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Chemistry LibreTexts

Partial symmetry

The symmetry operations of a space group are isometries operating on the whole crystal pattern and are also called total operations or global operations. More generally, the crystal space can be divided in N components S1 to SN, and a coincidence operation φ(Si)→Sj can act on just the i-th component Si to bring it to coincide with the j-th component Sj. Such an operation is not one of the operations of the space group of the crystal because it is not a coincidence operation of the whole crystal space; it is not even defined, in general, for any component k different from i. It is called a partial operation: from the mathematical viewpoint, partial operations are space-groupoid operations.

When i = ji.e. when the operation is φ(Si)→Si and brings a component to coincide with itself, the partial operation is of special type and is called local. A local operation is in fact a symmetry operation, which is defined only on a part of the crystal space: local operations may constitute a subperiodic group.