Department of Physics, University of Maryland, College Park, MD 20742-4111

Physics 731 HOMEWORK ASSIGNMENT #4 Due: Tuesday, Oct. 3, 2000

Read Ashcroft & Mermin (A&M), chaps. 21 & 22.

1. A&M 22–1

2. A&M 22–2

3. A&M 22–3

4. A&M 22–5, parts a, b, and c only. Hint for part a: Use the chain rule to show

f / Rm = f ¢ R/ Rm = f ¢ Rm /R etc. (where R2 = S m Rm 2),

and thence get f m n = 2f / Rm Rn in eq. (22.11).

5. [Essentially Kittel (7th) 4–7]: Consider a simple model of soft phonon modes: Consider a line of ions of equal mass but alternating in charge, with qm = (-1)me as the charge on the mth ion. Their interatomic potential is the sum of two contributions: 1) a short-range interaction of force constant K1R = 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 Kmc = 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 Km(1 - cos mka) [eqn. (22.90)], show that the dispersion relation for this specific system can be written as

    3. 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.

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Also look carefully at problem 22–4. The three results are interesting and important, but somewhat tedious to derive.

Solutions will also be provided to the following problem, which you should not turn in but may find interesting:

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.