Q6 please

4. Using the Ansatz B XA = 2na + u u4eilwt-2kna) = (2n + 1)a 8+uBei(wt-2kna) 7 n obtain the matrix equation for the frequency of the normal modes of the chain: mw2 w ()-( K1 + K2 -(K1 + K2e-2ika) -(K1 + K2e2ika) Ki + K2 ")) 5. Solve the matrix equation in Q.4 to obtain two dispersion relations w+(k) and w-(k), with w+(k) > W-(k). Sketch the dispersion relations in the first Brillouin zone. 6. Your result for Q.5 should show that at the edge of the first Brillouin zone, w+ does not depend on K2, and w does not depend on K1. To understand this result, solve the eigenvalue equation for the amplitudes uA and uB for each branch of solution at the Brillouin zone edge. Sketch the associated atomic displacements and hence explain your result for the frequencies. Hint: use the eigenvalue equation in Q.4, replacing w on the right hand side by w+ at the edge Brillouin zone, and k in the matrix by its value at the edge of the Brillouin zone. 4. Using the Ansatz B XA = 2na + u u4eilwt-2kna) = (2n + 1)a 8+uBei(wt-2kna) 7 n obtain the matrix equation for the frequency of the normal modes of the chain: mw2 w ()-( K1 + K2 -(K1 + K2e-2ika) -(K1 + K2e2ika) Ki + K2 ")) 5. Solve the matrix equation in Q.4 to obtain two dispersion relations w+(k) and w-(k), with w+(k) > W-(k). Sketch the dispersion relations in the first Brillouin zone. 6. Your result for Q.5 should show that at the edge of the first Brillouin zone, w+ does not depend on K2, and w does not depend on K1. To understand this result, solve the eigenvalue equation for the amplitudes uA and uB for each branch of solution at the Brillouin zone edge. Sketch the associated atomic displacements and hence explain your result for the frequencies. Hint: use the eigenvalue equation in Q.4, replacing w on the right hand side by w+ at the edge Brillouin zone, and k in the matrix by its value at the edge of the Brillouin zone