By G. Koster (Auth.)

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KOSTER a. General Point For a general point in the Brillouin zone the group of the k vector contains primitive translations alone. Aside from the identity, no operation of the point group sends the k vector into itself or into an equivalent point. The representation of the group of the k vector is merely a one by one representation of the group of pure translations corresponding to the k vector in question. The irreducible representation of the space group is 48 by 48. , the regular representation of the point group).

We shall assume that we are given an irreducible representation of a space group and shall study its properties. We denote the space group by g and a typical element of the group by {a|a}. This group has one of pure translations as an invariant subgroup. We shall call the subgroup 3 ; an element of the subgroup is {e|R„}. Let us assume that we have an irreducible representation of the group 9 °f dimension n. The matrices in the irreducible representation will be denoted by D({a|a}). We can assume, without loss of generality, that D({a|a}) forms a unitary representation of the group 9.

Since the Brillouin zone contains all the irreducible representations of the group of pure translation on or inside of its surface, and has the advantage of possessing the same symmetry as the lattice in k space, we will choose this to be the unit cell in k space from now on. We notice then that, for every Bravais lattice generated by a group of primitive translations, there is a corresponding Bravais lattice in k space whose basic primitive translations are given by (2-4). It is also true that, whenever a lattice is invariant under the operations of a certain point group about one of its lattice sites, the corresponding lattice 22 L.