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

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d 2 IdωdΩ = [ q 2 ∞∫16π 3 cǫ ]|−∞dt [ˆn × [(ˆn − ⃗ β) × ˙⃗ β](1 − ⃗ β · ˆn) 2 e iω[t−⃗ β·R/c] | 2For low frequencies, the exponential term is ≈ 1. Note that;ˆn × (ˆn − ⃗ β) × ˙⃗ β1 − ⃗ β · ˆn= d dtIntegrate by parts to obtain;d 2 IdωdΩ = [ q 2 ∞∫16π 3 cǫ ]|−∞d] [ˆn × ˆn × β ⃗1 − β ⃗ · ˆn[ˆn × ˆn × ⃗ β1 − ⃗ β · ˆnAlso assume non-relativistic motion, β → 0.d 2 IdωdΩ = [ q 216π 3 cǫ ]|ˆη · ∆⃗ β| 2]In the above ˆη is the polarization direction, and ∆ ⃗ β is the differential change in the velocityof the charge. In the soft photon limit, the energy of a photon is ω so the number of photonsinto dΩ and between frequencies between ω and ω + dω is approximately;q 2N ph = (1/ω)[16π 3 cǫ ]|ˆη · ∆⃗ β| 2The cross section for soft photon emission in a charged particle collision is;d 3 σ = [N dσdΩ q d(ω)dΩ ph ]gam dΩ qWhere dσdΩ qis the cross section for a Coulomb interaction (Rutherford scattering). Thensubstitute the various equations and integrate. Use figure 5 to obtain ∆ ⃗ β. The result forlow frequencies and non-relativistic motion is;dIdω =q24π 2 cv [ln(1.123γv wb min)]MvMvθQ = |Pin − Pout| < 2Mv∆ P ~ QFigure 5: The geometry for the momentum transfer12
q/2vvq/2φ oFigure 6: Radiation from sets of charge rotating in a circle∆ρ ∆p ≈ with ∆ρ = bb min (cut off value due to screening) = /2Mv = /Q max9 Radiation and coherenceWe have seen that an accelerated charge radiates energy. Suppose we set 2 charges in symmetricpositions on a loop and let them spin around the z axis as shown in figure 6. Thisresults in radiation because the charges are accelerated. Now suppose we consider the loopfilled with a continuium charge distribution that rotates. Contrary to intuition, there isno radiation. This occurs because of coherence of the radiation from each of the elementalcharge distributions cancels the radiation field. We see this as follows. The charge distributionof the two elements as shown in the figure can be written;δ(r − a)ρ = (q/2)a 2 δ(θ − π/2)[δ(φ + φ 0 ) + δ(φ − φ 0 )]We rewrite the charge per unit length so that we can add additional symmetric units ofcharge;λ =q(n + 1) πa δ(φ − πk/N − φ 0) k = 0, 1, 2, · · · , N − 1All of the charge elements rotate with the frequence ω. The charge density is then written;ρ k =q δ(r − a)(n + 1)π a 2 δ(θ − π/2) δ(φ − πk/N − ωt)13
Page 1 and 2: Scattering1 Classical scattering of
Page 3 and 4: sClosestApproachθxθθFigure 2: Th
Page 5 and 6: Now a properly normalized outgoing
Page 7 and 8: dσ ⊥dΩ = k4 η 62 |ǫ − 1ǫ +
Page 9 and 10: a spherical system, written in term
Page 11: IncidentDirectionzθn^ScatteredDire
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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