By J. Moser
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Additional info for Dynamical Systems, Theory and Applications
It is customary to define k within a symmetrical unit cell of the reciprocal lattice, called the "first Brillouin zone" or simply "Brillouin zone", corresponding to the Wigner-Seitz cell in the direct lattice. This definition is made as the symmetry properties of the k vectors are of relevant importance in the discussion of the irreducible representations of the full space group. Then, it is convenient to locate k vec- tors in the Brillouin zone which shows the full symmetry of the reciprocal lattice.
As it is known, there are only 14 possible lattice types, usually referred to as Bravais lattices grouped into 7 crystal systems, according to the point group symmetry which leaves the lattice invariant. 1. • However, the same lattice can be gene- rated by an infinite number of sets of three independent basis vectors. Among all possible unit cells, which can be defined in this way for a given lattice, one can select a symmetrical unit cell known to physicists as the "Wigner-Seitz primitive cell" .
This theorem states that 23 for h/l. 0 .. 0 l. 24 JlJl \Iv where the sum is over the h elements a of the group, and 1. is the dil. (a). 23a results when k'=O. e. 23b is obtained. 3 IRREDUCIBLE REPRESENTATIONS OF THE SPACE GROUPS In this paragraph the analysis previously initiated will be extened to space groups G which contain rotational symmetry. Since for a given wave vector kl we need to distinguish between elements that leave it invariant and elements which transform kl into a different point of the star, let the elements of the point group GO (k 1 ) be specified as .
Dynamical Systems, Theory and Applications by J. Moser