*Dennis Sherwood and Jon Cooper*

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199559046
- eISBN:
- 9780191595028
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199559046.003.0012
- Subject:
- Physics, Crystallography: Physics

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 ...
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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.Less

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.

*William Clegg*

- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199219469
- eISBN:
- 9780191722516
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219469.003.0009
- Subject:
- Physics, Crystallography: Physics

The Patterson synthesis is a reverse Fourier transform in which the amplitudes are replaced by their squares and the phases are omitted. Peaks in a Patterson map represent vectors between pairs of ...
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The Patterson synthesis is a reverse Fourier transform in which the amplitudes are replaced by their squares and the phases are omitted. Peaks in a Patterson map represent vectors between pairs of atoms; the Patterson function is the convolution of the electron density with itself. Patterson space groups are the combinations of Laue classes with unit cell centring arrangements. The large number of broad Patterson peaks leads to much overlap, with peaks due to pairs of heavy atoms (many electrons) dominating. Heavy atoms may thus often be found by Patterson map analysis, especially when they are related by symmetry, giving peaks in special positions (Harker sections and lines). Examples are shown in several space groups. Care is needed in some cases, when ambiguous solutions may be found. Geometrically rigid groups of atoms may also be located from a Patterson map by combined rotational and translational search methods.Less

The Patterson synthesis is a reverse Fourier transform in which the amplitudes are replaced by their squares and the phases are omitted. Peaks in a Patterson map represent vectors between pairs of atoms; the Patterson function is the convolution of the electron density with itself. Patterson space groups are the combinations of Laue classes with unit cell centring arrangements. The large number of broad Patterson peaks leads to much overlap, with peaks due to pairs of heavy atoms (many electrons) dominating. Heavy atoms may thus often be found by Patterson map analysis, especially when they are related by symmetry, giving peaks in special positions (Harker sections and lines). Examples are shown in several space groups. Care is needed in some cases, when ambiguous solutions may be found. Geometrically rigid groups of atoms may also be located from a Patterson map by combined rotational and translational search methods.