Illustrated Physics
physics, drawn
Quantum

The half-life that chemistry can change

The first rung of this ladder said a nucleus has no clock and that nothing outside it touches the rate. That is very nearly true and it is not exactly true. Electron capture takes an electron from the nucleus's own position, so its rate is proportional to how many electrons are there — which is chemistry, worth a per cent. And a nucleus stripped of every electron can gain a decay channel it did not have: rhenium-187 goes from forty-two billion years to thirty-three.
15 min read 6 figures Where the model stopsWho is measuring

Assumes: A nucleus with no clock · The chain that runs at its slowest member's rate

The first rung of this ladder makes a strong claim and makes it plainly: a nucleus has the same chance of decaying in the next second as it had on the day it formed, and nothing about its history, its temperature or its surroundings alters that. The claim is the foundation of radiometric dating, of activity standards, and of the whole of the ladder above it.

It is very nearly true. It is not exactly true, and the exceptions run from parts per million to a factor of a billion.

The half-life that depends on the electrons. Half-lives of 3 nuclides as neutral atoms and as bare nuclei, measured at a heavy-ion storage ring where fully stripped ions can be kept circulating for months. The changes are not corrections. ¹⁸⁷Re goes from 41.6 billion years to 32.9 — a factor of 1.26 billion — because a beta decay whose available energy is only 2.5 keV cannot put an electron into the continuum but can put one into an empty K orbital, and stripping the atom opens that channel. ¹⁶³Dy is stable as an atom and decays in 47 days as a bare nucleus, for the same reason. The rate is a property of the nucleus and of what surrounds it, and the rung below this one says otherwise.
Fig. 1 Half-lives of three nuclides as neutral atoms and as bare nuclei, measured at a heavy-ion storage ring. Rhenium-187 goes from 41.6 billion years to 32.9 — a factor of 1.26 billion. Dysprosium-163 is stable as an atom and decays in 47 days when stripped. Holmium-163 does the reverse: it decays by capturing one of its own electrons and cannot when it has none.

Where the nucleus can feel its surroundings

The rung below’s claim is right about the mechanism and the exceptions all exploit one loophole in it.

A nucleus is 10−1410^{-14} metres across and an atom is 10−1010^{-10} — a difference of scale the whole of atomic physics rests on, so the electrons are typically four orders of magnitude further out. The energies involved differ by six orders. Nothing a chemist can do moves an energy level of a nucleus by anything measurable, and that is why the claim holds as well as it does.

But one nuclear process does not act at nuclear range. Electron capture is a nucleus absorbing one of its own atomic electrons, converting a proton to a neutron, and the rate is proportional to the probability that an electron is at the nucleus — the modulus squared of its wavefunction evaluated at the origin.

That quantity is not a nuclear property at all. It is a property of the electron’s orbital, and orbitals are what chemistry rearranges — the same radial distributions whose overlap decides every chemical bond.

The same loophole works the other way for a certain kind of beta decay. If the available energy is too small to put an electron into a free state, the decay is forbidden — but putting it into a bound orbital costs less, because a bound electron has negative energy. That channel is open only when there is an empty orbital, so it is closed in a neutral atom and open in a stripped one.

How large the chemical effect is, and why it is small

Per cent effects, from chemistry. Measured fractional changes in the decay rate of nuclides that capture an orbital electron, when the chemistry, the pressure or the temperature around the nucleus is changed. The mechanism is direct: electron capture takes an electron from the nucleus's own position, so the rate is proportional to the electron density there, and that density is a chemical property. Beryllium-7 is the standard case because it has only four electrons, two of which are valence electrons whose arrangement changes completely between the metal and the fluoride — so the effect is nearly a per cent rather than the parts per million a heavy nuclide gives. These are small numbers and they are not zero, which is the whole content: the decay constant is not a property of the nucleus alone.
Fig. 2 Measured fractional changes in the decay rate of beryllium-7 when its chemical environment, its pressure or its temperature is changed. The largest is 0.84 per cent. These are not anomalies awaiting explanation: the mechanism is that electron capture samples the electron density at the nucleus, and every one of these treatments changes that density.

Beryllium-7 is the standard case and the reason is arithmetic rather than convenience.

Why only the lightest elements show it. The share of the electron density at the nucleus that belongs to the valence electrons rather than to the innermost shell, against atomic number, both logarithmic. A 1s electron's density at the origin goes as the cube of the nuclear charge, so the two K-shell electrons dominate more and more completely as the element gets heavier, and the outer electrons — the ones chemistry can rearrange — contribute a share falling as Z⁻³. The slope is exactly −3, measured off the curve. That is why beryllium-7, with four electrons and no filled inner shells to speak of, shows a change of nearly one per cent between two compounds, while a heavy electron-capturing nuclide shows parts per million and takes a dedicated experiment to see at all.
Fig. 3 The share of the electron density at the nucleus that belongs to the valence electrons rather than the innermost shell, against atomic number. A 1s electron’s density at the origin goes as the cube of the nuclear charge, so the two K-shell electrons dominate more and more completely and the share chemistry can rearrange falls as Z⁻³. The slope is exactly −3, measured off the curve.

Beryllium has four electrons, two in the 1s shell and two valence. Its nuclear charge is four, so the 1s density at the nucleus is modest and the valence electrons contribute a substantial share of the total. Rearranging them between the metal and the fluoride — where they are pulled well away toward the fluorines — changes the density at the nucleus by nearly a per cent, and the decay rate follows.

Lead has eighty-two. Its 1s density at the nucleus is larger by a factor of (82/4)3(82/4)^3, which is nearly nine thousand, so its valence electrons are a negligible part of the total and shifting them does nothing measurable. Every heavy electron-capturing nuclide has been checked and the shifts are parts per million.

That Z−3Z^{-3} is the whole reason the effect is a curiosity rather than a problem. If it went as Z−1Z^{-1}, every dating method involving an electron-capturing nuclide — potassium–argon, most obviously — would need a chemical correction, and the correction would depend on the mineral. It does not; the potassium-40 capture branch shifts by well under a part in ten thousand between silicates, which is below every other uncertainty in the method — including the one a chain’s disequilibrium introduces.

The measurements, and what they cost

The chemical shifts are hard to measure for a reason that has nothing to do with their size: the two samples being compared have to be identical in every other respect.

Two sources of beryllium-7 differ in geometry, in self-absorption, in how much has been lost to handling, and in how their gamma rays reach a detector. A per cent difference in counting efficiency is easy to produce accidentally, and it is indistinguishable from a per cent difference in half-life.

The solution used by the careful experiments is to alternate. Both samples are counted with the same detector in the same geometry, repeatedly, over months, and the ratio of their count rates is followed. A systematic error in the detector cancels; a real difference in half-life shows up as a drift in the ratio. The half-life difference between beryllium-7 in the metal and in the fluoride was established that way, and it took a year of counting.

Per cent effects, from chemistry. Measured fractional changes in the decay rate of nuclides that capture an orbital electron, when the chemistry, the pressure or the temperature around the nucleus is changed. The mechanism is direct: electron capture takes an electron from the nucleus's own position, so the rate is proportional to the electron density there, and that density is a chemical property. Beryllium-7 is the standard case because it has only four electrons, two of which are valence electrons whose arrangement changes completely between the metal and the fluoride — so the effect is nearly a per cent rather than the parts per million a heavy nuclide gives. These are small numbers and they are not zero, which is the whole content: the decay constant is not a property of the nucleus alone.
Fig. 4 Three nuclides rather than three treatments of one, and the spread is the Z⁻³ law appearing as data. Beryllium at Z = 4 gives 0.83 per cent; niobium at 41 gives 0.36; technetium’s isomeric transition, which is not electron capture at all and depends on the electron density only through internal conversion, gives 0.03. The effect is confined to the light end and to the processes that sample the origin.

The storage-ring measurements are a different kind of difficult. Producing bare rhenium-187 nuclei means accelerating them, stripping them by passing them through a foil, and then storing them in a ring at ultra-high vacuum for long enough to see a decay whose half-life is thirty-three years. What is watched is not decays but the change of charge: a bound-state beta decay turns the ion into a different element with the same charge state, and the two circulate at slightly different frequencies in the ring. The measurement is of a frequency.

The half-life that depends on the electrons. Half-lives of 2 nuclides as neutral atoms and as bare nuclei, measured at a heavy-ion storage ring where fully stripped ions can be kept circulating for months. The changes are not corrections. ¹⁸⁷Re goes from 41.6 billion years to 32.9 — a factor of 1.26 billion — because a beta decay whose available energy is only 2.5 keV cannot put an electron into the continuum but can put one into an empty K orbital, and stripping the atom opens that channel. ¹⁶³Dy is stable as an atom and decays in 47 days as a bare nucleus, for the same reason. The rate is a property of the nucleus and of what surrounds it, and the rung below this one says otherwise.
Fig. 5 Two cases of the same mechanism, with one of them a nuclide nobody would list as radioactive. Thallium-205 is a stable isotope making up seventy per cent of natural thallium; as a bare nucleus it decays to lead-205 with a half-life of 291 days. Stability is a statement about a nucleus and its electrons together, and removing the electrons removes it.

What “stable” turns out to mean

The dysprosium and thallium results are the ones worth pausing on, because they change a word rather than a number.

Dysprosium-163 is a stable isotope. It appears on every chart of the nuclides as stable, it makes up a quarter of natural dysprosium, and no decay of it has ever been observed in ordinary matter. Stripped of its sixty-six electrons it decays with a half-life of forty-seven days.

The reason is the energy bookkeeping this rung opened with. Dysprosium-163 would like to beta-decay to holmium-163, and as neutral atoms the holmium is heavier — so the decay is forbidden by energy conservation and the nuclide is stable. But the atomic masses include the electrons and their binding, and the nuclear masses tell a different story: the nucleus alone can decay, provided the emitted electron has somewhere to go that costs little enough. An empty K orbital is such a place.

So the stability of dysprosium-163 is a property of the atom rather than of the nucleus, and the same is true of thallium-205. There is no line on a chart of the nuclides marking which entries are stable only because they are dressed, and there could be: the criterion is that the nuclear decay is allowed and the atomic one is not, which is a computable condition on a mass table.

That reframing runs the other way too. Holmium-163 is radioactive as an atom, with a half-life of 4,570 years, and is stable as a bare nucleus — because its only decay is to capture an electron and it has none. Half of the pair is stable when dressed and half when stripped, and neither statement is about the nucleus alone.

The rate and the age of the Earth

The rhenium result is not a curiosity for its own sake, and the reason is that rhenium-187 decaying to osmium-187 is one of the standard geochronometers.

The neutral-atom half-life is what a rock uses, and it is 41.6 billion years. But rhenium in a hot enough plasma is partly ionised, and in the interior of a star it is very substantially so — which means the effective half-life during stellar nucleosynthesis is not the laboratory one. Estimates of the age of the elements from the rhenium–osmium pair have to model the ionisation state of rhenium in every environment the material has passed through, and the correction is not small.

That is a general point about this rung and it is the one worth carrying away. A decay constant measured in a laboratory is measured on neutral atoms at room temperature, and applying it to matter in a star, in a supernova, or in the interstellar medium is an extrapolation across nine orders of magnitude in the one variable that turns out to matter — the same hazard an approximation that holds almost everywhere always carries.

The s-process — the slow neutron capture that builds about half the elements heavier than iron — runs in stellar interiors at hundreds of millions of kelvin, where many of the nuclides involved are ionised enough for their beta decay rates to differ from the laboratory values by factors of several. Those corrected rates are inputs to every calculation of the resulting abundances, and they were measured at storage rings for precisely that reason.

Why only the lightest elements show it. The share of the electron density at the nucleus that belongs to the valence electrons rather than to the innermost shell, against atomic number, both logarithmic. A 1s electron's density at the origin goes as the cube of the nuclear charge, so the two K-shell electrons dominate more and more completely as the element gets heavier, and the outer electrons — the ones chemistry can rearrange — contribute a share falling as Z⁻³. The slope is exactly −3, measured off the curve. That is why beryllium-7, with four electrons and no filled inner shells to speak of, shows a change of nearly one per cent between two compounds, while a heavy electron-capturing nuclide shows parts per million and takes a dedicated experiment to see at all.
Fig. 6 The same law evaluated at the alkali metals, whose single valence electron is the cleanest case there is. Beryllium’s neighbours in the light half of the table give shares of a per cent or so; caesium’s is a part in three hundred thousand. Any experiment hoping to see a chemical effect on a decay has to be done at the top of the periodic table, and the choice of nuclide is made before anything else.

The internal-conversion cousin

There is a second process that samples the electron density at the nucleus, and including it fills out the picture of which decays can and cannot feel their surroundings.

An excited nucleus can shed its energy by emitting a gamma ray, or by handing the energy directly to one of its own electrons and ejecting it — internal conversion. The second route, like electron capture, requires an electron to be at the nucleus, so its rate carries the same ∣ψ(0)∣2|\psi(0)|^2 and the same sensitivity to chemistry.

The consequence is that an isomeric transition’s half-life can be chemically shifted even though no electron is captured. Technetium-99m is the case that has been measured, because it is made in quantity every day for medical imaging: its 6-hour half-life differs by about three parts in ten thousand between the pertechnetate ion and the sulphide, and the shift is in the conversion channel rather than in the gamma one.

The size is right for the Z−3Z^{-3} law at Z = 43 and it is a useful confirmation of the mechanism, because it is a different process with the same dependence. Two unrelated decay modes shifting by the amount one expression predicts is stronger evidence than either alone.

Niobium-90m gives the largest isomeric shift measured, at about 0.36 per cent between the metal and the fluoride, and it is the reason that nuclide keeps appearing in this literature: a low-energy, highly converted transition in a light-ish element is where all three factors line up.

What has been looked for and is not there

The exceptions above are real, understood, and confined to processes that sample the electron density. The literature also contains a set of claims that go much further, and the honest position on them is worth stating.

Several groups have reported annual variations of a fraction of a per cent in the decay rates of ordinary alpha and beta emitters — nuclides with no electron-capture branch at all — correlated with the Earth’s distance from the Sun, and proposed a neutrino-mediated influence. If real, the claim would overturn the rung below this one entirely.

The evidence does not survive scrutiny. Dedicated repeat measurements with temperature- and humidity-controlled apparatus find no annual variation at levels well below the claimed amplitude; the original signals correlate with laboratory environmental parameters that themselves vary annually; and a solar-flare test, in which a large flare should have produced a detectable step, produced none. The current limits are around a part in 10510^5 for a distance-dependent effect.

That is the state of it: a mechanism for environmental influence exists, is understood, and is confined to a specific class of decay; and a much broader claim was made, tested, and is not supported. The two are worth keeping apart, because the existence of the first is sometimes used as an argument for the plausibility of the second, and the Z−3Z^{-3} law and the requirement of an electron at the origin are precisely what prevent the mechanism from generalising.

What the exceptions are worth, practically

It is worth separating what the exceptions change and what they do not, because the answer is reassuring in one direction and important in the other.

They change nothing about dating. Every radiometric clock in use runs on alpha decay, ordinary beta decay, or electron capture in a heavy nuclide — potassium-40, rubidium-87, uranium, thorium, samarium. The first two have no environmental sensitivity above parts per million, and the third has a Z−3Z^{-3} suppression of four orders of magnitude against beryllium’s. A geological age computed with a laboratory decay constant is safe by a wide margin, and the uncertainty in those constants is dominated by how well they were measured rather than by where the sample sat.

They change a good deal about nucleosynthesis. Stellar interiors are hot enough to ionise substantially, and the s-process runs through nuclides whose beta-decay rates compete with neutron capture. Where the two rates are comparable the branching is sensitive to the decay rate, and the decay rate is sensitive to the ionisation. Rhenium-187 is the extreme case at a factor of a billion, and there are a dozen others at factors of two to ten — each of which moves a computed abundance.

And they are the reason a storage ring is a nuclear-physics instrument. Measuring a half-life on fully stripped ions was for decades impossible in principle, because the ions could not be kept. A ring that stores them at ultra-high vacuum and measures their revolution frequency turned an unmeasurable quantity into a routine one, and the results are among the few numbers in nuclear astrophysics that are measured rather than calculated.

Where this stops being right

The electron density has been treated as a single number. The rate depends on the density of the capturable orbitals, weighted by their overlap with the nucleus over its finite size, and the relativistic treatment of the innermost electrons matters at the per cent level for heavy nuclides — which is the same size as the effect being discussed.

The bound-state beta rates are calculated as well as measured. The storage-ring experiments measure a few cases; the rates used in stellar models are computed for hundreds of nuclides, from atomic structure and nuclear matrix elements, and the calculations are checked against the handful of measurements. Where they have been checked they agree; where they have not, they are theory.

The stripping is never quite complete. A stored beam picks up electrons from residual gas, so the measured rate is a mixture of charge states and the pure result is extrapolated. That extrapolation is one of the larger systematic uncertainties in the field.

And none of this touches alpha decay or ordinary beta decay. Both involve energies of megaelectronvolts and matrix elements confined to the nucleus, and no environmental effect on either has ever been established above the parts-per-million level.

What the pictures cannot show

The figures draw ratios of half-lives, and a half-life is inferred from a count rate over a finite time. For rhenium-187 as a bare nucleus that inference came from watching a stored beam for hours and extrapolating a thirty-three-year half-life — so the number on the figure is a fit to a small change in a frequency spectrum, not a decay curve anybody watched.

Nor can the Z−3Z^{-3} figure show what it is actually about. What matters is the electron density at a single point, the origin, and a drawing of a share of a density at a point has nothing spatial in it at all. The curve is a statement about a number that no picture of an atom would make visible.

Where this ladder stands

Five rungs stand on decay. The first found a rate with no memory. The second found the law is a transform. The third found the spectrum counts the bodies in the final state. The fourth put decays in series. This one goes back to the first and finds its one exception.

The habit worth carrying away is about the scale separation an idealisation rests on. A quantity is independent of its surroundings only to the extent that the surroundings are far away in the units that matter, and finding the process that reaches across the gap is how the exception is found. Here the gap is four orders of magnitude in distance and six in energy, and the one process that crosses it is the one that evaluates an atomic wavefunction at a nuclear position. The same test applied elsewhere finds the exceptions to every “independent of” statement in physics: something is always coupling, and the question is how weakly.

What is left on this ladder is the thing this rung has taken for granted. Every rate here has been a rate for a transition between two states, and which transitions are allowed at all is decided by angular momentum and parity rather than by energy — which is why some excited nuclear states last for years and others for femtoseconds, and why a selection rule can be worth twenty orders of magnitude in a lifetime.

Part 5 of 5

This essay is one argument about Decay. The others:

Everything in the chain decaying at the parent's rate. The activity of each member of ²³⁸U → ²³⁴Th → ²³⁴Pa, relative to the parent's, against time in days. The parent's half-life is far the longest, so its activity is effectively constant over the range drawn while each daughter rises to meet it. Once they have, every member of the chain is decaying at exactly the same rate — secular equilibrium, checked here to one per cent off the integrated solution — because each is being made as fast as it is disappearing. The amounts are not equal at all: the abundance of each member sits in the ratio of its half-life to the parent's, which for radium in uranium is one part in three million and is why radium had to be extracted from tonnes of ore. The chain that runs at its slowest member's rate Part 4 — Put decays in series and something happens that no single decay does. The population settles where every member is being made exactly as fast as it disappears, so every activity in the chain is equal — while the amounts differ by twelve orders of magnitude, in the ratio of the half-lives. A gram of uranium contains a third of a microgram of radium and two hundred million million million atoms fewer of radon, and both numbers are read off a list of half-lives.

Where this goes next

Decay, and the ensemble it is a property of. 400 nuclei followed for 4 half-lives. The smooth curve is the exponential; the stepped traces are 3 independent runs in which every nucleus was given its own decay time and told nothing about the others. The number surviving halves at each dashed line — 200, 100, 50, 25 — and it halves again over the next interval regardless of how long the sample has already been sitting there, which is the property no ordinary clock has. The traces wander further from the curve as the numbers get small: at the end only about 25 are left and the scatter is a visible fraction of that. A nucleus with no clock A half-life is a precise number and no individual nucleus has one. Each has the same chance of decaying in the next second as it had on the day it formed, and the exponential curve is a property of the population rather than of any member of it. Everything in the chain decaying at the parent's rate. The activity of each member of ²³⁸U → ²³⁴Th → ²³⁴Pa, relative to the parent's, against time in days. The parent's half-life is far the longest, so its activity is effectively constant over the range drawn while each daughter rises to meet it. Once they have, every member of the chain is decaying at exactly the same rate — secular equilibrium, checked here to one per cent off the integrated solution — because each is being made as fast as it is disappearing. The amounts are not equal at all: the abundance of each member sits in the ratio of its half-life to the parent's, which for radium in uranium is one part in three million and is why radium had to be extracted from tonnes of ore. The chain that runs at its slowest member's rate Put decays in series and something happens that no single decay does. The population settles where every member is being made exactly as fast as it disappears, so every activity in the chain is equal — while the amounts differ by twelve orders of magnitude, in the ratio of the half-lives. A gram of uranium contains a third of a microgram of radium and two hundred million million million atoms fewer of radon, and both numbers are read off a list of half-lives. Where the electron actually is, by radius. The radial probability density of the hydrogen 1s, 2s, 2p states — the chance of finding the electron in a thin shell at each radius, in units of the Bohr radius. 1s is most likely at 1.00 Bohr radii and averages 1.50; 2s is most likely at 5.24 Bohr radii and averages 6.00; 2p is most likely at 4.00 Bohr radii and averages 5.00. Each curve integrates to one, and each has n − l − 1 radial nodes where the electron is never found. Where the electron probably is The Bohr atom put the electron on a circle of definite radius. What replaced it keeps the radius as the most likely place to find the electron and gives up the circle, the speed and the trajectory entirely.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Atomic structureBeta decayDecayDecay constantElectron captureHalf-lifeIonisationPhase spaceProbability densityRadioactive decayScreeningWavefunction