Radial distribution function: Difference between revisions

Revision as of 19:46, 28 February 2013

Revision as of 00:37, 8 March 2013

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===The structure factor===

The second-order correlation function <math>g^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2})</math> is of special importance, as it is directly related (via a [[Fourier transform]]) to the [[structure factor]] of the system and can thus be determined experimentally using [[X-ray diffraction]] or [[neutron diffraction]]. <ref>citation needed</ref>

If the system consists of spherically symmetric particles, <math>g^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2})</math> depends only on the relative distance between them, <math>\mathbf{r}_{12} = \mathbf{r}_{2} - \mathbf{r}_{1} </math>. We will drop the sub- and superscript: <math>\textstyle g(\mathbf{r})\equiv g^{(2)}(\mathbf{r}_{12})</math>. Taking particle 0 as fixed at the origin of the coordinates, <math>\textstyle \rho g(\mathbf{r}) \mathrm{d}\rm{r} = \mathrm{d} n (\mathbf{r})</math> is the number of particles (among the remaining <math>N-1</math>) to be found in the volume <math>\textstyle \mathrm{d}\mathbf{r}</math> around the position <math>\textstyle \mathbf{r}</math>.

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===The energy equation===

If the particles interact via identical pairwise potentials: <math>\textstyle U_{N} = \sum_{i > j = 1}^N u(\left | \mathbf{r}_i - \mathbf{r}_j \right |)</math>, the average internal energy per particle is:<ref>See Hansen & McDonald, section 2.5.</ref>:

{{NumBlk|:| <math>\frac{\left \langle E \right \rangle}{N} = \frac{3}{2} kT + \frac{\left \langle U_{N} \right \rangle}{N} = \frac{3}{2} kT + \frac{\rho}{2}\int_V \mathrm{d} \mathbf{r} \, u(r)g(r, \rho, T) </math>.|{{EquationRef|9}}}}