International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by E. Arnold, D. M. Himmel and M. G. Rossmann © International Union of Crystallography 2012 |
International Tables for Crystallography (2012). Vol. F, ch. 2.1, pp. 56-57
Section 2.1.4.6. The structure factor^{a}Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands |
For noncentrosymmetric structures, the structure factor,is an imaginary quantity and can also be written as^{2}
It is sometimes convenient to split the structure factor into its real part, A(S), and its imaginary part, B(S). For centrosymmetric structures, if the origin of the structure is chosen at the centre of symmetry.
The average value of the structure-factor amplitude decreases with increasing or, because , with increasing reflecting angle θ.
This is caused by two factors:
In protein crystal structures determined at high resolution, each atom is given its own individual thermal parameter B.^{3} Anisotropic thermal vibration is described by six parameters instead of one, and the evaluation of this anisotropic thermal vibration requires more data (X-ray intensities) than are usually available. Only at very high resolution (better than 1.5 Å) can one consider the incorporation of anisotropic temperature factors.
The value of can be regarded as the effective number of electrons per unit cell scattering in the direction corresponding to S. This is true if the values of are on an absolute scale; this means that the unit of scattering is the scattering by one electron in a specific direction. The experimental values of are normally on an arbitrary scale. The average value of the scattered intensity, , on an absolute scale is , where is the atomic scattering factor reduced by the temperature factor. This can be understood as follows:
For a large number of reflections, S varies considerably, and assuming that the angles are evenly distributed over the range 0–2π for , the average value for the terms with will be zero and only the terms with remain, giving
Because of the thermal vibrationswhere i denotes a specific atom and is the scattering factor for the atom i at rest.
It is sometimes necessary to transform the intensities and the structure factors from an arbitrary to an absolute scale. Wilson (1942) proposed a method for estimating the required scale factor K and, as an additional bonus, the thermal parameter B averaged over the atoms:
To determine K and B, equation (2.1.4.11) is written in the form
Because depends on , average intensities, , are calculated for shells of narrow ranges. is plotted against . The result should be a straight line with slope , intersecting the vertical axis at ln K (Fig. 2.1.4.10).
For proteins, the Wilson plot gives rather poor results because the assumption in deriving equation (2.1.4.11) that the angles, , are evenly distributed over the range 0–2π for is not quite valid, especially not in the ranges at low resolution.
As discussed above, the average value of the structure factors, F(S), decreases with the scattering angle because of two effects:
This decrease is disturbing for statistical studies of structure-factor amplitudes. It is then an advantage to eliminate these effects by working with normalized structure factors, E(S), defined by
The application of equation (2.1.4.14) to gives
The average value, , is equal to 1. The advantage of working with normalized structure factors is that the scaling is not important, because if equation (2.1.4.14) is written asa scale factor affects numerator and denominator equally.
In practice, the normalized structure factors are derived from the observed data as follows:where is a correction factor for space-group symmetry. For general reflections it is 1, but it is greater than 1 for reflections having h parallel to a symmetry element. This can be understood as follows. For example, if m atoms are related by this symmetry element, (with j from 1 to m) is the same in their contribution to the structure factor
They act as one atom with scattering factor rather than as m different atoms, each with scattering factor f. According to equation (2.1.4.11), this increases by a factor on average. To make the F values of all reflections statistically comparable, F(h) must be divided by . For a detailed discussion, see IT B (2008), Chapter 2.1 , by U. Shmueli and A. J. C. Wilson.
References
International Tables for Crystallography (2008). Vol. B. Reciprocal Space, edited by U. Shmueli, 3rd ed. Heidelberg: Springer.Wilson, A. J. C. (1942). Determination of absolute from relative X-ray intensity data. Nature (London), 150, 151–152.