5. [Essentially Kittel (7^th) 4–7]: Consider a simple model of soft phonon modes: Consider a line of ions of equal mass but alternating in charge, with e_m = (-1)^me as the charge on the m^th ion. Their interatomic potential is the sum of two contributions: 1) a short-range interaction of force constant K_1R = g that acts between nearest neighbors only, and 2) a Coulomb interaction between all ions.

1. Show that the contribution of the Coulomb interaction to the atomic force constants is K_mc = 2 (-1)^me²/m³a³, where a is the equilibrium nearest-neighbor distances.
2. Using the general 1–D dispersion relation w ² = (2/M)S _m³ 1 K_m(1 - cos mka) [eqn. (22.90)], show that the dispersion relation for this specific system can be written as

where w ₀² = 4g /M and s = e²/g a³.

3. Show that w ² is negative (i.e. the mode is unstable, or "soft") at the zone boundary ka = p if s > 0.475 [i.e. 4/{7z (3)}, where z is the Riemann zeta function].
4. Show that the speed of sound at small ka is imaginary if s > 1/(2 ln 2) @ 0.721.

Consider a monatomic chain of N+1 atoms with interatomic separation a, as discussed in class. Supposed rather than periodic [B-vK] boundary conditions, we use fixed boundary conditions: u(0) º 0 and u(Na) º 0. What are the allowed independent values of k? How many are there? (Note that these solutions are standing waves rather than traveling waves.) Compare and reconcile your findings with those for periodic boundary conditions.