General transformation

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. 78-85

Section 5.1.3. General transformation

H. Arnold^a ^‡

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

5.1.3. General transformation

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Here the crystal structure is considered to be at rest, whereas the coordinate system and the unit cell are changed. Specifically, a point X in a crystal is defined with respect to the basis vectors a, b, c and the origin O by the coordinates x, y, z, i.e. the position vector r of point X is given by $[\eqalign{{\bf r} &= x{\bf a} + y{\bf b} + z{\bf c}\cr &= ({\bf a},{\bf b},{\bf c}) \pmatrix{x\cr y\cr z\cr}.}]$ The same point X is given with respect to a new coordinate system, i.e. the new basis vectors a′, b′, c′ and the new origin O′ (Fig. 5.1.3.1), by the position vector $[{\bf r}' = x'{\bf a}' + y'{\bf b}' + z'{\bf c}'.]$ In this section, the relations between the primed and unprimed quantities are treated.

Figure 5.1.3.1 | top | pdf |

General affine transformation, consisting of a shift of origin from O to O′ by a shift vector p with components $[p_{1}]$ and $[p_{2}]$ and a change of basis from a, b to a′, b′. This implies a change in the coordinates of the point X from x, y to x′, y′.

The general transformation (affine transformation ) of the coordinate system consists of two parts, a linear part and a shift of origin. The $[(3 \times 3)]$ matrix P of the linear part and the $[(3 \times 1)]$ column matrix p, containing the components of the shift vector p, define the transformation uniquely. It is represented by the symbol (P, p).

(i) The linear part implies a change of orientation or length or both of the basis vectors a, b, c, i.e. $[\eqalign{({\bf a}', {\bf b}', {\bf c}') &= ({\bf a}, {\bf b}, {\bf c}) {\bi P}\cr &= ({\bf a}, {\bf b}, {\bf c}) \pmatrix{P_{11} &P_{12} &P_{13}\cr P_{21} &P_{22} &P_{23}\cr P_{31} &P_{32} &P_{33}\cr}\cr &= (P_{11}{\bf a} + P_{21}{\bf b} + P_{31}{\bf c},\cr &\qquad \ P_{12}{\bf a} + P_{22}{\bf b} + P_{32}{\bf c},\cr &\qquad \ P_{13}{\bf a} + P_{23}{\bf b} + P_{33}{\bf c}).}]$ For a pure linear transformation, the shift vector p is zero and the symbol is (P, o).

The determinant of P, $[\hbox{det} ({\bi P})]$ , should be positive. If $[\hbox{det} ({\bi P})]$ is negative, a right-handed coordinate system is transformed into a left-handed one (or vice versa). If det $[({\bi P}) = 0]$ , the new basis vectors are linearly dependent and do not form a complete coordinate system.

In this chapter, transformations in three-dimensional space are treated. A change of the basis vectors in two dimensions, i.e. of the basis vectors a and b, can be considered as a three-dimensional transformation with invariant c axis. This is achieved by setting $[P_{33} = 1]$ and $[P_{13} = P_{23} = P_{31} = P_{32} = 0]$ .
(ii) A shift of origin is defined by the shift vector $[{\bf p} = p_{1}{\bf a} + p_{2}{\bf b} + p_{3}{\bf c.}]$ The basis vectors a, b, c are fixed at the origin O; the new basis vectors are fixed at the new origin O′ which has the coordinates $[p_{1}, p_{2}, p_{3}]$ in the old coordinate system (Fig. 5.1.3.1).

For a pure origin shift, the basis vectors do not change their lengths or orientations. In this case, the transformation matrix P is the unit matrix I and the symbol of the pure shift becomes (I, p).

Also, the inverse matrices of P and p are needed. They are $[{\bi Q} = {\bi P}^{-1}]$ and $[{\bi q} = - {\bi P}^{-1}{\bi p.}]$ The matrix q consists of the components of the negative shift vector q which refer to the coordinate system a′, b′, c′, i.e. $[{\bf q} = q_{1}{\bf a}' + q_{2}{\bf b}' + q_{3}{\bf c}'.]$ Thus, the transformation (Q, q) is the inverse transformation of (P, p). Applying (Q, q) to the basis vectors a′, b′, c′ and the origin O′, the old basis vectors a, b, c with origin O are obtained.

For a two-dimensional transformation of a′ and b′, some elements of Q are set as follows: $[Q_{33} = 1]$ and $[Q_{13} = Q_{23} = Q_{31} = Q_{32} = 0]$ .

The quantities which transform in the same way as the basis vectors a, b, c are called covariant quantities and are written as row matrices. They are:

the Miller indices of a plane (or a set of planes), (hkl), in direct space and the coordinates of a point in reciprocal space, h, k, l.

Both are transformed by $[(h', k', l') = (h, k, l){\bi P.}]$ Usually, the Miller indices are made relative prime before and after the transformation.

The quantities which are covariant with respect to the basis vectors a, b, c are contravariant with respect to the basis vectors $[{\bf a}^{*},{\bf b}^{*},{\bf c}^{*}]$ of reciprocal space.

The basis vectors of reciprocal space are written as a column matrix and their transformation is achieved by the matrix Q: $[\eqalign{\pmatrix{{\bf a}^{*}{}'\cr {\bf b}^{*}{}'\cr {\bf c}^{*}{}'\cr} &= {\bi Q} \pmatrix{{\bf a}^{*}\cr {\bf b}^{*}\cr {\bf c}^{*}\cr}\cr &= \pmatrix{Q_{11} &Q_{12} &Q_{13}\cr Q_{21} &Q_{22} &Q_{23}\cr Q_{31} &Q_{32} &Q_{33}\cr} \pmatrix{{\bf a}^{*}\cr {\bf b}^{*}\cr {\bf c}^{*}\cr}\cr &= \pmatrix{Q_{11}{\bf a}^{*} + Q_{12}{\bf b}^{*} + Q_{13}{\bf c}^{*}\cr Q_{21}{\bf a}^{*} + Q_{22}{\bf b}^{*} + Q_{23}{\bf c}^{*}\cr Q_{31}{\bf a}^{*} + Q_{32}{\bf b}^{*} + Q_{33}{\bf c}^{*}\cr}.}]$ The inverse transformation is obtained by the inverse matrix $[{\bi P} = {\bi Q}^{-1}]$ : $[\pmatrix{{\bf a}^{*}\cr {\bf b}^{*}\cr {\bf c}^{*}\cr} = {\bi P} \pmatrix{{\bf a}^{*}{}'\cr {\bf b}^{*}{}'\cr {\bf c}^{*}{}'\cr}.]$ These transformation rules apply also to the quantities covariant with respect to the basis vectors $[{\bf a}^{*},{\bf b}^{*},{\bf c}^{*}]$ and contravariant with respect to a, b, c, which are written as column matrices. They are the indices of a direction in direct space, [uvw], which are transformed by $[\pmatrix{u'\cr v'\cr w'\cr} = {\bi Q} \pmatrix{u\cr v\cr w\cr}.]$ In contrast to all quantities mentioned above, the components of a position vector r or the coordinates of a point X in direct space x, y, z depend also on the shift of the origin in direct space. The general (affine) transformation is given by $[\eqalign{\pmatrix{x'\cr y'\cr z'\cr} &= {\bi Q} \pmatrix{x\cr y\cr z\cr} + {\bi q}\cr &= \pmatrix{Q_{11}x + Q_{12}y + Q_{13}z + q_{1}\cr Q_{21}x + Q_{22}y + Q_{23}z + q_{2}\cr Q_{31}x + Q_{32}y + Q_{33}z + q_{3}\cr}.}]$

Example

If no shift of origin is applied, i.e. $[{\bi p} = {\bi q} = {\bi o}]$ , the position vector r of point X is transformed by $[{\bf r}' = ({\bf a},{\bf b},{\bf c}){\bi P}{\bi Q} \pmatrix{x\cr y\cr z\cr} = ({\bf a}',{\bf b}',{\bf c}') \pmatrix{x'\cr y'\cr z'\cr}.]$ In this case, $[{\bf r} = {\bf r}']$ , i.e. the position vector is invariant, although the basis vectors and the components are transformed. For a pure shift of origin, i.e. $[{\bi P} = {\bi Q} = {\bi I}]$ , the transformed position vector r′ becomes $[\eqalign{{\bf r}' &= (x + q_{1}){\bf a} + (y + q_{2}){\bf b} + (z + q_{3}){\bf c}\cr &= {\bf r} + q_{1}{\bf a} + q_{2}{\bf b} + q_{3}{\bf c}\cr &= (x - p_{1}){\bf a} + (y - p_{2}){\bf b} + (z - p_{3}){\bf c}\cr &= {\bf r} - p_{1}{\bf a} - p_{2}{\bf b} - p_{3}{\bf c}.}]$ Here the transformed vector r′ is no longer identical with r.

It is convenient to introduce the augmented $[(4 \times 4)]$ matrix $[\specialfonts{\bbsf Q}]$ which is composed of the matrices Q and q in the following manner (cf. Chapter 8.1 ): $[\specialfonts{\bbsf Q} = \pmatrix{{\bi Q} &{\bi q}\cr {\bi o} &1\cr} = \pmatrix{Q_{11} &Q_{12} &Q_{13} &q_{1}\cr Q_{21} &Q_{22} &Q_{23} &q_{2}\cr Q_{31} &Q_{32} &Q_{33} &q_{3}\cr 0 &0 &0 &1\cr}]$ with o the $[(1 \times 3)]$ row matrix containing zeros. In this notation, the transformed coordinates x′, y′, z′ are obtained by $[\specialfonts\eqalign{\pmatrix{x'\cr y'\cr z'\cr 1\cr} &= {\bbsf Q} \pmatrix{x\cr y\cr z\cr 1\cr}\cr &=\pmatrix{Q_{11} &Q_{12} &Q_{13} &q_{1}\cr Q_{21} &Q_{22} &Q_{23} &q_{2}\cr Q_{31} &Q_{32} &Q_{33} &q_{3}\cr 0 &0 &0 &1\cr} \pmatrix{x\cr y\cr z\cr 1\cr}\cr &= \pmatrix{Q_{11}x + Q_{12}y + Q_{13}z + q_{1}\cr Q_{21}x + Q_{22}y + Q_{23}z + q_{2}\cr Q_{31}x + Q_{32}y + Q_{33}z + q_{3}\cr 1\cr}.}]$ The inverse of the augmented matrix $[\specialfonts{\bbsf Q}]$ is the augmented matrix $[\specialfonts{\bbsf P}]$ which contains the matrices P and p, specifically, $[\specialfonts{\bbsf P} = {\bbsf Q}^{\rm -1} = \pmatrix{{\bi P} &{\bi p}\cr {\bi o} &1\cr} = \pmatrix{{\bi Q}^{-1} &- {\bi Q}^{-1}{\bi q}\cr {\bi o} &1\cr}.]$ The advantage of the use of $[(4 \times 4)]$ matrices is that a sequence of affine transformations corresponds to the product of the corresponding matrices. However, the order of the factors in the product must be observed. If $[\specialfonts{\bbsf Q}]$ is the product of n transformation matrices $[\specialfonts{\bbsf Q}_{i}]$ , $[\specialfonts{\bbsf Q} = {\bbsf Q}_{n} \ldots {\bbsf Q}_{\rm 2} {\bbsf Q}_{\rm 1},]$ the sequence of the corresponding inverse matrices $[\specialfonts{\bbsf P}_{i}]$ is reversed in the product $[\specialfonts{\bbsf P} = {\bbsf P}_{\rm 1} {\bbsf P}_{\rm 2} \ldots {\bbsf P}_{n}.]$

The following items are also affected by a transformation:

(i) The metric matrix of direct lattice G [more exactly: the matrix of geometrical coefficients (metric tensor)] is transformed by the matrix P as follows: $[{\bi G}' = {\bi P}^{t} {\bi G}{\bi P}]$ with $[{\bi P}^{t}]$ the transposed matrix of P, i.e. rows and columns of P are interchanged. Specifically, $[\eqalign{{\bi G}' &= \pmatrix{{\bf a}'\cdot {\bf a}' &{\bf a}'\cdot {\bf b}' &{\bf a}'\cdot {\bf c}'\cr {\bf b}'\cdot {\bf a}' &{\bf b}'\cdot {\bf b}' &{\bf b}'\cdot {\bf c}'\cr {\bf c}'\cdot {\bf a}' &{\bf c}'\cdot {\bf b}' &{\bf c}'\cdot {\bf c}'\cr}\cr &= \pmatrix{P_{11} &P_{21} &P_{31}\cr P_{12} &P_{22} &P_{32}\cr P_{13} &P_{23} &P_{33}\cr} \pmatrix{{\bf a}\cdot {\bf a} &{\bf a}\cdot {\bf b} &{\bf a}\cdot {\bf c}\cr {\bf b}\cdot {\bf a} &{\bf b}\cdot {\bf b} &{\bf b}\cdot {\bf c}\cr {\bf c}\cdot {\bf a} &{\bf c}\cdot {\bf b} &{\bf c}\cdot {\bf c}\cr}\cr &\quad \times \pmatrix{P_{11} &P_{12} &P_{13}\cr P_{21} &P_{22} &P_{23}\cr P_{31} &P_{32} &P_{33}\cr}.}]$
(ii) The metric matrix of reciprocal lattice $[{\bi G}^{*}]$ [more exactly: the matrix of geometrical coefficients (metric tensor)] is transformed by $[{\bi G}^{*}{'} = {\bi Q}{\bi G}^{*}{\bi Q}^{t}.]$ Here, the transposed matrix $[{\bi Q}^{t}]$ is on the right-hand side of $[{\bi G}^{*}]$ .

(iii) The volume of the unit cell V changes with the transformation. The volume of the new unit cell V′ is obtained by $[V' = \det ({\bi P})V = \left|\matrix{P_{11} &P_{12} &P_{13}\cr P_{21} &P_{22} &P_{23}\cr P_{31} &P_{32} &P_{33}\cr}\right| V]$ with $[\hbox{det}({\bi P})]$ the determinant of the matrix P. The corresponding equation for the volume of the unit cell in reciprocal space $[V^{*}]$ is $[V^{*}{'} = \det ({\bi Q})V^{*}.]$ Matrices P and Q that frequently occur in crystallography are listed in Table 5.1.3.1.

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 )

Figure 5.1.3.2 | top | pdf |

Monoclinic centred lattice, projected along the unique axis. Origin for all cells is the same. (a) Unique axis b. Cell choice 1: C-centred cell [a_1,b,c_1] . Cell choice 2: A-centred cell [a_2,b,c_2] . Cell choice 3: I-centred cell [a_3,b,c_3] . (b) Unique axis c. Cell choice 1: A-centred cell [a_1,b_1,c] . Cell choice 2: B-centred cell [a_2,b_2,c] . Cell choice 3: I-centred cell [a_3,b_3,c] . (c) Unique axis a. Cell choice 1: B-centred cell [a,b_1,c_1] . Cell choice 2: C-centred cell [a,b_2,c_2] . Cell choice 3: I-centred cell [a,b_3,c_3] .

Figure 5.1.3.3 | top | pdf |

Body-centred cell I with a, b, c and a corresponding primitive cell P with a′, b′, c′. Origin for both cells O. A cubic I cell with lattice constant $[a_{c}]$ can be considered as a primitive rhombohedral cell with $[a_{r} = a_{c} {1 \over 2} \surd{3}]$ and $[\alpha = 109.47^{\circ}]$ (rhombohedral axes) or a triple hexagonal cell with $[a_{h} = a_{c} \surd{2}]$ and $[c_{h} = a_{c} {1 \over 2} \surd{3}]$ (hexagonal axes).

Figure 5.1.3.4 | top | pdf |

Face-centred cell F with a, b, c and a corresponding primitive cell P with a′, b′, c′. Origin for both cells O. A cubic F cell with lattice constant $[a_{c}]$ can be considered as a primitive rhombohedral cell with $[a_{r} = a_{c} {1 \over 2} \surd{2}]$ and $[\alpha = 60^{\circ}]$ (rhombohedral axes) or a triple hexagonal cell with $[a_{h} = a_{c} {1 \over 2} \surd{2}]$ and $[c_{h} = a_{c} \surd{3}]$ (hexagonal axes).

Figure 5.1.3.5 | 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 $[a_{2}, b_{2}, c]$ . 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 $[a_{2}, b_{2}, c]$ . Origin for all three cells is the same.

Figure 5.1.3.6 | top | pdf |

Unit cells in the rhombohedral lattice: same origin for all cells. The basis of the rhombohedral cell is labelled a, b, c. 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] . (a) Obverse setting of triple hexagonal cell $[a_{1},b_{1},c_{1}]$ in relation to the primitive rhombohedral cell a, b, c. (b) Reverse setting of triple hexagonal cell $[a_{2},b_{2},c_{2}]$ in relation to the primitive rhombohedral cell a, b, c.(c) Primitive rhombohedral cell (- - - lower edges), a, b, c 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), a, b, c 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 5.1.3.7 | top | pdf |

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

Figure 5.1.3.8 | 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 $[a_{1},b_{1},c]$ ; $[a_{2}, b_{2}, c]$ ; $[a_{3}, b_{3}, c]$ . Origin for all cells is the same.

Figure 5.1.3.9 | 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. 5.1.3.6c) 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. 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. Origin for all four cells is the same.

Figure 5.1.3.10 | top | pdf |

Rhombohedral lattice with primitive rhombohedral cell a, b, 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. 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. Origin for all four cells is the same.

International Tables for Crystallography (2006). Vol. A. ch. 5.1, pp. 78-85