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Energy spectrum of backward scattered photons in the Compton process.

As already mentioned in the previous section, the effect of Compton scattering of photons through $180^\circ$ (backward scattering) opens the possibility to generate intensive high energy photon beams. In section we describe the $\gamma e$ and $\gamma\gamma$ collider projects in some detail. Here we use our results for Compton scattering to calculate the energy spectrum of backward scattered photons in the case of unpolarized incident laser photons and electrons.

We proceed again from Eqs. (1) and (2) and as before in the derivation of the Klein-Nishina formula, we express the Mandelstam variables in terms of the kinematical variables of the appropriate reference frame. If we denote the 4-momenta of the incident photon, incident electron and scattered photon, respectively, by $p_\gamma=(E_\gamma, 0,0, E_\gamma)$, $p_e=(E_e, 0,0, -P_e)$ and $p_\gamma^\prime =(E_\gamma^\prime, p_{\gamma x}^\prime, p_{\gamma y}^\prime, E_\gamma^\prime\cos\theta_\gamma)$, then we get

$$s = (p_\gamma + p_e)^2 = m_e^2 + 2 E_\gamma (E_e+P_e)$$

$$t = (p_\gamma - p_\gamma^\prime)^2 = -2 E_\gamma E_\gamma^\prime (1-\cos\theta_\gamma)$$

$$u = (p_e - p_\gamma^\prime)^2 = m_e^2 - 2 E_\gamma^\prime (E_e +P_e\cos\theta_\gamma)$$ (3)

and using the identity $s+t+u=2 m_e^2$ we find

$$\cos\theta_\gamma = \frac{s-m_e^2 - 2E_\gamma^\prime (E_e+E_\gamma)} {2E_\gamma^\prime (P_e-E_\gamma)}$$ (4)

and substituting this into the above expression for $u$ we get

$$m_e^2-u=(s-m_e^2)\frac{P_e-E_\gamma^\prime}{P_e-E_\gamma}$$

and hence ${\displaystyle du =(s-m_e^2)(P_e-E_\gamma)^{-1} dE_\gamma^\prime}$.

Finally we define dimensionless kinematical variables of photon backward scattering

$$x=\frac{4 E_{\gamma} E_e}{m_e^2} \qquad\mbox{and} \qquad y=\frac{E'_{\gamma}}{E_e},$$

and hence we get the following result for the photon spectrum:

$$\frac{d\sigma}{dy}= \frac{\pi \alpha^2}{2 E_eE_\gamma} \left... ...c{4y^2}{x^2(1-y)^2}-\frac{4y}{x(1-y)}+1-y+\frac{1}{1-y}\right]$$ (5)

where we have used the approximations

$$\frac{2 E_\gamma^\prime E_e}{m_e^2}(\cos \theta_\gamma +1) \; \approx \; x(1-y) - y,$$

$$s \approx m_e^2 (1+x), \qquad u \approx m_e^2 [1-x(1-y)].$$

From Eq. (4) we can express $E_\gamma^\prime$ in terms of $\cos\theta_\gamma$, and putting $\theta_\gamma=\pi$ we get the maximum value of $E_\gamma^\prime$ and hence also of $y$:

$$y_{\mbox{max}}=1-m_e^2/s$$

where we have assumed that $P_e\gg m_e$. Thus $y_{\mbox{max}}$ is close to one and increases with increasing electron beam energy $E_e$. The formula for the photon energy spectrum, Eq. (5), shows that the spectrum is strongly peaking at $y_{\mbox{max}}$. This fact, that the maximum intensity of the scattered photons occurs at the position of their maximum energy, makes it favourable to use this arrangement to produce high energy photon beams of sharply defined energy.