(International Tables for Crystallography) Printed symbols for crystallographic items

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

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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. Hahn^a ^*

^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 symbol	Explanation
a, b, c; or a_i	Basis vectors of the direct lattice
a, b, c	Lengths of basis vectors, lengths of cell edges	$[\Bigg\}]$ Lattice or cell parameters
α, β, γ	Interaxial (lattice) angles $[{\bf b} \wedge {\bf c}]$ , $[{\bf c} \wedge {\bf a}]$ , $[{\bf a} \wedge {\bf b}]$	$[\Bigg\}]$ Lattice or cell parameters
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
xa, yb, zc	Components of the position vector r
x, y, z; or $[x_{i}]$	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
$[{\bi x} = \pmatrix{x\cr y\cr z\cr} = \pmatrix{x_{1}\cr x_{2}\cr x_{3}\cr}]$	Column of point coordinates or vector coefficients
t	Translation vector
t	Length of the translation vector t
$[t_{1},\ t_{2},\ t_{3}]$ ; or $[t_{i}]$	Coefficients of translation vector t
$[{\bi t} = \pmatrix{t_{1}\cr t_{2}\cr t_{3}\cr}]$	Column of coefficients of translation vector t
u	Vector with integral coefficients
u, v, w; or $[u_{i}]$	Integers, coordinates of a (primitive) lattice point; coefficients of vector u
$[{\bi u} = \pmatrix{u\cr v\cr w\cr} = \pmatrix{u_{1}\cr u_{2}\cr u_{3}\cr}]$	Column of integral point coordinates or vector coefficients
o	Zero vector
o	Column of zero coefficients
a′, b′, c′; or $[{\bf a}_{i}']$	New basis vectors after a transformation of the coordinate system (basis transformation)
r′; or x′; x′, y′, z′; or $[x_{i}']$	Position vector and point coordinates after a transformation of the coordinate system (basis transformation)
$[\tilde{{\bf r}}]$ ; or $[\tilde{{\bf x}}]$ ; $[\tilde{x}]$ , $[\tilde{y}]$ , $[\tilde{z}]$ ; or $[\tilde{x}_{i}]$	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)
$[\langle uvw \rangle]$	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 $[{\bf a}_{1}]$ , $[{\bf a}_{2}]$ , $[{\bf a}_{3}]$ , 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 $[{\bf a}_{1}]$ , $[{\bf a}_{2}]$ , $[{\bf a}_{3}]$ , 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
$[{\bf a}^{}]$ , $[{\bf b}^{}]$ , $[{\bf c}^{}]$ ; or $[{\bf a}^{}_{i}]$	Basis vectors of the reciprocal lattice
$[a^{}]$ , $[b^{}]$ , $[c^{*}]$	Lengths of basis vectors of the reciprocal lattice
$[\alpha^{}]$ , $[\beta^{}]$ , $[\gamma^{*}]$	Interaxial (lattice) angles of the reciprocal lattice $[{\bf b}^{} \wedge {\bf c}^{}]$ , $[{\bf c}^{} \wedge {\bf a}^{}]$ , $[{\bf a}^{} \wedge {\bf b}^{}]$
$[{\bf r}^{*}]$ ; or h	Reciprocal-lattice vector
h, k, l; or $[h_{i}]$	Coordinates of a reciprocal-lattice point, expressed in units of $[a^{}]$ , $[b^{}]$ , $[c^{}]$ , coefficients of the reciprocal-lattice vector $[{\bf r}^{}]$
$[V^{*}]$	Cell volume of the reciprocal lattice
$[{\bf G}^{*}]$	Matrix of the geometrical coefficients (metric tensor) of the reciprocal lattice

1.1.4. Functions

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Printed symbol	Explanation
$[\rho (xyz)]$	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
$[\alpha (hkl)]$ ; or α	Phase angle of the structure factor

1.1.5. Spaces

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Printed symbol	Explanation
n	Dimension of a space
X	Point
$[\tilde{X}]$	Image of a point X after a symmetry operation (motion)
$[E^{n}]$	(Euclidean) point space of dimension n
$[{\bf V}^{n}]$	Vector space of dimension n
L	Vector lattice
L	Point lattice

1.1.6. Motions and matrices

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

1.1.7. Groups

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