The scattering of phonons in a two-dimensional lattice can be written as w= (q) = W=(4x, qy) = By sen- + sen 2 2 and let's suppose that there are three different acoustic branches s=1,2,3 with s characterizing each branch. Using the assumptions of the Debye model, find the behavior of branches if q-0. Calculate the contribution to the state density of each branch and the total state density. Determine the characteristic Debye frequency of the system including all modes within the first Brillouin zone. Determine c, at low temperatures for this 2D solid