View source for Physics:Klein–Nishina formula

[[Image:Klein-Nishina distribution-en.svg|thumb|Klein–Nishina distribution of scattering-angle cross sections over a range of commonly encountered energies.]]
{{short description|Electron-photon scattering cross section}}

In [[Physics:Particle physics|particle physics]], the '''Klein–Nishina formula''' gives the differential cross section (i.e. the "likelihood" and angular distribution) of [[Physics:Photons|photons]] [[Physics:Scattering|scattered]] from a single free [[Physics:Electron|electron]], calculated in the lowest order of [[Physics:Quantum electrodynamics|quantum electrodynamics]]. It was first derived in 1928 by [[Biography:Oskar Klein|Oskar Klein]] and Yoshio Nishina, constituting one of the first successful applications of the [[Physics:Dirac equation|Dirac equation]].<ref>{{cite journal |doi=10.1007/BF01366453 |author1=Klein, O |author2=Nishina, Y |title=Über die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac|journal=Z. Phys.|volume=52 |issue=11–12 |year=1929|pages=853 and 869|bibcode = 1929ZPhy...52..853K |s2cid=123905506 }}</ref> The formula describes both the [[Physics:Thomson scattering|Thomson scattering]] of low energy photons (e.g. [[Physics:Visible light|visible light]]) and the [[Physics:Compton scattering|Compton scattering]] of high energy photons (e.g. [[Physics:X-ray|x-ray]]s and gamma-rays), showing that the total cross section and expected deflection angle decrease with increasing photon energy.

== Formula ==
For an incident unpolarized photon of energy <math>E_\gamma</math>, the differential cross section is:<ref>{{cite book| author=Weinberg, Steven | title=The Quantum Theory of Fields|volume= I|year=1995|pages=362–9}}
</ref>

: <math> \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] </math>

where

* <math>r_e</math> is the [[Physics:Classical electron radius|classical electron radius]] (~2.82 [[Femtometre|fm]], <math>r_e^2</math> is about 7.94 × 10<sup>−30</sup> m<sup>2</sup> or 79.4 [[Physics:Barn (unit)|mb]])
* <math>\lambda/\lambda'</math> is the ratio of the wavelengths of the incident and scattered photons
* <math>\theta</math> is the scattering angle (0 for an undeflected photon).

The angular dependent photon wavelength (or energy, or frequency) ratio is
: <math>
\frac{\lambda}{\lambda'} = \frac{E_{\gamma'}}{E_\gamma} = \frac{\omega'}{\omega} = \frac{1}{1 + \epsilon(1-\cos\theta)}
</math>

as required by the conservation of [[Physics:Four-momentum|relativistic energy and momentum]] (see [[Physics:Compton scattering|Compton scattering]]). The dimensionless quantity <math>\epsilon = E_\gamma/m_e c^2</math> expresses the energy of the incident photon in terms of the electron rest energy (~511 [[Physics:Electronvolt|keV]]), and may also be expressed as <math>
\epsilon = \lambda_c/\lambda
</math>, where <math>\lambda_c = h/m_e c</math> is the [[Physics:Compton wavelength|Compton wavelength]] of the electron (~2.42 pm). Notice that the scatter ratio <math>
\lambda'/\lambda
</math> increases [[Monotonic function|monotonically]] with the deflection angle, from <math>
1
</math> (forward scattering, no energy transfer) to <math>
1+2\epsilon
</math> (180 degree backscatter, maximum energy transfer). 

In some cases it is convenient to express the classical electron radius in terms of the Compton wavelength: <math>r_e=\alpha \bar\lambda_c = \alpha \lambda_c/2\pi</math>, where <math>\alpha</math> is the fine structure constant (~1/137) and <math>\bar\lambda_c=\hbar/m_e c</math> is the ''reduced'' Compton wavelength of the electron (~0.386 pm), so that the constant in the cross section may be given as:
: <math>
\frac{1}{2}r_e^2 = \frac{1}{2}\alpha^2\bar\lambda_c^2 = \frac{\alpha^2\lambda_c^2}{8\pi^2} = \frac{\alpha^2\hbar^2}{2m_e^2c^2}
</math>

== Polarized photons ==
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:
: <math> \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] </math>
where <math>\phi</math> is the azimuthal scattering angle. Note that the unpolarized differential cross section can be obtained by averaging over <math>\cos^2 (\phi)</math>.

== Limits ==

=== Low energy ===
For low energy photons the wavelength shift becomes negligible (<math>\lambda/\lambda'\approx 1</math>) and the Klein–Nishina formula reduces to the classical [[Physics:Thomson scattering|Thomson expression]]:
: <math> \frac{d\sigma}{d\Omega} \approx \frac{1}{2} r_e^2  \left(1 + \cos^2(\theta)\right) \qquad (\epsilon \ll 1) </math>
which is symmetrical in the scattering angle, i.e. the photon is just as likely to scatter backwards as forwards. With increasing energy this symmetry is broken and the photon becomes more likely to scatter in the forward direction.

=== High energy ===
For high energy photons it is useful to distinguish between small and large angle scattering. For large angles, where <math>\epsilon(1-\cos\theta) \gg 1</math>, the scatter ratio <math>\lambda'/\lambda</math> is large and
: <math> \frac{d\sigma}{d\Omega} \approx \frac{1}{2} r_e^2 \frac{\lambda}{\lambda'} \approx \frac{1}{2} r_e^2 \frac{1}{1+\epsilon(1-\cos\theta)} \qquad (\epsilon \gg 1, \theta \gg \epsilon^{-1/2}) </math>
showing that the (large angle) differential cross section is inversely proportional to the photon energy.

The differential cross section has a constant peak in the forward direction:
: <math> \left(\frac{d\sigma}{d\Omega}\right)_{\theta=0} = r_e^2 </math>
independent of <math>\epsilon</math>. From the large angle analysis it follows that this peak can only extend to about <math>\theta_c \approx \epsilon^{-1/2}</math>. The forward peak is thus confined to a small solid angle of approximately <math>\pi\theta_c^2</math>, and we may conclude that the total small angle cross section decreases with <math>\epsilon^{-1}</math>.

== Total cross section ==
The differential cross section may be integrated to find the [[Physics:Gamma ray cross section#Compton scattering cross section|total cross section]].

In the low energy limit there is no energy dependence and we recover the [[Physics:Thomson scattering|Thomson cross section]] (~66.5 fm<sup>2</sup>):
: <math> \sigma \approx \frac{8}{3} \pi r_e^2 \qquad (E_\gamma \ll m_e c^2) </math>

== History ==
The Klein–Nishina formula was derived in 1928 by [[Biography:Oskar Klein|Oskar Klein]] and Yoshio Nishina, and was one of the first results obtained from the study of [[Physics:Quantum electrodynamics|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 [[Physics:Electron|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.{{Citation needed|date=March 2022}}

==See also==
* [[Physics:Synchrotron radiation|Synchrotron radiation]]
* Yoshio Nishina
* [[Biography:Oskar Klein|Oskar Klein]]

==References==
{{reflist}}

==Further reading==
*{{cite book |first=R. D. |last=Evans |title=The Atomic Nucleus |publisher=McGraw-Hill |location=New York |year=1955 |pages=674–676 |oclc=542611 }}
*{{cite book |first=A. C. |last=Melissinos |title=Experiments in Modern Physics |publisher=Academic Press |location=New York |year=1966 |pages=252–265 |isbn=0-12-489850-5 }}
*{{cite book |first1=O. |last1=Klein |first2=Y. |last2=Nishina |chapter=On the Scattering of Radiation by Free Electrons According to Dirac's New Relativistic Quantum Dynamics |title=The Oskar Klein Memorial Lectures, Vol. 2: Lectures by Hans A. Bethe and Alan H. Guth with Translated Reprints by Oskar Klein |editor-first=Gösta |editor-last=Ekspong |publisher=World Scientific |location=Singapore |year=1994 |pages=113–139 |bibcode=1994okml.book.....E }}

{{QED}}

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[[Category:Quantum electrodynamics]]
[[Category:Scattering]]

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