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

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

This equation can then be rewritten as:

u(na,t)=cos(knaωt)+isin(knaωt)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 ω\omega. By setting the proper conditions (periodic boundary), and assuming harmonic interactions (like springs), k and ω\omega become the following:

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

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