Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 35E / b / Solution

For $V(r)\sim r^{-n}$, the small-$q$ transform scales as $q^{n-3}$. Since $d\Omega$ contributes $qdq$ near forward scattering, the total Born cross section behaves as ${\int }_0{q}^{2n-5}dq$, finite exactly for $n>2$. Coulomb decay $n=1$ gives the familiar divergent forward Rutherford cross section and requires long-range treatment.