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[tex] ^A_ZX \rightarrow ^A_{Z+1} Y + e^{-} + \nu_e [/tex]

The Fermi golden rule is given by:

[tex] \Gamma = \frac{2\pi}{\hbar} |A_{fi}|^2 \frac{dN}{dE_f} [/tex]

Reaction amplitude is given by ##A_{fi} = G_F M_{nucl} ## while density of states is given by ##dN = \frac{4 \pi p^2 dp}{(2\pi)^3} ## and ##\frac{dp}{dE} = \frac{1}{2}##.

Putting all these together, it gives :

[tex]\Gamma = \frac{2\pi}{\hbar} G_p^2M_{nucl}^2 \frac{p^2}{2\pi^2} \frac{1}{2} = \frac{\pi}{2\hbar}G_p^2 M_{nucl}^2 p^2 [/tex]

[tex] d\Gamma = \frac{\pi}{\hbar}G_p^2 M_{nucl}^2 p~\ dp [/tex]

I'm not sure how the text derived this.