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

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zn^θyvtρxvρFigure 11: The geometry used for synchrotron radiation of an ultra-relativistic particleChoose a coordinate system as shown in fig. 11 and substitute for the velocity and the distancefrom the retarded point to the present point, rAlso;ˆn × (ˆn × ⃗ β) = βˆn × (ˆn × [ˆǫ parallel sin(vt/ρ) + ˆx cos(vt/ρ)])ˆn × (ˆn × ⃗ β) = β(−sin(vt/ρ) ˆǫ ‖ + cos(vt/ρ) sin(θ) ˆǫ ⊥ )ˆn · ⃗r = (ρ/c)sin(vt/ρ) cos(θ)Integration proceeds in the ultra-relativistic approximation, β ≈ 1. The result isd 2 Iω dΩ = 27 e2 c λ c32π 3 cρ λ γ8 [1 + (γψ) 2 ] 2 [K2/3 2 (η) +In this equationλ c = 4πργ−33η = λ c[1 + (γψ) 2 ] 3/22λK j i/2 are modified Bessel functions of the 2nd kind.(γψ)21 + (γψ) 2 K 2 1/3 (η)]13 Virtual photonA charged particle of momentum ⃗p I and energy E I cannot decay into a photon and a chargedparticle having the same mass. Such a process is shown in fig. 12. If we attempt of conserve20
oth energy and momentum, write;⃗p I = ⃗p F + ⃗p γE I = E F + E γWhen combining the equationsM 2 γ = E 2 γ − p 2 γ = 2[M 2 + p I p F cos(θ) − E I E F ]In the above M is the mass of the particle and θ is the angle between the incident andoutgoing momentum. Now M 2 γ < 0 so that M γ is imaginary. However, in some cases it isuseful to think that this photon has some of the the properties of a real photon. Indeed inQM such a virtual particle can live within the uncertainty relation ∆E ∆t ≈ . For laternotation we let this photon have energy ω = E I − E F and momentum ⃗q = ⃗ P I − ⃗p F . ThenM 2 γ = ω2 − q 2 < 014 Scattering of virtual quantaThis technique is an approximation to the interaction of charged particles, by viewing thefield as a collection of virtual photons. The approximation works for photons that are almost“on the mass shell”, ie virtual photons that have mass nearly zero.Consider the scattering of a charged particle from another charge system. The figure 12 isa schematic of the model to be described. Indeed the figure illustrates how bremsstrahlungcould be described as the scatering of a virtual photon into one of real mass. Generally wewould have a light particle scattering from a heavier one, but we use a frame in which theheavier mass is in motion and the lighter mass is at rest so that the motion of the heaviermass can be assumed to move with constant velocity during the collision. The lighterparticle recoils and emitts radiation. The relativistic factor for the movement of the chargeQ is γ ≫ 1. Now if γ ≫ 1 the E and B fields are almost transverse. The fields at point P are;E 1 = Q4πǫE 2 = Q4πǫγvt(b 2 + (γvt) 2 ) 3/2γb(b 2 + (γvt) 2 ) 3/2⃗B = ⃗ V × ⃗ EThe equation for the ⃗ B is due to the fact there is no magnetic field in the rest frame of Q.The E field is just the static field of a charge Q a distance r ′ away from the point, Q. Apply21
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 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: cos(θ) = 1/(1.5β)and gives a cut-
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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