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

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

Section 1.1.1. Vectors, coefficients and coordinates

Th. Hahna*

aInstitut 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 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)








































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