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I want to understand the concepts behind the thermionic emission. In thermionic emission, the energy randomization occurs and the energy may be split to electronic or roto-vibrational states. If this energy randomization happens, I didn't understand how the electron will get enough energy to be detached from the molecule/cluster! The randomized energy could be dissipated via radiation (IR or visible). So, to make clear, I am wondering how this randomzed energy can cause electron emission.

As far as I know (from readings), delayed emission of electrons is an indication for thermionic emission (Y Zhao J. Chem. Phys., 1996, 105(12), 4905). How the electron finally collect these randomized energy so that it eventually get detached?

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Consider the analogy of the velocity distribution in an ideal gas. On average the atoms have an energy of 3/2$kT$, but due to the random nature of atomic collisions some atoms have more energy and some less. The atom velocities end up obeying the Maxwell-Boltzman distribution.

The same argument applies to the electrons in a metal. In fact it's an often used approximation to consider the electrons in the conduction band as a gas of free electrons. The electrons will exchange energy with each other and with lattice vibrations, and the electron velocities will end up with something approximating a Maxwell-Boltzman distribution. The high energy tail of the distribution will have more energy than the work function and can therefore escape the metal surface. This is thermionic emission. In principle it occurs even at room temperature, but the fraction of electrons with energy higher than the work function is negligable at room temperature and increases rapidly as you heat the metal.

Your reference to "molecule/cluster" suggests you're also considering non-metals, but I'm not sure non-metals show thermionic emission in the sense I understand the term. You can certainly ionise molecules in a non-metal, and again it will be due to local concentrations of energy obeying a Maxwell-Boltzman type distribution. For example, you've probably seen the colours generated by metal ions in fireworks. These are due to electronic excitations even though in the firework $kT$ is well below the energies required. Again it's due to the small fraction of molecules colliding with enough energy to cause an electronic excitation.

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