A Model for Nuclear Fission — Question 10

Question 10

Shell corrections can reverse a smooth-model preference for a split

A smooth liquid-drop model predicts final potential energies 0 MeV for symmetric channel S and +4 MeV for asymmetric channel A, relative to a common reference. Shell corrections are -1 MeV for S and -8 MeV for A.

1. Calculate the corrected channel energies and identify which is lower.

2. For a supplied statistical weight model $w_i\propto e^{-U_i/\Theta}$ with $\Theta=1.50$ MeV, find $w_A/w_S$ before and after the corrections.

Question 10 - Solution

Shell corrections can reverse a smooth-model preference for a split

(a) Method and calculation

$$\boxed{U_S=-1\,\mathrm{MeV},\quad U_A=4-8=-4\,\mathrm{MeV}}.$$$$\boxed{U_A-U_S=-3\,\mathrm{MeV}}\quad\text{so A is lower}.$$

(b) Interpretation and further calculation

$$\boxed{(w_A/w_S)_{\mathrm{smooth}}=e^{-4/1.5}=.069483}.$$$$\boxed{(w_A/w_S)_{\mathrm{corrected}}=e^{-(-3)/1.5}=e^2=7.38906}.$$

Physical check

This supplied weight law illustrates sensitivity to structure. Real fission yields also depend on dynamical paths and cannot be predicted from endpoint energy alone.