E-X2 · Separate charge and magnetism

Proton Form Factors

How can scattered electrons separate a proton’s electric and magnetic responses? At the same momentum transfer, changing the electron energy and angle changes the relative electric and magnetic contributions. The reduced cross section follows a line in virtual-photon polarization. Its slope gives the squared electric response and its intercept gives the magnetic contribution.

Drag the electron detector around a proton target. Record a scan at fixed momentum transfer, then use its slope and intercept to separate electric and magnetic responses.

Current momentum transfer squared $0$

Fitted electric response squared $0$

Fitted magnetic response squared $0$

Model assumptions

Synthetic elastic electron–proton scattering uses one-photon Born kinematics, proton recoil and the Rosenbluth cross section. Energies use an ultra-relativistic electron and a stationary proton. Electric response has a declared dipole shape set by its zero-transfer charge-radius slope; magnetic response shares that shape with a specified magnetic-moment scale. A scan changes incident energy and angle together to hold momentum transfer fixed. Raw Poisson counts include known luminosity/solid-angle exposure and known background. Reduced samples and the ordinary least-squares line fit use the saved counts; noisy fitted squared responses can be negative. The unweighted fit omits optimal weighting and precision parameter-uncertainty estimation. No measured Hofstadter data, three-dimensional proton density reconstruction, two-photon exchange, radiative corrections, acceptance smearing or precision proton-radius fit is supplied. Proton size and beam travel time are enlarged.

Saved synthetic observations

One transfer, a line of responses

Default state

From saved electron counts

Fit to saved samples

Analytic source reference

Beam energy and angle vary together across the scan. Reduced samples use saved raw counts, known background and Mott normalization. The line is unweighted; sample bars show the square-root count scale.

\sigma_R=\epsilon G_E^2+\tau G_M^2

Start with the essentials

Focus question: How can scattered electrons separate a proton’s electric and magnetic responses?
One-sentence intuition: At fixed momentum transfer, the reduced cross section is linear in virtual-photon polarization.

Core mathematical model

Elastic energy and momentum transfer

E'=\frac{E}{1+(2E/M)\sin^2(\theta/2)},\qquad Q^2=4EE'\sin^2(\theta/2)

The proton starts at rest. Energies and the proton mass use GeV; momentum transfer squared uses GeV squared.

The reduced cross section

\sigma_R=\epsilon G_E^2+\tau G_M^2,\quad \tau=\frac{Q^2}{4M^2},\quad\epsilon=\frac1{1+2(1+\tau)\tan^2(\theta/2)}

The electric term controls the slope. At fixed transfer the magnetic term gives the intercept.

From counts to a reduced sample

\widehat\sigma_R=\frac{n-B}{\mathcal L\Delta\Omega\,\sigma_{\rm Mott}/[\epsilon(1+\tau)]}

The stored count is background-subtracted and divided by the known exposure and recoil-corrected Mott factor. The ideal curve is a separate source-model reference.

Common difficulties

A radius parameter is not a density image

Typical misconceptionThe dipole parameter draws a measured three-dimensional proton.

Better mental modelIt specifies the slope of a declared electric form factor. This scan contains synthetic counts and no spatial reconstruction.

Keep the transfer unchanged

Typical misconceptionAny angular scan gives a single Rosenbluth line.

Better mental modelThe line requires the same momentum transfer. Both angle and beam energy change during the saved scan.

Run the experiment

01
Turn the detector
Drag the detector around the target. Watch the outgoing electron and recoil direction.
What to observe: The angle changes the electron energy loss and momentum transfer together.
02
Record a fixed-transfer scan
Record a new run and inspect the line of reduced samples.
What to observe: Each saved sample uses a different angle and incident energy, selected to keep the same momentum transfer.
03
Remove the magnetic term
Choose Remove magnetic term and record a run.
What to observe: The model reference passes through zero intercept. A noisy fitted intercept can fall slightly below zero.