Klein-Nishina formula

The Klein-Nishina formula provides an accurate prediction of the angular distribution of x-rays and gamma-rays which are incident upon a single electron. The Klein-Nishina formula describes incoherent or Compton scattering.

More precisely, the Klein-Nishina formula provides the differential cross section with respect to solid angle of scattering, and it accounts for factors such as radiation pressure and relativistic quantum mechanics.

For an incident photon of energy $$E_gamma, the differential cross section is:
$$frac{dsigma}{dOmega} = frac{1}{2} r_e^2 (P(E_gamma, heta) - P(E_gamma, heta)^2 sin^2( heta) + P(E_gamma, heta)^3)
where $$heta is the scattering angle; $$r_e is the classical electron radius; $$m_e is the mass of an electron; and $$P(E_gamma, heta) is the ratio of photon energy after and before the collision:
$$P(E_gamma, heta) = frac{1}{1 + frac{E_gamma}{m_e c^2}(1-cos heta)}
The value $$dsigma is the differential cross section for a photon scattering into the solid angle defined by $$dOmega = 2 pi sin heta d heta.

The Klein-Nishina formula was derived in 1929 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 $$E_gamma << m_ec^2, $$frac {E_gamma}{m_ec^2} ightarrow 0, the Klein-Nishina formula reduces to the classical Thomson expression.

The final energy of the scattered photon, $$E_gamma', is entirely dependent upon the scattering angle and the original photon energy, and so it can be computed without the use of the Klein-Nishina formula:

$$E_gamma'(E_gamma, heta) = E_gamma cdot P(E_gamma, heta)

Notes and references

R. D. Evans, The Atomic Nucleus, McGraw-Hill, New York, 1955, pp. 674–676.

A. C. Melissinos, Experiments in Modern Physics, Academic Press, New York, 1966.

O. Klein and Y. Nishina, On the Scattering of Radiation by Free Electrons According to Dirac's New Relativistic Quantum Dynamics,The Oskar Klein Memorial Lectures, Vol. 2: Lectures by Hans A. Bethe and Alan H. Guth with Translated Reprints by Oskar Klein, Ed. Gösta Ekspong, World Scientific Publishing Co. Pte. Ltd., Singapore, 1994, pp. 113-139.