Space-group representations

The action of the space-group operator (R v) c G (with R c P), the point group of the space group G on the Bloch function kfr) gives the transformed function vJk(r). To find the transformed wave vector k we need the eigenvalue exp ( ik -t) of the translation operator ( t). 

The space-group operator (R v) acts on functions of r, and therefore the exponential factor in eq. (2), which is not a function of r, is unaffected by (R v). 

Exercise 16.4-1 (a) Verify the equahty of the operator products on each side of eq. (1). [Hint Use the multiplication rule for Seitz operators.] (b) Verify the equality of the RS of eqs. (2) and (3). (c) Find the transformed Bloch function i//k(r) when (R v) is (7 0). 

Equation (3) shows that the space-group operator (R v) transforms a Bloch function with wave vector k BZ into one with wave vector R k, which either also lies in the BZ or is equivalent to ( ) a wave vector k in the first BZ. (The case Id = k is not excluded.) Therefore, as R runs over the whole R = P, the isogonal point group of G, it generates a basis ( 0kl for a representation of the space group G, 

On introducing the notation (k, meaning the whole set of Bloch functions that form a basis for a representation of G, 

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Repeat steps (i)-(vi) for each k vector for which the space-group representations Tk(R w) rk( t) are required. 

Finally, the matrices Tk(if w) in Table 16.10 have to be multiplied by the appropriate representation rk(/i t) of the translation subgroup to give the space-group representations... 

On using eqs. (16.6.9) and (16.6.10) the matrices of the required representations found in (viii) give the elements Tk of the supermatrix as in step (vi) of Section 16.6, and these matrices, when multiplied by Tk(C t), are the space-group representations. 

Tables of space-group representations are given by Bradley and Cracknell (1972), Kovalev (1993), Miller and Love (1967), Zak (1969). Stokes and Hatch (1988) describe various errors in these compilations and discuss the different settings and labels used.

The IRs and their characters are given in Table 16.21. The space-group representations are obtained by multiplying these 1-D IRs in Table 16.21 by the MRs of T(k). 

For the space group 225 (Fmim or Ojj) write down the coset expansion of the little group G(k) on T. Hence write down an expression for the small representations. State the point group of the k vector P(k) at the symmetry points L(/4 V-i Vi) and X(q a 2a). Work out also the Cartesian coordinates of L and X. Finally, list the space-group representations at L and X. 

Table 2.7. Common Symbols Used for Space Group Representations...

Let us now consider the necessary conditions for the appearance of phonons in impurity-ion electronic spectra. The presence of a substitutional defect in an otherwise perfect crystal removes the translational symmetry of the system and reduces the symmetry group of the system from the crystal space group to the point group of the lattice site. Loudon [26] has provided a table for the reduction of the space group representations of a face-centered cubic lattice into a sum of cubic point-group representations. A portion of that table is shown in Table 1 here. Consider an impurity ion that undergoes a vibronic electric-dipole allowed transition, with T and Tf the irreducible representations of the initial and final electronic states. Since the electric dipole operator transforms as Tj in the cubic point group, Oh, the selection rule for participation of a phonon is that one of its site symmetry irreducible representations is contained in the direct product T x Ti X Tf. 

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