Subatomic Physics

156 Structure of Subatomic Particles
form factors, respectively, and they are both functions of q 2 . The designations
electric and magnetic stem from the fact that for q 2 = 0, the static limit, they are
given by
GE(q 2 =0)= Q
e , GM (q 2 =0)= µ
µN
, (6.40)
where Q and µ are the charge and magnetic moment, respectively, of the nucleon.
Specifically, GE(0) and GM (0) for the proton and the neutron are
G p
E (0) = 1, GnE (0) = 0,
G p
M (0) = 2.79, GnM (0) = −1.91.
(6.41)
Early electron–proton scattering experiments, (36) performedwithanelectron
energy of 188 MeV, were analyzed by fitting the observed differential cross section
with an expression of the form of Eq. (6.38) with fixed values of the parameters G.
An example is shown in Fig. 6.11. Comparison of the various theoretical curves with
the experimental one indicates that the proton is not a point particle. The conclusion
based on the discussion of the anomalous magnetic moment is consequently
verified by a direct measurement. However, an electron energy of about 200 MeV
is too small to permit studies at significant values of the momentum transfer and
to get information on the q 2 dependence of GE and GM . Since 1956, many experiments
have been performed at accelerators with much higher electron energies.
To extract the form factors from the measured elastic scattering cross sections, the
cross section for a fixed value of q 2 is normalized by division by the Mott cross
section and plotted against tan2 θ/2, as shown in Fig. 6.12. Such a plot should
is obtained. The intersection
yield a straight line; from the slope, the value of G2 M
with the y axis then yields G2 E .
Figure 6.13 gives the magnetic form factor of the proton. For convenience,
GM /(µ/µN) is plotted, where µ is the proton magnetic moment. For comparison
we show also a plot of the function:
GD(q 2 )=
1
(1 + |q| 2 /q2 , (6.42)
0 )2
with q 2 0 =0.71(GeV/c)2 . This function in conjunction with Table 6.1 can help the
reader picture the distribution of magnetism in the proton. Although it is clear that
at values of |q| 2 > 10 (GeV/c) 2 the dipole function does not reproduce the data very
well, it has become customary to compare the form factors to GD (37) . Initially both
the electric and magnetic form factors were determined by the procedure sketched
in Fig. 6.12. This method has the disadvantage that, as |q| 2 gets larger it becomes
more difficult to extract GE as is apparent from Eq. 6.38. Recently there has
36 R.W. McAllister and R. Hofstadter, Phys. Rev. 102, 851 (1956).
37 GD goes down as |q| −4 as |q| 2 →∞, a behavior that is predicted by QCD.