(a) Starting from the incident flux in channel and the outgoing flux in channel ,…

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Nuclear Physics · Nuclear reactions

The Nucleus Resolved, Ch Reactions: four-vectors, amplitudes and partial waves

Problem

(a) Starting from the incident flux in channel

a

and the outgoing flux in channel

b

, with speed

v = \hbar k/\mu

in each channel, derive

\mathrm{d}\sigma_{ab}/\mathrm{d}\Omega = (k_b/k_a)|f_{ba}(\theta)|^2

, stating explicitly where the reduced masses

\mu_a

\mu_b

enter. (b) For a nucleus of sharp radius

R

that is a perfect absorber up to

\ell_{\max} \approx kR

(that is,

\eta_\ell = 0

for

\ell \le kR

and

\eta_\ell = 1

above), use the partial-wave sums for

\sigma_{\rm el}

and

\sigma_{\rm reac}

to show

\sigma_{\rm el} = \sigma_{\rm reac} = \pi R^2

. (c) A detector counts

\mathrm{d}N

events per second into a small laboratory solid angle

\mathrm{d}\Omega_{\rm lab}

, which correspond to a small centre-of-mass solid angle

\mathrm{d}\Omega_{\rm cm}

around the same physical events. Using the fact that

\mathrm{d}N

is the same in both descriptions, show that

\sigma_{\rm tot} = \int (\mathrm{d}\sigma/\mathrm{d}\Omega)_{\rm lab}\,\mathrm{d}\Omega_{\rm lab} = \int (\mathrm{d}\sigma/\mathrm{d}\Omega)_{\rm cm}\,\mathrm{d}\Omega_{\rm cm}

, and explain why this permits computing

\sigma_{\rm el}

and

\sigma_{\rm reac}

from partial waves in the centre-of-mass frame and comparing them directly with a laboratory-measured total cross section.