2.1

Template of an fcc Brillouin Zone

Construct a model of the Brillouin zone of the fcc lattice by pasting a copy of the template shown in Fig. 2.27 on cardboard and cutting along the solid lines. Score along the broken lines. Tape the edges together.

Figure 2.27: Template for constructing a model of the Brillouin zone of the fcc lattice. Paste this sheet on thin cardboard and cut along solid lines. Score along broken lines and tape the joints

2.2

Group Theory Exercises

1. Verify the character table of the T _d point group as given in Table 2.3.

2. By applying the symmetry operations of T _d to the basis functions in Table 2.3 show that the functions transform according to their respective irreducible representations.

2.3

Group Theory Exercises

1. By using the character table of T _d show that T ₂ ? T ₂ = T ₁ ? T ₂ ? E ? A ₁.

2. Verify that the symmetrized linear combinations of the matrix elements of a second-rank tensor given in Sect. 2.3.4 transform according to the irreducible representations T ₁, T ₂, E, and A ₁.

2.4

Symmetrized Wave Functions: Transformation Properties

Verify that the symmetrized wave functions in Tables 2.9 and 2.10 transform according to their respective irreducible representations.

2.5

Characters of C _3v and C _2v Point Groups

Deduce by inspection the characters for the C _{3 v}