Alpha Decay & Quantum Tunneling

Definition

Alpha decay is the emission of an alpha particle — a helium-4 nucleus, two protons and two neutrons — from a heavy nucleus. The alpha particle does not have enough energy to climb over the electrostatic (Coulomb) barrier that confines it. Instead it leaks through that barrier by quantum tunneling:

A         A-4      4
 X    →      Y  +    He   (the alpha particle)
 Z         Z-2      2

For example, uranium-238 alpha-decays to thorium-234:

238            234         4
   U      →       Th    +    He      Q ≈ 4.27 MeV
 92            90          2

The released energy Q is shared between the alpha (which carries away most of the kinetic energy) and the recoiling daughter nucleus.

How it works — the barrier you can't climb

Picture the potential energy of the alpha particle as a function of its distance r from the centre of the nucleus. There are two competing forces:

• Inside the nucleus (r < R, where R is the nuclear radius, a few femtometres), the strong nuclear force dominates and the potential is a deep attractive well. The alpha rattles around in here.
• Outside the nucleus (r > R), only the long-range Coulomb repulsion between the positive alpha (+2e) and the positive daughter (+Ze) remains. The potential is a barrier that falls off as 1/r.

At the nuclear surface the Coulomb potential peaks. For a heavy emitter this peak is around 25–30 MeV. But the alpha emerges with only 4–9 MeV of kinetic energy. Classically, a ball rolling around in a bowl whose rim is 30 MeV high can never get out with only 5 MeV of energy. It is trapped forever.

Quantum mechanics breaks the deadlock. The alpha is not a ball; it is a wavefunction. When that wave hits the barrier, it does not stop — it decays exponentially through the classically forbidden region (where E < V) and re-emerges on the far side with a small surviving amplitude. The probability of finding the alpha outside is non-zero. Over enough barrier assaults, it escapes. This is quantum tunneling applied to the nucleus.

The Gamow factor — exponential suppression

George Gamow (and independently Ronald Gurney and Edward Condon) worked out the tunneling probability in 1928 using the WKB approximation. The transmission probability through the barrier is dominated by an exponential:

P ≈ e^(−2G)

         1   r₂
G   =   ───  ∫   √( 2m·(V(r) − E) )  dr
         ħ   r₁

G is the Gamow exponent: the integral of the imaginary wave number across the forbidden region, from the nuclear radius r₁ = R out to the classical turning point r₂ where the Coulomb potential has fallen back to the alpha energy E. The mass m is the alpha's (≈ 4 u), and ħ is the reduced Planck constant.

For a typical alpha emitter, 2G ≈ 60 to 90, so the per-assault tunneling probability e^(−2G) is somewhere around 10⁻³⁰ to 10⁻⁴⁰. Astronomically small. The crucial feature is that G sits in an exponent: a small change in alpha energy E (which sets the turning point r₂ and the barrier width) produces a gigantic change in e^(−2G). That exponential sensitivity is the whole story.

The decay constant is the assault frequency times the tunneling probability:

λ ≈ f · e^(−2G)          t½ = ln(2) / λ

f ≈ v / (2R) ≈ 10²¹ barrier collisions per second

The Geiger-Nuttall law

Hans Geiger and John Nuttall noticed in 1911 — seventeen years before Gamow explained it — that alpha emitters obey a strikingly simple empirical rule. Plot the logarithm of the half-life against the alpha energy and you get a straight line. In modern form:

log₁₀(t½) ≈ a · Z / √E  +  b

where Z is the charge of the daughter nucleus and E is the alpha kinetic energy in MeV. The constants a and b are nearly universal across a decay chain. This is exactly what the Gamow factor predicts: evaluating the WKB integral for a Coulomb barrier gives 2G ∝ Z/√E, and since log t½ ∝ 2G, half-life is linear in Z/√E. Higher-energy alphas (larger E) face a thinner barrier, tunnel far more readily, and live far shorter lives.

A worked example — uranium-238 vs polonium-212

Let's see the exponential at work with two real nuclei whose alpha energies differ by only a factor of about 2, yet whose half-lives differ by 24 orders of magnitude.

Quantity	Uranium-238	Polonium-212
Alpha energy E	4.27 MeV	8.95 MeV
Daughter charge Z	90 (Th)	82 (Pb)
Barrier peak (approx)	~28 MeV	~26 MeV
Gamow exponent 2G (approx)	~89	~62
Tunneling probability e^(−2G)	~2×10⁻³⁹	~1×10⁻²⁷
Half-life t½	4.5×10⁹ years	0.3 µs (3×10⁻⁷ s)

Both nuclei strike their barriers about 10²¹ times per second. The U-238 alpha has a tunneling probability around 10⁻³⁹ per hit, so even at 10²¹ attempts per second the expected wait is billions of years. The Po-212 alpha, just twice as energetic, sees its tunneling probability rocket up by roughly a dozen orders of magnitude — and escapes in microseconds. A factor-of-two change in energy bought a factor-of-10²⁴ change in half-life. That is the Gamow exponential made visible.

Variants and regimes

Process	What tunnels	Direction	Governing factor
Alpha decay	He-4 cluster	Out of nucleus	e^(−2G), Geiger-Nuttall
Cluster decay (e.g. C-14 emission)	Heavier cluster (C, O, Ne)	Out of nucleus	Larger 2G → far rarer than alpha
Proton emission	Single proton	Out of nucleus	Lower barrier, no preformation cost
Spontaneous fission	Nucleus splits in two	Through fission barrier	Competes with alpha in heaviest nuclei
Stellar fusion	Two charged nuclei	Toward each other	Same e^(−2G), inverted geometry
Scanning tunneling microscope	Electron	Across vacuum gap	e^(−2κd), same WKB form

The unifying thread is the WKB exponential e^(−2G). Whether it is an alpha leaving uranium, two protons fusing in the Sun, or an electron crossing the gap in a scanning tunneling microscope, the same imaginary-wave-number integral controls the rate, and the same brutal exponential sensitivity applies.

Estimating a half-life from the Gamow model

// Toy Gamow-model half-life estimator for an alpha emitter.
// Units: energies in MeV, lengths in fm. Constants folded into k.
// 2G ≈ (4 Z e² / (ħ v)) · [arccos√x − √(x(1−x))],  x = E / B
// We use the common compact approximation 2G ≈ a·Z/√E − c·√(Z·R).

function gamowExponent2G(Z, E_MeV, R_fm = 7.4) {
  // a and c calibrated to reproduce textbook alpha half-lives (MeV, fm).
  const a = 3.97;   // MeV^(1/2) per unit Z
  const c = 2.97;   // fm^(-1/2) per unit (Z·R)^(1/2)
  return a * Z / Math.sqrt(E_MeV) - c * Math.sqrt(Z * R_fm);
}

function halfLifeSeconds(Z, E_MeV) {
  const f = 1e21;                        // ~assault frequency, hits/second
  const twoG = gamowExponent2G(Z, E_MeV);
  const P = Math.exp(-twoG);             // tunneling probability per hit
  const lambda = f * P;                  // decay constant, 1/s
  return Math.LN2 / lambda;              // t½ in seconds
}

const yr = 3.156e7;
const cases = [
  ["U-238",  90, 4.27],
  ["Th-232", 90, 4.08],
  ["Ra-226", 88, 4.87],
  ["Po-210", 84, 5.41],
  ["Po-212", 82, 8.95],
];
for (const [name, Z, E] of cases) {
  const t = halfLifeSeconds(Z, E);
  const human = t > yr ? (t / yr).toExponential(2) + " yr" : t.toExponential(2) + " s";
  console.log(`${name}: 2G≈${gamowExponent2G(Z, E).toFixed(1)},  t½≈${human}`);
}
// Reproduces the ~24-order-of-magnitude spread from a single exponential.

The model is crude — a single tuned line — yet it captures the headline result: feed it five real alpha energies and it spits out half-lives ranging from microseconds to longer than the age of the universe. No fitting per nucleus; just Z/√E in an exponent.

Where it shows up

• Radiometric dating. The slow, fixed alpha half-lives of uranium and thorium are the clocks behind uranium-lead dating of zircons and the determination of Earth's 4.5-billion-year age. The Gamow exponential is why those clocks tick at constant, geologically convenient rates.
• Smoke detectors. The americium-241 in an ionization smoke detector is an alpha emitter (t½ ≈ 432 yr) whose tunneling alphas ionize the air gap; smoke disrupts the current and triggers the alarm.
• Radioisotope power (RTGs). Plutonium-238 (t½ ≈ 88 yr) alpha-decays steadily, and its heat powers deep-space probes like Voyager and the Mars rovers, where solar panels fail.
• Targeted alpha therapy. Short-range, high-energy alphas from isotopes like actinium-225 deliver lethal dose to cancer cells while sparing neighbours just a few cell-widths away.
• Stellar fusion. The Sun shines because protons tunnel through their mutual Coulomb barrier at only 15 million kelvin — the same Gamow factor, run in reverse, sets the solar burning rate. See nuclear fusion.
• Superheavy element synthesis. Whether a freshly made nucleus survives long enough to detect is set by the competition between alpha decay and spontaneous fission, both tunneling processes.

Common pitfalls and misconceptions

• "The alpha gets a sudden energy boost to jump the wall." No. Energy is conserved throughout. The alpha never has more than its 4–9 MeV; it passes through the forbidden region, never over the top. Tunneling is not borrowing-and-repaying energy.
• "A small barrier change gives a small rate change." The opposite. Because G is an exponent, a 10% change in barrier width or alpha energy can change the half-life by many orders of magnitude. This non-linearity is the entire reason for the 24-decade spread.
• "The alpha definitely exists pre-formed inside the nucleus." The four nucleons must first cluster into an alpha. A "preformation factor" (≈ 0.01–0.3) multiplies the rate. It matters for absolute rates but barely affects the Geiger-Nuttall slope, which is pure tunneling.
• "Heavier nuclei always alpha-decay faster." Not the mass but the energy Q and daughter charge Z set the rate via Z/√E. Two nuclei of similar mass can differ wildly in half-life if their alpha energies differ.
• "Tunneling violates energy conservation." It does not. The classically forbidden region is forbidden for a localized particle, but a wave has no such restriction. The escaped alpha carries exactly the energy Q allows.
• "The half-life depends on temperature or chemistry." To excellent approximation it does not. The nuclear barrier dwarfs any chemical or thermal energy scale, so an alpha emitter ticks at the same rate whether it is in ice, plasma, or a chemical compound.

Derivation analysis — why 24 orders of magnitude is inevitable

Evaluate the Gamow integral for a pure Coulomb barrier. With V(r) = 2Ze²/(4πε₀ r) and turning point r₂ = 2Ze²/(4πε₀ E), the WKB integral has a closed form. In the thick-barrier limit (E much less than the barrier peak) it reduces to:

2G ≈  (π · 2Z · e²) / (4πε₀ · ħ) · √(2m / E)   −   (correction term)

    ∝  Z / √E

So 2G scales as Z/√E — precisely the Geiger-Nuttall variable. Now propagate the numbers. For real alpha emitters, 2G ranges from about 62 (Po-212) to about 89 (U-238). Since log₁₀(t½) ≈ 2G/ln(10) + const, that span of 2G ≈ 27 translates directly into:

Δlog₁₀(t½) ≈ Δ(2G) / ln(10) ≈ 27 / 2.303 ≈ 12 decades from 2G alone

Adding the variation in the prefactor (assault frequency, turning-point distance, and the second WKB term, which itself depends on √(ZR)) roughly doubles this, delivering the observed ~24 orders of magnitude from microseconds to ~10¹⁰ years. The remarkable part is the leverage: alpha energies vary by barely a factor of 2 across all known emitters, yet the exponential turns that mild input range into the widest dynamic range of any single physical process you will meet — wider than the range of densities in the universe.