Subatomic Physics

6.11. More Details on Scattering and Structure 185
The Glauber Approximation (69,74) So far we have treated diffraction scattering
from a single object. We shall now turn to the coherent scattering of a projectile
from a target made up of several subunits, for instance, a nucleus built from nucleons.
An incoming high-energy particle can collide with a single nucleon, with many
in succession, or it can interact strongly with several at once. The treatment of
such a multiscattering process is difficult, but diffraction theory makes the problem
manageable; it leads to the Glauber approximation. (74)
To arrive at the Glauber approximation, we consider first the optical analog,
the passage of a light wave with momentum p = �k through a medium with index
of refraction n and thickness d. The electric vector, E1, after passage of the wave
through the absorber is related to the electric vector of the incident wave, E0, by (75)
E1 = E0 exp(iχ1), χ1 = k(1 − n) d. (6.105)
If the index of refraction is complex, then its imaginary part describes the absorption
of the wave. If the wave traverses successive absorbers, each characterized by a phase
χi, the end result is
En = E0 exp(iχ1)exp(iχ2) ···exp(iχn)
= E0 exp[i(χ1 + ···+ χn)] (6.106)
The phases of the various absorbers add. The same technique can be applied to
the scattering of high-energy particles. Equation (6.93) shows that the wave behind
a single scatterer is related to the incident wave as the electric waves are related
in Eq. (6.105). In the Glauber approximation it is assumed that the phases from
the individual scatterers in a compound system, such as a nucleus, also add. To
formulate the approximation, we assume that the individual scatterers are arranged
as shown in Fig. 6.33. The distance of the center of each scatterer to the axis
perpendicular to the shadow plane is denoted by si. The distance that determines
the profile function for each nucleon is no longer ρ but ρ − si, and the phase factor
for the ith nucleon is given by Eq. (6.96) as
e iχi =1− Γi(ρ − si).
For the total phase factor, additivity of the individual phases gives
exp(iχ) =exp(iχ1)exp(iχ2) ···exp(iχA)
A�
= [1 − Γi(ρ − si)],
i=1
74 R.J. Glauber, Phys. Rev. 100, 242 (1955).
75 The Feynman Lectures 1-31-3.