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

For a random distribution of a large number of scattering centers the phase difference betweenthe points causes the structure factor,F(q), to average to zero unless j = n. Whenj = n, the sum results in F(q) = N, the number of scattering centers. On the other handif the scattering centers are distributed in an ordered array then F(q) is small unless ⃗q ≈ 0.This is coherent scattering. Thus consider the form in 1-D;∑e i⃗q·(⃗x j−⃗x n) → ∑ e iqx je iqxnjnjnNow let x j = jd where d is the spacing of the centers. The sum is converted to an integral by;1 = [(j + 1) − j] = (δj) = [x j+1 − x j ]/d = ∆x/d∞∑j=−∞e iqx k =∑ (δj)eiqx k = (1/d)∞∫−∞dxe iqx k=δ(q)2πdFinally suppose the centers are distributed by some function f(⃗r) so that ⃗pf(⃗r) representsthe electric dipole moment per unit volume (use the electric moment here but it is obvioushow to include the magnetic moment). The scattering due to this distribution is ;dσdΩ =k4(4πǫ) 2 | ∫ d 3 x ′ (â · ⃗p) f(⃗r ′ ) e i⃗q·⃗x′ | 2which we re-write as previously;anddσ sdΩ =k4(4πǫ) 2 |[â · ⃗p] 2F(q) = | ∫ d 3 x ′ f(⃗r ′ )e i⃗q·⃗x′ | 2dσdΩ = dσ sdΩ F(q)The structure function (form factor) is the Fourier transform of the spatial distribution ofthe scattering centers.6 Expansion of a plane wave in spherical harmonicsThe scattering problem is solved naturally in spherical coordinates, as the center of the coordiantesystem is chosen to lie at the center of the scatttering source. Also the scatteringsolution is to be obtained in a multipole series as the scattering amplitude is observed as thedistance from the scattering center → ∞. However, a plane wave incident on the scatteringcenter is formulated in Cartesian coordinates. Thus we must formulate the incident wave in8