International
Tables for Crystallography Volume E Subperiodic groups Edited by V. Kopský and D. B. Litvin © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. E. ch. 1.2, pp. 22-27
Section 1.2.17. Symbols
a
Department of Physics, University of the South Pacific, Suva, Fiji, and Institute of Physics, The Academy of Sciences of the Czech Republic, Na Slovance 2, PO Box 24, 180 40 Prague 8, Czech Republic, and bDepartment of Physics, Penn State Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610-6009, USA |
The following general criterion was used in selecting the sets of symbols for the subperiodic groups: consistency with the symbols used for the space groups given in IT A (2005). Specific criteria following from this general criterion are as follows:
A survey of sets of symbols that have been used for the subperiodic groups is given below. Considering these sets of symbols in relation to the above criteria leads to the sets of symbols for subperiodic groups used in Parts 2 , 3 and 4 .
A list of sets of symbols for the frieze groups is given in Table 1.2.17.1. The information provided in this table is as follows:
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Sets of symbols which are of a non-Hermann–Mauguin (international) type are the set of symbols of the `black and white' symmetry type (column 3) and the sets of symbols in columns 6 and 7. The sets of symbols in columns 4, 5 and 11 do not follow the sequence of symmetry directions used for two-dimensional space groups. The sets of symbols in columns 3, 4, 5 and 10 do not use a lower-case script to denote a one-dimensional lattice. The set of symbols in column 9 uses parentheses and square brackets to denote specific symmetry directions. The symbol g is used in Part 1 to denote a glide line, a standard symbol for two-dimensional space groups (IT A , 2005). A letter identical with a basis-vector symbol, e.g. a or c, is not used to denote a glide line, as is done in the symbols of columns 5, 6, 7, 9 and 11, as such a letter is a standard notation for a three-dimensional glide plane (IT A , 2005).
Columns 2 and 3 show the isomorphism between frieze groups and one-dimensional magnetic space groups. The one-dimensional space groups are denoted by and . The list of symbols in column 3, on replacing r with , is the list of one-dimensional magnetic space groups. The isomorphism between these two sets of groups interexchanges the elements and 1′ of the one-dimensional magnetic space groups and, respectively, the elements and , mirror lines perpendicular to the [10] and [01] directions, of the frieze groups.
A list of sets of symbols for the rod groups is given in Table 1.2.17.2. The information provided in the columns of this table is as follows:
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Sets of symbols which are of a non-Hermann–Mauguin (international) type are the set of symbols in column 6 and the Niggli-type set of symbols in column 9. The set of symbols in column 8 does not use the lower-case script letter , as does IT A (2005), to denote a one-dimensional lattice. The order of the characters indicating symmetry elements in the set of symbols in column 7 does not follow the sequence of symmetry directions used for three-dimensional space groups. The set of symbols in column 4 have the characters indicating symmetry elements along non-lattice directions enclosed in parentheses, and do not use a lower-case script letter to denote the one-dimensional lattice. Lastly, the set of symbols in column 4, without the parentheses and with the one-dimensional lattice denoted by a lower-case script , are identical with the symbols in Part 3 , or in some cases are the second setting of rod groups whose symbols are given in Part 3 . These second-setting symbols are included in the symmetry diagrams of the rod groups.
A list of sets of symbols for the layer groups is given in Table 1.2.17.3. The information provided in the columns of this table is as follows:
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There is also a notation for layer groups, introduced by Janovec (1981), in which all elements in the group symbol which change the direction of the normal to the plane containing the translations are underlined, e.g. p4/m. However, we know of no listing of all layer-group types in this notation.
Sets of symbols which are of a non-Hermann–Mauguin (international) type are the sets of symbols of the Schoenflies type (columns 11 and 12) and symbols of the `black and white' symmetry type (columns 16, 17, 18, 20, 21, 22, 24 and 25). Additional non-Hermann–Mauguin (international) type sets of symbols are those in columns 14 and 23.
Sets of symbols which do not begin with a letter indicating the lattice centring type are the sets of symbols of the Niggli type (columns 13 and 15). The order of the characters indicating symmetry elements in the sets of symbols in columns 4 and 9 does not follow the sequence of symmetry directions used for three-dimensional space groups. The set of symbols in column 6 uses parentheses to denote a symmetry direction which is not a lattice direction. In addition, the set of symbols in column 6 uses upper-case letters to denote the two-dimensional lattice of the layer group, where as in IT A (2005) upper-case letters denote three-dimensional lattices.
The symbols in column 8 are either identical with or, in some monoclinic and orthorhombic cases, are the second-setting or alternative-cell-choice symbols of the layer groups whose symbols are given in Part 4 . These second-setting and alternative-cell-choice symbols are included in the symmetry diagrams of the layer groups.
The isomorphism between layer groups and two-dimensional magnetic space groups can be seen in Table 1.2.17.3. The set of symbols which we use for layer groups is given in column 2. The sets of symbols in columns 16, 17 and 22 are sets of symbols for the two-dimensional magnetic space groups. The basic relationship between these two sets of groups is the interexchanging of the magnetic symmetry element 1′ and the layer symmetry element mz. A detailed discussion of the relationship between these two sets of groups has been given by Opechowski (1986).
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