Subatomic Physics

6.11. More Details on Scattering and Structure 183
� � � �
2
ρ
Γ(ρ) =Γ(0)exp−
. (6.99)
The Fourier–Bessel transform then becomes (70)
f(θ) = 1
2 ikΓ(0)ρ2 � � � �
2
kθρ0
0 exp −
.
2
With −t = |q 2 |≈(�kθ) 2 , the corresponding differential cross section is
− dσ
dt
ρ0
π
=
4�2 Γ2 (0)ρ 4 0 exp
� � 2 ρ0 −
2�2 � �
|t| . (6.100)
A Gaussian profile function leads to an exponentially decreasing cross section dσ/dt.
The physical interpretation of the profile function becomes clear by considering
the total cross section. The optical theorem, Eq. (6.78), with Eq. (6.97) for θ =0◦ ,
yields
�
σtot =2 d 2 ρReΓ(ρ). (6.101)
For a black scatterer, Γ(ρ) =1isreal,andf(θ) is purely imaginary. If we assume
that in the limit of very high energy the amplitude is imaginary, (71) then Γ is real,
and Eq. (6.101) becomes
�
σtot =2 d 2 ρΓ(ρ). (6.102)
2Γ(ρ) can consequently be interpreted as the probability that scattering occurs in
the element d 2 ρ at the distance ρ from the center (see Fig. 6.31.) Γ(ρ) isthe
scattering probability density distribution in the shadow plane; hence the name
profile function.
As an application of these considerations, we return to elastic pp scattering. (69)
Figure 6.29 shows that the diffraction peak drops exponentially for many orders
of magnitude. This behavior suggests that the cross section in the region of the
forward peak can be approximated by
dσ dσ
(s, t) =
dt dt (s, t =0)e−b(s)|t| , (6.103)
where s is the conventional symbol for the square of the total energy of the colliding
protons in their c.m. and b(s) is called the slope parameter. It is remarkable that
the experimental data over a wide range of s and t can indeed be fitted by such a
simple expression. The slope parameter turns out to be a slowly varying logarithmic
function of the total energy s, as shown in Fig. 6.32. The exponential drop of dσ/dt
71 The ratio between the real and the imaginary part of the proton–proton forward scattering
amplitude is expected to become small at high incident momenta.