Klein–Nishina formula

The Klein–Nishina formula gives the differential cross section of photons scattered from a single free electron in lowest order of quantum electrodynamics.[1] At low frequencies (e.g., visible light) this yields Thomson scattering; at higher frequencies (e.g., x-rays and gamma-rays) this yields Compton scattering.

Klein–Nishina distribution of scattering-angle cross sections over a range of commonly encountered energies.

For an incident unpolarized photon of energy ${\displaystyle E_{\gamma }}$, the differential cross section is:[2]

${\displaystyle {\frac {d\sigma }{d\Omega }}={\frac {1}{2}}\alpha ^{2}r_{c}^{2}P(E_{\gamma },\theta )^{2}[P(E_{\gamma },\theta )+P(E_{\gamma },\theta )^{-1}-\sin ^{2}(\theta )]}$

where ${\displaystyle {\frac {d\sigma }{d\Omega }}}$ is a differential cross section, ${\displaystyle d\Omega }$ is an infinitesimal solid angle element, ${\displaystyle \alpha }$ is the fine structure constant (~1/137.04), ${\displaystyle \theta }$ is the scattering angle; ${\displaystyle r_{c}=\hbar /m_{e}c}$ is the "reduced" Compton wave length of the electron (~0.38616 pm); ${\displaystyle m_{e}}$ is the mass of an electron (~511 keV${\displaystyle /c^{2}}$); and ${\displaystyle P(E_{\gamma },\theta )}$ is the ratio of photon energy after and before the collision:

${\displaystyle P(E_{\gamma },\theta )={\frac {1}{1+(E_{\gamma }/m_{e}c^{2})(1-\cos \theta )}}={\frac {\lambda }{\lambda '}}}$

Note that this result may also be expressed in terms of the classical electron radius ${\displaystyle r_{e}=\alpha r_{c}}$:

${\displaystyle {\frac {d\sigma }{d\Omega }}={\frac {1}{2}}r_{e}^{2}\left({\frac {\lambda }{\lambda '}}\right)^{2}\left[{\frac {\lambda }{\lambda '}}+{\frac {\lambda '}{\lambda }}-\sin ^{2}(\theta )\right]}$

While this classical quantity is not particularly relevant in quantum electrodynamics, it is easy to appreciate: in the forward direction (for ${\displaystyle \theta }$ ~ 0), photons scatter off electrons as if these were about ${\displaystyle r_{e}=\alpha r_{c}}$ (~2.8179 fm) in linear dimension, and ${\displaystyle r_{e}^{2}}$ (~ 7.9406x10⁻³⁰ m² or 79.406 mb) in size.

If the incoming photon is polarized, the scattered photon is no longer isotropic with respect to the azimuthal angle. For a linearly polarized photon scattered with a free electron at rest, the differential cross section is instead given by:

${\displaystyle {\frac {d\sigma }{d\Omega }}={\frac {1}{2}}r_{e}^{2}\left({\frac {\lambda }{\lambda '}}\right)^{2}\left[{\frac {\lambda }{\lambda '}}+{\frac {\lambda '}{\lambda }}-2\sin ^{2}(\theta )\cos ^{2}(\phi )\right]}$

where ${\displaystyle \phi }$ is the azimuthal scattering angle. Note that the unpolarized differential cross section can be obtained by averaging over ${\displaystyle \cos ^{2}(\phi )}$.

The Klein–Nishina formula was derived in 1928 by Oskar Klein and Yoshio Nishina, and was one of the first results obtained from the study of quantum electrodynamics. Consideration of relativistic and quantum mechanical effects allowed development of an accurate equation for the scattering of radiation from a target electron. Before this derivation, the electron cross section had been classically derived by the British physicist and discoverer of the electron, J.J. Thomson. However, scattering experiments showed significant deviations from the results predicted by the Thomson cross section. Further scattering experiments agreed perfectly with the predictions of the Klein–Nishina formula.

Note that if ${\displaystyle E_{\gamma }\ll m_{e}c^{2}}$, ${\displaystyle P(E_{\gamma },\theta )\rightarrow 1}$ and the Klein–Nishina formula reduces to the classical Thomson expression.

The final energy of the scattered photon, ${\displaystyle E_{\gamma }'}$, depends only on the scattering angle and the original photon energy, and so it can be computed without the use of the Klein–Nishina formula:

${\displaystyle E_{\gamma }'(E_{\gamma },\theta )=E_{\gamma }\cdot P(E_{\gamma },\theta )\,}$