A shape isomer of sits above the ground state

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Nuclear Physics · Fission, fusion and nuclear energy

The Nucleus Resolved, Ch Fission: the double-humped barrier and the shells the drop cannot see

Problem

A shape isomer of

^{242}\mathrm{Am}

sits

2.3\,\mathrm{MeV}

above the ground state, trapped in the second minimum of the double-humped barrier. The top of the outer barrier lies

E_b = 3.4\,\mathrm{MeV}

above this isomeric state. Model the second well as a parabola with curvature

\hbar\omega_{II} = 0.80\,\mathrm{MeV}

and the outer barrier as a parabola with curvature

\hbar\omega_B

. Carry over the parabolic-barrier WKB penetrability,

P \approx \exp(-2\pi E_b/\hbar\omega_B)

, and the assault-frequency estimate

\nu = \hbar\omega_{II}/h

(with

h = 2\pi\hbar

) to estimate the fission decay constant

\lambda_f = \nu P

and hence the half-life

T_{1/2} = \ln 2/\lambda_f

. ((a)) Estimate

T_{1/2}

for

\hbar\omega_B = 0.50\,\mathrm{MeV}

. ((b)) Repeat for

\hbar\omega_B = 0.45\,\mathrm{MeV}

, all else unchanged, and state by what factor the half-life has moved.