1D Acoustical Relations beyond nearest neighbor couplings

Question

Consider some 1D Lattice of atoms with nth neighbor coupling of strength k_{n}. I'm looking for the dispersion relation for acoustical phonons under these conditions.

I start with the Lagrangian, $$L = K- V$$ $$L = \sum^{\infty}_{n} \frac{1}{2}m \dot{x}_{n}^{2} - \sum^{\infty}_{p=1} \frac{1}{2}k_{p} \{(x_n-x_{n+p})^2 + (x_n - x_{n-p})^2\}$$

Mass is the same for each atom. The Lagrange equation should be

$$m \ddot{x}_{n}=\sum_{p=1} k_p(x_{n-p}+x_{x+p}-2x_n)$$

Now, if I use a travelling wave solution as an ansatz, I should get my dispersion relation as some infinite series. Is this correct? If so, help me out because I can't make it work. Thanks!

Answer 1

A dispersion relation tells you the conditions under which a certain solution holds, which means that in order to get a dispersion relation you need to assume a solution of some general form. This is a system of harmonic oscillators coupled over a long range, so it is natural to assume a plane wave solution. Using your notation, try

$x_n= e^{i(kna - \omega t)}$

where $\omega$ is the frequency of phonon oscillation. You want to specify which $\omega$ satisfy the equations of motion. It should only take a couple of steps to arrive at the dispersion relation from this.

I hope this helps.