How come a photon cannot "give" partial energy while interacting with an electron fundamentally, as apposed to electron to electron interaction?

Question

Particle Interactions: How come a photon cannot "give" partial energy while interacting with an electron fundamentally (with high probability) (as apposed to electron-electron interaction)?
for example:
An electron in an atom has 5 energy levels : $E_0$, $3E_0$, $5E_0$, $7E_0$, $9E_0$
light in certain wave length $frac{hc}{5.5E_0}$ has $5.5E_0$, and thus will not transfer any energy to the electron.
But when an electron has de Broglie wavelength of $frac{h}{sqrt{13mR_0}}$, it has the energy of $6.5E_o$, and can transfer energy to the electron in the atom.
That's just an example, but i think it makes more sense into the question.

Vadim · Answer

Non-resonant inelastic scattering between photons and atoms is possible. Raman scattering is the most known example of it. E.g., a high energy incident photon can transfer part of its energy to atom, becoming a low energy photon:
$$hbaromega_{high} = hbarOmega_{atom} + hbaromega_{low}.$$
What may be confusing in comparing the electron-electron scattering and the atom-photon scattering is that the atom-photon scattering is usually discussed within the dipole approximation, since the dipole moment between two electron states in the atom is by far the strongest atom-photon coupling term. Thus, the non-resonant processes, like the Raman scattering, appear in higher orders of the perturbataion theory and/or after adding some other interactions. For electron-electron interaction it is however more appropriate to consider direct charge-charge coupling, which allows arbitrary energy transfer in a first-order scattering event.

How come a photon cannot "give" partial energy while interacting with an electron fundamentally, as apposed to electron to electron interaction?

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