Probing internal structure: Armed with this concept, we now return to Thomson scattering but replace the photon beam by π-mesons. First, we note that the Thomson cross section for the scattering of long-wavelength photons off a proton target is
$$ σ_{TH} = \frac{2}{3} α^2 (4 π R_p^2)$$
where Rp = 1/mp, in natural units. Our analyzer photon beam sees an effective area (4 π R_p^2) of the target proton, and α is, as before, the probability of absorbing and emitting a photon. The factor 2/3 is immaterial for present considerations; it actually reflects the fact that for a photon, one of the three polarization states expected for a particle of spin 1 is missing.