\rm{r}_{1},\ldots,\rm{r}_{n}) =\frac{N!}{(N-n)!} \cdot P^{(n)}(\rm{r}_{1},\ldots,\rm{r}_{n}) \, </math> (9)

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<math> \frac{1}{V} \int \rho^{(1)}(\rm{r}_{1})d \rm{r}_{1}=\rho^{(1)}=\frac{N}{V}=\rho \,</math> (10)

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<math>\rho^{(n)}(\rm{r}_{1},\ldots,\rm{r}_{n})=\rho^{n}g^{(n)}(\rm{r}_{1},\ldots,\rm{r}_{n}) \, </math> (11)

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<math>g^{(n)}(\rm{r}_{1},\ldots,\rm{r}_{n})=\frac{V^{n}N!}{N^{n}(N-n)!}\cdot\frac{\int \cdots \int e^{-\beta U_{N}}d\rm{r}_{n+1} \cdots d \rm{r}_{N}}{Z_{N}} \, </math> (12)

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<math>\int_{0}^{\infty}\rho g(r) 4\pi r^{2} dr = N-1 \approx N </math> (13)

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In fact, equationthis 13equation gives us the number of atoms between r and r + d r about a central atom.

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]].