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§ Phase vi · Tunneling · Chapter 02

Alpha decay

Uranium has been quietly spitting helium nuclei into the world for four and a half billion years, and until 1928 nobody had any idea how. A 24-year-old visitor at Göttingen, working with the brand-new Schrödinger equation in his hotel room, realised that the alpha particle was not climbing a mountain. It was leaking through it. From that one image he derived a law that spans twenty-five orders of magnitude in half-life.

Before there was a theory, there was a smell. In the late 1890s, the rooms at the Curies’ shed on the rue Lhomond glowed a faint blue at night, and the air around the open tubes of radium had an acrid, ozone-like tang. The radium was warm to the touch. It hissed, very softly, on a sensitive electrometer. Marie Curie weighed grams of pitchblende residue and concluded that something inside the heavy elements was pouring out energy with no visible source, no chemical change, and no obvious end. Ernest Rutherford, working in Cambridge and then in Montreal, sorted the emanations into two species: a heavy, easily-stopped ray he called alpha, and a lighter, more penetrating ray he called beta. He published the distinction in 1899. For the next decade the alpha was a mystery wrapped in a brass slug of lead foil.

The first decisive question was simple: what is an alpha particle? Rutherford had a guess by 1907, but a guess is not a measurement. In 1909, with his collaborator Thomas Royds, he ran the cleanest experiment of his career. A thin-walled glass tube was loaded with radon gas, a vigorous alpha emitter, and sealed inside a second, evacuated outer tube. The alphas could punch through the inner wall but not the outer one. After a week, Rutherford pulled the outer tube down to a tiny capillary and discharged it through a Geissler spectroscope. The spectrum that lit up on his eyepiece was unmistakable: the bright yellow line of helium, plus its blue and red companions, exactly as if he had ordered a fresh tank from the gas works. The alpha was a helium nucleus. Two protons and two neutrons, bound together so tightly that they travelled the universe as a single unit, and the heaviest elements were quietly leaking them.

That left the harder question. A uranium nucleus that is going to throw off a helium nucleus has, classically, no business doing so. The helium and the rest of the uranium attract each other electrically, with a Coulomb repulsion that becomes a wall of roughly 25 MeV at the edge of the nuclear surface. The alpha particle that uranium emits has a kinetic energy of about 4 MeV. Four. It comes out at one-sixth of the energy required, by the standard textbook argument, to even reach the lip of the cliff. And yet out it comes, on average, after four and a half billion years. Other isotopes do it in days, or in seconds, or in microseconds, and the variation in half-life across the periodic table spans more than twenty-five orders of magnitude. A theory that did not explain that range was not a theory at all.

The empirical fact ahead of any theory was a curious regularity discovered by Geiger and his colleague Nuttall in 1911. Hans Geiger, by then in Manchester and counting alpha tracks night and day with the brass scintillation apparatus he had built with Marsden, noticed that the faster an alpha emerged from a nucleus, the shorter the half-life of the parent. The relationship was not subtle. When Geiger plotted the logarithm of the half-life against the logarithm of the alpha’s range in air, he got a straight line, one straight line for each radioactive series. A few years later the German chemist Otto Hahn refined the same plot using kinetic energy instead of range. The result, soon called the Geiger-Nuttall law, looked like this: the logarithm of the half-life is a linear function of one over the square root of the alpha’s kinetic energy. Speeding the alpha up by a factor of two could shrink the half-life by a factor of ten billion.

For seventeen years nobody knew why. The Geiger-Nuttall law sat in every textbook of radioactivity as a brute empirical fact, like a chemist’s rule of thumb. Then in the summer of 1928, a freshly minted Soviet doctorate landed in Göttingen with a few hundred Reichsmarks of Rockefeller Fellowship money in his pocket and a copy of Atkinson and Houtermans’ new paper on stellar energy generation under his arm.

George Gamow was twenty-four. He had grown up in Odessa under a sky lit by red flares and shipping fires, had read Einstein’s special relativity in a Russian translation as a teenager, and had defended his doctorate at Leningrad under Friedmann, the man who had first found the expanding-universe solutions to Einstein’s equations. He had come to Göttingen for the summer because Max Born’s institute was where everyone went, and because he wanted to learn the new wave mechanics from the people who had built it. He stayed in a cheap pension and read in the library all day.

What Gamow noticed, sitting in the Göttingen reading room with Atkinson and Houtermans’ paper on protons fusing inside stars in front of him, was that the inverse problem was already half-solved. Atkinson and Houtermans had argued that two protons in the Sun’s core could tunnel through their mutual Coulomb repulsion and fuse, releasing energy. Quantum mechanics, they pointed out, did not forbid a particle from being on the wrong side of a wall it lacked the energy to climb. Schrödinger’s equation, the brand-new bible of the matter waves, has solutions that are exponentially small but not zero inside a classically forbidden region. Penetrate far enough and you emerge on the other side. The probability is tiny, but it is not zero.

Gamow turned the picture upside down. If two protons could leak together into a star, then a single alpha particle could leak out of a uranium nucleus. The model was almost insultingly simple. Inside the nucleus, the alpha sits in a deep, square-bottomed well, held there by the strong force at a depth of forty or fifty MeV. Outside the nucleus, the alpha sees a Coulomb potential 2Ze²/r dropping off as one over the distance, where Z is the charge of the daughter nucleus that remains after the alpha is gone. Inside the well, the alpha rattles back and forth, hitting the wall about 10²¹ times per second. Each hit, the Schrödinger equation says, has a tiny probability of slipping through. The probability is the squared modulus of the wavefunction on the far side of the barrier, and the wavefunction inside the forbidden region decays exponentially. Do the integral over the Coulomb barrier and you find that the log of the per-hit escape probability is proportional to minus the integral of √(V - E) across the barrier. Crunch through the integral for the 1/r shape, take the logarithm, and what emerges is exactly the Geiger-Nuttall straight line.

The whole calculation, Gamow later confessed, took him a few weeks of evenings, sketched out in the margins of borrowed lecture notes. By August 1928 he had a paper. By coincidence, two physicists at Princeton, Ronald Gurney and Edward Condon, had reached the same conclusion independently, and their note appeared in Nature almost simultaneously. The two groups agreed on essentials, but Gurney and Condon’s phrasing has been quoted ever since for its calm exactness: “It has hitherto been necessary to postulate some special arbitrary ‘instability’ of the nucleus, but in the following note, it is pointed out that disintegration is a natural consequence of the laws of quantum mechanics without any special hypothesis.” No special hypothesis. No instability. Just Schrödinger’s equation, a barrier, and time. “Much has been written of the explosive violence with which the alpha particle is hurled from its place in the nucleus,” they added. “But from the process pictured above, one would rather say that the alpha particle almost slips away unnoticed.”

The Geiger-Nuttall law now had a derivation. The slope of the log-half-life-versus-one-over-square-root-of-energy line was not a fitted parameter; it came out of the integral over the Coulomb barrier as a simple combination of fundamental constants and the charge of the daughter nucleus. The intercept depended on the nuclear radius, and from that intercept Gamow could read off a nuclear size of a few femtometres, in agreement with Rutherford’s gold-foil estimates. The same formula extended smoothly across twenty-five orders of magnitude in half-life. Polonium-212 emits an alpha at 8.95 MeV and lives for 0.3 microseconds. Uranium-238 emits an alpha at 4.27 MeV and lives for 4.5 billion years. Bismuth-209, the king of stability, was long thought to be the heaviest stable nuclide; in 2003 it was shown to undergo alpha decay with a half-life of 2 × 10¹⁹ years, more than a billion times the age of the universe. All three sit on the same Gamow line. A factor of two in energy, a factor of 10²⁵ in lifetime. The mathematics underneath is the exponential of an integral, and the exponent is exquisitely sensitive to its input.

κ(r) = √( 2m (V(r) − E) ) / ℏ

b = 2Ze² / (4πε₀ E)

G = ∫_R^b κ(r) dr

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

λ = f · exp(−2G) = f · exp(−A · Z / √E)

log T½  =  C₁  +  C₂ · Z / √E_α

There is a footnote to all of this that is worth recording, because it is the moment a piece of physics graduated from theory to applied engineering. In the summer of 1942, Eugene Wigner walked into Robert Oppenheimer’s office in Berkeley with a draft memorandum. The Manhattan Project had begun to wonder, in private, what happened if you packed enough alpha-emitters together to short-circuit the Geiger-Nuttall clock. Gamow’s formula was their tool. The whole apparatus of plutonium chemistry and weapons design rests, at one important step, on knowing exactly how a single nucleus leaks. Twenty-five orders of magnitude in half-life is not an academic curiosity. It is the difference between a chunk of uranium that warms your hand and one that ignites the sky.

Gamow himself, by the time of that memo, was a long-naturalised American who had spent the 1930s pivoting from nuclear physics to stellar nucleosynthesis to the early universe. He proposed Big Bang nucleosynthesis in 1948 with Ralph Alpher and Hans Bethe, an idea that was vindicated by the discovery of the cosmic microwave background eighteen years later. He wrote popular books about Mr Tompkins, a clerk who dreamed himself into universes where the speed of light was thirty miles per hour or where Planck’s constant was the size of a billiard ball. The bridge between his alpha-decay paper and Mr Tompkins is short: in both, he asked what would happen if you allowed quantum mechanics to run free in a place where classical physics insisted it had no business. The answer, in 1928, was a uranium nucleus quietly losing a helium ion. The answer, in his children’s books, was the visible world growing fuzzy at its edges.

It is worth pausing on what this chapter has actually achieved. We started with a phenomenon (heavy nuclei spit out helium nuclei) that the entire 19th century could not have predicted and that no classical theory of matter could explain. We worked through one image (a particle leaking through a wall it cannot climb) and one equation (the Schrödinger equation, applied to a Coulomb barrier), and we derived a quantitative law that organises every alpha emitter in the periodic table. The law works because tunneling is real. The exponential dependence of the lifetime on the square root of the alpha’s energy is not a polite analogy. It is what the wavefunction’s amplitude does as it crosses a forbidden zone, raised to the power of one over a small number, and then divided into the rattle frequency of a nucleon at relativistic speeds. The mathematics is unromantic, but the conclusion is breathtaking: the slow decay of uranium that drives the heat of the Earth’s core, the warmth of geothermal springs, and the existence of plate tectonics is the same physics as a photon hopping across a thin film of air between two prisms. Tunneling is one thing, applied at vastly different scales.