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5.4. The Glauber Approach 45
shape of the elastic nucleon-nucleon cross section. The elastic scattering amplitude
of Eq. (5.33) is directly related to the elastic differential cross section through the
following expression
d 2 σ el
d∆
= 1 k 2 f
|f el (∆)| 2 = σ2 tot(1 + ɛ 2 )
16π 2 exp(−∆ 2 β 2 ) , (5.34)
which is the standard high-energy approximation of the elastic differential cross
section. Integrating this differential cross section leaves us with
σ el =
∫ d 2 σ el
d∆ d∆ = σ2 tot(1 + ɛ 2 )
16π 2 β 2 . (5.35)
This expression allows to derive the value of the slope parameter from the experimentally
known values of the elastic cross section and the ratios of the real to
imaginary part of the forward elastic scattering amplitudes. An example of such
a calculation can be found in Fig. 5.8, that has been taken from Ref. [65]. In this
reference use is made of a widely spread partial-wave analysis of nucleon-nucleon
elastic scattering data by Arndt et al. [66, 67, 68, 69, 70]. We use the experimental
values for ɛ pp and ɛ pn of Ref. [64] however; they can be found in Fig. 5.9.
By analogy with the central amplitude, the spin-orbit amplitude can be parametrized
as
fpN s = γ k f σpN
tot
(
4π
(ɛs pN + i)exp − ∆2β pN
s2 )
. (5.36)
2
Let us now consider the case of central scattering scattering. Inverting Eq. (5.29)
leaves us with
Γ(k f , ⃗ b) = 2π
ik f
∫
d ⃗ ∆
(2π) 2 exp(−i⃗ ∆ ·⃗b)f E ( ⃗ ∆) . (5.37)
Inserting the parametrization of Eq. (5.33) into this expression, gives us then the
following expression for the profile function :
Γ(k f , ⃗ b) = σtot pN (1 − iɛ (
pN)
exp −
4πβ 2 pN
b2
2β 2 pN
)
. (5.38)
In the analysis of the proton-nucleus scattering data, the spinless version of
Glauber theory was very succesfull [5, 71]. As can be inferred from Fig. 5.6, the
spin-dependent contributions to the proton scattering process are relatively weak
for higher kinetic energies. All results that are presented in this work are obtained
within the framework of the spinless version of the Glauber theory.