Am I understanding the Geiger-Nuttall law correctly

[says]

Geiger–Nuttall rule relates the decay constant of a radioactive isotope with the energy of the alpha particles emitted. Roughly speaking, it states that short-lived isotopes emit more energetic alpha particles than long-lived ones. - wiki

I am trying to understand why there is an inverse relationship between alpha decay half-life and the decay energy.

The heavier an unstable nuclei is, the higher it's potential barrier and quantum tunnelling distance is. This reduces the probability an alpha particle can tunnel through the potential barrier, and thus increases the half life of the unstable nuclei.

I'm trying to understand where decay energy fits in here though...

 

[mfb]

The potential barrier close to the nucleus looks similar for many nuclides. A larger decay energy means "the alpha particle" has more energy inside and the barrier is lower for it.

 

[vanhees71]

Have a look at this nice article about Gamow's theory of $\alpha$ decay

https://fenix.tecnico.ulisboa.pt/downloadFile/3779576931836/Ex8Serie1/