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

ψ in = e ikz = e ikr cos(θ) .The scattered wave has the form of an outgoing spherical wave;ψ out → f(θ, φ) eikrrThen the asymptotic form of the solution as r → ∞ of the scattering equation isψ = ψ in + ψ outwith R = |⃗r −⃗r ′ | and L operating on the wave amplitude, ψ, which equals the source. Thenif we formally define an operator L −1 . The solution we seek has the form at a time t → ∞;ψ = ψ in + ψ outwhere ψ in is the form of the incoming wave and ψ out is the outgoing spherical wave. Thewave equation can be written in operator form as;ψ out = L −1 Sψ.Formally we can then write;ψ out =11 − (L −1 S) L−1 (Sψ in )Then substitute for ψ = ψ in + ψ out and collect terms. The formally solution is then writtenas;ψ out = [1 + L −1 S + L −1 SL −1 S + · · ·] ψ inThe series converges if the source term is sufficiently small. Generaly the scattered EM waveis much less than the incident wave so we take the first term, called the Born term, as theapproximate solution. The differential scattering cross section is then defined by the scatteredpower into a solid angle dω divided by the incident flux. The incident flux is obtainedfrom the Poynting vector of the incident plane wave.S in = |E| 2 /µcThus the cross section in the Born approximation can be written;dσdΩ=(1/cµ)|f(θ, φ)|2(1/cµ)|e ikz | 2 = |f(θ, φ)| 2 4