16 May 2015

The structure factor of a liquid - part II

[Continuing the preliminary discussion started in part I.]
We are now interested in an explicit form for \(g(\mathbf{r})\) (we return here to the general case —where \(g\) depends on the full vector \(\mathbf{r}\), and not only on its modulus— simply to avoid the radial integrals). Taking particle 0 as fixed in \(\mathbf{r}_0\), \(\rho g(\mathbf{r}) {\text{d}}^D \mathbf{r} = \text{d} n (\mathbf{r} - \mathbf{r}_0)\) is the number of particles (among the remaining \(N-1\)) found in the volume \({\text{d}}^D \mathbf{r}\) positioned at \(\mathbf{r}\) with respect to the reference particle. One can formally count these particles by writing:

\[ \text{d} n (\mathbf{r}) = \left \langle \sum_{i \neq 0} \delta \left [ \mathbf{r} - (\mathbf{r}_i - \mathbf{r}_0) \right ] \right \rangle {\text{d}}^D \mathbf{r} \]
leading to:
g(\mathbf{r}) = \frac{1}{\rho} \left \langle \sum_{i \neq 0} \delta \left [ \mathbf{r} - (\mathbf{r}_i - \mathbf{r}_0) \right ] \right \rangle = V \frac{N-1}{N} \left \langle \delta \left [ \mathbf{r} - (\mathbf{r}_1 - \mathbf{r}_0) \right ] \right \rangle
where the second step derives from the equivalence between particles \(1,\ldots,N-1\).
The radial distribution function is accessible indirectly, through scattering techniques, which yield the static structure factor \(S(q)\), proportional to the intensity scattered in point \(\mathbf{q}\) of reciprocal space:
S(\mathbf{q}) &=& \frac{1}{N} \left \langle \sum_{ij} \mathrm{e}^{-i \mathbf{q} (\mathbf{r}_i - \mathbf{r}_j)}\right \rangle = 1 + \frac{1}{N} \left \langle \sum_{i \neq j} \mathrm{e}^{-i \mathbf{q} (\mathbf{r}_i - \mathbf{r}_j)}\right \rangle \nonumber\\
&=& 1 + \frac{1}{N} \left \langle \int_V {\text{d}}^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 {\text{d}}^D \mathbf{r}\, \mathrm{e}^{-i \mathbf{q} \mathbf{r}} \left \langle \delta \left [ \mathbf{r} - (\mathbf{r}_1 - \mathbf{r}_0) \right ] \right \rangle = 1 + \rho \int {\text{d}}^D \mathbf{r}\, \mathrm{e}^{-i \mathbf{r} \mathbf{q}} \, g(\mathbf{r})\nonumber
where we invoked once more the equivalence between particles.

The structure factor is simply the Fourier transform (in the sense of distributions) of the radial distribution function. The "distribution" caveat is due to \(\displaystyle \lim_{r \rightarrow \infty} g(r) = 1\), which results in a Dirac peak at \(S(\mathbf{q} = 0)\). This contribution is generally inaccessible in experiments (I will discuss in a future post when and why it can become important), and one generally subtracts it from both sides of equation (2), turning the structure factor into a regular function:
S'(\mathbf{q}) = S(\mathbf{q}) - \rho \delta (\mathbf{q})= 1 + \rho \int {\text{d}}^D \mathbf{r}\, [g(\mathbf{r}) - 1] \mathrm{e}^{-i \mathbf{r} \mathbf{q}}
Finally, one can drop the prime symbol and bring back the isotropy of the system:
S(q) -1 = \rho \int {\text{d}}^D \mathbf{r}\, [g(r) - 1] \mathrm{e}^{i \mathbf{r} \mathbf{q}} = \rho h(q)
with \(h(q)\) the Fourier transform of \(h(r) = g(r) - 1\). \(h(r)\) is often called the total correlation function. Relation (\ref{eq:Sqfinal}) is now a Fourier transform over regular functions, provided \(h(r)\) is integrable, a condition fulfilled in the case of liquids, where the correlation range is finite.

The final result (\ref{eq:Sqfinal}) was derived rather quickly, without developing e.g. the ensemble average \(\left \langle \cdot \right \rangle\). For more details, refer to [1]. Alternatively, the same relation can be obtained via the general formalism of response functions [2], noting that \(g(r)\) is a correlation and \(S(q)\) the associated spectral density. They are paired by a Fourier transform via the Wiener-Khinchine theorem.

1. D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University Press, New York, 1987), section 7.5.
2. P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, 2000) section 2.3), section 2.3.

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