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Elastic Scattering in Electromagnetism In this lecture we will practice some more with calculating cross sections, playing special attention to the differential cross section in the laboratory frame. In Lab frame pb (M ,0,0,0) We assume that me<< M (a good assumption for the cases of physical interest) so that we can ignore me in what follows If E M the target is able to absorb momentum without effectively changing its energy and one can neglect the effects of target recoil. Aram Kotzinian 1 2004,Torino In the LAB frame with (essentially) massless electrons we have We will often use E, E’ and L as our variables. However, note that they are not independent. The last expression makes explicit the point that, when E M , E E ' 2004,Torino Aram Kotzinian 2 Next we need to derive the form of the differential cross section in the LAB frame. In the last lecture we had Noting that the variable t has both an explicit cos L dependence and an implicit dependence through E’, we find that Finally 2004,Torino Aram Kotzinian 3 For comparison the CM result is The remaining issue is to evaluate the matrix element M . The simplest case, and the one studied first historically, is that of a scalar scattering on scalar, i.e., two spin 0 bosons scattering via the exchange of a photon. The amplitude squared (no spin summing or averaging needed here) 2004,Torino Aram Kotzinian 4 Thus the differential cross section looks like The first factor in this last expression is called the “Rutherford cross section”, while the last factor is clearly a “recoil” correction, which becomes unity in the limit E M 2004,Torino Aram Kotzinian 5 The next step is to scatter “real” spin ½ electrons but still from a scalar target Thus the spin averaged and summed squared amplitude becomes Using the appropriate trace identity we find 2004,Torino Aram Kotzinian 6 Thus the differential cross section can be expressed as and Note, that 2004,Torino d ( L ) 0 d L Aram Kotzinian 7 We recognize the first factor as our previous result for scalar-scalar scattering (the Rutherford result) and the last factor as a (slightly different) recoil correction. Historically the above result (modulo the recoil correction) was referred to as the Mott cross section. We can easily work out the form of this cross section in CM variables, where we have used the relation Note that the cross sections in the 2 reference frames have very similar structure. 2004,Torino Aram Kotzinian 8 Recall that the standard fermion-QED vertex is helicity conserving. 1 2 1 R (1 5 ) ; R 2 1 (1 5 ) 2 1 (1 5 ) 2 where L (1 5 ) ; L Thus, if the incident electron is right handed, the scattered electron will be right handed also For backward scattering the exchanged photon must have J3 = +1, helicity +1, in order to conserve angular momentum at the electron vertex. Such a photon cannot be absorbed by a spin zero particle and conserve total angular momentum at the scalar vertex. The vanishing of the factor cos2 L 2 in this limit accounts for this fact. 2004,Torino Aram Kotzinian 9 The next level of complexity is illustrated by the case of fermion-fermion scattering. Ignoring the mass of the electron This is identical to the electron-scalar result except for the final term. In terms of LAB frame variables we have Note, that in this case d ( L ) 0 d L Aram Kotzinian 10 2004,Torino Compared to our previous result for an electron scattering from a scalar this result for scattering from another spin ½ particle exhibits a new possible type of process. In this case the spin of the “stationary” target particle can flip, a process not present in the scalar case. This configuration allows the absorption of the helicity ±1 photons described above. It is this process that leads to the sin 4 l 2 term in the numerator, which dominates at L Note that this term vanishes in the no recoil limit where E M 2004,Torino Aram Kotzinian 11 The final step of complexity is the elastic scattering of an electron from a proton, which is not a point particle. In general, the matrix element M J ( t arg et ) J ( beam) , where J ( p p' ) for pointlike scalars J u (p' ) u(p) for poinlike fermions Now, for objects with structure J ( p p' ) F(q 2 ) for scalars q 2 q q , q p p ' four-momentum transfer Object with extended charge distribution d d pointlike 2 2 F q d d 2004,Torino Aram Kotzinian 12 In nonrelativistic (three-dimensional) case F q 2 d 3 r e iqr r F ( 0) 1 d 3 r r 1 where (r) – charge distribution (q r ) 2 1 2 2 2 3 F (q ) d r (1 iq r ) (r ) 1 q r 2 6 Mean square charge radius: dF (q 2 ) r 2 6 2 dq q 2 0 2004,Torino Aram Kotzinian 13 A conventional normalization for the electromagnetic current of the proton in relativistic case is where k is the normalized anomalous magnetic moment of the proton, k 1.79 N and the form factors are normalized to have limits F1(0) = F2(0) = 1. The usual limits for a point Dirac particle are F1 = 1, F2 = 0 (all q 2 ). Other definitions often seen in the literature include “electric” and “magnetic” form factors which describe the full complexity of the (separate) elastic electric and magnetic couplings of the proton. 2004,Torino Aram Kotzinian 14 Using the above expression for the photon-proton coupling we find a LAB frame differential cross section of the form In terms of the electric and magnetic form factors this is often written (the Rosenbluth form) in terms of the variable The last term, spin flip arises from the magnetic interaction 2004,Torino Aram Kotzinian 15 Measurement of the elastic electron-proton scattering McAllister and Hofstadter 1956 - 188 MeV and 236 MeV electron beam from linear accelerator at Stanford Hydrogen and helium (gas) targets Magnetic spectrometer Measurements of scattering angle and energy 2004,Torino Aram Kotzinian 16 Elastic profiles Typical elastic profiles obtained in the scattering experiment with hydrogen gas The elastic profiles were used to obtain cross-section of scattering the cross-section at a particular scattering angle is proportional to the numb of observed events at this angle (area under the curve) 2004,Torino Aram Kotzinian 17 Energy estimates Energy of scattered electrons as a function of scattering angle in the laboratory frame Energy was measured as peak value of the elastic profile (from spectrometer data) Theoretical curve was obtained using relativistic kinematics 2004,Torino Aram Kotzinian 18 Cross-section of elastic scattering a - Mott curve for spinless point-like proton b - Rosenbluth curve for a point-like proton with the Dirac magnetic moment (without anomalous magnetic moment) F1 (q2) = 1, F2 (q2) = 0 c - Rosenbluth curve with contribution from anomalous magnetic moment for point-like proton F1 (q2) = 1, F2 (q2) = The deviation of experimental data from curve (c) were interpreted as an effect from proton form-factors - finite size proton 2004,Torino Aram Kotzinian 19 The size of the proton The experiment was not sensitive enough to the q2 dependence of the form factors. All that could be determined was a mean square radius of the nucleon. The best fit was achieved for <r2>1/2 = (0.70 ±0.24)10-13 cm. With 236 MeV electrons <r2>1/2 = (0.78±0.20)10-13cm. Combined result - <r2>1/2 = (0.74±0.24)10-13 cm. 2004,Torino Aram Kotzinian 20 Sometime in literature another normalization is used: Q2 2 4mN 2004,Torino Aram Kotzinian 21 2004,Torino Aram Kotzinian 22 No free neutron: extract from e-D elastic scattering 2004,Torino Aram Kotzinian 23