The Metastable Solitude

In the summer of 1998, batches of ritonavir began failing dissolution tests. The drug was a treatment for HIV, already on the market, formulated and manufactured. Capsules that had dissolved as designed stopped dissolving. For a time the cause was not known.¹

The molecule had not changed

What happened is now understood, and the explanation is finished. The compound was still ritonavir. Chemists could write the same formula, with the atoms bonded in the same order. What had changed was the shape the molecule took inside the solid, and the lattice in which many copies of it were packed.

Two crystal forms turned out to be possible. The form that had been manufactured dissolved well enough to be a medicine. A second form appeared, lower in energy, with a different conformation and a different packing, and much less willing to dissolve in water. Once that second form existed, it could seed further material. The useful form had been real. It had been made and stored. It was not the form the crystal most favored, and once a path opened into the favored lattice, the old process could no longer be counted on to produce the old behavior. The failure was a formulation that could not be manufactured reliably.

Nothing in that story required a new law of chemistry. The same compound had found another way to sit next to copies of itself. The depth of the new valley favored the less soluble crystal. A ridge had kept the old crystal in use until a path across the ridge was found.

That is a closed case. We can name the compound, the two lattices, and the solubility that was lost. The rest of this essay asks a question of the same kind, and that question is open. Ritonavir shows what a finished answer looks like. It does not supply the answer for a nucleus.

Two lattices

An arrangement can last because leaving it costs something. Picture the possible arrangements as valleys on a map. A system can sit in one valley even when another valley is deeper, because every path between them crosses a ridge. The depth of a valley says which arrangement would win if the system were given unlimited time and an open road. The height of the ridge, and which roads actually exist, say how long the present arrangement lasts.

That difference is metastability. The arrangement in front of you is the one that has not yet found its way out. Equilibrium has a favorite. History decides whether the favorite has been reached. A crystal can be warmed, a seed of the new form can be introduced, or a fluctuation can open a path, and the same substance then behaves as a different material.²

Ritonavir is one lattice, and the lattice is the whole story of that event. The molecules occupy a grid. Two grids are available to the same compound. One dissolves well enough to be a medicine. The other is more stable, and it does not. Describing the event means describing the grid: which neighbors each molecule has, how those neighbors are angled, and which grid is deeper. The shift from the first grid to the second is what the dissolution tests recorded.

Diamond and graphite are the second lattice. Both are made of carbon atoms, and only carbon atoms. In diamond, each atom is tied to four neighbors at the corners of a tetrahedron, and that pattern repeats in every direction. The result is a rigid solid. In graphite, the same atoms form sheets of hexagons. Within a sheet the ties are strong. Between sheets they are weak, which is why the sheets slide and why graphite writes on paper. At the pressures of ordinary life, graphite is the deeper valley. A diamond on a hand is already in the less favored lattice. It remains a diamond because the ridge between the two grids is enormous. Left alone, the atoms do not rearrange on any timescale a person will witness. The atoms did not have to become a different element. The lattice they occupy is the material.

A living system asks for one more distinction. A cell can hold its organization by spending energy, taking in matter, and repairing what breaks. The lifetimes that matter there are maintained. They are not only the lifetimes of a structure trapped above a deeper valley. Metastability explains how an arrangement can outlast the valley it sits over. It does not, by itself, explain a metabolism. An organism is not a ritonavir crystal that has not yet found its more stable form.³

Lasting long enough

The thought that connects a crystal to a nucleus is a relationship among times. A useful arrangement has to last long enough to be used, and it has to remain able to change when change is required.

A bond that never lets go cannot take part in a reaction. A fold in a protein has to hold its shape long enough to do its job, and it has to be able to open again. A nucleus that falls apart before a star can build on it never becomes part of a later planet. The parts of a living cell do not all have to last the same amount of time. They have to last in a way that lets the other parts work: construction and loss, damage and repair, a fuel that waits and a fuel that burns. I think of this as persistence without permanence. Lasting forever is a different property from lasting long enough.

The examples so far are arrangements under the interactions we already use. Ritonavir found another crystal form. Diamond persists in the less favored lattice. A cell maintains itself by spending energy. None of those facts required a new law.

The conjecture takes one further step. Suppose a background field could occupy another long-lived state, and that state changed a dimensionless property of matter. Could the change move nuclear reactions, the elements a star produces, or the conditions under which organized chemistry can keep itself going?

The crystal is the reason for asking. A finished account of ritonavir names a lattice and a lost solubility. It does not name a new interaction, and it is not evidence that a field history of that kind exists.

The largest extension of the question is the silence in the title. Suitable histories might be rare, or too short for observers to arise. That extension can wait. It is only as good as the account of what another history would change in matter.

The nucleus

An element is fixed by its protons. Hydrogen has one. Carbon has six. Iron has twenty-six. Uranium has ninety-two. Neutrons can be added or removed and the element stays the same until a decay changes the proton count. Electrons occupy the outer structure. Almost all of chemistry is the behavior of those electrons: which bonds form, which molecules dissolve, which colors a material has. The nucleus, by contrast, is only the protons and the neutrons. Nearly all of the atom's mass is there, in a volume far smaller than the electron cloud.

The electrons still have one nuclear job, and it is worth seeing before setting them aside. A nucleus in an excited state can give up its extra energy as a gamma ray. It can also hand that energy to a nearby electron and throw the electron out of the atom. The second route is called internal conversion. In iron-57, a state sitting 14.4 keV above the ground state uses both routes. In the neutral atom its half-life is 98.3 nanoseconds. The conversion route is about 8.6 times more common than the gamma ray. Take the electrons away, so that only the gamma ray remains, and the half-life becomes about 0.94 microseconds, if the radiative rate itself is unchanged. The nucleus is the same nucleus. The protons and neutrons have not been rearranged. The half-life changed because the electrons that had been carrying most of the decay are gone. Surroundings can decide how long a state lasts, with no new nuclear force at all.⁴

Three kinds of arrangement

The word configuration is about to do a lot of work, and the measurements do not treat every use of it as the same fact. Three meanings have to be separated before any result.

A different number of neutrons is a different isotope of the same element. Carbon-12 has six protons and six neutrons. Carbon-13 has six protons and seven neutrons. Both are carbon, because chemistry follows the proton count. They are different nuclei. Their masses differ, the energy required to pull a neutron away differs, and a star that builds one of them is not automatically building the other. Most of the public nuclear record is a catalog of these differences. It is a large catalog, and it is easy to mistake it for the question this essay is asking.

The same protons and the same neutrons can also occupy a relatively long-lived excited state. That state is an isomer. The particles have not been added or removed. They occupy a different pattern, with a different energy and often a different half-life. Not every excited state earns the name. The iron-57 state above lives for a fraction of a microsecond. An isomer lasts long enough to be treated as its own arrangement.

Natural tantalum is the sharp example. Tantalum-180 contains 73 protons and 107 neutrons. NUBASE2020 lists the ground state's half-life as 8.154 hours. About 75.3 keV above that ground state, a second pattern is listed as stable, and that is the pattern found in minerals, at about 0.012 percent of natural tantalum. Seventy-five keV is a small step on a nuclear scale. Nuclear bindings are measured in millions of electronvolts. The half-life listed for the ground state is part of a day. The excited state has no measured decay in that evaluation. A listing of "stable" records that absence. It is not a proof of an infinite lifetime, and it is not a history of every such atom from the star that made it. We have the surviving pattern in the crust. We do not have a record of one of those atoms in the ground state at an earlier date.⁵

The third meaning is a channel. A proton and a neutron can bind. Two neutrons, in the ordinary way their spins align, do not. The bound pair is the deuteron, the nucleus of heavy hydrogen. Pulling its neutron away costs 2.2246 MeV. Two neutrons attract each other strongly. Scattering experiments show that the attraction misses a bound state by a small margin: there is a nearby virtual state, which is a way of saying "almost," and there is no dineutron sitting on the table with a decay clock. The particles are the same kind of matter. The channel, meaning the combination of charge and spin, decides whether a nucleus exists.⁶

A particularly clear comparison would take the second of these three and repeat it. Same protons, same neutrons, two times, with a property of the underlying particles written beside the state. A catalog of different isotopes is the first meaning. It is the baseline. Other records can test a model without being that comparison.

A graph with three views

Once the three meanings are separate, the protons and neutrons can be drawn as a graph with three views. One view contains every nucleon, protons and neutrons together. A second view contains only the protons. A third contains only the neutrons. The split follows the questions the channels raise. A proton and a neutron can bind. Two protons feel the strong force and their electric repulsion. Two neutrons feel the strong force and carry no charge. A drawing that gave every pair the same kind of edge would hide that difference.

The views are a way to inspect the data. They are not three separately measured forces, and they are not a partition of the nucleus into three independent pieces. The mixed view contains every pair the other two views contain, together with the pairs between protons and neutrons. The three drawings overlap because they describe one coupled system. Removing a neutron changes its meetings with protons as well as with neutrons, and the nucleus that remains can rearrange.

The graph is still not a crystal lattice, and the difference matters. In ritonavir, or in diamond, the units are distinct objects. A particular molecule has particular neighbors. An edge can carry a length, an angle, and a share of the lattice energy. In a nucleus the protons are indistinguishable from one another, and the neutrons are indistinguishable from one another. Quantum mechanics does not let us point at one neutron, name its neighbor, and store a measured bond between those two labeled particles. Pairwise operators can still be written. What the picture cannot support is a set of permanently labeled classical bonds. The nodes of the graph are counts.

Carbon-12 makes the bookkeeping concrete. The proton view has six nodes. The neutron view has six nodes. The mixed view has twelve. Six neutrons can be paired in fifteen ways, so the neutron view contains fifteen neutron-neutron pairs in the accounting. The mass tables do not list fifteen energies. They list one mass for the whole nucleus, and from the mass of the nucleus left behind they list the energy required to remove one neutron. Removing one neutron from carbon-12 costs 18.72 MeV. That number is the difference between the binding of carbon-12 and the binding of the daughter. Dividing it by fifteen and writing the result on every edge would hand each pair an equal share of a number that was never a pair energy. Those shares would add back to 18.72 MeV. They would still be an assignment, not a measurement.⁷

Two further facts belong on the graph even though the tables do not supply numbers for them. The force between two nucleons depends on an angle: the angle between their spins and the line that joins them. That tensor force is why the deuteron is not a round pair. Part of its wavefunction is stretched, a D-wave mixed into the simpler S-wave, and the nucleus has a measurable deformation because of it. Angles are doing real work, as bond angles do in a drug crystal. They are just not stored, pair by pair, in the mass evaluation.

The other fact is that pairs are not enough. The triton, the nucleus of tritium, has one proton and two neutrons. A force written only between pairs, fitted to the deuteron and to scattering, does not land on the measured triton binding unless a three-nucleon term is added and adjusted. Every nucleus heavier than the deuteron inherits that requirement. The three-view account assembled for this essay records the angle and the three-nucleon term as open. It uses the measured removal energies. It does not pretend to have fit the missing pieces.⁸

What would move those views

Ordinary nuclear physics already changes what these views display. Add a neutron and the nucleus is a different isotope, with a different removal energy. Promote a nucleus into a long-lived excited state and the same counts occupy a second pattern, often with a different half-life. A star, while it burns, takes one set of nuclei and builds another, under the strong, electromagnetic, and weak forces as we already use them. The tables of binding energies are a map of that ordinary motion. Across one element, the energy holding the last neutron commonly changes by millions of electronvolts from isotope to isotope, while every one of those isotopes is still a bound nucleus. In the catalog, that spread is typically about 9 MeV among isotopes that remain bound against losing a neutron, and about 24 MeV for neon.

Those millions of electronvolts are differences between different nuclei. They are the ordinary map. They are not an uncertainty attached to one specified nucleus, and they are not a size a new effect must exceed before it can matter. A shift much smaller than the gap between two isotopes of neon can still move a reaction that sits near a threshold. The comparison that decides a proposed shift is the ordinary prediction for the observable being tested, together with the uncertainties and the competing explanations.

The further question is a change in the interactions, not a relabeling of the nodes. Suppose a persistent background occupied another state, and that state changed a dimensionless property of matter: the strength of the electromagnetic coupling, or the masses of the light quarks relative to the strong-interaction scale. The nucleus could keep the same protons and the same neutrons while the energies required to remove them moved. Quarks and gluons set the strong force. Electric charge adds a repulsion between protons. Those responses have to be calculated together, from one change in the background, rather than chosen separately for whichever nucleus gives the largest effect. A calculation of the response is not a detection that the parameter has changed. A drawing of two field states does not show that both are occupied, or that either survives inside a star. The field has to be present in the material that is burning, and it has to supply the parameter change the nuclear calculation uses.⁹¹⁰

The three views are where such a shift would be read. They remain readings of the whole nucleus.

On the neutron view the bookkeeping is the uncharged pairs. Two neutrons in ordinary matter attract and do not bind. If that attraction were stronger, the near miss could become a bound pair, and the proton count would not have to change for that to be true. The energies required to remove one neutron, or two, would respond, and so would the extra binding that appears when the neutron count is even. The response would still belong to the parent nucleus and the daughter. The neutron view can display those removal energies. It does not isolate the neutron-neutron force.

On the proton view the strong force is still there, and so is the electric repulsion, because both particles are charged. A change in the electromagnetic coupling reaches charged particles directly. The neutron-removal energies can move as well, because the protons' charge rearranges the whole nucleus. The two views are allowed to respond by different amounts. That is a reason to keep the drawings separate. It is not a measurement that has already taken the forces apart.

On the mixed view the strong force acts in the spin arrangement that binds the deuteron. The carbon resonance is the same kind of fact one step larger than a single pair. In a massive star, after the hydrogen is gone, helium nuclei meet. Two of them form beryllium-8, which is itself unbound and does not last. A third helium nucleus can meet that temporary pair while the three together pass through an excited state of carbon-12, the Hoyle state. That state sits only a few hundred keV above the energy at which the three helium nuclei come apart again. A few hundred keV is small beside the millions of electronvolts stored in the nucleus. Because the state is perched there, a small shift in its placement can change how often the meeting sticks, and therefore how much carbon the core contains when helium burning ends.

The weak interaction does an additional job. It can turn a neutron into a proton, or a proton into a neutron, so the nucleus leaves one isotope and appears as another. That is a real downstream effect, and it is how many nuclei decay. The shift this essay is tracking is the one that happens first, while the label of the isotope is unchanged: the measured energies move, and a later star or a later rock might inherit a different material history of the same nucleus.

That is the mechanism, stated as a question. Whether any calculation or any archive has seen it operate is the next question, and it should not be answered inside the drawing.

The row that would make the comparison sharp

An especially clear record would be a short table. One imaginary row is worth writing, so the later archives have something specific to be compared with. It is one informative comparison. It is not the only evidence a model could meet, and a completed row would not, by itself, establish a biosphere or a silent sky.

Take carbon-12. The row would name six protons and six neutrons, twice. Once might be a laboratory sample from this year. Once might be the same isotope as it was in a mineral that closed two billion years ago, or in a gas cloud whose light has been traveling for most of the age of the universe. At each time the row would record the state, not merely the name of the isotope. Spin and parity would say which pattern the nucleons occupied. A radius, or a quadrupole moment, would say how the charge was distributed. The energy required to remove one neutron, and the energy required to remove one proton, would say where those removal energies sat. Beside those numbers, the row would record the particle property suspected of pushing them: the mass of the light quarks, the strength of the electromagnetic coupling, or whichever parameter the theory actually uses. A change in the nuclear numbers without a change in that parameter is an ordinary nuclear event. A change in the parameter without a change in the nuclear numbers is a particle measurement with no downstream effect on that observable. The row is sharpest when both ends are filled, for the same nucleus, at two times.

A field, or any other underlying interaction, needs its own line in the same account. The interaction has to be the one operating where the nucleus burns or decays, and it has to be the same model that predicted the nuclear shift. A field that differs in the space around a star, and has its ordinary value throughout the gas that is actually burning, does not move the fire. The ritonavir case was settled by identifying the lattice that held the molecules, not by measuring a property of the room in which the bottle was stored.

Other records can test a model without filling this row. Their usefulness depends on which part of the history they preserve, and on whether their responses and uncertainties are connected through the same physical assumptions.

What we can actually read

The sources assembled for this project do not provide that complete paired record. What exists fills columns of it, one at a time.

Present arrangements, masses, and removal energies are in the 2020 atomic mass evaluation and in NUBASE. Level schemes, spins, and half-lives are in ENSDF, browsable through NuDat. From those tables I assembled the three views for every element: binding energy kept on the whole nucleus, a proton view, and a neutron view, for the most tightly bound isotope and for the longest-lived one, and for the other measured isotopes behind them. There are 118 elements in the account and a few thousand nuclei. One hundred and ten elements have a laboratory mass for the most tightly bound choice. From roentgenium through oganesson the masses are estimated, and the views are flagged that way. This is the map of ordinary motion. It is one time: now.⁷⁵⁸

The force between nucleons, as it is today, is in the Nijmegen analysis and the SAID database. Proton-proton scattering and proton-neutron scattering are measured directly, over a range of energies and angles, in the spin channels that matter. Neutron-neutron scattering has no target of free neutrons, so it is inferred from reactions that produce two neutrons together and record how they share energy as they leave. The result is a scattering length on the unbound side of zero. Those databases constrain the interaction as it is now. Knowledge of that interaction does not, by itself, specify how the interaction would respond to a new field.⁶

How a quark mass would move a nucleon mass is a calculated slope. Lattice QCD averages the dependence of the nucleon mass on the masses of the quarks. Separate nuclear calculations propagate a change of that kind into the deuteron and into the carbon resonance. A slope is what the first half of the mechanism would have to pass through. If the quark mass moved by a stated amount, the nuclear energy would move by a stated amount. Publishing the slope does not mean anyone watched the quark mass move.¹⁰

The long records are not all the same kind of clock.

Laboratory atomic clocks compare optical and microwave frequencies over years. A drift in the electromagnetic coupling would pull those frequencies apart. The limits from the best comparisons are very tight on the drift happening now, and the baseline is a human career, not a geological age.

Quasar absorption spectra look farther, and they look at a moment. Gas along the line of sight imprints atomic lines, and those lines can be compared with the same lines in a laboratory. The comparison belongs to the epoch when the light was absorbed. The released ESPRESSO exposures of the cloud toward HE0515−4414 are a public example, and the fit made for this project did not produce a detection. An atomic frequency is not a nuclear arrangement. It is a particle property, sampled once, at a cosmological distance.¹¹

Oklo, in Gabon, is a set of natural fission reactors that burned about two billion years ago. The neutron flux altered the isotopes of samarium. How much samarium-149 was destroyed depends on where a resonance in that nucleus sat relative to the energy of the neutrons. From the isotopic residue, the position of the resonance at that one epoch can be inferred, within a model of the reactor. There is no spectrum of a samarium nucleus from before the reactor started and another from after it cooled. There is a fingerprint in the proportions of different isotopes. That fingerprint is a real constraint. Its assumptions, including the reactor model and the way several parameters can move together, are part of the result.¹²

Rhenium-osmium ratios in iron meteorites are a different kind of record again. They integrate how much rhenium has become osmium since the mineral closed. The project's reanalysis reproduces the published regression. It also shows the trade that keeps the measurement from being the wished-for row. A large shift in the decay rate that lasts a short time, and a small shift that lasts a long time, can leave the same daughter abundance, and the date at which the mineral closed can trade against both. Under the reference laboratory half-life, the joint clocks allow no change at all. Use a different published half-life and the long excursion can be pushed away from zero. The rock remembers an average rate over an interval. It does not remember the arrangement of the rhenium nucleus at two separated moments. A quasar line and a meteorite ratio can be combined only to the extent that their responses and their windows of time supply different information.¹³¹⁴

The first minutes of the universe supply one more check on rates. The helium and deuterium we measure today were set by nuclear reactions while the cosmos was still hot. Those abundances agree, within their uncertainties, with the rates we measure now. They are a check that the channels were not wildly different then. They are not a carbon-12 nucleus inspected twice.

Set beside the imaginary row, the empty cells are specific. We have the arrangement now. We have the force now. We have slopes for how a particle mass would move a nucleus. We have constraints on particular histories: a frequency compared at the epoch of absorption, a resonance inferred from a reactor's ash, a decay rate integrated since a mineral closed. We do not have spin, radius, and removal energy for one isotope at two times, written beside a particle property at those same times. A model can still be tested with the records that exist, when the responses and the uncertainties are connected through the same assumptions. A null result on one parameter would leave other nuclear changes still to be explained. An unchanged nuclear observable would leave open the possibility that a varying parameter shows up in some other consequence.

What the results show

The calculations that bear on the question are many. Five of them change what the question is allowed to mean. The others are in the figures and the linked research pages, and they should stay there until a single history joins them.

Binding energy per nucleon is not a half-life. The three-view catalog's most-bound choice is the isotope with the greatest binding energy per nucleon among the entries its rules select. Different isotopes have different numbers of particles. Their binding energies per nucleon are not the depths of alternative states of one fixed nucleus, and a nucleus does not fall from one of them into another by sliding downhill. Which transformation is favored depends on a specified reaction and its full energy accounting. How long a nucleus lives depends, further, on the decays that are open and on their rates.

Hydrogen makes the separation plain. Its most tightly bound nucleus is tritium: one proton, two neutrons, 2.83 MeV of binding for each nucleon, a half-life of 12.32 years. The hydrogen that lasts is the proton. Its neutron count is zero. Its nuclear binding is zero. It is stable. The ranking by binding and the ranking by survival are different rankings.

Iron is quieter, and the quietness is the point. Iron-58 is more bound, per nucleon, than iron-56 by about 2 keV. Both are stable. Two keV a nucleon is a small difference on the ranking. The last neutron in iron-56 is held by about 1.2 MeV more than the last neutron in iron-58. Same element, both isotopes effectively permanent, and the energy required to change the neutron count already differs by a million electronvolts. That difference is a difference between two nuclei.

Uranium separates the ranking from the half-life completely. Uranium-218 has 92 protons and 126 neutrons. One hundred and twenty-six is a closed shell, and this is the most tightly bound uranium isotope in the mass table, at 7.64 MeV per nucleon. Its half-life is 354 microseconds. Uranium-238 has twenty more neutrons, is less tightly bound per nucleon, holds its last neutron by 6.15 MeV rather than 9.15 MeV, and has a half-life of 4.46 billion years. Binding energy per nucleon alone does not determine how long a nucleus lasts. The arrangement a person means by the word uranium is the isotope that had a long enough interval.⁸

The route can matter after the interaction returns. The catalog compares different nuclei. The deuteron calculations compare one nucleus with itself, under a prescribed change of the force.

One calculation weakens the interaction until the pair is unbound, holds it there, and restores the original interaction. For three speeds of that passage, about 23 percent, 20 percent, and 11 percent of the wavefunction is bound at the end. Restoring the original Hamiltonian does not restore the original state.

A separate calculation follows a path that remains bound and ends at a different binding energy, 0.3 MeV, rather than back at the force it started with. There, slowing the change raises the probability left in the final bound state from about 79 percent, to 95 percent, to 99.95 percent.

The two paths establish history dependence under prescribed changes. A slower change is not always the one that preserves more of the bound state: in the passage through the unbound regime, the slower return leaves less of it. Neither path is a field history of the cosmos. History dependence of this kind is also wider than metastability in the narrow sense. A metastable state, in that sense, is a long-lived state together with a way of leaving it. What these calculations establish is the fact the crystal suggested at a smaller scale. The preparation can matter after the interaction has been put back.¹⁵