A nucleus of masss \(M\) emits \(\gamma\)-ray photon of frequency \('v'.\) The loss of internal energy by the nucleus is:

Momentum of \(\gamma\) ray \(\left[P_{\gamma}\right]=\frac{h}{\gamma}=\frac{h v}{C}\)

Total momentum is conserved

\(\vec{P}_{\gamma}+\vec{P}_{N u}=0\)

Where \(\vec{P}_{N u}=\) Momentum of decayed nuclei

\(\Rightarrow P_{\gamma}=P_{N u} \Rightarrow \frac{h v}{C}=P_{N u}\)

\(\Rightarrow \text { K.E of nuclei }\)

\(=\frac{1}{2} M v^{2}=\frac{\left(P_{N u}\right)^{2}}{2 M}=\frac{1}{2 M}\left[\frac{h v}{C}\right]^{22}\)

Loss in internal energy \(=E_{\gamma}+K \cdot E_{N u}\)

\(=h v+\frac{1}{2 M}\left[\frac{h v}{C}\right]^{2}=h v\left[1+\frac{h v}{2 M C^{2}}\right]\)