Editing Pair production (section)

=== 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
 |access-date=2015-04-14 |url-status=dead
 |archive-url=https://web.archive.org/web/20160311042609/http://www.physics.usyd.edu.au/~kuncic/lectures/RP3_slides.pdf
 |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]].