Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section 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 Helical symmetry and associated selection rules

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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 [\varphi /2\pi] and z. The integer lattice with basis (i, j) is a period lattice for the [(\varphi, z)] dependence of the electron density ρ of an axially periodic fibre considered in Section[link]:[\rho (r, \varphi + 2 \pi k_{1}, z + k_{2}) = \rho (r, \varphi, z).]

Suppose the fibre now has helical symmetry, with u copies of the same molecule in t turns, where g.c.d. [(u, t) = 1]. Using the Euclidean algorithm, write [u = \lambda t + \mu] with λ and μ positive integers and [\mu \,\lt\, t]. The period lattice for the [(\varphi, z)] 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 basis[{\bf I} = {t \over u} {\bf i} + {1 \over u} {\bf j},\quad {\bf J} = {-\mu \over u} {\bf i} + {\lambda \over u} {\bf j.}]If α and β are coordinates along I and J, respectively,[\pmatrix{{\displaystyle{\varphi/2\pi}}\cr\noalign{\vskip 3pt} z\cr} = {1 \over u} \pmatrix{t &-\mu\cr 1 &\lambda\cr} \pmatrix{\alpha\cr \beta\cr}]or equivalently[\pmatrix{\alpha\cr \beta\cr} = \pmatrix{\lambda &\mu\cr -1 &t\cr} \pmatrix{{\displaystyle{\varphi /2\pi}}\cr\noalign{\vskip 3pt} z\cr}.]By Fourier transformation,[\eqalign{ \left({\varphi \over 2\pi}, z\right) &\Leftrightarrow (-n, l)\cr (\alpha, \beta) &\Leftrightarrow (m, p)}]with the transformations between indices given by the contragredients of those between coordinates, i.e.[\pmatrix{n\cr l\cr} = \pmatrix{-\lambda &1\cr -\mu &t\cr} \pmatrix{m\cr p\cr}]and[\pmatrix{m\cr p\cr} = {1 \over u} \pmatrix{-t &1\cr \mu &\lambda\cr} \pmatrix{n\cr l\cr}.]It follows that[l = tn + um,]or alternatively that[\mu n = up - \lambda l,]which 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 obtain[h_{m, \, p} (r) = {\textstyle\int\limits_{0}^{1}} {\textstyle\int\limits_{0}^{1}} \rho (r, \alpha, \beta) \exp [2 \pi i(m\alpha + p\beta)] \hbox{ d}\alpha \hbox{ d}\beta]and then reindexing to get only the allowed [g_{nl}]'s by[g_{nl} (r) = uh_{-\lambda m + p, \, \mu m + tp} (r).]This is u times faster than analysing u subunits with respect to the [(\varphi, z)] coordinates.

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