Subatomic Physics

186 Structure of Subatomic Particles
Figure 6.33: Arrangement of the individual scatterers in a nucleus.
and for the complete profile function
Γ(ρ) =1−
A�
[1 − Γi(ρ − si)]. (6.107)
i=1
This relation describes the Glauber approximation. If the profile functions for the
individual nucleons are known, the profile function for the entire nucleus can be
calculated. One more step is needed to arrive at the Glauber expression for the
scattering amplitude. Nucleons are not fixed, as shown in Fig. 6.33; they move
around and their probability distribution is given by the relevant wave function.
For elastic scattering, initial and final wave functions are identical, and Γ(ρ) in
Eq. (6.97) must be replaced by
�
d 3 x1 ···d 3 xAψ ∗ (x1,...,xA)Γ(ρ)ψ(x1,...,xA)
≡〈i|Γ(ρ)|i〉.
The scattering amplitude equation (6.97) thus becomes
f(q) = ik
�
2π
d 2 � �
iq·ρ
ρ exp 〈i|Γ(ρ)|i〉,
�
(6.108)
with an inverse which is
〈i|Γ(ρ)|i〉 = 1
2πik
�
� �
iq · ρ
exp − f(q) d
�
2 q.
As an example, we consider the elastic scattering of a high-energy projectile from
the simplest nucleus, the deuteron (Fig. 6.34). When the energy of the incident
particle is so high that its wavelength is much smaller than the deuteron radius
(R ≈ 4 fm), one could at first assume that neutron and proton scatter independently
and that the total cross section is simply the sum of the individual ones. Use of the