We can square the conservation equation:  However, in most cases the recoil of the nuclei is much smaller compared to the energy of the photon and can be neglected. Taking this approximation of formula_8 to simplify and expanding the remaining relation:  Therefore, this approximation can only be satisfied if the electron and positron are emitted in very nearly the same direction, that is, formula_12. 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. The energy transfer to electron and positron in pair production interactions is given by:  where formula_14 is Planck's constant, formula_15 is the frequency of the photon and the formula_16 is the combined rest mass of the electron–positron. In general, ignoring the nuclei recoil, the electron and positron can be emitted with different kinetic energies, but the average transferred to each is:  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:  where formula_19 is the fine structure constant, formula_20 is the classical electron radius, formula_21 is the atomic number of the material and formula_22 is some complex 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. Pair production is invoked to predict the existence of hypothetical Hawking radiation. According to quantum mechanics, particle pairs are constantly appearing and disappearing as a quantum foam. In a region of strong gravitational tidal forces, the two particles in a pair may sometimes be wrenched apart before they have a chance to mutually annihilate. When this happens in the region around a black hole, one particle may escape while its antiparticle partner is captured by the black hole. Pair production is also the mechanism behind the hypothesized pair-instability supernova type of stellar explosion, where pair production suddenly lowers the pressure inside a supergiant star, leading to a partial implosion, and then explosive thermonuclear burning. Supernova SN 2006gy is hypothesized to have been a pair production type supernova. Pair production does not occur in X-ray imaging because the machines are usually rated ~150 kV (photons with energy 150 keV) which is much smaller than the minimum of 1022 keV needed to create the smallest possible particle pair, the electron and positron.