Symbols for plane groups (two-dimensional space groups)

Bertaut, E. F.

doi:10.1107/97809553602060000508

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. 4.2, p. 61
https://doi.org/10.1107/97809553602060000508

Chapter 4.2. Symbols for plane groups (two-dimensional space groups)

E. F. Bertaut^a ^‡

^a Laboratoire de Cristallographie, CNRS, Grenoble, France

Chapters 4.1 , 4.2 and 4.3 contain extensive explanations and tabulations of the various types of space-group symbols. Chapter 4.2 presents a table with the short, full and extended symbols for the 17 plane groups, including symbols for the tetragonal c and the hexagonal h cell.

Keywords: plane groups; symbols; symmetry elements; group–subgroup relations.

4.2.1. Arrangement of the tables

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Comparative tables for the 17 plane groups first appeared in IT (1952). The classification of plane groups is discussed in Chapter 2.1 . Table 4.2.1.1 lists for each plane group its system, lattice symbol, point group and the plane-group number, followed by the short, full and extended Hermann–Mauguin symbols. Short symbols are included only where different from the full symbols. The next column contains the full symbol for another setting which corresponds to an interchange of the basis vectors a and b; it is only needed for the rectangular system. Multiple cells c and h for the square and the hexagonal system are introduced in the last column.

Table 4.2.1.1 | top | pdf |
Index of symbols for plane groups

System and lattice symbol	Point group	No. of plane group		Hermann–Mauguin symbol			Full symbol for other setting	Multiple cell
System and lattice symbol	Point group	No. of plane group		Short	Full	Extended	Full symbol for other setting	Multiple cell
Oblique	1		1		p 1
p	2		2		p 2
Rectangular p, c	m	$[\Bigg\{]$	3	pm
			4	pg
			5	cm		$[\matrix{c1m1\hfill\cr g\hfill\cr}]$
	2mm	$[\Bigg\{]$	6
			7
			8
			9			$[\matrix{c2mm\hfill\cr g\ g\hfill\cr}]$
Square p	4		10
	4mm	$[\Bigg\{]$	11			$[\matrix{p4mm\hfill\cr g\hfill\cr}]$		$[\matrix{c4mm\hfill\cr g\hfill\cr}]$
	4mm	$[\Bigg\{]$	12			$[\matrix{p4gm\hfill\cr g\hfill\cr}]$		$[\matrix{ c4mg\hfill\cr g\hfill\cr}]$
Hexagonal p	3		13		p 3			h 3
	3m	$[\Bigg\{]$	14			$[\matrix{p3m1\hfill\cr g\hfill}]$		$[\matrix{h31m\hfill\cr g\hfill\cr}]$
	3m	$[\Bigg\{]$	15			$[\matrix{p31m\hfill\cr g\hfill\cr}]$		$[\matrix{ h3m1\hfill\cr g\hfill\cr}]$
	6		16		p 6			h 6
	6mm		17		p 6mm	$[\matrix{p6mm\hfill\cr g\;g\hfill\cr}]$		$[\matrix{h6mm\hfill\cr g\;g\hfill\cr}]$

4.2.2. Additional symmetry elements and extended symbols

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`Additional symmetry' elements are

(i) rotation points 2, 3 and 4, reproduced in the interior of the cell (cf. Table 4.1.2.1 and plane-group diagrams in Part 6 );
(ii) glide lines g which alternate with mirror lines m.

In the extended plane-group symbols, only the additional glide lines g are listed: they are due either to c centring or to `inclined' integral translations, as shown in Table 4.1.2.2 .

4.2.3. Multiple cells

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The c cell in the square system is defined as follows: $[{\bf a}' = {\bf a} \mp {\bf b}{\hbox{;}} \quad {\bf b}' = \pm {\bf a} + {\bf b},]$ with `centring points' at 0, 0; $[{1 \over 2},{1 \over 2}]$ . It plays the same role as the three-dimensional C cell in the tetragonal system (cf. Section 4.3.4 ).

Likewise, the triple cell h in the hexagonal system is defined as follows: $[{\bf a}' = {\bf a} - {\bf b}{\hbox{;}} \quad {\bf b}' = {\bf a} + 2{\bf b},]$ with `centring points' at 0, 0; $[{2 \over 3}, {1 \over 3}{\hbox{;}} {1 \over 3}, {2 \over 3}]$ . It is the two-dimensional analogue of the three-dimensional H cell (cf. Chapter 1.2 and Section 4.3.5 ).

4.2.4. Group–subgroup relations

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The following example illustrates the usefulness of multiple cells.

Example: p3m1 (14)

The symbol of this plane group, described by the triple cell h, is h31m, where the symmetry elements of the secondary and tertiary positions are interchanged. `Decentring' the h cell gives rise to maximal non-isomorphic k subgroups p31m of index [3], with lattice parameters $[a\sqrt{3}, a\sqrt{3}]$ (cf. Section 4.3.5 ).

References

International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]Google Scholar

International Tables for Crystallography (2006). Vol. A. ch. 4.2, p. 61
https://doi.org/10.1107/97809553602060000508