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At what energies and momentum transfers? Also do you have a polarized beam or target? – dmckee Jan 23 '12 at 15:59
Any energy/momentum transfer for which data is available. – Computist Jan 23 '12 at 20:41
The point is that data are available over very wide ranges in energy. SLAC ran at up to $50\text{ GeV}$ electron beams. My dissertation project to $\mathrm{H}(e,e'p)$ at $5.0$ and $5.5\text{ GeV}$ beams and $Q^2$s between $3.3$ and $8.1\text{ GeV}^2$. – dmckee Jan 23 '12 at 21:57
up vote 4 down vote accepted

you can search the HEPDATA database at with the query string

[reac = e- p --> e- p]

and the first result will be:

"Jefferson Lab. Measurement of the elastic electron-proton cross section in the Q*2 range from 0.4 to 5.5 GeV*2"

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The answer is highly dependent on the scale of the momentum transfer.

The figure of merit is $Q^2$ is the squared momentum transfer, and in some regimes the missing energy $\omega = \epsilon' - \epsilon$. The formalism is usually developed in the lab frame with a stationary proton target and a energetic electron beam. We write $Q = \mathbf{k}' - \mathbf{k}$ and $\mathbf{k} = (\epsilon,\vec{k})$ and $\mathbf{k}' = (\epsilon',\vec{k}')$ are the four momentum of the incident and scattered electron respectively.

  • If $Q^2 \ll m_p^2$ then you can treat the proton as a point particle to first order and you can simple look this up. In the upolarized case you use the Mott cross-section.

  • For Q^2 on the same order of magnitude at the proton mass squared the problem is complicated enough that one typically uses a parameterized experiment results in the shape of a set of "form factors" (note that the formalism typically used at medium energies is different from but equivalent to that used at high energies). The JLAB results that luksen links to are among the highest precision available at this time.

    In nuclear physics parlance you get $$\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} = \left( \frac{\mathrm{d}\sigma}{\mathrm{d}\Omega} \right)_{\text{Mott}} \frac{Q^2}{\left|\vec{q}\right|^2} \left[ G^2_\mathrm{E}(Q^2) + \tau \epsilon^{-1}G^2_\mathrm{M}(Q^2)\right] .$$

    To a first approximation you can use the dipole form for the form factors $$G_\mathrm{M} \approx \mu G_\mathrm{E} \approx \mu \left( 1 + \frac{Q^2}{0.71\text{ GeV}^2}\right)^{-2} ,$$ where $\mu$ is the magnetic moment of the proton.

  • For $Q^2 \gg m_p^2$ you are in the deep inelastic scattering regime and can treat the proton at a collection of bound partons. The phrase you're looking for is "structure functions" (PDF link).

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