Difference between revisions of "Bravais flock"
From Online Dictionary of Crystallography
m (→See also: 6th edition of ITA) |
BrianMcMahon (talk | contribs) m (Edits to align with printed edition) |
||
| Line 1: | Line 1: | ||
| − | <font color="blue">Troupeau de Bravais </font> (''Fr'') | + | <font color="blue">Troupeau de Bravais </font> (''Fr''). <Font color="red">Bravais Herde</Font> (''Ge''). <Font color="green">Rebaño de Bravais</Font> (''Sp''). <Font color="black">Gregge di Bravais</Font> (''It''). <Font color="purple">ブラベー群れ</font> (''Ja''). |
== Definition == | == Definition == | ||
Space groups that are assigned to the same [[Bravais class]] belong to the same '''Bravais flock of space groups'''. | Space groups that are assigned to the same [[Bravais class]] belong to the same '''Bravais flock of space groups'''. | ||
| − | |||
The introduction of the concept of Bravais flock is necessary in order to classify space groups on the basis of their Bravais type of lattice, independently from any accidental metric of the lattice. | The introduction of the concept of Bravais flock is necessary in order to classify space groups on the basis of their Bravais type of lattice, independently from any accidental metric of the lattice. | ||
| − | An orthorhombic crystal may have accidentally ''a'' = ''b'' have a tetragonal lattice not because of symmetry restrictions but just by accident. If the Bravais types of lattices were used directly to classify space | + | An orthorhombic crystal may have accidentally ''a'' = ''b'' and have a tetragonal lattice not because of symmetry restrictions but just by accident. If the Bravais types of lattices were used directly to classify space groups, such a crystal would belong to another category with respect to an orthorhombic crystal without specialized metric. The concept of Bravais flock solves this ambiguity. |
| − | A space group G is ''assigned'' to a [[Bravais class]] on the basis of the corresponding point group P and the [[arithmetic crystal class]] associated with it: | + | A space group ''G'' is ''assigned'' to a [[Bravais class]] on the basis of the corresponding point group ''P'' and the [[arithmetic crystal class]] associated with it: |
| − | * If the arithmetic crystal class of G is a Bravais class, then G is assigned to that Bravais class; | + | * If the arithmetic crystal class of ''G'' is a Bravais class, then ''G'' is assigned to that Bravais class; |
| − | * if the arithmetic crystal class of G is not a Bravais class, then the Bravais class to which G is assigned is obtained as follows: | + | * if the arithmetic crystal class of ''G'' is not a Bravais class, then the Bravais class to which ''G'' is assigned is obtained as follows: |
| − | ** those Bravais classes are retained whose point group | + | ** those Bravais classes are retained whose point group ''B'' is such that ''P'' is a subgroup of ''B''; |
| − | ** G is assigned to that Bravais class, among those selected above, for which the ratio of the order of the point groups of B and of P is minimal. | + | ** ''G'' is assigned to that Bravais class, among those selected above, for which the ratio of the order of the point groups of ''B'' and of ''P'' is minimal. |
| − | For example, a space group of type ''I''4<sub>1</sub> belongs to the arithmetic crystal class 4''I'', to which two Bravais | + | For example, a space group of type ''I''4<sub>1</sub> belongs to the arithmetic crystal class 4''I'', to which two Bravais classes can be associated, 4/''mmmI'' and <math>m{\bar 3}mI</math>. The second condition uniquely assigns G to the Bravais flock of 4/''mmmI'', despite the fact that the Bravais class of the lattice may be <math>m{\bar 3}mI</math> as a result of accidental symmetry. |
== See also == | == See also == | ||
| − | + | *Chapter 1.3.4.4.1 in of ''International Tables for Crystallography, Volume A'', 6th edition | |
[[category: Fundamental crystallography]] | [[category: Fundamental crystallography]] | ||
Revision as of 09:39, 13 May 2017
Troupeau de Bravais (Fr). Bravais Herde (Ge). Rebaño de Bravais (Sp). Gregge di Bravais (It). ブラベー群れ (Ja).
Definition
Space groups that are assigned to the same Bravais class belong to the same Bravais flock of space groups.
The introduction of the concept of Bravais flock is necessary in order to classify space groups on the basis of their Bravais type of lattice, independently from any accidental metric of the lattice.
An orthorhombic crystal may have accidentally a = b and have a tetragonal lattice not because of symmetry restrictions but just by accident. If the Bravais types of lattices were used directly to classify space groups, such a crystal would belong to another category with respect to an orthorhombic crystal without specialized metric. The concept of Bravais flock solves this ambiguity.
A space group G is assigned to a Bravais class on the basis of the corresponding point group P and the arithmetic crystal class associated with it:
- If the arithmetic crystal class of G is a Bravais class, then G is assigned to that Bravais class;
- if the arithmetic crystal class of G is not a Bravais class, then the Bravais class to which G is assigned is obtained as follows:
- those Bravais classes are retained whose point group B is such that P is a subgroup of B;
- G is assigned to that Bravais class, among those selected above, for which the ratio of the order of the point groups of B and of P is minimal.
For example, a space group of type I41 belongs to the arithmetic crystal class 4I, to which two Bravais classes can be associated, 4/mmmI and [math]m{\bar 3}mI[/math]. The second condition uniquely assigns G to the Bravais flock of 4/mmmI, despite the fact that the Bravais class of the lattice may be [math]m{\bar 3}mI[/math] as a result of accidental symmetry.
See also
- Chapter 1.3.4.4.1 in of International Tables for Crystallography, Volume A, 6th edition