Consider a face-centered cubic monatomic bravais lattice in which each ion interacts only with its (twelve) nearest neighbors. assume that the interaction between a pair of neighboring ions is described by a pair potential depends only distance the pair of ions. show that the frequencies of the three normal modes with wave vector k are given by (a) a/m (22.960 where the are the eigenvalues of the 3 x 3 matrix: (22.97) here the sum is over the twelve nearest neighbors of r 0: (ty t z), (ti t r); (22.98) ow), and rr is the diadic formed from the unit vectors r r/r he, rr),v the constants a and b are: a 2g (d)/d, b 20p"(d) p'(d)/d], where