Introduction to Classical Mechanics · Problem 7.7

Angular momentum, Part I (Constant L) — Morin

Physics — Mechanics · gravity-kepler-laws

Problem

Rutherford scattering *** A particle of mass

m

travels in a hyperbolic orbit past a mass

M

, whose position is assumed to be fixed. The speed at infinity is

v_0

, and the impact parameter is

b

(see Exercise 7.14). ((a)) Show that the angle through which the particle is deflected is

\phi = \pi - 2\tan^{-1}(\gamma b) \implies b = \frac{1}{\gamma}\cot\!\left(\frac{\phi}{2}\right),

where

\gamma \equiv v_0^2/GM

. ((b)) Let

d\sigma

be the cross-sectional area (measured when the particle is initially at infinity) that gets deflected into a solid angle of size

d\Omega

at angle

\phi

. Show that

\frac{d\sigma}{d\Omega} = \frac{1}{4\gamma^2 \sin^4(\phi/2)}.

This quantity is called the differential cross section.