Pair production

The probability of pair production in photon–matter interactions increases with photon energy and also increases approximately as the square of the atomic number (number of protons) of the nearby atom.

Photon to electron and positron

At 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 Blackett's counter-controlled cloud chamber, leading to the 1948 Nobel Prize in Physics. If the photon is near an atomic nucleus, the energy of a photon can be converted into an electron–positron pair:

(Z+)γ → e⁻
+ e⁺

The photon's energy is converted to particle mass in accordance with Einstein's equation, E = mc²; where E is energy, m is mass and 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 × 511 keV = 1.022 MeV, resulting in a photon wavelength of 1.2132 pm) for the production to occur. (Thus, pair production does not occur in medical X-ray imaging because these X-rays only contain ~ 150 keV.) The photon must be near a nucleus, or collide with another photon, in order to satisfy conservation of momentum, as an electron–positron pair produced in free space cannot satisfy conservation of both energy and momentum. Because of this, when pair production occurs, the atomic nucleus receives some 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:

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

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

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

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

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

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

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

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

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

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.