Question

Particles of mass m are elastically scattered *** target nuclei of mass M initially at rest. Assuming that the scattering in The CMS is isotropic show that the angular distribution of M in the LS has cos ϕ dependence.

Answer 1

As the scattering is isotropic in the CMS the differential cross-section of the recoiling nuclei is constant and is given by σ (ϕ∗) = σ/4π = constant.

Now the differential cross-sections in the LS and CMS are related by

σ(ϕ) = (sinϕ∗dϕ∗/sinϕdϕ). σ(ϕ∗)

But ϕ∗ = 2ϕ and dϕ∗ = 2dϕ

σ(ϕ) = (sin2ϕ.2dϕ /sinϕdϕ)(σ/4π) = σ/π cos ϕ

Thus, σ(ϕ) has cos ϕ dependence. It is of interest to note that

∫σ(ϕ)dΩ = ∫σ cosϕ.2πsinϕdϕ/π for ϕ ∈ [0, π/2] = σ

as it should. The upper limit for the integration is confined to 90◦ as the target nucleus can not recoil in the backward sphere in the LS.