Scattering off a Central Potential

Fire a parallel beam at a fixed center and watch it fan out. How far a particle bends depends only on its impact parameter $b$, the sideways miss distance: aim near the center and it scatters hard, aim wide and it barely turns. That mapping from $b$ to the deflection angle $\chi$ is the deflection function, and it is the whole story of a scattering experiment. A hard sphere bounces specularly; an inverse-square (Coulomb) center backscatters the closest particles, the Rutherford result that revealed the nucleus; a screened Yukawa center is softer at long range. The scene shows the beam scattering; the diagnostic plots $\chi(b)$, with a cursor on the impact parameter you select.

Figure 1. Scattering off a central potential. Top: a parallel beam, colored by impact parameter, fanning out after the encounter. Bottom: the deflection function, scattering angle versus impact parameter, with a cursor at the selected b. Method: closed form for the hard sphere and Coulomb cases, velocity-Verlet for Yukawa.

WHAT TO TRY

Drag the impact parameter: the highlighted ray and the cursor on the deflection curve move together, small b bends hard, large b barely turns.
Switch to Coulomb: the closest rays come almost straight back, the Rutherford backscattering that implied a tiny dense nucleus.
Raise the energy: faster particles spend less time near the center, so every deflection shrinks and the beam fans out less.
Compare Yukawa to Coulomb: screening kills the long-range tail, so wide-b particles pass nearly undeflected.