The structure factor

Drenth, J.

doi:10.1107/97809553602060000658

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 2.1, pp. 56-57 | 1 | 2 |

Section 2.1.4.6. The structure factor

J. Drenth^a ^*

^a Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Correspondence e-mail: j.drenth@chem.rug.nl

2.1.4.6. The structure factor

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For noncentrosymmetric structures , the structure factor, $[F({\bf S}) = \textstyle\sum\limits_{j=1}^{n} f_{j} \exp (2\pi i{\bf r}_{j}\cdot {\bf S}),]$ is an imaginary quantity and can also be written as ² $[F({\bf S}) = \textstyle\sum\limits_{j=1}^{n} f_{j} \cos (2\pi {\bf r}_{j}\cdot {\bf S}) + i \textstyle\sum\limits_{j=1}^{n} f_{j} \sin (2\pi {\bf r}_{j}\cdot {\bf S}) = A({\bf S}) + iB({\bf S}).]$

It is sometimes convenient to split the structure factor into its real part, A(S), and its imaginary part, B(S). For centrosymmetric structures , $[B({\bf S}) = 0]$ if the origin of the structure is chosen at the centre of symmetry.

The average value of the structure-factor amplitude $[|F({\bf S})|]$ decreases with increasing $[|{\bf S}|]$ or, because $[|{\bf S}| = 2\sin \theta /\lambda]$ , with increasing reflecting angle θ.

This is caused by two factors:

(1) A stronger negative interference between the electrons in the atoms at a larger scattering angle; this is expressed in the decrease of the atomic scattering factor as a function of S.
(2) The temperature-dependent vibrations of the atoms. Because of these vibrations, the apparent size of an atom is larger during an X-ray exposure, and the decrease in its scattering as a function of S is stronger. If the vibration is equally strong in all directions, it is called isotropic, and the atomic scattering factor must be multiplied by a correction factor, the temperature factor , $[\exp[-B(\sin^{2} \theta)/\lambda^{2}]]$ . It can be shown that the parameter B is related to the mean-square displacement of the atomic vibrations, $[\overline{u^{2}}]$ : $[B = 8\pi^{2} \overline{u^{2}}.]$

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 $[|F({\bf S})|]$ 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 $[|F({\bf S})|]$ 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 $[|F({\bf S})|]$ are normally on an arbitrary scale. The average value of the scattered intensity, $[\overline{I(\hbox{abs}., {\bf S})}]$ , on an absolute scale is $[\overline{I(\hbox{abs}., {\bf S})} = \overline{|F({\bf S})|^{2}} = \textstyle\sum_{i}{f_{i}}^{2}]$ , where $[f_{i}]$ is the atomic scattering factor reduced by the temperature factor. This can be understood as follows: $[\eqalignno{I(\hbox{abs}., {\bf S}) &= F({\bf S})\cdot F^{*}({\bf S}) = |F({\bf S})|^{2}&\cr&= \textstyle\sum\limits_{i}\sum\limits_{j}f_{i}\;f_{j} \exp \left[2\pi i ({\bf r}_{i} - {\bf r}_{j})\cdot {\bf S}\right].&(2.1.4.10)\cr}]$

For a large number of reflections, S varies considerably, and assuming that the angles $[[2\pi ({\bf r}_{i} - {\bf r}_{j})\cdot {\bf S}]]$ are evenly distributed over the range 0–2π for $[i \neq j]$ , the average value for the terms with $[i \neq j]$ will be zero and only the terms with [i = j] remain, giving $[\overline{|F({\bf S})|^{2}} = \overline{I(\hbox{abs}., {\bf S})} = \textstyle\sum\limits_{i}{f_{i}}^{2}. \eqno(2.1.4.11)]$

Because of the thermal vibrations $[{f_{i}}^{2} = \exp \left(-2B_{i} \sin^{2} \theta/\lambda^{2}\right) ({f_{i}}^{o})^{2},]$ where i denotes a specific atom and $[{f_{i}}^{o}]$ 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: $[\overline{I({\bf S})} = K \overline{I(\hbox{abs}., {\bf S})} = K \exp (-2 B \sin^{2} \theta /\lambda^{2}) \textstyle\sum\limits_{i} ({f_{i}}^{o})^{2}. \eqno(2.1.4.12)]$

To determine K and B, equation (2.1.4.11) is written in the form $[\ln [\overline{I({\bf S})} / \textstyle\sum\limits_{i} ({f_{i}}^{o})^{2}] = \ln K - 2B \sin^{2} \theta /\lambda^{2}. \eqno(2.1.4.13)]$

Because $[{f_{i}}^{o}]$ depends on $[\sin \theta /\lambda]$ , average intensities, $[\overline{I({\bf S})}]$ , are calculated for shells of narrow $[\sin \theta /\lambda]$ ranges. $[\ln [\overline{I({\bf S})} /\textstyle\sum_{i} ({f_{i}}^{o})^{2}]]$ is plotted against $[\sin^{2} \theta /\lambda^{2}]$ . The result should be a straight line with slope [-2B] , intersecting the vertical axis at ln K (Fig. 2.1.4.10).

Figure 2.1.4.10 | top | pdf |

The Wilson plot for phospholipase A₂ with data to 1.7 Å resolution. Only beyond 3 Å resolution is it possible to fit the curve to a straight line. Reproduced with permission from Drenth (1999). Copyright (1999) Springer-Verlag.

For proteins, the Wilson plot gives rather poor results because the assumption in deriving equation (2.1.4.11) that the angles, $[[2\pi ({\bf r}_{i} - {\bf r}_{j})\cdot {\bf S}]]$ , are evenly distributed over the range 0–2π for $[{i \neq j}]$ is not quite valid, especially not in the $[\sin \theta /\lambda]$ 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:

(1) the decrease in the atomic scattering factor f;
(2) the temperature factor.

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 $[\eqalignno{E({\bf S}) &= F({\bf S}) \bigg/ \left(\textstyle\sum\limits_{j} {f_{j}}^{2}\right)^{1/2}& \cr&= F({\bf S}) \exp \left(B \sin^{2} \theta /\lambda^{2}\right) \bigg/ \left[\textstyle\sum\limits_{j} ({f_{j}}^{o})^{2}\right]^{1/2}.&(2.1.4.14)}]$

The application of equation (2.1.4.14) to $[\overline{|E({\bf S})|^{2}}]$ gives $[\overline{|E({\bf S})|^{2}} = \overline{|F({\bf S})|^{2}} \Big/ \textstyle\sum\limits_{j} {f_{j}}^{2} = \overline{|F({\bf S})|^{2}} \Big/\ \overline{|F({\bf S})|^{2}} = 1. \eqno(2.1.4.15)]$

The average value, $[\overline{|E({\bf S})|^{2}}]$ , 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 as $[E({\bf S}) = {F({\bf S}) \over (\overline{|F({\bf S})|^{2}})^{1/2}},]$ a scale factor affects numerator and denominator equally.

In practice, the normalized structure factors are derived from the observed data as follows: $[E({\bf S}) = F({\bf S}) \exp \left(B \sin^{2} \theta /\lambda^{2}\right) \big/ \left(\varepsilon |F({\bf S})|^{2}\right)^{1/2}, \eqno(2.1.4.16)]$ where $[\varepsilon]$ 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, $[{\bf r}_{j}\cdot {\bf S}]$ (with j from 1 to m) is the same in their contribution to the structure factor $[F({\bf h}) = \textstyle\sum\limits_{j=1}^{m} f_{j} \exp (2\pi i {\bf r}_{j}\cdot {\bf S}).]$

They act as one atom with scattering factor $[m\times f]$ rather than as m different atoms, each with scattering factor f. According to equation (2.1.4.11), this increases $[F({\bf h})]$ by a factor $[m^{1/2}]$ on average. To make the F values of all reflections statistically comparable, F(h) must be divided by $[m^{1/2}]$ . For a detailed discussion, see IT B (2001), Chapter 2.1 , by U. Shmueli and A. J. C. Wilson.

References

International Tables for Crystallography (2001). Vol. B. Reciprocal space, edited by U. Shmueli. Dordrecht: Kluwer Academic Publishers.Google Scholar

Wilson, A. J. C. (1942). Determination of absolute from relative X-ray intensity data. Nature (London), 150, 151–152.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 2.1, pp. 56-57