Scattering 1 Classical scattering of a charged particle (Rutherford ...

dσ ⊥dΩ = k4 η 62 |ǫ − 1ǫ + 2 |2The total differential cross section is;dσdΩ = dσ ‖dΩ + dσ ⊥dΩThe total cross section is obtained by integration over dΩ.σ T = 8πk4 η 63| ǫ − 1ǫ + 2 |2The polarization of the scattered wave is;P(θ) = dσ ⊥/dΩ − dσ ‖ /dΩdσ ⊥ /dΩ + dσ ‖ /dΩ =sin2 (θ)1 + cos 2 (θ)5 Form factorSuppose the scatterers are a system of charges with fixed locations in space. The resultis a superposition of scattering amplitudes from all the point scattering centers. We havepreviously seen that coherence must be included when one adds the contributions from eachscattering point. Assume that the positions of the scatters are located by the vectors, ⃗x j .The incident wave will have the phase factor, e ik 0ˆn 0·⃗x j, and the phase for the scattered waveat large distances will be e ikˆn·⃗x j. Here ˆn 0 is the direction of the scattered wave. We assumedipole scattering and write;dσdΩ =k4(4πǫ) 2 | ∑ j[â · ⃗p j + (1/c)(ˆn × â) · ⃗m j ] e i⃗q·⃗x j| 2In the above write ⃗q = kˆn 0 − kˆn = k(ˆn 0 − ˆn) for elastic scattering from massive points,(ie no recoil). Then suppose that the scatterers are identical so that ⃗p j = ⃗p and ⃗m j = ⃗m.Rewrite the above as;where;dσdΩ = [dσ sdΩ ] F(q)dσ sdΩ =F(q) = | ∑ jk4(4πǫ) 2 [â · ⃗p j + (1/c)(ˆn × â) · ⃗m j ] 2e i⃗q·⃗x j| 2 = ∑ jne i⃗q·(⃗x j−⃗x n)7