Impact parameter

Impact parameter b and scattering angle θ.

The impact parameter $b$ is defined as the perpendicular distance between the path of a projectile and the center of a potential field $U(r)$ created by an object that the projectile is approaching (see diagram). It is often referred to in nuclear physics (see Rutherford Scattering), as well as in Classical Mechanics.

The impact parameter is related to the scattering angle $\theta$ by^[1]

\theta=\pi-2b\int_{r_\mathrm{min}}^\infty \frac{dr}{r^2\sqrt{1-(b/r)^2-2U/mv_\infty^2}}

where $v_\infty$ is the velocity of the projectile when it is far from the center, and $r_\mathrm{min}$ is its closest distance from the center.

Scattering from a hard sphere

The most simple example illustrating the use of the impact parameter is in the case of scattering from a hard sphere. Here, the object that the projectile is approaching is a hard sphere with radius $R$ . In the case of a hard sphere, $U(r) = 0$ when $r > R$ , and $U(r) = \infty$ for $r \leq R$ . When $b > R$ , the projectile misses the hard sphere. We immediately see that $\theta = 0$ . When $b \leq R$ , we find that $\theta = R \cos\left(\frac{\theta}{2}\right)$ .