Neutrons are fired at natural gold ( )

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Nuclear Physics · Nuclear reactions

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

Problem

14.1\,\mathrm{MeV}

neutrons are fired at natural gold (

^{197}\mathrm{Au}

). The measured total cross section is

\sigma_{\rm tot} = 4.90\,\mathrm{b}

. Take the neutron wavenumber as

k = 0.82\,\mathrm{fm^{-1}}

(the same value used for lead at this energy, since the reduced mass is within a fraction of a per cent of the neutron mass for both nuclei) and the sharp-radius formula

R = 1.2A^{1/3}\,\mathrm{fm}

. ((a)) Estimate the black-disc radius

R

and the black-disc total cross section

\sigma_{\rm tot}^{\rm bd} = 2\pi R^2

for

^{197}\mathrm{Au}

. ((b)) Find the effective radius

R_{\rm eff}

that reproduces the measured

\sigma_{\rm tot}

under the same black-disc formula, and compare

R_{\rm eff}-R

with

1/k

. ((c)) Use the optical theorem

\sigma_{\rm tot} = (4\pi/k)\,\mathrm{Im}\,f(0)

to find

\mathrm{Im}\,f(0)

from the measured cross section, and compare it with the ideal black-disc value

\mathrm{Im}\,f(0) = kR^2/2

Related theory

Browse the problem library