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2.1 Consider a linear chain in which alternate ions have masses M₁ and M2, and only nearest neighbors interact. (1) K /2 a K M₁ Show that the dispersion relation for normal modes is: M₂ ( ² (K) = K (1/₂ + ²) + K √( ₁ + ₂)² - 4 K -sin M₁M₂ ka 2 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. (iii) Use the results of part (ii), to evaluate vg for k at the Brillouin Zone boundary, [k = ±¹/a], and briefly discuss the physical significance of this Brillouin Zone boundary group velocity.(Specifically, what do you say about propagation of longitudinal waves in this lattice at frequency w(k = ±"/a)? (iv) Suppose that we allow the two masses M₁ and M2 in a one-dimensional diatomic lattice to become equal. What happens with the frequency gap? Draw the dispersion relation curve to support your answer (show your calculations).

2.1 Consider a linear chain in which alternate ions have masses M₁ and M2, and only nearest neighbors interact. (1) K /2 a K M₁ Show that the dispersion relation for normal modes is: M₂ ( ² (K) = K (1/₂ + ²) + K √( ₁ + ₂)² - 4 K -sin M₁M₂ ka 2 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. (iii) Use the results of part (ii), to evaluate vg for k at the Brillouin Zone boundary, [k = ±¹/a], and briefly discuss the physical significance of this Brillouin Zone boundary group velocity.(Specifically, what do you say about propagation of longitudinal waves in this lattice at frequency w(k = ±"/a)? (iv) Suppose that we allow the two masses M₁ and M2 in a one-dimensional diatomic lattice to become equal. What happens with the frequency gap? Draw the dispersion relation curve to support your answer (show your calculations).

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2.1 Consider a linear chain in which alternate ions have masses M₁ and M2, and only nearest neighbors interact. (1) a/2 a M₂ K M₁ Show that the dispersion relation for normal modes is: w ² (K) = K (1/₂ + 1/₂) + K 2 1 1 (₁+₂) C M₂ K W 4 M₁M₂ -sin (ka) 2 Where, K is the spring constant, and a, is the size of the unit cell (so the spacing between atoms is a/2). P (ii) 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 = ±¹], and briefly discuss the physical significance of this Brillouin Zone boundary group velocity.(Specifically, what do you say about propagation of longitudinal waves in this lattice at frequency w(k = ±"/a)? (iv) Suppose that we allow the two masses M₁ and M2 in a one-dimensional diatomic lattice to become equal. What happens with the frequency gap? Draw the dispersion relation curve to support your answer (show your calculations).
Transcribed Image Text:2.1 Consider a linear chain in which alternate ions have masses M₁ and M2, and only nearest neighbors interact. (1) a/2 a M₂ K M₁ Show that the dispersion relation for normal modes is: w ² (K) = K (1/₂ + 1/₂) + K 2 1 1 (₁+₂) C M₂ K W 4 M₁M₂ -sin (ka) 2 Where, K is the spring constant, and a, is the size of the unit cell (so the spacing between atoms is a/2). P (ii) 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 = ±¹], and briefly discuss the physical significance of this Brillouin Zone boundary group velocity.(Specifically, what do you say about propagation of longitudinal waves in this lattice at frequency w(k = ±"/a)? (iv) Suppose that we allow the two masses M₁ and M2 in a one-dimensional diatomic lattice to become equal. What happens with the frequency gap? Draw the dispersion relation curve to support your answer (show your calculations).
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