Question

In a chain of identical atoms the vibration frequency ω depends on wave number k as ω= ω_max sin (ka/2), where ω_max is the maximum vibration frequency, k = 2π/λ, is the wave number corresponding to frequency ω, a is the distance between neighbouring atoms. Making use of this dispersion relation, find the dependence of the number of longitudinal vibrations per unit frequency interval on ω, i.e. dN/dω, if the length of the chain is l. Having obtained dN/dω, find the total number N of possible longitudinal vibrations of the chain.

Answer 1

If the chain has N atoms, we can assume atom number 0 and N +1 held ficed. Then the displacement of the n^th atom has the form