Cross Section for Attractive Potentials

I want to derive a cross section for objects falling into a potential (such as a $U(r) = -1/r^n$ potential). I defined the total “falling onto” cross section $σ_F$ as the ratio of the total number of
particles that fall onto a certain object per unit time to the incident intensity. Now consider a beam
of particles incident upon a central potential $V (r) = −k/r^n$, where $k > 0$ and $n ≥ 2$, with a sphere of radius $R$ situated exactly at the center of the potential. My first goal is to determine the range of values of the impact parameter for which the particles will hit the sphere. Any hints or suggestions?