What is the threshold energy for gamma+e→3e and gamma→e⁺e⁻ reactions?

[Starwanderer1]

Can someone help me out with this problem(not homework ofcourse)..

Find the threshold energy of gamma quantum required to form
- an electron-positron pair in the field of a stationary electron;
- a pair of pions of opposite signs in the field of a stationary proton..

Thank you for reading

 

[Meir Achuz]

Use $$E^2-P^2=W^2$$, where W= total cm energy.
For gamma+e-->3e, this becomes $$(k+m)^2-k^2=(3m)^2.$$