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:^[5]

${\displaystyle p_{\gamma }=p_{{\text{e}}^{-}}+p_{{\text{e}}^{+}}+p_{\text{ʀ}}}$

where ${\displaystyle p_{\text{ʀ}}}$ is the recoil of the nucleus. Note the modulus of the four vector

${\displaystyle A\equiv (A^{0},\mathbf {A} )}$

is:

${\displaystyle A^{2}=A^{\mu }A_{\mu }=-(A^{0})^{2}+\mathbf {A} \cdot \mathbf {A} }$

which implies that ${\displaystyle (p_{\gamma })^{2}=0}$ for all cases and ${\displaystyle (p_{{\text{e}}^{-}})^{2}=-m_{\text{e}}^{2}c^{2}}$. We can square the conservation equation:

${\displaystyle (p_{\gamma })^{2}=(p_{{\text{e}}^{-}}+p_{{\text{e}}^{+}}+p_{\text{ʀ}})^{2}}$

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 ${\displaystyle p_{R}\approx 0}$ and expanding the remaining relation:

${\displaystyle (p_{\gamma })^{2}\approx (p_{{\text{e}}^{-}})^{2}+2p_{{\text{e}}^{-}}p_{{\text{e}}^{+}}+(p_{{\text{e}}^{+}})^{2}}$

${\displaystyle -2\,m_{\text{e}}^{2}c^{2}+2\left(-{\frac {E^{2}}{c^{2}}}+\mathbf {p} _{{\text{e}}^{-}}\cdot \mathbf {p} _{{\text{e}}^{+}}\right)\approx 0}$

${\displaystyle 2\,(\gamma ^{2}-1)\,m_{\text{e}}^{2}\,c^{2}\,(\cos \theta _{\text{e}}-1)\approx 0}$

Therefore, this approximation can only be satisfied if the electron and positron are emitted in very nearly the same direction, that is, ${\displaystyle \theta _{\text{e}}\approx 0}$.

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:

${\displaystyle (E_{k}^{pp})_{\text{tr}}=h\nu -2\,m_{\text{e}}c^{2}}$

where ${\displaystyle h}$ is Planck's constant, ${\displaystyle \nu }$ is the frequency of the photon and the ${\displaystyle 2\,m_{\text{e}}c^{2}}$ 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:

${\displaystyle ({\bar {E}}_{k}^{pp})_{\text{tr}}={\frac {1}{2}}(h\nu -2\,m_{\text{e}}c^{2})}$

Cross section

The exact analytic form for the cross section of pair production must be calculated through quantum electrodynamics in the form of Feynman diagrams and results in a complicated function. To simplify, the cross section can be written as:

${\displaystyle \sigma =\alpha \,r_{\text{e}}^{2}\,Z^{2}\,P(E,Z)}$

where ${\displaystyle \alpha }$ is the fine-structure constant, ${\displaystyle r_{\text{e}}}$ is the classical electron radius, ${\displaystyle Z}$ is the atomic number of the material, and ${\displaystyle P(E,Z)}$ 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 millimeter-thick gold target, was used to generate positron–electron pairs in large numbers.^[6]