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. Aroyoa

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 ai 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 αj interaxial angles [\angle ({\bf b}, {\bf c})], [\angle ({\bf c}, {\bf a})], [\angle ({\bf a}, {\bf b})]
G, gik 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 xi 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 ti 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)
dhkl 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 hi 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
[P(uvw)] Patterson function for a vector with coefficients u, v, w
[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)]
[\alpha(hkl)], or α phase angle of the structure factor [F(hkl)]

Mappings, symmetry operations and their matrix–column presentation
A, B, W (3 × 3) matrices describing the linear part of a mapping
Aik, Wik matrix coefficients
I (3 × 3) unit matrix
AT 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 wi describing the translation part of a mapping
wg intrinsic translation part of a symmetry operation
wl 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}]








































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