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36 Chapter 5. The Eikonal Final State
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k f
K
q
incident
direction
Figure 5.2 Illustration of the vectors involved in the eikonal trajectory.
As a matter of fact, this feature is generally not considered as a serious deficiency.
Indeed, the transition matrix elements, which involve bound state wave functions,
are only sensitive to the radial range in which the potentials are non-vanishing.
Another point of concern is the angular range in which the eikonal method is
a valid approximation. In Eq. (5.16) the eikonal phase is calculated by performing
a straight line integration along the direction of the average momentum ⃗ K, as in
Fig. 5.2. A more accurate evaluation of the scattering wave function would in fact
involve the calculation of its phase along the actual curved classical trajectory. This
is exactly the small-angle approximation that was imposed. It must be remembered
that the semi-classical approximation is not valid for large scattering angles. But, as
already mentioned, high energy pA scattering is diffractive and extremely forwardly
peaked.
The calculation of the eikonal phase of Eq. (5.16) involves a transformation to
a reference frame other than the usual laboratory or center-of-mass frame, namely
the frame where the average momentum is pointing along the z axis. As the eikonal
phase has to be reevaluated for every ( ⃗ b, z) point in space, the Dirac eikonal (e, e ′ p)
calculations are very demanding as far as computing power is concerned. In evaluating
the matrix elements, the radial integrations were performed on a 0.1 fm mesh.
5.3 Optical Potentials and the Eikonal Method
The potentials that are used in relativistic Hartree calculations are real potentials.
This results in a purely imaginary eikonal phase (that is, apart from a tiny contribution
in the spin-orbit channel). Although this consistent treatment minimizes the
effects of spurious states, it can also only take elastic contributions into account. In
general, strength from the incident beam is drained into other inelastic channels,
and one needs to incorporate this local absorption in the description of the reaction
process. This is commonly done in DWIA by adopting a complex or optical potential
that is able to describe elastic scattering accompanied by absorption. With
such potentials one obtains an eikonal phase that contains both an imaginary and
real part. This reflects the fact that part of the strength will be removed from the