In classical scattering theory, the differential scattering cross section, σ(Ω), is defined such that σ(Ω)dΩ is the number of particles scattered into the differential solid angle dΩ in the direction Ω per unit time, divided by the incident intensity.
Now, I understand that in the case of a long range force, for which all particles arbitrarily far from the scatterer are deflected by some finite angle, the total scattering cross section is infinite. In this case, the integral of σ(Ω) over 4π sr simply equals the total number of incident particles (infinity) divided by the finite intensity.
My question is, in the case of a short range force, where the force goes to zero at some finite distance from the scatterer, why does this lead to a cross section which is finite? There are still an infinite number of particles being scattered at some angle. It's just that in this case there are an infinite number of them scattering into the angle ϴ=0 (i.e. those beyond the range of force that are not deflected).
Why is it that integrating σ(Ω) over the full 4π sr does not always equal the total number incident particles, resulting in an infinite total scattering cross section for every force, short or long range?