Difference between revisions of "Partial symmetry"
From Online Dictionary of Crystallography
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| − | <font color="blue">Symétrie partielle</font> (<i>Fr</i>) | + | <font color="blue">Symétrie partielle</font> (<i>Fr</i>). <font color="black">Simmetria parziale</font> (<i>It</i>). |
| − | The symmetry operations of a [[space group]] are [[Euclidean mapping|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 S<sub>1</sub> to S<sub>''N''</sub>, and a coincidence operation φ(S<sub>''i''</sub>)→S<sub>''j''</sub> can act on just the ''i'' | + | The symmetry operations of a [[space group]] are [[Euclidean mapping|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 S<sub>1</sub> to S<sub>''N''</sub>, and a coincidence operation φ(S<sub>''i''</sub>)→S<sub>''j''</sub> can act on just the ''i''th component S<sub>''i''</sub> to bring it to coincide with the ''j''th component S<sub>''j''</sub>. 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 [[Groupoid|space-groupoid operations]]. |
| − | When ''i'' = ''j'', ''i | + | When ''i'' = ''j'', ''i.e.'' when the operation is φ(S<sub>''i''</sub>)→S<sub>''i''</sub> and brings a component to coincide with itself, the partial operation is of special type and is called '''[[local symmetry|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]]. |
[[Category: Fundamental crystallography]] | [[Category: Fundamental crystallography]] | ||
Revision as of 12:18, 16 May 2017
Symétrie partielle (Fr). Simmetria parziale (It).
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 ith component Si to bring it to coincide with the jth 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 = j, i.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.