International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2011). Vol. A1, ch. 2.1, pp. 77-79   | 1 | 2 |

Section 2.1.3.3. Basis transformation and origin shift

Hans Wondratscheka* and Mois I. Aroyob

aInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and bDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain
Correspondence e-mail:  wondra@physik.uni-karlsruhe.de

2.1.3.3. Basis transformation and origin shift

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Each t-subgroup [{\cal H} \,\lt\, {\cal G}] is defined by its representatives, listed under `sequence' by numbers each of which designates an element of [{\cal G}]. These elements form the general position of [{\cal H}]. They are taken from the general position of [{\cal G}] and, therefore, are referred to the coordinate system of [{\cal G}]. In the general position of [{\cal H}], however, its elements are referred to the coordinate system of [{\cal H}]. In order to allow the transfer of the data from the coordinate system of [{\cal G}] to that of [{\cal H}], the tools for this transformation are provided in the columns `matrix' and `shift' of the subgroup tables. The designation of the quantities is that of IT A Part 5[link] and is repeated here for convenience. The transformation described in this section is not restricted to translationengleiche subgroups but is applied to klassengleiche subgroups as well.

In the following, columns and rows are designated by boldface italic lower-case letters. Point coordinates [{\bi x},{\bi x}'], translation parts [{\bi w}, {\bi w}'] of the symmetry operations and shifts [{\bi p}, {\bi q}=-{\bi P}^{-1}{\bi p}] are represented by columns. The sets of basis vectors [({\bf a},\,{\bf b},\,{\bf c})=({\bi a})^{\rm T}] and [({\bf a}',\,{\bf b}',\,{\bf c}')=({\bi a}')^{\rm T}] are represented by rows [indicated by [(\ldots)^{\rm T}], which means `transposed']. The quantities with unprimed symbols are referred to the coordinate system of [{\cal G}], those with primes are referred to the coordinate system of [{\cal H}].

The following columns will be used ([{\bi w}'] is analogous to w): [{\bi w} = \left(\matrix { w_{1}\cr w_{2} \cr w_{3} } \right); \, \, {\bi x} = \left(\matrix { x \cr y \cr z } \right); \,\, {\bi x}\,' =\left(\matrix { x' \cr y' \cr z' } \right); \,\, {\bi p} = \left(\matrix { p_{1}\cr p_{2} \cr p_{3} } \right); \,\, {\bi q} = \left(\matrix { q_{1}\cr q_{2} \cr q_{3} } \right).]

The [(3\times3)] matrices W and [{\bi W}'] of the symmetry operations, as well as the matrix P for a change of basis and its inverse [{\bi Q}={\bi P}^{-1}], are designated by boldface italic upper-case letters ([{\bi W}'] is analo­gous to W): [{\bi W} = \left(\matrix {W_{11} &W_{12} &W_{13} \cr W_{21} & W_{22} & W_{23} \cr W_{31} & W_{32} & W_{33} } \right)\semi {\bi P} = \left(\matrix{P_{11} & P_{12} & P_{13} \cr P_{21} & P_{22} & P_{23} \cr P_{31} & P_{32} & P_{33} } \right)\semi][{\bi Q} = \left(\matrix { Q_{11} & Q_{12} & Q_{13} \cr Q_{21} & Q_{22} & Q_{23} \cr Q_{31} & Q_{32} & Q_{33} } \right).]Let [{\bf a}, {\bf b}, {\bf c} = ({\bi a})^{\rm T}] be the row of basis vectors of [{\cal G}] and [{\bf a}', {\bf b}', {\bf c}'\,=\, ({\bi a}')^{\rm T}] the basis of [{\cal H}], then the basis [({\bi a}')^{\rm T}] is expressed in the basis [({\bi a})^{\rm T}] by the system of equations[\eqalignno{ {\bf a}' &=P_{11}{\bf a} +P_{21}{\bf b} + P_{31}{\bf c}& \cr {\bf b}' &=P_{12}{\bf a} +P_{22}{\bf b} + P_{32}{\bf c} &(2.1.3.1)\cr {\bf c}' &=P_{13}{\bf a} +P_{23}{\bf b} +P_{33}{\bf c}&}]or [({\bf a}', {\bf b}', {\bf c}') ^{\rm T} = ({\bf a}, {\bf b}, {\bf c}) ^{\rm T} \left(\matrix { P_{11} & P_{12} & P_{13} \cr P_{21} & P_{22} & P_{23} \cr P_{31} & P_{32} & P_{33} } \right). \eqno (2.1.3.2)]

In matrix notation, this is [({\bi a}')^{\rm T} = ({\bi a})^{\rm T} \, {\bi P}. \eqno (2.1.3.3)]

The column p of coordinates of the origin [O'] of [{\cal H}] is referred to the coordinate system of [{\cal G}] and is called the origin shift. The matrix–column pair (P, p) describes the transformation from the coordinate system of [{\cal G}] to that of [{\cal H}], for details, cf. IT A, Part 5[link] . Therefore, P and p are listed in the subgroup tables in the columns `matrix' and `shift', cf. Section 2.1.3.2[link]. The column `matrix' is empty if there is no change of basis, i.e. if P is the unit matrix I. The column `shift' is empty if there is no origin shift, i.e. if p is the column o consisting of zeroes only.

A change of the coordinate system, described by the matrix–column pair [({\bi P},\, {\bi p})], changes the point coordinates from the column x to the column [{\bi x}\,']. The formulae for this change do not contain the pair [({\bi P},\, {\bi p})] itself, but the related pair [({\bi Q},\, {\bi q})=] [({\bi P}^{-1},\, -{\bi P}^{-1}\,{\bi p})]: [ {\bi x}\,'={\bi Q}{\bi x}+{\bi q}={\bi P}^{-1}{\bi x}- {\bi P}^{-1}{\bi p}={\bi P}^{-1}({\bi x}-{\bi p}). \eqno (2.1.3.4)]

Not only the point coordinates but also the matrix–column pairs for the symmetry operations are changed by a change of the coordinate system. A symmetry operation [\sf W] is described in the coordinate system of [{\cal G}] by the system of equations2[\eqalignno{ \tilde{x}&=W_{11}\,x+W_{12}\,y+W_{13}\,z+w_1 &\cr \tilde{y}&=W_{21}\,x+W_{22}\,y+W_{23}\,z+w_2 & (2.1.3.5)\cr \tilde{z}&=W_{31}\,x+W_{32}\,y+W_{33}\,z+w_3, &}]or[{\tilde {\bi x}}={\bi W}{\bi x}+{\bi w}=({\bi W,w})\,{\bi x}, \eqno (2.1.3.6)]i.e. by the matrix–column pair (W, w). The symmetry operation [\sf W] will be described in the coordinate system of the subgroup [{\cal H}] by the equation [\tilde{\bi{x}}\,'=\bi{W}\,'\,\bi{x}\,'+\bi{w}\,'= (\bi{W}\,',\,\bi{w}\,')\,\bi{x}\,', \eqno (2.1.3.7)]and thus by the pair [(\bi{W}\,',\,\bi{w}\,')]. This pair can be calculated from the pair [(\bi{W},\ \bi{w})] by the equations [ {\bi W}\,'={\bi Q}\,{\bi W}\,{\bi P}={\bi P}^{-1}{\bi W}{\bi P} \eqno (2.1.3.8)]and [ {\bi w}\,'={\bi q}+ {\bi Q}{\bi w}+{\bi Q}{\bi W}{\bi p}={\bi P}^{-1}({\bi w}+ {\bi W}{\bi p}-{\bi p})={\bi P}^{-1}({\bi w}+({\bi W}-{\bi I}) {\bi p}). \eqno (2.1.3.9)]

These equations are rather complicated and unpleasant. They become simple when using augmented matrices and columns. In this case the formulae are reduced formally to normal matrix multiplication [the formalism is simpler but the necessary calculations are not, because the inversion of a (4 × 4) matrix is tedious if done by hand].

The matrices P, Q, W and W′ may be combined with the corresponding columns p, q, w and w′ to form (4 × 4) matrices (callled augmented matrices):3[\specialfonts\let\normalbaselines\relax\openup-3pt\eqalign{{\bbsf P}&=\pmatrix{P_{11} & P_{12} &P_{13}{\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt}p_1\cr P_{21} & P_{22} &P_{23}{\hskip -4pt}&{\vrule height 10pt depth 6pt}\hfill&{\hskip -4pt}p_2\cr P_{31} &P_{32} & P_{33}{\hskip -4pt}&{\vrule height 10pt depth 6pt}\hfill&{\hskip -4pt}p_3\cr\noalign{\hrule}\cr 0 & 0 & 0{\hskip -4pt}&{\vrule height 11pt depth 4pt}\hfill&{\hskip -4pt} 1}\semi\cr {\bbsf Q}={\bbsf P}^{-1}&=\pmatrix{Q_{11} & Q_{12} &Q_{13}{\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt}q_1\cr Q_{21} & Q_{22} &Q_{23}{\hskip -4pt}&{\vrule height 10pt depth 6pt}\hfill&{\hskip -4pt}q_2\cr Q_{31} &Q_{32} & Q_{33}{\hskip -4pt}&{\vrule height 10pt depth 6pt}\hfill&{\hskip -4pt}q_3\cr\noalign{\hrule}\cr 0 & 0 & 0{\hskip -4pt}&{\vrule height 11pt depth 4pt}\hfill&{\hskip -4pt} 1}\semi\cr\specialfonts{\bbsf W}&=\pmatrix{W_{11} & W_{12} &W_{13}{\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt}w_1\cr W_{21} & W_{22} &W_{23}{\hskip -4pt}&{\vrule height 10pt depth 6pt}\hfill&{\hskip -4pt}w_2\cr W_{31} &W_{32} & W_{33}{\hskip -4pt}&{\vrule height 10pt depth 6pt}\hfill&{\hskip -4pt}w_3\cr\noalign{\hrule}\cr 0 & 0 & 0{\hskip -4pt}&{\vrule height 11pt depth 4pt}\hfill&{\hskip -4pt} 1}\semi\cr \qquad{\bbsf W}'&=\pmatrix{W'_{11} & W'_{12} &W'_{13}{\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt}w'_1\cr W'_{21} & W'_{22} &W'_{23}{\hskip -4pt}&{\vrule height 10pt depth 6pt}\hfill&{\hskip -4pt}w'_2\cr W'_{31} &W'_{32} & W'_{33}{\hskip -4pt}&{\vrule height 10pt depth 6pt}\hfill&{\hskip -4pt}w'_3\cr\noalign{\hrule}\cr 0 & 0 & 0{\hskip -4pt}&{\vrule height 11pt depth 4pt}\hfill&{\hskip -4pt} 1}.}]The coefficients of these augmented matrices are integer, rational or real numbers.

The (3 × 1) rows (a)T and (a′)T must be augmented to (4 × 1) rows by appending some vectors [{\bf s}_{\cal G}] and [{\bf s}'_{\cal H}], respectively, as fourth entries in order to enable matrix multiplication with the aug­mented matrices:[\specialfonts({\bbsf a})^{\rm T}=({\bf a}, {\bf b}, {\bf c}\ |\ {\bf s}_{\cal G})^{\rm T}\quad {\rm and}\quad ({\bbsf a}')^{\rm T}=({\bf a}', {\bf b}', {\bf c}'\ |\ {\bf s}'_{\cal H})^{\rm T}]with [{\bf s}'_{\cal H} ={\bf p} + {\bf s}_{\cal G}]. As the vector [{\bf s}_{\cal G}] one can take the zero vector [{\bf s}_{\cal G} ={\bf o}], which results in [{\bf s}'_{\cal H} ={\bf p} =p_1{\bf a}+p_2{\bf b} +p_3{\bf c}], i.e.[\specialfonts({\bbsf a})^{\rm T}=({\bf a}, {\bf b}, {\bf c}\ |\ {\bf o})^{\rm T}\quad {\rm and}\quad ({\bbsf a}')^{\rm T}=({\bf a}', {\bf b}', {\bf c}'\ |\ {\bf p})^{\rm T}.]The relation between [\specialfonts({\bbsf a}')^{\rm T}] and [\specialfonts({\bbsf a})^{\rm T}] is given by equation (2.1.3.10)[link], which replaces equation (2.1.3.3)[link],[\specialfonts({\bbsf a}')^{\rm T}= ({\bbsf a})^{\rm T}{\bbsf P}.\eqno(2.1.3.10)]

Analogously, the (3 × 1) columns x and x′ must be augmented to (4 × 1) columns by a `1' in the fourth row in order to enable matrix multiplication with the augmented matrices:[\specialfonts{\bbsf x}=\pmatrix{x\cr y\cr z\vphantom{_{l_l}}\cr\noalign{\hrule}\cr 1}\semi\quad{\bbsf x}'=\pmatrix{x'\cr y'\cr z'\vphantom{_{l_l}}\cr\noalign{\hrule}\cr 1}.]The three equations (2.1.3.4)[link], (2.1.3.8)[link] and (2.1.3.9)[link] are replaced by the two equations[\specialfonts{\bbsf x}'={\bbsf Q}{\bbsf x}={\bbsf P}^{\rm -1}{\bbsf x}\eqno(2.1.3.11)]and[\specialfonts{\bbsf W}'={\bbsf Q}{\bbsf W}{\bbsf P}={\bbsf P}^{\rm -1}{\bbsf W}{\bbsf P}.\eqno(2.1.3.12)]

Example 2.1.3.3.1

Consider the data listed for the t-subgroups of [Pmn2_1], No. 31:[\matrix{{\rm Index} & {\rm HM\,\, \& \,\,No.} &{\rm sequence} &{\rm matrix} &{\rm shift} \cr [2] &P1n1\,\, (7, P1c1)\hfill &1\semi3 &{\bf c},{\bf b}, -{\bf a} -{\bf c}\hfill &\cr [2] &Pm11\,\,(6, P1m1)\hfill &1\semi4 &{\bf c}, {\bf a}, {\bf b}\hfill&\cr [2] &P112_{1}\,\,(4)\hfill &1\semi2 & &1/4, 0, 0\cr}]This means that the transformation matrices and origin shifts are[\quad{\bi P}_{1}= \left(\matrix { 0 & 0 & \overline{1} \cr 0 & 1 & 0 \cr 1 & 0 & \overline{1} } \right), {\bi P}_{2} =\left(\matrix { 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 } \right), {\bi P}_{3} = \left(\matrix { 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 } \right)][\quad{\bi p}_{1} = \left(\matrix { 0 \cr 0 \cr 0 } \right), \,\,{\bi p}_{2} = \left(\matrix {0 \cr 0 \cr 0 }\right), \,\,{\bi p}_{3} = \left(\matrix { {{1}\over{4}} \cr 0 \cr 0 } \right).]

The first subgroup is monoclinic, the symmetry direction is the b axis, which is standard. However, the glide direction [{{1}\over{2}}({\bf a}+{\bf c})] is nonconventional. Therefore, the basis of [{\cal G}] is transformed to a basis of the subgroup [{\cal H}] such that the b axis is retained, the glide direction becomes the [c'] axis and the [a'] axis is chosen such that the basis is a right-handed one, the angle [\beta'\ge90^{\circ}] and the transformation matrix P is simple. This is done by the chosen matrix [{\bi P}_1]. The origin shift is the o column.

With equations (2.1.3.8)[link] and (2.1.3.9)[link], one obtains for the glide reflection [x,\overline{y},z-{{1}\over{2}}], which is [x,\overline{y},z+{{1}\over{2}}] after standardization by [0\le w_j \,\lt\, 1].

For the second monoclinic subgroup, the symmetry direction is the (nonconventional) a axis. The rules of Section 2.1.2.5[link] require a change to the setting `unique axis b'. A cyclic per­mutation of the basis vectors is the simplest way to achieve this. The reflection [\overline{x}, y, z] is now described by [x, \overline{y}, z]. Again there is no origin shift.

The third monoclinic subgroup is in the conventional setting `unique axis c', but the origin must be shifted onto the [2_1] screw axis. This is achieved by applying equation (2.1.3.9)[link] with [{\bi p}_3], which changes [\overline{x}+{{1}\over{2}},\overline{y},z+{{1}\over{2}}] of [Pmn2_1] to [\overline{x},\overline{y},z+{{1}\over{2}}] of [P112_1].

Example 2.1.3.3.2

Evaluation of the t-subgroup data of the space group [P3_112], No. 151, started in Example 2.1.3.2.5[link]. The evaluation is now continued with the columns `sequence', `matrix' and `shift'. They are used for the transformation of the elements of [{\cal H}] to their conventional form. Only the monoclinic t-subgroups are of interest here because the trigonal subgroup is already in the standard setting.

One takes from the tables of subgroups in Chapter 2.3[link] [\quad\matrix{{\rm Index} \quad\quad{\rm HM\,\, \& \,\, No.}\quad{\rm sequence} \quad\quad{\rm matrix} \quad\quad\quad\quad{\rm shift}\hfill\cr \Biggl\{\matrix{[3] &P112\,\,(5,\,C121) &\,\,1\semi6 &\quad{\bf b},-2{\bf a}- {\bf b},{\bf c} \hfill & \cr [3] &P112\,\,(5,\,C121) & \,\,1\semi4 &\quad -{\bf a}-{\bf b}, {\bf a}-{\bf b}, {\bf c}\hfill & 0,0,1/3 \cr [3] &P112\,\,(5,\,C121) & \,\,1\semi5 &\quad {\bf a},{\bf a} + 2{\bf b},{\bf c}\hfill & 0,0,2/3\cr}}]Designating the three matrices by [{\bi P}_6], [{\bi P}_4], [{\bi P}_5], one obtains[\quad{{\bi P}_{6} = \left(\matrix { 0 & -2 & 0 \cr 1 & -1 & 0 \cr 0 & {\phantom -}0 & 1 } \right), \ {\bi P}_{4} = \left(\matrix {-1 & {\phantom -}1 & 0 \cr -1 & -1 & 0 \cr {\phantom -}0 & {\phantom -}0 & 1 } \right), \ {\bi P}_{5} = \left(\matrix {1 & 1 & 0 \cr 0 & 2 & 0 \cr 0 & 0 & 1 } \right)}]with the corresponding inverse matrices[\quad{\bi Q}_{6} = \left(\matrix { -{{1}\over{2}} & 1 & 0 \cr -{{1}\over{2}} & 0 & 0 \cr {\phantom -}0 & 0 & 1 } \right),\ {\bi Q}_{4} = \left(\matrix { -{{1}\over{2}} & -{{1}\over{2}} & 0 \cr {\phantom -}{{1}\over{2}} & -{{1}\over{2}} & 0 \cr {\phantom -}0 & {\phantom -}0 & 1 } \right),\ {\bi Q}_{5} = \left(\matrix { 1 & -{{1}\over{2}} & 0 \cr 0 & {\phantom -}{{1}\over{2}} & 0 \cr 0 & {\phantom -}0 & 1 } \right)]and the origin shifts[\quad{\bi p}_{6} = \left(\matrix { 0 \cr 0 \cr 0 } \right), \, \, {\bi p}_{4} = \left(\matrix { 0 \cr 0 \cr {{1}\over{3}} } \right),\, \, {\bi p}_{5} = \left(\matrix { 0 \cr 0 \cr {{2}\over{3}}} \right).]For the three new bases this means [\quad\eqalign{ &{\bf a}_{6}' = {\bf b},\,\, {\bf b}_{6}'=-2{\bf a}-{\bf b}, \,\,{\bf c}_6'={\bf c} \cr &{\bf a}_4'=-{\bf a}-{\bf b},\,\, {\bf b}_4'={\bf a}-{\bf b}, \,\,{\bf c}_4'={\bf c}\,\, {\rm and} \cr& {\bf a}_5'={\bf a}, \,\,{\bf b}_5'={\bf a}+2{\bf b}, \,\,{\bf c}_5'={\bf c}. }]All these bases span ortho-hexagonal cells with twice the volume of the original hexagonal cell because for the matrices [\det({\bi P}_{i})=2] holds.

In the general position of [{\cal G}=P3_{1}12], No.151, one finds[\quad(1)\,\, x,y,z; \ (4)\,\, \overline{y},\overline{x}, \overline{z}+{\textstyle{{2}\over{3}}};\ (5) \,\,\overline{x}+y,y, \overline{z}+{\textstyle{{1}\over{3}}};\ (6) \,\,x,x-y,\overline{z}.]These entries represent the matrix–column pairs [({\bi W},\,{\bi w})]: [\quad\matrix{(1)\, \left(\matrix {1 &0 &0 \cr 0 &1 &0 \cr 0 &0 &1 } \right), \left(\matrix { 0 \cr 0 \cr 0 } \right)\semi \quad (4)\, \left(\matrix { 0&{\overline 1}&0 \cr {\overline 1}&0&0 \cr 0&0&\overline{1}} \right), \left(\matrix { 0 \cr 0 \cr {{2}\over{3}} } \right)\semi\cr\cr\cr (5)\, \left(\matrix {\overline{1} & 1 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & \overline{1}} \right), \left(\matrix { 0 \cr 0 \cr {{1}\over{3}} } \right) \semi \quad (6)\, \left(\matrix { 1&0&0 \cr 1&\overline{1}&0 \cr 0&0&\overline{1} } \right), \left(\matrix { 0 \cr 0 \cr 0 } \right).}]Application of equations (2.1.3.8)[link] on the matrices [{\bi W}_k] and (2.1.3.9)[link] on the columns [{\bi w}_k] of the matrix–column pairs results in [\quad{\bi W}'_{4} = {\bi W}'_{5} = {\bi W}'_{6} = \left(\matrix { \overline{1} & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & \overline{1} } \right)\semi \,\, {\bi w}'_{4} = {\bi w}'_{6} = {\bi o}\semi \, {\bi w}'_{5} = \left(\matrix { 0 \cr 0 \cr \overline{1} } \right).]All translation vectors of [{\cal G}] are retained in the subgroups but the volume of the cells is doubled. Therefore, there must be centring-translation vectors in the new cells. For example, the application of equation (2.1.3.9)[link] with [({\bi P}_{6},\, {\bi p}_{6})] to the translation of [{\cal G}] with the vector [-{\bf a}], i.e. [{\bi w} = -(1,0,0)], results in the column [{\bi w}' = ({{1}\over{2}}, {{1}\over{2}}, 0)], i.e. the centring translation [{{1}\over{2}}({\bf a}'+{\bf b}')] of the subgroup. Either by calculation or, more easily, from a small sketch one sees that the vectors [-{\bf b}] for [{\bi P}_4], [{\bf a}+{\bf b}] for [{\bi P}_5] (and [-{\bf a}] for [{\bi P}_6]) correspond to the cell-centring translation vectors of the subgroup cells.

Comments:

This example reveals that the conjugation of conjugate sub­groups does not necessarily imply the conjugation of the representatives of these subgroups in the general positions of IT A. The three monoclinic subgroups [C121] in this example are conjugate in the group [{\cal G}] by the [3_1] screw rotation. Conjugation of the representatives (4) and (6) by the [3_1] screw rotation of [{\cal G}] results in the column [{\bi w}_{5} = 0,0,\textstyle{{4}\over{3}}], which is standardized according to the rules of IT A to [{\bi w}_{5} = 0,0,\textstyle{{1}\over{3}}]. Thus, the conjugacy relation is disturbed by the standardization of the representative (5).








































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