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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 89   | 1 | 2 |

Section 1.3.4.3.6.2. Monoclinic groups

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

1.3.4.3.6.2. Monoclinic groups

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A general monoclinic transformation is of the form[S_{g}: {\bf x} \,\longmapsto\, {\bf R}_{g} {\bf x} + {\bf t}_{g}]with [{\bf R}_{g}] a diagonal matrix whose entries are [+1] or [-1], and [{\bf t}_{g}] a vector whose entries are 0 or [{1 \over 2}]. We may thus decompose both real and reciprocal space into a direct sum of a subspace [{\bb Z}^{n_{+}}] where [{\bf R}_{g}] acts as the identity, and a subspace [{\bb Z}^{n_{-}}] where [{\bf R}_{g}] acts as minus the identity, with [n_{+} + n_{-} = n = 3]. All usual entities may be correspondingly written as direct sums, for instance:[\matrix{\eqalign{{\bf R}_{g} &= {\bf R}_{g}^{+} \oplus {\bf R}_{g}^{-},\cr {\bf t}_{g} &= {\bi t}_{g}^{+} \oplus {\bf t}_{g}^{-},\cr {\bf m} &= {\bf m}^{+} \oplus {\bf m}^{-}, \cr{\bf h} &= {\bf h}^{+} \oplus {\bf h}^{-},} &\eqalign{{\bf N} &= {\bf N}^{+} \oplus {\bf N}^{-}, \cr {\bf t}_{g}^{(1)} &= {\bf t}_{g}^{(1) +} \oplus {\bf t}_{g}^{(1) -},\cr {\bf m}_{1} &= {\bf m}_{1}^{+} \oplus {\bf m}_{1}^{-},\cr{\bf h}_{1} &= {\bf h}_{1}^{+} \oplus {\bf h}_{1}^{-}, } &\eqalign{{\bf M} &= {\bf M}^{+} \oplus {\bf M}^{-},\cr {\bf t}_{g}^{(2)} &= {\bf t}_{g}^{(2) +} \oplus {\bf t}_{g}^{(2) -},\cr {\bf m}_{2} &= {\bf m}_{2}^{+} \oplus {\bf m}_{2}^{-},\cr {\bf h}_{2} &= {\bf h}_{2}^{+} \oplus {\bf h}_{2}^{-}.}}]

We will use factoring by 2, with decimation in frequency when computing structure factors, and decimation in time when computing electron densities; this corresponds to [{\bf N} = {\bf N}_{1}{\bf N}_{2}] with [{\bf N}_{1} = {\bf M}], [{\bf N}_{2} = 2{\bf I}]. The nonprimitive translation vector [{\bf Nt}_{g}] then belongs to [{\bf M}{\bb Z}^{n}], and thus[{\bf t}_{g}^{(1)} = {\bf 0} \hbox{ mod } {\bf M}{\bb Z}^{n},\quad {\bf t}_{g}^{(2)} \in {\bb Z}^{n} / 2{\bb Z}^{n}.]The symmetry relations obeyed by [\rho\llap{$-\!$}] and F are as follows: for electron densities[\rho\llap{$-\!$} ({\bf m}^{+}, {\bf m}^{-}) = \rho\llap{$-\!$} ({\bf m}^{+} + {\bf N}^{+}{\bf t}_{g}^{+}, -{\bf m}^{-} - {\bf N}^{-}{\bf t}_{g}^{-})]or, after factoring by 2,[\eqalign{ &\rho\llap{$-\!$} ({\bf m}_{1}^{+}, {\bf m}_{2}^{+}, {\bf m}_{1}^{-}, {\bf m}_{2}^{-}) \cr &\phantom{\rho\llap{$-\!$}} = \rho\llap{$-\!$} ({\bf m}_{1}^{+}, {\bf m}_{2}^{+} + {\bf t}_{g}^{(2) +}, {\bf M}^{-}{\boldzeta} ({\bf m}_{1}^{-}) - {\bf m}_{1}^{-} - {\bf m}_{2}^{-}, {\bf m}_{2}^{-} + {\bf t}_{g}^{(2) -})\hbox{\semi}}]while for structure factors[F ({\bf h}^{+}, {\bf h}^{-}) = \exp [2 \pi i({\bf h}^{+} \cdot {\bf t}_{g}^{+} + {\bf h}^{-} \cdot {\bf t}_{g}^{-})] F ({\bf h}^{+}, -{\bf h}^{-})]with its Friedel counterpart[F ({\bf h}^{+}, {\bf h}^{-}) = \exp [2 \pi i({\bf h}^{+} \cdot {\bf t}_{g}^{+} + {\bf h}^{-} \cdot {\bf t}_{g}^{-})] \overline{F (-{\bf h}^{+}, {\bf h}^{-})}]or, after factoring by 2,[\eqalign{ F({\bf h}_{1}^{+}, {\bf h}_{2}^{+}, {\bf h}_{1}^{-}, {\bf h}_{2}^{-}) =\ &(-1)^{{\bf h}_{2}^{+} \cdot {\bf t}_{g}^{(2) +} + {\bf h}_{2}^{-} \cdot {\bf t}_{g}^{(2) -}}\cr &\times F({\bf h}_{1}^{+}, {\bf h}_{2}^{+}, {\bf M}^{-}{\boldzeta} ({\bf h}_{1}^{-}) - {\bf h}_{1}^{-} - {\bf h}_{2}^{-}, {\bf h}_{2}^{-})}]with Friedel counterpart[\eqalign{&F({\bf h}_{1}^{+}, {\bf h}_{2}^{+}, {\bf h}_{1}^{-}, {\bf h}_{2}^{-})\cr &\quad= (-1)^{{\bf h}_{2}^{+} \cdot {\bf t}_{g}^{(2) +} + {\bf h}_{2}^{-} \cdot {\bf t}_{g}^{(2) -}} \overline{F [{\bf M}^{+}{\boldzeta} ({\bf h}_{1}^{+}) - {\bf h}_{1}^{+} - {\bf h}_{2}^{+}, {\bf h}_{2}^{+}, {\bf h}_{1}^{-}, {\bf h}_{2}^{-}].}}]

When calculating electron densities, two methods may be used.

  • (i) Transform on [{\bf h}^{-}] first.

    The partial vectors defined by [X_{{\bf h}^{+}, \, {\bf h}_{2}^{-}} = F({\bf h}^{+}, {\bf h}_{1}^{-}, {\bf h}_{2}^{-})] obey symmetry relations of the form[X({\bf h}_{1}^{-} - {\bf h}_{2}^{-}) = \varepsilon X[{\bf M}^{-}{\boldzeta} ({\bf h}_{1}^{-}) - {\bf h}_{1}^{-}]]with [\varepsilon = \pm 1] independent of [{\bf h}_{1}^{-}]. This is the same relation as for the same parity class of data for a (complex or real) symmetric [(\varepsilon = + 1)] or antisymmetric [(\varepsilon = - 1)] transform. The same techniques can be used to decrease the number of [F({\bf M}^{-})] by multiplexing pairs of such vectors and demultiplexing their transforms. Partial vectors with different values of [epsilon] may be mixed in this way (Section 1.3.4.3.5.6[link]).

    Once [F({\bf N}^{-})] is completed, its results have Hermitian symmetry with respect to [{\bf h}^{+}], and the methods of Section 1.3.4.3.5.1[link] may be used to obtain the unique electron densities.

  • (ii) Transform on [{\bf h}^{+}] first.

    The partial vectors defined by [X_{{\bf h}^{-}, \, {\bf h}_{2}^{+}} = F({\bf h}_{1}^{+}, {\bf h}_{2}^{+}, {\bf h}^{-})] obey symmetry relations of the form[X({\bf h}_{1}^{+} - {\bf h}_{2}^{+}) = \overline{\varepsilon X[{\bf M}^{+}{\boldzeta} ({\bf h}_{1}^{+}) - {\bf h}_{1}^{+}]}]with [\varepsilon = \pm 1] independent of [{\bf h}_{1}^{+}]. This is the same relation as for the same parity class of data for a Hermitian symmetric [(\varepsilon = + 1)] or antisymmetric [(\varepsilon = - 1)] transform. The same techniques may be used to decrease the number of [F({\bf M}^{+})]. This generalizes the procedure described by Ten Eyck (1973)[link] for treating dyad axes, i.e. for the case [n_{+} = 1, {\bf t}_{g}^{(2) -} = {\bf 0}], and [{\bf t}_{g}^{(2) +} = {\bf 0}] (simple dyad) or [{\bf t}_{g}^{(2) +} {\ne {\bf 0}}] (screw dyad).

    Once [F({\bf N}^{+})] is completed, its results have Hermitian symmetry properties with respect to [{\bf h}^{-}] which can be used to obtain the unique electron densities.

    Structure factors may be computed by applying the reverse procedures in the reverse order.

References

Ten Eyck, L. F. (1973). Crystallographic fast Fourier transforms. Acta Cryst. A29, 183–191.








































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