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Fourier Techniques in X-Ray Structure Determination

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Structure Determination by X-ray Crystallography

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Abstract

We have reached the stage where we can consider how to attack the solving of a crystal structure. After the earliest trial and error determinations in the 1920s with very simple and highly symmetrical structures, it was found that the application of Fourier series, initially in one dimension, led to the electron density function, in which peak maxima in the electron density corresponded to atomic positions. As we have seen in the previous chapters, it is necessary to have the phases of the structure factors for a Fourier synthesis to be carried out meaningfully. One way in which phase information may be obtained is through the Patterson function of vector density, a function of interatomic vectors in the crystal structure.

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Notes

  1. 1.

    If the forward direction of this vector is used, the structure obtained would, in general, be inverted through the origin. This does not happen with the example under study because the molecule possesses only twofold symmetry.

  2. 2.

    See Bibliography, Buerger (1959).

  3. 3.

    In the original paper, the origin in Imma was chosen on a center of symmetry displaced by \( \tfrac{1}{4} \), \( \tfrac{1}{4} \), \( \tfrac{1}{4} \) from this origin.

  4. 4.

    In the work of Duwell and Baenziger [6], the positions listed are 8 (i), with x = 0.186 and z = 0.089, each being \( \tfrac{1}{4} \) minus the value given here.

  5. 5.

    See Bibliography, Buerger (1959).

  6. 6.

    The resolution of a protein X-ray analysis is loosely defined as d min, where d min = λ/2 sin θ max, θ max being the maximum Bragg angle associated with the analysis: initially θ max may be temporarily restricted in order to limit the work required, but at the expense of the quality of the electron density image.

  7. 7.

    Single Isomorphous Replacement with Anomalous Scattering.

  8. 8.

    These omissions give rise to the proportionality sign.

  9. 9.

    Lipson H, Beevers CA (1936) Proc Phys Soc 48:772.

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Author information

Authors and Affiliations

  1. University of Surrey, Guildford, England

    Mark Ladd (Formerly Head of Chemical Physics)

  2. Birkbeck College, University of London, London, England

    Rex Palmer (Reader Emeritus in Structural Crystallography, Visiting Professor in X-ray Crystallography, Senior Visiting Research Fellow)

  3. University of Greenwich, London, England

    Rex Palmer (Reader Emeritus in Structural Crystallography, Visiting Professor in X-ray Crystallography, Senior Visiting Research Fellow)

  4. Christ Church University, Canterbury, England

    Rex Palmer (Reader Emeritus in Structural Crystallography, Visiting Professor in X-ray Crystallography, Senior Visiting Research Fellow)

Authors
  1. Mark Ladd
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  2. Rex Palmer
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Ladd, M., Palmer, R. (2013). Fourier Techniques in X-Ray Structure Determination. In: Structure Determination by X-ray Crystallography. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-3954-7_7

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