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| [[Image:Klein-Nishina distribution.png|thumb|Klein–Nishina distribution of scattering-angle cross sections over a range of commonly encountered energies.]]
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| The '''Klein–Nishina formula'''<ref>{{cite journal |
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| doi=10.1007/BF01366453 |
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| author=Klein, O; Nishina, Y |
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| title=Über die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac|journal=Z. Phys.|volume=52 |
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| issue=11-12 |year=1929|pages=853 and 869}}</ref>
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| gives the [[differential cross section]] of photons scattered from a single free electron in lowest order of
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| [[quantum electrodynamics]]. At low frequencies (e.g., [[visible light]]) this is referred to as
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| [[Thomson scattering]]; at higher frequencies (e.g., [[x-ray]]s and [[gamma-ray]]s) this is referred
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| to as [[Compton scattering]].
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| For an incident photon of energy <math>E_\gamma</math>, the differential [[Cross section (physics)|cross section]] is:<ref>{{cite book| author=Weinberg, Steven | title=The Quantum Theory of Fields|volume= I|year=1995|pages=362–9}}
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| </ref>
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| : <math> \frac{d\sigma}{d\Omega} = \alpha^2 r_c^2 P(E_\gamma,\theta)^2 [P(E_\gamma,\theta) + P(E_\gamma,\theta)^{-1} -1 + \cos^2(\theta)]/2 </math>
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| where <math>\frac{d\sigma}{d\Omega}</math> is a [[differential cross section]], <math>d\Omega</math> is an infinitesimal solid angle[[volume element| element]], <math>\alpha</math> is the [[fine structure constant]] (~1/137.04), <math>\theta</math> is the scattering angle; <math>r_c=\hbar/m_e c</math> is the "reduced" [[Compton wave length]] of the electron (~0.38616 pm); <math>m_e</math> is the mass of an electron (~511 keV<math>/c^2</math>); and <math>P(E_\gamma,\theta)</math> is the ratio of photon energy after and before the collision:
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| : <math>
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| P(E_\gamma,\theta) = \frac{1}{1 + (E_\gamma/m_e c^2)(1-\cos\theta)}
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| </math>
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| Note that this result may also be expressed in terms of the
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| [[classical electron radius]] <math>r_e=\alpha r_c</math>. While this classical quantity is not particularly relevant
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| in quantum electrodynamics, it is easy to appreciate: in the forward direction (for <math>\theta</math> ~ 0), photons scatter off electrons as if these were about <math>r_e=\alpha r_c</math> (~2.8179 fm) in linear dimension, and <math>r_e^2</math> (~ 7.9406x10<sup>−30</sup> m<sup>2</sup> or 79.406 mb) in size.
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| 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.
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| Note that if <math>E_\gamma \ll m_ec^2</math>,
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| <math>P(E_\gamma,\theta)\rightarrow 1</math> and the Klein–Nishina formula reduces to the classical Thomson expression.
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| The final energy of the scattered photon, <math>E_\gamma'</math>, 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:
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| : <math> | |
| E_\gamma'(E_\gamma,\theta) = E_\gamma \cdot P(E_\gamma, \theta) \,
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| </math>
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| ==Notes==
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| {{reflist}}
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| ==External references==
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| * R. D. Evans, The Atomic Nucleus, McGraw-Hill, New York, 1955, pp. 674–676.
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| * A. C. Melissinos, Experiments in Modern Physics, Academic Press, New York, 1966, pp. 252–265
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| * 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.
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| {{DEFAULTSORT:Klein-Nishina formula}}
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| [[Category:Quantum field theory]]
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