# Pair production near an electron

A high energy photon (E>~MeV) after interacting with an atomic nucleus produces an electron positron pair. The threshold of this process is the sum of the rest mass energies of these two leptons. My question is: is it possible for a photon to produce an electron-positron pair when interacting with an electron?

High energy photons when interacting with a field, can produce electron positron pairs, because the interaction allows energy and momentum conservation in the center of mass of the produced pair, which will have an invariant mass of the sum of the masses. It makes no difference how the field is attained.

The probability of scattering off the field of a nucleus is higher because the field of a nucleus is stronger than the field of a single electron, and its mass with respect to the leptons does not require a lot of momentum for conservation of momentum.

What you are asking is really a particle interaction,

$γ + e^- -> e^- e^+e^-$

a legitimate feynman diagram of first order can be written, where the three leptons will be on equal footing as far as energy and momentum behavior as they are of the same mass.

Yes, it's possible (see answer by @annav ), but the threshold energy will be larger. A quick invariant mass calculation will show that the threshold energy of the incoming photon is $$2m_e c^2\left(1+\frac{m_e}{M}\right),$$ where $m_e$ is the electron mass, and $M$ is the interacting mass.

If the interacting mass is $m_e$, then the threshold photon energy would be $4m_e c^2$!

If the interacting mass is a proton, the threshold is about $2.001 m_e c^2$.

If there is no interacting mass, pair-production can't happen (energetically).