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

International Tables for Crystallography (2015). Vol. A, ch. 1.5, p. 90

Section Transformation between the two origin-choice settings of I41/amd

G. Chapuis,d H. Wondratscheka and M. I. Aroyob Transformation between the two origin-choice settings of I41/amd

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The zircon example of Section[link] illustrates how the atomic coordinates change under an origin-choice transformation. Here, the case of the two origin-choice descriptions of the same space group I41/amd (141) will be used to demonstrate how the rest of the crystallographic quantities are affected by an origin shift.

The two descriptions of I41/amd in the space-group tables of this volume are distinguished by the origin choices of the reference coordinate systems: the origin statement of the origin choice 1 setting indicates that its origin [{O}_1] is taken at a point of [\bar{4}m2] symmetry, which is located at [0, {\textstyle{1 \over 4}},-{\textstyle{1 \over 8}}] with respect to the origin [{O}_2] of origin choice 2, taken at a centre (2/m). Conversely, the origin [{O}_2] is taken at a centre (2/m) at [0, -{\textstyle{1 \over 4}},{\textstyle{1 \over 8}}] from the origin [{O}_1]. These origin descriptions in fact specify explicitly the origin-shift vector p necessary for the transformation between the two settings. For example, the shift vector listed for origin choice 2 expresses the origin [{O}_2] with respect to [{O}_1], i.e. the corresponding transformation matrix[({\bi P},{\bi p})=({\bi I},{\bi p})=(\pmatrix{1& 0 & 0\cr 0 & 1 & 0 \cr 0 & 0 & 1 }, \pmatrix{ \hfill 0 \cr \hfill -{\textstyle{1 \over 4}}\cr \hfill {\textstyle{1 \over 8}} })]transforms the crystallographic data from the origin choice 1 setting to the origin choice 2 setting.

  • (i) Transformation of point coordinates. In accordance with the discussion of Section[link] [cf. equation ([link]], the transformation of point coordinates [{\bi x}_1=\pmatrix{ x_1 \cr y_1\cr z_1 }] of the origin choice 1 setting of [I4_1/amd] to [{\bi x}_2=\pmatrix{ x_2 \cr y_2\cr z_2 }] of the origin choice 2 setting is given by[{\bi x}_2=\pmatrix{ x_2 \cr y_2\cr z_2 }=({\bi P}, {\bi p})^{-1}{\bi x}_1=({\bi I}, -{\bi p}){\bi x}_1=\pmatrix{ x_1 \cr y_1+{\textstyle{1 \over 4}}\cr z_1-{\textstyle{1 \over 8}} }. \eqno(]

  • (ii) Metric tensors and the data for the reflection conditions. The metric tensors and the data for the reflection conditions are not affected by an origin shift as [{\bi P}={\bi I}], cf. equations ([link] and ([link].

  • (iii) Transformation of the matrix–column pairs [({\bi W}, {\bi w})] of the symmetry operations. The origin-shift transformation [({\bi I},{\bi p})] relates the matrix–column pairs [({\bi W}_1, {\bi w}_1)] of the symmetry operations of the origin choice 1 setting of [I4_1/amd] to [({\bi W}_2, {\bi w}_2)] of the origin choice 2 setting [cf. equation ([link]]:[({\bi W}_2, {\bi w}_2)=({\bi I},-{\bi p})({\bi W}_1, {\bi w}_1)({\bi I},{\bi p})=({\bi W}_1,{\bi w}_1 +[{\bi W}_1-{\bi I}]\,{\bi p}).\eqno(]The rotation part of the symmetry operation is not affected by the origin shift, but the translation part is affected, i.e. [{\bi W}_2={\bi W}_1] and [{\bi w}_2={\bi w}_1+[{\bi W}_1-{\bi I}]\,{\bi p}]. For example, the translation and unit element generators of [I4_1/amd] are not changed under the origin-shift transformation, as [{\bi W}_1={\bi I}]. The first non-translation generator given by the coordinate triplet [\bar{y},x+ {\textstyle{1 \over 2}},z+{\textstyle{1 \over 4}}] and represented by the matrix[(\openup2pt\pmatrix{0& \bar{1} & 0\cr 1 & 0 & 0 \cr 0 & 0 & 1 },\ \pmatrix{0 \cr {\textstyle{1 \over 2}}\cr {\textstyle{1 \over 4}} }) \ {\rm transforms\ to\ } (\pmatrix{0& \bar{1} & 0\cr 1 & 0 & 0 \cr 0 & 0 & 1 }, \pmatrix{{\textstyle{1 \over 4}} \cr {\textstyle{3 \over 4}}\cr {\textstyle{1 \over 4}} }),]which corresponds to the coordinate triplet [\bar{y}+{\textstyle{1 \over 4}},x+ {\textstyle{3 \over 4}},] [z+{\textstyle{1 \over 4}}].

    The second non-translation generator [\bar{x},\bar{y}+{\textstyle{1 \over 2}},\bar{z}+{\textstyle{1 \over 4}}], represented by the matrix[\quad\quad(\openup2pt\pmatrix{\bar{1}& 0 & 0\cr 0 & \bar{1} & 0 \cr 0 & 0 & \bar{1} },\pmatrix{0 \cr {\textstyle{1 \over 2}}\cr {\textstyle{1 \over 4}} }) \ {\rm transforms\ to\ } (\pmatrix{\bar{1}& 0 & 0\cr 0 & \bar{1} & 0 \cr 0 & 0 & \bar{1}}, \pmatrix{0 \cr 1\cr 0 }),]which under the normalization [0\leq w_i\,\lt\, 1] is written as the coordinate triplet [\bar{x},\bar{y},\bar{z}]. The coordinate triplets of the transformed symmetry operations are the entries of the corresponding generators of the origin choice 2 setting of [I4_1/amd] (cf. the space-group tables of [I4_1/amd] in Chapter 2.3[link] ).

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