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

International Tables for Crystallography (2006). Vol. A, ch. 5.2, pp. 87-89

Section 5.2.3. Example: low cristobalite and high cristobalite

H. Arnolda

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany

5.2.3. Example: low cristobalite and high cristobalite

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The positions of the silicon atoms in the low-cristobalite structure (Nieuwenkamp, 1935[link]) are compared with those of the high-cristobalite structure (Wyckoff, 1925[link]; cf. Megaw, 1973[link]). At low temperatures, the space group is [P4_{1}2_{1}2] (92). The four silicon atoms are located in Wyckoff position 4(a) ..2 with the coordinates x, x, 0; [\bar{x}], [\bar{x}], [{1 \over 2}]; [{1 \over 2} - x,{1 \over 2} + x,{1 \over 4}]; [{1 \over 2} + x,{1 \over 2} - x,{3 \over 4}]; [x = 0.300]. During the phase transition, the tetragonal structure is transformed into a cubic one with space group [Fd\bar{3}m] (227). It is listed in the space-group tables with two different origins. We use `Origin choice 1' with point symmetry [\bar{4}3m] at the origin. The silicon atoms occupy the position 8(a) [\bar{4}3m] with the coordinates 0, 0, 0; [{1 \over 4},{1 \over 4},{1 \over 4}] and those related by the face-centring translations. In the diamond structure, the carbon atoms occupy the same position.

In order to compare the two structures, the conventional P cell of space group [P4_{1}2_{1}2] (92) is transformed to an unconventional C cell (cf. Section 4.3.4[link] ), which corresponds to the F cell of [Fd\bar{3}m] (227). The P and the C cells are shown in Fig. 5.2.3.1[link]. The coordinate system [{\bf a}',{\bf b}',{\bf c}'] with origin [O'] of the C cell is obtained from that of the P cell, origin O, by the linear transformation[{\bf a}' = {\bf a} + {\bf b},\qquad {\bf b}' = - {\bf a} + {\bf b},\qquad {\bf c}' = {\bf c}] and the shift[{\bi p} = {\textstyle{1 \over 4}}{\bf a} + {\textstyle{1 \over 4}}{\bf b}.] The matrices P, p and [\specialfonts{\bbsf P}] are thus given by[\specialfonts{\bi P} = \pmatrix{1 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &1\cr},\quad {\bi p} = \pmatrix{{1 \over 4}\cr {1 \over 4}\cr 0\cr},\quad {\bbsf P} = \pmatrix{1 &\bar{1} &0 &{1 \over 4}\cr 1 &1 &0 &{1 \over 4}\cr 0 &0 &1 &0\cr 0 &0 &0 &1\cr}.] From Fig. 5.2.3.1[link], we derive also the inverse transformation[{\bf a} = {\textstyle{1 \over 2}}{\bf a}' - {\textstyle{1 \over 2}}{\bf b}',\quad {\bf b} = {\textstyle{1 \over 2}}{\bf a}' + {\textstyle{1 \over 2}}{\bf b}',\quad {\bf c} = {\bf c}',\quad {\bf q} = - {\textstyle{1 \over 4}}{\bf a}'.] Thus, the matrices Q, q and [\specialfonts{\bbsf Q}] are[\specialfonts\eqalignno{{\bi Q} = {\bi P}^{-1} &= \pmatrix{{1 \over 2} &{1 \over 2} &0\cr {\bar{1} \over 2} &{1 \over 2} &0\cr 0 &0 &1\cr},\quad {\bi q} = - {\bi P}^{-1} {\bi p} = \pmatrix{{\bar{1} \over 4}\cr 0\cr 0\cr}, &\cr {\bbsf Q} = {\bbsf P}^{-1} &= \pmatrix{{1 \over 2} &{1 \over 2} &0 &{\bar{1} \over 4}\cr {\bar{1} \over 2} &{1 \over 2} &0 &0\cr 0 &0 &1 &0\cr 0 &0 &0 &1\cr}.}] The coordinates x, y, z of points in the P cell are transformed by [\specialfonts{\bbsf Q}]:[{\pmatrix{\noalign{\vskip3pt}x'\cr\noalign{\vskip5pt} y'\cr\noalign{\vskip3pt} z'\cr\noalign{\vskip4pt} 1\cr}} {\raise5pt\hbox{=}} {\openup3pt\pmatrix{{1 \over 2} &{1 \over 2} &0 &{\bar{1} \over 4}\cr {\bar{1 \over 2}} &{1 \over 2} &0 &0\cr 0 &0 &1 &0\cr 0 &0 &0 &1\cr}} {\pmatrix{\noalign{\vskip5pt}x\cr\noalign{\vskip4pt} y\cr\noalign{\vskip5pt} z\cr\noalign{\vskip4pt} 1\cr}} {\raise5pt\hbox{=}} \openup4pt{\pmatrix{\noalign{\vskip2pt}{1 \over 2}(x + y) - {1 \over 4}\cr\noalign{\vskip2pt} {1 \over 2}(- x + y)\cr\noalign{\vskip-2pt} z\cr 1\cr}}.] The coordinate triplets of the four silicon positions in the P cell are 0.300, 0.300, 0; 0.700, 0.700, [{1 \over 2}]; 0.200, 0.800, [{1 \over 4}]; 0.800, 0.200, [{3 \over 4}]. Four triplets in the C cell are obtained by inserting these values into the equation just derived. The new coordinates are 0.050, 0, 0; 0.450, 0, [{1 \over 2}]; 0.250, 0.300, [{1 \over 4}]; 0.250, −0.300, [{3 \over 4}]. A set of four further points is obtained by adding the centring translation [{1 \over 2}][{1 \over 2}], 0 to these coordinates.

[Figure 5.2.3.1]

Figure 5.2.3.1 | top | pdf |

Positions of silicon atoms in the low-cristobalite structure, projected along [[00\bar{1}]]. Primitive tetragonal cell abc; C-centred tetragonal cell [a',b',c']. Shift of origin from O to [O'] by the vector [{\bf p} = {1 \over 4}{\bf a} + {1 \over 4}{\bf b}].

The indices h, k, l are transformed by the matrix P:[(h',k',l') = (h,k,l) \pmatrix{1 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &1\cr} = (h + k, - h + k,l),] i.e. the reflections with the indices h, k, l of the P cell become reflections [h + k, - h + k, l] of the C cell.

The symmetry operations of space group [P4_{1}2_{1}2] are listed in the space-group tables for the P cell as follows:[\openup3pt\matrix{(1) &x,y,z;\quad\hfill &(2) &\bar{x},\bar{y},{1 \over 2} + z;\hfill\cr (3) &{1 \over 2} - y,{1 \over 2} + x,{1 \over 4} + z;\quad &(4) &{1 \over 2} + y,{1 \over 2} - x,{3 \over 4} + z;\cr (5) &{1 \over 2} - x,{1 \over 2} + y,{1 \over 4} - z;\quad &(6) &{1 \over 2} + x,{1 \over 2} - y,{3 \over 4} - z;\cr (7) &y,x,\bar{z};\quad\hfill &(8) &\bar{y},\bar{x},{1 \over 2} - z.\hfill\cr}] The corresponding matrices [\specialfonts{\bbsf W}] are[\openup3pt\matrix{(1) \pmatrix{1 &0 &0 &0\cr 0 &1 &0 &0\cr 0 &0 &1 &0\cr 0 &0 &0 &1\cr}; &(2) \pmatrix{\bar{1} &0 &0 &0\cr 0 &\bar{1} &0 &0\cr 0 &0 &1 &{1 \over 2}\cr 0 &0 &0 &1\cr}; &(3) \pmatrix{0 &\bar{1} &0 &{1 \over 2}\cr 1 &0 &0 &{1 \over 2}\cr 0 &0 &1 &{1 \over 4}\cr 0 &0 &0 &1\cr};\cr (4) \pmatrix{0 &1 &0 &{1 \over 2}\cr \bar{1} &0 &0 &{1 \over 2}\cr 0 &0 &1 &{3 \over 4}\cr 0 &0 &0 &1\cr}; &(5) \pmatrix{\bar{1} &0 &0 &{1 \over 2}\cr 0 &1 &0 &{1 \over 2}\cr 0 &0 &\bar{1} &{1 \over 4}\cr 0 &0 &0 &1\cr}; &(6) \pmatrix{1 &0 &0 &{1 \over 2}\cr 0 &\bar{1} &0 &{1 \over 2}\cr 0 &0 &\bar{1} &{3 \over 4}\cr 0 &0 &0 &1\cr};\cr (7) \pmatrix{0 &1 &0 &0\cr 1 &0 &0 &0\cr 0 &0 &\bar{1} &0\cr 0 &0 &0 &1\cr}; &(8) \pmatrix{0 &\bar{1} &0 &0\cr \bar{1} &0 &0 &0\cr 0 &0 &\bar{1} &{1 \over 2}\cr 0 &0 &0 &1\cr}.}] These matrices of the P cell are transformed to the matrices [\specialfonts{\bbsf W}'] of the C cell by[\specialfonts{\bbsf W}' = {\bbsf Q} {\bbsf W}{\bbsf P}.] For matrix (2), for example, this results in[\specialfonts\eqalignno{{\bbsf W}' &= \pmatrix{{1 \over 2} &{1 \over 2} &0 &{\bar{1} \over 4}\cr {\bar{1} \over 2} &{1 \over 2} &0 &0\cr 0 &0 &1 &0\cr 0 &0 &0 &1\cr} \pmatrix{\bar{1} &0 &0 &0\cr 0 &\bar{1} &0 &0\cr 0 &0 &1 &{1 \over 2}\cr 0 &0 &0 &1\cr} \pmatrix{1 &\bar{1} &0 &{1 \over 4}\cr 1 &1 &0 &{1 \over 4}\cr 0 &0 &1 &0\cr 0 &0 &0 &1\cr}\cr &= \pmatrix{\bar{1} &0 &0 &{\bar{1} \over 2}\cr 0 &\bar{1} &0 &0\cr 0 &0 &1 &{1 \over 2}\cr 0 &0 &0 &1\cr}.}] The eight transformed matrices [\specialfonts{\bbsf W}'], derived in this way, are[\openup3pt\matrix{(1) \pmatrix{1 &0 &0 &0\cr 0 &1 &0 &0\cr0 &0 &1 &0\cr 0 &0 &0 &1\cr}; &(2) \pmatrix{\bar{1} &0 &0 &{\bar{1} \over 2}\cr 0 &\bar{1} &0 &0\cr 0 &0 &1 &{1 \over 2}\cr 0 &0 &0 &1\cr}; &(3) \pmatrix{0 &\bar{1} &0 &{1 \over 4}\cr 1 &0 &0 &{1 \over 4}\cr 0 &0 &1 &{1 \over 4}\cr 0 &0 &0 &1\cr};\cr (4) \pmatrix{0 &1 &0 &{1 \over 4}\cr \bar{1} &0 &0 &{\bar{1} \over 4}\cr 0 &0 &1 &{3 \over 4}\cr 0 &0 &0 &1\cr}; &(5) \pmatrix{0 &1 &0 &{1 \over 4}\cr 1 &0 &0 &{1 \over 4}\cr 0 &0 &\bar{1} &{1 \over 4}\cr 0 &0 &0 &1\cr}; &(6) \pmatrix{0 &\bar{1} &0 &{1 \over 4}\cr \bar{1} &0 &0 &{\bar{1} \over 4}\cr 0 &0 &\bar{1} &{3 \over 4}\cr 0 &0 &0 &1\cr};\cr (7) \pmatrix{1 &0 &0 &0\cr 0 &\bar{1} &0 &0\cr 0 &0 &\bar{1} &0\cr 0 &0 &0 &1\cr}; &(8) \pmatrix{\bar{1} &0 &0 &{\bar{1} \over 2}\cr 0 &1 &0 &0\cr 0 &0 &\bar{1} &{1 \over 2}\cr 0 &0 &0 &1\cr}. &\cr}] Another set of eight matrices is obtained by adding the C-centring translation [{1 \over 2},{1 \over 2},0] to the w's.

From these matrices, one obtains the coordinates of the general position in the C cell, for instance from matrix (2)[\pmatrix{\tilde{x}\cr \tilde{y}\cr \tilde{z}\cr 1\cr} = \pmatrix{\bar{1} &0 &0 &{\bar{1} \over 2}\cr 0 &\bar{1} &0 &0\cr 0 &0 &1 &{1 \over 2}\cr 0 &0 &0 &1\cr} \pmatrix{x\cr y\cr z\cr 1\cr} = \pmatrix{-x - {1 \over 2}\cr -y\cr z + {1 \over 2}\cr 1\cr}.] The eight points obtained by the eight matrices [\specialfonts{\bbsf W}'] are[\openup3pt\matrix{(1) &x,y,z;\quad\hfill &(2) &-{1 \over 2} - x,\bar{y},{1 \over 2} + z;\hfill\cr (3) &{1 \over 4} - y,{1 \over 4} + x,{1 \over 4} + z;\quad &(4) &{1 \over 4} + y, - {1 \over 4} - x,{3 \over 4} + z;\cr (5) &{1 \over 4} + y,{1 \over 4} + x,{1 \over 4} - z;\quad &(6) &{1 \over 4} - y, - {1 \over 4} - x,{3 \over 4} - z;\cr (7) &x,\bar{y},\bar{z};\quad\hfill &(8) &- {1 \over 2} - x,y,{1 \over 2} - z.\hfill \cr}] The other set of eight points is obtained by adding [{1 \over 2}][{1 \over 2}], 0.

In space group [P4_{1}2_{1}2], the silicon atoms are in special position 4(a) ..2 with the coordinates xx, 0. Transformed into the C cell, the position becomes[\eqalignno{&(0,0,0) +\quad ({\textstyle{1 \over 2}},{\textstyle{1 \over 2}},0) +\cr &x,0,0;\qquad {\textstyle{1 \over 2}} - x,0,{\textstyle{1 \over 2}};\quad {\textstyle{1 \over 4},{1 \over 4}} + x,{\textstyle{1 \over 4}};\quad {\textstyle{1 \over 4}},\;{\textstyle{3 \over 4}} - x,{\textstyle{3 \over 4}}.}] The parameter [x = 0.300] of the P cell has changed to [x = 0.050] in the C cell. For [x = 0], the special position of the C cell assumes the same coordinate triplets as Wyckoff position 8(a) [\bar{4}3m] in space group [Fd\bar{3}m] (227), i.e. this change of the x parameter reflects the displacement of the silicon atoms in the cubic to tetragonal phase transition.

References

Megaw, H. D. (1973). Crystal structures: a working approach, pp. 259–262. Philadelphia: Saunders.
Nieuwenkamp, W. (1935). Die Kristallstruktur des Tief-Cristobalits SiO2. Z. Kristallogr. 92, 82–88.
Wyckoff, R. W. G. (1925). Die Kristallstruktur von β-Cristobalit SiO2 (bei hohen Temperaturen stabile Form). Z. Kristallogr. 62, 189–200.








































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