A slow-neutron capture resonance in an even-even nuclide of mass number (target spin…

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

The Nucleus Resolved, Ch Reaction mechanisms: resonances, the compound nucleus, direct reactions and the optical model

Problem

A slow-neutron capture resonance in an even-even nuclide of mass number

A = 114

(target spin

0

) is studied by measuring the radiative-capture cross section

\sigma_\gamma

at several neutron energies:

E\,(\mathrm{eV})

1.30,\ 1.40,\ 1.45,\ 1.50,\ 1.55,\ 1.60,\ 1.70

\sigma_\gamma\,(\mathrm{b})

70.6,\ 240,\ 600,\ 1200,\ 600,\ 240,\ 70.6

((a)) From the table, read off the resonance energy

E_r

and the total width

\Gamma

, using the points where the cross section has fallen to half its peak value. ((b)) Given that for this resonance

\Gamma_n \ll \Gamma_\gamma

(as for the

6.67\,\mathrm{eV}

level of

^{238}\mathrm{U}

), so that

\Gamma \approx \Gamma_\gamma

, state

\Gamma_\gamma

. ((c)) Using

\sigma_{\gamma,\rm peak} = 4\pi\bar\lambda^2 g_{\rm stat}\dfrac{\Gamma_n\Gamma_\gamma}{\Gamma^2}

with

g_{\rm stat}=1

and the reduced neutron wavelength

\bar\lambda = \hbar c/\sqrt{2m_nc^2E_r}

(non-relativistic), with

m_nc^2 = 939.57\,\mathrm{MeV}

and

\hbar c = 197.327\,\mathrm{MeV\,fm}

, find

\Gamma_n