= Solution
Model <alpha decay> as a preformed <alpha particle> making repeated escape attempts from a short-range nuclear potential well, then undergoing <quantum tunnelling> through the external <Coulomb potential>. Write the daughter <atomic number> as $Z_D$, the nuclear radius as $R$, and the relative <reduced mass> as $\mu$. For zero <orbital angular momentum>, the <WKB approximation> gives
$$
P_{\mathrm{tun}}\simeq e^{-G},\qquad
G=\frac2\hbar\int_R^{r_2}\sqrt{2\mu(V(r)-Q)}\,dr,
\qquad V(r)=\frac{k}{r},\quad
k=\frac{2Z_De^2}{4\pi\epsilon_0},\quad r_2=\frac{k}{Q}.
$$
Set $v=\sqrt{2Q/\mu}$ and $r=r_2\sin^2\theta$. Then
$$
G=\frac{2\sqrt{2\mu Q}\,r_2}{\hbar}
\left[\arccos\sqrt{\frac R{r_2}}-
\sqrt{\frac R{r_2}\left(1-\frac R{r_2}\right)}\right].
$$
When $R/r_2\ll1$, the leading term is
$$
G\simeq\frac{2\pi k}{\hbar v}
=\frac{Z_De^2}{\epsilon_0\hbar v},\qquad
\boxed{\tau\simeq(P_\alpha\nu_{\mathrm{attempt}})^{-1}
\exp\left(\frac{Z_De^2}{\epsilon_0\hbar v}\right).}
$$
Here $P_\alpha$ is the <alpha preformation probability> and $\nu_{\mathrm{attempt}}$ the attempt frequency. The stated proportionality retains only the dominant barrier exponent. It assumes a thick classically forbidden barrier, nonrelativistic relative motion, a spherical unscreened external <Coulomb potential>, and slowly varying prefactors. Finite $R$, changes in <alpha preformation probability>, and a <centrifugal barrier> for nonzero <orbital angular momentum> produce significant corrections. The precise charge in the barrier is $Z_D=Z_P-2$; using the parent charge is an approximation for large $Z_P$. The measured <alpha particle> speed approximates $v$ only when daughter recoil is negligible.
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