Subatomic Physics

6.6. Nucleon Elastic Form Factors 155
Figure 6.10: A physical proton is pictured as a superposition of many states, for instance a bare
proton or three quarks, a bare neutron plus a pion, and so forth.
factor. The proton also possesses, in addition to its charge, a magnetic moment. It
is unlikely that it behaves like a point moment and sits at the center of the proton.
It is to be expected that the magnetization is also distributed over the volume of the
nucleon and this distribution will be described by a magnetic form factor. (33) The
detailed computation indeed proves that elastic electromagnetic scattering from a
particle with structure must be described by two form factors; the laboratory
cross section can be written as
spin- 1
2
where
dσ
dΩ =
� �
dσ
� 2 GE + bG
dΩ Mott
2 M
1+b
+2bG 2 M tan2
� ��
θ
, (6.38)
2
b = −q2
4m2 . (6.39)
c2 Equation (6.38) is called the Rosenbluth formula; (34) m is the mass of the nucleon, θ
the scattering angle, and q the four-momentum transferred to the nucleon. (35) The
Mott cross section is given by Eq. (6.11). GE and GM are the electric and magnetic
33 Nuclei with spin J ≥ 1/2 also possess magnetic moments, and the magnetization is also
distributed over the volume of the nucleus. For such nuclei, the discussion given in Section 6.4
must be generalized.
34 M.N. Rosenbluth, Phys. Rev. 79, 615 (1950).
35 Here a word of explanation is in order: The variable q is the four-momentum transfer. It is
defined as
�
E
q =
c
�
E′
− , p − p′ .
c
Its square,
q 2 = 1
c2 (E − E′ ) 2 − (p − p ′ ) 2 = 1
c2 (E − E′ ) 2 − q 2 ,
is a Lorentz-invariant quantity. Since q2 is a Lorentz scalar, its use is preferred in high-energy
physics. For elastic scattering in the c.m. or at low energies, q2 = −q2 .