So I've been looking into particle-antiparticle pair production from a gamma ray and don't understand one thing.
Let's say I have a 1,1 MeV photon and it hits a nucleus – electron-positron pair with some momentum will be created and the nucleus will probably get some momentum as well because of the impact.
But why does the photon need the nucleus at all? Why can't it just fly through space and suddenly, with some probability, change into a electron-positron pair with momentum? I see that the momentum of the system wouldn't be conserved but I don't really understand how the nucleus helps it.
Best Answer
look at energy-momentum conservation:
$$p_\gamma = p_1+p_2$$
the photon has invariant mass 0 wheras the electron and positron have mass $m_e$
$$p_\gamma^2 = (p_1+p_2)^2 = p_1^2+p_2^2+2p_1\cdot p_2$$ $$0 = 2m_e^2 + 2p_1\cdot p_2$$ $$-m_e^2= p_1\cdot p_2 = E_1E_2-|\vec{p_1}||\vec{p_2}|cos\theta > E_1E_2-|\vec{p_1}||\vec{p_2}| = E_1E_2(1-\beta_1\beta_2) > 0$$
The betas cannot be greater than one. So the right hand side always stays positive. The nucleon helps. because it changes the initial state to one with a invariant mass greater than zero.