4.3 Consider the model one-dimensional monatomic chain of N atoms, equally spaced with separation a, and each with the same mass m. The force constant coupling each atom to its nearest-neighbors is K. The normal mode vibrational frequency (k) of a mode with wave-vector k for this model is: @3 4K sin m ka - (π/a) ≤k ≤ (¹/a) (a) Derive an expression for the group velocity vg as a function of k. (b) Using the results of part a, evaluate vg for k at a very small value of k (k→ 0). Briefly discuss the physical significance of this low k group velocity. (c) Using the results of part a, evaluate vg for k at Brillouin Zone boundary (k = π/a). Briefly discuss the physical significance of this Brillioun zone boundary group velocity.

4.3 Consider the model one-dimensional monatomic chain of N atoms, equally spaced with separation a, and each with the same mass m. The force constant coupling each atom to its nearest-neighbors is K. The normal mode vibrational frequency (k) of a mode with wave-vector k for this model is: @3 4K sin m ka - (π/a) ≤k ≤ (¹/a) (a) Derive an expression for the group velocity vg as a function of k. (b) Using the results of part a, evaluate vg for k at a very small value of k (k→ 0). Briefly discuss the physical significance of this low k group velocity. (c) Using the results of part a, evaluate vg for k at Brillouin Zone boundary (k = π/a). Briefly discuss the physical significance of this Brillioun zone boundary group velocity.