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.

Answer 1

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: $$\hbar\omega_{high} = \hbar\Omega_{atom} + \hbar\omega_{low}.$$

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.