= Rutherford impact-parameter relation
{c}
{title2=$b=\frac{\kappa}{2E}\cot\frac\theta2$}
Let $\theta$ be the scattering <angle> between incoming and outgoing <velocities>, so forward scattering is $\theta=0$ and exact reversal is $\theta=\pi$. Define the <length> $a=\kappa/(2E)$. Then
$$
\boxed{b=a\cot\frac\theta2},\qquad
\theta=2\arctan\frac{a}{b}.
$$
For $b>0$, this gives $0<\theta<\pi$; $b=0$ is the head-on limiting case $\theta=\pi$. Large deflections select small <impact parameters>, rather than requiring every incident <alpha particle> to reverse.
A short derivation uses <conservation of angular momentum> and <conservation of energy>. Choose the polar <angle> $\phi=0$ at closest approach, and write $u=1/r$. The <Binet equation> for the repulsive <Coulomb potential> gives
$$
\frac{d^2u}{d\phi^2}+u=-\frac{m\kappa}{L^2},\qquad
u=\frac{m\kappa}{L^2}(\epsilon\cos\phi-1),\qquad
\epsilon^2=1+\frac{2EL^2}{m\kappa^2}.
$$
The last equality follows by substituting this <orbit> into $E=m\dot r^2/2+L^2/(2mr^2)+\kappa/r$. The <hyperbola> reaches infinity at $\phi=\pm\phi_\infty$, with $\cos\phi_\infty=1/\epsilon$. Its scattering <angle> is $\theta=\pi-2\phi_\infty$, hence
$$
\cot\frac\theta2=\sqrt{\epsilon^2-1}
=\frac{\sqrt{2E/m}\,L}{\kappa}
=\frac{2Eb}{\kappa}.
$$
This proves the relation without assuming that the deflection is small. The symbol $\epsilon$ here is a dimensionless orbital parameter, distinct from $\varepsilon_0$.
Back to article page
