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Most derivations of Planck's Radiation use a classical model of the atom and don't really explain the mechanism behind it. How exactly does a vibrating atom or a lattice of atoms(like in a solid), release photons? Electrons moving to different energy levels release photons in the process but how is this related to atoms vibrating?

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Planck radiation is a thermodynamic effect - more precisely, it is a photon gas in thermal equilibrium. Ubiquitous (but old-fashioned) derivations often derive it as radiation in equilibrium with a hypothetical black body, which consists of very large number of atoms and hence can be described thermodynamically. Applying thermodynamic concepts to a single atom doesn't make sense (exception is an atom coupled to a heat bath, but this is not related to the subject.)

See also:
How does radiation become black-body radiation?

Roger V.
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