Table 5.1.3.1

Arnold, H.

doi:10.1107/97809553602060000510

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

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International Tables for Crystallography (2006). Vol. A. ch. 5.1, pp. 80-83

Table 5.1.3.1

H. Arnold^a ^‡

^a Institut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany

Table 5.1.3.1 | 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.

Transformation	P	$[{\bi Q} = {\bi P}^{-1}]$	Crystal system
$[{\bf c} \rightarrow {1 \over 2}{\bf c}]$	$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &{1 \over 2}\cr}]$	$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &2\cr}]$	All systems
$[{\bf b} \rightarrow {1 \over 2}{\bf b}]$	$[\pmatrix{1 &0 &0\cr 0 &{1 \over 2} &0\cr 0 &0 &1\cr}]$	$[\pmatrix{1 &0 &0\cr 0 &2 &0\cr 0 &0 &1\cr}]$	All systems
$[{\bf a} \rightarrow {1 \over 2}{\bf a}]$	$[\pmatrix{{1 \over 2} &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$	$[\pmatrix{2 &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$	All systems
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. Section 2.2.16 )
Cell choice 2 $[\rightarrow]$ cell choice 3: $[\cases{P \rightarrow P\cr A \rightarrow I\cr}]$ $[\matrix{\cr \hbox{Unique axis }{\bf b}\hfill\cr \hbox{invariant}\hfill\cr}]$
Cell choice 3 $[\rightarrow]$ cell choice 1: $[\cases{P \rightarrow P\cr I \rightarrow C\cr}]$
(Fig. 5.1.3.2a)
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. Section 2.2.16 )
Cell choice 2 $[\rightarrow]$ cell choice 3: $[\cases{P \rightarrow P\cr B \rightarrow I\cr}]$ $[\matrix{\hbox{Unique axis }{\bf c}\hfill\cr \hbox{invariant}\hfill\cr}]$
Cell choice 3 $[\rightarrow]$ cell choice 1: $[\cases{P \rightarrow P\cr I \rightarrow A\cr}]$
(Fig. 5.1.3.2b)
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. Section 2.2.16 )
Cell choice 2 $[\rightarrow]$ cell choice 3: $[\cases{P \rightarrow P\cr C \rightarrow I\cr}]$ $[\matrix{\hbox{Unique axis } {\bf a}\hfill\cr \hbox{invariant}\hfill\cr}]$
Cell choice 3 $[\rightarrow]$ cell choice 1: $[\cases{P \rightarrow P\cr I \rightarrow B\cr}]$
(Fig. 5.1.3.2c)
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. Section 2.2.16 )
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. Section 2.2.16 )
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. Section 2.2.16 )
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. 5.1.3.3) $[{\hbox to 2pc{}}{1 \over 2}({\bf a} + {\bf b} + {\bf c}) \rightarrow ({\bf a}' + {\bf b}' + {\bf c}')]$	$[\openup2pt\pmatrix{\bar{{1 \over 2}} &{1 \over 2} &{1 \over 2}\cr {1 \over 2} &\bar{{1 \over 2}} &{1 \over 2}\cr {1 \over 2} &{1 \over 2} &\bar{{1 \over 2}}\cr}]$	$[\pmatrix{0 &1 &1\cr 1 &0 &1\cr 1 &1 &0\cr}]$	$[\matrix{{\rm Orthorhombic}\hfill\cr {\rm Tetragonal}\hfill\cr {\rm Cubic}\hfill}]$
$[F \rightarrow P]$ (Fig. 5.1.3.4) $[{\hbox to 1.9pc{}}({\bf a} + {\bf b} + {\bf c})]$ invariant vector	$[\openup2pt\pmatrix{0 &{1 \over 2} &{1 \over 2}\cr {1 \over 2} &0 &{1 \over 2}\cr {1 \over 2} &{1 \over 2} &0\cr}]$	$[\pmatrix{\bar{1} &1 &1\cr 1 &\bar{1} &1\cr 1 &1 &\bar{1}\cr}]$	$[\matrix{{\rm Orthorhombic}\hfill\cr {\rm Tetragonal}\hfill\cr {\rm Cubic}\hfill}]$
$[({\bf b}, {\bf a}, \bar{{\bf c}}) \rightarrow ({\bf a}, {\bf b}, {\bf c})]$	$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$	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. 5.1.3.5) 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 4.3.4 )
$[\left.\matrix{P\rightarrow C_{2}\cr I\rightarrow F_{2}\cr}\right\}]$ (Fig. 5.1.3.5) 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 4.3.4 )
Primitive rhombohedral cell $[\rightarrow]$ triple hexagonal cell $[R_{1}]$ , obverse setting (Fig. 5.1.3.6c)	$[\pmatrix{1 &0 &1\cr \bar{1} &1 &1\cr 0 &\bar{1} &1\cr}]$	$[\openup2pt\pmatrix{{2 \over 3} &\bar{{1 \over 3}} &\bar{{1 \over 3}}\cr {1 \over 3} &{1 \over 3} &\bar{{2} \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow]$ triple hexagonal cell $[R_{2}]$ , obverse setting (Fig. 5.1.3.6c)	$[\pmatrix{0 &\bar{1} &1\cr 1 &0 &1\cr \bar{1} &1 &1\cr}]$	$[\openup2pt\pmatrix{\bar{{1 \over 3}} &{2 \over 3} &\bar{{1 \over 3}}\cr \bar{{2} \over 3} &{1 \over 3} &{1 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow]$ triple hexagonal cell $[R_{3}]$ , obverse setting (Fig. 5.1.3.6c)	$[\pmatrix{\bar{1} &1 &1\cr 0 &\bar{1} &1\cr 1 &0 &1\cr}]$	$[\openup2pt\pmatrix{\bar{{1 \over 3}} &\bar{{1 \over 3}} &{2 \over 3}\cr {1 \over 3} &\bar{{2} \over 3} &{1 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow]$ triple hexagonal cell $[R_{1}]$ , reverse setting (Fig. 5.1.3.6d)	$[\pmatrix{\bar{1} &0 &1\cr 1 &\bar{1} &1\cr 0 &1 &1\cr}]$	$[\openup2pt\pmatrix{\bar{{2 \over 3}} &{1 \over 3} &{1 \over 3}\cr \bar{{1} \over 3} &\bar{{1} \over 3} &{2 \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow]$ triple hexagonal cell $[R_{2}]$ , reverse setting (Fig. 5.1.3.6d)	$[\pmatrix{0 &1 &1\cr \bar{1} &0 &1\cr 1 &\bar{1} &1\cr}]$	$[\openup2pt\pmatrix{{1 \over 3} &\bar{{2 \over 3}} &{1 \over 3}\cr {2 \over 3} &\bar{{1} \over 3} &\bar{{1} \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow]$ triple hexagonal cell $[R_{3}]$ , reverse setting (Fig. 5.1.3.6d)	$[\pmatrix{1 &\bar{1} &1\cr 0 &1 &1\cr \bar{1} &0 &1\cr}]$	$[\openup2pt\pmatrix{{1 \over 3} &{1 \over 3} &\bar{{2 \over 3}}\cr \bar{{1} \over 3} &{2 \over 3} &\bar{{1} \over 3}\cr {1 \over 3} &{1 \over 3} &{1 \over 3}\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Hexagonal cell $[P \rightarrow]$ orthohexagonal centred cell $[C_{1}]$ (Fig. 5.1.3.7)	$[\pmatrix{1 &1 &0\cr 0 &2 &0\cr 0 &0 &1\cr}]$	$[\openup2pt\pmatrix{1 &\bar{{1 \over 2}} &0\cr 0 &{1 \over 2} &0\cr 0 &0 &1\cr}]$	Trigonal Hexagonal (cf. Section 4.3.5 )
Hexagonal cell $[P \rightarrow]$ orthohexagonal centred cell $[C_{2}]$ (Fig. 5.1.3.7)	$[\pmatrix{1 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &1\cr}]$	$[\openup2pt\pmatrix{{1 \over 2} &{1 \over 2} &0\cr \bar{{1} \over 2} &{1 \over 2} &0\cr 0 &0 &1\cr}]$	Trigonal Hexagonal (cf. Section 4.3.5 )
Hexagonal cell $[P \rightarrow]$ orthohexagonal centred cell $[C_{3}]$ (Fig. 5.1.3.7)	$[\pmatrix{0 &\bar{2} &0\cr 1 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\openup2pt\pmatrix{\bar{{1 \over 2}} &1 &0\cr \bar{1 \over 2} &0 &0\cr 0 &0 &1\cr}]$	Trigonal Hexagonal (cf. Section 4.3.5 )
Hexagonal cell $[P \rightarrow]$ triple hexagonal cell $[H_{1}]$ (Fig. 5.1.3.8)	$[\pmatrix{1 &1 &0\cr \bar{1} &2 &0\cr 0 &0 &1\cr}]$	$[\openup2pt\pmatrix{{2 \over 3} &\bar{{1 \over 3}} &0\cr {1 \over 3} &{1 \over 3} &0\cr 0 &0 &1\cr}]$	Trigonal Hexagonal (cf. Section 4.3.5 )
Hexagonal cell $[P \rightarrow]$ triple hexagonal cell $[H_{2}]$ (Fig. 5.1.3.8)	$[\pmatrix{2 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &1\cr}]$	$[\openup2pt\pmatrix{{1 \over 3} &{1 \over 3} &0\cr \bar{{1} \over 3} &{2 \over 3} &0\cr 0 &0 &1\cr}]$	Trigonal Hexagonal (cf. Section 4.3.5 )
Hexagonal cell $[P \rightarrow]$ triple hexagonal cell $[H_{3}]$ (Fig. 5.1.3.8)	$[\pmatrix{1 &\bar{2} &0\cr 2 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\openup2pt\pmatrix{\bar{{1 \over 3}} &{2 \over 3} &0\cr \bar{{2 \over 3}} &{1 \over 3} &0\cr 0 &0 &1\cr}]$	Trigonal Hexagonal (cf. Section 4.3.5 )
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 4.3.5 )
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 4.3.5 )
Triple hexagonal cell R, obverse setting $[\rightarrow]$ C-centred monoclinic cell, unique axis b, cell choice 1 (Fig. 5.1.3.9a) c and b axes invariant	$[\openup3pt\pmatrix{{2 \over 3} &0 &0\cr {1 \over 3} &1 &0\cr \bar{{2 \over 3}} &0 &1\cr}]$	$[\openup3pt\pmatrix{{3 \over 2} &0 &0\cr \bar{{1 \over 2}} &1 &0\cr 1 &0 &1\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Triple hexagonal cell R, obverse setting $[\rightarrow]$ C-centred monoclinic cell, unique axis b, cell choice 2 (Fig. 5.1.3.9a) c axis invariant	$[\openup2pt\pmatrix{\bar{{1 \over 3}} &\bar{1} &0\cr {1 \over 3} &\bar{1} &0\cr \bar{{2 \over 3}} &0 &1\cr}]$	$[\openup2pt\pmatrix{\bar{{3 \over 2}} &{3 \over 2} &0\cr \bar{{1 \over 2}} &\bar{{1 \over 2}} &0\cr \bar{1} &1 &1\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Triple hexagonal cell R, obverse setting $[\rightarrow]$ C-centred monoclinic cell, unique axis b, cell choice 3 (Fig. 5.1.3.9a) $[{\bf a}_{\rm h} \rightarrow {\bf b}_{\rm m}, {\bf c}]$ axis invariant	$[\openup3pt\pmatrix{\bar{{1 \over 3}} &1 &0\cr \bar{{2 \over 3}} &0 &0\cr \bar{{2 \over 3}} &0 &1\cr}]$	$[\openup3pt\pmatrix{0 &\bar{{3 \over 2}} &0\cr 1 &\bar{{1 \over 2}} &0\cr 0 &\bar{1} &1\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Triple hexagonal cell R, obverse setting $[\rightarrow]$ A-centred monoclinic cell, unique axis c, cell choice 1 (Fig. 5.1.3.9b) $[{\bf b}_{\rm h} \rightarrow {\bf c}_{\rm m}, {\bf c}_{\rm h} \rightarrow {\bf a}_{\rm m}]$	$[\openup2pt\pmatrix{0 &{2 \over 3} &0\cr 0 &{1 \over 3} &1\cr 1 &\bar{{2 \over 3}} &0\cr}]$	$[\openup3pt\pmatrix{1 &0 &1\cr {3 \over 2} &0 &0\cr \bar{{1 \over 2}} &1 &0\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Triple hexagonal cell R, obverse setting $[\rightarrow]$ A-centred monoclinic cell, unique axis c, cell choice 2 (Fig. 5.1.3.9b) $[{\bf c}_{\rm h} \rightarrow {\bf a}_{\rm m}]$	$[\openup3pt\pmatrix{0 &\bar{{1 \over 3}} &\bar{1}\cr 0 &{1 \over 3} &\bar{1}\cr 1 &\bar{{2 \over 3}} &0\cr}]$	$[\openup3pt\pmatrix{\bar{1} &1 &1\cr \bar{{3 \over 2}} &{3 \over 2} &0\cr \bar{{1 \over 2}} &\bar{{1 \over 2}} &0\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Triple hexagonal cell R, obverse setting $[\rightarrow]$ A-centred monoclinic cell, unique axis c, cell choice 3 (Fig. 5.1.3.9b) $[{\bf a}_{\rm h} \rightarrow {\bf c}_{\rm m}, {\bf c}_{\rm h} \rightarrow {\bf a}_{\rm m}]$	$[\openup3pt\pmatrix{0 &\bar{{1 \over 3}} &1\cr 0 &\bar{{2 \over 3}} &0\cr 1 &\bar{{2 \over 3}} &0\cr}]$	$[\openup3pt\pmatrix{0 &\bar{1} &1\cr 0 &\bar{{3 \over 2}} &0\cr 1 &\bar{{1 \over 2}} &0\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow C]$ -centred monoclinic cell, unique axis b, cell choice 1 (Fig. 5.1.3.10a) $[[111]_{\rm r} \rightarrow {\bf c}_{\rm m}]$	$[\pmatrix{0 &0 &1\cr \bar{1} &1 &1\cr \bar{1} &\bar{1} &1\cr}]$	$[\openup2pt\pmatrix{1 &\bar{{1 \over 2}} &\bar{{1 \over 2}}\cr 0 &{1 \over 2} &\bar{{1 \over 2}}\cr 1 &0 &0\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow C]$ -centred monoclinic cell, unique axis b, cell choice 2 (Fig. 5.1.3.10a) $[[111]_{\rm r} \rightarrow {\bf c}_{\rm m}]$	$[\pmatrix{\bar{1} &\bar{1} &1\cr 0 &0 &1\cr \bar{1} &1 &1\cr}]$	$[\openup3pt\pmatrix{\bar{{1 \over 2}} &1 &\bar{{1 \over 2}}\cr \bar{{1 \over 2}} &0 &{1 \over 2}\cr 0 &1 &0\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow C]$ -centred monoclinic cell, unique axis b, cell choice 3 (Fig. 5.1.3.10a) $[[111]_{\rm r} \rightarrow {\bf c}_{\rm m}]$	$[\pmatrix{\bar{1} &1 &1\cr \bar{1} &\bar{1} &1\cr 0 &0 &1\cr}]$	$[\pmatrix{\bar{{1 \over 2}} &\bar{{1 \over 2}} &1\cr {1 \over 2} &\bar{{1 \over 2}} &0\cr 0 &0 &1\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow A]$ -centred monoclinic cell, unique axis c, cell choice 1 (Fig. 5.1.3.10b) $[[111]_{\rm r} \rightarrow {\bf a}_{\rm m}]$	$[\pmatrix{1 &0 &0\cr 1 &\bar{1} &1\cr 1 &\bar{1} &\bar{1}\cr}]$	$[\openup3pt\pmatrix{1 &0 &0\cr 1 &\bar{{1 \over 2}} &\bar{{1 \over 2}}\cr 0 &{1 \over 2} &\bar{{1 \over 2}}\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow A]$ -centred monoclinic cell, unique axis c, cell choice 2 (Fig. 5.1.3.10b) $[[111]_{\rm r} \rightarrow {\bf a}_{\rm m}]$	$[\pmatrix{1 &\bar{1} &\bar{1}\cr 1 &0 &0\cr 1 &\bar{1} &1\cr}]$	$[\openup3pt\pmatrix{0 &1 &0\cr \bar{{1 \over 2}} &1 &\bar{{1 \over 2}}\cr \bar{{1 \over 2}} &0 &{1 \over 2}\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )
Primitive rhombohedral cell $[\rightarrow A]$ -centred monoclinic cell, unique axis c, cell choice 3 (Fig. 5.1.3.10b) $[[111]_{\rm r} \rightarrow {\bf a}_{\rm m}]$	$[\pmatrix{1 &\bar{1} &1\cr 1 &\bar{1} &\bar{1}\cr 1 &0 &0\cr}]$	$[\openup3pt\pmatrix{0 &0 &1\cr \bar{{1 \over 2}} &\bar{{1 \over 2}} &1\cr {1 \over 2} &\bar{{1 \over 2}} &0\cr}]$	Rhombohedral space groups (cf. Section 4.3.5 )