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

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*

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

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 symbolExplanation
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 , ,
V Cell volume of the direct lattice
G Matrix of the geometrical coefficients (metric tensor) of the direct lattice
Element of metric matrix (tensor) G
r; or x Position vector (of a point or an atom)
r Length of the position vector r
xa, yb, zc Components of the position vector r
x, y, z; or 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
Column of point coordinates or vector coefficients
t Translation vector
t Length of the translation vector t
; or Coefficients of translation vector t
Column of coefficients of translation vector t
u Vector with integral coefficients
u, v, w; or Integers, coordinates of a (primitive) lattice point; coefficients of vector u
Column of integral point coordinates or vector coefficients
o Zero vector
o Column of zero coefficients
a′, b′, c′; or New basis vectors after a transformation of the coordinate system (basis transformation)
r′; or x′; x′, y′, z′; or Position vector and point coordinates after a transformation of the coordinate system (basis transformation)
; or ; , , ; or New position vector and point coordinates after a symmetry operation (motion)

### 1.1.2. Directions and planes

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Printed symbolExplanation
[uvw] Indices of a lattice direction (zone axis)
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 , , , c (Bravais–Miller indices)
Indices of a set of all symmetrically equivalent crystal faces (crystal form'), or net planes
Indices of a set of all symmetrically equivalent crystal faces (crystal form'), or net planes, for the hexagonal axes , , , c
hkl Indices of the Bragg reflection (Laue indices) from the set of parallel equidistant net planes (hkl)
Interplanar distance, or spacing, of neighbouring net planes (hkl)

### 1.1.3. Reciprocal space

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Printed symbolExplanation
, , ; or Basis vectors of the reciprocal lattice
, , Lengths of basis vectors of the reciprocal lattice
, , Interaxial (lattice) angles of the reciprocal lattice , ,
; or h Reciprocal-lattice vector
h, k, l; or Coordinates of a reciprocal-lattice point, expressed in units of , , , coefficients of the reciprocal-lattice vector
Cell volume of the reciprocal lattice
Matrix of the geometrical coefficients (metric tensor) of the reciprocal lattice

### 1.1.4. Functions

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

### 1.1.5. Spaces

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Printed symbolExplanation
n Dimension of a space
X Point
Image of a point X after a symmetry operation (motion)
(Euclidean) point space of dimension n
Vector space of dimension n
L Vector lattice
L Point lattice

### 1.1.6. Motions and matrices

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

### 1.1.7. Groups

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Printed symbolExplanation
Space group
Group of all translations of
Supergroup; also used for site-symmetry group
Subgroup
Group of all motions (Euclidean group)
Group of all affine mappings (affine group)
; or Euclidean or affine normalizer of a space group
Point group
Eigensymmetry (inherent symmetry) group
[i] Index i of sub- or supergroup
Element of a space group