Structure of Subatomic Particles
6.2 Rutherford and Mott Scattering
The classical picture of elastic scattering of an alpha particle by the Coulomb field of a nucleus of charge Ze is shown in Fig. 6.1. This event is called Rutherford scattering if the nucleus is spinless; the alpha particle also has spin 0. The cross section for scattering of a spin-0 particle by a spinless nucleus can be computed classically or quantum mechanically, with the same result. The Rutherford scattering formula is one of the few equations that can be taken over into quantum mechanics without change, and this fact was a source of great pride to Rutherford.(2)
A fast way to derive the differential cross section for Rutherford scattering is based on the first Born approximation. In general, the differential cross section is written as
dσ
dΩ=|f(q)|2, (6.2)
where f (q) is called the scattering amplitude and q is the momentum transfer,
q = p − p. (6.3)
p is the momentum of the incident and p that of the scattered particle. For elastic scattering, Fig. 6.1(b) shows that the magnitude of the momentum transfer is connected to the scattering angle θ by
1E. Rutherford, Phil. Mag.21, 669 (1911).
2Rutherford scorned complicated theories and used to say that a theory is good only if it could be understood by a barmaid. (G. Gamow, My World Line, Viking, New York, 1970.)
6.2. Rutherford and Mott Scattering 137
Figure 6.1: Rutherford scattering. (a) Classical trajectory of a particle with charge Z1e in the field of a heavy nucleus with charge Ze. (b) Representation of the collision in momentum space.
q = 2p sin12θ. (6.4)
In the first Born approximation it is assumed that the incident and the scattered particle can be described by plane waves. The scattering amplitude can then be written as(3)
V (x) is the scattering potential. If it is spherically symmetric, integration over angles can be performed, and the scattering amplitude becomes, with x =|x|,
f (q2) =−2m
Since f no longer depends on the direction of q but only on its magnitude, it is now written as f (q2).
For Rutherford scattering, the potential V (x) is the Coulomb potential.(4) Or-dinarily, the Coulomb interaction between two charges q1q2 at a distance x is writ-ten as
V (x) = q1q2
x .
3We introduce Eq. (6.2) and the Born approximation here without derivation. This omission will be rectified later, in Section 6.11 and, with a different approach, in Problem 10.3. The student who has not yet encountered Eqs. (6.2) and (6.5) should simply use them as a tool here and then study their derivation later. Derivations are also given in Merzbacher, Section 13.4; and Park, Section 9.3.
4In the original Rutherford experiments, the probing particles were α particles. These are hadrons, and if they get close to the nucleus, the hadronic force must also be taken into account.
The experiments discussed here are performed with electrons, and no problems from hadronic forces arise.
In the scattering experiment shown in Fig. 6.1, the nucleus is surrounded by its electron cloud, and the nuclear charge Ze is shielded. Shielding is taken into account by writing where a is a length characteristic of atomic dimension. Eq. (6.7) enables the integral in Eq. (6.6) to be done, and the scattering amplitude becomes
f (q2) =− 2mZ1Ze2
q2+ (/a)2. (6.8)
In all collisions exploring the structure of nuclei, the momentum transfer q is at least of the order of a few MeV/c, and the term (/a)2can be neglected completely.
With Eqs. (6.8) and (6.2) the Rutherford differential cross section becomes
dσ
The Rutherford scattering formula, Eq. (6.9), is based on a number of assump-tions. The four most important ones are
1. The Born approximation.
2. The target particle is very heavy and does not take up energy (no recoil).
3. The incident and target particle have spin 0.
4. The incident and target particle have no structure; they are assumed to be point particles.
These four restrictions have to be justified or removed. We shall retain and justify the first two and partially remove the second two.
1. The Born approximation assumes that the incident and the outgoing particle can be described by plane waves. Such an assumption is allowed as long as
Z1Ze2
c 1. (6.10)
If condition (6.10) is not satisfied, a more detailed calculation is necessary (phase-shift analysis or higher Born approximations).(5) The essential physical aspects can, however, be understood by using the first Born approximation, and we shall not go beyond it.
5D.R. Yennie, D.G. Ravenhall, and R.N. Wilson, Phys. Rev. 95, 500 (1954).
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-12 particles with charge |Z1| = 1 from spinless target particles has been treated by Mott, and the cross section for Mott scattering is(6)
dσ
E is the energy of the incident electron and v = βc its velocity. The term β2sin2θ/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 12, 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.
6A 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.