Consider one dimensional chain consists of many atoms with identical masses m and separated by a distance of a (lattice parameter). At equilibrium the abscissa of the nth atom is x=na. The atoms can move only in a direction parallel to the chain and we note u,, the displacement with respect to its equilibrium position. For harmonic approximation.. ei(kx) We consider the first and second nearest neighbours interaction. 1- Determine the dispersion relation co(k) for normal modes. 2- Write (k) near the center and at the limit of the 1st Brillouin zone. 3- Represent oo(k) -