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40 Chapter 5. The Eikonal Final State
5.4 The Glauber Approach
Glauber theory is a multiple-scattering extension of the standard (non-relativistic)
eikonal approximation that relates the ejectile’s distorted wave function to the proton
elastic scattering data through the introduction of a profile function. In the past,
Glauber theory has been highly succesful in describing small-angle proton-nucleus
scattering in the energy domain T p > 500 MeV [6, 9]. It has been suggested that
Glauber theory may also provide a good starting basis for the description of the
final state interactions in A(e, e ′ p) reactions at high energies.
5.4.1 Non-relativistic Glauber theory
In deriving the non-relativistic eikonal wave function, one starts from the timeindependent
Schrödinger equation
(
)
− ¯h2 ∇
2m ⃗ 2 + V (⃗r) Ψ(⃗r) = EΨ(⃗r) , (5.24)
for a particle with mass m moving in the field of a potential V (⃗r). As in Sec. 5.2,
one arrives, after some straightforward manipulations, at the following eikonal wave
function [63] :
[
Ψ⃗ E kf
(⃗r) = (2π) −3/2 exp i ⃗ k f · ⃗r −
i ∫ ]
z
dz ′ U(x, y, z ′ ) , (5.25)
2k f −∞
where U(⃗r) ≡ 2mV (⃗r)/¯h 2 is the reduced potential.
formally written as
The scattering amplitude is
f E = −2π 2 < Φ ⃗ki |U|Ψ E ⃗ kf
> , (5.26)
with Φ ⃗ki a plane wave corresponding to the momentum ⃗ k i . Inserting the eikonal
wave of Eq. (5.25) into Eq. (5.26) leaves us with the following expression :
f E = − 1 ∫
[
d⃗r e i(⃗ k f − ⃗ k i )·⃗r U(⃗r) exp − i ∫ ]
z
U(x, y, z ′ )dz ′ . (5.27)
4π
2k f −∞
Since only small-angle scattering is considered here, on the basis of Fig. 5.2 one can
write that
( ⃗ k f − ⃗ k i ) · ⃗r ≃ ( ⃗ k f − ⃗ k i ) ·⃗b = ⃗ ∆ ·⃗b . (5.28)
The integration over the variable z is straightforward. With the aid of Eq. (5.28)
the eikonal scattering amplitude can be rewritten as
f E = ik ∫
f
d
2π
⃗ b exp(i∆ ⃗ ·⃗b)Γ(k f , ⃗ b) (5.29)