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

International Tables for Crystallography (2006). Vol. A, ch. 2.2, pp. 38-39

Section 2.2.16.2. Settings

Th. Hahna* and A. Looijenga-Vosb

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:  hahn@xtl.rwth-aachen.de

2.2.16.2. Settings

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The term setting of a cell or of a space group refers to the assignment of labels (a, b, c) and directions to the edges of a given unit cell, resulting in a set of basis vectors a, b, c. (For orthorhombic space groups, the six settings are described and illustrated in Section 2.2.6.4[link].)

The symbol for each setting is a shorthand notation for the transformation of a given starting set abc into the setting considered. It is called here `setting symbol'. For instance, the setting symbol bca stands for [{\bf a}' = {\bf b},\quad{\bf b}' = {\bf c},\quad{\bf c}' = {\bf a}] or [({\bf a}' {\bf b}' {\bf c}') = ({\bf abc}) \pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr} = ({\bf bca}),] where a′, b′, c′ is the new set of basis vectors. (Note that the setting symbol bca does not mean that the old vector a changes its label to b, the old vector b changes to c, and the old c changes to a.) Transformation of one setting into another preserves the shape of the cell and its orientation relative to the lattice. The matrices of these transformations have one entry [+]1 or −1 in each row and column; all other entries are 0.

In monoclinic space groups, one axis, the monoclinic symmetry direction, is unique. Its label must be chosen first and, depending upon this choice, one speaks of `unique axis b', `unique axis c' or `unique axis a'.12 Conventionally, the positive directions of the two further (`oblique') axes are oriented so as to make the monoclinic angle non-acute, i.e. [\geq 90^{\circ}], and the coordinate system right-handed. For the three cell choices, settings obeying this condition and having the same label and direction of the unique axis are considered as one setting; this is illustrated in Fig. 2.2.6.4[link].

Note: These three cases of labelling the monoclinic axis are often called somewhat loosely b-axis, c-axis and a-axis `settings'. It must be realized, however, that the choice of the `unique axis' alone does not define a single setting but only a pair, as for each cell the labels of the two oblique axes can be interchanged.

Table 2.2.16.1[link] lists the setting symbols for the six monoclinic settings in three equivalent forms, starting with the symbols a b c (first line), a b c (second line) and a b c (third line); the unique axis is underlined. These symbols are also found in the headline of the synoptic Table 4.3.2.1[link] , which lists the space-group symbols for all monoclinic settings and cell choices. Again, the corresponding transformation matrices are listed in Table 5.1.3.1[link] .

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Monoclinic setting symbols (unique axis is underlined)

Unique axis bUnique axis cUnique axis a 
[Scheme scheme1] [{\bf c}{\underline{\bar{\bf b}}}{\bf a}] [{\bf c}{\bf a}\underline{\bf b}] [{\bf a}{\bf c}\underline{\bar{\bf b}}] [{\underline{\bf b}{\bf c}{\bf a}}] [{\underline{\bar{\bf b}}{\bf a}{\bf c}}] [Scheme scheme4]
[{{\bf b}\underline{\bf c}{\bf a}}] [{{\bf a}\underline{\bar{\bf c}}{\bf b}}] [Scheme scheme2] [{{\bf b}{\bf a}\underline{\bar{\bf c}}}] [{\underline{\bf c}{\bf a}{\bf b}}] [{\underline{\bar{\bf c}}{\bf b}{\bf a}}] [Scheme scheme5]
[{{\bf c}\underline{\bf a}{\bf b}}] [{{\bf b}\underline{\bar{\bf a}}{\bf c}}] [{{\bf b}{\bf c}\underline{\bf a}}] [{{\bf c}{\bf b}\underline{\bar{\bf a}}}] [Scheme scheme3] [{\underline{\bar{\bf a}}{\bf c}{\bf b}}] [Scheme scheme6]

Note: An interchange of two axes involves a change of the handedness of the coordinate system. In order to keep the system right-handed, one sign reversal is necessary.

In the space-group tables, only the settings with b and c unique are treated and for these only the left-hand members of the double entries in Table 2.2.16.1[link]. This implies, for instance, that the c-axis setting is obtained from the b-axis setting by cyclic permutation of the labels, i.e. by the transformation [({\bf a}'{\bf b}'\underline{{\bf c}}') = ({\bf a}\underline{{\bf b}}{\bf c}) \pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr} = ({\bf ca}\underline{{\bf b}}).] In the present discussion, also the setting with a unique is included, as this setting occurs in the subgroup entries of Part 7[link] and in Table 4.3.2.1[link] . The a-axis setting [\underline{{\bf a}}'{\bf b}'{\bf c}' = \underline{{\bf c}} {\bf a b}] is obtained from the c-axis setting also by cyclic permutation of the labels and from the b-axis setting by the reverse cyclic permutation: [\underline{{\bf a}}'{\bf b}'{\bf c}' = \underline{{\bf b}} {\bf c a}].

By the conventions described above, the setting of each of the cell choices 1, 2 and 3 is determined once the label and the direction of the unique-axis vector have been selected. Six of the nine resulting possibilities are illustrated in Fig. 2.2.6.4[link].








































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