The spin a photon has to carry away

The half-life that chemistry can change ends by pointing at an assumption that every account of decay it builds on has made. Each rate in them has been a rate for a transition between two states, and the rate has been taken as given. But which transitions happen at all, and how fast, is decided by something the energy of the transition hardly touches: angular momentum and parity. The consequence, that essay says, is that “some excited nuclear states last for years and others for femtoseconds, and a selection rule can be worth twenty orders of magnitude in a lifetime”. The drawings here make that number and trace it to its cause.

The cause is a mismatch of sizes. A nucleus is a few femtometres across. The gamma ray it emits has a wavelength of hundreds of femtometres at 1 MeV, and thousands at 100 keV. A source much smaller than its wavelength is excellent at radiating the simplest kind of wave, the dipole — a charge sloshing back and forth — and very poor at radiating anything that twists more, because to launch a wave carrying more angular momentum its charges must be arranged so that the field varies across the source, and the field varies across something so small by only a tiny fraction. When the source is not heard all at once makes the point for antennas: the dipole approximation is exactly the statement that every part of the source is heard at the same moment, and everything beyond it is a correction proportional to the source’s size over the wavelength.

Each unit costs a factor of tens of thousands #

In 1951 Victor Weisskopf estimated the rate of each kind of gamma transition by assuming the simplest possible nucleus: one proton changing its orbit inside a uniform sphere. The estimates are crude — real transitions routinely differ from them by factors of ten or a hundred, for reasons that are themselves informative — but they get the scale right, and the scale is the point. The drawing plots them for a medium-weight nucleus.

The lines fan out enormously. At one megaelectronvolt, a transition that can shed its energy as an electric dipole takes about a femtosecond. One that must carry away two units of angular momentum takes ten picoseconds; three, a couple of microseconds; four, a third of a second; five, most of a day. At a hundred kiloelectronvolts the fan is wider still: the dipole takes a picosecond, and five units take longer than the universe has existed. The half-life of a gamma transition is decided, to within a few orders of magnitude, by a small integer — the multipole order — and only within that by the energy.

Where the energy does enter, it enters steeply. The rate of a multipole of order $L$ grows as the photon energy to the power $2L + 1$, so a transition that must carry off five units slows by a factor of about two thousand every time its energy halves. Low energy and high angular momentum together are what make a long-lived excited nucleus, which is called an isomer.

The nucleus is small compared with the light it makes #

The factor behind the fan is drawn here on its own. A photon of energy $E$ has a reduced wavelength $\lambda/2\pi = \hbar c/E$, which is 197 femtometres at one megaelectronvolt. A nucleus of mass 100 has a radius of about 5.6 femtometres. The ratio is about one in thirty-five, and its square, eight parts in ten thousand, is the price of each extra unit of angular momentum: a charge distribution that radiates a multipole of order $L$ couples to the photon field in proportion to $(kR)^L$, and the rate goes as the square of the coupling. The rest of the factor between successive estimates is numerical — combinatorial factors that grow with the order, the same ones that make a quadrupole take five numbers to describe — and it adds another factor of a hundred or so.

The same reasoning applies to anything that radiates. A radio antenna comparable in size to its wavelength radiates high multipoles readily; that is how directional antennas work. An atom emitting visible light is a few tenths of a nanometre across against a wavelength of five hundred nanometres — its $(kR)^2$ is about a millionth — so atoms are even more strongly confined to dipole transitions than nuclei are. And a magnetic multipole of a given order is weaker than the electric one by a further factor that, for atoms, is the square of the fine-structure constant, about one in twenty thousand, because magnetic effects of moving charges are smaller than their electric effects by the ratio of their speed to light’s.

One transition, ten ways it could have gone #

The selection rules that decide which multipole a transition uses are conservation laws. The photon must carry off at least the difference between the two states’ spins, $|J_i - J_f|$, and at most their sum; and the change of parity — whether the state’s wavefunction changes sign when space is reflected — decides whether an odd or even multipole of the electric or the magnetic kind is needed. In practice the lowest allowed order dominates completely, because each higher one is tens of thousands of times slower.

Barium-137 in its excited state is a standard case. It is made by the beta decay of caesium-137, which is why every caesium source glows with its 662 kiloelectronvolt gamma ray, and it has spin 11/2 and negative parity against the ground state’s 3/2 and positive parity. The photon must carry off at least four units and must change the parity, which makes it magnetic of order four. The drawing estimates the half-life as if the transition could have been each of ten multipoles, and only one of them lands near the measured value: the M4 estimate of 5.6 minutes against a measured gamma half-life of about 2.8. Every other estimate is off by factors of thousands or more.

That is how multipolarities are assigned in practice. A nuclear physicist who measures a half-life of minutes for a gamma ray of several hundred kiloelectronvolts knows, before measuring anything else, that the transition carries four units of angular momentum, and the spins of the two states follow. Selection rules are stated as prohibitions — a transition with $\Delta J = 4$ cannot go as a dipole — but in use they are a price list, and the price is paid in time.

The transition no single photon can make #

The rules have one absolute prohibition. A photon always carries at least one unit of angular momentum — it has no state of zero angular momentum, which is the same fact as light having only two polarisations — so a transition between two states that both have zero spin cannot emit a single photon at all, however much energy it releases. There is no multipole of order zero in radiation. Such transitions happen anyway, by routes that avoid a real photon: the nucleus hands its energy to one of the atom’s electrons and ejects it, or, if the energy exceeds twice an electron’s rest energy, it creates an electron and a positron directly out of its own field. Oxygen-16 has exactly such a state, six megaelectronvolts up with spin zero and the same parity as the ground state, and it decays by emitting a pair — matter made from a nucleus’s energy because light was not allowed to carry it, the same trade a hot enough gas makes with its heat.

The parity half of the rule is subtler and just as strict. An electric multipole of order $L$ changes the parity of the state it leaves if $L$ is odd, a magnetic one if $L$ is even. So the same change of spin can require an electric or a magnetic photon depending on whether the states have the same parity or opposite, and since magnetic multipoles are weaker, the parity of two states can change a lifetime by a further factor of tens or hundreds. The hydrogen 21 centimetre transition is magnetic because the two spin states of the ground level have the same parity; it could not be electric at any order.

What the dipole approximation throws away #

The electric dipole is the classical radiator of a charge that turns and must glow: a charge accelerating back and forth radiates with a strength set by its acceleration, in a doughnut pattern round the direction of oscillation. Every classical antenna shorter than its wavelength radiates essentially that pattern, whatever its shape, because only the dipole part of its current distribution survives at a distance. The distance where a field changes its mind marks the boundary, at a reduced wavelength from the source, beyond which the radiation’s pattern is set. A nucleus’s quadrupole, octupole and higher moments are real — the shape that takes five numbers measures them in charge distributions — but their contributions to the radiation are suppressed by powers of the source’s size over the wavelength, and it is only when the dipole is forbidden by a conservation law that the suppressed terms become the whole story.

Forbidden lines, and where they can be seen #

Atoms show the same price list with different units. Hydrogen’s 2p state decays to the ground state by an electric-dipole photon, the Lyman-alpha line, in 1.6 nanoseconds. Its 2s state has the same energy and cannot: both it and the ground state have zero angular momentum and the same parity, and a single photon must carry at least one unit. It waits, and eventually emits two photons at once, sharing the energy, after about a tenth of a second — nearly a hundred million times longer. Hydrogen’s ground state is split by the interaction between the electron’s and the proton’s spins into two levels a fraction of a microelectronvolt apart, and the transition between them is a magnetic dipole at a wavelength of 21 centimetres, so slow that it takes eleven million years on average.

In a laboratory such transitions are invisible. At ordinary densities an atom in a long-lived state is struck by another atom long before it radiates, and the collision carries the energy away without light — the line is “forbidden”. In thin enough gas it is not. In the upper atmosphere, a hundred kilometres up, oxygen atoms excited by the solar wind survive for most of a second before emitting the green light of the aurora. In a planetary nebula, where the density is a few thousand atoms per cubic centimetre, doubly ionised oxygen emits green light at 500.7 nanometres after tens of seconds. When the line was first seen in the 1860s it matched no known element and was attributed to a new one, “nebulium”; in 1927 Ira Bowen identified it as a forbidden line of ordinary oxygen, visible only because a nebula is a better vacuum than any laboratory could make. And the 21 centimetre line, too slow for any laboratory, is visible from every direction in the sky because there is so much hydrogen that even an eleven-million-year lifetime gives a bright glow.

Survival on a clock that spans forty-five decades #

Drawn on a logarithmic clock, every exponential decay is the same shape — a flat plateau, a cliff about a decade wide, and nothing after — and a nucleus with no clock explains why the shape never changes: an excited nucleus has no memory of how long it has waited. What changes is only where the cliff stands, and the drawing shows cliffs spread over forty-five decades of time, from the femtosecond of an allowed dipole to beyond ten to the seventeenth years.

The last curve is the most remarkable object in the drawing. Tantalum-180 has a ground state that beta-decays with a half-life of eight hours, and an excited state 77 kiloelectronvolts higher with spin 9 against the ground state’s 1. To reach the ground state the excited state would have to emit a photon carrying eight units of angular momentum, and at that energy the estimated half-life for such a photon is absurdly long. The excited state has never been observed to decay by any route. Experiments searching for its decay in large samples of tantalum have set a lower limit of about $3 \times 10^{17}$ years, twenty million times the age of the universe. It is the only nuclear isomer that occurs in nature: a small fraction of all the tantalum on Earth is in this excited state, made in stars billions of years ago and still waiting to shed its energy, held by a conservation law.

Isomers as batteries, and the argument about triggering them #

An isomer stores energy — tantalum-180m holds 77 kiloelectronvolts per nucleus, and hafnium-178m2, with a 31-year half-life and 2.4 megaelectronvolts stored, holds some three hundred thousand times more energy per gram than a chemical explosive. That has made isomers a recurring subject of speculation: if the energy could be released on demand, by a photon or an electron that knocks the nucleus into a state from which it can decay quickly, an isomer would be an extraordinarily dense battery. In 1999 a group reported triggering hafnium-178m2 with dental X-rays; repeated attempts at synchrotrons to reproduce the result found nothing, and the claim is generally regarded as refuted. The multipole arithmetic explains the difficulty. The same selection rules that make the isomer long-lived make it hard to couple to from outside, because any photon that could lift it into a fast-decaying state must itself supply the missing angular momentum.

The one form of controlled release that has been demonstrated uses atoms rather than photons. Molybdenum-93m was reported in 2018 to be de-excited by nuclear excitation by electron capture — an electron captured into the atom’s shell, handing its energy to the nucleus — though the size of the reported effect has been disputed by later theory. The half-life that chemistry can change describes the one way the electrons around a nucleus routinely alter a nuclear rate, and isomer depletion is the frontier of the same idea.

Where Weisskopf’s estimates stop #

One nucleon. The estimates assume a single proton changing its orbit. Real transitions often involve many nucleons moving together, and a collective transition — a whole deformed nucleus rotating from one state to the next — can be a hundred times faster than the estimate. Others are slower, because the two states’ structures overlap poorly. The estimates are a yardstick against which real rates are measured, and the ratio, in “Weisskopf units”, is itself a measurement of nuclear structure.

Gamma emission only. An excited nucleus can also hand its energy directly to one of the atom’s own electrons, which is ejected — internal conversion — and for high multipoles at low energy this can be thousands of times faster than emitting a photon. The measured half-lives of isomers include it; the barium comparison removes the eleven per cent of its decays that go that way.

Other hindrances. In deformed nuclei an additional quantum number, the projection of the angular momentum on the nucleus’s own axis, is nearly conserved, and transitions that change it by much are hindered further — the origin of hafnium-178m2’s long life, and a selection rule that the spherical estimate does not know about.

What the pictures cannot show #

The drawings show rates and times. What they cannot show is the photon’s angular momentum itself, which is the whole of the story: a gamma ray carrying four units of angular momentum has a radiation pattern with four lobes’ worth of structure, and an emission probability that depends on direction relative to the nucleus’s spin. Aligning the nuclei and measuring the angular distribution of the gamma rays is how the multipole order is confirmed directly — the pattern of an angular momentum that is not a rotation, carried off by light. The half-lives are the price; the angular distributions are the goods.

Still open: how long the longest isomer lives #

Tantalum-180m’s decay has been sought with increasing sensitivity, most recently by experiments built to search for extremely rare processes in germanium detectors, placing large tantalum samples next to them underground. Theory predicts half-lives for its various possible decay routes — a gamma transition to the ground state, beta decay to tungsten or hafnium — that range over many orders of magnitude, with the beta branches depending on nuclear matrix elements that are hard to compute because of exactly the angular-momentum mismatch that protects the state. Whether its decay will be observed with foreseeable sensitivity is unknown, and it bears on how the tantalum-180m in the Solar System was made, since the amount that survived depends on how quickly hot stellar plasma could shuttle nuclei between the isomer and the ground state through higher states.

The habit worth carrying away is to ask what a transition must carry off, not only how much energy it releases. A source much smaller than its wavelength pays a factor of roughly its size over the wavelength, squared, for every unit of angular momentum it must shed — which is why the rates of nuclear and atomic transitions are sorted by a small integer before they are sorted by energy, and why a state that must shed many units can outlast the stars that made it.