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

More documents

Recommendations

Info

E = √ 1 ∞∫2π−∞E(t) = √ 1 ∞∫2πdt e iωt E(t)−∞dt e −iωt EThe energy radiated is the time integral of the power. Therefore;dWdΩ= 12πµc∞∫−∞dt∞∫−∞dω∞∫−∞dω ′ ⃗ E∗ · ⃗E e i(ω′ −ω)tInterchange the time and frequency integration and differentiate with respect to ω.dWdΩ= 12µc |E(ω)|2 R 2The above is the energy radiated per unit solid angle. The equation has been multiplied bythe differential area element, R 2 dΩ and a factor of (1/2) is included so that we define ω > 0always. The radiation field (Gaussian units) is;E(t) = (q/c)[ ˆN × (ˆn − β) ⃗ × ˙⃗ βκ 3 ] = (e/cκ) d Rdt [ˆn × (ˆn × β)] ⃗The right side of the energy equation has the form G ∗ G.G = 1(4µc) 1/2 E(ω)R = 1(8πµc) 1/2∞∫−∞dt e iωt [ˆn × (ˆn − ⃗ β) × ˙⃗ βκ 3 ]Change to present time units, κt, and approximate the time in the phase component byt ′ ≈ t − ˆn · ⃗r/c for large r. Integrate by parts over the time, and then put this value for Gback into the equation for the differential power above to obtain;d 2 Idω dΩ = q2 ω 216π 2 c 3 |∞∫−∞dt[ˆn × (ˆn × ⃗ β)] e iω(t−ˆn·⃗r/c) | 2In a dielectric medium the velocity of the EM wave is c → c/n = c/ √ ǫ with n the index ofrefraction. Assume straight line motion so that ⃗r = ⃗ V t .d 2 Idω dΩ = µq2 cω 216π 2 |ˆn × V ⃗ ∞∫(ω 2 /2π) dt e iωt(1−ˆn·⃗r/nc) | 2−∞The integral is a δ function and results in the expression for the intensity of the Cherenkovradiation (Gaussian units);d 2 Idω dΩ = q2 β 2nc sin2 (θ)|δ(1 − (β/n) cos(θ)| 2The above δ function gives the Cherenkov angle as previously obtained. As an example,glass has an index of refraction of 1.5. The Cherenkov angle is then;18
cos(θ) = 1/(1.5β)and gives a cut-off velocity of β = 0.666 An electron with this velocity has a momentum of0.45 MeV.12 Synchrotron radiationSynchrotron radiation occurs when a particle is acclerated perpendicular to the velocity asin a circular orbit. It occurs naturally for electrons moving around the magentic field lines inthe upper atmosphere of the earth, and in electron accelerators used to produce EM radiationfor studies of material strucutre and biological samples. Suppose a relativistic particle in acircular orbit as shown in figure 10. The radius of the orbit is ρ, and we assume a pulsedbeam to avoid coherence because a uniform, continuious beam no radiation occurs.ψρρ2/γ BdAObserverRadiationconeθFigure 10: Synchrotron radiation beams from a circular acceleratorThe arc distance A to B is d = 2ρ/γ. The time for the particle to travel this distance ist = d/v. The first photon leaves the initial pulse from A and travels a distance t/c duringthis time interval. Thus the distance between the 1 st and last photon in the emission coneis ∆L;∆L = 2ργ [1/β − 1] ≈ ρ/γ3The time for the EM pulse to travel this distance is ρ/(γ 3 c) which determines the frequencycut-off for the radiation; ω c = γ 3 c/ρ . We obtain the radiated power from the expression forthe power per solid angle per frequency obtained in the Cherenkov radiation section;d 2 IdωdΩ = q2 ω 216π 2 c 3 |∞∫−∞dt[ˆn × (ˆn × ⃗ β)] e iω(t−ˆn·⃗r/c) | 219
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: RetardedPointn^R =c(t − t’)θβ
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 ...

Create successful ePaper yourself

Delete template?

Save as template?