1. 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
We assume that me<< M (a good assumption for the cases of physical interest) so that we can ignore
me in what follows If
the target is able to absorb momentum without effectively changing its energy and one can neglect the effects of target recoil.
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Aram Kotzinian

2. In the LAB frame with (essentially) massless electrons we have
We will often use E, E’ and as our variables.  However, note that they are not independent. 
The last expression makes explicit the point that, when
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Aram Kotzinian

3. 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 dependence and an implicit dependence through E’, we find that
Finally
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Aram Kotzinian

4. 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

5. 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
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Aram Kotzinian

6. Thus the spin averaged and summed squared amplitude becomes
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
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Aram Kotzinian

7. Thus the differential cross section can be expressed as
and
Note, that
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Aram Kotzinian

8. We can easily work out the form of this cross section in CM variables,
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

9. Recall that the standard fermion-QED vertex is helicity conserving.
where
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   in this limit accounts for this fact.
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Aram Kotzinian

10. 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
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Aram Kotzinian

11. Note that this term vanishes in the no recoil limit where
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 term in the numerator, which dominates at  
Note that this term vanishes in the no recoil limit where
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Aram Kotzinian

12. Now, for objects with structure
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 where
Now, for objects with structure
four-momentum transfer
Object with extended charge distribution
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Aram Kotzinian

13. In nonrelativistic (three-dimensional) case
where r(r) – charge distribution
Mean square charge radius:
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Aram Kotzinian

14. where k is the normalized anomalous magnetic moment of the proton,
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,
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 ).  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

15. The last term, spin flip arises from the magnetic interaction
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

16. Measurement of the elastic electron-proton scattering
McAllister and Hofstadter MeV and 236 MeV electron beam from linear accelerator at Stanford
Hydrogen and helium (gas) targets
Magnetic spectrometer
Measurements of scattering angle and energy
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Aram Kotzinian

17. 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 number
of observed events at this angle (area under the curve)
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Aram Kotzinian

18. 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
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Aram Kotzinian

19. 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
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Aram Kotzinian

20. 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.
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Aram Kotzinian

21. Sometime in literature another normalization is used:
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22. 2004,Torino
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23. No free neutron: extract from e-D elastic scattering
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Aram Kotzinian