Bravais flocks of space groups

Wondratschek, H.

doi:10.1107/97809553602060000515

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 8.2, p. 729

Section 8.2.6. Bravais flocks of space groups

H. Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: hans.wondratschek@physik.uni-karlsruhe.de

8.2.6. Bravais flocks of space groups

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Another plausible classification of space groups and space-group types, as well as of arithmetic crystal classes, is based on the lattice of the space group. One is tempted to use the definition: `Two space groups are members of the same class if their lattices belong to the same Bravais type'. There is, however, a difficulty which will become apparent by an example.

It was shown in Section 8.2.5 with the two examples of space groups $[P6_{3}mc]$ and $[P6_{3}/m]$ that the lattice L of the space group $[{\cal G}]$ may systematically have higher symmetry than the point group $[{\cal P}]$ of $[{\cal G}]$ . The lattice L, however, may also accidentally have higher symmetry than $[{\cal P}]$ because the lattice parameters may have special metrical values.

Example

For a monoclinic crystal structure at some temperature $[T_{1}]$ , the monoclinic angle β may be equal to $[91^{\circ}]$ , whereas, for the same monoclinic crystal structure at some other temperature [T_2] , $[\beta = 90^{\circ}]$ may hold. In this case, the lattice L at temperature $[T_{2}]$ , if considered to be detached from the crystal structure and its space group, has orthorhombic symmetry, because all the symmetry operations of an orthorhombic lattice map L onto itself. The lattice L at other temperatures, however, has always monoclinic symmetry.

This is of importance for the practising crystallographer, because quite often difficulties arise in the interpretation of X-ray powder diagrams if no single crystals are available. In some cases, changes of temperature or pressure may enable one to determine the true symmetry of the substance. The example shows, however, that the lattices of different space groups of the same space-group type may have different symmetries. The possibility of accidental lattice symmetry prevents the direct use of lattice types for a rigorous classification of space-group types.

Such a classification is possible, however, via the point group $[{\cal P}]$ of the space group $[{\cal G}]$ and its matrix groups. Referred to a primitive basis, the point group $[{\cal P}]$ of $[{\cal G}]$ is represented by a finite group of integral $[(n \times n)]$ matrices which belongs to some arithmetic crystal class. This matrix group can be uniquely assigned to a Bravais class: It either belongs already to a Bravais class, e.g. for space groups Pmna and [C2/c] , or it may be uniquely connected to a Bravais class by the following two conditions:

(i) The matrix group of $[{\cal P}]$ is a subgroup of a matrix group of the Bravais class.
(ii) The order of the matrix group of the Bravais class is the smallest possible one compatible with condition (i).

Example

A space group of type $[I4_{1}]$ belongs to the arithmetic crystal class 4I. The Bravais classes fulfilling condition (i) are [4/mmmI] and $[m\bar{3}mI]$ . With condition (ii), the Bravais class $[m\bar{3}mI]$ is excluded. Thus, the space group $[I4_{1}]$ is uniquely assigned to the Bravais class . Even though, with accidental lattice parameters $[a = b = c = 5\;\hbox{\AA}]$ , the symmetry of the lattice alone is higher, namely $[Im\bar{3}m]$ , this does not change the Bravais class of $[I4_{1}]$ .

This assignment leads to the definition:

Definition: Space groups that are assigned to the same Bravais class belong to the same Bravais flock of space groups.

By this definition, the space group $[I4_{1}]$ mentioned above belongs to the Bravais flock of [4/mmmI] , despite the fact that the Bravais class of the lattice may be $[m\bar{3}mI]$ as a result of accidental symmetry.

Obviously, to each Bravais class a Bravais flock corresponds. Thus, there exist five Bravais flocks of plane groups and 14 Bravais flocks of space groups, see Fig. 8.2.1.1, and the Bravais flocks may be denoted by the symbols of the corresponding Bravais classes; cf. Section 8.2.5.

Though Bravais flocks themselves are of little practical importance, they are necessary for the definition of crystal families and lattice systems, as described in Sections 8.2.7 and 8.2.8.

References

International Tables for Crystallography (2006). Vol. A. ch. 8.2, p. 729