International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A. ch. 1.1, pp. 2-3
https://doi.org/10.1107/97809553602060000500

Chapter 1.1. Printed symbols for crystallographic items

Th. Hahna*

a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany
Correspondence e-mail: [email protected]

This chapter lists the printed symbols used for crystallographic items in this volume.

Keywords: symbols; vectors; reciprocal space; groups; crystallography.

1.1.1. Vectors, coefficients and coordinates

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Printed symbol Explanation
a , b, c; or ai Basis vectors of the direct lattice
a , b, c Lengths of basis vectors, lengths of cell edges Lattice or cell parameters
α, β, γ Interaxial (lattice) angles b∧c, c∧a, a∧b
V Cell volume of the direct lattice
G Matrix of the geometrical coefficients (metric tensor) of the direct lattice
g ij Element of metric matrix (tensor) G
r ; or x Position vector (of a point or an atom)
r Length of the position vector r
x a , yb, zc Components of the position vector r
x , y, z; or xi Coordinates of a point (location of an atom) expressed in units of a, b, c; coordinates of end point of position vector r; coefficients of position vector r
x = ⎛⎝ xyz ⎞⎠ = ⎛⎝ x 1 x 2 x 3 ⎞⎠ Column of point coordinates or vector coefficients
t Translation vector
t Length of the translation vector t
t 1 , t 2 , t 3 ; or ti Coefficients of translation vector t
t = ⎛⎝ t 1 t 2 t 3 ⎞⎠ Column of coefficients of translation vector t
u Vector with integral coefficients
u , v, w; or ui Integers, coordinates of a (primitive) lattice point; coefficients of vector u
u = ⎛⎝ uvw ⎞⎠ = ⎛⎝ u 1 u 2 u 3 ⎞⎠ Column of integral point coordinates or vector coefficients
o Zero vector
o Column of zero coefficients
a ′, b′, c′; or ai New basis vectors after a transformation of the coordinate system (basis transformation)
r ′; or x′; x′, y′, z′; or xi Position vector and point coordinates after a transformation of the coordinate system (basis transformation)
˜ r ; or ˜x; ˜x, ˜y, ˜z; or ˜xi New position vector and point coordinates after a symmetry operation (motion)

1.1.2. Directions and planes

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Printed symbol Explanation
[uvw] Indices of a lattice direction (zone axis)
uvw Indices of a set of all symmetrically equivalent lattice directions
(hkl) Indices of a crystal face, or of a single net plane (Miller indices)
(hkil) Indices of a crystal face, or of a single net plane, for the hexagonal axes a1, a2, a3, c (Bravais–Miller indices)
{ hkl } Indices of a set of all symmetrically equivalent crystal faces (`crystal form'), or net planes
{ hkil } Indices of a set of all symmetrically equivalent crystal faces (`crystal form'), or net planes, for the hexagonal axes a1, a2, a3, c
hkl Indices of the Bragg reflection (Laue indices) from the set of parallel equidistant net planes (hkl)
d hkl Interplanar distance, or spacing, of neighbouring net planes (hkl)

1.1.3. Reciprocal space

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Printed symbol Explanation
a ∗ , b∗, c∗; or a∗i Basis vectors of the reciprocal lattice
a ∗ , b∗, c∗ Lengths of basis vectors of the reciprocal lattice
α ∗ , β∗, γ∗ Interaxial (lattice) angles of the reciprocal lattice b∗∧c∗, c∗∧a∗, a∗∧b∗
r ∗ ; or h Reciprocal-lattice vector
h , k, l; or hi Coordinates of a reciprocal-lattice point, expressed in units of a∗, b∗, c∗, coefficients of the reciprocal-lattice vector r∗
V ∗ Cell volume of the reciprocal lattice
G ∗ Matrix of the geometrical coefficients (metric tensor) of the reciprocal lattice

1.1.4. Functions

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Printed symbol Explanation
ρ ( xyz ) Electron density at the point x, y, z
P ( xyz ) Patterson function at the point x, y, z
F ( hkl ) ; or F Structure factor (of the unit cell), corresponding to the Bragg reflection hkl
| F ( hkl ) | ; or |F| Modulus of the structure factor F(hkl)
α ( hkl ) ; or α Phase angle of the structure factor F(hkl)

1.1.5. Spaces

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Printed symbol Explanation
n Dimension of a space
X Point
˜ X Image of a point X after a symmetry operation (motion)
E n (Euclidean) point space of dimension n
V n Vector space of dimension n
L Vector lattice
L Point lattice

1.1.6. Motions and matrices

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Printed symbol Explanation
W ; M Symmetry operation; motion
(W, w) Symmetry operation W, described by an (n×n) matrix W and an (n×1) column w
W Symmetry operation W, described by an (n+1)×(n+1) `augmented' matrix
I ( n × n ) unit matrix
T Translation
(I, t) Translation T, described by the (n×n) unit matrix I and an (n×1) column t
T Translation T, described by an (n+1)×(n+1) `augmented' matrix
I Identity operation
(I, o) Identity operation I, described by the (n×n) unit matrix I and the (n×1) column o
I Identity operation I, described by the (n+1)×(n+1) `augmented' unit matrix
r , or x Position vector (of a point or an atom), described by an (n+1)×1 `augmented' column
(P, p); or (S, s) Transformation of the coordinate system, described by an (n×n) matrix P or S and an (n×1) column p or s
P ; or S Transformation of the coordinate system, described by an (n+1)×(n+1) `augmented' matrix
(Q, q) Inverse transformation of (P, p)
Q Inverse transformation of P

1.1.7. Groups

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Printed symbol Explanation
G Space group
T Group of all translations of G
S Supergroup; also used for site-symmetry group
H Subgroup
E Group of all motions (Euclidean group)
A Group of all affine mappings (affine group)
N E ( G ) ; or NA(G) Euclidean or affine normalizer of a space group G
P Point group
C Eigensymmetry (inherent symmetry) group
[i] Index i of sub- or supergroup
G Element of a space group G








































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