The parabola that decides which way a nucleus decays

The mass that is missing weighed nuclei and found them lighter than their parts, by an amount that is their binding energy divided by $c^2$, and drew the binding per nucleon against mass number as the curve that decides whether fission or fusion releases energy. That curve is drawn at the most strongly bound proton number for each mass number, and its last paragraphs named what it hides: at each mass number there are several nuclei with different numbers of protons, the isobars, and which of them is lightest decides which of them survive.

The question of which way a nucleus decays turns out to be a question about those isobars alone. The weak interaction can turn a neutron into a proton, or a proton into a neutron, and nothing else; it leaves the total number of nucleons $A$ alone. So a weak decay moves a nucleus sideways among the isobars with its mass number, one proton at a time. Whether a move is allowed is decided by the masses of the two atoms at its ends, and by nothing else. Mass is a form of energy, which is not the same as a source of it made the point in general: a system can release energy only by becoming a system of smaller mass. Here that general statement becomes a precise rule, with a twist for positrons that comes from counting electrons.

Why the masses lie on a parabola #

The semi-empirical mass formula, whose five terms the mass that is missing took apart, gives the binding energy of a nucleus with $Z$ protons and $N$ neutrons. At fixed $A = Z + N$, only two of its terms change with $Z$. The Coulomb term, the electrical repulsion of the protons, grows as $Z^2/A^{1/3}$ and penalises protons. The asymmetry term grows as $(N - Z)^2/A$ and penalises any imbalance between protons and neutrons, in either direction; it is the exclusion principle at work, because protons and neutrons each fill their own ladder of states, and no two in the same state found that filling one ladder higher than another costs energy. Add the small difference between the masses of a neutron and a hydrogen atom, and the atomic mass at fixed $A$ is a quadratic in $Z$.

The first figure draws it for $A = 91$. The points sit on a parabola whose bottom is zirconium-91, which is in fact the stable nucleus of that mass. (The formula does not always manage that. It has no shells in it, and at mass numbers near the magic numbers, around 125 to 137 for instance, its floor sits one or two protons above the real one.) To the left of the floor the nuclei have too many neutrons. Each can turn one of them into a proton, emitting an electron and an antineutrino, and step one place to the right; each step releases the mass difference, by the formula 8.2 MeV from krypton-91, 5.8 from rubidium, 3.4 from strontium and 1.0 from yttrium. To the right of the floor the nuclei have too many protons and step the other way.

This is the history of every fission fragment. Uranium splits into two pieces that inherit its neutron-rich proportions, far up the left wall of their parabolas, and each then falls step by step to the floor. The steps shrink as the floor approaches, because the slope of a parabola does, and smaller steps mean slower decays: the measured chain of mass 91 runs krypton-91 in 8.6 seconds, rubidium-91 in 58 seconds, strontium-91 in 9.6 hours and yttrium-91 in 58 days. The same pattern at mass 137 ends with caesium-137, whose last step releases only 1.18 MeV and takes thirty years, which is why it is caesium and not its short-lived parents that contaminated the ground around Chernobyl and Fukushima for decades.

Downhill by any amount, with one exception #

The rule that decides the direction is stated with atomic masses, the masses of whole neutral atoms, because those are what is measured and what the parabola draws. A $\beta^-$ decay, $Z \to Z + 1$, emits an electron and leaves the daughter nucleus with one more proton; the daughter atom needs one more electron to be neutral, and the emitted electron supplies it. So the decay is possible whenever the parent atom is heavier than the daughter atom by any amount. Electron capture, $Z \to Z - 1$, takes one of the atom’s own electrons into the nucleus, and again the electron count works out: possible whenever the parent atom is heavier, apart from the small binding energy of the captured electron.

Positron emission is the exception. It also takes $Z \to Z - 1$, but by creating a positron rather than absorbing an electron. Afterwards the daughter atom has one electron too many — the parent’s full set — and a positron has been made from nothing. Both cost an electron’s mass, so the parent atom must outweigh the daughter by more than $2m_ec^2 = 1{,}022$ keV before a positron can be emitted at all.

The second figure shows the rule at work on five real nuclei. Beryllium-7 is heavier than lithium-7 by 862 keV, enough to capture an electron and too little to make a positron, and it decays only by capture — which is why, as the half-life that chemistry can change found, its half-life depends slightly on its chemical surroundings: the decay needs an electron at the nucleus, and chemistry decides how many are there. The others clear the threshold, and the share of their decays that emit positrons grows with the energy above it, from a vanishing fraction in potassium-40, 482 keV over the line, to nearly all of fluorine-18’s and carbon-11’s. Those two are the workhorses of positron-emission tomography, chosen because almost every decay produces a positron, and so a pair of gamma rays a scanner can trace back to it.

Two parabolas when the nucleons are even #

The fifth term of the mass formula breaks the single parabola in two. Nucleons of the same kind bind slightly more strongly in pairs, so a nucleus with an even number of protons and an even number of neutrons gets a bonus, and one with odd numbers of both pays a penalty. For odd $A$ one of the two numbers is always odd and the term cancels, which is why the first figure needed only one parabola. For even $A$, the even–even and odd–odd isobars alternate as $Z$ steps along, and they lie on two parabolas, the odd–odd one raised above the even–even by twice the pairing energy.

Every single weak decay changes $Z$ by one, so for even $A$ every step jumps from one parabola to the other. That has two consequences, and the third figure shows the first. An odd–odd nucleus near the floor of the upper parabola can sit above both of its even–even neighbours on the lower one, and then it can decay in either direction. Potassium-40 is the classic case. Its atom is heavier than calcium-40 by 1.311 MeV and than argon-40 by 1.504 MeV, and it decays both ways: to calcium by $\beta^-$ decay in 89 per cent of decays, and to argon, nearly always by capturing an electron, in the rest. Its half-life of 1.25 billion years has left enough of it in the Earth to make it, with uranium and thorium, one of the three sources of the planet’s internal heat, and it is the largest single source of radioactivity in the human body — about four thousand decays a second in an adult. The argon branch is the clock of potassium–argon dating, the method the clock read by counting atoms belongs with: the argon is a gas trapped in a rock only after it solidifies, so counting it against the remaining potassium dates the rock.

Two steps at once #

The second consequence is the stranger one. An even–even nucleus near the floor of the lower parabola may be heavier than the even–even nucleus two places along, and still unable to reach it, because the odd–odd nucleus in between, on the upper parabola, is heavier than both. The fourth figure draws the textbook case. Germanium-76 outweighs selenium-76 by 2.039 MeV, but the step through arsenic-76 would go uphill by 923 keV, and a nucleus cannot borrow energy for a real intermediate state.

What it can do is take both steps in one event: two neutrons become two protons at once, two electrons and two antineutrinos leaving together. The process is second order in the weak interaction, so it is extraordinarily slow, and it was seen in a laboratory only in 1987. In germanium-76 the half-life is $1.9\times10^{21}$ years, a hundred billion times the age of the universe, and it is measured by watching a few tens of kilograms of germanium enriched in that isotope for years. About a dozen nuclei are stranded this way, and for all of them it is the pairing gap that strands them. Calcium-40, in the previous figure, is in the mirror-image position: heavier than argon-40 by 194 keV, but able to reach it only by capturing two electrons at once, a process allowed by the masses and never yet seen.

The decay that would not need the neutrinos #

Double β decay has made germanium-76 and xenon-136 the subject of some of the largest low-background experiments ever built, and the reason is a possibility the masses allow but the rate does not settle. If the neutrino is its own antiparticle, then the antineutrino emitted at one step could be absorbed as a neutrino at the other, and the decay could happen with no neutrinos leaving at all. Then the two electrons would carry the entire mass difference, 2,039 keV for germanium-76, and the sum of their energies would be a sharp line instead of a broad spectrum. The energy that did not all arrive found the continuous spectrum of ordinary β decay to be the evidence that a neutrino carries off part of the energy; the neutrinoless version would show its absence by a spike at the endpoint.

The measured mass difference is what tells the experiments where to look. A germanium detector made of the source itself measures the summed electron energy with a resolution of a few keV, and a peak at exactly 2,039 keV would be a discovery about the nature of the neutrino and the conservation of lepton number. None has been seen: the half-life of the neutrinoless mode in germanium-76 is longer than $10^{26}$ years, which bounds the combination of neutrino masses that would drive it to below roughly a tenth of an electronvolt.

Why stable odd–odd nuclei are almost unknown #

The two parabolas have a consequence that can be read off a chart of the nuclides at a glance. An odd–odd nucleus lies on the upper parabola, and both its neighbours lie on the lower one, two pairing energies down. Unless the upper parabola’s floor happens to fall exactly between two even–even nuclei that are both well up their own walls, at least one of those neighbours will be lighter than the odd–odd nucleus, and it will decay. So stable odd–odd nuclei should be rare, and they are: there are four, deuterium, lithium-6, boron-10 and nitrogen-14, all among the lightest nuclei, where the asymmetry term is so strong that $N = Z$ wins outright and the pairing term is not yet small. Above nitrogen every odd–odd nucleus decays, and the handful that survive in nature — potassium-40, vanadium-50, lanthanum-138, lutetium-176 — do so only because a large change of spin slows their decay to billions of years or more.

Even–even nuclei are the opposite case. Sitting on the lower parabola, an even–even nucleus near the floor often has two odd–odd neighbours above it and cannot move, and for many even mass numbers two or even three even–even isobars are stable together, separated by odd–odd nuclei that each one would have to climb to. Tin, with ten stable isotopes, owes part of its record to this and part to its magic number of protons. The arithmetic that strands germanium-76 is the same arithmetic that lets the even–even nuclei crowd the stable list.

The smallest step #

At the other extreme from a fission fragment is the decay with almost nothing to release. Tritium, hydrogen with two neutrons, is heavier than the helium-3 atom it decays to by 18.6 keV, among the smallest β-decay energies known. Every one of its decays shares that 18.6 keV between the electron and the antineutrino, and the electron can reach the full amount only if the antineutrino takes away no kinetic energy at all — only its rest mass. So the shape of the electron spectrum in the last few electronvolts below the endpoint depends on the neutrino’s mass, and a tritium source watched closely enough weighs the neutrino by the mass difference the parabola encodes.

The KATRIN experiment at Karlsruhe does that with a seventy-metre beamline ending in a spectrometer that lets through only electrons above a set energy, and by 2024 it had shown that the neutrino mass is below about 0.45 electronvolts. It is the most direct of all neutrino-mass measurements, because it uses nothing but the conservation of energy and the masses of two atoms, and it is a long way from the scale at which double β decay or cosmology now probe the same quantity indirectly.

The valley the parabolas trace #

Each parabola has a floor, and the last figure draws the floors for every mass number in the plane of protons and neutrons. For light nuclei the floor lies on the diagonal of equal numbers, where the asymmetry penalty vanishes. As $A$ grows, the protons’ mutual repulsion grows as $Z^2$ while the asymmetry penalty for a given fractional imbalance grows only as $A$, so the floor drifts towards the neutron side; by uranium the stable proportion is about three neutrons for every two protons. That drift is why a heavy nucleus that splits has too many neutrons for its new mass, and why fission fragments start so far up the left walls of their parabolas.

Every weak decay in nature is a step across this valley along a line of constant $A$, towards its floor. Followed far enough out, the same arithmetic governs matter at densities no nucleus reaches. In a neutron star the electrons are squeezed to such high energies that turning a proton and an electron into a neutron becomes the downhill direction, and the protons a star cannot afford found the star settling at a proton fraction of a few per cent. That is the same balance of masses that puts the floor of each parabola where it is, extended to a single nucleus a few kilometres across.

Where the parabola is a guide and not an answer #

The parabola comes from a formula with five terms, and the formula is a smooth fit to a quantity that is not smooth. It knows nothing of nuclear shells, so near the magic numbers — 28, 50, 82 and 126 — real masses depart from it by an MeV or more, and that is enough to move the floor by a place; that is why the even-$A$ figures here draw only the formula’s curvature and take the positions of the parabolas from measured masses. The pairing term is a single number with a rough $1/\sqrt{A}$ dependence; the gaps measured from argon, potassium and calcium, and from germanium, arsenic and selenium, are 4.0 and 3.4 MeV, against 3.5 and 2.6 in the formula.

The parabola also says only what is allowed, not how fast. When any one nucleus decays is random, as a nucleus with no clock found, and only the rate is a property of the species. The rate of a decay depends on the energy released — roughly as its fifth power for an allowed β decay — and also on how different the parent and daughter states are in spin and shape. A decay that needs a large change of spin can be slower by many factors of ten than its energy suggests, and that, not the energy alone, is why caesium-137 lasts thirty years and potassium-40 more than a billion. The masses decide the direction; the nuclear structure decides the clock.

Ground states only #

The figures draw masses as points and decays as arrows between them. They do not show that a decay usually leads not to the daughter’s ground state but to one of its excited states, which then emits gamma rays; potassium-40’s capture branch nearly always does, and its 1.46 MeV gamma ray is a background in every sensitive detector on Earth. They do not show the neutrinos, which carry off a share of every β decay’s energy, nor the recoil of the daughter atom. And they draw the parabolas as if the formula’s curvature were exact, which the measured gaps show it is not; what the drawings carry faithfully is the ordering of the masses, which is all the direction of a decay depends on.

Still open: whether a neutrino is its own antiparticle #

The masses permit neutrinoless double β decay in about a dozen nuclei, and the next generation of experiments — tonne-scale germanium, xenon and tellurium detectors — aims at half-lives near $10^{28}$ years, enough to test the inverted ordering of neutrino masses completely. A detection would show that lepton number is not conserved and that the neutrino is a Majorana particle, its own antiparticle, which no other experiment can establish. Interpreting a measured rate as a neutrino mass needs the nuclear matrix element, the overlap of the parent and daughter nuclei’s structures, and the calculations of it differ between methods by factors of two to three, which is now as large an uncertainty as any experimental one.

The habit worth carrying away is to weigh the atoms before asking about the forces. A weak decay moves a nucleus one proton sideways among its isobars, and it goes wherever the atom ends up lighter — by any margin for $\beta^-$ decay or capture, by more than 1,022 keV for a positron — so the masses lying on a parabola, or on two for an even number of nucleons, decide that fission fragments fall step by step to the floor, that potassium-40 decays both ways, and that germanium-76, 2.039 MeV above selenium-76 with arsenic-76 between them 923 keV higher, can only take both steps at once.