. 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+) → + 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. The photon's energy is converted to particle mass in accordance with
Einstein's equation,; where is
energy, is
mass and 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 ) 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 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: : p_\gamma = p_{\text{e}^-} + p_{\text{e}^+} + p_{\text{ʀ}} where p_\text{ʀ} is the recoil of the nucleus. Note the modulus of the four vector : A \equiv (A^0,\mathbf{A}) is : A^2 = A^{\mu} A_{\mu} = - (A^0)^2 + \mathbf{A} \cdot \mathbf{A} which implies that (p_\gamma)^2 = 0 for all cases and (p_{\text{e}^-})^2 = -m_\text{e}^2 c^2 . We can square the conservation equation : (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 p_{R} \approx 0 and expanding the remaining relation : (p_\gamma)^2 \approx (p_{\text{e}^-})^2 + 2 p_{\text{e}^-} p_{\text{e}^+} + (p_{\text{e}^+})^2 : -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 : 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, \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 : (E_k^{pp})_\text{tr} = h \nu - 2\, m_\text{e} c^2 where h is the
Planck constant, \nu is the frequency of the photon and the 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 : (\bar E_k^{pp})_\text{tr} = \frac{1}{2} (h \nu - 2\, m_\text{e} c^2)
Cross section 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 diagrams and results in a complicated function. To simplify, the cross section can be written as: : \sigma = \alpha \, r_\text{e}^2 \, Z^2 \, P(E,Z) where \alpha is the
fine-structure constant, r_\text{e} is the
classical electron radius, Z is the
atomic number of the material, and 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. == Laboratory production ==