Analytical Solution

An analytical solution has been developed that describes Lattice Vibrations of 1D atom chains. The solution comes in the form:

$$u(na,t) \propto e^{i(kna-\omega t)}$$

where $u(na,t)$ is displacement from position na at time t.

This equation can then be rewritten as:

$$u\left (na,t\right ) = \cos{(k na - \omega t)} + i\sin{(k na - \omega t)}$$

From this you can find the displacement of any atom, at position na and at time t, given k and $\omega$. By setting the proper conditions (periodic boundary), and assuming harmonic interactions (like springs), k and $\omega$ become the following:

$k = \frac{2\pi s}{aN}$ (s integer)

$\omega = \sqrt{\frac{2K[1-\cos{(ka)}]}{M}} = \sqrt{\frac{4K}{M}}\mid\sin{(ka/2)\mid}$