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I know gamma-ray photon can only give its momentum energy to the electrons of an atom.

My question is: Can a photon give some of its momentum to the atom (including its nucleus) to give it heat or speed?

If yes, can you tell me how much energy can it give?

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It is not very clear to me if you are asking about energy or momentum. You should also ask about a specific interaction process as there are many, this is required especially to answer your last, quantitative, question.

However, generally speaking, a $\gamma$ photon cannot give some of its energy to anything else: it is all or nothing. Even in the Compton scattering, in which you get a less energetic photon, the initial photon is destroyed. The momentum must be conserved as well, so yes: when a photon hits another particle this is accelerated, you can even generate some measurable pressure with a very intense radiation!

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  • $\begingroup$ can the photon accelerate every particle with mass on the atom? if yes its possible for the photon to accelerate the entire atom (for example hydrogen atom)? $\endgroup$
    – Qu4terni0n
    Sep 7 '14 at 10:14
  • $\begingroup$ @Qu4terni0n Momentum conservation is a physical rule. It does not depend on the mass. When the photon interacts its momentum is transferred, this is not just a possibility, but how things work. Of course if you shoot a highly energetic gamma to a particle in a complex and delicate structure as an atom, most of the times that particle is ejected from the atom, without accelerating all of it. $\endgroup$
    – DarioP
    Sep 7 '14 at 10:48
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Not merely can it transfer its momentum as well as its energy when it interacts, but it must.

If the target atom is in a fluid context (liquid, gas, plasma), then that energy and momentum must end up in the target or some other reaction products(s).

In a solid context the Mössbauer effect can be an issue, allowing the transfer of that momentum to a much more massive collection of mass resulting in lower recoil.

It is also worth noting that this requirement to transfer both energy and momentum is the reason a photon can not be absorbed on a isolated point target in free-space: $$ \gamma + e \to e \tag{forbidden} \,,$$ though the same reaction is possible in the field of a massive partner $$ \gamma + e + A \to e + A \tag{allowed}$$ or if the target is a compound object with excited states $$ \gamma + A \to A^* \tag{allowed} \,.$$

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