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) are compared with those of the high-cristobalite structure (Wyckoff, 1925; cf. Megaw, 1973). At low temperatures, the space group is (92). The four silicon atoms are located in Wyckoff position 4(a) ..2 with the coordinates x, x, 0; , , ; ; ; . During the phase transition, the tetragonal structure is transformed into a cubic one with space group (227). It is listed in the space-group tables with two different origins. We use `Origin choice 1' with point symmetry at the origin. The silicon atoms occupy the position 8(a) with the coordinates 0, 0, 0; 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 (92) is transformed to an unconventional C cell (cf. Section 4.3.4 ), which corresponds to the F cell of (227). The P and the C cells are shown in Fig. 5.2.3.1. The coordinate system with origin of the C cell is obtained from that of the P cell, origin O, by the linear transformation and the shift The matrices P, p and are thus given by From Fig. 5.2.3.1, we derive also the inverse transformation Thus, the matrices Q, q and are The coordinates x, y, z of points in the P cell are transformed by : The coordinate triplets of the four silicon positions in the P cell are 0.300, 0.300, 0; 0.700, 0.700, ; 0.200, 0.800, ; 0.800, 0.200, . 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, ; 0.250, 0.300, ; 0.250, −0.300, . A set of four further points is obtained by adding the centring translation , 0 to these coordinates.

 Figure 5.2.3.1 | top | pdf |Positions of silicon atoms in the low-cristobalite structure, projected along . Primitive tetragonal cell a, b, c; C-centred tetragonal cell . Shift of origin from O to by the vector .

The indices h, k, l are transformed by the matrix P: i.e. the reflections with the indices h, k, l of the P cell become reflections of the C cell.

The symmetry operations of space group are listed in the space-group tables for the P cell as follows: The corresponding matrices are These matrices of the P cell are transformed to the matrices of the C cell by For matrix (2), for example, this results in The eight transformed matrices , derived in this way, are Another set of eight matrices is obtained by adding the C-centring translation to the w's.

From these matrices, one obtains the coordinates of the general position in the C cell, for instance from matrix (2) The eight points obtained by the eight matrices are The other set of eight points is obtained by adding , 0.

In space group , the silicon atoms are in special position 4(a) ..2 with the coordinates xx, 0. Transformed into the C cell, the position becomes The parameter of the P cell has changed to in the C cell. For , the special position of the C cell assumes the same coordinate triplets as Wyckoff position 8(a) in space group (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.