Photon absorption by free electron during Inverse bremsstrahlung: Why is a heavy particle needed?

Question

I read about the process of inverse bremsstrahlung where a free electron can gain kinetic energy by absorbing a photon.

However, I'm having some trouble to understand why exactly a heavy particle must take part in this process, c.f.

The momentum conservation law requires this process can proceed only in the presence of an ion, which carries the extra momentum...

from google books

Which extra momentum is the author talking about? The photon's?

btw: sorry for the useless tags, I'm not a physicist..

EDIT: I kind of get it, I guess:

before absorption: electron energy : $0.5 m_e v_1^2$

electron momentum : $m_e v_1$

photon energy:$\hbar\omega$

photon momentum: $\hbar\omega/c$

total momentum: $\hbar\omega/c + m_e v_1$

After absorption:

electron energy: $1/2 m_e v_2^2$ where $v_2^2=v_1^2 + 2 \hbar\omega/m_e$

electron momentum: $m_e v_2=m_e \sqrt{v_1^2 + 2 \hbar\omega/m_e}$

which is probably more than $\hbar\omega/c + m_e v_1$ ? or isn't it?

Answer 1

Let's consider the inverse bremsstrahlung in vacuum: this process can be written out as

$$\gamma + e^- \to e^-$$

A process such as this does not conserve 4-momentum. One can easily see this by going in the center of mass frame: the photon and the electron hit head to head and the outgoing particle goes somewhere at an angle with respect to the collision axis. In the center of mass frame the total momentum is zero and, by conservation, the total momentum of the outgoing particles should be zero. But if the particle is only one it is clearly impossible.

The statement

The momentum conservation law requires this process can proceed only in the presence of an ion, which carries the extra momentum.

implies exactily this: the ion takes away the momentum in such a way that the total momentum after the collision is again zero, in the center of mass frame. The right process is then

$$\gamma + X \to e^-+X^+$$

in this way the colliding particles are the photon and the atom and the outgoing one are an electron and the ion. Obviously the atom will recoil very little since it is very massive, while the electron will take most of the energy.

Answer 2

There are two main conservation laws to obey here:

1. energy

Electrons and photons are QM entities, they obey QM laws. If the electron is accelerated then the laws can be obeyed through interaction with a third party, that might be the external EM field (through virtual particles) or in your case, a nucleus (ion). A free electron does have a rest mass and does not have excited states it could go to, so it cannot take up the energy of the photon.

Total absorption would mean an incoming photon+ electron , and outgoing only an electron. This cannot happen because the electron has a fixed mass and does not have excited states to absorb all the energy of the photon. What can happen is that most of the energy of the photon becomes kinetic energy of the electron, in any inertial frame, and correspondingly the photon can have very small energy , tending to zero but never zero. If the outgoing (or incoming) photon becomes virtual, connecting with an electric or magnetic field, then the kinematics has to include the originator of the field in energy momentum considerations, and the electron can absorb all the energy of the incoming photon the energy/momentum balance in its rest mass system taken up by the generator of the field that gave the virtual photon.

Can an accelerated "free" electron absorb a photon?

2. Momentum

The real reason this is prohibited with a free electron is that all the four momenta of the electron before, photon before and electron after cannot lie on their mass shell simultaneously.

The four-momentum of any real particle satisifies the relationship pμpμ=−m2. This defines a 3-D surface in the 4-D space of all possible four-momenta; this surface is called the mass shell for the particle. A virtual particle, on the other hand, can have any four-momentum vector that you want; a virtual particle is usually "off-shell", because its four-momentum doesn't lie on the mass shell.

Can a free electron absorb a virtual photon even though it cannot absorb an ordinary photon?

The answer to your question is that the nucleus (ion) will recoil (take the momentum of the photon).