In particle physics, the Klein–Nishina formula gives the differential cross section (i.e. the "likelihood" and angular distribution) of photons scattered from a single free electron, calculated in the lowest order of quantum electrodynamics. It was first derived in 1928 by Oskar Klein and Yoshio Nishina, constituting one of the first successful applications of the Dirac equation. The formula describes both the Thomson scattering of low energy photons (e.g. visible light) and the Compton scattering of high energy photons (e.g. x-rays and gamma-rays), showing that the total cross section and expected deflection angle decrease with increasing photon energy. In quantum field theory it is known as Klein–Nishina–Tamm formula, adding the name of Igor Tamm who derived the formula from field quantization.
Formula For an incident unpolarized photon of energy E γ {\displaystyle E_{\gamma }} , the differential cross section is:
d σ d Ω = 1 2 r e 2 ( λ λ ′ ) 2 [ λ λ ′ + λ ′ λ − sin 2 ( θ ) ] {\displaystyle {\frac {d\sigma }{d\Omega }}={\frac {1}{2}}r_{e}^{2}\left({\frac {\lambda }{\lambda '}}\right)^{2}\left[{\frac {\lambda }{\lambda '}}+{\frac {\lambda '}{\lambda }}-\sin ^{2}(\theta )\right]}
where
r e {\displaystyle r_{e}} is the classical electron radius (~2.82 fm, r e 2 {\displaystyle r_{e}^{2}} is about 7.94 × 10−30 m2 or 79.4 mb)
λ / λ ′ {\displaystyle \lambda /\lambda '} is the ratio of the wavelengths of the incident and scattered photons
θ {\displaystyle \theta } is the scattering angle (0 for an undeflected photon). The angular dependent photon wavelength (or energy, or frequency) ratio is
λ λ ′ = E γ ′ E γ = ω ′ ω = 1 1 + ϵ ( 1 − cos θ ) {\displaystyle {\frac {\lambda }{\lambda '}}={\frac {E_{\gamma '}}{E_{\gamma }}}={\frac {\omega '}{\omega }}={\frac {1}{1+\epsilon (1-\cos \theta )}}}
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