Geiger-Nuttall law
= Geiger-Nuttall law
{c}
{title2=$\log_{10}T_{1/2}\simeq aZ_d/\sqrt{Q_\alpha}+b$}
The <Geiger-Nuttall law> expresses the extreme sensitivity of an <alpha decay> <half-life> to its <Q value>. In a restricted range of heavy <nuclei>, a useful local form is $\log_{10}[T_{1/2}/T_*]\simeq aZ_d/\sqrt{Q_\alpha}+b$, where $Z_d$ is the daughter <atomic number>, and the constants depend on the chosen units and nuclear family. It is named for <Hans Geiger> and <John Mitchell Nuttall>.
The <Coulomb barrier> outside the daughter has $V(r)=2Z_d\alpha/r$ in <natural units>. Its outer turning point is $r_t=2Z_d\alpha/Q_\alpha$. The <WKB approximation> gives a penetration probability
$$
P\simeq\exp\left[-2\int_R^{r_t}\sqrt{2\mu\,[V(r)-Q_\alpha]}\,dr\right],
$$
where $\mu$ is the daughter-alpha <reduced mass>. The leading large-barrier exponent is $2\pi(2Z_d\alpha)\sqrt{\mu/(2Q_\alpha)}$, producing the approximate $Z_d/\sqrt{Q_\alpha}$ dependence. Finite nuclear size modifies the exponent and the slowly varying intercept. The <nuclear decay constant> also contains an assault frequency and an alpha-cluster formation probability. Extra <orbital angular momentum> raises the barrier further, and odd-nucleon structure can introduce substantial hindrance. Thus extrapolation from even-even decays gives an order-of-magnitude reference rather than a precision prediction for an odd-odd <nucleus>.