Research Article: Estimating the structure factors in X-ray diffraction

Date Published: September 01, 2018

Publisher: International Union of Crystallography

Author(s): Paul F. Fewster.


The meaning of the structure factor and how it impacts on the determination of structures are reassessed. A route to obtaining the structure factors is presented for several data collection methods and crystal qualities.

Partial Text

X-ray diffraction analysis has relied to an increasing extent on the accuracy of intensity measurements to reveal important structural information in complex molecules, e.g. functional groups in proteins etc. It is therefore crucial to ensure that the derived structural model closely resembles reality, which can only be achieved if the description of diffraction is sufficiently complete. Also, the reliability of the measured data can only be estimated with confidence if a complete interpretation of the diffraction pattern is available.

An X-ray plane wave incident on a plane of atoms will create a spherical wave from each atom, and at large radii along a specific direction these waves will appear planar. All the amplitude contributions that travel along a specific direction 2θ will combine and form a resultant amplitude depending on all their relative phase relationships. At the Bragg condition the maximum amplitude from each plane in a stack is in-phase with all the others, resulting in maximum intensity. As the stack of planes is rotated, the phase is no longer optimal, and the amplitude falls but still forms a peak at the specular scattering angle1 for the incident wave angle Ω to these planes. This results in the characteristic Bragg peak and fringes. In the conventional theory, there is one scattered wave kH for each incident wave k0 that is related to the scattering vector S to give the familiar relation kH = k0 + S, where both k vectors have a magnitude 1/λ and S has a magnitude of 1/dhkl at the Bragg angle.2 This can be graphically represented by the Ewald sphere with the conclusion that only features touching its surface can form intensity.

The conventional theory requires the crystal to be mosaic to account for data and to account for the suppression of dynamical effects. Crystals might be mosaic, but if this is a requirement then we are taking a risk in accepting that the kinematical approximation is valid. It is certainly reasonable to accept that crystal planes could be bent to accommodate point defects, dislocations and precipitates, but to assume that the mosaic blocks must be sufficiently small to suppress dynamical effects is difficult to accept.

A single crystal diffracting in a random orientation is very unlikely to satisfy the Bragg condition for any reflections, but it will contain diffraction peaks that are very weak. This is observed at XFELs because of the very high incident-beam intensity. As the number of randomly orientated crystals increases more regions of the full diffraction pattern are explored which may contain a few Bragg peaks. The mean intensity of each diffraction feature becomes more representative of the full diffraction pattern. In polycrystalline diffraction all these individual patterns from each crystal are superimposed on each other to give the characteristic Debye–Scherrer rings with fluctuating intensity (Fewster & Andrew, 1993 ▸, 1999 ▸). This description explains the fluctuating intensity at XFELs. The diffraction pattern either from polycrystalline samples as the sum of contributions around the Debye–Scherrer rings or combined intensity contributions from individually indexed XFEL snapshots will stabilize when the full distribution of intensities has been explored and more contributions just confirm this (the central limit theorem). This is reached quite quickly in polycrystalline diffraction because of the large number of crystals in a typical experiment. In XFELs the intensity from each crystal is captured sequentially and will follow the same principle, making it possible to estimate the reliability in the intensities from the snapshot data. Structure analysis with single crystals collects the intensity near the Bragg condition for each reflection. This will not capture the full intensity distribution and could be subject to errors. In all cases the true mean of the intensity distribution will represent the intensity Ihkl.

The calculation of the resultant amplitudes in every direction for a very large number of atoms is currently impractical (even considering a crystal composed of structure-factor amplitudes). The approach by Fewster (2014 ▸) considers the scattering in terms of an ordered array of unit cells.

The full calculation as in the work of Fewster (2014 ▸) is very computer intensive and a short-cut procedure is given here. As shown in Fig. 8 ▸(a) most of the intensity is along X = 0 and the ‘banana-shaped’ contribution where ΩXj = θ. ΩXj (< Ω0j) is the projected angle where X ≠ 0, such that as Ω0j is increased there will be a point when ΩXj = θ and this occurs when The X = 0 contribution corresponds to the term(2) in equation (3) with ΩXj = Ω0j, and term(3) = 1. The two ‘banana-shaped’ contributions above and below X = 0 correspond to term(3) in equation (3), with the locus of the maximum intensity, given in equation (5), occurring when term(2) = 1. To calculate the total intensity at a specific 2θ value that can be associated with the structure factor, the whole intensity over all of X and Ω needs to be estimated by independently determining the mean values of term(2) and term(3). The above discussion relates to the intensity determined at one 2θ position, whereas any measurement will integrate the intensity around the diffraction peak. But how much of the intensity should be included either side of the peak along 2θ? To apply conventional structure determination methods a representative value for each of these individual peaks requires an estimate for the range over which the intensity should be isolated. The intensity maps in Fig. 8 ▸ assume the sample is perfect; however this is unlikely to be the case and we need a method of generating the intensities from imperfect crystals. This section describes various scenarios that indicate the importance of reconsidering the conventional approaches. This new theory does have an impact on the assignment of intensity to the structure factors in polycrystalline diffraction, serial crystallography and single-crystal analysis.   Source:


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