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
doi: 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.

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 [\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}]
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 symbolExplanation
[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 symbolExplanation
[{\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 symbolExplanation
[\rho (xyz)] Electron density at the point x, y, z
[P(xyz)] Patterson function at the point x, y, z
[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)]

1.1.5. Spaces

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Printed symbolExplanation
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 symbolExplanation
[{\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 symbolExplanation
[{\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}]








































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