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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. A. ch. 1.1, p. 2

Section 1.1.1. Vectors, coefficients and coordinates

Th. Hahna*

a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

1.1.1. Vectors, coefficients and coordinates

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Printed symbol Explanation
a , b, c; or ai Basis vectors of the direct lattice
a , b, c Lengths of basis vectors, lengths of cell edges Lattice or cell parameters
α, β, γ Interaxial (lattice) angles b∧c, c∧a, a∧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
x a , yb, zc Components of the position vector r
x , y, z; or xi 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
x = ⎛⎝ xyz ⎞⎠ = ⎛⎝ x 1 x 2 x 3 ⎞⎠ Column of point coordinates or vector coefficients
t Translation vector
t Length of the translation vector t
t 1 , t 2 , t 3 ; or ti Coefficients of translation vector t
t = ⎛⎝ t 1 t 2 t 3 ⎞⎠ Column of coefficients of translation vector t
u Vector with integral coefficients
u , v, w; or ui Integers, coordinates of a (primitive) lattice point; coefficients of vector u
u = ⎛⎝ uvw ⎞⎠ = ⎛⎝ u 1 u 2 u 3 ⎞⎠ Column of integral point coordinates or vector coefficients
o Zero vector
o Column of zero coefficients
a ′, b′, c′; or ai New basis vectors after a transformation of the coordinate system (basis transformation)
r ′; or x′; x′, y′, z′; or xi Position vector and point coordinates after a transformation of the coordinate system (basis transformation)
˜ r ; or ˜x; ˜x, ˜y, ˜z; or ˜xi New position vector and point coordinates after a symmetry operation (motion)








































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