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. 88-90

Section 1.5.3.2.1. Transformations between different settings of P21/c

G. Chapuis,d H. Wondratscheka and M. I. Aroyob

1.5.3.2.1. Transformations between different settings of P21/c

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In the space-group tables of this volume, the monoclinic space group [P2_1/c] (14) is described in six different settings: for each of the `unique axis b' and `unique axis c' settings there are three descriptions specified by different cell choices (cf. Section 2.1.3.15[link] ). The different settings are identified by the appropriate full Hermann–Mauguin symbols. The basis transformations [({\bi P},{\bi p})] between the different settings are completely specified by the linear part of the transformation, the 3 × 3 matrix P [cf. equation (1.5.1.4)[link]], as all settings of [P2_1/c] refer to the same origin, i.e. [{\bi p}={\bi o}]. The transformation matrices P necessary for switching between the different descriptions of [P2_1/c] can either be read off directly or constructed from the transformation-matrix data listed in Table 1.5.1.1[link].

(A) Transformation from [P12_1/c1] (unique axis b, cell choice 1) to [P112_1/a] (unique axis c, cell choice 1). The change of the direction of the screw axis [2_1] indicates that the unique direction b transforms to the unique direction c, while the glide vector along c transforms to a glide vector along a. These changes are reflected in the transformation matrix P between the basis [{\bf a}_b,{\bf b}_b,{\bf c}_b] of [P12_1/c1] and [{\bf a}_c,{\bf b}_c,{\bf c}_c] of [P112_1/a], which can be read directly from Table 1.5.1.1[link]:[({\bf a}_c,{\bf b}_c,{\bf c}_c)=({\bf a}_b,{\bf b}_b,{\bf c}_b){\bi P}=({\bf a}_b,{\bf b}_b,{\bf c}_b)\pmatrix{0& 1 &0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 }.]

  • (i) Transformation of point coordinates. From [{\bi x'}={\bi P}^{-1}{\bi x}], cf. equation (1.5.1.5)[link], it follows that[\pmatrix{ x_c \cr y_c \cr z_c }=\pmatrix{0& 0 & 1\cr 1 & 0 & 0 \cr 0 & 1 & 0 }\pmatrix{ x_b \cr y_b \cr z_b }=\pmatrix{ z_b \cr x_b \cr y_b }.]For example, the representative coordinate triplets of the special Wyckoff position 2d [\bar{1}] of [P12_1/c1] transform exactly to the representative coordinate triplets of the special Wyckoff position 2d [\bar{1}] of [P112_1/a]: [\pmatrix{ {\textstyle{1 \over 2}} \cr 0 \cr {\textstyle{1 \over 2}} }] and [\pmatrix{ {\textstyle{1 \over 2}} \cr {\textstyle{1 \over 2}} \cr 0 }] transform to [\pmatrix{ {\textstyle{1 \over 2}} \cr {\textstyle{1 \over 2}} \cr 0 }] and [\pmatrix{ 0 \cr {\textstyle{1 \over 2}} \cr {\textstyle{1 \over 2}} }].

  • (ii) Transformation of the indices in the `Reflection conditions' block. Under a coordinate transformation specified by a matrix P, the indices of the reflection conditions (Miller indices) transform according to [(h'k'l')=(hkl){\bi P}], cf. equation (1.5.2.2)[link]. The transformation under[{\bi P}=\pmatrix{0& 1 &0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 }]of the set of general or special reflection conditions [h_bk_bl_b] for [P12_1/c1] should result in the set of general or special reflection conditions [h_ck_cl_c] of [P112_1/a]:[(h_ck_cl_c)=(h_bk_bl_b)\pmatrix{0& 1 &0 \cr 0 & 0 & 1 \cr 1 & 0 & 0}=(l_bh_bk_b),]i.e. [h_c=l_b,k_c=h_b,l_c=k_b] (see Table 1.5.3.1[link]).

    Table 1.5.3.1| top | pdf |
    Transformation of reflection-condition data for P121/c1 to P1121/a

     P121/c1P1121/a
     [h_bk_bl_b][h_ck_cl_c]
    General conditions h0l: l = 2n hk0: h = 2n
      0k0: k = 2n 00l: l = 2n
      00l: l = 2n h00: h = 2n
         
    Special conditions for the inversion centres hkl: k + l = 2n hkl: h + l = 2n
  • (iii) Transformation of the matrix–column pairs [({\bi W}, {\bi w})] of the symmetry operations. The matrices of the representatives of the symmetry operations of [P12_1/c1] can be constructed from the coordinate triplets listed in the general-position block of the group:[(1)\ x,y,z\quad (2)\ \bar{x},y+{\textstyle{1 \over 2}},\bar{z}+{\textstyle{1 \over 2}} \quad (3)\ \bar{x},\bar{y},\bar{z}\quad (4)\ x,\bar{y}+{\textstyle{1 \over 2}},z+{\textstyle{1 \over 2}}]Their transformation is more conveniently performed using the augmented-matrix formalism. According to equation (1.5.2.17)[link], the matrices [\specialfonts{\bbsf W}_c] of the symmetry operations of [P112_1/a] are related to the matrices [\specialfonts{\bbsf W}_b] of [P12_1/c1] by the equation [\specialfonts{\bbsf W}_c={\bbsf Q}{\bbsf W}_b{\bbsf P}], where[\specialfonts{\bbsf P}=\openup-2pt\pmatrix{0 & 1 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 0 & 0 & 1 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 1 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr\noalign{\hrule}\cr 0 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 1}\quad {\rm and}\quad{\bbsf Q}=\pmatrix{0 & 0 & 1 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 1 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 0& 1 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr\noalign{\hrule}\cr 0 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 1}.]

    The unit matrix representing the identity operation (1) is invariant under any basis transformation, i.e. [x,y,z] transforms to [x,y,z]. Similarly, the matrix of inversion [\bar{1}] (3) (the linear part of which is a multiple of the unit matrix) is also invariant under any basis transformation, i.e. [\bar{x}, \bar{y}, \bar{z}] transforms to [\bar{x}, \bar{y}, \bar{z}]. The symmetry operation (2) [\bar{x},\,y+ \textstyle{{1}\over{2}},\,\bar{z}+ \textstyle{{1}\over{2}}], represented by the matrix[\openup-4pt\pmatrix{\bar{1} & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 0 & 1 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} {\textstyle{1 \over 2}} \cr 0 & 0 & \bar{1} {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} {\textstyle{1 \over 2}} \cr\noalign{\hrule}\cr 0 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 1}]transforms to[\eqalign{& \openup-4pt\pmatrix{0 & 0 & 1 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 1 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 0& 1 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr\noalign{\hrule}\cr 0 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 1}\pmatrix{\bar{1} & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 0 & 1 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} {\textstyle{1 \over 2}} \cr 0 & 0 & \bar{1} {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} {\textstyle{1 \over 2}} \cr\noalign{\hrule}\cr 0 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 1}\pmatrix{0 & 1 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 0 & 0 & 1 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 1 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr\noalign{\hrule}\cr 0 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 1}\cr &=\openup-4pt\pmatrix{\bar{1} & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} {\textstyle{1 \over 2}} \cr 0 & \bar{1} & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 0 & 0 & 1 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} {\textstyle{1 \over 2}} \cr\noalign{\hrule}\cr 0 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 1},}]which corresponds to [\bar{x}+{\textstyle{1 \over 2}},\bar{y},z+{\textstyle{1 \over 2}}].

    Finally, the symmetry operation [(4)\ x,\bar{y}+ {\textstyle{1 \over 2}},z+ {\textstyle{1 \over 2}}] represented by the matrix[\openup-4pt\pmatrix{1 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 0 & \bar{1} & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} {\textstyle{1 \over 2}} \cr 0 & 0 & 1 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} {\textstyle{1 \over 2}}\cr\noalign{\hrule}\cr 0 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 1}\ {\rm transforms\ to}\ \pmatrix{1 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} {\textstyle{1 \over 2}} \cr 0 & 1 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 0 \cr 0 & 0 & \bar{1} {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} {\textstyle{1 \over 2}} \cr\noalign{\hrule}\cr 0 & 0 & 0 {\hskip -4pt}&{\vrule height 10pt depth6pt}\hfill&{\hskip -4pt} 1},]corresponding to the coordinate triplet [x+{\textstyle{1 \over 2}},y,\bar{z}+{\textstyle{1 \over 2}}] [the matrices of (4) and its transformed are those of (2) and its transformed, multiplied by [\bar{1}]].

    The coordinate triplets of the transformed symmetry operations correspond to the entries of the general-position block of [P112_1/a] (cf. the space-group tables of [P2_1/c] in Chapter 2.3[link] ).

(B) Transformation from [P112_1/b] (unique axis c, cell choice 3) to [P12_1/c1] (unique axis b, cell choice 1): [({\bf a}_{b,1},{\bf b}_{b,1},{\bf c}_{b,1})] = [({\bf a}_{c,3},{\bf b}_{c,3},{\bf c}_{c,3}){\bi P}]. A transformation matrix from [P112_1/b] directly to [P12_1/c1] is not found in Table 1.5.1.1[link], but it can be constructed in two steps from transformation matrices that are listed there. For example:

Step 1. Unique axis c fixed: transformation from `cell choice 3' to `cell choice 1':[({\bf a}_{c,1},{\bf b}_{c,1},{\bf c}_{c,1})=({\bf a}_{c,3},{\bf b}_{c,3},{\bf c}_{c,3}){\bi P}_1=({\bf a}_{c,3},{\bf b}_{c,3},{\bf c}_{c,3})\pmatrix{0& \bar{1} & 0\cr 1 & \bar{1} & 0 \cr 0 & 0 & 1}.\eqno(1.5.3.1)]

  • Step 2. Cell choice 1 invariant: transformation from unique axis c to unique axis b:[({\bf a}_{b,1},{\bf b}_{b,1},{\bf c}_{b,1})=({\bf a}_{c,1},{\bf b}_{c,1},{\bf c}_{c,1}){\bi P}_2=({\bf a}_{c,1},{\bf b}_{c,1},{\bf c}_{c,1})\pmatrix{0& 0 & 1\cr 1 & 0 & 0 \cr 0 & 1 & 0 }.\eqno(1.5.3.2)]The transformation matrix P for the change from [P112_1/b] to [P12_1/c1] is obtained by starting from equation (1.5.3.2)[link] and replacing the expression for [{\bf a}_{c,1},{\bf b}_{c,1},{\bf c}_{c,1}] with that from equation (1.5.3.1)[link]:[\eqalign{({\bf a}_{b,1},{\bf b}_{b,1},{\bf c}_{b,1})&=({\bf a}_{c,1},{\bf b}_{c,1},{\bf c}_{c,1}){\bi P}_2=({\bf a}_{c,3},{\bf b}_{c,3},{\bf c}_{c,3}){\bi P}_1{\bi P}_2\cr&=({\bf a}_{c,3},{\bf b}_{c,3},{\bf c}_{c,3})\pmatrix{0& \bar{1} & 0\cr 1 & \bar{1} & 0 \cr 0 & 0 & 1 }\pmatrix{0& 0 & 1\cr 1 & 0 & 0 \cr 0 & 1 & 0}\cr&=({\bf a}_{c,3},{\bf b}_{c,3},{\bf c}_{c,3})\pmatrix{\bar{1}& 0& 0\cr \bar{1} & 0 & 1 \cr 0 & 1 & 0}.}]The inverse matrix [{\bi Q}={\bi P}^{-1}] can be obtained either by inversion or by the product of the factors [{\bi Q}_1={\bi P}_1^{-1}] and [{\bi Q}_2={\bi P}_2^{-1}] but in reverse order:[\eqalign{{\bi Q}&=({\bi P}_1{\bi P}_2)^{-1}={\bi P}_2^{-1}{\bi P}_1^{-1}={\bi Q}_2{\bi Q}_1=\pmatrix{0& 1 & 0\cr 0 & 0 & 1 \cr 1 & 0 & 0}\pmatrix{\bar{1}& 1 & 0\cr \bar{1} & 0 & 0 \cr 0 & 0 & 1}\cr&=\pmatrix{\bar{1}& 0 & 0\cr 0 & 0 & 1 \cr \bar{1} & 1 & 0 }.}]The transformation matrix P determined above and its inverse Q permit the transformation of crystallographic data for the change from [P112_1/b] to [P12_1/c1].








































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