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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: 13 4K 771 sin (1). ("/a) ≤k ≤ ("/a) (a) Derive an expression for the group velocity vg as a function of k. (b) Using the results of part a, evaluate vo 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 vo for k at Brillouin Zone boundary (k = x/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: 13 4K 771 sin (1). ("/a) ≤k ≤ ("/a) (a) Derive an expression for the group velocity vg as a function of k. (b) Using the results of part a, evaluate vo 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 vo for k at Brillouin Zone boundary (k = x/a). Briefly discuss the physical significance of this Brillioun zone boundary group velocity.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Please assist with 4. 3 a, b and c

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: W3 4K m sin -("/a) ≤k ≤ (π/a) (a) Derive an expression for the group velocity vg as a function of k. (b) Using the results of part a, evaluate vo 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 vo for k at Brillouin Zone boundary (k = x/a). Briefly discuss the physical significance of this Brillioun zone boundary group velocity.
Transcribed Image Text: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: W3 4K m sin -("/a) ≤k ≤ (π/a) (a) Derive an expression for the group velocity vg as a function of k. (b) Using the results of part a, evaluate vo 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 vo for k at Brillouin Zone boundary (k = x/a). Briefly discuss the physical significance of this Brillioun zone boundary group velocity.
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