Sinceδℓ(k) is dimensionless, we need indeed two dimensionful parameters ER, the position of
the resonance, and Γ, the width of the resonance, to characterize the expansion to first order.
Computing now the partial cross section, we use
sin^2 δℓ(k) =1
1 + cotg^2 δℓ(k)=
Γ^2 / 4
(E−ER)^2 + Γ^2 / 4
(12.124)
and thus the partial cross section
σ(totℓ)=
4 π
k^2(2ℓ+ 1)Γ^2 / 4
(E−ER)^2 + Γ^2 / 4
(12.125)
It turns out that this kind of curves fit experimental observations of resonance scattering processes
very well.