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5.4. The Glauber Approach 51
so that Eq. (5.43) becomes
f Multi ( ⃗ ∆) = ik f
2π
∫
d ⃗ be i⃗ ∆·⃗b < f|Γ tot ( ⃗ b, ⃗ b 1 , . . . ⃗ b N |i > . (5.46)
We introduce the functions
Γ i ( ⃗ b − ⃗ b i ) = 1 − exp[iχ i ( ⃗ b − ⃗ b i )] , (5.47)
and by combining Eqs. (5.44), (5.45) and (5.47), we can write that
or
N∏
Γ tot = 1 − [1 − Γ i ( ⃗ b − ⃗ b i )] , (5.48)
i=1
N∑
Γ tot = Γ i − ∑ Γ i Γ j + . . . + (−1) N−1 ∏ N Γ i . (5.49)
i=1 i≠j
i=1
This last equation, when substituted into Eq. (5.46), leads directly to an interpretation
of the collision process in terms of a multiple scattering expansion. The terms
linear in Γ i on the right-hand side of Eq. (5.49) account for the single scattering,
whereas the next terms provide double, triple,... scattering contributions. As is
mostly done in many-body calculations, all Glauber calculations reported in this
work are performed in the approximation
N∑
Γ tot ≈ Γ i . (5.50)
i=1
which amounts to retaining all single rescattering processes in the calculation.
In calculating the transition strength in the Glauber approach one finds himself
confronted with matrix elements of the type
A−1 ∑
< A − 1| Γ j ( ⃗ b − ⃗ b j )θ(z − z j )|A − 1 > , (5.51)
j=1
where θ(z − z j ) expresses the fact that the proton only interacts (collides) with the
other nucleons if they are localised in its forward propagation path. The above
expression easily reduces to the following sum :
∫
∫
d⃗r 1 ρ 1 (⃗r 1 )Γ 1 ( ⃗ b − ⃗ b 1 )θ(z − z 1 ) + d⃗r 2 ρ 2 (⃗r 2 )Γ 2 ( ⃗ b − ⃗ b 2 )θ(z − z 2 )
∫
+... + d⃗r A−1 ρ A−1 (⃗r A−1 )Γ A−1 ( ⃗ b − ⃗ b A−1 )θ(z − z A−1 ) , (5.52)