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In Compton Scattering, does the electron that is scattered leaves the atom in which it orbited around, i.e. is it ionized from its atom?

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I would add to Victor's answer that Compton scattering refers to one particular regime of photon-electron scattering, in which the energy of the photon becomes comparable to the mass-equivalent energy of the electron. A resting electron has a mass of $511\ \textrm{keV}/c^2$, which means you basically need at least x-rays to see the effect. Any lower energy photons will give you Thomson, Raman, Rayleigh, resonant, or other scattering, depending on the situation. Now, compare the mass energy of the electron to the scale of electron binding energies (typically ~1 eV). Binding energies are so much smaller that Compton scattering will certainly result in ionization.

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Based on the first paragraph in this article:

In scattering of a photon by a bound electron, in contrast to the case of scattering by a free electron, three processes are distinguished; 1) Rayleigh or coherent scattering, in which the state of the electron is not changed, 2) Raman scattering, as a result of which the scattering system goes over to another bound state, and 3) Compton scattering, which is accompanied by ionization.

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