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When a metal surface is illuminated with light of appropriate frequency so as to cause photoelectric emission, when does the work function of the metal come into play? Is it the energy required to bring an electron to the metal surface or is it the energy required to liberate the electron from the metal surface?

Also, is it possible for any other substance( except metal) to emit photoelectrons?

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The work function of a metallic surface is the minimum amount of energy required to liberate an electron. The rough, classical idea is that electrons are free to move around inside a metal, but require extra energy in order to leave.

To answer your second question, yes, non-conductors can emit photoelectrons. This is the basis of x-ray photoelectron spectroscopy.

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