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{{refimprove|date=May 2013}}
{{Too technical|date=September 2013}}
{{Light–matter interaction}}

'''Pair production''' refers to the creation of an [[elementary particle]] and its [[antiparticle]], usually when a [[photon]] (or another neutral [[boson]]) interacts with a nucleus or another boson. For example an electron and its antiparticle, the positron, may be created. This is allowed, provided there is enough [[energy]] available to create the pair{{spaced ndash}} at least the total [[rest mass energy]] of the two particles{{spaced ndash}} and that the situation allows both energy and momentum to be conserved. Other pairs produced could be a muon and anti-muon or a tau and anti-tau. However all other conserved quantum numbers ([[angular momentum]], [[electric charge]], lepton number) of the produced particles must sum to zero{{spaced ndash}} thus the created particles shall have opposite values of each other. For instance, if one particle has electric charge of +1 the other must have electric charge of &minus;1, or if one particle has [[Strangeness (particle physics)|strangeness]] of +1 then another one must have strangeness of &minus;1. The probability of pair production in photon-matter interactions increases with increasing photon energy and also increases with atomic number approximately as Z<sup>2</sup>.

== Examples ==

:{{SubatomicParticle|Gamma|link=yes}} + {{SubatomicParticle|Gamma|link=no}} &nbsp;&rarr;&nbsp;{{SubatomicParticle|Electron|link=yes}}&nbsp;+&nbsp;{{SubatomicParticle|Positron|link=yes}}

In [[nuclear physics]], this occurs when a high-energy [[photon]] interacts with a [[atomic nucleus|nucleus]]. The energy of this photon can be converted into mass through [[Mass–energy equivalence|Einstein’s equation, {{math|''E''{{=}}''mc''<sup>2</sup>}}]]; where {{math|''E''}} is [[energy]], {{math|''m''}} is [[mass]] and {{math|''c''}} is the [[speed of light]]. The photon must have enough energy to create the mass of an [[electron]] plus a [[positron]].  The [[Invariant mass|rest mass]] of an electron is 9.11 × 10<sup>&minus;31</sup> kg (0.511 MeV), the same as a positron. Without a nucleus to absorb [[momentum]], a photon decaying into electron-positron pair (or other pairs for that matter) can never conserve energy and momentum simultaneously.<ref>{{cite journal
 | last=Hubbell | first=J. H. | title=Electron positron pair production by photons: A historical overview
 | journal=Radiation Physics and Chemistry
 |date=June 2006 | volume=75 | issue=6
 | pages=614–623 | doi=10.1016/j.radphyschem.2005.10.008
 | bibcode=2006RaPC...75..614H
}}</ref>

== Photon–nucleus interaction ==
There are different processes how an electron-positron pair can be produced. In air (e.g. in lightning discharges) the most important one is the scattering of photons at the nuclei of atoms or molecules.
Quantum mechanically, the process of pair production can be described by the quadruply differential cross section:<ref>Bethe, H.A., Heitler, W., 1934. On the stopping of fast particles and on the creation of positive electrons. Proc. Phys. Soc. Lond. 146, 83–112</ref>

<math>
\begin{align}
d^4\sigma &=
\frac{Z^2\alpha_{fine}^3c^2}{(2\pi)^2\hbar}|\mathbf{p}_+||\mathbf{p}_-|
\frac{dE_+}{\omega^3}\frac{d\Omega_+ d\Omega_- d\Phi}{|\mathbf{q}|^4}\times \\
&\times\left[-
\frac{\mathbf{p}_-^2\sin^2\Theta_-}{(E_--c|\mathbf{p}_-|\cos\Theta_-)^2}\left
(4E_+^2-c^2\mathbf{q}^2\right)\right.\\
&-\frac{\mathbf{p}_+^2\sin^2\Theta_+}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)^2}\left
(4E_-^2-c^2\mathbf{q}^2\right)  \\
&+2\hbar^2\omega^2\frac{\mathbf{p}_+^2\sin^2\Theta_++\mathbf{p}_-^2\sin^2\Theta_-}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)(E_--c|\mathbf{p}_-|\cos\Theta_-)} \\
&+2\left.\frac{|\mathbf{p}_+||\mathbf{p}_-|\sin\Theta_+\sin\Theta_-\cos\Phi}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)(E_--c|\mathbf{p}_-|\cos\Theta_-)}\left(2E_+^2+2E_-^2-c^2\mathbf{q}^2\right)\right]. \\
\end{align}
</math>

with

<math>
\begin{align}
d\Omega_+&=\sin\Theta_+\ d\Theta_+,\\
d\Omega_-&=\sin\Theta_-\ d\Theta_-.
\end{align}
</math>

This expression can be derived by using a quantum mechanical symmetry between pair production and [[Bremsstrahlung]]. <br>
<math>Z</math> is the [[atomic number]], <math>\alpha_{fine}\approx 1/137</math> the [[fine structure constant]], <math>\hbar</math> the reduced [[Planck's constant]] and <math>c</math> the [[speed of light]]. The kinetic energies <math> E_{kin,+/-} </math> of the positron and electron relate to their total energies <math> E_{+,-}</math> and [[momenta]] <math> \mathbf{p}_{+,-} </math> via 

<math>
E_{+,-}=E_{kin,+/-}+m_e c^2=\sqrt{m_e^2 c^4+\mathbf{p}_{+,-}^2 c^2}.
</math>

[[Conservation of energy]] yields

<math>
\hbar\omega=E_{+}+E_{-}.
</math>

The momentum <math> \mathbf{q} </math> of the [[virtual photon]] between incident photon and nucleus is:

<math>
\begin{align}
-\mathbf{q}^2&=-|\mathbf{p}_+|^2-|\mathbf{p}_-|^2-\left(\frac{\hbar}{c}\omega\right)^2+2|\mathbf{p}_+|\frac{\hbar}{c}
\omega\cos\Theta_+ +2|\mathbf{p}_-|\frac{\hbar}{c} \omega\cos\Theta_- \\
&-2|\mathbf{p}_+||\mathbf{p}_-|(\cos\Theta_+\cos\Theta_-+\sin\Theta_+\sin\Theta_-\cos\Phi),
\end{align}
</math>

where the directions are given via:

<math>
\begin{align}
\Theta_+&=\sphericalangle(\mathbf{p}_+,\mathbf{k}),\\
\Theta_-&=\sphericalangle(\mathbf{p}_-,\mathbf{k}),\\
\Phi&=\text{Angle between the planes } (\mathbf{p}_+,\mathbf{k}) \text{ and } (\mathbf{p}_-,\mathbf{k}),
\end{align}
</math>

where <math> \mathbf{k} </math> is the momentum of the incident photon.

In order to analyse the relation between the photon energy <math> E_+ </math> and the emission angle <math> \Theta_+ </math> between photon and positron, Köhn and Ebert integrated <ref>Koehn, C., Ebert, U., Angular distribution of Bremsstrahlung photons and of positrons for calculations of terrestrial gamma-ray flashes and positron beams, Atmos. Res. (2013), http://dx.doi.org/10.1016/j.atmosres.2013.03.012</ref> the quadruply differential cross section over <math> \Theta_- </math> and <math> \Phi </math>. The double differential cross section is:

<math>
\begin{align}
\frac{d^2\sigma (E_+,\omega,\Theta_+)}{dE_+d\Omega_+} =
\sum\limits_{j=1}^{6} I_j 
\end{align}
</math>

with

<math>
\begin{align}
I_1&=\frac{2\pi A}{\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+}} \\
&\times
\ln\left(\frac{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+-\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2
\Theta_+}(\Delta^{(p)}_1+\Delta^{(p)}_2)+\Delta^{(p)}_1\Delta^{(p)}_2}{-(\Delta^{(p)}_2)
^2-4p_+^2p_-^2\sin^2\Theta_+
-\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2 \Theta_+}(\Delta^{(p)}_1-\Delta^{(p)}_2)+\Delta^{(p)}_1\Delta^{(p)}_2
}\right) \\
&\times\left[-1-\frac{c\Delta^{(p)}_2}{p_-(E_+-cp_+\cos\Theta_+)}+\frac{p_+^2c^2\sin^2\Theta_+}
{(E_+-cp_+\cos\Theta_+)^2}-\frac{2\hbar^2\omega^2p_-\Delta^{(p)}_2}{c(E_+-cp_+\cos
\Theta_+)((\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+)}\right], \\
I_2&=\frac{2\pi Ac}{p_-(E_+-cp_+\cos\Theta_+)}\ln\left(
\frac{E_-+p_-c}{E_--p_-c}\right), \\
I_3&=\frac{2\pi A}{\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+
}}  \\
&\times\ln\Bigg(\Big((E_-+p_-c)(4p_+^2p_-^2\sin^2\Theta_+(E_--p_-c)+(\Delta^{(p)}_1+\Delta^{(p)}_2)
((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c) \\
&-\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}))\Big)\Big((E_--p_-c)
(4p_+^2p_-^2\sin^2\Theta_+(-E_--p_-c) \\
&+(\Delta^{(p)}_1-\Delta^{(p)}_2)
((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)-\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}))\Big)^{-1}\Bigg)  \\
&\times\left[\frac{c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)}{p_-(E_+-cp_+\cos\Theta_+)}\right.\\
&+\Big[((\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+)(E_-^3+E_-p_-c)+p_-c(2
((\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+)E_-p_-c \\
&+\Delta^{(p)}_1\Delta^{(p)}_2(3E_-^2+p_-^2c^2))\Big]\Big[(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1} \\
&+\Big[-8p_+^2p_-^2m^2c^4\sin^2\Theta_+(E_+^2+E_-^2)-2\hbar^2\omega^2p_+^2\sin^2\Theta_+p_-c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c) \\
&+2\hbar^2\omega^2p_- m^2c^3(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)\Big]
\Big[(E_+-cp_+\cos\Theta_+)((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)\Big]^{-1} \\
&+\left.\frac{4E_+^2p_-^2(2(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2-4m^2c^4p_+^2p_-^2\sin^2\Theta_+)(\Delta^{(p)}_1E_-+\Delta^{(p)}_2p_-c)}{((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)^2}\right], \\
I_4&=\frac{4\pi Ap_-c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)}{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}+\frac{16\pi E_+^2p_-^2
A(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2}{((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)^2}, \\
I_5&=\frac{4\pi A}{(-(\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+)
((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)} \\
&\times\left[\frac{\hbar^2\omega^2p_-^2}{E_+cp_+\cos\Theta_+}
\Big[E_-[2(\Delta^{(p)}_2)^2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)+8p_+^2p_-^2\sin^2\Theta_+((\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2)]
\right.\\
&+p_-c[2\Delta^{(p)}_1\Delta^{(p)}_2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)+16\Delta^{(p)}_1\Delta^{(p)}_2p_+^2p_-^2\sin^2\Theta_+]\Big]\Big[(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1}\\
&+ \frac{2\hbar^2\omega^2 p_{+}^2 \sin^2\Theta_+(2\Delta^{(p)}_1\Delta^{(p)}_2
p_-c+2(\Delta^{(p)}_2)^2E_-+8p_+^2p_-^2\sin^2\Theta_+ E_-)}{E_+-cp_+\cos\Theta_+}\\
&-\Big[2E_+^2p_-^2\{2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2
+8p_+^2p_-^2\sin^2\Theta_+[((\Delta^{(p)}_1)^2+(\Delta^{(p)}_2)^2)(E_-^2+p_-^2c^2)\\
&+4\Delta^{(p)}_1\Delta^{(p)}_2E_-p_-c]\}\Big]\Big[(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1}\\
&-\left.\frac{8p_+^2p_-^2\sin^2\Theta_+(E_+^2+E_-^2)(\Delta^{(p)}_2p_-c +\Delta^{(p)}_1
E_-)}{E_+-cp_+\cos\Theta_+}\right], \\
I_6&=-\frac{16\pi E_-^2p_+^2\sin^2\Theta_+ A}{(E_+-cp_+\cos\Theta_+)^2
(-(\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+)} 
\end{align}
</math>

and

<math>
\begin{align}
A&=\frac{Z^2\alpha_{fine}^3c^2}{(2\pi)^2\hbar}\frac{|\mathbf{p}_+||\mathbf{p}_-|}{\omega^3},\\
\Delta^{(p)}_1&:=-|\mathbf{p}_+|^2-|\mathbf{p}_-|^2-\left(\frac{\hbar}{c}\omega\right)
+ 2\frac{\hbar}{c}\omega|\mathbf{p}_+|\cos\Theta_+,\\
\Delta^{(p)}_2&:=2\frac{\hbar}{c}\omega|\mathbf{p}_i|-2|\mathbf{p}_+||\mathbf{p}_-|
\cos\Theta_+ + 2.
\end{align}
</math>

This cross section can be applied in Monte Carlo simulations. An analysis of this expression shows that positrons are mainly emitted in the direction of the incident photon.

== Energy ==
Photon-nucleus pair production can only occur if the photons have an energy exceeding twice the rest energy ({{math|''m<sub>e</sub>c''<sup>2</sup>}}) of an electron ({{val|1.022|ul=MeV}}).  These interactions were first observed in [[Patrick Maynard Stuart Blackett|Patrick Blackett]]'s counter-controlled [[cloud chamber]], leading to the 1948 [[Nobel Prize in Physics]]. The same conservation laws apply for the generation of other higher energy particles such as the [[muon]] and [[tau (particle)|tau]]. 

In semiclassical [[general relativity]], pair production was invoked to predict hypothetical [[Hawking radiation]]. According to [[quantum mechanics]], particle pairs are constantly appearing and disappearing as a [[quantum foam]]. In a region of strong gravitational [[tidal forces]], the two particles in a pair may sometimes be wrenched apart before they have a chance to mutually [[annihilation|annihilate]]. When this happens in the region around a [[black hole]], one particle may escape while its antiparticle partner is captured by the black hole.

Pair production is also the mechanism behind the hypothesized [[pair instability supernova]] type of stellar explosion, where pair production suddenly lowers the pressure inside a supergiant star, leading to a partial implosion, and then explosive thermonuclear burning.  [[Supernova]] [[SN 2006gy]] is hypothesized to have been a pair production type supernova.

In 2008 the [[Titan laser]] aimed at a 1-millimeter-thick [[gold]] target was used to generate positron–electron pairs in large numbers.<ref>{{cite news |first= |last= |authorlink= |coauthors= |title=Laser technique produces bevy of antimatter |url=http://www.msnbc.msn.com/id/27998860/ |quote=The LLNL scientists created the positrons by shooting the lab's high-powered Titan laser onto a one-millimeter-thick piece of gold. |work=[[MSNBC]] |year=2008 |accessdate=2008-12-04 }}</ref>

==See also==
*[[Electron–positron annihilation]]
*[[Meitner–Hupfeld effect]]
*[[Pair-instability supernova]]
*[[Two-photon physics]]
*[[Dirac equation]]
*[[Matter creation]]

==References==
{{reflist}}

==External links==
* [http://www.modspil.dk/agger/speciale.pdf Theory of photon-impact bound-free pair production]

{{DEFAULTSORT:Pair Production}}
[[Category:Particle physics]]
[[Category:Nuclear physics]]