How to find the correct transformation

Bilbao Crystallographic Server Transformation matrix

The transformation matrix

The relation between an arbitrary setting of a space group (given by a set of basis vectors (a, b, c) and an origin O) and a reference (default) coordinate system, defined by the set (a', b', c') and the origin O^', is determined by a (3x4) matrix - column pair (P,p). The (3x3) linear matrix P

P =

P₁₁ P₁₂ P₁₃

P₂₁ P₂₂ P₂₃

P₃₁ P₃₂ P₃₃

describes the transformation of the row of basis vectors (a, b, c) to the reference basis vectors (a', b', c').

a' = P₁₁a + P₂₁b + P₃₁c

b' = P₁₂a + P₂₂b + P₃₂c

c' = P₁₃a + P₂₃b + P₃₃c

which is often written as

(a', b', c') = (a, b, c)P

The (3x1) column p = (p₁, p₂, p₃) determines the origin shift of O^' with respect the origin O:

O^' = O + p₁a+p₂b+p₃c

[*] For more information: International Tables for Crystallography. Vol. A, Space Group Symmetry. Ed. Theo Hahn (3rd ed.), Dordrecht, Kluwer Academic Publishers, Section "Transformations in crystallography", 1995.

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