Answer to Question #87259 in Physical Chemistry for Anand

Question

Show that the group velocity is zero at edge of the first Briclouin zone for a linear monoatomic chain.

Answer 1

The range of wave numbers from –π/a to π/a is known as the first Brillouin zone. Any wave vectors outside the first Brillouin zone can be mapped to it by adding an integer multiple of the

reciprocal lattice vector; π/a in a one-dimensional example. The dispersion relation creates familiar arches with maxima and minima in the center and at the edges of the first Brillouin zone. An acoustic branch behaves like a solution for a primitive (monatomic) linear circuit, but the optical branch has a location where its maximum and minimum permutation is. This becomes apparent when calculating the frequencies for the two limiting cases. In the center of the first Brillouin zone, k = 0