Alternative derivation of the dispersion relation

We have seen in the previous section that, if the system is excited into a particular normal mode, the general time dependent solution for the displacement $x_n$ of a particle on the chain can be expressed as:

$$x_n(t) = A e^{-i(q n a -\omega t)},$$ (7.36)

where $n a$ is the position of the particle on the chain, and $q$ is some wavenumber that together with $\omega$ needs to be determined. We now use Newton's equation of motion:

$$M \ddot{x}_n = - \frac{dU}{d x_n},$$ (7.37)

which for the potential energy function  gives

$$-M \omega^2 x_n = - k [ 2 x_n - x_{n-1} - x_{n+1}],$$ (7.38)

and so

$$M \omega^2 = k [ 2 - e^{iq a} - e^{-iq a} ] = 2k [ 1 - \cos (q a) ],$$ (7.39)

from which we obtain again the dispersion relation:

$$\omega = 2\left (\frac{k}{M} \right )^\frac{1}{2} \left \vert\sin\left(\frac{a q}{2}\right) \right \vert.$$ (7.40)

This has the same form as . To determine the allowed wavenumbers $q$ we impose the boundary condition $x_{n+N} = x_n$, which requires

$$q [N+n] a = qna + 2l \pi,$$ (7.41)

with $l$ any integer, which gives

$$q_l = \frac{ 2 \pi l }{N a}; \quad; \quad -\frac{N}{2} < l \le \frac{N}{2},$$ (7.42)

where we have placed limits on $l$ to restrict $q_l$ to the first Brillouin zone, as we only need to define $q_l$ over one period.