Harmonic1DMD:Background:Harmonic 1DMD Analytical Solution

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



where is displacement from position na at time t.

This equation can then be rewritten as:



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

(s integer)