Radial distribution function: Difference between revisions

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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]].{{citation needed|date=December 2013}}

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}^3r = \mathrm{d} n (\mathbf{r})</math> is the ''average'' number of particles (among the remaining <math>N-1</math>) to be found in the volume <math>\textstyle \mathrm{d}\mathbf{r}^3r</math> around the position <math>\textstyle \mathbf{r}</math>.

We can formally count these particles and take the average via the expression <math>\textstyle \frac{\mathrm{d} n (\mathbf{r})}{d^3r} = \langle \sum_{i \neq 0} \delta ( \mathbf{r} - \mathbf{r}_i) \rangle</math>., with <math>\textstyle \langle \cdot \rangle</math> the ensemble average, yielding:

Taking the average we arrive at the expression <math>\textstyle \mathrm{d} n (\mathbf{r}) = \langle \sum_{i \neq 0} \delta ( \mathbf{r} - \mathbf{r}_i) \rangle</math>, with <math>\textstyle \langle \cdot \rangle</math> the ensemble average, yielding:

{{NumBlk|:| <math>g(\mathbf{r}) = \frac{1}{\rho} \langle \sum_{i \neq 0} \delta ( \mathbf{r} - \mathbf{r}_i) \rangle = V \frac{N-1}{N} \left \langle \delta ( \mathbf{r} - \mathbf{r}_1) \right \rangle</math>|{{EquationRef|5}}}}