Symbols and terms used in Parts 1–4

Kopskyacute, V.; Litvin, D. B.

doi:10.1107/97809553602060000646

International
Tables for
Crystallography
Volume E
Subperiodic groups
Edited by V. Kopský and D. B. Litvin

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International Tables for Crystallography (2006). Vol. E. ch. 1.1, pp. 2-4 | 1 | 2 |
https://doi.org/10.1107/97809553602060000646

Chapter 1.1. Symbols and terms used in Parts 1–4

V. Kopský^a and D. B. Litvin^b ^*

^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
Correspondence e-mail: u3c@psu.edu

In this chapter the crystallographic symbols and terms that occur in the tables and the text of Parts 1 to 4 of this volume are defined. These symbols and definitions follow those given in Part 1 of Volume A of International Tables for Crystallography.

Keywords: crystallography; subperiodic groups; rod groups; layer groups; frieze groups; symmetry planes; symmetry axes; symmetry elements; nomenclature.

In this chapter the crystallographic symbols and terms that occur in the tables and the text of Parts 1–4 of this volume are defined. The symbols and definitions given below in Tables 1.1.1 to 1.1.3 follow those given in Part 1 of Volume A of International Tables for Crystallography (2005 ).

Table 1.1.1| top | pdf |
Printed symbols for crystallographic items

Printed symbol	Explanation
a; b; c	Basis vectors of direct lattice
a; b; c	Length of basis vectors
α; β; γ	Interaxial (lattice) angles b $[\land]$ c, c $[\land]$ a, a $[\land]$ b
a′; b′; c′	New basis vectors after a transformation of the basis vectors
(abc)	Setting symbol, notation for the transformation of the basis vectors, e.g. (b $[\bar{a}]$ c) means a′ = b, b′ = −a and c′ = c
r	Position vector of a point or an atom
x, y, z	Coordinates of a point or location of an atom expressed in units of a, b and c; coordinates of the end point of the position vector r
xa; yb; zc	Components of the position vector r
[uvw]	Indices of a three-dimensional lattice direction
[uv]	Indices of a two-dimensional lattice direction
(hkl)	Miller indices

Table 1.1.2| top | pdf |
Printed symbols for symmetry elements and for the corresponding symmetry operations

Printed symbol	Symmetry element and its orientation	Generating symmetry operation with glide or screw vector
m	Reflection plane, mirror plane (three dimensions)	Reflection through a plane
m	Reflection line, mirror line (two dimensions)	Reflection through a line
a, b or c	`Axial' glide plane	Glide reflection through a plane, with glide vector
a	⊥[010] or ⊥[001]	½a
b	⊥[100] or ⊥[001]	½b
c	⊥[100] or ⊥[010]	½c
	⊥[1 $[\bar{1}]$ 0] or ⊥[110]	½c
	⊥[100] or ⊥[010] or ⊥[ $[\bar{1}]$ $[\bar{1}]$ 0]	½c, hexagonal coordinate system
	⊥[1 $[\bar{1}]$ 0] or ⊥[120] or ⊥[ $[\bar{2}]$ $[\bar{1}]$ 0]	½c, hexagonal coordinate system
n	`Diagonal' glide plane (in noncentred cells only)	Glide reflection through a plane, with glide vector
n	⊥[001]	½(a + b)
e	`Double' glide plane ⊥[001] (in centred cells only)	Two glide reflections through planes with glide vectors ½a and ½b
g	Glide line (two dimensions)	Glide reflection through a line, with glide vector
g	⊥[01]; ⊥[10]	½a; ½b
1	None	Identity
2, 3, 4, 6	n-fold rotation axis, n (three dimensions)	Counterclockwise rotation of 360/n degrees about an axis
2, 3, 4, 6	n-fold rotation point, n (two dimensions)	Counterclockwise rotation of 360/n degrees about a point
$[\bar{1}]$	Centre of symmetry, inversion centre	Inversion through a point
$[\bar{2}]$ = m, $[\bar{3}]$ , $[\bar{4}]$ , $[\bar{6}]$	Rotoinversion axis, $[\bar{n}]$	Counterclockwise rotation of 360/n degrees around an axis, followed by inversion through a point on the axis
2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄, 6₅	n-fold screw axes, n_p	Right-handed screw rotation of 360/n degrees around an axis, with screw vector (p/n)t; t is the shortest translation vector parallel to the axis in the direction of the screw

Table 1.1.3| top | pdf |
Graphical symbols

(a) Symmetry planes normal to the plane of projection (three dimensions) and symmetry lines in the plane of the figure (two dimensions).

Symmetry plane or symmetry line	Glide vectors in units of lattice translation vectors parallel and normal to the projection plane	Printed symbol
Mirror plane, mirror line	None	m
Glide plane, glide line	½ along line parallel to projection plane; ½ along line in plane	a, b or c; g
Glide plane	½ normal to projection plane	c

(b) Symmetry planes parallel to plane of projection.

Symmetry plane	Glide vector in units of lattice translation vectors parallel to the projection plane	Printed symbol
Mirror plane	None	m
Glide plane	½ in the direction of arrow	a, b or c
`Double' glide plane	Two glide vectors; ½ in either of the directions of the two arrows	e
`Diagonal' glide plane	½ in the direction of the arrow	n

(c) Symmetry axes normal to the plane of projection (three dimensions) and symmetry points in the plane of the figure (two dimensions).

Symmetry axis or symmetry point	Screw vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axis	Printed symbol
Twofold rotation axis, twofold rotation point	None	2
Twofold screw axis: `2 sub 1'	$[\textstyle{1 \over 2}]$	2₁
Threefold rotation axis	None	3
Threefold screw axis: `3 sub 1'	$[\textstyle{1 \over 3}]$	3₁
Threefold screw axis: `3 sub 2'	$[\textstyle{2 \over 3}]$	3₂
Fourfold rotation axis	None	4
Fourfold screw axis: `4 sub 1'	$[\textstyle{1 \over 4}]$	4₁
Fourfold screw axis: `4 sub 2'	$[\textstyle{1 \over 2}]$	4₂
Fourfold screw axis: `4 sub 3'	$[\textstyle{3 \over 4}]$	4₃
Sixfold rotation axis	None	6
Sixfold screw axis: `6 sub 1'	$[\textstyle{1 \over 6}]$	6₁
Sixfold screw axis: `6 sub 2'	$[\textstyle{1 \over 3}]$	6₂
Sixfold screw axis: `6 sub 3'	$[\textstyle{1 \over 2}]$	6₃
Sixfold screw axis: `6 sub 4'	$[\textstyle{2 \over 3}]$	6₄
Sixfold screw axis: `6 sub 5'	$[\textstyle{5 \over 6}]$	6₅
Centre of symmetry, inversion centre: `1 bar'	None	$[\bar{1}]$
Twofold rotation axis with centre of symmetry	None	2/m
Twofold screw axis with centre of symmetry	$[\textstyle{1 \over 2}]$	2₁/m
Inversion axis: `3 bar'	None	$[\bar{3}]$
Inversion axis: `4 bar'	None	$[\bar{4}]$
Fourfold rotation axis with centre of symmetry	None	4/m
`4 sub 2' screw axis with centre of symmetry	$[\textstyle{1 \over 2}]$	4₂/m
Inversion axis: `6 bar'	None	$[\bar{6}]$
Sixfold rotation axis with centre of symmetry	None	6/m
`6 sub 3' screw axis with centre of symmetry	$[\textstyle{1 \over 2}]$	6₃/m

(d) Symmetry axes parallel to plane of projection.

Symmetry axis	Graphical symbol	Screw vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axis	Printed symbol
Twofold rotation axis		None	2
Twofold screw axis		$[\textstyle{1 \over 2}]$	2₁

References

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer. [Previous editions: 1983, 1987, 1992, 1995 and 2002. Abbreviated as IT A (2005).]Google Scholar

International Tables for Crystallography (2006). Vol. E. ch. 1.1, pp. 2-4
https://doi.org/10.1107/97809553602060000646