(a) Considering only the nearest-neighbor interaction, find the dispersion relation for the diatomic linear chain of a silicon lattice along the (111) crystal axis. Note that the masses of the silicon atoms are identical, and the positions of the nearest-neighbor atoms on the (111) axis are located at (0,0,0), (a/4, a/4, a/4), and (a, a, a) inside the unit cell. Assume that the force constant between the nearest-neighbor atoms is equal to β.
(b) Plot the dispersion curves (ω vs. q) from the result obtained in (a).
(c) Sketch the atomic displacement for the longitudinal and transverse optical lattice vibration modes at q = π/4 √3a,π/2 √3a, and π/√3a.
