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Quantum Field Theory

From Coulomb Scattering to Deep Inelastic Scattering

Coulomb scattering is the scattering of charged particles off one another as a result of the electrostatic force between them. Such process belongs to the first order graphs in the Feynman diagrams. Since there is no virtual particles involved, classical mechanics can be used to show that for a given kinetic energy Ek of the incident particle, the scattering angle is related inversely to the impact parameter b as depicted in Figure 02a. It also shows that for a given the incident particle can probe closer to the target (smaller b) with higher Ek. Since there is no way to measure b experimentally, the cross section = b2 is defined such that the incident particle is initially directed anywhere within such area. The Rutherford scattering formula is derived by identifying the alpha particles to be the incident beam:

d(,Ek)/d = (Ze2/4Ek)2[1/sin4(/2)] ---------- (38c)

where Z is the number of charges in the "nucleus", which contains all the positive charges and most of the mass in an atom.
Impact Parameter Rutherford Scattering This formula is related to the probability of an incident particle with kinetic energy Ek scattered into an solid angle d in the direction . The experimental apparatus to verify the Rutheford scattering is shown in Figure 02b. Figure 02c is the measured angular distribution of the scattered particles with incident energy of 15 Mev.

Figure 02a Impact Parameter [view large image]

Figure 02b Rutherford Scattering
[view large image]
Rutherford Scattering Cross Section, Vary Angles Rutherford Scattering Cross Section, Varying Energy The agreement between theoretical prediction and experimental data overruled J. J. Thomson's "Plum Pudding Model", in which electrons are suspended in a pudding-like positively charged substance that contains most of atom's mass. The success of Rutherford's theory gives him the distinction of having discovered the atomic nucleus. However, there is a problem. When the incident energy is over about 27.5 Mev, the experimental data start to deviate from the theoretical curve. Figure 02d shows the deviation at a fixed angle of 60o.

Figure 02c Cross Section vs Angles [view large image]

Figure 02d Cross Section vs Energy [view large image]

The relativistic correction has been obtained by N. F. Mott within the framework of QED but without recourse to perturbation theory. The improved formula has the form:

d(,E)/d = [(Ze2E)/(2|p|2c2)]2 {[(1 - 2sin2(/2)]/sin4(/2)} ---------- (38d)

where p is the momentum of the incident particle, E = (mo2c4+p2c2)1/2, and =|p|c/E.

When the incident energy increases to over 1 Gev, the collision becomes inelastic. Under the impact of such high energy, the target nucleon is likely to disintegrate. High energy scatterings are used to probe the structure of nucleons. According to the parton model, at high enough energy the probe would see only the point-like parton inside the structure instead of seeing the whole nucleon as a coherent object (see Figure 02e). In general, the inelastic scattering cross section takes on the form:

d2/dq2d = (42/q4) (E'/EM) [(M/)W2(q2,)cos2(/2) + 2W1(q2,)sin2(/2)] ---------- (38e)

where in the limit when the mass of the incident particle is negligible, q2 = (k - k')24EE'sin2(/2), = (pq)/ME - E'. They are the 4-momentum transfer squared, and the energy transfer respectively representing such transfers from the electron to the "off mass-shell" photon with q2 0 (see Figure 02e for the definition of the various notations).

In Eq.(38e) the cross section is expressed in term of the simple Rutherford scattering formula. The W1 and W2 in the rest of the equation are dimensionless form factors. It is used to correct the point particle description. In general the precise structure of the nucleon is unknown, but the form of W1,2 is severely restricted by Lorentz invariance and electromagnetic current conservation. This is an extremely general and important
Parton Model Form Factor tool in the absence of a complete theory. In the deep inelastic region: , q2, and 0 x = q2/(2M) 1 (elastic scattering corresponds to x = 1), W1F1(x), W2F2(x). The F1(x) and F2(x) are finite and depend only on x. This relation is called Bjorken scaling. The usefulness of the parton model is that we can compare the scaling behavior of F1 and F2 against the various predictions for spin-0 and spin-1/2 partons, e.g., F1(x) = 0, and 2xF1(x) = F2(x) respectively for each case. Experimentally, the spin-1/2 model is reasonably satisfied (see Figure 02f). This result suggests that the partons are, in fact, just the quarks. Thus we have progressed from the discovery of atomic nucleus to unravelling the structure of the nucleon in about 100 years.

Figure 02e Deep Inelastic Scattering [view large image]

Figure 02f Form Factor [view large image]

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