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

International Tables for Crystallography (2006). Vol. A, ch. 5.2, pp. 86-89
https://doi.org/10.1107/97809553602060000511

Chapter 5.2. Transformations of symmetry operations (motions)

H. Arnolda

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

In this chapter, matrix notation is used to describe transformations of symmetry operations. Some important invariants, especially the metric tensor, are discussed and, as an example, the structures of low and high cristobalite are compared.

5.2.1. Transformations

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Symmetry operations are transformations in which the coordinate system, i.e. the basis vectors a, b, c and the origin O, are considered to be at rest, whereas the object is mapped onto itself. This can be visualized as a motion' of an object in such a way that the object before and after the motion' cannot be distinguished.

A symmetry operation transforms every point X with the coordinates xyz to a symmetrically equivalent point with the coordinates . In matrix notation, this transformation is performed byThe matrix W is the rotation part and the column matrix w the translation part of the symmetry operation . The pair (W, w) characterizes the operation uniquely. Matrices W for point-group operations are given in Tables 11.2.2.1 and 11.2.2.2 .

Again, we can introduce the augmented matrix (cf. Chapter 8.1 ) The coordinates of the point , symmetrically equivalent to X with the coordinates xyz, are obtained by or, in short notation, A sequence of symmetry operations can be obtained as a product of matrices .

An affine transformation of the coordinate system transforms the coordinates of the starting point as well as the coordinates of a symmetrically equivalent pointThus, the affine transformation transforms also the symmetry-operation matrix and the new matrix is obtained by

Example

Space group (85) is listed in the space-group tables with two origins; origin choice 1 with , origin choice 2 with as point symmetry of the origin. How does the matrix of the symmetry operation 0, 0, z; 0, 0, 0 of origin choice 1 transform to the matrix of symmetry operation z; , 0 of origin choice 2?

In the space-group tables, origin choice 1, the transformed coordinates are listed. The translation part is zero, i.e. . In Table 11.2.2.1, the matrix W can be found. Thus, the matrix is obtained:

The transformation to origin choice 2 is accomplished by a shift vector p with components , 0. Since this is a pure shift, the matrices P and Q are the unit matrix I. Now the shift vector q is derived: . Thus, the matrices and are By matrix multiplication, the new matrix is obtained: If the matrix is applied to x′, y′, z′, the coordinates of the starting point in the new coordinate system, we obtain the transformed coordinates , By adding a lattice translation a, the transformed coordinates are obtained as listed in the space-group tables for origin choice 2.

5.2.2. Invariants

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A crystal structure and its physical properties are independent of the choice of the unit cell. This implies that invariants occur, i.e. quantities which have the same values before and after the transformation. Only some important invariants are considered in this section. Invariants of higher order (tensors) are treated by Altmann & Herzig (1994), second cumulant tensors, i.e. anisotropic temperature factors, are given in International Tables for Crystallography (2004), Vol. C.

The orthogonality of the basis vectors a, b, c of direct space and the basis vectors , , of reciprocal space, is invariant under a general (affine) transformation. Since both sets of basis vectors are transformed, is always perpendicular to the plane defined by b and c and perpendicular to b′ and cetc.

5.2.2.1. Position vector

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The position vector r in direct space, is invariant if the origin of the coordinate system is not changed in the transformation (see example in Section 5.1.3 ).

5.2.2.2. Modulus of position vector

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The modulus r of the position vector r gives the distance of the point xyz from the origin. Its square is obtained by the scalar product with the transposed representation of r; a, b, c the moduli of the basis vectors a, b, c (lattice parameters); G the metric matrix of direct space; and α, β, γ the angles of the unit cell.

The same considerations apply to the vector in reciprocal space and its modulus . Here, is applied. Note that and are independent of the choice of the origin in direct space.

5.2.2.3. Metric matrix

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The metric matrix G of the unit cell in the direct lattice changes under a linear transformation, but G is invariant under a symmetry operation of the lattice. The volume of the unit cell V is obtained by The same considerations apply to the metric matrix of the unit cell in the reciprocal lattice and the volume of the reciprocal-lattice unit cell. Thus, there are two invariants under an affine transformation, the product and the product

5.2.2.4. Scalar product

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The scalar product of the vector in reciprocal space with the vector r in direct space is invariant under a linear transformation but not under a shift of origin in direct space.

A vector r in direct space can also be represented as a product of augmented matrices: As stated above, the basis vectors are transformed only by the linear part, even in the case of a general affine transformation. Thus, the transformed position vector is obtained by The shift p is set to zero. The shift of origin is contained in the matrix only.

Similarly, a vector in reciprocal space can be represented by The coordinates h, k, l in reciprocal space transform also only linearly. Thus, The reader can see immediately that the scalar product transforms correctly.

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

Altmann, S. L. & Herzig, P. (1994). Point-Group Theory Tables. Oxford Science Publications.
International Tables for Crystallography (2004). Vol. C, edited by E. Prince, Table 8.3.1.1. Dordrecht: Kluwer Academic Publishers.
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.