# When and why spontaneous nuclear fission is preferred to alpha emission?

I reasoned as follow:

the probability of an $\alpha$ emission

$^A_ZX\to^{A-4}_{Z-2}X^{'}+\alpha$

is given by:

$T=e^{-G}$

where the Gamow factor is given by:

$G\simeq \pi \sqrt{\frac{2\mu c^2}{E}}Z_1Z_2\frac{e^2}{\hbar c}$

so $G\propto \sqrt{\mu}$

where $\mu=\frac{m_{\alpha}m_{X^{'}}}{m_{X^{'}}+m_{\alpha}}\simeq m_{\alpha}$ because usually $m_{X^{'}}>>m_{\alpha}$

In the case of a spontaneous fission I have that

$^A_ZX\to^{A_1}_{Z_1}X_1+^{A_2}_{Z_2}X_2$

The only thing I can say is that the probability of tunneling is less than in the previous case because now the reduced mass is higher. Then I have some qualitative knolwledges like "the fission happen when the energy associated to proton repulsion become higher than the binding energy, so because of the repulsion energy is $\propto Z^2$ the fission is a mechanism that involves heavy nucleus"

What is the correct way to calculate the spontaneous fission probability and so compare it with the $\alpha$ emission probability?