International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B, ch. 1.3, pp. 8485

Crystals of proteins and nucleic acids contain large amounts of mother liquor, often in excess of 50% of the unitcell volume, occupying connected channels. The well ordered electron density corresponding to the macromolecule thus occupies only a periodic subregion of the crystal. Thus implying the convolution identity between structure factors (Main & Woolfson, 1963): which is a form of the Shannon interpolation formula (Sections 1.3.2.7.1, 1.3.4.2.1.7; Bricogne, 1974; Colman, 1974).
It is often possible to obtain an approximate `molecular envelope' from a poor electrondensity map , either interactively by computer graphics (Bricogne, 1976) or automatically by calculating a moving average of the electron density within a small sphere S. The latter procedure can be implemented in real space (Wang, 1985). However, as it is a convolution of with , it can be speeded up considerably (Leslie, 1987) by computing the moving average as
This remark is identical in substance to Booth's method of computation of `bounded projections' (Booth, 1945a) described in Section 1.3.4.2.1.8, except that the summation is kept threedimensional.
The iterative use of the estimated envelope for the purpose of phase improvement (Wang, 1985) is a submethod of the previously developed method of molecular averaging, which is described below. Sampling rules for the Fourier analysis of envelopetruncated maps will be given there.
References
Booth, A. D. (1945a). Two new modifications of the Fourier method of Xray structure analysis. Trans. Faraday Soc. 41, 434–438.Bricogne, G. (1974). Geometric sources of redundancy in intensity data and their use for phase determination. Acta Cryst. A30, 395–405.
Bricogne, G. (1976). Methods and programs for directspace exploitation of geometric redundancies. Acta Cryst. A32, 832–847.
Colman, P. M. (1974). Noncrystallographic symmetry and the sampling theorem. Z. Kristallogr. 140, 344–349.
Leslie, A. G. W. (1987). A reciprocalspace method for calculating a molecular envelope using the algorithm of B. C. Wang. Acta Cryst. A43, 134–136.
Main, P. & Woolfson, M. M. (1963). Direct determination of phases by the use of linear equations between structure factors. Acta Cryst. 16, 1046–1051.
Wang, B. C. (1985). Resolution of phase ambiguity in macromolecular crystallography. In Diffraction methods for biological macromolecules (Methods in enzymology, Vol. 115), edited by H. Wyckoff, C. H. W. Hirs & S. N. Timasheff, pp. 90–112. New York: Academic Press.