International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2015). Vol. A, ch. 1.5, pp. 75-83

Section 1.5.1. Origin shift and change of the basis1

H. Wondratscheka and M. I. Aroyob

1.5.1. Origin shift and change of the basis1

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1.5.1.1. Origin shift

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Let a coordinate system be given with a basis [{\bf a},{\bf b}, {\bf c}] and an origin O. Referred to this coordinate system, the column of coordinates of a point X is [{\bi x}= \pmatrix{x_1 \cr x_2 \cr x_3 }] and the corresponding vector is [{\bf x}=x_1{\bf a}+x_2{\bf b}+x_3 {\bf c}]. Referred to a new coordinate system, specified by the basis [{\bf a}',{\bf b}',{\bf c}'] and the origin [O'], the column of coordinates of the point X is [{\bi x}'=\pmatrix{ x_1' \cr x_2' \cr x_3' }]. Let [{\bi p}=\ \buildrel{\longrightarrow}\over{OO'}=\pmatrix{ p_1 \cr p_2 \cr p_3 }] be the column of coefficients for the vector p from the old origin O to the new origin [O'], see Fig. 1.5.1.1[link].

Table 1.5.1.1| top | pdf |
Selected 3 × 3 transformation matrices [{\bi P}] and [{\bi Q} = {\bi P}^{ -1}]

For inverse transformations (against the arrow) replace P by Q and vice versa.

TransformationP[{\bi Q} = {\bi P}^{-1}]Crystal system
Cell choice 1 [\rightarrow] cell choice 2: [\cases{P \rightarrow P\cr C \rightarrow A\cr}] [\pmatrix{\bar{1} &0 &1\cr 0 &1 &0\cr \bar{1} &0 &0\cr}] [\pmatrix{0 &0 &\bar{1}\cr 0 &1 &0\cr 1 &0 &\bar{1}\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2 [\rightarrow] cell choice 3: [\cases{P \rightarrow P\cr A \rightarrow I\cr}] Unique axis b invariant
Cell choice 3 [\rightarrow] cell choice 1: [\cases{P \rightarrow P\cr I \rightarrow C\cr}]
(Fig. 1.5.1.2[link]a)      
Cell choice 1 [\rightarrow] cell choice 2: [\cases{P \rightarrow P\cr A \rightarrow B\cr}] [\pmatrix{0 &\bar{1} &0\cr 1 &\bar{1} &0\cr 0 &0 &1\cr}] [\pmatrix{\bar{1} &1 &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2 [\rightarrow] cell choice 3: [\cases{P \rightarrow P\cr B \rightarrow I\cr}] Unique axis c invariant
Cell choice 3 [\rightarrow] cell choice 1: [\cases{P \rightarrow P\cr I \rightarrow A\cr}]
(Fig. 1.5.1.2[link]b)      
Cell choice 1 [\rightarrow] cell choice 2: [\cases{P \rightarrow P\cr B \rightarrow C\cr}] [\pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &\bar{1}\cr}] [\pmatrix{1 &0 &0\cr 0 &\bar{1} &1\cr 0 &\bar{1} &0\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2 [\rightarrow] cell choice 3: [\cases{P \rightarrow P\cr C \rightarrow I\cr}] Unique axis a invariant
Cell choice 3 [\rightarrow] cell choice 1: [\cases{P \rightarrow P\cr I \rightarrow B\cr}]
(Fig. 1.5.1.2[link]c)      
Unique axis b [\rightarrow] unique axis c      
Cell choice 1: [\cases{P \rightarrow P\cr C \rightarrow A\cr}] [\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}] [\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2: [\cases{P \rightarrow P\cr A \rightarrow B\cr}] Cell choice invariant
Cell choice 3: [\cases{P \rightarrow P\cr I \rightarrow I\cr}]
Unique axis b [\rightarrow] unique axis a      
Cell choice 1: [\cases{P \rightarrow P\cr C \rightarrow B\cr}] [\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}] [\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2: [\cases{P \rightarrow P\cr A \rightarrow C\cr}] Cell choice invariant
Cell choice 3: [\cases{P \rightarrow P\cr I \rightarrow I\cr}]
Unique axis c [\rightarrow] unique axis a      
Cell choice 1: [\cases{P \rightarrow P\cr A \rightarrow B\cr}] [\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}] [\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}] Monoclinic (cf. Sections 1.5.4.3 and 2.1.3.15)
Cell choice 2: [\cases{P \rightarrow P\cr B \rightarrow C\cr}] Cell choice invariant
Cell choice 3: [\cases{P \rightarrow P\cr I \rightarrow I\cr}]
[I \rightarrow P] (Fig. 1.5.1.3[link]) [\openup2pt\pmatrix{\bar{{1 \over 2}} &{1 \over 2} &{1 \over 2}\cr {1 \over 2} &\bar{{1 \over 2}} &{1 \over 2}\cr {1 \over 2} &{1 \over 2} &\bar{{1 \over 2}}\cr}] [\pmatrix{0 &1 &1\cr 1 &0 &1\cr 1 &1 &0\cr}] Orthorhombic
Tetragonal
Cubic
[F \rightarrow P] (Fig. 1.5.1.4[link]) [\openup2pt\pmatrix{0 &{1 \over 2} &{1 \over 2}\cr {1 \over 2} &0 &{1 \over 2}\cr {1 \over 2} &{1 \over 2} &0\cr}] [\pmatrix{\bar{1} &1 &1\cr 1 &\bar{1} &1\cr 1 &1 &\bar{1}\cr}] Orthorhombic
Tetragonal
Cubic
[({\bf b}, {\bf a}, \bar{{\bf c}}) \rightarrow ({\bf a}, {\bf b}, {\bf c})] [\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}] [\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}] Unconventional orthorhombic setting
[({\bf c}, {\bf a}, {\bf b}) \rightarrow ({\bf a}, {\bf b}, {\bf c})] [\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}] [\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}] Unconventional orthorhombic setting
[(\bar{{\bf c}}, {\bf b}, {\bf a}) \rightarrow ({\bf a}, {\bf b}, {\bf c})] [\pmatrix{0 &0 &\bar{1}\cr 0 &1 &0\cr 1 &0 &0\cr}] [\pmatrix{0 &0 &1\cr 0 &1 &0\cr \bar{1} &0 &0\cr}] Unconventional orthorhombic setting
[({\bf b}, {\bf c}, {\bf a}) \rightarrow ({\bf a}, {\bf b}, {\bf c})] [\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}] [\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}] Unconventional orthorhombic setting
[({\bf a}, \bar{{\bf c}}, {\bf b}) \rightarrow ({\bf a}, {\bf b}, {\bf c})] [\pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &0\cr}] [\pmatrix{1 &0 &0\cr 0 &0 &1\cr 0 &\bar{1} &0\cr}] Unconventional orthorhombic setting
[\left.\matrix{P\rightarrow C_{1}\cr I\rightarrow F_{1}\cr}\right\}] (Fig. 1.5.1.5[link]), c axis invariant [\pmatrix{1 &1 &0\cr \bar{1} &1 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{{1 \over 2} &\bar{{1 \over 2}} &0\cr {1 \over 2} &{1 \over 2} &0\cr 0 &0 &1\cr}] Tetragonal (cf. Section 1.5.4.3[link])
[\left.\matrix{P\rightarrow C_{2}\cr I\rightarrow F_{2}\cr}\right\}] (Fig. 1.5.1.5[link]), c axis invariant [\pmatrix{1 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{{1 \over 2} &{1 \over 2} &0\cr \bar{1 \over 2} &{1 \over 2} &0\cr 0 &0 &1\cr}] Tetragonal (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{1}], obverse setting (Fig. 1.5.1.6[link]a,c) [\pmatrix{1 &0 &1\cr \bar{1} &1 &1\cr 0 &\bar{1} &1\cr}] [\openup2pt\pmatrix{{2 \over 3} &\bar{{1 \over 3}} &\bar{{1 \over 3}}\cr {1 \over 3} &{1 \over 3} &\bar{{2} \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{2}], obverse setting (Fig. 1.5.1.6[link]c) [\pmatrix{0 &\bar{1} &1\cr 1 &0 &1\cr \bar{1} &1 &1\cr}] [\openup2pt\pmatrix{\bar{{1 \over 3}} &{2 \over 3} &\bar{{1 \over 3}}\cr \bar{{2} \over 3} &{1 \over 3} &{1 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{3}], obverse setting (Fig. 1.5.1.6[link]c) [\pmatrix{\bar{1} &1 &1\cr 0 &\bar{1} &1\cr 1 &0 &1\cr}] [\openup2pt\pmatrix{\bar{{1 \over 3}} &\bar{{1 \over 3}} &{2 \over 3}\cr {1 \over 3} &\bar{{2} \over 3} &{1 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{1}], reverse setting (Fig. 1.5.1.6[link]d) [\pmatrix{\bar{1} &0 &1\cr 1 &\bar{1} &1\cr 0 &1 &1\cr}] [\openup2pt\pmatrix{\bar{{2 \over 3}} &{1 \over 3} &{1 \over 3}\cr \bar{{1} \over 3} &\bar{{1} \over 3} &{2 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{2}], reverse setting (Fig. 1.5.1.6[link]b,d) [\pmatrix{0 &1 &1\cr \bar{1} &0 &1\cr 1 &\bar{1} &1\cr}] [\openup2pt\pmatrix{{1 \over 3} &\bar{{2 \over 3}} &{1 \over 3}\cr {2 \over 3} &\bar{{1} \over 3} &\bar{{1} \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow] triple hexagonal cell [R_{3}], reverse setting (Fig. 1.5.1.6[link]d) [\pmatrix{1 &\bar{1} &1\cr 0 &1 &1\cr \bar{1} &0 &1\cr}] [\openup2pt\pmatrix{{1 \over 3} &{1 \over 3} &\bar{{2 \over 3}}\cr \bar{{1} \over 3} &{2 \over 3} &\bar{{1} \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] orthohexagonal centred cell [C_{1}] (Fig. 1.5.1.7[link]) [\pmatrix{1 &1 &0\cr 0 &2 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{1 &\bar{{1 \over 2}} &0\cr 0 &{1 \over 2} &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] orthohexagonal centred cell [C_{2}] (Fig. 1.5.1.7[link]) [\pmatrix{1 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{{1 \over 2} &{1 \over 2} &0\cr \bar{{1} \over 2} &{1 \over 2} &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] orthohexagonal centred cell [C_{3}] (Fig. 1.5.1.7[link]) [\pmatrix{0 &\bar{2} &0\cr 1 &\bar{1} &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{\bar{{1 \over 2}} &1 &0\cr \bar{1 \over 2} &0 &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] triple hexagonal cell [H_{1}] (Fig. 1.5.1.8[link]) [\pmatrix{1 &1 &0\cr \bar{1} &2 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{{2 \over 3} &\bar{{1 \over 3}} &0\cr {1 \over 3} &{1 \over 3} &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] triple hexagonal cell [H_{2}] (Fig. 1.5.1.8[link]) [\pmatrix{2 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{{1 \over 3} &{1 \over 3} &0\cr \bar{{1} \over 3} &{2 \over 3} &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] triple hexagonal cell [H_{3}] (Fig. 1.5.1.8[link]) [\pmatrix{1 &\bar{2} &0\cr 2 &\bar{1} &0\cr 0 &0 &1\cr}] [\openup2pt\pmatrix{\bar{{1 \over 3}} &{2 \over 3} &0\cr \bar{{2 \over 3}} &{1 \over 3} &0\cr 0 &0 &1\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] triple rhombohedral cell [D_{1}] [\pmatrix{1 &0 &\bar{1}\cr 0 &1 &\bar{1}\cr 1 &1 &1\cr}] [\openup2pt\pmatrix{{2 \over 3} &\bar{{1 \over 3}} &{1 \over 3}\cr \bar{1 \over 3} &{2 \over 3} &{1 \over 3}\cr \bar{1 \over 3} &\bar{1 \over 3} &{1 \over 3}\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Hexagonal cell [P \rightarrow] triple rhombohedral cell [D_{2}] [\pmatrix{\bar{1} &0 &1\cr 0 &\bar{1} &1\cr 1 &1 &1\cr}] [\openup2pt\pmatrix{\bar{{2 \over 3}} &{1 \over 3} &{1 \over 3}\cr {1 \over 3} &\bar{2 \over 3} &{1 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}] Trigonal
Hexagonal (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] C-centred monoclinic cell, unique axis b, cell choice 1
(Fig. 1.5.1.9[link]a)
[\openup3pt\pmatrix{{2 \over 3} &0 &0\cr {1 \over 3} &1 &0\cr \bar{{2 \over 3}} &0 &1\cr}] [\openup3pt\pmatrix{{3 \over 2} &0 &0\cr \bar{{1 \over 2}} &1 &0\cr 1 &0 &1\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] C-centred monoclinic cell, unique axis b, cell choice 2
(Fig. 1.5.1.9[link]a)
[\openup2pt\pmatrix{\bar{{1 \over 3}} &\bar{1} &0\cr {1 \over 3} &\bar{1} &0\cr \bar{{2 \over 3}} &0 &1\cr}] [\openup2pt\pmatrix{\bar{{3 \over 2}} &{3 \over 2} &0\cr \bar{{1 \over 2}} &\bar{{1 \over 2}} &0\cr \bar{1} &1 &1\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] C-centred monoclinic cell, unique axis b, cell choice 3
(Fig. 1.5.1.9[link]a)
[\openup3pt\pmatrix{\bar{{1 \over 3}} &1 &0\cr \bar{{2 \over 3}} &0 &0\cr \bar{{2 \over 3}} &0 &1\cr}] [\openup3pt\pmatrix{0 &\bar{{3 \over 2}} &0\cr 1 &\bar{{1 \over 2}} &0\cr 0 &\bar{1} &1\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] A-centred monoclinic cell, unique axis c, cell choice 1
(Fig. 1.5.1.9[link]b)
[\openup2pt\pmatrix{0 &{2 \over 3} &0\cr 0 &{1 \over 3} &1\cr 1 &\bar{{2 \over 3}} &0\cr}] [\openup3pt\pmatrix{1 &0 &1\cr {3 \over 2} &0 &0\cr \bar{{1 \over 2}} &1 &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] A-centred monoclinic cell, unique axis c, cell choice 2
(Fig. 1.5.1.9[link]b)
[\openup3pt\pmatrix{0 &\bar{{1 \over 3}} &\bar{1}\cr 0 &{1 \over 3} &\bar{1}\cr 1 &\bar{{2 \over 3}} &0\cr}] [\openup3pt\pmatrix{\bar{1} &1 &1\cr \bar{{3 \over 2}} &{3 \over 2} &0\cr \bar{{1 \over 2}} &\bar{{1 \over 2}} &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Triple hexagonal cell R, obverse setting [\rightarrow] A-centred monoclinic cell, unique axis c, cell choice 3
(Fig. 1.5.1.9[link]b)
[\openup3pt\pmatrix{0 &\bar{{1 \over 3}} &1\cr 0 &\bar{{2 \over 3}} &0\cr 1 &\bar{{2 \over 3}} &0\cr}] [\openup3pt\pmatrix{0 &\bar{1} &1\cr 0 &\bar{{3 \over 2}} &0\cr 1 &\bar{{1 \over 2}} &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow C]-centred monoclinic cell, unique axis b, cell choice 1
(Fig. 1.5.1.10[link]a)
[\pmatrix{0 &0 &1\cr \bar{1} &1 &1\cr \bar{1} &\bar{1} &1\cr}] [\openup2pt\pmatrix{1 &\bar{{1 \over 2}} &\bar{{1 \over 2}}\cr 0 &{1 \over 2} &\bar{{1 \over 2}}\cr 1 &0 &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow C]-centred monoclinic cell, unique axis b, cell choice 2
(Fig. 1.5.1.10[link]a)
[\pmatrix{\bar{1} &\bar{1} &1\cr 0 &0 &1\cr \bar{1} &1 &1\cr}] [\openup3pt\pmatrix{\bar{{1 \over 2}} &1 &\bar{{1 \over 2}}\cr \bar{{1 \over 2}} &0 &{1 \over 2}\cr 0 &1 &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow C]-centred monoclinic cell, unique axis b, cell choice 3
(Fig. 1.5.1.10[link]a)
[\pmatrix{\bar{1} &1 &1\cr \bar{1} &\bar{1} &1\cr 0 &0 &1\cr}] [\pmatrix{\bar{{1 \over 2}} &\bar{{1 \over 2}} &1\cr {1 \over 2} &\bar{{1 \over 2}} &0\cr 0 &0 &1\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow A]-centred monoclinic cell, unique axis c, cell choice 1
(Fig.1.5.1.10b)
[\pmatrix{1 &0 &0\cr 1 &\bar{1} &1\cr 1 &\bar{1} &\bar{1}\cr}] [\openup3pt\pmatrix{1 &0 &0\cr 1 &\bar{{1 \over 2}} &\bar{{1 \over 2}}\cr 0 &{1 \over 2} &\bar{{1 \over 2}}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow A]-centred monoclinic cell, unique axis c, cell choice 2
(Fig. 1.5.1.10[link]b)
[\pmatrix{1 &\bar{1} &\bar{1}\cr 1 &0 &0\cr 1 &\bar{1} &1\cr}] [\openup3pt\pmatrix{0 &1 &0\cr \bar{{1 \over 2}} &1 &\bar{{1 \over 2}}\cr \bar{{1 \over 2}} &0 &{1 \over 2}\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
Primitive rhombohedral cell [\rightarrow A]-centred monoclinic cell, unique axis c, cell choice 3
(Fig. 1.5.1.10[link]b)
[\pmatrix{1 &\bar{1} &1\cr 1 &\bar{1} &\bar{1}\cr 1 &0 &0\cr}] [\openup3pt\pmatrix{0 &0 &1\cr \bar{{1 \over 2}} &\bar{{1 \over 2}} &1\cr {1 \over 2} &\bar{{1 \over 2}} &0\cr}] Rhombohedral space groups (cf. Section 1.5.4.3[link])
[Figure 1.5.1.1]

Figure 1.5.1.1 | top | pdf |

The coordinates of the points X (or Y) with respect to the old origin O are x (y), and with respect to the new origin [O'] they are [{\bi x}'] [({\bi y}')]. From the diagram one reads [{\bf p}+{\bf x}'={\bf x}] and [{\bf p}+{\bf y}'={\bf y}].

For the columns [{\bi p}+{\bi x}'={\bi x}] holds, i.e.[{\bi x}'={\bi x}-{\bi p}\quad{\rm or} \quad\pmatrix{ x_1' \cr x_2' \cr x_3' }=\pmatrix{ x_1 \cr x_2 \cr x_3 }-\pmatrix{ p_1 \cr p_2 \cr p_3 }=\pmatrix{ x_1-p_1 \cr x_2-p_2 \cr x_3-p_3 }.\eqno(1.5.1.1)]This can be written in the formalism of matrix–column pairs (cf. Section 1.2.2.3[link] for details of the matrix–column formalism) as[{\bi x}'=({\bi I},-{\bi p}) \,{\bi x} \quad {\rm or} \quad {\bi x}'=({\bi I},{\bi p})^{-1}{\bi x},\eqno(1.5.1.2)]where [({\bi I},{\bi p})] represents the translation corresponding to the vector p of the origin shift.

The vector r determined by the points X and Y (also known as a `distance vector'), [{\bf x}+{\bf r}={\bf y}] (cf. Fig. 1.5.1.1[link]), and thus with coefficients[{\bi r}={\bi y}-{\bi x}=\pmatrix{y_1-x_1 \cr y_2-x_2\cr y_3-x_3},]shows a different transformation behaviour under the origin shift. From the diagram one reads the equations [{\bf p} + {\bf x}' = {\bf x}], [{\bf x} + {\bf r} = {\bf y}], [{\bf x}' + {\bf r} = {\bf y}'], and thus [{\bf r}={\bf y}'-{\bf x}'={\bf y}-{\bf x},\eqno(1.5.1.3)]i.e. the vector coefficients of r are not affected by the origin shift.

Example

The description of a crystal structure is closely related to its space-group symmetry: different descriptions of the underlying space group, in general, result in different descriptions of the crystal structure. This example illustrates the comparison of two structure descriptions corresponding to different origin choices of the space group.

To compare the two structures it is not only necessary to apply the origin-shift transformation but also to adjust the selection of the representative atoms of the two descriptions.

In the Inorganic Crystal Structure Database (2012[link]) (abbreviated as ICSD) one finds the following two descriptions of the mineral zircon ZrSiO4:

  • (a) Wyckoff & Hendricks (1927[link]), ICSD No. 31101, space group [I4_1/amd=D_{4h}^{19}], No. 141, cell parameters a = 6.61 Å, c = 5.98 Å.

    The coordinates of the atoms in the unit cell are (normalized so that [0\leq x_i \,\lt\, 1]):[\quad\quad\ \let\normalbaselines\relax\openup2pt\matrix{{\rm Zr}{:}\hfill& 4a\hfill & 0,0,0\semi\ 0,{\textstyle{1 \over 2}},{\textstyle{1 \over 4}}\ \ [{\rm and\ the\ same\ with}\ ({\textstyle{1 \over 2}},{\textstyle{1 \over 2}},{\textstyle{1 \over 2}})+]\hfill \cr {\rm Si}{:}\hfill& 4b\hfill & 0,0,{\textstyle{1 \over 2}}\semi\ 0,{\textstyle{1 \over 2}},{\textstyle{3 \over 4}}\ \ [{\rm and\ the\ same\ with}\ ({\textstyle{1 \over 2}},{\textstyle{1 \over 2}},{\textstyle{1 \over 2}})+]\hfill\cr {\rm O}{:}\hfill & 16h\hfill &0,0.2,0.34\semi\ 0.5,0.3,0.84\semi\ 0.8,0.5,0.59\semi\ \hfill\cr&& 0.7,0,0.09\semi\ 0.5,0.2,0.41\semi\ 0,0.3,0.91\semi\ \hfill\cr&&0.7,0.5,0.16\semi\ 0.8,0,0.66\hfill\cr&& [{\rm and\ the\ same\ with}\ ({\textstyle{1 \over 2}},{\textstyle{1 \over 2}},{\textstyle{1 \over 2}})+].\hfill}]The coordinates of Zr and Si atoms indicate that the space-group setting corresponds to the origin choice 1 description of [I4_1/amd] given in this volume, i.e. origin at [\bar{4}m2] (cf. the space-group tables for [I4_1/amd] in Chapter 2.3[link] ).

  • (b) Krstanovic (1958[link]), ICSD No. 45520, space group [I4_1/amd=D_{4h}^{19}], No. 141, cell parameters a = 6.6164 (5) Å, c = 6.0150 (5) Å.

    The coordinates of the atoms in the unit cell are (normalized so that [0\leq x_i \,\lt\, 1]):[\quad\quad\let\normalbaselines\relax\openup2pt\matrix{{\rm Zr}{:}\hfill& 4a\hfill &0,{\textstyle{3 \over 4}},{\textstyle{1 \over 8}}\semi\ {\textstyle{1 \over 2}},{\textstyle{3 \over 4}},{\textstyle{3 \over 8}}\ \ [{\rm and\ the\ same\ with}\ ({\textstyle{1 \over 2}},{\textstyle{1 \over 2}},{\textstyle{1 \over 2}})+]\hfill \cr {\rm Si}{:}\hfill& 4b\hfill & 0,{\textstyle{1 \over 4}},{\textstyle{3 \over 8}}\semi\ 0,{\textstyle{3 \over 4}},{\textstyle{5 \over 8}}\ \ [{\rm and\ the\ same\ with}\ ({\textstyle{1 \over 2}},{\textstyle{1 \over 2}},{\textstyle{1 \over 2}})+]\hfill \cr {\rm O}{:}\hfill & 16h\hfill & 0, 0.067, 0.198\semi\ 0.5, 0.933, 0.698\semi\ \hfill\cr&& 0.183, 0.75,0.448\semi\ 0.317,0.25,0.948\semi\ \hfill\cr&& 0.5, 0.067,0.302\semi\ 0, 0.933, 0.802\semi\ \hfill\cr&& 0.317, 0.75, 0.052\semi\ 0.183,0.25,0.552\hfill\cr && [{\rm and\ the\ same\ with}\ ({\textstyle{1 \over 2}},{\textstyle{1 \over 2}},{\textstyle{1 \over 2}})+].\hfill }]The structure is described with respect to the origin choice 2 setting of [I4_1/amd] specified in this volume as `Origin at centre (2/m) at [0,\bar{{\textstyle{1 \over 4}}},{\textstyle{1 \over 8}}] from [\bar{4}m2]' (cf. the space-group tables for [I4_1/amd] in Chapter 2.3[link] ).

In order to compare the different structure descriptions, the atomic coordinates of the origin choice 1 description are to be transformed to `Origin at centre 2/m', i.e. origin choice 2.

Origin choice 2 has coordinates [0,\bar{{\textstyle{1 \over 4}}}, {\textstyle{1 \over 8}}] referred to origin choice 1. Therefore, the change of coordinates consists of subtracting [{\bi p}=\openup2pt\pmatrix{ 0 \cr \bar{{\textstyle{1 \over 4}}} \cr {\textstyle{1 \over 8}} }] from the origin choice 1 values, i.e. leave the x coordinate unchanged, add [{\textstyle{1 \over 4}}=0.25] to the y coordinate and subtract [{\textstyle{1 \over 8}}=0.125] from the z coordinate [cf. equation (1.5.1.1)[link]].

The transformed and normalized coordinates (so that [0\le x_i\,\lt\,1]) are

  • (i) Zr: 4a [0,{\textstyle{1 \over 4}}, {\textstyle{7 \over 8}}]; [0,{\textstyle{3 \over 4}},{\textstyle{1 \over 8}}]; [{\textstyle{1 \over 2}},{\textstyle{1 \over 4}}, {\textstyle{5 \over 8}}]; [{\textstyle{1 \over 2}},{\textstyle{3 \over 4}},{\textstyle{3 \over 8}}];

  • (ii) Si: 4b [0,{\textstyle{1 \over 4}},{\textstyle{3 \over 8}}]; [0,{\textstyle{3 \over 4}},{\textstyle{5 \over 8}}]; [{\textstyle{1 \over 2}},{\textstyle{1 \over 4}},{\textstyle{1 \over 8}}]; [{\textstyle{1 \over 2}},{\textstyle{3 \over 4}},{\textstyle{7 \over 8}}];

  • (iii) O: 16h [0,0.20+0.25,0.34 - 0.125=0, 0.45,0.215].

    This oxygen atom obviously does not correspond to the representative [0, 0.067, 0.198] given by Krstanovic (1958[link]), but by adding the centring vector [({\textstyle{1 \over 2}},{\textstyle{1 \over 2}},{\textstyle{1 \over 2}})] it is seen to correspond to the second position with coordinates [0.5, 0.933, 0.698]. The transformed (and normalized) coordinates of the rest of the oxygen atoms in the unit cell are:[\quad\quad\eqalign{&0.5, 0.55, 0.715\semi\ 0.8, 0.75, 0.465\semi\ 0.7, 0.25, 0.965\semi\hfill\cr &0.5, 0.45, 0.285\semi\ 0, 0.55, 0.785\semi\ 0.7, 0.75, 0.035\semi\quad\hfill\cr &0.8, 0.25, 0.535\semi\hfill}]all also with [({\textstyle{1 \over 2}},{\textstyle{1 \over 2}},{\textstyle{1 \over 2}})+]. The difference in the coordinates of the two descriptions could be explained by the difference in the accuracy of the two refinements.

1.5.1.2. Change of the basis

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A change of the basis is described by a (3 × 3) matrix:[{\bi P}= \pmatrix{P_{11} & P_{12} & P_{13} \cr P_{21} & P_{22} & P_{23} \cr P_{31} & P_{32} & P_{33}}.]The matrix P relates the new basis [{\bf a}',{\bf b}',{\bf c}'] to the old basis [{\bf a},{\bf b},{\bf c}] according to[\eqalignno{&\left({\bf a}', {\bf b}', {\bf c}'\right)= \left ({\bf a}, {\bf b},{\bf c}\right){\bi P}= \left ({\bf a},{\bf b},{\bf c}\right)\pmatrix{P_{11} & P_{12} & P_{13} \cr P_{21} & P_{22} & P_{23} \cr P_{31} & P_{32} & P_{33}}&\cr &\!\!=\left ({\bf a}P_{11}+ {\bf b}P_{21}+{\bf c}P_{31},\, {\bf a}P_{12}+ {\bf b}P_{22}+{\bf c}P_{32},\, {\bf a}P_{13}+ {\bf b}P_{23}+{\bf c}P_{33}\right).&\cr&&(1.5.1.4)}]The matrix P is often referred to as the linear part of the coordinate transformation and it describes a change of direction and/or length of the basis vectors. It is preferable to choose the matrix P in such a way that its determinant is positive: a negative determinant of P implies a change from a right-handed coordinate system to a left-handed coordinate system or vice versa. If det(P) = 0, then the new vectors [{\bf a}',{\bf b}',{\bf c}'] are linearly dependent, i.e. they do not form a complete set of basis vectors.

For a point X (cf. Fig. 1.5.1.1[link]), the vector [\buildrel{\longrightarrow}\over{OX}\ ={\bf x}] is[\eqalign{{\bf x}&={\bf a}x_1+{\bf b}x_2+{\bf c}x_3= {\bf a}'x_1'+{\bf b}'x_2'+{\bf c}'x_3'\ {\rm or}\cr {\bf x}&=({\bf a},{\bf b},{\bf c})\pmatrix{ x_1 \cr x_2 \cr x_3 }=({\bf a}',{\bf b}',{\bf c}')\pmatrix{ x_1' \cr x_2' \cr x_3' }.}]By inserting equation (1.5.1.4)[link] one obtains[{\bf x}=({\bf a}',{\bf b}',{\bf c}')\pmatrix{ x_1' \cr x_2' \cr x_3' }=({\bf a},{\bf b},{\bf c}) \pmatrix{P_{11} & P_{12} & P_{13} \cr P_{21} & P_{22} & P_{23} \cr P_{31} & P_{32} & P_{33}}\pmatrix{ x_1' \cr x_2' \cr x_3' }]or[\pmatrix{ x_1 \cr x_2 \cr x_3 }=\pmatrix{P_{11} & P_{12} & P_{13} \cr P_{21} & P_{22} & P_{23} \cr P_{31} & P_{32} & P_{33} }\pmatrix{ x_1' \cr x_2' \cr x_3' },]i.e. [{\bi x}={\bi P}{\bi x}'] or [{\bi x}'={\bi P}^{-1}{\bi x}= ({\bi P},{\bi o})^{-1}{\bi x}], which is often written as[{\bi x}'={\bi Q}{\bi x}=({\bi Q},{\bi o}){\bi x}.\eqno(1.5.1.5)]Here the inverse matrix [{\bi P}^{-1}] is designated by Q, while o is the (3 × 1) column vector with zero coefficients. [Note that in equation (1.5.1.4)[link] the sum is over the row (first) index of P, while in equation (1.5.1.5)[link], the sum is over the column (second) index of Q.]

A selected set of transformation matrices P and their inverses [{\bi P}^{-1}={\bi Q}] that are frequently used in crystallographic calculations are listed in Table 1.5.1.1[link] and illustrated in Figs. 1.5.1.2[link] to 1.5.1.10[link][link][link][link][link][link][link][link].

Example

Consider an F-centred cell with conventional basis [{\bf a}_F,{\bf b}_F,{\bf c}_F] and a corresponding primitive cell with basis [{\bf a}_P,{\bf b}_P,{\bf c}_P], cf. Fig. 1.5.1.4[link]. The transformation matrix P from the conventional basis to a primitive basis can either be deduced from Fig. 1.5.1.4[link] or can be read directly from Table 1.5.1.1[link]: [{\bf a}_P] = [{\textstyle{1 \over 2}}({\bf b}_F + {\bf c}_F)], [{\bf b}_P={\textstyle{1 \over 2}}({\bf a}_F + {\bf c}_F)], [{\bf c}_P={\textstyle{1 \over 2}}({\bf a}_F + {\bf b}_F)], which in matrix notation is[({\bf a}_P,{\bf b}_P,{\bf c}_P)=({\bf a}_F,{\bf b}_F,{\bf c}_F){\bi P}= ({\bf a}_F,{\bf b}_F,{\bf c}_F)\openup2pt\pmatrix{0 & {\textstyle{1 \over 2}} &{\textstyle{1 \over 2}} \cr {\textstyle{1 \over 2}} & 0 & {\textstyle{1 \over 2}} \cr {\textstyle{1 \over 2}} & {\textstyle{1 \over 2}} & 0 }.]The inverse matrix [{\bi P}^{-1}={\bi Q}] is also listed in Table 1.5.1.1[link] or can be deduced from Fig. 1.5.1.4[link]. It is the matrix that describes the conventional basis vectors [{\bf a}_F,{\bf b}_F,{\bf c}_F] by linear combinations of [{\bf a}_P,{\bf b}_P,{\bf c}_P]: [{\bf a}_F] = [-{\bf a}_P + {\bf b}_P + {\bf c}_P], [{\bf b}_F] = [{\bf a}_P - {\bf b}_P + {\bf c}_P], [{\bf c}_F] = [{\bf a}_P + {\bf b}_P - {\bf c}_P], or[({\bf a}_F,{\bf b}_F,{\bf c}_F)=({\bf a}_P,{\bf b}_P,{\bf c}_P){\bi P}^{-1}=({\bf a}_P,{\bf b}_P,{\bf c}_P)\pmatrix{\hfill -1 & \hfill 1 &\hfill 1 \cr \hfill 1 & \hfill -1 & \hfill 1 \cr \hfill 1 & \hfill 1 & \hfill -1 }.]Correspondingly, the point coordinates transform as[\eqalign{\pmatrix{ x_P \cr y_P \cr z_P }&={\bi P}^{-1}\pmatrix{ x_F \cr y_F \cr z_F }=\pmatrix{\hfill -1 & \hfill 1 &\hfill 1 \cr \hfill 1 & \hfill -1 & \hfill 1 \cr \hfill 1 & \hfill 1 & \hfill-1 } \pmatrix{ x_F \cr y_F \cr z_F }\cr&=\pmatrix{\hfill -x_F+y_F+ z_F\cr \hfill x_F-y_F+z_F \cr \hfill x_F+y_F-z_F }.}]For example, the coordinates [\pmatrix{1\cr 0\cr 0}_{\!\!F}] of the end point of [{\bf a}_F] with respect to the conventional basis become [\pmatrix{-1\cr 1\cr 1}_{\!\!P}] in the primitive basis, the centring point [\pmatrix{{\textstyle{1 \over 2}}\cr {\textstyle{1 \over 2}}\cr 0}_{\!\!F}] of the [{\bf a}_F,{\bf b}_F] plane becomes the end point [\pmatrix{0\cr 0\cr 1}_{\!\!P}] of [{\bf c}_P] etc.

[Figure 1.5.1.2]

Figure 1.5.1.2 | top | pdf |

Monoclinic centred lattice, projected along the unique axis. The origin for all the cells is the same. The fractions [{\textstyle{1\over 2}}] indicate the height of the lattice points along the axis of projection.

(a) Unique axis b: (b) Unique axis c: (c) Unique axis a:
Cell choice 1: C-centred cell [a_1,b,c_1]. Cell choice 1: A-centred cell [a_1,b_1,c]. Cell choice 1: B-centred cell [a,b_1,c_1].
Cell choice 2: A-centred cell [a_2,b,c_2]. Cell choice 2: B-centred cell [a_2,b_2,c]. Cell choice 2: C-centred cell [a,b_2,c_2].
Cell choice 3: I-centred cell [a_3,b,c_3]. Cell choice 3: I-centred cell [a_3,b_3,c]. Cell choice 3: I-centred cell [a,b_3,c_3].

[Figure 1.5.1.3]

Figure 1.5.1.3 | top | pdf |

Body-centred cell I with aI, bI, cI and a corresponding primitive cell P with aP, bP, cP. The origin for both cells is O. A cubic I cell with lattice constant [a_{c}] can be considered as a primitive rhombohedral cell with [a_{\rm rh} = a_{c} {1 \over 2} \sqrt{3}] and [\alpha = 109.47^{\circ}] (rhombohedral axes) or a triple hexagonal cell with [a_{\rm hex} = a_{c} \sqrt{2}] and [c_{\rm hex} = a_{c} {1 \over 2} \sqrt{3}] (hexagonal axes).

[Figure 1.5.1.4]

Figure 1.5.1.4 | top | pdf |

Face-centred cell F with aF, bF, cF and a corresponding primitive cell P with aP, bP, cP. The origin for both cells is O. A cubic F cell with lattice constant [a_{c}] can be considered as a primitive rhombohedral cell with [a_{\rm rh} = a_{c} {1 \over 2} \sqrt{2}] and [\alpha = 60^{\circ}] (rhombohedral axes) or a triple hexagonal cell with [a_{\rm hex} = a_{c} {1 \over 2} \sqrt{2}] and [c_{\rm hex} = a_{c} \sqrt{3}] (hexagonal axes).

[Figure 1.5.1.5]

Figure 1.5.1.5 | top | pdf |

Tetragonal lattices, projected along [[00\bar{1}]]. (a) Primitive cell P with a, b, c and the C-centred cells [C_{1}] with [a_{1}, b_{1}, c] and [C_{2}] with [a_{2}, b_{2}, c]. The origin for all three cells is the same. (b) Body-centred cell I with a, b, c and the F-centred cells [F_{1}] with [a_{1}, b_{1}, c] and [F_{2}] with [a_{2}, b_{2}, c]. The origin for all three cells is the same. The fractions [{\textstyle{1\over 2}}] indicate the height of the lattice points along the axis of projection.

[Figure 1.5.1.6]

Figure 1.5.1.6 | top | pdf |

Unit cells in the rhombohedral lattice: same origin for all cells. The basis of the rhombohedral cell is labelled arh, brh, crh. Two settings of the triple hexagonal cell are possible with respect to a primitive rhombohedral cell: The obverse setting with the lattice points 0, 0, 0; [{2 \over 3}, {1 \over 3}, {1 \over 3}]; [{1 \over 3}, {2 \over 3}, {2 \over 3}] has been used in International Tables since 1952. Its general reflection condition is [- h + k + l = 3n]. The reverse setting with lattice points 0, 0, 0; [{1 \over 3}, {2 \over 3}, {1 \over 3}]; [{2 \over 3}, {1 \over 3}, {2 \over 3}] was used in the 1935 edition. Its general reflection condition is [h - k + l = 3n]. The fractions indicate the height of the lattice points along the axis of projection. (a) Obverse setting of triple hexagonal cell [a_{1},b_{1},c_{1}] in relation to the primitive rhombohedral cell arh, brh, crh. (b) Reverse setting of triple hexagonal cell [a_{2},b_{2},c_{2}] in relation to the primitive rhombohedral cell arh, brh, crh. (c) Primitive rhombohedral cell (- - - lower edges), arh, brh, crh in relation to the three triple hexagonal cells in obverse setting [a_{1},b_{1},c']; [a_{2},b_{2},c']; [a_{3},b_{3},c']. Projection along c′. (d) Primitive rhombohedral cell (- - - lower edges), arh, brh, crh in relation to the three triple hexagonal cells in reverse setting [a_{1},b_{1},c']; [a_{2},b_{2},c']; [a_{3},b_{3},c']. Projection along c′.

[Figure 1.5.1.7]

Figure 1.5.1.7 | top | pdf |

Hexagonal lattice projected along [[00\bar{1}]]. Primitive hexagonal cell P with a, b, c and the three C-centred (orthohexagonal) cells [a_{1},b_{1},c]; [a_{2},b_{2},c]; [a_{3},b_{3},c]. The origin for all cells is the same.

[Figure 1.5.1.8]

Figure 1.5.1.8 | top | pdf |

Hexagonal lattice projected along [[00\bar{1}]]. Primitive hexagonal cell P with a, b, c and the three triple hexagonal cells H with [a_{1},b_{1},c]; [a_{2}, b_{2}, c]; [a_{3}, b_{3}, c]. The origin for all cells is the same.

[Figure 1.5.1.9]

Figure 1.5.1.9 | top | pdf |

Rhombohedral lattice with a triple hexagonal unit cell a, b, c in obverse setting (i.e. unit cell a1, b1, c in Fig. 1.5.1.6[link]c) and the three centred monoclinic cells. (a) C-centred cells [C_{1}] with [a_{1}, b_{1}, c]; [C_{2}] with [a_{2}, b_{2}, c]; and [C_{3}] with [a_{3}, b_{3}, c]. The unique monoclinic axes are [b_{1}, b_{2}] and [b_{3}], respectively. The origin for all four cells is the same. (b) A-centred cells [A_{1}] with [a', b_{1}, c_{1}]; [A_{2}] with [a', b_{2}, c_{2}]; and [A_{3}] with [a', b_{3}, c_{3}]. The unique monoclinic axes are [c_{1}], [c_{2}] and [c_{3}], respectively. The origin for all four cells is the same. The fractions indicate the height of the lattice points along the axis of projection.

[Figure 1.5.1.10]

Figure 1.5.1.10 | top | pdf |

Rhombohedral lattice with primitive rhombohedral cell arh, brh, crh and the three centred monoclinic cells. (a) C-centred cells [C_{1}] with [a_{1}, b_{1}, c']; [C_{2}] with [a_{2}, b_{2}, c']; and [C_{3}] with [a_{3}, b_{3}, c']. The unique monoclinic axes are [b_{1}], [b_{2}] and [b_{3}], respectively. The origin for all four cells is the same. (b) A-centred cells [A_{1}] with [a', b_{1}, c_{1}]; [A_{2}] with [a', b_{2}, c_{2}]; and [A_{3}] with [a', b_{3}, c_{3}]. The unique monoclinic axes are [c_{1}], [c_{2}] and [c_{3}], respectively. The origin for all four cells is the same. The fractions indicate the height of the lattice points along the axis of projection.

1.5.1.3. General change of coordinate system

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A general change of the coordinate system involves both an origin shift and a change of the basis. Such a transformation of the coordinate system is described by the matrix–column pair [({\bi P}, {\bi p})], where the (3 × 3) matrix P relates the new basis [{\bf a}',{\bf b}',{\bf c}'] to the old one [{\bf a},{\bf b},{\bf c}] according to equation (1.5.1.4)[link]. The origin shift is described by the shift vector [{\bf p}=p_1{\bf a}+p_2{\bf b}+p_3{\bf c}]. The coordinates of the new origin [O'] with respect to the old coordinate system [{\bf a},{\bf b},{\bf c}] are given by the (3 × 1) column [{\bi p} =\pmatrix{ p_1 \cr p_2\cr p_3}].

The general coordinate transformation can be performed in two consecutive steps. Because the origin shift p refers to the old basis [{\bf a},{\bf b},{\bf c}], it has to be applied first (as described in Section 1.5.1.1[link]), followed by the change of the basis (cf. Section 1.5.1.2[link]):[{\bi x}'=({\bi P},{\bi o})^{-1} ({\bi I},{\bi p})^{-1}{\bi x}=\left(({\bi I},{\bi p}) ({\bi P},{\bi o})\right)^{-1}{\bi x}=({\bi P},{\bi p})^{-1}{\bi x}.\eqno(1.5.1.6)]Here, I is the three-dimensional unit matrix and o is the (3 × 1) column matrix containing only zeros as coefficients.

The formulae for the change of the point coordinates from x to [{\bi x}'] uses [({\bi Q},\ {\bi q})=({\bi P},{\bi p})^{-1}=({\bi P}^{-1},\ -{\bi P}^{-1}{\bi p})], i.e.[\eqalignno{\pmatrix{ x_1' \cr x_2' \cr x_3' }&=\pmatrix{Q_{11} & Q_{12} & Q_{13} \cr Q_{21} & Q_{22} & Q_{23} \cr Q_{31} & Q_{32} & Q_{33} } \pmatrix{ x_1 \cr x_2\cr x_3 } +\pmatrix{ q_1 \cr q_2\cr q_3 }& \cr &{\rm with }\ {\bi Q}={\bi P}^{-1} \ {\rm and}\ {\bi q}=-{\bi P}^{-1}{\bi p}, &\cr &{\rm thus}\ {\bi x}'= {\bi P}^{-1}{\bi x} - {\bi P}^{-1}{\bi p} = {\bi P}^{-1}({\bi x}-{\bi p}).&(1.5.1.7)}]The effect of a general change of the coordinate system [({\bi P},{\bi p})] on the coefficients of a vector r is reduced to the linear transformation described by P, as the vector coefficients are not affected by the origin shift [cf. equation (1.5.1.3)[link]].

Hereafter, the data for the matrix–column pair[({\bi P}, {\bi p})=(\pmatrix{P_{11} & P_{12} & P_{13} \cr P_{21} & P_{22} & P_{23} \cr P_{31} & P_{32} & P_{33}},\ \pmatrix{ p_1 \cr p_2 \cr p_3 })]are often written in the following concise form:[\eqalignno{&P_{11}{\bf a}+P_{21}{\bf b}+P_{31}{\bf c}, \ P_{12}{\bf a}+P_{22}{\bf b}+P_{32}{\bf c},\ P_{13}{\bf a}+P_{23}{\bf b}+P_{33}{\bf c}\semi&\cr&\quad \ p_1,p_2,p_3.&(1.5.1.8)}]The concise notation of the transformation matrices is widely used in the tables of maximal subgroups of space groups in International Tables for Crystallography Volume A1 (2010[link]), where [({\bi P},{\bi p})] describes the relation between the conventional bases of a group and its maximal subgroups. For example, the expression [({\bi P},{\bi p})=({\bf a}-{\bf b}, {\bf a}+{\bf b},2{\bf c}\semi 0,0,{\textstyle{1 \over 2}})] (cf. the table of maximal subgroups of [P\bar{4}2m], No. 111, in Volume A1) stands for[{\bi P}=\pmatrix{1& 1 & 0 \cr \bar{1}& 1 & 0 \cr 0 & 0 & 2 }\ {\rm and}\ {\bi p}=\pmatrix{ 0 \cr 0 \cr {\textstyle{1 \over 2}} }.]Note that the matrix elements of P in equation (1.5.1.8)[link] are read by columns since they act on the row matrices of basis vectors, and not by rows, as in the shorthand notation of symmetry operations which apply to column matrices of coordinates (cf. Section 1.2.2.1[link] ).

References

Inorganic Crystal Structure Database (2012). Release 2012/2. Fach­informationszentrum Karlsruhe and National Institute of Standards and Technology. http://www.fiz-karlsruhe.de/icsd.html . (Abbreviated as ICSD.)
International Tables for Crystallography (2010). Vol. A1, Symmetry Relations between Space Groups. Edited by H. Wondratschek & U. Müller, 2nd ed. Chichester: John Wiley & Sons.
Krstanovic, I. R. (1958). Redetermination of the oxygen parameters in zircon (ZrSiO4). Acta Cryst. 11, 896–897.
Wyckoff, R. W. G. & Hendricks, S. B. (1927). Die Kristallstruktur von Zirkon und die Kriterien fuer spezielle Lagen in tetragonalen Raumgruppen. Z. Kristallogr. Kristallgeom. Kristallphys. Kristallchem. 66, 73–102.








































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