Projections of centred cells (lattices)

Hahn, Th.; Looijenga-Vos, A.

doi:10.1107/97809553602060000505

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

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

Section 2.2.14.2. Projections of centred cells (lattices)

Th. Hahn^a ^* and A. Looijenga-Vos^b ^‡

^a Institut 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 [], [ $[\bar{1}11]$ ], [ $[1\bar{1}1]$ ] 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 [ $[\bar{1}11]$ ], [] or [ $[\bar{1}\bar{2}1]$ ] for the obverse setting. For the reverse setting, the corresponding directions are [ $[1\bar{1}1]$ ], [ $[\bar{2}\bar{1}1]$ ], []; cf. Chapter 1.2 .

(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 [] 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 [] of an orthorhombic I-centred cell leads to a rectangular c-centred plane cell, which is conventional.
(3) Projection along [] 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 ). A simple cell transformation leads to the conventional hexagonal p cell.

References

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