A Model for Nuclear Fission — Question 9

Question 9

Post-scission Coulomb energy sets a fragment kinetic-energy scale

Two separated spherical fragments have charges $40e$ and $52e$ and center separation d=18.0 fm. Neglect nuclear attraction and initial radial motion. Use $e^2/(4\pi\varepsilon_0)=1.44$ MeV fm.

1. Calculate their mutual Coulomb energy and eventual total relative kinetic energy at infinite separation.

2. If that kinetic energy is measured as 150 MeV in this model, infer the corresponding separation and explain the model dependence.

Question 9 - Solution

Post-scission Coulomb energy sets a fragment kinetic-energy scale

(a) Method and calculation

$$\boxed{U_C(d)=1.44\frac{40(52)}{18.0}=166.4\,\mathrm{MeV}}.$$$$U_C(d)+K(d)=U_C(\infty)+K(\infty)\Rightarrow\boxed{K(\infty)=166.4\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{d_{\mathrm{inferred}}=\frac{1.44(40)(52)}{150}=19.968\,\mathrm{fm}}.$$$$K_\infty=K_{\mathrm{initial}}+U_C+\Delta U_{\mathrm{other}}\quad\Rightarrow\quad d\text{ is model dependent}.$$

Physical check

The inferred d is not a direct measurement of either fragment radius. Deformation, initial motion and emission can change this estimate.