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2.1 Consider a linear chain in which alternate ions have masses M₁ and M2, and only neares neighbors interact. (i) K K a/2 a K ww w² (k)= M₁ K Show that the dispersion relation for normal modes is: K wow M₂ 2 1 1 1 1 4 ka = K < (+) ± √(-+)²-, M₂ sin (22) +K += M₁ M₂ M₂ M₁M₂ Where, K is the spring constant, and a, is the size of the unit cell (so the spacing between atoms is a/2). Derive an expression for the group velocity vg as a function of k. (iii) Use the results of part (ii), to evaluate vg for k at the Brillouin Zone boundary.[k = + and briefly discuss the physical significance of this Brillouin Zone boundary group

2.1 Consider a linear chain in which alternate ions have masses M₁ and M2, and only neares neighbors interact. (i) K K a/2 a K ww w² (k)= M₁ K Show that the dispersion relation for normal modes is: K wow M₂ 2 1 1 1 1 4 ka = K < (+) ± √(-+)²-, M₂ sin (22) +K += M₁ M₂ M₂ M₁M₂ Where, K is the spring constant, and a, is the size of the unit cell (so the spacing between atoms is a/2). Derive an expression for the group velocity vg as a function of k. (iii) Use the results of part (ii), to evaluate vg for k at the Brillouin Zone boundary.[k = + and briefly discuss the physical significance of this Brillouin Zone boundary group

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2.1 Consider a linear chain in which alternate ions have masses M₁ and M2, and only nearest neighbors interact. (i) K K www a/2 a w²(k) = K K M₁ K Show that the dispersion relation for normal modes is: K Ow M₂ (+) ± √(-+)-M₁M₂ Sin (7) Where, K is the spring constant, and a, is the size of the unit cell (so the spacing between atoms is a/2). (ii) Derive an expression for the group velocity vg as a function of k. H (iii) Use the results of part (ii), to evaluate và for k at the Brillouin Zone boundary, [k and briefly discuss the physical significance of this Brillouin Zone boundary group volocity (Specifically what do you say about propagation of longitudinal waves in the ( ±
Transcribed Image Text:2.1 Consider a linear chain in which alternate ions have masses M₁ and M2, and only nearest neighbors interact. (i) K K www a/2 a w²(k) = K K M₁ K Show that the dispersion relation for normal modes is: K Ow M₂ (+) ± √(-+)-M₁M₂ Sin (7) Where, K is the spring constant, and a, is the size of the unit cell (so the spacing between atoms is a/2). (ii) Derive an expression for the group velocity vg as a function of k. H (iii) Use the results of part (ii), to evaluate và for k at the Brillouin Zone boundary, [k and briefly discuss the physical significance of this Brillouin Zone boundary group volocity (Specifically what do you say about propagation of longitudinal waves in the ( ±
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