A compound nucleus of mass number is formed at excitation energy

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 compound nucleus of mass number

A = 60

is formed at excitation energy

E^* = 12.0\,\mathrm{MeV}

. Take the Fermi-gas level-density parameter

a_{\rm ld} = A/8 = 7.50\,\mathrm{MeV^{-1}}

and a spin cut-off parameter

\sigma_J = 3.0

. Consider compound levels of spin and parity

J^\pi = 1^-

. ((a)) Using

\omega(E^*) = \dfrac{\sqrt\pi}{12}a_{\rm ld}^{-1/4}E^{*-5/4}\exp\!\left(2\sqrt{a_{\rm ld}E^*}\right)

, compute the total state density

\omega(E^*)

. ((b)) Compute the level density of all spins,

\rho_{\rm lev} = \omega/(\sqrt{2\pi}\,\sigma_J)

, then the density of

1^-

levels,

\rho(J^\pi) = \tfrac{1}{2}\rho_{\rm lev}\,\dfrac{2J+1}{2\sigma_J^2}\exp\!\left[-\dfrac{(J+\frac{1}{2})^2}{2\sigma_J^2}\right]

, and the mean spacing

D = 1/\rho(1^-)

of those levels. ((c)) If the summed transmission coefficients of all open exit channels at this excitation are

\sum_c\mathcal{T}_c = 4.5

, estimate the total width

\Gamma = (D/2\pi)\sum_c\mathcal{T}_c

and decide whether the statistical (compound-nucleus) treatment is fully justified here, using the criterion that levels overlap once

\sum_c\mathcal{T}_c \gt 2\pi