Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, p. 392   | 1 | 2 |

Section Convolution techniques

D. L. Dorsete* Convolution techniques

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One of the first relationships ever derived for phase determination is the Sayre (1952)[link] equation: [F_{{\bf h}} = {\theta \over V} \sum\limits_{{\bf k}} F_{{\bf k}} F_{{\bf h - k}},]which is a simple convolution of phased structure factors multiplied by a function of the atomic scattering factors. For structures with nonoverlapping atoms, consisting of one atomic species, it is an exact expression. Although the convolution term resembles part of the tangent formula above, no statistical averaging is implied (Sayre, 1980[link]). In X-ray crystallography this relationship has not been used very often, despite its accuracy. Part of the reason for this is that it requires relatively high resolution data for it to be useful. It can also fail for structures comprised of different atomic species.

Since, relative to X-ray scattering factors, electron scattering factors span a narrower range of magnitudes at [\sin \theta/\lambda = 0], it might be thought that the Sayre equation would be particularly useful in electron crystallography. In fact, Liu et al. (1988)[link] were able to extend phases for simulated data from copper per­chloro­phthalocyanine starting at the image resolution of 2 Å and reaching the 1 Å limit of an electron-diffraction data set. This analysis has been improved with a 2.4 Å basis set obtained from the Fourier transform of an electron micrograph of this material at 500 kV and extended to the 1.0 Å limit of a 1200 kV electron-diffraction pattern (Dorset et al., 1995[link]). Using the partial phase sets for zonal diffraction data from several polymers by symbolic addition (see above), the Sayre equation has been useful for extending into the whole hk0 set, often with great accuracy. The size of the basis set is critical but the connectivity to access all reflections is more so. Fan and co-workers have had considerable success with the analysis of incommensurately modulated structures. The average structure (basis set) is found by high-resolution electron microscopy and the `superlattice' reflections, corresponding to the incommensurate modulation, are assigned phases in hyperspace by the Sayre convolution. Examples include a high Tc superconductor (Mo et al., 1992[link]) and the mineral ankangite (Xiang et al., 1990[link]). Phases of regular inorganic crystals have also been extended from the electron micrograph to the electron-diffraction resolution by this technique (Hu et al., 1992[link]).

In an investigation of how direct methods might be used for phase extension in protein electron crystallography, low-resolution phases from two proteins, bacteriorhodopsin (Henderson et al., 1986[link]) and halorhodopsin (Havelka et al., 1993[link]) were extended to higher resolution with the Sayre equation (Dorset et al., 1995[link]). For the noncentrosymmetric bacteriorhodopsin hk0 projection a 10 Å basis set was used, whereas a 15 Å set was accepted for the centrosymmetric halorhodopsin projection. In both cases, extensions to 6 Å resolution were reasonably successful. For bacteriorhodopsin, for which data were available to 3.5 Å, problems with the extension were encountered near 5 Å, corresponding to a minimum in a plot of average intensity versus resolution. Suggestions were made on how a multisolution procedure might be successful beyond this point.


Dorset, D. L., Kopp, S., Fryer, J. R. & Tivol, W. F. (1995). The Sayre equation in electron crystallography. Ultramicroscopy, 57, 59–89.
Havelka, W., Henderson, R., Heymann, J. A. W. & Oesterhelt, D. (1993). Projection structure of halorhodopsin from Halobacterium halobium at 6 Å resolution obtained by electron cryomicroscopy. J. Mol. Biol. 234, 837–846.
Henderson, R., Baldwin, J. M., Downing, K. H., Lepault, J. & Zemlin, F. (1986). Structure of purple membrane from Halobacterium halobium: recording, measurement and evaluation of electron micrographs at 3.5 Å resolution. Ultramicroscopy, 19, 147–178.
Hu, H. H., Li, F. H. & Fan, H. F. (1992). Crystal structure determination of K2O·7Nb2O5 by combining high resolution electron microscopy and electron diffraction. Ultramicroscopy, 41, 387–397.
Liu, Y.-W., Fan, H.-F. & Zheng, C.-D. (1988). Image processing in high-resolution electron microscopy using the direct method. III. Structure-factor extrapolation. Acta Cryst. A44, 61–63.
Mo, Y. D., Cheng, T. Z., Fan, H. F., Li, J. Q., Sha, B. D., Zheng, C. D., Li, F. H. & Zhao, Z. X. (1992). Structural features of the incommensurate modulation in the Pb-doped Bi-2223 high-Tc phase by defect method electron diffraction analysis. Supercond. Sci. Technol. 5, 69–72.
Sayre, D. (1952). The squaring method: a new method for phase determination. Acta Cryst. 5, 60–65.
Sayre, D. (1980). Phase extension and refinement using convolutional and related equation systems. In Theory and Practice of Direct Methods in Crystallography, edited by M. F. C. Ladd & R. A. Palmer, pp. 271–286. New York: Plenum Press.
Xiang, S.-B., Fan, H.-F., Wu, X.-J., Li, F.-H. & Pan, Q. (1990). Direct methods in superspace. II. The first application to an unknown incommensurate modulated structure. Acta Cryst. A46, 929–934.

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