International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 2.2, p. 34

Section 2.2.14.2. Projections of centred cells (lattices)

Th. Hahna* and A. Looijenga-Vosb

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:  hahn@xtl.rwth-aachen.de

2.2.14.2. Projections of centred cells (lattices)

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For centred lattices, two different cases may occur:

  • (i) The projection direction is parallel to a lattice-centring vector. In this case, the projected plane cell is primitive for the centring types A, B, C, I and R. For F lattices, the multiplicity is reduced from 4 to 2 because c-centred plane cells result from projections along face diagonals of three-dimensional F cells.

    Examples

    • (1) A body-centred lattice with centring vector [{1 \over 2}({\bf a} + {\bf b} + {\bf c})] gives a primitive net, if projected along [[111]], [[\bar{1}11]], [[1\bar{1}1]] or [[11\bar{1}]].

    • (2) A C-centred lattice projects to a primitive net along the directions [110] and [[1\bar{1}0]].

    • (3) An R-centred lattice described with `hexagonal axes' (triple cell) results in a primitive net, if projected along [[\bar{1}11]], [[211]] or [[\bar{1}\bar{2}1]] for the obverse setting. For the reverse setting, the corresponding directions are [[1\bar{1}1]], [[\bar{2}\bar{1}1]], [[121]]; cf. Chapter 1.2[link] .

  • (ii) The projection direction is not parallel to a lattice-centring vector (general projection direction). In this case, the plane cell has the same multiplicity as the three-dimensional cell. Usually, however, this centred plane cell is unconventional and a transformation is required to obtain the conventional plane cell. This transformation has been carried out for the projection data in this volume.

    Examples

    • (1) Projection along [[010]] of a cubic I-centred cell leads to an unconventional quadratic c-centred plane cell. A simple cell transformation leads to the conventional quadratic p cell.

    • (2) Projection along [[010]] of an orthorhombic I-centred cell leads to a rectangular c-centred plane cell, which is conventional.

    • (3) Projection along [[001]] of an R-centred cell (both in obverse and reverse setting) results in a triple hexagonal plane cell h (the two-dimensional analogue of the H cell, cf. Chapter 1.2[link] ). A simple cell transformation leads to the conventional hexagonal p cell.








































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