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I will present the result for the one-dimensional case only. It is readily generalized to your three-dimensional situation. Knowing the phonon dispersion $\omega(q)$, the DOS can be defined as $$D(\omega) = \frac{1}{2\pi}\int\mathrm{d}q\,\delta(\omega-\omega(q)).$$ Using the formal identity $$\delta(g(x)) = \sum_{i}\frac{\delta(x-x_i)}{|g'(x_i)|}$$ where ...


Careful! The essence of your question is a good thought but I think you are having several misconceptions. First of all, it doesn't make sense to talk about a phonon being directed towards a single atom. Phonons are delocalized. Secondly, the "input of temperature required to eject an electron" is a dangerous idea. You need to input energy to eject an ...

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