A Model for Nuclear Fission — Question 8

Question 8

Barrier action controls relative tunneling lifetimes exponentially

Two schematic fission paths have dimensionless under-barrier actions $G_1=20$ and $G_2=23$, with penetrability $P=e^{-2G}$. Their attempt frequencies and all other prefactors are identical.

1. Find the ratio of penetrabilities and half-lives.

2. An improved calculation changes $G_1$ by $+.50$. Find the resulting factor in half-life and the logarithmic sensitivity.

Question 8 - Solution

Barrier action controls relative tunneling lifetimes exponentially

(a) Method and calculation

$$\boxed{P_2/P_1=e^{-2(23-20)}=e^{-6}=.00247875}.$$$$T_{1/2}\propto P^{-1}\Rightarrow\boxed{T_2/T_1=e^6=403.43}.$$

(b) Interpretation and further calculation

$$\boxed{T_{1,\mathrm{new}}/T_{1,\mathrm{old}}=e^{2(.50)}=2.71828}.$$$$\boxed{\frac{d\ln T_{1/2}}{dG}=2}.$$

Physical check

The comparison needs no numerical assault frequency because it cancels. A small action error can create a large lifetime error.