4.6 In a linear chain all the atoms have equal mass but are connected alternatively by springs of force constants, f1 and f2. Derive the frequency-wavevector relation for this chain. Are there still two branches? Explain.
4.7 In Problem 4.6, if ω1, ω2 and ω3 are, respectively, the maximum frequency of the acoustic branch and minimum and maximum frequencies of the optical branch, show that for f1 > f2,
ω1 = √(2 f2 / M), ω2 = √(2 f1 / M) and ω3 = √(2(f1 + f2) / M)