(International Tables for Crystallography) Symbols for crystallographic items used in this volume

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, Symbols for crystallographic items.

Symbols for crystallographic items used in this volume

M. I. Aroyo^a

^aDepartamento de Física de la Materia Condensada, Universidad del País Vasco (UPV/EHU), Bilbao, Spain

Direct space: points and vectors
$[{\bb E}^n]$	n-dimensional Euclidean point space
$[{\bb V}^n]$	n-dimensional vector space
$[{\bb R}]$ , $[{\bb Q}]$ , $[{\bb Z}]$	the field of real numbers, the field of rational numbers, the ring of integers
L	lattice in $[{\bb V}^3]$
$[{\sf L}]$	line in $[{\bb E}^3]$
a, b, c; or a_i	basis vectors of the lattice
a, b, c; or \|a\|, \|b\|, \|c\|	lengths of basis vectors, lengths of cell edges	$[\left\}\matrix{\cr{\rm lattice}\hfill\cr{\rm parameters}\hfill\cr\vphantom{1}}\right.]$
α, β, γ; or α_j	interaxial angles $[\angle ({\bf b}, {\bf c})]$ , $[\angle ({\bf c}, {\bf a})]$ , $[\angle ({\bf a}, {\bf b})]$
G, g_ik	fundamental matrix (metric tensor) and its coefficients
V	cell volume
X, Y, Z, P	points
r, d, x, v, u	vectors, position vectors
r, \|r\|	norm, length of a vector
x = xa + yb + zc	vector with coefficients x, y, z
x, y, z; or x_i	point coordinates expressed in units of a, b, c; coefficients of a vector
$[{\bi x} = \pmatrix{x\cr y\cr z\cr} \equiv \pmatrix{x_{1}\cr x_{2}\cr x_{3}\cr}]$	column of point coordinates or vector coefficients
t	translation vector
t₁, t₂, t₃; 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
O	origin
o	zero vector (all coefficients zero)
o	(3 × 1) 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 $[x_{i}']$	vector and point coordinates after a transformation of the coordinate system (basis transformation)
$[{\bi x}' = \pmatrix{x'\cr y'\cr z'\cr}]$	column of coordinates after a transformation of the coordinate system (basis transformation)
$[\tilde{X}]$	image of a point X after the action of a symmetry operation
$[\tilde{x}, \tilde{y}, \tilde{z}]$ ; or $[\tilde{x}_i]$	coordinates of an image point $[\tilde{X}]$
$[\tilde{\bi x} = \pmatrix{\tilde{x}\cr \tilde{y}\cr \tilde{z}\cr}]$	column of coordinates of an image point $[\tilde{X}]$
$[\specialfonts{\bbsf x}]$ , or $[\specialfonts{\bbsf r}]$	(3 + 1) × 1 `augmented' columns of point coordinates or vector coefficients

Directions and planes
[uvw]	indices of a lattice direction (zone axis)
〈uvw〉	indices of a set of all symmetry-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 symmetry-equivalent crystal faces (`crystal form'), or net planes
{hkil}	indices of a set of all symmetry-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)

Reciprocal space
L*	reciprocal lattice
a, b, c; or $[{\bf a}_i^]$	basis vectors of the reciprocal lattice
a, b, c; or \|a\|, \|b\|, \|c\|	lengths of basis vectors of the reciprocal lattice
$[\alpha^,\beta^,\gamma^]$ ; or $[\alpha_j^]$	interaxial angles $[\angle ({\bf b}^, {\bf c}^)]$ , $[\angle ({\bf c}^, {\bf a}^)]$ , $[\angle ({\bf a}^, {\bf b}^)]$ of the reciprocal lattice
r*, or h	vector in reciprocal space, or vector of reciprocal lattice
r, or \|r\|	length of a vector in reciprocal space
h, k, l; or h_i	coefficients of a reciprocal-lattice vector
h = (h, k, l)	(1 × 3) row of coefficients of a reciprocal-lattice vector
V*	cell volume of the reciprocal lattice
G, $[{g}^_{ik}]$	fundamental matrix (metric tensor) of the reciprocal lattice and its coefficients

Functions
$[\rho(xyz)]$	electron density at the point x, y, z
	Patterson function for a vector with coefficients u, v, w
, 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

Mappings, symmetry operations and their matrix–column presentation
A, B, W	(3 × 3) matrices describing the linear part of a mapping
A_ik, W_ik	matrix coefficients
I	(3 × 3) unit matrix
A^T	matrix A transposed
det(A), tr(A)	determinant of matrix A, trace of matrix A

$[{\bi w}=\pmatrix{w_1\cr w_2\cr w_3}]$	(3 × 1) column of coefficients w_i describing the translation part of a mapping
w_g	intrinsic translation part of a symmetry operation
w_l	location translation part of a symmetry operation
$[ \ispecialfonts{\sfi A}, {\sfi I}, {\sfi W}]$	mappings, symmetry operations
t	translation symmetry operation
(W, w)	matrix–column pair of a symmetry operation given by a (3 × 3) matrix W and a (3 × 1) column w
(I, t)	matrix–column pair of a translation
(I, o)	matrix–column pair of the identity
(P, p)	transformation of the coordinate system, described by a (3 × 3) matrix P and a (3 × 1) column p
(Q, q)	inverse transformation of (P, p): (Q, q) = (P, p)⁻¹
$[\specialfonts{\bbsf W}]$	symmetry operation $[ \ispecialfonts{\sfi W}]$ , described by a (3 + 1) × (3 + 1) `augmented' matrix
$[\specialfonts{\bbsf P}]$	transformation of the coordinate system, described by a (3 + 1) × (3 + 1) `augmented' matrix
$[\specialfonts{\bbsf Q}]$	inverse transformation of $[\specialfonts{\bbsf P}]$ : $[\specialfonts{\bbsf Q}]$ = $[\specialfonts{\bbsf P}]$ ⁻¹
$[\{{\bi R}\|{\bi v}\}]$	Seitz symbol of a symmetry operation

Groups
$[{\cal G}]$	group, space group
$[{\cal H}, {\cal U}]$	subgroups
$[{\cal I}]$	trivial group, consisting of the unit element $[\ispecialfonts{\sfi e}]$ only
$[{\cal P}, {\cal S}, {\cal F}, {\cal D}, {\cal R}]$	groups
$[\|{\cal G}\|]$	order of the group $[{\cal G}]$
i, or [i]	index of a subgroup in a group
$[{\cal T}]$ , or $[{\cal T_G}]$	group of all translations of a space group, or of the space group $[{\cal G}]$
$[{\cal P}]$ , or $[{\cal P_G}]$	point group of a space group, or of the space group $[{\cal G}]$
$[{\cal M}]$	Hermann's group
$[{\cal A}]$	group of all affine mappings (affine group)
$[{\cal E}]$	group of all isometries (motions) (Euclidean group)
$[{\cal E}^+]$	group of chirality-preserving isometries
$[\varphi]$ , $[ker(\varphi)]$	homomorphic mapping (homomorphism), kernel of homomorphism $[\varphi]$
$[{\cal G}/{\cal H}]$	factor group or quotient group of $[{\cal G}]$ by $[{\cal H}]$
$[{\cal N_G(H)}]$	normalizer of $[{\cal H}]$ in $[{\cal G}]$
$[{\cal N_E(G)}]$ , or $[{\cal N_{E^+}(G)}]$	Euclidean or chirality-preserving Euclidean normalizer of the space group $[{\cal G}]$
$[{\cal N_A(G)}]$	affine normalizer of $[{\cal G}]$
$[{\cal G}(\omega)]$	orbit of $[\omega]$ under the group $[{\cal G}]$
$[{\cal S_G}(\omega)]$ , $[{\cal S_H}(\omega)]$	stabilizer of $[\omega]$ in the group $[{\cal G}]$ , or $[{\cal H}]$
$[{\cal O}={\cal G}(X)]$	orbit of point X under the group $[{\cal G}]$
$[{\cal S}_X={\cal S_G}(X)]$	site-symmetry group of point X
$[{\scr E}]$	eigensymmetry group of an orbit $[{\cal O}]$
$[\ispecialfonts{\sfi a}, {\sfi b}, {\sfi g}, {\sfi h}, {\sfi m}, {\sfi t}]$	group elements
$[ \ispecialfonts{\sfi e}]$	unit element of a group
$[\ispecialfonts{\sfi t}]$	element of the translation group $[{\cal T}]$