Compton Scattering Kinematics

What you are seeing: a photon of incident wavelength $\lambda$ scatters off a free electron at scattering angle $\theta$, and emerges with wavelength $\lambda' = \lambda + (h/m_e c)(1 - \cos\theta)$. The Compton wavelength $h/m_e c \approx 2.426$ pm is the universal length that sets the shift. Forward scatter ($\theta = 0$) gives no shift; backscatter ($\theta = \pi$) gives the maximum shift $2\lambda_C$. Right-angle scatter gives exactly $\lambda_C$.

The recoiling electron carries the missing energy $T = hc(\lambda^{-1} - \lambda'^{-1})$. Its recoil angle $\phi$ relative to the incident photon satisfies $\cot\phi = (1 + \alpha)\tan(\theta/2)$ with $\alpha = \lambda_C/\lambda$, so for $\theta \to \pi$ the electron goes straight forward, and for $\theta \to 0$ the electron barely moves.