Alpha Decay and the Gamow Theory of Tunneling

Alpha decay is the emission of a helium-4 nucleus from a heavier parent,

Two facts about it demand explanation. The energy released is only a few MeV, yet the process is universal among the heaviest nuclei; and the half-life is fantastically sensitive to that energy, changing by twenty-four orders of magnitude while the decay energy changes by a factor of two. Both follow from a single mechanism: the alpha particle is bound inside a potential well but must tunnel through a Coulomb barrier that classically it has nowhere near enough energy to cross.¹

The alpha Q-value and why heavy nuclei decay

The energy release is the difference in rest energies of the initial and final systems. Written in nuclear masses,

and the electron masses and binding cancel to a good approximation when atomic masses are substituted. Expressing each mass through its binding energy $m{c}^2=Z{m}_p{c}^2+N{m}_n{c}^2-B$ collapses the nucleon rest energies and leaves a difference of binding energies,

Decay is energetically allowed when $Q_{\alpha }>0$, that is when the binding lost by removing four nucleons from the parent is less than the $28.3\mathrm{MeV}$ recovered by assembling them into an alpha particle. The alpha is exceptionally tightly bound because it is doubly magic, $N=Z=2$, with a binding energy per nucleon of $7.07\mathrm{MeV}$ far above its neighbors. No other light fragment is competitive: the corresponding $Q$ for emitting a proton, a deuteron, or a triton is negative across the same region of the chart, so the alpha is the fragment that heavy nuclei actually release.

Evaluating $Q_{\alpha }$ from the semi-empirical mass formula shows where the process switches on. The Coulomb term grows as $Z^2/A^{1/3}$ while the volume and surface terms scale more slowly, so the binding per nucleon falls beyond the iron peak, and $Q_{\alpha }$ crosses zero near $A\approx 150$. Above that mass almost every nuclide is unstable to alpha emission; below it the decay is energetically forbidden. The rare-earth alpha emitters near $A=150$ (for example ${}^{147}\mathrm{Sm}$, with $Q_{\alpha }=2.31\mathrm{MeV}$) sit right at the threshold and have half-lives comparable to the age of the universe.²

The barrier and the tunneling picture

Model the alpha as a preformed particle moving in the field of the daughter. Inside the nuclear radius $R$ the strong force holds it in a flat attractive well of depth $V_0$; outside $R$ the only interaction is the Coulomb repulsion between the alpha charge $2e$ and the daughter charge $(Z-2)e$,

The barrier reaches its maximum at contact,

which for a uranium daughter ($Z-2=90$, $R\approx 9.3\mathrm{fm}$) is about $28\mathrm{MeV}$, roughly six times the alpha's kinetic energy of $4$–$5\mathrm{MeV}$. Classically the particle is trapped: it lacks the energy to reach $r=R$ from outside, or to cross the barrier from inside. The alpha nonetheless escapes because its wavefunction does not vanish in the classically forbidden region. It leaks through, emerging at the outer turning point $b$ where the Coulomb potential has fallen back to the alpha energy,

The Gamow factor

The tunneling probability follows from the WKB approximation, in which the wavefunction decays across the forbidden region as $\exp (-\int |k|\mathbf{d}r)$ with the local wavenumber $|k(r)|=\sqrt{2m[V(r)-Q_{\alpha }]}/\hslash$. The transmission coefficient is $P=e^{-2G}$, where the Gamow factor is the barrier integral

using $V(r)=Q_{\alpha }b/r$ on the Coulomb tail. Here $m$ is the alpha mass, or more precisely the reduced mass of the alpha-daughter pair, which for a heavy daughter is within a percent of $m_{\alpha }$. The integral is elementary. With $x=R/b$,

For a real actinide the barrier is thick, $R\ll b$, so $x$ is small. Expanding $\arccos \sqrt{x}\approx \frac{\pi }{2}-\sqrt{x}$ and $\sqrt{x(1-x)}\approx \sqrt{x}$, the bracket becomes $b(\frac{\pi }{2}-2\sqrt{x})$, and

The first term, proportional to $(Z-2)Q_{\alpha }^{-1/2}$, dominates and carries the entire sensitivity to the decay energy. The second, proportional to $\sqrt{(Z-2)R}$, is a slowly varying correction that depends on the radius. The steep $Q^{-1/2}$ dependence in the exponent is the origin of the enormous spread in half-lives: a small increase in $Q_{\alpha }$ lowers $b$, thins the barrier, and multiplies the penetrability by a large factor.

From penetrability to half-life

The decay constant is the penetrability times the rate at which the alpha strikes the barrier. Treating the preformed alpha as bouncing inside the well with velocity $v_{\text{in}}=\sqrt{2(Q_{\alpha }+V_0)/m}$, it presents itself at the wall with an assault frequency

and the decay constant and half-life are

The prefactor $f$ varies by less than an order of magnitude across the alpha emitters, while $e^{-2G}$ ranges over more than forty powers of ten. The half-life is therefore controlled almost entirely by the Gamow exponent, and a crude estimate of the preformation probability and the assault frequency still lands within a couple of orders of magnitude of the measured lifetime.

The Geiger-Nuttall relation

Keeping only the dominant term of $G$ and absorbing the slowly varying prefactor and radius correction into a constant, the logarithm of the half-life is linear in $(Z-2)Q_{\alpha }^{-1/2}$:

The theoretical slope comes straight from $2G_1/\ln 10$:

with $Q_{\alpha }$ in MeV. This is the Geiger-Nuttall relation, observed empirically in 1911 and explained by Gamow, and independently Condon and Gurney, in 1928 as the first application of quantum tunneling.³ Plotting ${\log }_{10}{t}_{1/2}$ against $Q_{\alpha }^{-1/2}$ for the isotopes of a single element gives a straight line, and the lines for different elements are nearly parallel, their spacing set by the daughter charge $Z-2$. Across the natural emitters the relation compresses a range from $10^{-7}\mathrm{s}$ to $10^{17}\mathrm{s}$ onto one line.

The same steep dependence explains a systematic feature of the chart: within an isotopic chain the alpha half-life grows rapidly as $Q_{\alpha }$ falls toward the stability line, and the shortest-lived, highest-$Q$ emitters lie farthest from stability. The relation is quantitative enough to predict an unknown half-life from a measured $Q$ to within about a factor of ten, and conversely a measured lifetime fixes $Q$, which is why alpha spectroscopy became an early tool for mapping the actinide masses.

The one-dimensional barrier model treats every transition as feeding the daughter ground state with the alpha carrying zero orbital angular momentum. Real spectra show several alpha groups of slightly different energy, feeding excited states of the daughter, and some transitions run far slower than the Gamow estimate. Those departures, governed by angular-momentum and parity selection rules and by the overlap of the parent and daughter wavefunctions, are the subject of the next lesson on fine structure and hindrance.