Problem -

The diagram shoves the track (red) of a particle of mass m as it is scattered elastically from a particle of mass M (blue).

(a) For this part of the question, assume that m << M so that the heavy particle remains at rest during the scattering process.

i. Suppose that for this particular process, the assymptotic angle Φ shown in the diagram is related the impact parameter b. the mew E, and a constant C according to

cos Φ = (1 + (2CEb)²)^-½  

Express this equation in terms the impact parameter b for this process as a function of the angle Φ.

ii. Consider this process in the usual scattering geometry ark determine the scattering angle θ in terms of the angle Φ shown in the diagram so that you can express the impact parameter in terms of the scattering angle (b(θ)).

iii. From this result, determine an expression for the differential cross section

dσ(θ)/dΩ

for this process. Express, your answer in terms of the constant C, the energy E, and the scattering angle θ.

(b) Now consider the same system but now in the laboratory reference frame with m = M. Assume particle with mass M is initially at rest and far from the target, the particle with mass m has a kinetic energy E.

i. Write an expression for the differential cross section for this system evaluated in the center of mass frame of reference.

ii. Show a diagram for the scattering process in the laboratory reference frame, where the scattering angle k called ψ.

iii. How are the laboratory (ψ) and center of mass (θ) scattering angles related?

iv. Write an expression for the scattering cross section evaluated in the laboratory halite of reference. Evaluate the expression for the lab angle and cross section for the case that θ = 0 and for the case that θ = 180^o.