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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 90   | 1 | 2 |

Section 1.3.4.3.6.5. Cubic groups

G. Bricognea

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1.3.4.3.6.5. Cubic groups

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These are usually treated as their orthorhombic or tetragonal subgroups, as the body-diagonal threefold axis cannot be handled by ordinary methods of decomposition.

The three-dimensional factorization technique of Section 1.3.4.3.4.1[link] allows a complete treatment of cubic symmetry. Factoring by 2 along all three dimensions gives four types (i.e. orbits) of parity classes:[\eqalign{&(000) \qquad \qquad \qquad \hbox{ with residual threefold symmetry,}\cr &(100), (010), (001) \quad \,\hbox{related by threefold axis,}\cr &(110), (101), (011) \quad \,\hbox{related by threefold axis,}\cr &(111) \qquad \qquad \qquad \hbox{ with residual threefold symmetry.} }]Orbit exchange using the threefold axis thus allows one to reduce the number of partial transforms from 8 to 4 (one per orbit). Factoring by 3 leads to a reduction from 27 to 11 (in this case, further reduction to 9 can be gained by multiplexing the three diagonal classes with residual threefold symmetry into a single class; see Section 1.3.4.3.5.6[link]). More generally, factoring by q leads to a reduction from [q^{3}] to [{1 \over 3}(q^{3} - q) - q]. Each of the remaining transforms then has a symmetry induced from the orthorhombic or tetragonal subgroup, which can be treated as above.

No implementation of this procedure is yet available.








































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