Content deleted Content added
Mahdisadjadi (talk | contribs) aligning a long equation |
|||
Line 58:
where the second equality requires the equivalence of particles <math>\textstyle 1, \, \ldots, \, N-1</math>. The formula above is useful for relating <math>g(\mathbf{r})</math> to the static structure factor <math>S(\mathbf{q})</math>, defined by <math>\textstyle S(\mathbf{q}) = 1/N \langle \sum_{ij} \mathrm{e}^{-i \mathbf{q} (\mathbf{r}_i - \mathbf{r}_j)} \rangle</math>, since we have:
: <math>
: <math>S(\mathbf{q}) = 1 + \frac{1}{N} \langle \sum_{i \neq j} \mathrm{e}^{-i \mathbf{q} (\mathbf{r}_i - \mathbf{r}_j)} \rangle = 1 + \frac{1}{N} \left \langle \int_V \mathrm{d} \mathbf{r} \, \mathrm{e}^{-i \mathbf{q} \mathbf{r}} \sum_{i \neq j} \delta \left [ \mathbf{r} - (\mathbf{r}_i - \mathbf{r}_j) \right ] \right \rangle = 1+ \frac{N(N-1)}{N} \int_V \mathrm{d} \mathbf{r}\, \mathrm{e}^{-i \mathbf{q} \mathbf{r}} \left \langle \delta ( \mathbf{r} - \mathbf{r}_1 ) \right \rangle </math>, and thus:▼
\begin{align}
▲
\end{align}
</math>
, and thus:
<math>S(\mathbf{q}) = 1 + \rho \int_V \mathrm{d} \mathbf{r} \, \mathrm{e}^{-i \mathbf{q} \mathbf{r}} g(\mathbf{r})</math>, proving the Fourier relation alluded to above.
| |||