Illustrated Physics
physics, drawn
Electromagnetism

The atom whose nucleus is a reflection

An electron set down on liquid helium cannot go in, because the helium atoms' closed shells repel it, and it cannot go far, because the liquid polarises and pulls it back with the field of an image charge. The image is weak, three-hundredths of the electron's charge. So the electron sits in a one-dimensional hydrogen atom whose nucleus is its own reflection: its levels follow hydrogen's 1/n² series at 0.66 milli-electronvolts instead of 13.6 electronvolts, and it hovers eleven nanometres above the surface. Microwaves at 126 gigahertz lift it to the next level, and a field pressing down on it tunes the line.
15 min read 5 figures The same equation againWhat happens at the edge

Assumes: The charge that has to be somewhere else · Where the electron probably is

The charge that has to be somewhere else found that a charge held above an earthed metal sheet feels exactly the field of a second charge, of opposite sign, buried at its mirror position. The partner is not there, and it does not need to be: it is a guess that satisfies the equations and the boundary, and in electrostatics a guess that does both is the answer. The same trick works above a surface that is not a conductor. An insulator polarises when a charge comes near it, and the polarisation, seen from above, is the field of an image whose strength depends on how easily the material polarises. For a material that polarises fully, a metal, the image is as strong as the charge. For one that barely polarises at all, the image is faint.

Liquid helium polarises less than anything else that is liquid. Its atoms hold their two electrons more tightly than any other atom does, and its relative permittivity is 1.0572: almost a vacuum. An electron above it feels an image of a few hundredths of its own charge. That would be a curiosity, but for a second property of helium: the electron cannot go in. Pressed against the surface, it meets a barrier of about one electronvolt, because the exclusion principle forbids it the states the helium atoms’ own electrons occupy. A free electron above helium is therefore attracted towards a surface it cannot cross, by a force that falls off as the inverse square of its height. That arrangement has a name, and the name is the hydrogen atom.

An electron held above helium by its own reflection. The potential energy of an electron at height z above liquid helium (ε = 1.0572), attracted by its image, −Λe²/4πε₀z with Λ = 0.00695; the surface itself is a wall the electron cannot cross. The lowest three levels (horizontal lines) are −0.657, −0.164, −0.073 meV — hydrogen's 1/n² series at 0.657 meV instead of 13.6 eV — and each is drawn with its probability density above it. The ground state's electron hovers 11.4 nm above the surface on average, about thirty helium atoms' diameters; the first excited state 45.7 nm.
Fig. 1 The potential energy of an electron at height z above liquid helium, pulled down by its image, −Λe²/4πε₀z with Λ = 0.00695, and stopped at the surface. Its lowest three levels are −0.657, −0.164 and −0.073 meV, hydrogen’s levels at a twenty-thousandth of the scale, each drawn with its probability density; the ground state hovers 11.4 nm above the surface.

An image of three hundredths of a charge

A charge ee at a height zz above a flat dielectric of permittivity ε induces a surface polarisation whose field above the surface is that of a charge −e(ε−1)/(ε+1)-e(\varepsilon-1)/(\varepsilon+1) at depth zz below it. For a metal, with ε effectively infinite, the factor is 1 and the image is the full opposite charge. For helium it is 0.0278. The image is a distance 2z2z from the electron, and the energy of the pair is half what it would be for two real charges, because the image is not independent: it follows the electron and is made by it. Altogether

V(z)=−Λe24πε0 z,Λ=ε−14(ε+1)=0.00695 for helium.V(z) = -\frac{\Lambda e^2}{4\pi\varepsilon_0\, z}, \qquad \Lambda = \frac{\varepsilon-1}{4(\varepsilon+1)} = 0.00695 \text{ for helium.}

This is the Coulomb potential of a nucleus carrying 0.00695 of the electron’s charge, felt only on one side of a wall. The radial equation for an electron in the s-states of hydrogen, written for u=rRu = rR, is exactly the equation for an electron moving in one dimension in a −1/r-1/r potential with a wall at r=0r = 0 where uu must vanish. The electron above helium obeys that equation with rr replaced by its height and the nuclear charge replaced by Λ. Its levels are hydrogen’s, scaled:

En=−Λ2 R∞n2,a=a0Λ,E_n = -\frac{\Lambda^2\, R_\infty}{n^2}, \qquad a = \frac{a_0}{\Lambda},

a Rydberg of 0.657 millielectronvolts instead of 13.6 electronvolts, and a Bohr radius of 7.61 nanometres instead of 0.053. Why an atom is the size it is traced the size of hydrogen to a balance between the kinetic energy confinement costs and the potential energy it gains; weaken the attraction by a factor Λ and the balance moves out by 1/Λ1/\Lambda and down by Λ2\Lambda^2. The first figure is drawn from the numerical solution of the equation, not from the formula, and the levels it finds are hydrogen’s to three parts in a thousand.

The wall the electron meets

The other half of the arrangement, the wall, is as unusual as the image. A helium atom’s two electrons fill its only shell, and an extra electron approaching it would have to occupy a state orthogonal to theirs, which costs kinetic energy; no two in the same state is the whole reason. Averaged over the liquid, that cost is a repulsive step of about one electronvolt at the surface. It is short-ranged, a matter of ångströms, so above the surface the electron feels only the long-range image; at the surface it meets a wall nearly two thousand times higher than its binding energy.

What happens to an electron that is pushed through the wall anyway shows how strong the repulsion is. Rather than sit among the atoms, it clears a cavity for itself. A bubble of radius rr confines the electron, at a kinetic cost falling as 1/r21/r^2, and costs the liquid a surface energy growing as r2r^2, the tension that the small bubble blows up the big one set against the pressure inside a bubble. Liquid helium’s surface tension is tiny, about 0.35 millinewtons per metre, two hundred times less than water’s, and the balance falls at a radius of about 1.9 nanometres: a hole some fifteen atoms across, containing nothing but one electron’s wavefunction. These electron bubbles drift through the liquid as heavy, slow ions, and they are one of the standard probes of superfluid helium.

The surface the electron floats over is also exceptionally clean. Every other substance freezes solid long before helium’s temperature is reached and sticks to the walls of the cell, so the only atoms at the surface are helium’s, and at a kelvin the vapour above is so thin that an electron rarely meets one. The one-dimensional atom of the first figure therefore sits over a mirror with almost no flaws in it.

A hydrogen atom with no nucleus in it

The electron in its ground state hovers on average 11.4 nanometres above the surface — one and a half times the Bohr radius, exactly as the s-electron of hydrogen sits at one and a half Bohr radii from the proton on average. That is about thirty helium atoms’ diameters, so the electron is well clear of the liquid and sees it as a smooth, uniform dielectric, which is what the image calculation assumed. The first excited state hovers four times as high, at 45.7 nanometres. Horizontally the electron is not bound at all: it glides along the surface, free in two dimensions and bound in the third.

The nearest thing to it elsewhere in physics is the atom the hydrogen atom inside a crystal found when an electron in a semiconductor is bound to a positive charge: there the Coulomb attraction is weakened by the crystal’s permittivity and the electron’s effective mass is changed by the band structure, so the atom is hydrogen scaled to a few meV and a few nanometres, as here. The difference is that a donor atom has a real nucleus and the image atom has none. Its nucleus is the electron’s own reflection, and it moves when the electron does. The image atom is the simplest one-electron system that is not a particle bound to another particle.

Where the electron probably is drew the probability clouds of hydrogen in three dimensions. The densities in the first figure are their one-dimensional relatives: the s-state radial densities of hydrogen, laid out along a height instead of a radius, with one node in the first excited state and two in the second.

Faint reflections make large atoms

The weaker the reflection, the larger and looser the atom. The binding energy of the image atom's ground state (solid, meV) and its Bohr radius (dashed, nm), against the permittivity of the surface it floats on, on a logarithmic scale; both follow from Λ = (ε − 1)/[4(ε + 1)], the binding as Λ² and the radius as 1/Λ. Liquid helium-3 (ε = 1.0426): 0.370 meV, 10.1 nm; liquid helium-4 (ε = 1.0572): 0.657 meV, 7.6 nm; solid neon (ε = 1.2440): 10.1 meV, 1.9 nm. Helium is the extreme case: its atoms hold their electrons so tightly that it barely polarises, which makes the reflection faint and the atom it makes enormous and fragile.
Fig. 2 The binding energy of the image atom’s ground state (solid) and its Bohr radius (dashed) against the permittivity of the surface, on a logarithmic scale: liquid helium-3, 0.370 meV and 10.1 nm; liquid helium-4, 0.657 meV and 7.6 nm; solid neon, 10.1 meV and 1.9 nm. The binding goes as Λ2\Lambda^2 and the radius as 1/Λ1/\Lambda.

The second figure follows the image atom across materials. Because Λ is roughly proportional to ε − 1 for a weak dielectric, the binding grows as the square of the material’s polarisability, and the atom shrinks in proportion. On helium-3, whose atoms are lighter and whose liquid is less dense, the permittivity is 1.0426 and the atom is looser still: 0.370 millielectronvolts, ten nanometres. On solid neon, with a permittivity of about 1.24, it binds at ten millielectronvolts and is under two nanometres across.

A more polarisable surface does not simply give a better atom. The image atom needs a surface that is both a dielectric and a barrier — that keeps the electron out — and a surface that polarises strongly usually also has states the electron can fall into. Helium and neon, closed-shell atoms whose electrons are held tightly, are the materials that do both, and the faint image is the price of the high wall.

Lines that a field tunes

Transitions a pressing field tunes. The frequencies of the image atom's transitions from its ground state to the first and second excited states, against a field pressing the electron towards the surface, solved numerically with the surface as a hard wall. At zero field they are 119.2 and 141.3 GHz; measured on real helium they are 125.9 and 149.4 GHz (points), in the same ratio, 0.844 against 0.843, both higher by about 6% than the sharp-surface model allows. The field pushes the excited state, which sits further out, up more than the ground state, so the line rises — by about 774 MHz per V/cm at first, and faster as the field begins to dominate the image. In the model a source fixed at 130 GHz (dotted) is brought into resonance at 14.3 V/cm, which is how the line was first found: the frequency held still and the atom tuned to it.
Fig. 3 The 1→2 and 1→3 transition frequencies against a field pressing the electron towards the surface, solved numerically. At zero field the model gives 119.2 and 141.3 GHz; on real helium they are measured at 125.9 and 149.4 GHz, in the same ratio, both about 6% higher. The line rises by about 774 MHz per V/cm at first; in the model a 130 GHz source is brought into resonance at 14.3 V/cm.

The transitions from the ground state to the first and second excited states fall at 119.2 and 141.3 gigahertz in this model, in the millimetre-wave band between microwaves and the far infrared. Charles Grimes, Thomas Brown, Michael Burns and Clifford Zipfel measured them in 1976, by shining millimetre waves across a sheet of electrons floating on helium in a cell cooled to about a kelvin and watching for absorption. They found the lines at 125.9 and 149.4 gigahertz. The absolute values are about six per cent above the sharp-surface model’s, but their ratio, 0.843, matches the hydrogenic (1−14)/(1−19)=0.844(1-\tfrac14)/(1-\tfrac19) = 0.844. The discrepancy is the surface: a real liquid surface is not a perfect step, its density falls from liquid to vapour over a few ångströms, and the electron’s barrier is about one electronvolt rather than infinite. Models that include a diffuse surface of finite height reproduce the measured lines.

Grimes and his colleagues did not sweep their source’s frequency, which at those wavelengths was hard to do. They held it fixed and tuned the atom. A small electric field applied across the cell, pressing the electrons towards the surface, adds to their energy in proportion to their height. The excited states sit higher, so they are lifted more, and the line rises: by about 774 megahertz for each volt per centimetre, at first. A field of fourteen volts per centimetre in the model brings the 1→2 line up to a fixed 130 gigahertz source, and the absorption appears as the field passes through resonance. This is the Stark effect, and on an ordinary atom it is minute. The field an atom calls strong compared applied fields with the field inside an atom and found the internal one larger by factors of millions; here the internal field, the image’s pull at the electron’s height, is only about a thousand volts per centimetre, so a few tens of volts per centimetre is already a substantial fraction of it. By a hundred volts per centimetre the 1→2 line has moved from 119 to 178 gigahertz and is no longer anything like a small shift.

Pressing the electron down

How hard the field presses the electron down. The ground state's mean height above the helium against the pressing field, on a logarithmic field axis (points, solved numerically; the left-hand point is zero field). With no field the electron hovers at 11.4 nm, 1.5 Bohr radii, and fields of the size used to tune its spectrum barely move it: 11.3 nm at 10 V/cm, 10.4 nm at 100. The field begins to dominate near 864 V/cm, where it changes the electron's energy across one Bohr radius by the binding energy: 7.7 nm at 1000 V/cm and 4.4 nm at 10⁴. Dashed: the electron in the field alone against a hard wall, a triangular well, whose height falls as the inverse cube root of the field — 5.2 nm at 10⁴ V/cm, still higher than with the image's pull added.
Fig. 4 The ground state’s mean height against the pressing field. Fields used to tune the spectrum barely move it: 11.4 nm with no field, 11.3 at 10 V/cm, 10.4 at 100. Near 860 V/cm the field begins to dominate, and the electron is at 7.7 nm at 1000 V/cm and 4.4 nm at 10⁴. Dashed: the field alone against a hard wall, a triangular well, 5.2 nm at 10⁴ V/cm.

The fourth figure follows the electron’s height as the field grows. At the fields that tune the spectrum it barely moves; the line shifts because the excited states move, not the ground state. The natural scale of field is the one that changes the electron’s energy across a Bohr radius by the binding energy itself, R/eaR/ea, about 860 volts per centimetre. Above that, the image’s pull matters less and less, and the electron approaches the state of a particle pressed against a wall by a uniform force — a triangular well, whose states are Airy functions and whose height falls as the inverse cube root of the field.

That limit is the same problem as the bounce that comes in fixed heights, where neutrons falling onto a mirror under gravity occupy quantised heights of tens of micrometres. The force there is the neutron’s weight and the scale is micrometres; here it is an electric field on an electron and the scale is nanometres. The triangular well appears in both because both are a particle pushed towards a surface it cannot enter, and the same Airy function gives the shape of both ground states.

A sheet that freezes

A sheet of floating electrons that freezes into a lattice. The temperature below which the electrons floating on helium arrange themselves into a triangular lattice, against how many there are per square centimetre: the sheet crystallises when the Coulomb energy between neighbours exceeds their thermal energy by a factor of about 137, the value measured for this system. At 10⁸ electrons per cm² — about 1.0 μm apart, 88 times the height they float at — the sheet freezes at 0.22 K; at 10⁹, 0.68 K. Beyond about 2 × 10⁹ per cm² (shaded) the field needed to hold that much charge down dimples the helium surface until it gives way, and no denser sheet can be made on bulk helium.
Fig. 5 The temperature below which the floating electrons form a triangular lattice, against their number per cm²: the sheet freezes when the Coulomb energy between neighbours exceeds their thermal energy by about 137, the measured value. At 10⁸ per cm², electrons 1.0 μm apart, it freezes at 0.22 K; at 10⁹, 0.68 K. Beyond about 2 × 10⁹ per cm² (shaded) the helium surface gives way under the charge.

The electrons on helium are bound vertically and free horizontally, so a sheet of them is a two-dimensional gas of electrons with no lattice, no impurities and no nuclei beneath it — the cleanest such gas there is. Its electrons repel one another by the ordinary Coulomb force, at spacings of a micrometre, nearly a hundred times the height they float at, so to each other they are point charges in a plane. Eugene Wigner predicted in 1934 that electrons whose mutual repulsion overwhelms their kinetic energy should freeze into a lattice. In a metal that needs densities far lower than any real metal has. On helium it is a matter of cooling: the ratio of the Coulomb energy between neighbours to the thermal energy grows as the temperature falls, and when it passes about 137 the sheet orders into a triangular lattice. Grimes and Gregory Adams saw the transition in 1979, through a change in how the sheet absorbed radio waves: the lattice of electrons dimpled the helium surface beneath each electron and coupled to its ripples. It was the first Wigner crystal observed.

The figure gives the melting temperature, which grows as the square root of the density, under a kelvin for every sheet that can be made on bulk helium. There is a ceiling, and it too is electrostatic. To hold a sheet of charge against the surface, a field must press it down, and the pressure of that field on the liquid scales as the square of the charge density. Beyond about two thousand million electrons per square centimetre the surface dimples and the sheet falls through.

Where the picture holds

The image potential treats the helium as a continuous medium with a sharp edge, which is accurate at the heights the electron occupies, where the nearest helium atoms are tens of their own diameters away, and wrong within a few ångströms of the surface, which is why the measured lines sit six per cent higher than the model. The wall is taken as infinite where it is about one electronvolt; since the electron’s binding is less than a thousandth of that, the difference is small but not nothing. The calculation also leaves out the surface’s motion. Liquid helium’s surface ripples with thermal waves, and the electron’s coupling to those ripples, and to helium atoms in the vapour above, sets how freely it glides. At the lowest temperatures the vapour is gone and the ripples are weak, and electrons on helium have been measured to move more freely than electrons in any semiconductor.

What the figures cannot show

The figures show an electron bound in one dimension and free in the other two, and they do not show the price of that freedom: the electron’s horizontal motion, combined with any roughness of the surface or any variation in the pressing field, mixes into its vertical levels and broadens the lines. They do not show the cell in which the electrons are held — the electrodes beneath the helium that supply the pressing field, and the guard electrodes that stop the sheet from escaping sideways. And they do not show where the electrons come from. In practice a hot filament above the liquid sprays them down onto the surface, and they stay because there is nowhere else for them to go.

Still open: whether a floating electron can hold a qubit

A single electron above helium has two properties that recommend it as a quantum bit: a pair of well-separated levels, the ground and first excited states of the image atom, and a surface so clean that little should disturb them. Philip Platzman and Mark Dykman proposed in 1999 to build a quantum computer from arrays of such electrons, each held over a small electrode that controls its Stark shift, and the proposal has been pursued since in many forms: with the vertical states, with the electron’s motion in the plane, and with its spin, which on helium is expected to be very well isolated. In 2022 a related system, a single electron trapped on the surface of solid neon, was made into a qubit, and its coherence has since been extended to tens of microseconds. What remains unsettled is what ultimately limits the coherence of an electron on helium. Ripplons, the quanta of the surface’s thermal waves, couple to its height; vapour atoms collide with it; charge noise in the electrodes beneath shifts its levels; and which of these sets the floor at the temperatures and fields a working device would use is being measured rather than predicted.

The habit worth carrying away is to read a potential for its form before its source. A dielectric that an electron cannot enter attracts it with an image of (ε − 1)/(ε + 1) of its charge, and a potential −Λe²/4πε₀z against a wall is hydrogen with its charge scaled by Λ: on helium, Λ = 0.00695, a Rydberg of 0.657 meV, a Bohr radius of 7.61 nm, an electron hovering 11.4 nm high, and lines at 125.9 and 149.4 GHz in hydrogen’s own ratio, tunable by fields of tens of volts per centimetre because the atom’s whole internal field is only about a thousand. An atom does not need a nucleus. It needs an inverse-distance pull and somewhere the electron cannot go.

Part 8 of 8

This essay is one argument about Potential. The others:

Water drops that charge two cans to a spark. The voltage between the two cans of a Kelvin water dropper against time, on a logarithmic axis, starting from a stray difference of 1 V, for 10, 20, 40 drops a second in each stream; drops 3 mm across, each with a capacitance of 0.17 pF to its ring, cans of 20 pF. Each drop carries a charge induced by the opposite can's potential into its own can, which raises the opposite can's induced charge on the next drop: the difference grows exponentially, multiplying by e every 6.0 s at 20 drops a second. It passes the 10 kV that will jump a few millimetres of air after 55 s — from one volt to a spark, with nothing but falling water. The water drops that charge themselves to a spark Part 7 — Two streams of water drip from one earthed pipe through two metal rings into two metal cans, and each ring is wired to the opposite can. Nothing is charged to begin with, nothing is rubbed, nothing is plugged in. Within a minute a spark cracks between the cans at ten thousand volts. A stray volt between them makes the drops forming in one ring pick up a slight charge, which they carry to the can that makes the other ring's drops pick up the opposite charge — a loop in which every drop strengthens the imbalance that charged it. Kelvin built it in 1867; it is a dynamo made of water and gravity.

Where this goes next

The plane deleted, and one charge put in its place. A charge of 1 nC held 20 mm above an earthed conducting plane. The lines are traced through the field of the real charge plus an equal and opposite one at the mirror position, and then cut at the plane, because below it there is metal and no field whatever. Nothing in the tracing knows about the surface: each line follows the local field direction and stops where it arrives. That every one of them arrives perpendicular — the worst departure among the 9 drawn is 2.2° away from square — is the boundary condition showing itself rather than a rule imposed on the drawing. The image charge is drawn faint because it is not there: it is a way of writing a function that happens to satisfy the equation and the boundary values, which by the uniqueness theorem makes it the field and not a model of the field. The charge that has to be somewhere else Hold a charge above an earthed metal sheet and the field above it is exactly the field of two charges — the real one and an imaginary partner buried at the mirror position. The partner is not an analogy or an approximation. It is a legal guess, and a legal guess is a proof. Absorption below and above the gap. The light absorbed by a direct-gap crystal against photon energy measured from the gap, in units of the exciton's Rydberg, from Elliott's formula with each line broadened to a half-width of 0.06 Rydberg (solid), beside the absorption the same crystal would have if electron and hole did not attract (dashed), which rises from zero as the square root of the energy above the gap. Below the gap the attraction adds a series of lines at −1, −1/4, −1/9 … Rydberg, the hydrogen series scaled down, with strengths falling as 1/n³, crowding towards the gap. Above it the continuum is lifted: the attraction pulls each free pair together, and at the gap itself the absorption is a finite step rather than zero. The hydrogen atom inside a crystal Light absorbed by a semiconductor lifts an electron across the gap and leaves a hole behind, and the two attract. In an ordinary crystal they bind into a copy of the hydrogen atom, with the same series of levels and the same 1/n² — but the crystal screens their attraction and lends them light masses, so the copy is a few thousand times shallower and several hundred times larger. That rescaling is the whole theory, it predicts lines below the gap and a finite step at it where no attraction would give a smooth rise from zero, and it says which crystals keep their excitons at room temperature. The heights a neutron is allowed to bounce to. A neutron above a horizontal mirror in the Earth's gravity: the potential energy mgz (straight line), the first four allowed energies and, drawn on each, the probability density of the neutron's height. The natural length is z₀ = (ħ²/2m²g)^(1/3) = 5.87 μm and the natural energy mgz₀ = 0.602 peV. The allowed energies are 1.41, 2.46, 3.32, 4.08 peV — pico-electronvolts — and a classical neutron with each energy would rise to 13.7, 24.0, 32.4, 39.8 μm. Each density is an Airy function, with n − 1 nodes, leaking a little above the classical turning height and vanishing at the mirror. The lowest state never touches the mirror and never rises above about 25 μm. The bounce that comes in fixed heights A ball dropped on a floor can bounce to any height. A neutron cannot. Slow enough, and resting on a flat mirror in the Earth's gravity, it has a lowest state about fifteen micrometres high, with an energy of 1.41 pico-electronvolts, and a staircase of higher states above it — the first quantum states ever seen in a gravitational field, measured in 2002 with a slit narrower than a hair. Shaking the mirror at 255 hertz drives a neutron from the first state to the second. The staircase is now a laboratory for gravity at distances where nothing else can test it.

The objects named here

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

Bohr radiusDielectricHydrogen atomImage chargeMethod of imagesPermittivityStark effectWigner crystal