Question \#5: Let's consider a model for lattice vibrations that is as follows. Atoms are placed in a 1D lattice. They have the same mass. Now the equilibrium spacing between atoms alternates between (a−d) and (d); this means that the interatomic forces change depending on the equilibrium spacing, which we account for by imagining springs of different spring constants G and K connecting the atoms: 1) Write Newton's third law for each atom in the "unit cell" of this lattice. Use the displacements u_2n 2x) and u−​(2n+1)(x) as seen in class. 2) State the result of circular boundary conditions on u(x), assuming a very large number of atomic sites N. K being the wavenumber associated with such boundary conditions, how many possible values of K are there? Which are they? What is the interval over which K is periodic (i.e., give one possible Brillouin zone for K) ? 3) Propose the shape of the solution for the u_2n(x) and u_ (2n+1)(x). Write out the resulting system of equations that will eventually help us solve for ω(K). 4) Solve for the possible values of ω(K). 5) Plot the two branches of solution for ω(K). 6) In the limits K=0 and K at the edge of the Brillouin zone, what are the values of ω(K) in those two branches? Add those values to your plot. 7) In the limit of K=0, solve for the amplitudes of oscillation for u_2n(x) and u_(2n+1)(x). What is the physical significance of these results?