Subatomic Physics

162 Structure of Subatomic Particles
Figure 6.16: Typical double differential cross section, normalized by dividing through by the Mott
cross section, for inelastic electron scattering from a nucleus. The final rise shown is due to the
onset of pion production.
but in the laboratory system where p h = 0. For quasi-elastic scattering, on the
other hand, the energy loss is taken up by a single nucleon that is usually ejected
from the nucleus; ν is
ν = |q2 |
, (6.52)
2m
where m is the mass of a nucleon. The peak is not sharp because the nucleon is
bound in the nucleus and therefore has a momentum spread of order of magnitude
given by the uncertainty principle, namely �/R ∼ 100 MeV/c, where R is the
nuclear radius. Finally, one reaches a characterless continuum region where many
broad states are excited. For the measurement of the differential cross section in this
continuum region and for broad resonances it is necessary to determine the double
differential cross section d 2 σ/dE ′ dΩ, which is proportional to the probability of a
scattering occurring in a given solid angle dΩ and into an energy interval between
E ′ and E ′ + dE ′ . At still higher energies, barely shown in Fig. 6.16 pion production
occurs and new features appear.
A scattering spectrum on a proton target is sketched in Fig. 6.17. Its appearance
resembles that of Fig. 6.16 except that it is plotted as a function of E ′ rather than
ν and there is no quasi-elastic peak. The reason for this absence is that quarks are
permanently confined inside the proton and cannot be ejected. The elastic cross
section, already discussed in Section 6.6, is shown in Fig. 6.18 normalized by division
through the Mott cross section, Eq. (6.11). The differential cross sections for the
production of particular resonances can also be studied; their angular distributions
have features similar to the elastic case. Like the nucleus, the nucleon in its excited
states has a spatial extension similar to that in its ground state.