In a fixed target experiment a particle of mass M and kinetic energy T strikes a stationary particle of mass M. By evaluating s, t and u in the laboratory frame and using the above relation, or otherwise, show that the kinetic energy T' of the particle scattered elastically at an angle θ is given by
T' = T cos^2 θ / (1 + ((T/2M)*sin^2 θ))
We know that:
We define P1 as the four momentum of the incoming proton:
P1 = (T+M,0,0,T)
P2 as the stationary proton:
P2 = (M,0,0,0)
P3 as the initially incoming proton after the scattering event:
P3 = (T'+M, 0, T'sinθ, T'cosθ)
P4 as the initially stationary proton after the scattering event:
P4 = (E4+M, P4) where P4 in the bracket is a 3 vector not 4.
We also know:
s = (P1 + P2)2 , t = (P1 − P3)2 and u = (P1 − P4)2
s + t + u = ∑ mi2 = 4m2
The Attempt at a Solution
I have attempted this multiple times over a few days, I realize that P4 and therefore u need to be eliminated as they aren't in the solution but I am unsure how.
Any help/ suggestions would be appreciated. Thanks, John.