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

International Tables for Crystallography (2010). Vol. E, ch. 1.1, pp. 2-4   | 1 | 2 |
https://doi.org/10.1107/97809553602060000782

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

V. Kopskýa and D. B. Litvinb*

aFreelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and bDepartment of Physics, The Eberly College of Science, 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.

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 symbolExplanation
a; b; c Basis vectors of direct lattice
a; b; c Length of basis vectors
α; β; γ Interaxial (lattice) angles b c, c a, a 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 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 symbolSymmetry element and its orientationGenerating symmetry operation with glide or screw vector
m Reflection plane, mirror plane (three dimensions) Reflection through a plane
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 ⊥ or ⊥ ½a
b ⊥ or ⊥ ½b
c ⊥ or ⊥ ½c
⊥[1 0] or ⊥ ½c
⊥ or ⊥ or ⊥[  0] ½c, hexagonal coordinate system
⊥[1 0] or ⊥ or ⊥[  0] ½c, hexagonal coordinate system
n Diagonal' glide plane (in noncentred cells only) Glide reflection through a plane, with glide vector
⊥ ½(a + b)
e Double' glide plane ⊥ (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
⊥; ⊥ 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
n-fold rotation point, n (two dimensions) Counterclockwise rotation of 360/n degrees about a point Centre of symmetry, inversion centre Inversion through a point = m, , , Rotoinversion axis, 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, np 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 lineGraphical symbolGlide vectors in units of lattice translation vectors parallel and normal to the projection planePrinted 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 planeGraphical symbolGlide vector in units of lattice translation vectors parallel to the projection planePrinted 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 pointGraphical symbolScrew vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axisPrinted symbol
Twofold rotation axis, twofold rotation point None 2
Twofold screw axis: 2 sub 1'  21
Threefold rotation axis None 3
Threefold screw axis: 3 sub 1'  31
Threefold screw axis: 3 sub 2'  32
Fourfold rotation axis None 4
Fourfold screw axis: 4 sub 1'  41
Fourfold screw axis: 4 sub 2'  42
Fourfold screw axis: 4 sub 3'  43
Sixfold rotation axis None 6
Sixfold screw axis: 6 sub 1'  61
Sixfold screw axis: 6 sub 2'  62
Sixfold screw axis: 6 sub 3'  63
Sixfold screw axis: 6 sub 4'  64
Sixfold screw axis: 6 sub 5'  65
Centre of symmetry, inversion centre: 1 bar' None Twofold rotation axis with centre of symmetry None 2/m
Twofold screw axis with centre of symmetry  21/m
Inversion axis: 3 bar' None Inversion axis: 4 bar' None Fourfold rotation axis with centre of symmetry None 4/m
4 sub 2' screw axis with centre of symmetry  42/m
Inversion axis: 6 bar' None Sixfold rotation axis with centre of symmetry None 6/m
6 sub 3' screw axis with centre of symmetry  63/m
 (d) Symmetry axes parallel to plane of projection.
Symmetry axisGraphical symbolScrew vector of a right-handed screw rotation in units of the shortest lattice translation vector parallel to the axisPrinted symbol
Twofold rotation axis None 2
Twofold screw axis  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).]