International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 5.2, pp. 8789

The positions of the silicon atoms in the lowcristobalite structure (Nieuwenkamp, 1935) are compared with those of the highcristobalite 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 spacegroup 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 facecentring 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.

Positions of silicon atoms in the lowcristobalite structure, projected along . Primitive tetragonal cell a, b, c; Ccentred 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 spacegroup 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 Ccentring 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 x, x, 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 TiefCristobalits SiO_{2}. Z. Kristallogr. 92, 82–88.
Wyckoff, R. W. G. (1925). Die Kristallstruktur von βCristobalit SiO_{2} (bei hohen Temperaturen stabile Form). Z. Kristallogr. 62, 189–200.