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How do phonons affect the work function? Could phonons that are directed towards a single atom, such as in a body-centered cubic crystal, increase the amount of thermal excitation and thus decrease the input of temperature required to eject an electron from that atom?

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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 electron. Temperature is not energy, though we can often write down an equation which connects average energies with the temperature.

However, your basic idea should be correct. If you heat up a metal then, on average, the electrons should be more thermally excited and the work function should decrease. For an example in the research literature see here.

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