Editing Pair production (section)

== Photon to electron and positron ==
[[File:Pair production Cartoon.gif|thumb|Diagram showing the process of electron–positron pair production. In reality the produced pair are nearly collinear. The black dot labelled 'Z' represents an adjacent atom, with [[atomic number]] {{mvar|Z}}.]]
For photons with high [[photon energy]] ([[MeV]] scale and higher), pair production is the dominant mode of photon interaction with matter. 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]].<ref>
{{cite web
 |last=Bywater |first=Jenn
 |date=29 October 2015
 |title=Exploring dark matter in the inaugural Blackett Colloquium
 |website=Imperial College London
 |url=http://www3.imperial.ac.uk/newsandeventspggrp/imperialcollege/newssummary/news_27-10-2015-14-30-15
 |access-date=29 August 2016
}}
</ref> If the photon is near an atomic nucleus, the energy of a photon can be converted into an electron–positron pair:

(Z+){{SubatomicParticle|Gamma|link=yes}} →&nbsp;{{SubatomicParticle|Electron|link=yes}}&nbsp;+&nbsp;{{SubatomicParticle|Positron|link=yes}}

[[File:Dominant Photon-Matter Interaction.svg|thumb|Plot of photon energies calculated for a given element (atomic number Z) at which the [[Cross section (physics)|cross section]] value for the process on the right becomes larger than the cross section for the process on the left. For calcium (Z=20), Compton scattering starts to dominate at ''hυ''=0.08 MeV and ceases at 12 MeV.<ref>{{Cite journal |date=2009-09-17 |title=XCOM: Photon Cross Sections Database |url=https://dx.doi.org/10.18434/T48G6X |journal=NIST |doi=10.18434/T48G6X |language=en |last1=Seltzer |first1=Stephen }}</ref>]]
[[File:Subatomic particle pair production.png|thumb|alt=Subatomic particle pair production|Subatomic particle pair production]]

The photon's energy is converted to particle mass in accordance with [[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 higher energy than the sum of the rest mass energies of an electron and positron (2&nbsp;×&nbsp;511&nbsp;keV = 1.022&nbsp;MeV, resulting in a photon wavelength of {{val|1.2132|ul=pm}}) for the production to occur. (Thus, pair production does not occur in medical X-ray imaging because these X-rays only contain ~&nbsp;150&nbsp;keV.) 
The photon must be near a nucleus in order to satisfy conservation of momentum, as an electron–positron pair produced in free space cannot satisfy conservation of both energy and momentum.<ref>
{{cite journal
 | last=Hubbell | first=J.H.
 | date=June 2006
 | title=Electron positron pair production by photons: A historical overview
 | journal=Radiation Physics and Chemistry
 | volume=75 | issue=6 | pages=614–623
 | doi=10.1016/j.radphyschem.2005.10.008 | bibcode=2006RaPC...75..614H
 | url=https://zenodo.org/record/1259327
}}
</ref> Because of this, when pair production occurs, the atomic nucleus receives some [[Atomic recoil|recoil]]. The reverse of this process is [[electron–positron annihilation]].

=== Basic kinematics ===
These properties can be derived through the kinematics of the interaction. Using [[four vector]] notation, the conservation of energy–momentum before and after the interaction gives:<ref>
{{cite web
 |last=Kuncic |first=Zdenka, Dr.
 |date=12 March 2013
 |title=PRadiation Physics and Dosimetry
 |series=PHYS 5012
 |website=Index of Dr.&nbsp;Kuncic's Lectures
 |publisher=The University of Sydney
 |place=Sydney, Australia
 |url=http://www.physics.usyd.edu.au/~kuncic/lectures/RP3_slides.pdf
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 |archive-date=11 March 2016
}}
</ref>
: <math>p_\gamma = p_{\text{e}^-} + p_{\text{e}^+} + p_{\text{ʀ}}</math>
where <math>p_\text{ʀ}</math> is the recoil of the nucleus. Note the modulus of the four vector 
: <math>A \equiv (A^0,\mathbf{A}) </math>
is
: <math>A^2 = A^{\mu} A_{\mu} = - (A^0)^2 + \mathbf{A} \cdot \mathbf{A} </math>
which implies that <math>(p_\gamma)^2 = 0 </math> for all cases and <math>(p_{\text{e}^-})^2 = -m_\text{e}^2 c^2 </math>. We can square the conservation equation
: <math>(p_\gamma)^2 = (p_{\text{e}^-} + p_{\text{e}^+} + p_\text{ʀ})^2 </math>

However, in most cases the recoil of the nucleus is small compared to the energy of the photon and can be neglected. Taking this approximation of <math>p_{R} \approx 0</math> and expanding the remaining relation
: <math>(p_\gamma)^2 \approx (p_{\text{e}^-})^2 + 2 p_{\text{e}^-} p_{\text{e}^+} + (p_{\text{e}^+})^2 </math>
: <math>-2\, m_\text{e}^2 c^2 + 2 \left( -\frac{E^2}{c^2} + \mathbf{p}_{\text{e}^-} \cdot \mathbf{p}_{\text{e}^+} \right) \approx 0 </math>
: <math>2\,(\gamma^2 - 1)\,m_\text{e}^2\,c^2\,(\cos \theta_\text{e} - 1) \approx 0 </math>

Therefore, this approximation can only be satisfied if the electron and positron are emitted in very nearly the same direction, that is, <math>\theta_\text{e} \approx 0 </math>.

This derivation is a semi-classical approximation. An exact derivation of the kinematics can be done taking into account the full [[quantum mechanical scattering of photon and nucleus]].

=== Energy transfer ===
The energy transfer to electron and positron in pair production interactions is given by
: <math>(E_k^{pp})_\text{tr} = h \nu - 2\, m_\text{e} c^2</math>
where <math>h</math> is the [[Planck constant]], <math>\nu </math> is the frequency of the photon and the <math>2\, m_\text{e} c^2</math> is the combined rest mass of the electron–positron. In general the electron and positron can be emitted with different kinetic energies, but the average transferred to each (ignoring the recoil of the nucleus) is
: <math>(\bar E_k^{pp})_\text{tr} = \frac{1}{2} (h \nu - 2\, m_\text{e} c^2)</math>

=== Cross section ===
{{See also|Gamma ray cross section}}
[[File:Electron-Positron nuclear Pair production Feynman Diagram.svg|thumb|[[Feynman diagram]] of electron–positron pair production. One must calculate multiple diagrams to get the net cross section]]

The exact analytic form for the cross section of pair production must be calculated through [[quantum electrodynamics]] in the form of [[Feynman diagram]]s and results in a complicated function. To simplify, the cross section can be written as:
: <math>\sigma = \alpha \, r_\text{e}^2 \, Z^2 \, P(E,Z)</math>
where <math>\alpha</math> is the [[fine-structure constant]], <math>r_\text{e}</math> is the [[classical electron radius]], <math>Z</math> is the [[atomic number]] of the material, and <math>P(E,Z)</math> is some complex-valued function that depends on the energy and atomic number. Cross sections are tabulated for different materials and energies.

In 2008 the [[Titan laser]], aimed at a 1&nbsp;millimeter-thick [[gold]] target, was used to generate positron–electron pairs in large numbers.<ref>{{cite news|title=Laser technique produces bevy of antimatter|url=https://www.nbcnews.com/id/wbna27998860|website=[[MSNBC]]|year=2008|access-date=2019-05-27}}</ref>