International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 101-102   | 1 | 2 |

## Section 1.3.4.5.1.4. Helical symmetry and associated selection rules

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

#### 1.3.4.5.1.4. Helical symmetry and associated selection rules

| top | pdf |

Helical symmetry involves a clutching' between the two (hitherto independent) periodicities in ϕ (period 2π) and z (period 1) which causes a subdivision of the period lattice and hence a decimation (governed by selection rules') of the Fourier coefficients.

Let i and j be the basis vectors along and z. The integer lattice with basis (i, j) is a period lattice for the dependence of the electron density ρ of an axially periodic fibre considered in Section 1.3.4.5.1.3:

Suppose the fibre now has helical symmetry, with u copies of the same molecule in t turns, where g.c.d. . Using the Euclidean algorithm, write with λ and μ positive integers and . The period lattice for the dependence of ρ may be defined in terms of the new basis vectors:

 I, joining subunit 0 to subunit l in the same turn; J, joining subunit 0 to subunit λ after wrapping around.

In terms of the original basisIf α and β are coordinates along I and J, respectively,or equivalentlyBy Fourier transformation,with the transformations between indices given by the contragredients of those between coordinates, i.e.andIt follows thator alternatively thatwhich are two equivalent expressions of the selection rules describing the decimation of the transform. These rules imply that only certain orders n contribute to a given layer l.

The 2D Fourier analysis may now be performed by analysing a single subunit referred to coordinates α and β to obtainand then reindexing to get only the allowed 's byThis is u times faster than analysing u subunits with respect to the coordinates.