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

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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
Now a properly normalized outgoing Green’s function for a point source can be written ase ikRR(ie the Green function in a previous lecture). Therefore the solution in practical ratherthan in the above formal formal terms can be written ;ψ out = ∫ d⃗r eikRR S(⃗r′ )ψ in + ∫ d⃗r eikRR S(⃗r′ )ψ outThis is not a solution but an integral equation because ψ is not known until the solution isobtained, ie the source term in the intergand is unknown.3 Scattering of the EM waveScattering generally occurs when the EM wavelength is large compared to the scatteringsource. EM Scattering occurs when some of the incident wave is absorbed and re-radiated.As in radiation, we keep the lowest terms in the multipole expansion of the scattered fields.Almost always this results in dipole radiation as the incident wave polarizes the charges inthe scattering center. If the incident wave has large amplitude, then it is possible that thecharges will not move linearly with the incident E field, but will bend due to the ⃗v × ⃗ B forceof the magnetic component in the incident wave. An intense incident wave would probablyalso require terms of higher order than the Born term in the perturbation expansion forthe cross section. The scattering formulation occurs as follows (we always assume a timedependence of e iωt )A plane, monochromatic, linearly polarized wave is incident on a scattering center. Theincident wave is polarized in the ˆx direction and moves in the ẑ direction.;⃗E in = ˆxE 0 e ik inrcos(θ)⃗B in = ˆk in /c × ⃗ E inThese fields induce dipole moments (⃗p, ⃗m) in the scattering center. The dipoles radiate energy.This radiation can be calculated in the multipole approximation which uses the staticfields obtained from the multipole moments. The dipole fields are;⃗E sc =4πǫ k2 eikrr [ˆn × ⃗p × ˆn] − z 0k 24π eikrr [ˆn × ⃗m]⃗B sc = (1/c)ˆn × ⃗ E scIn the above, ⃗p and ⃗m are the induced electric and magnetic dipole moments.5
Page 1 and 2: Scattering1 Classical scattering of
Page 3: sClosestApproachθxθθFigure 2: Th
Page 7 and 8: dσ ⊥dΩ = k4 η 62 |ǫ − 1ǫ +
Page 9 and 10: a spherical system, written in term
Page 11 and 12: IncidentDirectionzθn^ScatteredDire
Page 13 and 14: q/2vvq/2φ oFigure 6: Radiation fro
Page 15 and 16: xEρyrPzFigure 7: A schematic of a
Page 17 and 18: RetardedPointn^R =c(t − t’)θβ
Page 19 and 20: cos(θ) = 1/(1.5β)and gives a cut-
Page 21 and 22: oth energy and momentum, write;⃗p
Page 23 and 24: ⃗B(k, ω) = i/c 2 ( ⃗ k × ⃗v

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

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