Classical Mechanics (Kibble & Berkshire) · Problem 10.22

The angular momentum principle — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · two-body-problem-scattering

Problem

Consider two-body elastic scattering in which the incident particles have energy

E_0

. Show that the energies of the recoiling particles lie in the interval

0 \leq E \leq E_{\max}

, where

E_{\max} = 4\gamma E_0/(1+\gamma)^2

. Show further that the energies of the recoiling particles are distributed over the interval

0 \leq E \leq E_{\max}

by the frequency distribution

f(E) = \left(\frac{4\pi}{E_{\max}}\right)\sigma^{ZM}(\psi),

where

\sigma^{ZM}

is defined by equation (10.32), and

\psi = 2\sin^{-1}\left(\frac{E}{E_{\max}}\right)^{1/2}.

In the elastic scattering of neutrons of energy

E_0

by protons at rest, the energies of the recoiling protons were found to be uniformly distributed over the interval

0 \leq E \leq E_0

, the total cross section being

A

. Find the angular distribution of the recoiling protons and the scattering cross section of the incident neutrons.