International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. F. ch. 19.5, p. 445
Section 19.5.3.3. Structure factors
aWhistler Center for Carbohydrate Research, Purdue University, West Lafayette, IN 47907, USA, and bDepartment of Molecular Biology, Vanderbilt University, Nashville, TN 37235, USA |
Cochran et al. (1952) showed that the structure factor on layer line l of a helix made up of repeating subunits is Diffraction occurs only for and are the real-space coordinates of atom j in the repeating unit of the helix; is the atomic scattering factor of that atom. is the Bessel function of the first kind of order n. The summation over n includes only those values of n that satisfy the selection rule where m is any integer. In practice, the summation may be limited to values of less than , where is the radius of the outermost atom in the polymer, because the value of a Bessel function is negligible for n greater than about x + 2. For low-order Bessel functions or applications requiring greater accuracy, slight variations of this limitation are used.
The structure factor F is a complex number with an amplitude and phase, and is fully equivalent to that derived using the trigonometric functions in crystallography. The expression for intensity holds good.
Equation (19.5.3.1) can be rewritten (Klug et al., 1958) as where the Fourier–Bessel structure factor is independent of ψ and is given by is 1 when and 0 otherwise. For this reason, the structure factors on the meridian are nonzero only on layer lines for which l is an integral multiple of u. Hence, a visual inspection of the diffraction pattern often helps to determine u.
References
Cochran, W., Crick, F. H. & Vand, V. (1952). The structure of synthetic polypeptides. I. The transform of atoms in a helix. Acta Cryst. 5, 581–586.Google ScholarKlug, A., Crick, F. H. C. & Wyckoff, H. W. (1958). Diffraction from helical structures. Acta Cryst. 11, 199–212.Google Scholar