Impact parameter
The impact parameter is the distance of closest approach if there were no forces; for a particle arriving with speed it fixes the angular momentum per unit mass, .
The impact parameter is the distance of closest approach if there were no forces.
Relation to angular momentum
The impact parameter is related to the angular momentum (per unit mass) as
Proof
A non-interacting particle has a conserved angular momentum, and its velocity does not change. At the closest point of its straight-line path,
This must also be the angular momentum at the start: the initial is the same for the interacting and non-interacting particles, and it is also conserved in the interacting case.
Scattering angle
Scattering by a repulsive interaction between two positively charged particles has
so the results from the attractive inverse-square problem apply with . The orbits are
with , and incoming and outgoing impact parameters are equal by conservation of and . Matching the conserved energy with its value on the orbit gives the angle through which the particle is scattered:
A small leads to large-angle scattering, so scattering experiments probe very small distances. Rutherford scattering (1911) showed that scattering experiments on atoms could be explained if all the positive charge in an atom was confined to a tiny nucleus.
Related
Stated in
- Definition 3.3 (Impact Parameter)§3.6 Repulsive Potentials and Scattering
- Proposition 3.4§3.6 Repulsive Potentials and Scattering