Matrix crystal symmetry operator representation

Because of the restrictions imposed on the values of the rotation angles (see Table 1.4), sincp and cos(p in Cartesian basis are 0, 1 or -1 for one, two and four-fold rotations, and they are 1/2 or Vs/2 for three and six-fold rotations. However, when the same rotational transformations are considered in the appropriate crystallographic coordinate system, all matrix elements become equal to 0, -1 or 1. This simplicity (and undeniably, beauty) of the matrix representation of symmetry operations is the result of restrictions imposed by the three-dimensional periodicity of crystal lattice. The presence of rotational symmetry of any other order (e.g. five-fold rotation) will result in the non-integer values of the elements of corresponding matrices in three dimensions. 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (c,y) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sjj) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cy,). As a result, cubic crystal has only three independent elastic constants (cu== C22=C33, C44= css= and Ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( c - C i)l2. 

The relations between degeneracy of crystal vibrations and dimensionality of irreducible representations was generally treated by Mara-dudin and Vosko (1968). The unitary matrices T( , rj assodatai with symmetry operations (R ) were derived by Matadudin and Vosko (1968). A similarity transformation of the dynamical matrix D (J) may be carried out ... 

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