Astroparticle Physics

3.5 Problems 47If j is the particle flux per cm 2 and s, the number of particlesdN scattered through an angle θ into the solid angle dΩ perunit time isdN(θ) = jσ(θ)dΩ , (3.58)whereσ(θ) = dσdΩis the differential scattering cross section, describing theprobability of scattering into the solid-angle element dΩ,wheredifferential scatteringcross sectiondΩ = sin θ dθ dϕ (3.59)(ϕ – azimuthal angle, θ – polar angle).For azimuthal symmetry one hasdΩ = 2π sin θ dθ =−2π d(cos θ) . (3.60)Apart from the angular dependence the cross section canalso depend on other quantities, so that a large number ofdifferential cross sections are known, for example,dσdE ,dσdp ,or even double differential cross sections such asd 2 σdE dθ . (3.61)Apart from the mentioned characteristic quantities there isquite a large number of other kinematical variables whichare used for the treatment of special processes and decays.double differentialcross section3.5 Problems1. What is the threshold energy for a photon, E γ1 ,toproducea µ + µ − pair in a collision with a blackbody photonof energy 1 meV?2. The mean free path λ (in g/cm 2 ) is related to the nuclearcross section σ N (in cm 2 )byλ = 1N A σ N,