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→The structure factor: added clarity about the intention of obtaining an averaged observable Tags: Mobile edit Mobile web edit |
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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} = \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}</math> around the position <math>\textstyle \mathbf{r}</math>.
We can formally count these particles
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) {{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}}}}
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