Impact parameters

Impact parameter b and scattering angle θ.

In physics of collisions between two particles, the impact parameter is the minimum distance between their centers of gravity if no force would act between the two particles , so that they would happen on straight paths. Geometrically, it is the distance ( normal vector ) of a particle to the current (relative) flight direction of the other particle. ${\displaystyle b}$

${\displaystyle {\vec {b}}={\hat {v}}\times ({\vec {r}}\times {\hat {v}})}$

with the unit vector of the relative speed v and the instantaneous distance between the two particles r ${\displaystyle {\hat {v}}={\tfrac {\vec {v}}{|v|}}}$

For the trajectory of a particle in a force field that is fixed in space and originates from a point, the scattering center , the impact parameter denotes the minimum distance to this point if it were not deflected by the force field. By transforming the equation of definition recognizes the one in the two-body problem constant specific angular momentum ρ :

${\displaystyle {\vec {r}}\times {\vec {v}}={\vec {b}}\times {\vec {v}}={\tfrac {\vec {L}}{m}}=r^{2}{\vec {\omega }}={\vec {\rho }}}$

With ${\displaystyle |{\vec {b}}\times {\vec {v}}|=b\ v}$

To simplify the mathematical treatment of collisions between two particles, what happens is usually viewed from the center of gravity system. The trajectory of a single particle with the reduced mass in a fixed-space force field can be assigned to it, whereby the force has the same dependence on the distance. The particle distance corresponds to the distance to the scattering center, so that the impact parameter remains the same for this consideration.

See also

• Scattering (physics)
• Scattering theory
• Cross section