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I am having a conceptual problem understanding the kinetic energy of thermionically emitted electrons.

I know that in order to escape the surface the electrons must have energies of at least the Fermi Level plus the work function ($\varepsilon \ge \varepsilon_F + \phi$). However, is all that kinetic energy converted to potential energy once it leaves the material so that once it leaves the surface an electron has kinetic energy $T = \varepsilon - \varepsilon_F + \phi$?

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The electron maintains the total energy $\epsilon$ that it had while inside the material. Inside the material it has some kinetic energy and some potential energy, but we don't really care exactly what those are since we know their sum. The point is, once it enters the vacuum, its total energy is still made up of kinetic and potential energy. Since the vacuum potential energy is $U = \epsilon_F + \phi$ just outside the surface, then indeed the kinetic energy is $T = \epsilon - U = \epsilon - \epsilon_F - \phi$. Of course if there are any electric fields in the vacuum, then the kinetic energy of the electron will change as it moves away from the surface.

A simple way to understand it is that the minimum kinetic energy is zero, since the electron can't go into the vacuum if that would give it less than zero kinetic energy. The actual distribution of energies is a complex topic from what I understand, and material dependent.

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  • $\begingroup$ Yes, you're right Nanite. I did some more research and it's the same idea as ionization. The distribution of energies is the Fermi-Dirac Distribution, $f(\varepsilon) = \frac{1}{1 + \exp(\frac{\varepsilon - \epsilon_F}{k_B T})}$ $\endgroup$ Nov 16 '13 at 2:15
  • $\begingroup$ Ah, I mean the distribution of the energies of the emitted electrons. As you say it depends on the Fermi-Dirac distribution, but also on the density of electronic states in the material and their associated velocity. Also, it depends on the probability of an electron in the material successful leaving when it hits the surface. What I mean is, although there are such-and-such number of electrons energetically able to leave the material, the number that actually do leave can be tricky to predict theoretically. I suppose the Richardson's constant adds up all these effects in the end, though. $\endgroup$
    – Nanite
    Nov 16 '13 at 16:33

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