Lesson plan
Patterson function
The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT. The Patterson function is defined as P(u,v,w) = \sum_{h, k, \ell\in\mathbb Z} \left|F_{h, k, \ell}\right|^2 \;e^{-2\pi i(hu + kv + \ell w)}. It is essentially the Fourier transform of the intensities rather than the structure factors. The Patterson function is also equivalent to the electron density convolved with its inverse: : P\left(\vec{u}\right) = \rho\left(\vec{r}\right) * \rho\left(-\vec{r}\right). Furthermore, a Patterson map of N points will have peaks, excluding the central (origin) peak and any overlap. The peaks' positions in the Patterson function are the interatomic distance vectors and the peak heights are proportional to the product of the number of electrons in the atoms concerned. Because for each vector between atoms i and j there is an oppositely oriented vector of the same length (between atoms j and i), the Patterson function always has centrosymmetry.
Article summary
The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT. The Patterson function is defined as P(u,v,w) = \sum_{h, k, \ell\in\mathbb Z} \left|F_{h, k, \ell}\right|^2 \;e^{-2\pi i(hu + kv + \ell w)}. It is essentially the Fourier transform of the intensities rather than the structure factors. The Patterson function is also equivalent to the electron density convolved with its inverse: : P\left(\vec{u}\right) = \rho\left(\vec{r}\right) * \rho\left(-\vec{r}\right). Furthermore, a Patterson map of N points will have peaks, excluding the central (origin) peak and any overlap. The peaks' positions in the Patterson function are the interatomic distance vectors and the peak heights are proportional to the product of the number of electrons in the atoms concerned. Because for each vector between atoms i and j there is an oppositely oriented vector of the same length (between atoms j and i), the Patterson function always has centrosymmetry.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.