The molecule that lives outside its own well

Every minimum is a parabola explained why the harmonic oscillator describes so much: near the bottom of any smooth well the potential is quadratic, and a quantum system in its lowest states samples only that bottom. The spectrum that weighs a bond climbed up the stack of vibrational levels and found the harmonic spacing shrinking as the levels approached the top, where the well’s outer wall, and not its curvature, decides where the stack ends. That essay closed on the last level of all — the one bound by almost nothing — and noted that the helium dimer, two helium atoms held together by the weakest attraction in chemistry, has exactly one.

This essay follows that last level to its limit. The result runs against the harmonic picture’s central assumption. A state bound by almost nothing is not near the bottom of its well, or near the well at all. It lives mostly outside, in the region where the force holding it together has fallen to nothing. And because it lives there, it forgets the well that made it. The well’s shape, depth and detailed form leave almost no trace on a weakly bound state’s size or behaviour, which are set by a single number, the energy by which it is bound.

A tail that outgrows the well #

The cleanest place to see it is a square well: a region of radius $R$ where the potential is $-V_0$, surrounded by nothing. Inside, the radial wavefunction of the lowest state oscillates; outside, where the particle’s energy is below the potential, it cannot oscillate and decays as $e^{-\kappa r}$, where $\hbar\kappa = \sqrt{2\mu E_b}$ and $E_b$ is the binding energy. That exponential is a wave in a region it is forbidden to travel through, the same object as the field beyond a totally reflecting surface. The only difference is that here the forbidden region extends to infinity and the tail has nowhere to leak to.

The figure draws the probability of finding the particle at each distance for four wells, each only slightly deeper than the depth at which a bound state first appears. Energies are measured in the natural unit of the problem, $\varepsilon = \hbar^2/2\mu R^2$, the kinetic energy it costs to confine the particle to the well’s size.

For the deepest of the four, bound by 1.4 of those units, the state looks like a conventional bound state: most of it inside the shaded well, 35 per cent spilling out. Make the well a little shallower and the binding falls quickly, because the state is near the top of the well and a small change of depth moves it a long way in relative terms. At a binding of 0.245 units, 62 per cent of the state is outside. At 0.040, 82 per cent. At 0.0059 units the state has 93 per cent of its probability outside the well, and its mean radius is seven times the well’s.

That last state is not a particle in a well with a little leakage. It is a long, thin exponential cloud in which the well is a small region near the centre that the particle visits rarely. The force that holds it together acts only during those visits, yet the state is bound: the brief moments inside the well are enough to keep the cloud from dispersing.

Where most of the state is not #

The fraction of a state outside the range of its force depends on the binding in a way that is universal near threshold.

The figure plots that fraction against the binding energy on a logarithmic axis. At bindings comparable with $\varepsilon$ the state is mostly inside. As the binding falls the fraction outside climbs, and it approaches one as the binding approaches zero. It does so for any short-range well, because the tail’s reach grows as $1/\kappa$ while the region the well occupies stays fixed.

Two real systems sit on the curve. The deuteron, the bound state of a proton and a neutron, is held by the nuclear force, whose range is about the size of the nucleons themselves. Modelled as a square well of radius 2 femtometres binding at its measured 2.22 MeV, it has 64 per cent of its probability outside the range of the force that holds it together. The deuteron is the only bound state of two nucleons, and it is bound by less than a tenth of the energy it would take to confine the nucleons to the range of their force. It is loosely assembled, and much of the time the neutron and proton are farther apart than the force between them reaches.

The helium dimer is the more extreme case. Two helium atoms attract through the weak van der Waals interaction, with a well whose minimum lies at about three ångströms and whose depth is about eleven kelvin. The two atoms are light and the well is shallow, so the zero-point energy that cannot be stopped almost exceeds the depth, and only one level survives, bound by about 1.76 millikelvin. Modelled crudely as a square well of radius 5 ångströms binding at that energy, it has 94 per cent of its probability outside. Most of the time, the helium dimer’s two atoms are so far apart that they are not interacting at all.

A size that forgets the well #

The size of the state follows the same logic, and it can be measured.

The figure plots the mean radius against the binding energy on logarithmic axes. At strong binding the radius is a fraction of the well’s range, set by the well. As the binding weakens, the radius grows as the inverse square root of the binding energy and joins the dashed line, which is the mean radius a pure exponential tail would have, $1/2\kappa$. Along that line the well’s radius has disappeared from the answer. A state bound by $E_b$ has a size of $\hbar/2\sqrt{2\mu E_b}$, whatever made it.

The helium dimer’s size has been measured, which is a considerable feat for a molecule held together by less than two millikelvin. In 2000, Grisenti, Toennies and colleagues sent a supersonic beam of helium through a transmission grating with slits a hundred nanometres wide. Dimers, being twice as heavy, diffract at half the angle of atoms and can be separated. They are also physically large, so the grating’s bars clip them more often than they clip atoms. The effective slit width is narrowed by the molecule’s size, and from the diffraction intensities the mean separation of the two atoms came out as 52 ± 4 ångströms. The crude square well of the figure gives 44; realistic potentials give close to the measured value. A later experiment, published in 2016, ionised both atoms of individual dimers with a laser and imaged the fragments, recording the wavefunction’s exponential tail directly out to more than two hundred ångströms. The molecule is seventeen times longer than the distance at which its atoms attract most strongly.

The deuteron’s case is milder but real. On the square-well model its mean separation is 3.2 femtometres against a force range of 2. Measured charge radii include the proton’s own size and give a larger number, and the conclusion is the same: the deuteron is large for the force that binds it, because it is barely bound.

This behaviour belongs to the lowest level of a very shallow well, and it is not the same as the atom the size of a bacterium, which is large because it is a high level of a long-range Coulomb potential. A Rydberg atom’s size grows as the square of its quantum number and its structure still reflects the $1/r$ force at every distance. A halo state’s size grows because the force has stopped acting.

Two wells that cannot be told apart #

The forgetting can be demonstrated directly. Take two wells with very different shapes, tune each so that it binds a state at the same energy, and compare the states.

The figure does this for a square well and a Gaussian well of the same range, each deepened until its single bound state sits at $\kappa R = 0.12$, a binding energy of 0.0144 in units of $\varepsilon$. The square well needs a depth of 2.72 of those units and the Gaussian 3.06, because its softer edges put less of its depth where it can act. Inside the range of the force the two wavefunctions differ visibly: the square well’s is a sine wave cut off sharply, the Gaussian’s a smoother curve. Beyond it they are exactly the same exponential. Since 89 per cent of the probability lies out there, most of the state cannot tell which well made it.

That is the practical meaning of universality here. Any quantity that depends mainly on the outside of the state — its size, its response to a distant perturbation, the way it scatters a slow particle — is fixed by the binding energy and is the same for every well that gives that binding. The corrections depend on the well’s shape through a second number, its effective range, and they are small when the state is much larger than the well. For the helium dimer they are a few per cent. For the deuteron they are larger, because the deuteron is less weakly bound, and nuclear physics treats the effective range as a measured parameter alongside the binding.

The harmonic approximation fails here in a specific way. The picture of a pendulum near its lowest point — a state determined by the curvature of the bottom — has been replaced by a state determined by how far the top of the well is below zero. Near the bottom, the physics was set by the second derivative of the potential. Near the top, it is set by a single energy difference, and the potential’s shape has almost no influence.

Why helium, and why only one isotope #

A square well binds a state only once it is deep enough: its depth parameter $\sqrt{2\mu V_0 R^2}/\hbar$ must exceed $\pi/2$, which says the well’s depth must exceed $\pi^2\hbar^2/8\mu R^2$, the kinetic energy of confining the particle to the well. A well that falls short of it binds nothing at all, however attractive it is — a result with no counterpart in one dimension, where any attractive well binds a state. In three dimensions, attraction has to beat confinement, and the reduced mass decides by how much.

That is why a halo state is rare and why the helium dimer is the celebrated example. Helium atoms are light and their mutual attraction is the weakest of any neutral atoms, so the ratio of well depth to confinement cost sits almost exactly at threshold. The ratio the motion that cannot be stopped called the de Boer parameter, which compares zero-point energy with well depth, is larger for helium than for any other substance, and helium’s refusal to freeze at ordinary pressure is the same fact seen in bulk. For the pair of atoms, the well only just wins.

The isotopes make the point sharply. A helium-3 atom is three-quarters as massive as a helium-4 atom, so a mixed ³He–⁴He pair has a smaller reduced mass than a ⁴He pair, and a ³He pair smaller still. The potential between them is identical — isotopes differ only in their nuclei, and the electrons that set the interaction do not notice. Yet the confinement cost rises as the reduced mass falls, and it rises past the depth. Neither the mixed pair nor the ³He pair has a bound state. The molecule ⁴He₂ exists, bound by a couple of millikelvin, and its lighter siblings do not, though the force between the atoms is exactly the same.

The deuteron is in the same position among nuclei. The nuclear force between a neutron and a proton binds one state and only in one spin arrangement; the two-neutron and two-proton systems, with a force of nearly the same strength, are not bound. The difference between existing and not existing is a few per cent in the strength of the attraction, and the states that do exist are halos because they are so close to not existing.

The length a collision measures #

The same number appears when the two particles are not bound but collide. A slow particle scattering from a short-range well is described, at low energy, by one quantity, the scattering length $a$. It is where the outside wavefunction, extrapolated inwards as a straight line, crosses zero, and it sets the size of the target the particle sees. The wall that is not quite a wall met it as an apparent displacement of a barrier seen from outside.

The figure plots the scattering length of a square well against its depth. For a shallow well the scattering length is small and negative: the well attracts, and pulls the wavefunction in. As the depth approaches the value that first binds a state, the scattering length runs off to minus infinity. Just past that depth it comes back from plus infinity, and on this side a bound state exists. The dashed curve is that state’s size, $1/\kappa$, and near the threshold it runs alongside the scattering length: at a depth parameter of 1.6 the scattering length is 22.4 well radii and the bound state’s size 21.9.

That coincidence is the universal relation $E_b \approx \hbar^2/2\mu a^2$. A large positive scattering length measured at zero energy is a weakly bound state seen from outside; a large negative one is a state that almost binds. The helium–helium scattering length is about 100 ångströms, twice the dimer’s mean separation, as the relation requires; the neutron–proton scattering length in the spin state of the deuteron is 5.4 femtometres. The neutron–proton scattering length in the other spin state is −23.7 femtometres: a state that almost binds and does not, which is why there is no bound state of a neutron and a proton with opposite spins.

Near the divergence a tiny change in the well produces an enormous change in the scattering length, and that is now an instrument. In ultracold atomic gases a magnetic field can shift a bound molecular level through the threshold — a Feshbach resonance — and so tune the atoms’ scattering length from large and negative through infinity to large and positive, turning an attracting gas into one that forms weakly bound molecules of exactly the kind drawn here.

What the square well cannot show #

The figures use a square well and a Gaussian well to make one point, and the models leave several things out.

The van der Waals tail. A real interatomic potential does not end at a radius; it falls as $1/r^6$. That tail is short-ranged enough for the universal results to hold for the helium dimer, with corrections, but it adds its own length, the van der Waals length, and for atoms with many bound levels the top of the stack is shaped by it, as the spectrum that weighs a bond found.

Higher angular momentum. Everything here is for s-waves, states with no angular momentum. A state with angular momentum has a centrifugal barrier that holds its wavefunction in, and it cannot become a halo in the same way; the universality is a property of the lowest partial wave.

Three bodies. Adding a third particle to a system with a large scattering length produces something no two-body argument predicts. Efimov showed in 1970 that three particles interacting with a resonant pair force have an infinite sequence of bound states, each larger than the last by a factor of about 22.7, even when no pair is bound. Helium trimers show two members of that sequence, and ultracold caesium gases have shown several. The three-body parameter that fixes where the sequence starts is not set by the pair’s scattering length, and why it is nearly the same for many different atoms — about nine times the van der Waals length — is a question that took years to answer and is still refined.

Still open: how universal the three-body limit really is #

The two-body results above are established: near threshold, one number sets the size, the binding and the scattering. The three-body story adds one more number, and measurements on caesium, lithium, potassium and helium suggest that number is itself nearly universal, set by the long-range $1/r^6$ tail rather than by the chemistry at short distances. Theoretical explanations based on a barrier that the tail creates at short range account for much of it, but measured values scatter by tens of per cent around the prediction, and whether the scatter reflects details of each atom’s potential, experimental systematics, or something missing from the theory is not settled. Four-body states tied to each Efimov trimer have also been predicted and seen, and how far up in particle number the tower of universal states extends is open.

The habit worth carrying away is to ask where a state spends its time before assuming it samples the well. A state bound by an energy much smaller than the well’s own scale lives mostly outside the well, and there the only thing it knows is how weakly it is bound. The harmonic picture describes a particle that sits at the bottom and feels the curvature. The last level of a shallow well sits near the top, and nothing about the curvature, the depth or the shape reaches the part of it that is measured.