Free electron can't absorb a photon [duplicate]

Why can't a free electron absorb a photon? But a one attached to an atom can.. Can you explain to me logically and by easy equations? Thank you..

• I didn't realize a free electron could not absorb a photon. Or at least I never though about it but I will now. That makes photon / electron interaction a little more interesting when you think about a double slit experiments with electrons and observations. Commented Dec 23, 2015 at 9:28
• Oh yes I have it written on my notebook but I didn't understand my teacher's explanation..
– user65035
Commented Dec 23, 2015 at 9:35
• I would like to mention that an electron in Bremsstrahlung process is not BOUND to ion but a photon impacts it and changes the electron's path.
– Amin
Commented Dec 23, 2015 at 11:28

It is because energy and momentum cannot be simultaneously conserved if a free electron were to absorb a photon. If the electron is bound to an atom then the atom itself is able to act as a third body repository of energy and momentum.

Details below:

Conservation of momentum when a photon ($$\nu$$) interacts with a free electron, assuming that it were absorbed, gives us $$$$p_{1} + p_{\nu} = p_{2}, \tag{1}$$$$ where $$p_1$$ and $$p_2$$ are the momentum of the electron before and after the interaction. Conservation of energy gives us $$$$\sqrt{(p_{1}^{2}c^{2} + m_{e}^{2}c^{4})} + p_{\nu}c = \sqrt{(p_{2}^{2}c^{2}+m_{e}^{2}c^{4})} \tag{2}$$$$ Squaring equation (2) and substituting for $$p_{\nu}$$ from equation (1), we have $$p_{1}^{2}c^{2} + m_{e}^{2}c^{4} + 2(p_{2}-p_{1})\sqrt{(p_{1}^{2}c^{2}+m_{e}^{2}c^{4})} + (p_{2}-p_{1})^{2}c^{2}=p_{2}^{2}c^{2}+m_{e}^{2}c^{4}$$ $$(p_{2}-p_{1})^{2}c^{2} - (p_{2}^{2}-p_{1}^{2})c^{2} + 2(p_{2}-p_{1})c\sqrt{(p_{1}^{2}c^{2}+m_{e}^{2}c^{4})} = 0$$ Clearly $$p_{2}-p_{1}=0$$ is a solution to this equation, but cannot be possible if the photon has non-zero momentum. Dividing through by this solution we are left with $$\sqrt{(p_{1}^{2}c^{2}+m_{e}^{2}c^{4})} - p_{1}c =0$$ This solution is also impossible if the electron has non-zero rest mass (which it does). We conclude therefore that a free electron cannot absorb a photon because energy and momentum cannot simultaneously be conserved.

NB: The above demonstration assumes a linear interaction. In general $$\vec{p_{\nu}}$$, $$\vec{p_1}$$ and $$\vec{p_2}$$ would not be aligned. However you can always transform to a frame of reference where the electron is initially at rest so that $$\vec{p_1}=0$$ and then the momentum of the photon and the momentum of the electron after the interaction would have to be equal. This then leads to the nonsensical result that either $$p_2=0$$ or $$m_e c^2 = 0$$. This is probably a more elegant proof.

• By george, I think he's got it. Commented Dec 23, 2015 at 10:22
• +1. Does this imply that in Compton scattering, the electron is 'off-mass shell'? I was wondering, how to avoid confusions (like this) between conservation laws of asymptotic scattering (well defined incoming fourvectors), and a virtual processes inside a diagram (it is commonly called absorption too inside a diagram, I believe -- "In compton scattering the electron is absorbed and emitted", people might say). Commented Dec 23, 2015 at 10:27
• This answer assumes that the mass of the electron is the same before and after the absorption. What justifies that? Commented Dec 24, 2015 at 14:04
• @WillO The rest mass of an electron is invariant. Commented Dec 24, 2015 at 14:50
• @RobJeffries: I apologize in advance for the fact that if I were better educated I wouldn't have to ask this, but: Why can't the combined electron-plus-photon have a new rest mass that allows both energy and momentum to be conserved? If electrons are defined in such a way that their mass is invariant, then we are not allowed to call this new particle an electron. But I still don't see why it can't exist. Commented Dec 24, 2015 at 14:57

The electrons in a Free-electron laser aren't actually free in the sense used for the question. They are tightly controlled by an oscillating magnetic field, so that they emit coherent radiation. An electron can't absorb a photon all by itself, because you can't get conservation of energy and momentum with the electron alone. (source)

The book is wrong. See Compton scattering.

• In Compton scattering a photon with a different momentum is emitted. Commented Dec 23, 2015 at 10:06
• ...but a photon is also absorbed? Maybe the point of the 'lecturers notes' is related to conservation of some quantity (angular momentum?), which can be broken by virtual particles. In any case, "Free electron can't absorb a photon" is a bad way to represent a thing like that. Commented Dec 23, 2015 at 10:10
• Mmm maybe he means that it can't be completely absorbed but partially such that the amount absorbed turns as a kinetic energy for the electron, right?
– user65035
Commented Dec 23, 2015 at 10:10
• 'inside an electron' is also ill defined :) If photon is absorbed, it just ceases to exists. It is a Boson, and the number of Bosons need not to be conserved (unlike charge, baryon number etc). Commented Dec 23, 2015 at 10:21
• This answer is not complete enough to be useful, in my opinion.
– Danu
Commented Dec 23, 2015 at 10:22