Subatomic Physics

6.2. Rutherford and Mott Scattering 139
2. Only elastic scattering is considered here. The target particle remains in
its ground state, and it does not accept excitation energy. Moreover, it is
assumed to be so heavy that its recoil energy can be neglected. However, as
Fig. 6.1(b) shows, a very large momentum can be transferred to the target
particle. At first the idea of a collision with large momentum transfer but with
negligible energy transfer seems unrealistic. A simple experiment will convince
an unbeliever that such a process is possible: take a car or motorcycle and
race straight into a concrete wall. If well constructed, the wall will take up
the entire momentum but will accept very little energy. Most of the later
discussion will be concerned with the scattering of electrons from nuclei and
nucleons. In this case, restriction 2 is satisfied as long as the ratio of incident
electron energy to target rest energy is small. At higher energy, the cross
section can be corrected for nucleon or nuclear recoil in a straightforward
manner. Essential results remain unaffected, and we shall therefore not treat
the recoil corrections.
3. As just pointed out, most experiments to be discussed concern the scattering
of electrons. In this case, the spin has to be taken into account. Scattering of
spin- 1
2 particles with charge |Z1| = 1 from spinless target particles has been
treated by Mott, and the cross section for Mott scattering is (6)
� �
dσ
=4(Ze
dΩ Mott
2 2 E2
)
(qc) 4
�
1 − β 2 �
2 θ
sin . (6.11)
2
E is the energy of the incident electron and v = βc its velocity. The term
β 2 sin 2 θ/2 comes from the interaction of the electron’s magnetic moment with
the magnetic field of the target. In the rest frame of the target, this field
vanishes, but in the electron’s rest frame, it is present. The term is peculiar
to spin 1
2 , it disappears as β → 0, and it is as important as the ordinary
electric interaction as β → 1 since the magnetic and electric forces are then of
equal strength. In the limit β → 0(E → mc2 ), the Mott cross section reduces
to the Rutherford formula, Eq. (6.9).
4. The aim of the present chapter is the exploration of the structure of subatomic
particles, and restriction 4 must consequently be removed. This task will be
performed in the following section.
6 A relatively easy-to-read derivation of Eq. (6.11) can be found in R. Hofstadter, Annu. Rev.
Nucl. Sci. 7, 231 (1958). A more sophisticated proof is given in J.D. Bjorken and S.D. Drell,
Relativistic Quantum Mechanics, McGraw-Hill, New York, 1964, p. 106, or in J. J. Sakurai,
Advanced Quantum Mechanics, Addison-Wesley, Reading, Mass., 1967, p. 193.