Patterson functions

Fourier transforms

We will use the following convention for the Fourier transforms:\begin{equation} \begin{split} \rho(\mathbf{q}) = \mathcal{F} [\rho(\mathbf{r})](\mathbf{q}) & \triangleq \int_{\mathcal{V}} \rho(\mathbf{r}) \exp(-i \mathbf{q} \mathbf{r}) {\textrm d} \mathbf{r} \\ \rho(\mathbf{r}) = \mathcal{F}^{-1} [\tilde{\rho}(\mathbf{q})](\mathbf{r}) & \triangleq \dfrac{1}{(2\pi)^3}\int_{\mathbb{R}^3} \tilde{\rho}(\mathbf{q}) \exp(i \mathbf{q} \mathbf{r}) {\textrm d} \mathbf{q} \end{split} \label{eq:Fourierdef} \end{equation} 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 \(P(\mathbf{r})\).