# If photons can be absorbed by electrons, why can photons not be absorbed by free electrons?

Because it is impossible to do so while simultaneously conserving energy, momentum, and energy.

The following system is lovingly created in Paint.

A photon of energy is incident upon a stationary electron* of mass. The photon is absorbed by the electron and it begins moving with momentum

Let’s now look at this in terms of a Four vector

. The momentum 4 vector contains energy in the zeroth and third elements. Elements 1 through 3 are your usual 3-momentum vectors.

If the unit is divided into units, the first element should always be

This is how the invariant product is obtained. However, if you calculate it explicitly, we get:

If the particle is stationary, then, as follows:

4-Momentum is conserved (which encompasses both energy and momentum conservation), so if we have a 4-vector describing the incident photon (), and one describing the stationary electron (), then their sum must be equal to the one describing the outgoing electron ():

We now know what they are.

(The electron is stationary so its energy is only. However, we are in units where

We don’t really know what it is, but we don’t have to!

We know that it must be square!

We get the following:

We can see again that photons are not massless by using the fact that we have already seen. This cancels out with the on-the-right hand side of equation, leaving only:

If we do the dot product between the two vectors we know, we get:

….

Well….bugger.

Yep — the only way that we can conserve both energy and momentum is if the photon has zero energy.

A photon that has zero energy is ….nothing.

A photon cannot be absorbed by a free electron.

This is all very formal mathematics — but there’s actually a more intuitive way to see this!

This is the reverse of what happens: An electron travels along and emits one photon with enough energy to stop it from moving.

The laws of physics at this level are time-invariant, so it is exactly the same as before, but in the opposite direction.

Consider the frame where the original electron was resting. Now consider this: A single electron with energy suddenly emits a photon in the foreward direction and begins moving in the reverse direction.

The same event is shown, but from different angles.

Except…where did the energy go make the photon and the electron move in the rest frame?

A stationary electron was the source of our energy, and then suddenly, a boatload of it appeared out of thin air. Because its mass hasn’t changed, the energy of the moving electron must be higher than.

This process cannot be done!

There are processes that can occur, however.

If the interaction results with a photon being scattered somewhere else, then Compton scattering

:

Here, because you have multiple components on both sides of the process, energy conservation isn’t violated — and the process is perfectly allowed!

Other such processes involve the transfer of momentum or energy to other particles. For example, Bremsstrahlung radiation is an example where a nearby nuclei plays this role.

However, atoms can absorb photons due to their internal structure, which stores the energy. An electron can be excited up to an energy level. We think that electrons have no such internal structure, so single-photon absorption/emission is not possible.

This would prove that photons do not exist as fundamental particles. However, the absence of such evidence severely limits what this internal structure might be.

*If its not stationary, find a reference frame where it is stationary — your photon energy will change, but the physics is the same in all frames.