I think I might have a serious misunderstanding of some concepts to do with scattering cross sections and would really appreciate any help.
As far as I can tell the differential cross section is basically a measure of what fraction of particles get scattered into some solid angle. To get the total cross section all we need to do is then integrate over all solid angles. However I can't determine why this wouldn't just be equal to 1 i.e. that we are guaranteed to find the scattered particle at some solid angle. An attempt in my mind to fix this would be to not include the infinitesimal on axis solid angle. However, unless the differential cross section was infinite their, it wouldn't make any difference.
Could anyone tell me what's gone wrong with my reasoning? Much appreciated!