(International Tables for Crystallography) List of symbols and abbreviations used in this volume

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2011). Vol. A1, List of symbols and abbreviations.

List of symbols and abbreviations used in this volume

Mois I. Aroyo,^a ^* Ulrich Müller^b and Hans Wondratschek^c

^aDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain,^bFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany, and ^cInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: wmpararm@lg.ehu.es

(1) Points and point space
P, Q, R, X	points
O	origin
$[A_n,{ \bb A}_n, P_n]$	n-dimensional affine space
$[E_n, {\bb E}_n]$	n-dimensional Euclidean point space
; or	point coordinates
x	column of point coordinates
$[\tilde{X}]$	image point
$[\tilde{\bi{x}}]$	column of coordinates of an image point
$[\tilde{x}_i]$	coordinates of an image point
$[\bi{x}']$	column of coordinates in a new coordinate system (after basis transformation)
	coordinates in a new coordinate system

(2) Vectors and vector space
a, b, c; or a_i	basis vectors of the space
r, x	vectors, position vectors
o	zero vector (all coefficients zero)
a, b, c	lengths of basis vectors	$[\left\}\matrix{ {\rm lattice}\hfill\cr {\rm parameters}\hfill}\right.]$
α, β, γ; or α_j	angles between basis vectors
r	column of vector coefficients
r_i	vector coefficients
(a)^T	row of basis vectors
V_n	n-dimensional vector space

(3) Mappings and their matrices and columns
A, W	(3 × 3) matrices
A^T	matrix A transposed
I	(3 × 3) unit matrix
A_ik, W_ik	coefficients
(A, a), (W, w)	matrix–column pairs
$[\specialfonts\bbsf W]$	augmented matrix
$[\specialfonts{\bbsf x}, \tilde{\bbsf x}, {\bbsf t}]$	augmented columns
P, $[\specialfonts\bbsf P]$	transformation matrices
$[{\sf A}, {\sf I}, {\sf W}]$	mappings
w	column of the translation part of a mapping
w_i	coefficients of the translation part of a mapping
G, G_ik	fundamental matrix and its coefficients
det(…)	determinant of a matrix
tr(…)	trace of a matrix

(4) Groups
$[\cal G]$	group; space group
$[\cal R]$	space group (Chapter 1.4 )
$[{\cal H}, {\cal U}]$	subgroups of $[\cal G]$
$[\cal M]$	maximal subgroup of $[\cal G]$ (Chapter 1.4 )
$[\cal M]$	Hermann's group (Chapters 1.2 , 1.7 , 2.1 )
$[{\cal P}, {\cal S}, {\cal V}, {\cal Z}]$	groups or sets of group elements, e.g. cosets
$[\cal T(G), T(R)]$	group of all translations of $[\cal G, R]$
$[\cal A]$	group of all affine mappings = affine group
$[\cal E]$	group of all isometries (motions) = Euclidean group
$[\cal F]$	factor group
$[\cal I]$	trivial group, consisting of the unit element $[\ispecialfonts\sfi e]$ only
$[\cal N]$	normal subgroup
$[\cal O]$	group of all orthogonal mappings = orthogonal group
$[\cal N_G(H)]$	normalizer of $[\cal H]$ in $[\cal G]$
$[\cal N_E(H)]$	Euclidean normalizer of $[\cal H]$
$[\cal N_A(H)]$	affine normalizer of $[\cal H]$
$[\cal P_G, P_H]$	point groups of the space groups $[\cal G]$ , $[\cal H]$
$[{\cal S_G}(X), {\cal S_H}(X)]$	site-symmetry groups of point X in the space groups $[\cal G]$ , $[\cal H]$
$[\ispecialfonts{\sfi a}, {\sfi b}, {\sfi g}, {\sfi h}, {\sfi m}, {\sfi t}]$	group elements
$[\ispecialfonts\sfi e]$	unit element
$[{\bi 2}, {\bi 2}_1, {\bi m}, {\bi \bar 1},\ldots]$	symmetry operations
i or [i]	index of $[\cal H]$ in $[\cal G]$

(5) Symbols used in the tables
p	prime number > 1
n, n′, n′′, n′′′	arbitrary positive integer numbers
q, r, u, v, w	arbitrary integer numbers in the given range
a, b, c	basis vectors of the space group
a′, b′, c′	basis vectors of the subgroup or supergroup
x, y, z	point coordinates in the space group
t(1, 0, 0), t(0, 1, 0),…	generating translations

(6) Abbreviations
HM symbol	Hermann–Mauguin symbol
IT A	International Tables for Crystallography Volume A
PCA	parent-clamping approximation
k-subgroup	klassengleiche subgroup
t-subgroup	translationengleiche subgroup

International Tables for Crystallography (2011). Vol. A1, List of symbols and abbreviations.