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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F904caeb1-cd31-4dcf-9af8-09f5b2d51865%2F85551bbd-3138-4075-8e20-c8b4ddb96877%2F8670bpg_processed.jpeg&w=3840&q=75)
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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