Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 4.5, p. 475   | 1 | 2 |

Section Helix symmetry, cell constants and space-group symmetry

R. P. Millanea* Helix symmetry, cell constants and space-group symmetry

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The first step in analysis of any fibre diffraction pattern is determination of the molecular helix symmetry [u_{v}]. Only the zero-order Bessel term contributes diffracted intensity on the meridian, and referring to equation ([link] shows that the zero-order term occurs only on layer lines for which l is a multiple of u. Therefore, inspection of the distribution of diffraction along the meridian allows the value of u to be inferred. This procedure is usually effective, but can be difficult if u is large, because the first meridional maximum may be on a layer line that is difficult to measure. This difficulty was overcome in one case by Franklin & Holmes (1958)[link] by noting that the second Bessel term on the equator is [n = u], estimating [G_{00}(R)] using data from a heavy-atom derivative (see Section[link]), subtracting this from [I_{0}(R)], and using the behaviour of the remaining intensity for small R to infer the order of the next Bessel term [using equation ([link]] and thence u.

Referring to equations ([link] and ([link] shows that the distribution of [R_{\min}] for [0  \lt  l  \lt  u] depends on the value of v. Therefore, inspection of the intensity distribution close to the meridian often allows v to be inferred. Note, however, that the distribution of [R_{\min}] does not distinguish between the helix symmetries [u_{v}] and [u_{u-v}]. Any remaining ambiguities in the helix symmetry need to be resolved by steric considerations, or by detailed testing of models with the different symmetries against the available data.

For a polycrystalline system, the cell constants are determined from the [(R, Z)] coordinates of the spots on the diffraction pattern as described in Section[link]. Space-group assignment is based on analysis of systematic absences, as in conventional crystallography. However, in some cases, because of possible overlap of systematic absences with other reflections, there may be some ambiguity in space-group assignment. However, the space group can always be limited to one of a few possibilities, and ambiguities can usually be resolved during structure determination (Section[link]).


First citationFranklin, R. E. & Holmes, K. C. (1958). Tobacco mosaic virus: application of the method of isomorphous replacement to the determination of the helical parameters and radial density distribution. Acta Cryst. 11, 213–220.Google Scholar

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