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34 Chapter 5. The Eikonal Final State
Figure 5.1 The 500 MeV p- 40 Ca elastic scattering cross section at T p = 500 MeV, calculated
using the Dirac eikonal method of Ref. [57] (solid line) compared to that using a Dirac
partial wave code (dashed line). This picture was taken from Ref. [57].
the solutions to the equation (5.8) determine the complete relativistic eigenvalue
problem.
So far no approximations have been made. Various groups [17, 18, 55] have
solved a Dirac equation of the type (5.8) for the final scattering state using Dirac
optical potentials derived from global fits to elastic proton scattering data [56].
Not only are global parametrizations of Dirac optical potentials usually restricted
to proton kinetic energies T p ≤ 1 GeV, calculations based on exact solutions of the
Dirac equation frequently become impractical at higher energies. This is particularly
the case for approaches that rely on partial-wave expansions in determining the
transition matrix elements. To overcome these complications, we solve the Dirac
equation (5.8) in the eikonal limit [57, 58, 59, 60]. In intermediate-energy proton
scattering (T p ≈ 500 MeV) the eikonal approximation was shown to reproduce fairly
well the exact Dirac partial wave results as can been seen from Fig. 5.1 [57].
Following the discussion of Ref. [57], we define the average momentum K ⃗ and
the momentum transfer ⃗q in terms of the projected initial ( ⃗ k i ) and final momentum
( ⃗ k f ) of the ejectile
⃗q = ⃗ k f − ⃗ k i , (5.11)
⃗K = 1 2 (⃗ k f + ⃗q) . (5.12)