Question

5. Phonon: a. Consider an infinite 1D chain of masses which are not only connect to nearest neighbors by a spring of constant f1 but also to next nearest neighbors by a spring of constant f2. Derive the dispersion relation assuming that f2<<f1. b. (F = f_1(u_{s+1} - 2u_s + u_{s-1}) + f_2(u_{s+2} - 2u_s + u_{s-2}), ?^2 = (2f_1/M)(1 - cos(ka)) + (2f_2/M)(1 - cos(2ka)) c. Sketch the dispersion relation (same periodicity and shape as before) and extract the group velocity for k -> 0 and k -> ?/2. v(k -> 0) = sqrt(((f_1+4f_2)a^2)/M), v(k -> ?/a) = 0

          5. Phonon:
a. Consider an infinite 1D chain of masses which are not only connect to nearest neighbors by a spring of constant f1 but also to next nearest neighbors by a spring of constant f2. Derive the dispersion relation assuming that f2<<f1.
b. (F = f_1(u_{s+1} - 2u_s + u_{s-1}) + f_2(u_{s+2} - 2u_s + u_{s-2}), ?^2 = (2f_1/M)(1 - cos(ka)) + (2f_2/M)(1 - cos(2ka))
c. Sketch the dispersion relation (same periodicity and shape as before) and extract the group velocity for k -> 0 and k -> ?/2.
v(k -> 0) = sqrt(((f_1+4f_2)a^2)/M), v(k -> ?/a) = 0

5. Phonon:
a. Consider an infinite 1D chain of masses which are not only connect to nearest neighbors by a spring of constant f1 but also to next nearest neighbors by a spring of constant f2. Derive the dispersion relation assuming that f2<<f1.
b. (F = f1(us+1 - 2us + us-1) + f2(us+2 - 2us + us-2), ?^2 = (2f1/M)(1 - cos(ka)) + (2f2/M)(1 - cos(2ka))
c. Sketch the dispersion relation (same periodicity and shape as before) and extract the group velocity for k -> 0 and k -> ?/2.
v(k -> 0) = sqrt(((f1+4f2)a^2)/M), v(k -> ?/a) = 0

Added by Cynthia S.

Question

Please give Ace some feedback