The toy Strutinsky ladder (levels bunched into degenerate groups of five
PHYSICS · Multi-part · Fermi University Physics · Nuclear Physics · The nuclear force and nuclear models · Fission, fusion and nuclear energy
The Nucleus Resolved, Ch Fission: the double-humped barrier and the shells the drop cannot see
Problem
The toy Strutinsky ladder (levels bunched into degenerate groups of five, one group every five level spacings, smoothed with a Gaussian of width one group spacing) gives a shell correction, relative to the evenly spaced ladder, in units of the local level spacing ϵ0, of −1 when the Fermi level sits exactly at a gap between two full groups (position 5 within a group) and +2 when it sits amid a bunch, three levels into the next group (position 3). Consider two hypothetical even-even actinide-like nuclides, P and Q, with the same liquid-drop fission barrier of 5.00MeV (their Z2/A is close enough that the smooth part of the barrier is indistinguishable) and the same shell-plus-pairing correction at their deformed ground states. Their outer saddle points sample different parts of the single-particle spectrum: nucleus P's Fermi level sits exactly at a gap at the saddle (position 5 in the ladder above), while nucleus Q's sits amid a bunch three levels into the next group (position 3). Take the local single-particle level spacing at the saddle to be ϵ0=0.70MeV for both, and the outer-barrier curvature to be ℏωB=0.65MeV for both.
((a)) Using the pattern above, give the sign and size, in MeV, of the saddle-point shell correction for P and for Q, and hence the difference in outer-barrier height between the two nuclides (state which is higher).
((b)) Using the parabolic-barrier WKB exponent's sensitivity to barrier height (T1/2∝exp[2πΔE/ℏωB] for a shift ΔE in barrier height at fixed curvature), estimate the ratio of spontaneous-fission half-lives T1/2(Q)/T1/2(P) implied by the barrier-height difference found in part (a).
((c)) In two or three sentences, explain why a mechanism of this kind, riding on top of the liquid drop's smooth trend of spontaneous-fission half-life with Z2/A, is needed to account for both the many-decade span of measured half-lives and their staggering between neighbouring nuclides, including the extra hindrance of odd-A systems.