Difference between revisions of "Bravais class"
From Online Dictionary of Crystallography
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Space groups that are assigned to the same [[Bravais arithmetic class]] belong to the same '''Bravais class of space groups'''. | Space groups that are assigned to the same [[Bravais arithmetic class]] belong to the same '''Bravais class of space groups'''. | ||
| − | The introduction of the concept of Bravais | + | The introduction of the concept of Bravais class 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 class solves this ambiguity. | 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 class solves this ambiguity. | ||
A space group ''G'' is ''assigned'' to a [[Bravais arithmetic 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 arithmetic 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 arithmetic class, then ''G'' is assigned to that Bravais arithmetic class; |
* if the arithmetic crystal class of ''G'' is not a Bravais arithmetic class, then the Bravais arithmetic class to which ''G'' is assigned is obtained as follows: | * if the arithmetic crystal class of ''G'' is not a Bravais arithmetic class, then the Bravais arithmetic class to which ''G'' is assigned is obtained as follows: | ||
** those Bravais arithmetic classes are retained whose point group ''B'' is such that ''P'' is a subgroup of ''B''; | ** those Bravais arithmetic classes are retained whose point group ''B'' is such that ''P'' is a subgroup of ''B''; | ||
** ''G'' is assigned to that Bravais arithmetic 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 arithmetic class, among those selected above, for which the ratio of the order of the point groups of ''B'' and of ''P'' is minimal. | ||
| − | |||
| − | In earlier editions of Volume A of | + | For example, a space group of type ''I''4<sub>1</sub> belongs to the arithmetic crystal class 4''I'', to which two Bravais arithmetic classes can be associated, 4/''mmmI'' and <math>m{\bar 3}mI</math>. The second condition uniquely assigns ''G'' to the Bravais class of 4/''mmmI'', despite the fact that the Bravais arithmetic class of the lattice may be <math>m{\bar 3}mI</math> as a result of accidental symmetry. |
| + | |||
| + | In earlier editions of Volume A of ''International Tables for Crystallography, Volume A'' Bravais classes were called ''Bravais flocks''. | ||
== See also == | == See also == | ||
| − | *Chapter 1.3.4.4.1 | + | *Chapter 1.3.4.4.1 of ''International Tables for Crystallography, Volume A'', 6th edition |
[[category: Fundamental crystallography]] | [[category: Fundamental crystallography]] | ||
Latest revision as of 08:27, 14 July 2021
Classe Bravais (Fr). Bravais-klasse (Ge). Classe di Bravais (It). ブラベー類 (Ja). Clase de Bravais (Sp).
Definition
Space groups that are assigned to the same Bravais arithmetic class belong to the same Bravais class of space groups.
The introduction of the concept of Bravais class 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 class solves this ambiguity.
A space group G is assigned to a Bravais arithmetic 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 arithmetic class, then G is assigned to that Bravais arithmetic class;
- if the arithmetic crystal class of G is not a Bravais arithmetic class, then the Bravais arithmetic class to which G is assigned is obtained as follows:
- those Bravais arithmetic classes are retained whose point group B is such that P is a subgroup of B;
- G is assigned to that Bravais arithmetic 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 arithmetic classes can be associated, 4/mmmI and [math]m{\bar 3}mI[/math]. The second condition uniquely assigns G to the Bravais class of 4/mmmI, despite the fact that the Bravais arithmetic class of the lattice may be [math]m{\bar 3}mI[/math] as a result of accidental symmetry.
In earlier editions of Volume A of International Tables for Crystallography, Volume A Bravais classes were called Bravais flocks.
See also
- Chapter 1.3.4.4.1 of International Tables for Crystallography, Volume A, 6th edition