Answer
A moving electron can not spontaneously change into an x-ray photon in free space. A third particle must be present in the interaction process to conserve the momentum of the system.
Work Step by Step
We first consider that a moving electron having energy $E$ and momentum $p_e$ spontaneously change into an x-ray photon in free space. For, the electron we can write,
$E^2=(p_ec)^2+(m_0c^2)^2$
or, $p_ec=\sqrt {E^2-(m_0c^2)^2}$
or, $p_e=\sqrt {(\frac{E}{c})^2-m_0^2c^2}$ ..............................$(1)$
According to the law of conservation of energy, the energy should be conserved. Therefore, the photon must have the energy $E$. If $E$ be the energy of the photon, the momentum of the photon will be
$p_p=\frac{E}{c}$ ..............................$(2)$
Eq. $1$ and Eq. $2$ imply that $p_e\lt p_p $, that is, the momentum of the system is not conserved, which is impossible. Therefore, a third particle must present in the interaction process to conserve the momentum.
Thus, a moving electron can not spontaneously change into an x-ray photon in free space. A third particle must be present in the interaction process to conserve the momentum of the system.
