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..
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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 \begin{equation} p_{1} + p_{\nu} = p_{2}, \tag{1} \end{equation} where $p_1$ and $p_2$ are the momentum of the electron before and after the interaction. Conservation of energy gives us \begin{equation} \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} \end{equation} 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.
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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)
