= Solution
An alpha emitter has an alpha-like cluster coupled to a daughter nucleus. Its positive decay <energy> is $Q=[M_P-M_D-M_\alpha]c^2$; atomic <masses> may be used consistently because the <electron> counts balance. The daughter and alpha share this <energy> according to <momentum> conservation. The attractive nuclear interior and repulsive exterior <Coulomb potential> produce a quasibound state with a barrier, so positive $Q$ does not imply rapid classical escape.
For daughter charge $Z_de$, the exterior barrier is approximately $C/r$, $C=2Z_de^2/(4\pi\epsilon_0)$, with outer turning radius $b=C/Q$. For zero <orbital angular momentum>, the leading WKB <probability> is
$$
T\sim\exp\left[-\frac2\hbar\int_R^b\sqrt{2\mu(C/r-Q)}\,dr\right],
\qquad\lambda\simeq\nu P_{\rm pre}T,\qquad t_{1/2}=\frac{\ln2}{\lambda}.
$$
Here $R$ is a nuclear matching radius, $\mu$ the <reduced mass>, $\nu$ an interior assault <frequency> and $P_{\rm pre}$ a preformation factor. The barrier factor alone cannot determine the latter two. Nonzero emitted <orbital angular momentum> adds a <centrifugal barrier> and can strongly hinder a transition.
The dominant <action> varies roughly as $Z_d/\sqrt Q$, explaining the Geiger-Nuttall relation and the enormous spread of alpha <half-lives>. <angular momentum> and <parity> constrain the emitted <partial wave>: a spin-zero positive-parity alpha requires $|J_P-J_D|\le L\le J_P+J_D$ and $\pi_P=\pi_D(-1)^L$. Fine-structure alpha groups populate different daughter levels and therefore have different kinetic energies and penetrabilities. Shell structure and alpha clustering also change preformation <probabilities>, so equal $Q$ values do not guarantee equal lifetimes. Escape conserves the total <energy> throughout tunnelling; the particle does not borrow a measured <energy> from the barrier.
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