Subatomic Physics

180 Structure of Subatomic Particles
be modified to take these complications into account, and the resulting theory fits
the experimental data reasonably well. (64,65)
Diffraction phenomena appear also in high-energy physics. (66,67) We restrict the
discussion to elastic proton–proton scattering because it already displays characteristic
diffraction features. Differential cross sections, dσ/d|t|, with|t| = |q| 2 ,for
elastic pp scattering at various momenta are shown in Fig. 6.29. (68) The spectacular
forward peak stands out clearly, and some other diffraction traits are also evident.
In particular, the value of dσ/d|t| at |t| = 0 is approximately independent of the
incident momentum, and this turns out to be a prediction of the simple dark-disk
model mentioned above. The total cross section can be extracted from these measurements
via the optical theorem, Eq. (6.78) and it is shown in Fig. 6.30.
Fig. 6.30 shows also the ¯pp cross section and confirms a prediction of high energy
physics, namely, that particle and antiparticle cross sections on a given target
should approach each other at very high energies because there are so many possible
reactions that the difference becomes blurred.
In nuclear physics, the most outstanding diffraction structure is the occurrence
of maxima and minima as shown in Fig. 6.28. In particle physics, the smooth
distribution of the electric charge and presumably also of nuclear matter washes out
the diffraction structure up to momenta of at least 20 GeV/c. At higher momenta,
however, the first minimum and the following maximum appear as shown in the
lowest curve in Fig. 6.29.
The Profile Function (69) The black-disk approximation reproduces the coarse
features, but not the finer details, of diffraction scattering. It can be improved by
assuming the scatterer to be gray. The shadow of a gray scatterer is not uniformly
black; its grayness (transmission) is a function of ρ, whereρ is the radius vector
in the shadow plane (Fig. 6.31). Knowing the shadow allows calculation of the
scattering amplitude, f(θ). In the black-disk approximation the total wave, ψ(r ′ ) ≡
ψ(ρ), in the shadow plane is zero behind the scatterer. For a gray scatterer it is
assumed that the total wave behind the scatterer in the shadow plane is given by
ψ(ρ) =e ik0·ρ e iχ(ρ) . (6.93)
66 F. Zachariasen, Phys. Rep. C2, 1 (1971); B. T. Feld, Models of Elementary Particles,
Ginn/Blaisdell, Waltham, Mass., 1969, Chapter 11. M. Kawasaki et al, Phys. Rev. D 70, 114024
(2004).
67 M. M. Islam, Phys. Today 25, 23 (May 1972); for details see Diffraction 2000, R.Fioreetal.
eds, North-Holland, Elsevier (2001), Nucl. Phys. B Proceedings, suplements; 99A (2001).
68 J. V. Allaby et al., Nucl. Phys. B52, 316 (1973); G. Barbiellini et al., Phys. Lett. 39B, 663
(1972); A. Böhm et al., Phys. Lett. 49B, 491 (1974).
69 R.J. Glauber, in Lectures in Theoretical Physics, Vol. 1 (W. E. Brittin et al., eds.), Wiley-
Interscience, New York, 1959, p. 315; R.J. Glauber, in High Energy Physics and Nuclear Structure
(G. Alexander, ed.), North-Holland, Amsterdam, 1967, p. 311; W. Czyz, in The Growth Points
of Physics, Rivista Nuovo Cimento 1, Special No., 42 (1969) (From Conf. European Physical
Society).