Inverse Compton Scattering

When an electron is not at rest, but colliding with a soft photon, the recoiling photon gains its energy by the transfer from the incoming electrons. This process called Inverse Compton Scattering, $e\gamma\to e\gamma$, is a representative radiation in many astronomical environments that produces high energy X-ray/$\gamma$-ray.

The colliding energy between a photon and an electron is often defined by the he Lorentz invariant center of mass (CM) energy squared $s$ as defined by

$$s = (P^{\mu}+k_{\mu})^2.$$ (26)

Here $P^{\mu}=(E,{\bf P})$ and $k^{\mu}=(E_{rad},{\bf k})$ are 4-momentum of the electron and the photon respectively. $s$ can be rewritten in the different system:

$$s=\left\{ \begin{array}{ll} (\ast{E}+\ast{\epsilon_{\gamma}}... ...\theta & {\sf (Universe lab system)}\\ \end{array} \right.$$ (27)

Here natural unit is used i.e. $c=\hbar=1$. In this expression, the photon energy $\epsilon_{\gamma}^{'}$ in the electron rest system in Eq. 25 is written by

$$\epsilon_{\gamma}^{'}={s-m_e^2\over 2m_e}$$ (28)

in the Lorentz invariant form. The Lorentz transformation gives the relations between energies in the electron rest system and those in Universe lab system when incoming electron energy is by far higher than the target photon. Then one can calculate energy of the recoiling electron $E_e^{rec}$ as a function of incoming electron energy $E_e$ in the lab system as

$$E_e^{rec} \simeq E_e-\epsilon_{\gamma}^{rec} = E_e-{E_e\ove... ...= {E_e\over 1+ {\epsilon^{'}_{\gamma}\over m_e}(1-\cos\theta)}$$ (29)

where $\epsilon_{\gamma}^{rec}$ is the recoiling energy of $\gamma$-ray in the Universe lab system. By using Eq. 29 we can modify the Kline-Nishina formula Eq. 25 to derive the energy differential cross section. We obtain

$${d\sigma\over d\chi} = {3\over 8}\sigma_T {2m_e^2\over s-m_e^2}[ \chi +{1\over \chi} + \left({2m_e^2\over s-m_e^2}\right)^2\left(1-{1\over\chi}\right)^2$$

$$+ 2\left({2m_e^2\over s-m_e^2}\right)\left(1-{1\over\chi}\right)].$$ (30)

Here $\chi\equiv E_e^{rec}/E_e$ is the dimensionless energy of the electron normalized by primary energy of incoming electron. The electron transfer most of its energy to the photon via the inverse Compton scattering as you see the cross section becomes larger in $\chi\ll 1$ regime in Eq. 30. Integrating $d\sigma/d\chi$ from $m_e^2/s$ to 1 (this range is given by Eq. 29 with $-1\leq\cos\theta\leq 1$), we obtain the total cross section:

$$\sigma_{IC} = {3\over 8}\sigma_T{m_e^2\over s\beta} [{2\over \beta(1+\beta)}(2+2\beta-\beta^2-2\beta^3)$$

$$-{1\over \beta^2}(2-3\beta^2-\beta^3)\ln{1+\beta\over 1-\beta}],$$ (31)

where $\beta\equiv (s-m_e^2)/(s+m_e^2)$ is the velocity of the outgoing electron in the CMS.