2.3. Interaction of Photons with Matter 95
Hereλ 0 is the wavelength of the incident photon,ais the radius of the target particle,
andm=ns/nmis the ratio of the index of refraction of the target particle to that
of the surrounding medium.
A.5 PairProduction
Pair production is the process that results in the conversion of a photon into an
electron-positron pair. Since photon has no rest mass, while both the electron and
the positron do, therefore we can say that this process converts energy into mass
according to Einstein’s mass energy relationE=mc^2. Earlier in the chapter we
discussed the process of electron-positron annihilation, in which mass converts into
energy. Hence we can think of pair production as the inverse process of the electron-
positron annihilation. However there is oneoperationaldifference between the two
processes: the pair production always takes part in a material while the electron
positron annihilation does not have any such requirement. To be more specific, for
the pair production to take place, there must be another particle in the vicinity
of the photon to ensure momentum conservation. The process in the vicinity of a
heavy nucleus can be represented as
γ+X→e+e++X∗, (2.3.24)whereXandX∗represent the ground and excited states of a heavy nucleus.
Intuitively thinking, we can say that since energy is being converted into two
particles that have discrete masses, therefore enough energy should be available for
this process to take place. That is, the photon must have the energy equivalent of
at least the rest masses of two electrons (electron and positron have equal masses).
Eγ,thresh ≥ 2 mec^2 (2.3.25)
⇒Eγ,thresh ≥ 1. 022 MeVHeremeis the mass of an electron or a positron. Hence a photon carrying energy
below 1. 022 MeVcan not convert into an electron-positron pair.
The actual threshold energy for the process in the vicinity of a heavy nucleus is
given by
Eγ,thresh≥ 2 mec^2 +2 m^2 ec^2
mnuc, (2.3.26)
Heremnucis the mass of the nucleus. We can also write the above equation in the
form
Eγ,thresh≥ 2 mec^2[
1+
me
mnuc]
. (2.3.27)
Now since the mass of a nucleus is much greater than the mass of an electron
(mnuc>> me), we can neglect the second term in the parenthesis on the right hand
side and get the threshold condition we derived earlier, such as
Eγ,thresh≥ 2 mec^2.The pair production can also occur in the vicinity of lighter particles, such as
electrons. The process in the vicinity of an electron is generally referred to astriplet
productionand can be written as
γ+e→e+e++e. (2.3.28)