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This chapter demonstrates that the experimental observable in diffraction analysis is the intensity of each diffraction spot which provides only the amplitude of the corresponding structure factor and not its phase — the fundamental phase problem in crystallography. To obtain an image of the molecule forming a crystal we need to calculate a Fourier transform, which requires that we know both the amplitude and phase of each structure factor. Nevertheless, very important information can be derived by calculating a ‘phase-less’ Fourier transform of the intensities alone, which is known as the Patterson function. Although this function is inherently more complex than an electron density map since it displays all inter-atomic vectors, certain sections of the Patterson function, known as Harker sections, can yield information on the positions of the most electron-rich atoms within the crystal. The Patterson function is exploited in most methods of the solving the phase problem for proteins and simple rules for the interpretation of a Patterson function are derived.

Starting from a definition of the relationship between a crystal structure and its Patterson function, this chapter considers the role of the Patterson function in the particular context of powder diffraction. The limitations of powder data and their effects upon the appearance of derived Patterson maps are discussed, as are ‘conventional’ and ‘non-conventional’ (e.g., maximum entropy) methods for improving their interpretability. The utility of the Patterson function in the context of improving intensity estimates for overlapping reflections is considered, as is the use of automated procedures for the location of atomic positions based on Patterson map superposition.

The two methods to be described here, the Patterson method and the isomorphous replacement method, have made it possible to determine the three-dimensional structures of large biological molecules such as proteins and nucleic acids. In addition, the Patterson function is still useful for small-molecule studies if problems are encountered during the structure analysis. If a crystal structure determination proves to be difficult, the Patterson map should be determined to see if it is consistent with the proposed trial structure. The Patterson method involves a Fourier series in which only the indices (h, k, l) and the |F (hkl)|2 value of each diffracted beam are required (Patterson, 1934, 1935). These quantities can be obtained directly by experimental measurements of the directions and intensities of the Bragg reflections. The Patterson function, P(uvw), is defined in Eqn. (9.1).