Nuclear Fission: The Basic Process — Question 3

Question 3

The binding-energy curve predicts the sign of a nuclear split

A schematic parent with A=240 has binding energy per nucleon 7.60 MeV. It splits into two A=120 fragments, each with binding energy per nucleon 8.40 MeV. In this model there are no free neutrons and proton and neutron totals are unchanged.

1. Compute the total binding before and after the split and the released energy.

2. A proposed reverse fusion joins the same fragments. Determine its Q value and explain why a barrier and an endothermic threshold are different concepts.

Question 3 - Solution

The binding-energy curve predicts the sign of a nuclear split

(a) Method and calculation

$$\boxed{B_i=240(7.60)=1824\,\mathrm{MeV},\quad B_f=2(120)(8.40)=2016\,\mathrm{MeV}}.$$$$Q=(M_i-M_f)c^2=B_f-B_i\Rightarrow\boxed{Q=192\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{Q_{\mathrm{reverse}}=B_i-B_f=-192\,\mathrm{MeV}}.$$$$E_{\mathrm{available,cm}}=K_{\mathrm{cm}}+Q_{\mathrm{reverse}}\ge0\Rightarrow\boxed{K_{\mathrm{cm}}\ge192\,\mathrm{MeV}}.$$

Physical check

This inequality is only an energy requirement. A dynamical Coulomb/deformation barrier can further suppress a reaction even when its Q value is positive.