The atom with no nucleus

Where the electron probably is found that the hydrogen atom’s levels come from a single attraction and a single mass, and why an atom is the size it is found its size as a competition between the kinetic energy of confinement and the attraction of the nucleus. In both, the proton is treated as a fixed centre, which is a very good approximation because it is 1,836 times heavier than the electron and barely moves.

Positronium has no fixed centre. It is an electron bound to its antiparticle, the positron, which has the same mass and the opposite charge. The attraction is the same as hydrogen’s; the arrangement is completely symmetric; and the atom exists only until the two particles meet and turn into light. It was predicted in 1934 by Stjepan Mohorovičić, named in 1945 by Arthur Ruark and found in 1951 by Martin Deutsch, who showed that positrons stopped in a gas sometimes lived for a hundred nanoseconds before annihilating — far longer than a free positron should.

That long life is the most striking fact about positronium, and its explanation is a symmetry that has nothing to do with the attraction. This essay builds the atom first and destroys it second.

Hydrogen with no centre #

When two bodies orbit each other, both move round their common centre of mass, and the problem separates into the motion of that centre and the motion of one body relative to the other. The relative motion behaves like a single particle with the reduced mass

$$\mu = \frac{m_1 m_2}{m_1 + m_2},$$

moving in the attraction between the two. For hydrogen that is $0.99946$ of the electron’s mass, and the correction to the levels is half a part in a thousand. For positronium the two masses are equal and the reduced mass is exactly half an electron’s.

Every energy in the hydrogen formula is proportional to the reduced mass and every length inversely proportional to it, so positronium’s levels are hydrogen’s halved — the ground state at −6.8 electronvolts instead of −13.6 — and its size is doubled. Muonium, an electron bound to a positive antimuon two hundred times heavier, sits between them and is almost indistinguishable from hydrogen; it is one of the cleanest tests of the two-body formula, since neither partner has internal structure. Positronium is the extreme case: two light partners of equal mass, each moving on an orbit of hydrogen’s size round a centre between them.

The spectrum is a hydrogen spectrum with every line at half the frequency. The Lyman-alpha line of hydrogen, at 121.6 nanometres, appears in positronium at 243 nanometres, in the near ultraviolet, and it was seen in emission from excited positronium in 1975, the first optical spectroscopy of an atom of antimatter and matter together.

Two spins, one atom, two kinds #

The electron and positron each have spin one-half, and four states and one of them is odd found what two spin-halves combine into: three states with total spin one, the triplet, and one with total spin zero, the singlet, antisymmetric under exchange of the two spins. In helium the same split produced two families of spectra that nineteenth-century spectroscopists took for two substances. In positronium it produces two kinds of atom, para-positronium with spins opposed in the singlet and ortho-positronium with spins aligned in the triplet.

They differ slightly in energy. The magnetic interaction between the two spins, which in hydrogen splits the ground state by the tiny amount behind the line that takes eleven million years, is in positronium about six hundred times larger, because a positron’s magnetic moment is that much larger than a proton’s; and a second contribution, unique to an atom made of a particle and its antiparticle, comes from the pair briefly annihilating into a virtual photon and re-forming. The two together split ortho from para by 203 gigahertz. But the important difference between them is not their energy. It is how they die.

Two ways to end, a thousand times apart #

An electron and a positron at the same point can annihilate, converting their rest energy — 511 keV each, the mass of mass is a form of energy made literal — into photons. In positronium they are not at the same point, but their wavefunction gives a probability of finding them there, $|\psi(0)|^2 = 1/\pi a^3$ for the ground state, and the annihilation rate is that probability times a rate per unit density.

The two kinds of atom have the same wavefunction for their relative position and the same probability of overlap. Yet para-positronium lives 125 picoseconds and ortho-positronium 142 nanoseconds. The atoms are identical in every respect but the alignment of two spins, and one lives more than a thousand times longer than the other.

The lowest-order rates are

$$\Gamma_{\text{para}\to2\gamma} = \frac{\alpha^5 mc^2}{2\hbar}, \qquad \Gamma_{\text{ortho}\to3\gamma} = \frac{2(\pi^2 - 9)}{9\pi}\,\frac{\alpha^6 mc^2}{\hbar},$$

with $\alpha \approx 1/137$ the fine-structure constant. The ortho rate carries one more power of $\alpha$, and its numerical factor is about an eighth of the para one’s; together they make the ratio. The formulas give 124 picoseconds and 138 nanoseconds, and the small remaining differences from measurement come from higher-order corrections in quantum electrodynamics, which have been computed and tested to parts in a thousand. But the formulas presuppose the decisive fact: that para decays to two photons and ortho to three. Why it should be so is the interesting part.

Why the overlap sets the clock #

Both rates are proportional to the probability of finding the electron and positron at the same point, and that probability is where positronium’s doubled size shows. For hydrogen’s ground state it is $1/\pi a_0^3$; for positronium’s, with twice the radius, it is eight times smaller. An atom built like positronium but with hydrogen’s size would die eight times faster.

The same dependence makes excited positronium live longer. The probability of contact at the origin falls as $1/n^3$ for the $s$ states and vanishes altogether for the others, whose wavefunctions are zero at the centre. Ortho-positronium in its $2S$ state lives about 1.1 microseconds, eight times longer than in its ground state, and in states with orbital angular momentum it cannot annihilate directly at all: it must first fall back to an $s$ state by emitting a photon. Positronium excited to a high Rydberg state — the same swollen orbits as the atom the size of a bacterium, at twice the size again — lives for microseconds to milliseconds, long enough to be steered by electric fields and sent through an interferometer. That is the route by which the experiments described at the end of this essay hope to drop it in a gravitational field.

The photon counts a symmetry allows #

Two photons is the obvious decay. A pair at rest has zero momentum, so its decay products must have zero total momentum, and one photon cannot — a single photon always carries momentum, as a photon with a momentum found. Two photons leaving back to back can, each with 511 keV. Three can as well, sharing the energy in many ways. What forbids ortho-positronium the easy route is charge conjugation.

Charge conjugation is the operation of swapping every particle for its antiparticle. Electromagnetism is unchanged by it: an atom of antihydrogen has exactly hydrogen’s spectrum. So it is a symmetry, and states can be classified by whether they are even or odd under it, with the classification conserved in any electromagnetic process. For an electron–positron pair the operation swaps the two particles, and the sign it produces depends on how the wavefunction behaves under the swap: $(-1)^L$ from the orbital part, $(-1)^{S+1}$ from the spin part, and a further $-1$ because the two are fermions. The product is

$$C = (-1)^{L+S}.$$

For the ground states, $L = 0$: para, with $S = 0$, is even; ortho, with $S = 1$, is odd. A photon is odd under charge conjugation — the electromagnetic field reverses sign when every charge does — so $n$ photons are $(-1)^n$. Conservation then says para can decay only to even numbers of photons and ortho only to odd numbers.

Ortho cannot use one photon, which momentum forbids, nor two, which charge conjugation forbids, so its fastest route is three, and each extra photon in a decay costs a factor of order the fine-structure constant. That single forbidden channel is the whole of the thousandfold difference.

The same rule, with one substitution, explains one of the great surprises of particle physics. A charm quark bound to its antiquark is an atom of the same kind, held by the strong force instead of the electric one, and its forces carry a charge-conjugation rule of their own: the gluons that replace photons as decay products must, for the spin-aligned state, come in threes. When that spin-aligned state was found in November 1974, simultaneously at Brookhaven and at Stanford, its most astonishing feature was its narrowness — it lived a thousand times longer than any comparable particle — and that long life was the strong-force copy of ortho-positronium’s. Three gluons are expensive for the same reason three photons are. The particle, named J at one laboratory and ψ at the other, settled the existence of the charm quark, and positronium was the model everyone reached for to understand it.

The rule is exact in electromagnetism. A search for ortho-positronium decaying into two photons, in violation of it, set an upper limit of a few parts in a million on the fraction that does, and every other forbidden channel has been looked for and not found. It is the conservation law a symmetry hands over in its plainest form: a symmetry of the forces, a quantity it preserves, and a process the quantity forbids, not suppresses.

Telling the two apart #

The two decays leave different signatures. Two-photon decay at rest gives two photons of exactly 511 keV, back to back. Three-photon decay gives photons whose energies can take any values that add up to 1,022 keV with zero total momentum; Aadne Ore and John Powell worked out their distribution in 1949, a smooth spectrum from zero to the maximum, rising towards the top. A gamma-ray detector watching positrons stop in a material sees the sharp 511 keV line from para-positronium and from free annihilation, and underneath it a continuum from ortho, whose height relative to the line measures how much ortho was formed. That is how Deutsch identified the atom in 1951, alongside its lifetime.

A magnetic field mixes the two. It cannot change the total spin of a state with $S_z = \pm1$, but it couples the ortho state with $S_z = 0$ to the para state, which also has $S_z = 0$, and the mixed state acquires some of para’s fast two-photon decay. In a field of a tesla the affected third of the ortho atoms annihilate in nanoseconds instead of a hundred; the effect was one of the first confirmations of the spin assignments, and it is now used to measure the ortho–para splitting by sweeping the field through the resonance.

Ortho-positronium as a ruler #

Positrons are easy to make — a sodium-22 source emits them steadily — and when one stops in a solid it often captures an electron and forms positronium. In an insulator with gaps between its molecules, the atom settles in a gap, because the repulsion between its electron and the surrounding electrons pushes it into whatever empty space there is.

There its long life is no longer protected. The positron in ortho-positronium has its spin aligned with its own electron, which forbids the two-photon decay with that electron. But it is also surrounded by the electrons of the pore’s walls, whose spins are random, and wherever its wavefunction overlaps them it can annihilate with one of those instead, in two photons, at a rate that does not care about its partner. This pick-off annihilation shortens the lifetime by an amount that depends on how much of the atom’s wavefunction reaches the wall, which depends on the size of the pore.

The Tao–Eldrup model makes that quantitative with a single empirical parameter, the thickness of the electron layer on the wall, and turns a measured lifetime into a pore radius. A lifetime of two nanoseconds means a pore about 0.3 nanometres across in radius; eight nanoseconds, 0.6. Positron annihilation lifetime spectroscopy is used to measure the free volume in polymers — the gaps between chains that control how gases diffuse through a plastic and how it ages — and the pore sizes in the porous films used as insulators in microchips, at a scale no microscope reaches inside a solid.

The scanner that counts back-to-back pairs #

Positron annihilation is in daily clinical use. A positron-emission tomography scan injects a tracer labelled with a positron-emitting nucleus, usually fluorine-18 attached to a sugar molecule, which collects where tissue is metabolically active. Each positron travels a millimetre or so, slows, and annihilates, mostly through para-positronium or directly with an electron, and the scanner looks for pairs of 511 keV photons arriving on opposite sides of the patient within a few hundred picoseconds of each other. Each pair defines a line through the point of annihilation, and many lines reconstruct the tracer’s distribution. The back-to-back geometry the scanner relies on is the momentum argument above; the energy it filters on is the electron’s rest mass.

The ortho-positronium formed in tissue carries extra information that conventional scanners throw away. Its lifetime, shortened by pick-off just as in a polymer, depends on the size of the gaps between molecules in the tissue and on how much dissolved oxygen is there to quench it, and both differ between healthy and cancerous tissue. Scanners able to record the third photon and the delay between a positron’s emission and its annihilation have produced the first images of positronium lifetime in a patient, in 2021, and whether that contrast proves clinically useful is being studied.

What the atom’s simplicity hides #

The hydrogen formulas are not exact. Positronium’s partners move at about $\alpha c/2$, and relativistic corrections, the spin–spin interaction and the virtual annihilation all shift its levels at relative order $\alpha^2$, much more than in hydrogen, because there is no heavy nucleus to suppress the recoil terms. Calculating them to the precision of modern measurements is a test of quantum electrodynamics for a bound system, and in one case, the $n = 2$ fine structure, measurement and theory disagree by several standard deviations.

Positronium does not form everywhere. In metals the electron density is too high and too mobile for a bound pair to survive; the positron annihilates with a conduction electron instead, with a lifetime of a few hundred picoseconds that measures the electron density. Positronium forms in insulators, in gases and in vacuum near surfaces.

Gravity has not been measured on it. Because positronium is made of a particle and an antiparticle, its fall in a gravitational field tests whether antimatter falls like matter, the question the antiatom that had to be weighed anyway asked of antihydrogen. Positronium in its long-lived excited Rydberg states lives long enough to fall measurably, and experiments to do it are being built.

Still open: whether positronium can be made to condense #

Positronium atoms are bosons, light ones — at twice an electron’s mass, a thousand times lighter than any atom — and a gas of them would become a Bose–Einstein condensate at a temperature of about 15 K at densities that intense positron sources can approach. A condensate of positronium would be the first made of matter and antimatter together. If it could be made dense and coherent enough, its annihilation photons would be emitted coherently as well, stimulated by each other, and it would be a gamma-ray laser, emitting at 511 keV.

The obstacles are severe. The atoms must be formed in large numbers in a small volume within a fraction of their 142-nanosecond lifetime, cooled from the electronvolt energies at which they form to a few kelvin, and kept in the ortho state, since collisions between ortho atoms can flip them into para, which annihilate at once. Each step has been demonstrated separately: dense positron bunches, positronium molecules formed from two atoms at a surface, laser cooling of positronium reported in 2024. Whether all can be made to work together, faster than the atom’s own clock runs out, has not been shown.

The atom itself sets that clock, and the clock is a symmetry. The spin a photon has to carry away found that conservation of angular momentum decides which transitions an atom can make. In positronium a second conservation law decides how the atom can stop existing, and it is the difference between a picosecond and a hundred nanoseconds — between an atom that vanishes before it can be studied and one that can be cooled, excited, trapped and dropped.