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Consider the one-dimensional linear chain in which the force constants between nearest-
neighbor atoms are alternately C and 4C. Let the masses be equal, and the nearest neighbor
separation is a. The dispersion relation for the frequency is given by
o =:(G, + c; + C + c} + 26,6,coska
w? =
C1 + C2 ±
(a) Find w² for C, = C and C2 = 4C
(b) Find frequencies w(k) at k = 0 and k
for optical and acoustical modes.

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