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 |
doi: 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[link] 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[link] to 1.1.3[link][link] follow those given in Part 1[link] of Volume A of International Tables for Crystallography (2005[link]).

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[\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 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 ⊥[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
⊥[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
  ⊥[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
  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, 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 [Scheme scheme1] None m
Glide plane, glide line [Scheme scheme2] ½ along line parallel to projection plane; ½ along line in plane a, b or c; g
Glide plane [Scheme scheme3] 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 [Scheme scheme4] None m
Glide plane [Scheme scheme5] ½ in the direction of arrow a, b or c
`Double' glide plane [Scheme scheme6] Two glide vectors; ½ in either of the directions of the two arrows e
`Diagonal' glide plane [Scheme scheme7] 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 [Scheme scheme8] None 2
Twofold screw axis: `2 sub 1' [Scheme scheme9] [\textstyle{1 \over 2}] 21
Threefold rotation axis [Scheme scheme10] None 3
Threefold screw axis: `3 sub 1' [Scheme scheme11] [\textstyle{1 \over 3}] 31
Threefold screw axis: `3 sub 2' [Scheme scheme12] [\textstyle{2 \over 3}] 32
Fourfold rotation axis [Scheme scheme13] None 4
Fourfold screw axis: `4 sub 1' [Scheme scheme14] [\textstyle{1 \over 4}] 41
Fourfold screw axis: `4 sub 2' [Scheme scheme15] [\textstyle{1 \over 2}] 42
Fourfold screw axis: `4 sub 3' [Scheme scheme16] [\textstyle{3 \over 4}] 43
Sixfold rotation axis [Scheme scheme17] None 6
Sixfold screw axis: `6 sub 1' [Scheme scheme18] [\textstyle{1 \over 6}] 61
Sixfold screw axis: `6 sub 2' [Scheme scheme19] [\textstyle{1 \over 3}] 62
Sixfold screw axis: `6 sub 3' [Scheme scheme20] [\textstyle{1 \over 2}] 63
Sixfold screw axis: `6 sub 4' [Scheme scheme21] [\textstyle{2 \over 3}] 64
Sixfold screw axis: `6 sub 5' [Scheme scheme22] [\textstyle{5 \over 6}] 65
Centre of symmetry, inversion centre: `1 bar' [Scheme scheme23] None [\bar{1}]
Twofold rotation axis with centre of symmetry [Scheme scheme24] None 2/m
Twofold screw axis with centre of symmetry [Scheme scheme25] [\textstyle{1 \over 2}] 21/m
Inversion axis: `3 bar' [Scheme scheme26] None [\bar{3}]
Inversion axis: `4 bar' [Scheme scheme27] None [\bar{4}]
Fourfold rotation axis with centre of symmetry [Scheme scheme28] None 4/m
`4 sub 2' screw axis with centre of symmetry [Scheme scheme29] [\textstyle{1 \over 2}] 42/m
Inversion axis: `6 bar' [Scheme scheme30] None [\bar{6}]
Sixfold rotation axis with centre of symmetry [Scheme scheme31] None 6/m
`6 sub 3' screw axis with centre of symmetry [Scheme scheme32] [\textstyle{1 \over 2}] 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 [Scheme scheme33] None 2
Twofold screw axis [Scheme scheme34] [\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).]








































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