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Chapter 5
The Eikonal Final State
In the description of photoinduced breakup reactions, semi-classical methods become
useful whenever the de Broglie wavelength λ = h/k of the ejected particle with
momentum k is sufficiently short compared with the distance in which the potential
varies appreciably. The eikonal approximation, which finds its origins in optics,
belongs to the class of semi-classical methods. If the potential varies smoothly and
has a range a, this short wavelength condition is equivalent to the requirement that
ka ≫ 1. Referring to the potentials of Fig. 3.2, it is clear that the potentials vary
smoothly and have an approximate range of 2-3 fm. The condition to be fulfilled
can then be written as k ≫ 500 MeV/c, what will be the case in our discussion of
the performed calculations.
For A(e, e ′ p) reactions with proton kinetic energies T p ≥ 500 MeV/c a relativistic
description is a prerequisite. Originally, the gap between the low and the
intermediate/high energy range was bridged by a distorted wave formalism, where
a broad basis of Dirac eigenstates are joined in a coherent manner to construct the
final distorted scattering state [49, 50, 51]. In principle, this method allows to solve
the Dirac equation for the scattered particle exactly. As already mentioned in the
introduction of this work this method can no longer be applied at higher energies,
say a few GeV. As one goes higher in energies, one needs an increasingly larger base
of plane wave solutions to reach a certain point of convergence, and, unfortunately,
this procedure gets increasingly more tedious and the numerical stability gets out
of hand. Since we want to develop a relativistic model that makes use of (optical)
scattering potentials, and still remains valid for higher values of Q 2 , we need another
way to describe the scattering problem.
Fortunately, an alternative is offered by the eikonal method, and its multiplescattering
extension, which was developed by Glauber [5]. This technique is primarily
based on the observation that at high energies, pp reactions become highly
diffractive, implying that they are extremely forwardly peaked.
In this chapter, we will first briefly review the relativistic distorted wave impulse
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