Fourier transforms
We will use the following convention for the Fourier transforms:ρ(q)=F[ρ(r)](q)≜ where we integrate in real space over the (as yet unspecified) volume of interest \mathcal{V} and in reciprocal space over the entire \mathbb{R}^3.
Wiener-Khinchine theorem
The autocorrelation of the real-space density function is \Gamma_{\rho \rho} = \int_{\mathcal{V}} \rho(\mathbf{r}') \rho(\mathbf{r}'+\mathbf{r}) {\textrm d} \mathbf{r}', which can be developed (using the second line of \eqref{eq:Fourierdef}) into:\begin{equation}
\begin{split}
\Gamma_{\rho \rho}(\mathbf{r}) & = \dfrac{1}{(2\pi)^6} \int_{\mathcal{V}} {\textrm d} \mathbf{r}' \rho(\mathbf{r}') \int_{\mathbb{R}^3} {\textrm d} \mathbf{q} \, \tilde{\rho}(\mathbf{q}) \exp(i \mathbf{q} \mathbf{r}') \int_{\mathbb{R}^3} {\textrm d} \mathbf{q}' \tilde{\rho}(\mathbf{q}') \exp[i \mathbf{q}' (\mathbf{r}' + \mathbf{r})] \\
& = \dfrac{1}{(2\pi)^6} \int_{\mathbb{R}^3} {\textrm d} \mathbf{q} \int_{\mathbb{R}^3} {\textrm d} \mathbf{q}' \tilde{\rho}(\mathbf{q}) \tilde{\rho}(\mathbf{q}') \exp(i \mathbf{q}' \mathbf{r}) \underbrace{\int_{\mathcal{V}} {\textrm d} \mathbf{r}' \exp[i (\mathbf{q} + \mathbf{q}') \mathbf{r} ]}_{(2\pi)^3 \delta (\mathbf{q} + \mathbf{q}')} \\
& = \dfrac{1}{(2\pi)^3} \int_{\mathbb{R}^3} {\textrm d} \mathbf{q}' \exp(i \mathbf{q}' \mathbf{r}) \tilde{\rho}(\mathbf{q}') \underbrace{\int_{\mathbb{R}^3} {\textrm d} \mathbf{q} \, \tilde{\rho}(\mathbf{q}) \delta (\mathbf{q} + \mathbf{q}')}_{\tilde{\rho}(-\mathbf{q}')}
\end{split}
\end{equation} where we assumed that everything converges, and thus we can interchange the integration order at will. Dropping the prime and noting that \tilde{\rho}(-\mathbf{q}) = \overline{\tilde{\rho}(\mathbf{q})} (Friedel's law) we finally prove the Wiener-Khinchine theorem: the autocorrelation function of the scattering length density is the inverse Fourier transform of its spectral density: \begin{equation}
\Gamma_{\rho \rho}(\mathbf{r}) = \dfrac{1}{(2\pi)^3} \int_{\mathbb{R}^3} {\textrm d} \mathbf{q} \exp(i \mathbf{q} \mathbf{r}) \left | \tilde{\rho}(\mathbf{q}) \right |^2 = \mathcal{F}^{-1} [|\tilde{\rho}(\mathbf{q})|^2]
\label{eq:WK}
\end{equation}
The Patterson function
As discussed during the lecture, the scattered intensity is precisely the spectral density of \rho(\mathbf{r}): I(\mathbf{q}) = \left | \tilde{\rho}(\mathbf{q}) \right |^2. Unlike \rho(\mathbf{r}) itself, its autocorrelation \Gamma_{\rho \rho}(\mathbf{r}) is directly accessible via Fourier transform from the experimental data, provided their quality and q-range are sufficient. Since it is frequently used in crystallography, it has a specific name: the Patterson function, denoted by P(\mathbf{r}).
