Out Through a Wall It Cannot Climb — Gamow and Quantum Tunnelling, 1928

Last time, the newborn quantum mechanics explained why atoms join together, in 1927 (Part 1 is here). Part 2 moves to the very centre of the atom, the nucleus. Every so often a nucleus throws out a small particle — and that particle turned up on the far side of a wall it should never have been able to cross. In 1928 the contradiction was resolved by an idea that only quantum mechanics could supply: the particle does not climb over the wall. It slips through it.

A particle weaker than the wall, found outside it

Left to themselves, the nuclei of heavy elements such as uranium and radium will, with some probability, throw out a small particle and turn into a different element. The particle is called an alpha particle — it is the nucleus of a helium atom — and the change is called alpha decay.

The nucleus carries positive electric charge, and so does the alpha particle. Like charges push each other away, so an alpha particle brought towards a nucleus from outside is pushed back, as if rolling up a slope. Here we will call that slope of repulsion "the wall". For an alpha particle inside the nucleus to get out, common sense says it must reach the top of the wall and go over.

In 1927 Rutherford set out an awkward fact about this wall. When alpha particles of 8.57 MeV from thorium C′ (today, polonium-212) were fired at uranium, they bounced off in exactly the clean way that the repulsion of two positive charges predicts. MeV — million electron volts — is the unit of energy used for nuclei; read a bigger number as a harder push. So the wall around uranium is at least 8.57 MeV high. Yet the alpha particles that uranium itself gives off carry only about 4.2 MeV — less than half the height of the wall.

How could a particle with half the push be on the outside? Rutherford proposed that the alpha particle picks up two electrons, leaves the nucleus electrically neutral, and sheds the electrons once it is outside. It was a stopgap assumption, invented to paper over the contradiction.

Summer 1928: Göttingen and Princeton

In the summer of 1928 George Gamow, a 24-year-old from the Soviet Union, was staying at the University of Göttingen in Germany. As the story goes, he read this paper of Rutherford's soon after he arrived, and that was where it began. But the tale of the discovery has been coloured as it passed through Gamow's own entertaining memoirs, and the physicist and historian Eugen Merzbacher warns that fact and fiction are hard to separate (the same kind of caution applied to Heisenberg's Heligoland story in Part 5 of the previous series).

What is certain are the dates on the record. Gamow's paper reached the journal on 2 August 1928. Around the same time, across the Atlantic at Princeton, Ronald Gurney from Britain and Edward Condon from the United States had reached the same answer entirely on their own. The short report they sent to Nature is dated 30 July. Neither group knew of the other's work; they arrived at the same conclusion within days of each other.

The wave seeps through the wall

The key was Schrödinger's equation, which describes where a particle is by means of a wave, together with Born's reading of that wave: its size tells you the probability of finding the particle there (both were covered in Part 5 of the previous series).

In the old physics, a ball rolled at a slope without enough speed stops partway up and rolls back. It never appears on the other side. But when quantum mechanics is used to work out the particle's wave, the wave does not snap off where it meets the wall. It seeps a little way into it. Inside the wall it dies away quickly the further it goes — but if the wall is thin enough, some of it reaches the far side before it fades out. Then there is a small but real probability of finding the particle outside the wall.

The idea itself was not new in 1928. In 1927 Friedrich Hund had used the same kind of calculation for electrons inside molecules, and in 1928 Ralph Fowler and Lothar Nordheim used it for electrons pulled out of a metal surface by a strong electric field. The name "tunnel effect" came into use a little later, around 1931; papers of the time spoke of particles "penetrating" or "leaking through" a barrier. What Gamow and the others did that was new was to carry the idea into the nucleus.

An alpha particle inside the nucleus strikes the wall and is turned back again and again, countless times, and each time there is a tiny chance that it seeps out. Try often enough, and sooner or later it leaves. Seen this way, Rutherford's contradiction disappears. The 4.2 MeV particle did not climb a wall of 8.57 MeV or more — it slipped through. And the fact that the moment any single nucleus breaks up is a matter of pure chance came out naturally, as a question of probability. Gurney and Condon ended their report by noting that much had been written about the explosive violence with which the alpha particle is hurled from the nucleus — but on this picture, they wrote, it "slips away almost unnoticed".

Less than twice the energy, 10²¹ times the lifetime

What won the explanation acceptance was that it accounted, numbers and all, for a strange rule known for seventeen years. In 1911 Hans Geiger and John Nuttall had found that the elements giving off the most energetic alpha particles are the shortest-lived. Lifetime here means the half-life: the time it takes for half of the nuclei to decay.

What is strange is the size of the differences. In the examples that Gurney and Condon set out in a figure in their 1929 paper, uranium-238 gives off alpha particles of 4.18 MeV and has a half-life of 4.5 billion years; radium A (today, polonium-218), about 6 MeV and three minutes; radium C′ (polonium-214), 7.7 MeV and 160 microseconds — 1.6 ten-thousandths of a second. The energies differ by less than a factor of two, yet the half-lives differ by a factor of about 10²¹ — a thousand billion billion.

On the tunnelling picture, this extreme behaviour comes about naturally, because the chance of slipping through is exquisitely sensitive to how thick the wall is and how far below its top the particle sits. A little more push means a thinner stretch of wall to get through, and the chance jumps by many powers of ten. Gamow drew the line given by his formula on the same graph as Geiger and Nuttall's measured values for the elements of the uranium series, and showed that they agreed well. It was the first clear evidence that quantum mechanics holds inside the nucleus too.

Through the wall the other way — fusion inside stars

If a particle can slip out, it should also be able to slip in. In March 1929 Robert Atkinson and Fritz Houtermans submitted a paper that used Gamow's method to calculate the chance of a proton — a hydrogen nucleus — getting into a light nucleus. Their conclusion: at the temperatures and densities at the centre of a star, protons that cannot climb the repulsive wall head-on can still slip through it and enter light nuclei often enough to matter. The energy that makes stars shine might be explained by light nuclei combining into heavier ones. It was the moment the question of what powers the Sun began to take the concrete shape of an answer called nuclear fusion (what came next is in the nuclear fusion timeline).

Tunnelling later took a leading role inside electronic components too. In 1958 Leo Esaki reported electrons tunnelling through an extremely thin junction made of germanium, work that would lead to a Nobel Prize. And the 2025 Nobel Prize in Physics went to John Clarke, Michel Devoret and John Martinis. In 1985 the three showed that a superconducting circuit big enough to hold in your hand could tunnel as a whole, as if it were a single particle. That discovery became part of the foundation for today's superconducting quantum computers.

What could be done at this point, and what could not

The tunnelling theory of 1928 achieved two things. First, it resolved the contradiction of a particle weaker than the wall turning up outside it, without adding any special assumption. Second, it derived from a single idea both the fact that the timing of alpha decay is a matter of chance and the extreme relationship between energy and lifetime.

What it could not do was just as clear. Nobody knew what force held the alpha particle in on the inside of the wall. Nobody even knew what nuclei were made of; the neutron would not be found until 1932. Gamow and the others assumed that the alpha particle already existed as a single lump inside the nucleus, and replaced the inside with a simple well for the calculation. What matched was the trend and the rough order of magnitude — not the exact lifetime of each nucleus.

One more problem remained, a problem of words. "Slipping through", "tunnelling" — the language suggests a particle burrowing its way through the wall. Yet all the calculation shows is that the probability of finding the particle outside the wall is small but not zero. Where inside the wall the particle passed, and when, the calculation does not say. Merzbacher points out that the familiar way of describing tunnelling carries an anachronistic echo of Newtonian mechanics — the assumption that a particle always follows an unbroken path. The answers come out right, but the picture used to tell them is still borrowed from the old physics. The gap we saw in Part 1, between getting the answer and understanding it, is here too.

Next time, quantum mechanics meets relativity. The single equation Paul Dirac wrote in 1928 was dogged by a strange set of answers with negative energy. In time, those answers would turn into the prediction of something nobody had yet seen: an electron with positive charge.