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Chapter 7 - electron-proton elastic scattering - Coggle Diagram
Chapter 7 - electron-proton elastic scattering
Types of e-p
elastic scattering
Proton target is usually liquid hydrogen at rest.
Observables: energy/momentum of initial and scattered electron.
Rutherford
scattering
: non-relativistic \(e^-\), negligible proton recoil.
\(\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{16 E_K^2\sin^4(\frac{\theta}{2})}\)
\(e^-\) only interacts with the \(V= e^2/r\) electric potential
Mott scattering
: relativistic \(e^-\), still ok to neglect proton recoil
\(\frac{d\sigma}{d\Omega}_{Mott} = \frac{\alpha^2}{4 E^2\sin^4(\frac{\theta}{2})} \cos^2(\frac{\theta}{2})\)
To account for
finite size
of the potential (either
electric
or
magnetic
), instead of treat proton as point-like particle, introduce
Form Factor
\(F(\mathbf{q^2}) = \int \rho(r) e^{i\mathbf{q}r} d^3r\)
\((\frac{d\sigma}{d\Omega})_{Mott} \rightarrow (\frac{d\sigma}{d\Omega}) |F(\mathbf{q^2})|^2\)
Kinematics
\(q^2 = (p1-p3)^3\), define \(Q^2 = -q^2 = 4E_1E_3\sin^2(\frac{\theta}{2})\)
Write everything in electron observables
Rosenbluth formula
\(\frac{d\sigma}{d\Omega} = \frac{\alpha^2}{4E_1^2\sin^4(\theta/2)}\frac{E_3}{E_1}\Big( \frac{G_E^2 + \tau G_M^2}{1 + \tau}\cos^2(\frac{\theta}{2}) + 2\tau G_M^2 \sin^2(\frac{\theta}{2})\Big)\)
\(\tau = \frac{Q^2}{4m_p^2}\)
\(G_E(Q^2), G_M(Q^2)\) describes the
charge distribution
/
magnetic moment distribution
of proton.
Proton has "
anomalous
"
magnetic moment:
instead of \(\mu = \frac{q}{m}S\) for point-like particle, \(\mu_p = 2.79 \frac{e}{m_p}S\)
Therefore, expect \(G_E(Q^2=0)=0, G_M(Q^2=0)=2.79\)
Measure \(G_E, G_M\) at low and high energy \((Q^2)\)respectively, where \(\tau \rightarrow 0\) and \(\tau \rightarrow \infty\)
Result 1: \(G_E, G_M\) decrease with increasing \(Q^2\) ->
proton is not a point particle
!
\(G_M(Q^2) ~ 2.79G_E(Q^2) ~ 2.79\frac{1}{1 + Q^2}\), a "dipole function".
Proton charge distribution
: \(\rho(r) \propto e^{\frac{-r}{a}}\), where \(a \sim 0.24 fm\)
Result 2: \(G_E, G_M\) at \(Q^2=0\) consistent with 1 and 2.79