Alpha Decay as Tunneling

Alpha decay is one of the most striking demonstrations that quantum tunneling is not merely a theoretical curiosity — it governs the lifetimes of atomic nuclei. The puzzle it solves is striking: an alpha particle inside a nucleus is classically forbidden from escaping, yet escape it does. Quantum mechanics explains exactly how, and even gives the right order of magnitude for half-lives that range from microseconds to billions of years.

The nuclear potential puzzle

An alpha particle ( ${}^4\text{He}$ nucleus, charge $+2e$ ) experiences two competing forces while inside a nucleus of charge $Ze$ :

The nuclear force — short-ranged and attractive. Inside the nuclear radius $R$ the net potential energy is strongly negative, trapping the alpha particle in a deep well.
The Coulomb repulsion — long-ranged. For $r > R$ the alpha particle sees a purely repulsive Coulomb barrier

V_C(r) = \frac{2(Z-2)e^2}{4\pi\varepsilon_0 r} \quad (r > R).

At the nuclear surface $r = R$ this barrier reaches its maximum value $V_C(R)$ . For a typical heavy nucleus the peak is on the order of tens of MeV, yet observed alpha energies are only a few MeV. Classically, an alpha particle with energy $E \approx 5\,\text{MeV}$ cannot escape a barrier peak of $\sim 25\,\text{MeV}$ . There is a classically forbidden region stretching from $r = R$ out to a classical turning point $r_2$ , where $V_C(r_2) = E$ .

Gamow's tunneling picture

In 1928 George Gamow (independently also Gurney and Condon) applied the WKB (Wentzel–Kramers– Brillouin) approximation to this barrier. The key quantity is the tunneling exponent — the integral of $|\kappa(r)|$ across the forbidden region, where

\kappa(r) = \frac{\sqrt{2m(V(r) - E)}}{\hbar}

is the real exponential decay rate at position $r$ , and $m$ is the reduced mass of the alpha particle. In the WKB approximation the transmission probability through the barrier is

T \approx e^{-2G}, \qquad G = \int_{R}^{r_2} \kappa(r)\, dr = \int_{R}^{r_2} \frac{\sqrt{2m(V_C(r) - E)}}{\hbar}\, dr.

The quantity $G$ is called the Gamow factor. Because $T$ falls exponentially with $G$ , even a moderate change in the barrier parameters leads to an enormous change in the decay rate — and hence in the half-life.

Evaluating the Gamow factor for a Coulomb barrier

Substituting $V_C(r) = k_e(2)(Z-2)e^2/r$ (where $k_e = 1/(4\pi\varepsilon_0)$ ) and noting that the turning point satisfies $V_C(r_2) = E$ , so $r_2 = k_e(2)(Z-2)e^2/E$ , lets us factor the integral. Letting $u = r/r_2$ , the lower limit $r = R$ maps to $u = R/r_2$ (not $0$ ):

G = \frac{k_e\,2(Z-2)e^2}{\hbar}\sqrt{\frac{2m}{E}} \int_{R/r_2}^{1} \sqrt{\frac{1}{u} - 1}\,du.

For a low-energy alpha the turning point lies well outside the nucleus, $R \ll r_2$ , so the lower limit is close to zero. Approximating $R/r_2 \to 0$ uses the closed form $\int_0^1 \sqrt{1/u - 1}\, du = \pi/2$ and gives the famous Gamow result

G \approx \frac{\pi k_e\, 2(Z-2)e^2}{\hbar} \sqrt{\frac{m}{2E}}.

Keeping the exact lower limit instead replaces $\pi/2$ by $\arccos\!\sqrt{x} - \sqrt{x(1-x)}$ with $x = R/r_2$ , a finite-size correction that lowers $G$ slightly; the expression above is the standard $R \ll r_2$ Gamow approximation, not an exact evaluation.

Writing this in SI-friendly form with the fine-structure constant $\alpha_{\rm em} = k_e e^2/(\hbar c) \approx 1/137$ :

G = \frac{2\pi\,\alpha_{\rm em}\,(Z-2)}{\beta},

where $\beta = v/c$ is the velocity of the emitted alpha particle in units of $c$ . Because $\beta$ is small for typical alpha energies ( $\beta \ll 1$ ), $G$ is large, and the tunneling probability $e^{-2G}$ is tiny — precisely why many heavy nuclei are stable on geological timescales.

The Geiger-Nuttall law

The decay constant $\lambda$ (probability of decay per unit time) is proportional to $T$ :

\lambda \propto e^{-2G} \propto e^{-\text{const}/\sqrt{E}}.

Taking logarithms:

\log_{10} \lambda = -\frac{A}{\sqrt{E}} + B,

where $A$ and $B$ are constants that depend on the daughter nucleus. This is the empirical Geiger-Nuttall law (1911), discovered purely from data before quantum mechanics existed. The Gamow tunneling model explains it from first principles.

Why tunneling and not something else?

A key test of the tunneling picture is that the same WKB integral that predicts the correct Gamow factor for one nucleus also predicts correct half-lives for other nuclei, across many decades of half-life, using only the measured alpha energy and the nuclear charge $Z$ . No other mechanism — thermal fluctuation, classical over-barrier escape — reproduces the systematic dependence on $Z$ and $E$ that the data show. The universality is the fingerprint of tunneling.