The energy that did not all arrive

The first rung of this ladder established that a decay has no clock, and the second that the law describing it is the Fourier transform of an energy distribution. Neither says anything about what comes out.

What comes out is measurable, and in the case of beta decay it was the first thing about nuclear physics that appeared to be impossible.

A nucleus decays from one definite state to another definite state, so the energy released is a definite number. If one particle carries it away, momentum conservation fixes how the energy divides between that particle and the recoiling nucleus, and every decay of that nuclide gives an electron of exactly the same energy. The spectrum is a line.

What was actually on the table #

Chadwick found the continuum in 1914, and it survived every attempt to explain it away.

The first explanation was that the electrons lose energy on the way out. Ellis and Wooster settled that in 1927 by the only decisive method available: they put a radium-E source inside a calorimeter thick enough to absorb everything and measured the heat. If the electrons started at a single energy and were degraded by the source, the calorimeter would record that single energy per decay. It recorded the mean of the observed spectrum — 350 keV against an endpoint near 1,050 — so the energy was not lost in transit. It never left.

That left two possibilities, and both were serious.

Energy is not conserved in individual nuclear events. Bohr held this position for several years and did not regard it as desperate: quantum mechanics had already required abandoning determinism, and a statistical conservation law was not obviously worse. He proposed it in print.

Something else carries the energy away and is not detected. Pauli proposed this in a letter in December 1930, addressed to a conference he was not attending because he was going to a ball. He called the proposal desperate himself. The particle had to be neutral, of very small mass, and had to interact so weakly that a calorimeter thick enough to stop the electrons stopped none of them.

What decided between the two was not a detection — that took twenty-six years. It was the shape, which is a count of arrangements in a setting nobody had thought to count.

The shape is a count #

Fermi wrote the theory in 1934, and its content is that the spectrum is a phase-space count with no adjustable parameters beyond an overall rate.

The decay produces an electron and a neutrino sharing an energy $Q$. The number of ways of doing that with the electron at kinetic energy $T$ is the product of the number of momentum states available to each: $p^2\,\mathrm{d}p$ for the electron, which is $pE\,\mathrm{d}T$, and $(Q-T)^2$ for a massless neutrino. One correction is needed — the departing electron is retarded by the daughter nucleus’s charge, which piles up the low-energy end — and it is the Fermi function $F(Z,T)$. So

$$N(T)\,\mathrm{d}T \;\propto\; F(Z,T)\,p\,E\,(Q-T)^2\,\mathrm{d}T.$$

Two things about that expression are worth separating, because they are usually conflated.

It contains no nuclear physics. Everything in it is kinematics — how many ways two particles can share a fixed energy — plus one electrostatic correction. The nuclear matrix element that decides how fast the decay goes is a constant multiplying the whole thing, so it changes the rate and not the shape. That is why three nuclides with completely different structures give nearly the same curve.

And it is the shape that carried the argument. A conservation law that failed statistically would produce some distribution of electron energies, and there is no reason for that distribution to be a two-particle phase-space count. Fermi’s expression fitted the measured spectra of many nuclides with one normalisation each. That is a great deal of agreement to obtain from a hypothesis about nothing.

The neutrino was finally detected in 1956, by Cowan and Reines, twenty-two years after the theory that used it and twenty-six after it was proposed — and the cross-section that made it hard is the same number Bethe and Peierls had computed as a reason not to try. In the meantime it was believed on the strength of a curve.

What the neutrino had to be, before it was seen #

Pauli’s proposal was not vague, and reading off its requirements from the spectrum alone is a good exercise in how much a curve can constrain.

It is electrically neutral, because charge is conserved and the electron carries away the whole change.

Its spin is a half. A nucleus’s spin changes by an integer or by nothing in beta decay, and the electron carries a half, so something else must carry a half or the angular momentum does not balance. That argument is independent of the energy spectrum and pointed to the same conclusion.

Its mass is small compared with the electron’s. The spectrum runs essentially to the full available energy, and it could not if a massive third body had to be created — the endpoint would sit lower by the mass, and it does not, to within the resolution of the measurements of the time.

And it interacts extraordinarily weakly. Ellis and Wooster’s calorimeter was thick enough to stop the electrons and absorbed nothing else, which puts an upper bound on the cross-section immediately. Bethe and Peierls turned that into a number in 1934 and concluded that the particle could pass through the Earth without being stopped — a statement they intended as a demonstration that it could never be detected. How far a neutrino gets is the arithmetic behind that conclusion, and it was right about the difficulty and wrong about the impossibility.

Every one of those properties came out of an energy spectrum and a conservation law. That is a considerable amount of information about an object nobody had seen, and it is the reason the proposal was accepted long before the detection.

The plot that makes the curve a straight line #

The spectrum’s shape is a product of three factors, two of which are known exactly. Dividing them out leaves the third, and the third is where the physics that is still unsettled lives.

$$K(T) = \sqrt{\frac{N(T)}{F(Z,T)\,p\,E}} \;\propto\; (Q - T).$$

That is the Kurie plot, and for a massless neutrino it is a straight line whose intercept on the energy axis is the endpoint.

The straightness is the useful property. A measured spectrum plotted this way is a straight line if the theory is right, and departures from straightness are visible in a way that departures from a curve are not. It is the same move as plotting a supposed exponential on a logarithmic axis, and for the same reason.

A neutrino with mass changes two things. The spectrum stops at $Q - m c^2$ rather than at $Q$, and the last few electronvolts curve downward instead of running straight into the axis.

Which of those is measured matters, and it is the second. The absolute endpoint of tritium is not known from any other source to better than an electronvolt or so, so a shift of the endpoint alone would be unobservable — there is nothing to compare it with. That is a distinction between a component and an invariant in a different currency: what can be measured is the shape, which is fixed, and not the offset, which is not. The shape over the last few electronvolts is a different matter: it is a prediction with no free parameters, and any deviation from the straight line is a signal.

The part of the spectrum the answer is in #

The difficulty is arithmetic, and it is severe enough to have shaped the whole field.

Everything about a neutrino-mass experiment follows from that number.

The source has to be enormous and thin at the same time, in the way a small antenna is both inefficient and narrowband for one reason rather than two. Enormous, because the rate near the endpoint is a part in ten million million of the total; thin, because an electron that loses energy in the source before leaving it is indistinguishable from one that started with less. Those two requirements pull in opposite directions, and every experiment is a different compromise between them.

The spectrometer has to reject almost everything. Of the decays, all but one in $10^{13}$ produce an electron that carries no information, and each of those is a background event if it reaches the detector. The KATRIN experiment’s spectrometer is a vessel ten metres across and twenty-four long, held at $10^{-11}$ millibar, working as a filter that passes only electrons above a set energy.

And the sensitivity improves very slowly. Statistical uncertainty on a rate falls as the square root of the counts, and the mass enters the count as a cube, so the reachable mass falls as the sixth root of the exposure. Doubling the sensitivity means sixty-four times the running.

The current limit from that programme is 0.45 electronvolts, and the improvement over the previous generation was a factor of two after a decade of work. The number is not a disappointment; it is what the cube in the phase space allows.

Why tritium #

The choice of nuclide is forced by the same arithmetic and is worth stating, because it explains why a single isotope has carried the whole subject.

The fraction of decays near the endpoint goes as $(\Delta E/Q)^3$, so a small $Q$ is worth a great deal — and tritium’s 18.6 keV is among the smallest of any beta emitter that can be handled. Halving $Q$ multiplies the useful fraction by eight.

Tritium also has the simplest possible atomic structure, which matters more than it sounds. The measurement is of the electron’s energy to a fraction of an electronvolt, and the final state of the molecule left behind is not unique: the daughter helium ion can be left rotating or vibrating, each such state removing a few tenths of an electronvolt from the electron. That distribution of final states has to be calculated, and it can be calculated for a two-atom molecule and not for anything larger. It is currently the leading systematic uncertainty in the field.

The alternatives being pursued are different in kind rather than better. Holmium-163 decays by electron capture with a $Q$ of about 2.8 keV, and the measurement is of the total energy deposited rather than of one particle’s — a calorimetric method with quite different systematics. Whether it overtakes the tritium programme is not yet clear.

The other thing the endpoint is used for #

A spectrum’s endpoint is an energy difference between two nuclear states, measured with a spectrometer, and that makes it a mass measurement — which turns beta decay into a metrological instrument quite apart from any question about neutrinos.

The difference between the parent’s and daughter’s atomic masses is $Q$ plus a known correction, so measuring $Q$ to an electronvolt measures a mass difference to an electronvolt, which is a part in $10^{10}$ of either mass. Before Penning-trap mass spectrometry reached that precision, beta endpoints were how the nuclear mass surface was tied together.

The relationship now runs the other way and is a consistency check the field depends on. A Penning trap measures the tritium–helium mass difference directly, without any spectrum; the endpoint measured from the spectrum must agree. Any discrepancy would be a systematic error in one of them, and finding out which is a considerable amount of work — which is why both are done.

The mass that is missing from a nucleus is the same quantity on a much larger scale, and the two measurements are the same measurement at different precisions: what a beta endpoint gives is a binding-energy difference between two neighbouring nuclides, resolved finely enough that the atomic electrons’ binding has to be accounted for.

The decay that would have no neutrino at all #

There is one experiment whose whole design is an attempt to see the spectrum’s third body fail to appear, and it is worth ending the physics on because it turns this rung’s argument round.

If the neutrino is its own antiparticle, then a nucleus that would ordinarily emit two electrons and two neutrinos can emit two electrons and nothing else, the two neutrinos annihilating each other inside the nucleus. The signature is unmistakable: the two electrons carry the entire available energy, always, so their summed spectrum is a line at $Q$ sitting on top of the continuum the ordinary two-neutrino process produces.

That is exactly the two-body signature this essay opened with, appearing where the three-body continuum is expected — and it is the reason the search is regarded as decisive rather than suggestive. A peak at the endpoint of a summed spectrum cannot be produced by anything else.

Nothing has been seen. The limits are on half-lives above $10^{26}$ years, which is sixteen orders of magnitude beyond the age of the universe and is reached by watching many kilograms of an enriched isotope for years underground. What those limits constrain is again the neutrino mass, through a different combination of the three states than the endpoint measures — so the two programmes are not competitors but different projections of the same unknown.

And the physics at stake is larger than a mass. A decay with two electrons and nothing else changes the number of leptons in the universe by two, which no observed process does. Seeing it once would establish that lepton number is not conserved, which is a requirement of most explanations for why there is more matter than antimatter. The whole of that argument rests on the difference between a line and a continuum, which is where this rung started.

Where this stops being right #

The Fermi function used here is the non-relativistic form. The exact treatment solves the Dirac equation in the daughter’s Coulomb field and includes screening by the atomic electrons and the finite size of the nucleus. Those corrections matter at the per cent level over the whole spectrum and are negligible in the last few electronvolts, which is why the figures use the simple form and the experiments do not.

The decay has been treated as allowed. Some beta decays require a change of angular momentum the leading term cannot supply, and their spectra have an extra energy-dependent factor — a shape factor — that changes the curve substantially. Tritium’s is allowed, which is another reason it is used.

The neutrino has been given one mass. There are three neutrino states and they mix, so what a beta spectrum measures is a weighted average of three masses, and oscillation experiments constrain the differences rather than the values. A beta endpoint is one of the very few measurements sensitive to the absolute scale at all.

And the shape near the endpoint assumes nothing else is going on. A sterile neutrino, if one exists, would put a second kink in the plot at its own mass. Searching for that is a second use of the same data and constrains a quite different question.

What the pictures cannot show #

Every figure here draws a smooth curve, and a spectrum is a histogram of a finite number of events. In the region that matters the count is a handful per day, so the actual measurement is a scatter of integers with error bars, and the straight line is what a fit to them produces rather than what any of them looks like.

Nor can any figure show the neutrino. Its entire contribution to these pictures is the factor $(Q-T)^2$, which is a statement about how many ways there are for something to have an energy — and a count of possibilities is the one thing a drawing of a particle could not represent honestly.

What the ladder has reached #

Three rungs stand on decay. The first found that a half-life is a property of a population and of no member of it. The second found that the exponential law is a Fourier transform and fails at both ends. This one leaves the timing alone and asks what comes out, and the answer required inventing a particle to keep a conservation law.

The habit worth carrying away is about what a distribution’s shape is evidence for. A rate says how likely something is; a spectrum says how many ways it can happen, which is a statement about what the final state contains. Two bodies give a line and three give a continuum, and counting the continuum’s shape counts the bodies. The same reasoning identifies the number of particles in any decay whose spectrum can be measured, and it is how the tau neutrino’s existence was inferred long before it was seen.

What is left on this ladder is what happens when decays are put in series. A single decay has a rate; a chain has several, and the interesting behaviour is not in any of them but in what the chain settles into — a state in which every member decays at exactly the same rate while their abundances differ by twelve orders of magnitude.