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

More documents

Recommendations

Info

zθn^^k ina^xx^φa^yFigure 3: The geometry used to obtain the polarization vectors for scattering from a dielectricsphere4 Dipole cross section for a dielectric sphereUsing the Poynting vector and the dipole fields obtained previously. We find the differentialradiated power for dipole.dσdΩ = r2 |â · ⃗E sc | 2ẑ · | E ⃗ in | 2The unit vector â defines directions perpendicular to the observation direction ˆn. Substitutionfor the field givesdσdΩ =k4(4πǫ) 2 |â · ⃗p + (1/c)(ˆn × â) · ⃗m| 2For a small dielectric sphere of radius, η, the electric dipole moment of the sphere is;⃗p = 4πǫ 0 ( ǫ − 1ǫ + 2 )η3 ⃗ EinUse the geometry shown in Figure. 3. The cross section is obtained for the scattered polarizationperpendicular â ⊥ and parallel, â ‖ to the scattering plane as defined by the planecontaining the incident and scattering vector directions. From the figure;â ‖ · ˆx = cos(θ) cos(φ)â ⊥ · ˆx = −sin(φ)Averaging over all possible initial polarizations (ie averaging over φ) we obtaindσ ‖dΩ = k4 η 62 |ǫ − 1ǫ + 2 |2 cos 2 (θ)6
dσ ⊥dΩ = k4 η 62 |ǫ − 1ǫ + 2 |2The total differential cross section is;dσdΩ = dσ ‖dΩ + dσ ⊥dΩThe total cross section is obtained by integration over dΩ.σ T = 8πk4 η 63| ǫ − 1ǫ + 2 |2The polarization of the scattered wave is;P(θ) = dσ ⊥/dΩ − dσ ‖ /dΩdσ ⊥ /dΩ + dσ ‖ /dΩ =sin2 (θ)1 + cos 2 (θ)5 Form factorSuppose the scatterers are a system of charges with fixed locations in space. The resultis a superposition of scattering amplitudes from all the point scattering centers. We havepreviously seen that coherence must be included when one adds the contributions from eachscattering point. Assume that the positions of the scatters are located by the vectors, ⃗x j .The incident wave will have the phase factor, e ik 0ˆn 0·⃗x j, and the phase for the scattered waveat large distances will be e ikˆn·⃗x j. Here ˆn 0 is the direction of the scattered wave. We assumedipole scattering and write;dσdΩ =k4(4πǫ) 2 | ∑ j[â · ⃗p j + (1/c)(ˆn × â) · ⃗m j ] e i⃗q·⃗x j| 2In the above write ⃗q = kˆn 0 − kˆn = k(ˆn 0 − ˆn) for elastic scattering from massive points,(ie no recoil). Then suppose that the scatterers are identical so that ⃗p j = ⃗p and ⃗m j = ⃗m.Rewrite the above as;where;dσdΩ = [dσ sdΩ ] F(q)dσ sdΩ =F(q) = | ∑ jk4(4πǫ) 2 [â · ⃗p j + (1/c)(ˆn × â) · ⃗m j ] 2e i⃗q·⃗x j| 2 = ∑ jne i⃗q·(⃗x j−⃗x n)7
Page 1 and 2: Scattering1 Classical scattering of
Page 3 and 4: sClosestApproachθxθθFigure 2: Th
Page 5: Now a properly normalized outgoing
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 ...

Create successful ePaper yourself

Delete template?

Save as template?