Representations and Basis Functions

A set of matrices transforming under the multiplication laws of a group constitutes a representation of this group. When this set is in the diagonal form and that it can be reduced into subsets that cannot be further reduced 

In the notation of [11], the IRs are noted by capital letters eventually with indices and/or primes, the convention being to label by A or B the ID IRs, by E (not to be confused with the identity operation E), the 2D ones and by T the 3D ones. In the notation of Bethe [2] used by Koster [9], the IRs are simply noted J] (i = 1, 2, 3, etc), eventually with indices and primes. 

This table can be used to determine the representation for a polar or axial vector in a given symmetry group. In some cases, these representations are irreducible, as for the T j group, but for the others, they are reducible and the character table of the IRs of the group must be used for the reductions into IRs. 

to go further and to provide conceptual tools that will be used in the interpretation of the electronic spectra of impurities in crystals, a new group has to be introduced, the 3D rotation group, noted here i +(3), which is the 

For the double representations, the basis functions that are eigenfunctions of angular momentum j and projection m on the z axis are noted f (j, m). For T7, the basis functions transform like the products 4 (j, m) of the basis functions of T6 and those of T2 and they are noted T6 x T2. 

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Table 17.1. Irreducible representations and basis functions for the symmetry point X in the BZ of the sc Bravais lattice.

The symmetry properties of spontaneous strains are most conveniently understood by referring to the irreducible representations and basis functions for the point group of the high symmetry phase of a crystal of interest. These are given in Table 2 for the point group Almmm as an example. Basis functions x + y ) and are associated with the identity representation and are equivalent to (ei + ei) and C3 respectively. This is the same as saying that both strains are consistent with Almmm symmetry e = ei). The shear strain e - ei) is equivalent to the basis function (x - y ) which is associated with the Big representation, the shear strain e is equivalent to xy (B2g) and shear strains e and e to xz, yz respectively (Eg). The combinations (ci + 62) and (ci - ei) are referred to as symmetry-adapted strains because they have the form of specific basis functions of the... 

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