PX156 - A5 - the compton effect

the inelastic scattering of a photon from a charged particle
the scattered photon loses energy to the particle
the degree of energy loss is dependent on the angle of scattering

\begin{matrix} E_{e} = e V \\ E_{p} = h f_{m a x} = \frac{h c}{λ_{m i n}} \end{matrix}

consider an x-ray photon scattering off a free stationary electron
conserve momentum:

\begin{aligned} {\vec{p}}_{1} & = {\vec{p}}_{2} + {\vec{p}}_{e} \\ p_{e}^{2} & = p_{1}^{2} + p_{2}^{2} - 2 p_{1} p_{2} \cos θ \end{aligned}

conserve energy:

\begin{matrix} p_{1} c + E_{0} = p_{2} c + \sqrt{(p_{e} c)^{2} + E_{0}^{2}} \\ (p_{1} c - p_{2} c + E_{0})^{2} = (p_{e} c)^{2} + E_{0}^{2} \\ p_{1}^{2} c^{2} + p_{2}^{2} c^{2} + E_{0}^{2} + 2 p_{1} c E_{0} - 2 p_{2} c E_{0} - 2 p_{1} p_{2} c^{2} = p_{e}^{2} c^{2} + E_{0}^{2} \\ \frac{E_{0}}{c} (p_{1} - p_{2}) = p_{1} p_{2} (1 - \cos θ) \\ m_{e} c h (\frac{c}{λ_{1}} - \frac{c}{λ_{2}}) = \frac{h^{2}}{λ_{1} λ_{2}} (1 - \cos θ) \\ Δ λ = λ_{2} - λ_{1} = \frac{h}{m_{e} c} (1 - \cos θ) \end{matrix}

compton wavelength:

λ_{c} = \frac{h}{m c} = 0.024 Å

Δ λ_{m a x} |_{θ = 180 °} = 2 λ_{c} \approx 0.05 Å

$λ_{v i s i b l e l i g h t} \approx (4 - 7) \times 10^{- 7} m$
$λ_{x - r a y s} \approx 0.1 - 100 Å : Δ λ$ could be significant