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

a spherical system, written in terms of spherical harmonics. We write;e ikz = e ikr cos(θ) = ∑ C n Y 0 nUse the orthonormal properties of the spherical harmonic to write;C n = 2π ∫ d(cos(θ)) Y 0 neikrcos(θ)Note that an integral representation of the spherical Bessel function, j l (kr), isj l (kr) = (−i) l /2π∫0d(cos(θ)) e ikr cos(θ) P n (cos(θ)We may also use the addition theorem which has the form;P l (cos(θ)) =4π2l + 1∑mYlm (θ 1 , φ 1 ) Yl ∗ m (θ 2 , φ 2 )where the angles (1) and (2) refer to the angles of vectors ⃗ k and ⃗r in the specified coordinatesystem. This gives the required form for the plane wave;e ikr cos(θ) = 4π ∑ i l j l (kr) Yl m (ˆk) Yl∗ m (ˆr)l,mThis form for the incident wave can then be matched at the scattering surface to the fieldequations inside and outside the scattering center by matching the boundary conditions atthe surface. Recall that a similar technique was used to obtain the reflection (scattering)and transmission coefficients of an EM wave incident on a dielectric and conducting surfaceplaner surfaces. For example, this technique can be used to develop Mie scattering (lightscattering from small dielectric particles).7 Thompson scattering (e.g.Classical EM scattering fromatomic electrons)The power radiated by an accelerated charge is obtained from the Poynting vector.⃗S = ⃗ E × ⃗ H = ǫc|E| 2 ˆnThe radiation field of an accelerated charge when β ≪ c is ;⃗E =4πǫ q × ˆn ×⃗˙β[ˆnR] ret9