# Proof no photoelectric effect on a free electron

## Homework Statement

The question asks me to prove that the photoelectric effect cannot occur with a free electron. ie. one not bound to an atom. A hint is also provided: Consider the reference frame in which the total momentum of the electron and incident photon are zero.

## The Attempt at a Solution

I've been thinking about how to prove this for like a week, but i cannot figure out how to do so. I know that in the case of a free electron, we will have a work function potential of 0V. As for the hint, I cannot really figure out what it is trying to get me to do. In the zero momentum frame, we are moving at a velocity that gives the electron a momentum equal to the momentum of the photon. This direction is the same direction that the photon moves in. The electron momentum will be the same, since p=E/c, which is a constant after an energy is predetermined in all frames (right?). After the collision (or absorption in this case), in this frame, the electron should stay stationary, since this is the zero momentum frame. If I were to move back to the lab frame, I would see the electron moving in the same direction as the photon was moving.

This actually brings me to more confusion. Why don't we just get Compton scattering at this point? Why should the photon be absorbed by the electron anyways?

If anyone could please push me into the right direction I would appreciate it greatly!

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Matterwave
Gold Member
I think the question is really asking whether a free electron can in fact absorb a photon, rather than just scatter it (e.g. Compton scattering or Thompson scattering). Can momentum be conserved in that frame of reference when the photon will disappear after absorption?

I actually haven't done these calculations myself, so if I'm leading you on the wrong path, I'm sorry. =P

ideasrule
Homework Helper
Matterwave is right in that that's what the question is asking, since the photoelectric effect is when a photon gets absorbed by an electron.

That said, you deduced correctly that the electron has to be stationary. But energy has to be conserved too, so where did all the energy go?

It wouldnt have anywhere to go if it is absorbed. so that is why a photoelectric effect is wrong.
so I can say:

Before in CM:
$$E_{CM,before}=E_{\gamma}+E_{e}$$

$$E_{e}=\sqrt{(pc)^{2}+(mc^{2})^2}$$

but, the momentum of the electron is the same as of the photon

$$E_{e}=\sqrt{E_{\gamma}^{2}+(mc^{2})^2}$$

$$E_{CM,before}=E_{\gamma}+\sqrt{E_{\gamma}^{2}+(mc^{2})^2}$$

After:

$$E_{CM,after}=mc^{2}$$

but
$$E_{CM,after}$$ does not equal $$E_{CM,before}$$ so a electron cannot not absorb a photon?

ideasrule
Homework Helper
Yes. Even simpler, you can say that the kinetic energy before the absorption is non-zero but the kinetic energy after is zero, which is impossible.

Although as soon as you break translation invariance this problem disappears!

Have you tried to see if the relativistic four momentum is conserved in this frame?

I think I got it. Before the encounter, the total four momentum in said frame is 0, after it it will be something not zero due to the rest mass of the electron. I haven't really thought about it, but maybe this helps.