GATE 2024 PH – Question 36
A two-nucleon bound state has binding energy B and rest energy $mc^2$. The minimum photon energy to dissociate it is
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Correct answer: (B) $B(1+B/(2mc^2))$
Explanation
The photon must supply the binding energy $B$ and also carry momentum, so a bit more than $B$ is needed. At the **threshold** the two nucleons are produced at rest relative to each other (they move together).
**Use the invariant mass.** The bound state has rest energy $mc^2$ (initially at rest). The final state has invariant mass $mc^2+B$ (the sum of the free-nucleon rest energies):
$$(E_\gamma+mc^2)^2-E_\gamma^2=(mc^2+B)^2 .$$
**Expand:**
$$2E_\gamma mc^2+m^2c^4=m^2c^4+2mc^2B+B^2\;\Rightarrow\;E_\gamma=B+\frac{B^2}{2mc^2}=B\left(1+\frac{B}{2mc^2}\right).$$
Answer **option B**.