International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 2.1, pp. 277-281
Section 2.1.3.1.1. Example
a
Institut für Physikalische Chemie, Universität Göttingen, Tammannstrasse 6, D-37077 Göttingen, Germany |
As an example, we consider a crystal of tetragonal symmetry, space group , with lattice parameters a and c. The primitive cell spanned by the three mutually orthogonal vectors a, b and c contains ten atoms at the positions listed in Table 2.1.3.1 and shown in Fig. 2.1.3.4. Consequently, the dynamical matrix has elements.
|
The space group contains eight symmetry operations, namely
Obviously, atoms No. 3 to 10 are chemically identical and have the same mass.
For the reduction of the dynamical matrix, we need the function , yielding the label of that atom into which κ is sent by the symmetry operation S. This function can be represented by the atom transformations shown in Table 2.1.3.2. This table displays the labels of atoms κ and K related by a particular symmetry operation and also the relative position of the primitive cells l and L where both atoms are located. This information is needed for the calculation of phase factors in the expression for the matrix operators T. Via the twofold axis, atom 6, for example, is transformed into atom 9 located within the cell which is shifted by the vector .
|
Let us first consider the case of phonons with infinite wavelengths and, hence, the symmetry reduction of the dynamical matrix at zero wavevector (the Γ point). Here, the point group of the wavevector is equivalent to the point group of the lattice. According to equation (2.1.3.19a), we can immediately write down the transformation matrix for any of these symmetry operations. Using the notationfor the three-dimensional vector representation of the symmetry elements, we obtain the T matrix operatorsandSince each of these matrices commutes with the dynamical matrix (, with ), the following relations are obtained for the submatrices: and so on for the other submatrices.
For nonzero wavevectors q along (), the point group contains the identity and the mirror plane only. The respective T matrix operators are the same as for the Γ point: There are, however, symmetry elements that invert the wavevector, namely and . Hence the enlarged group consists of the elements E, , and . Inspection of the atom transformation table yields the remaining matrix operators: andBeing anti-unitary, the corresponding inverse operators are7The invariance of the dynamical matrix with respect to the similarity transformation () using any of these operators leads to the following relations for wavevectors along :8andFor the submatrix (and similarly also for and ) we can combine the three relations and obtainHenceObviously, the symmetry considerations lead to a remarkable reduction of the independent elements of the dynamical matrix.