Illustrated Physics
physics, drawn
Thermodynamics

The pressure that ionises

Squeeze a hot gas of hydrogen and Saha's equation says it recombines: a free electron has less room to be in, and room was what paid for its freedom. Squeeze it far enough and it comes apart again, completely, at any temperature — not because the room has grown but because the atom has nowhere left to be. Crowded to within a few atomic radii of each other, hydrogen atoms have no bound state, and their electrons, freed by pressure alone, form a metal. Deep inside Jupiter most of the planet's hydrogen is in that state.
14 min read 5 figures The same equation againWhere the model stops

Assumes: The electron that leaves because there is room · The product doping cannot move

The electron that leaves because there is room followed Saha’s equation through the stars and the early universe and found hydrogen coming apart at a small fraction of its binding temperature, because a freed electron has the whole gas to be in and a bound one has one atom. Density entered as the thing that took the room away: a denser gas gives its free electrons less room, and at a given temperature it recombines. That essay ended with the case where this stops being true — the interiors of the giant planets, where hydrogen is compressed until its atoms overlap and becomes a metal “without any temperature having ionised it”.

This essay follows Saha’s equation to that end of the density axis. The answer is that the equation, left to itself, predicts recombination for ever, and the reason it is wrong is that it counts only the free electron’s room. The bound electron needs room too. An atom is a certain size, and a gas compressed until the atoms are closer than that size has nowhere to put a bound electron. Somewhere along the way the gas must come apart again, and it does so at a density that depends hardly at all on the temperature.

Ionised at one end by room and at the other by crowding

Hydrogen ionised by heat at one end and by crowding at the other. The ionised fraction of hydrogen against the density of nuclei, at 5,000 K, 10,000 K, 20,000 K, from Saha's equation with the binding energy lowered by the neighbouring charges (solid) and without the lowering (dashed). Thin gas is ionised by the room a free electron gains; compression takes the room away and the gas recombines, as the dashed curves continue to do for ever. The solid curves turn round: at 5,000 K the gas is least ionised, 2.5·10⁻⁴ per cent, near 10²⁶ per cubic metre; at 10,000 K the gas is least ionised, 0.41 per cent, near 10²⁷ per cubic metre; at 20,000 K the gas is least ionised, 12 per cent, near 10²⁸ per cubic metre, and by 6·10²⁸ per cubic metre — 0.10 g/cm³ — every curve is fully ionised, because the ground state no longer exists. That density does not depend on the temperature at all.
Fig. 1 The ionised fraction of hydrogen against density at 5,000, 10,000 and 20,000 K, from Saha’s equation with the binding lowered by the neighbouring charges (solid) and without (dashed). The dashed curves fall for ever; the solid ones reach a minimum — 0.41 per cent near 1027 m−310^{27}\ \text{m}^{-3} at 10,000 K — and rise to full ionisation by 6×1028 m−36\times10^{28}\ \text{m}^{-3}, 0.1 g/cm³, at every temperature.

The drawing puts the two effects on one axis. The dashed curves are Saha’s equation as it was used before: at each temperature the gas is fully ionised when thin and recombines steadily as it is compressed, because the room a free electron gains, measured by the ratio of the quantum concentration to the density, shrinks. The solid curves add one physical fact. A hydrogen atom in a dense gas is surrounded by other charges — protons, electrons, other atoms — and they lower the energy needed to free its electron, because the electron, once free, is already partly bound to the neighbours. The lowering grows as the neighbours crowd in, and the simplest estimate of it, which counts the charge in a sphere of the volume each atom has, removes the ground state entirely when that sphere’s radius has shrunk to three Bohr radii.

With that one addition the curves turn round. At ten thousand kelvin the gas is least ionised — less than half a per cent — at about 102710^{27} nuclei per cubic metre, a hundredth of the density of water. Squeezed further it ionises again, and by 6×10286\times10^{28} per cubic metre, a tenth of a gram per cubic centimetre, it is fully ionised at every temperature drawn. The rise at the high-density end does not depend on the temperature: it is where the atom stops existing, and heat has nothing to do with it.

The shape is a U, and the bottom of the U is the most atomic that hydrogen can be at that temperature. At five thousand kelvin the bottom is very deep — a few parts in a million ionised — and at twenty thousand kelvin it is shallow: twelve per cent ionised even at its most atomic. Hot enough, and there is no density at which hydrogen is mostly made of atoms.

Two costs that both shrink

Room and binding, both shrinking with density. For hydrogen at 10,000 K, the energy it costs to free an electron — 13.6 eV, lowered by the neighbours as the gas is compressed — and the free energy its freedom is worth, kT ln(nQ/n), both against density. Compression shrinks both. The room a free electron gains falls steadily, as the logarithm of the density, and they cross near 3.5·10²⁰ m⁻³: thinner than that, room wins and the gas is ionised; denser, binding wins and it is atomic. The binding cost holds nearly constant and then collapses as the atoms crowd, reaching zero at 6·10²⁸ m⁻³. By then the room term has itself gone negative — the free electrons would be packed tighter than their own thermal wavelength, and Saha's counting of room has stopped applying — so what ionises the dense gas is not room winning a contest but the contest disappearing: there is no bound state left to pay for.
Fig. 2 For hydrogen at 10,000 K, the energy to free an electron — 13.6 eV lowered by the neighbours — and the free energy its room is worth, kTln⁡(nQ/n)kT\ln(n_Q/n), against density. They cross near 3.5×1020 m−33.5\times10^{20}\ \text{m}^{-3}; the binding cost then collapses to zero at 6×1028 m−36\times10^{28}\ \text{m}^{-3}, by which point the room term has itself gone negative.

The balance that decides ionisation, as the essay on room framed it, is between an energy cost and an entropy gain: the binding energy against kTkT times the logarithm of the room a free electron gains. The drawing plots both against density at ten thousand kelvin. The room’s worth falls steadily as the gas is compressed, and it crosses the binding cost near 102010^{20} per cubic metre: thinner than that, room wins and the gas is ionised; denser, binding wins and it is atomic. That is the recombination the dashed curves showed.

The binding cost stays close to 13.6 electronvolts for a long way, then falls steeply as the atoms crowd, and reaches zero at 6×10286\times10^{28}. By then something has happened to the other curve too: the room term has gone negative. The quantum concentration is the number of electrons that can be accommodated per unit volume before their thermal wavelengths overlap, and at these densities there are more electrons than that. Saha’s way of counting room, which treats free electrons as a thin classical gas, has stopped applying. So what ionises the dense gas is not the room winning a contest with the binding. It is the contest disappearing: there is no binding left to pay for, and the electron is free because it has nowhere else to be.

An atom squeezed in a box

There is a second, independent way to see where the bound state must vanish, and it agrees with the first. An atom is the size it is because of a balance: confining the electron more tightly raises its kinetic energy, by the uncertainty principle, while bringing it closer to the proton lowers its potential energy, and the Bohr radius is where the sum is least. Put the atom in a box smaller than its natural size and the balance is forced. The electron’s kinetic energy rises as the box shrinks, the potential energy cannot compensate, and at some box size the total energy of the lowest state reaches zero — the energy of an electron at rest far away. Below that size there is no bound state.

For a hydrogen atom at the centre of a spherical box with impenetrable walls, the calculation can be done exactly, and the ground state becomes unbound when the box’s radius is about 1.8 Bohr radii. A box of that radius per atom corresponds to about 3×10293\times10^{29} atoms per cubic metre, a few times denser than the ion-sphere estimate. The neighbours in a real gas are not impenetrable walls — their electrons and protons pull as well as push — so the true answer lies between these simple limits, which put it between a tenth and half a gram per cubic centimetre — a density hydrogen reaches only under pressures of a million atmospheres or more, because it takes that much to squeeze the lightest element to a fraction of the density of water at planetary temperatures. The pressure is large; the density it produces is modest; and the reason so much pressure buys so little density is the same kinetic energy of confinement that is doing the ionising. Two crude models from opposite directions bracketing a transition is about as well as a sharp estimate can do for a problem that is really about a dense quantum fluid.

Freed electrons that are crowded too

The freed electrons are crowded too. The density of hydrogen nuclei over the quantum concentration of electrons, nQ — the number of electrons that would share each cell of room the size of an electron's thermal wavelength if every atom were ionised — against density, at 5,000 K, 10,000 K, 20,000 K. Saha's counting of room treats the free electrons as a thin classical gas, which needs this ratio well below one. At the density where the ground state disappears, 6·10²⁸ m⁻³, it is 70, 25, 9 at the three temperatures: the electrons pressure has freed are packed many to a cell, and they form a degenerate Fermi sea unless the gas is hotter than about 8.5·10⁴ K. Pressure-ionised hydrogen is therefore not a hot plasma but a cold metal, its electrons stacked into states by the exclusion principle rather than spread by heat.
Fig. 3 The density of nuclei over the electrons’ quantum concentration, n/nQn/n_Q, against density at 5,000, 10,000 and 20,000 K. At the density where the ground state disappears it is 70, 25 and 9: the freed electrons are packed many to a cell of room, and they are a degenerate Fermi sea below about 85,000 K.

That negative room term is itself a statement about what the freed electrons become. At the density where the bound state vanishes, each cell of room the size of an electron’s thermal wavelength would hold tens of electrons if they were a classical gas. They cannot: they are fermions, and no two can share a state. They fill the available states from the bottom up, into a Fermi sea whose top lies well above the thermal energy, and the gas is degenerate — the regime in which the pressure is not a temperature, because it comes from the stacking of electrons into states rather than from their thermal motion.

Pressure-ionised hydrogen is therefore not what the word “ionised” suggests. It is not a hot plasma of fast, independent particles; it is a lattice-free fluid of protons immersed in a degenerate sea of electrons, conducting electricity as a metal does. Unless its temperature is near a hundred thousand kelvin, the electrons’ thermal energy is a small correction to the energy the exclusion principle gives them. That is why the transition in the giant planets is called metallisation: the hydrogen does not boil into a plasma, it turns into a liquid metal.

One criterion from hydrogen to doped silicon

One criterion from hydrogen to doped silicon. The density at which a gas of hydrogen-like centres turns metallic, from Mott's criterion n^(1/3)a = 0.26, against the size a of the centre's bound orbit, on logarithmic axes. Hydrogen: an orbit of 0.05 nm (the Bohr radius times 1), metallic above 1.2·10²³ per cm³; silicon, donors: an orbit of 2.06 nm (the Bohr radius times 39), metallic above 2·10¹⁸ per cm³; germanium, donors: an orbit of 7.06 nm (the Bohr radius times 133), metallic above 5·10¹⁶ per cm³; gallium arsenide, donors: an orbit of 10.19 nm (the Bohr radius times 193), metallic above 1.7·10¹⁶ per cm³. A donor atom in a semiconductor holds its extra electron in an orbit swollen by the crystal's permittivity and the electron's small effective mass, so the same crowding that turns hydrogen into a metal at a fifth of a gram per cubic centimetre turns silicon metallic at a few donors per million atoms. Measured transitions in these materials lie on the same line within a factor of two.
Fig. 4 Mott’s criterion n^(1/3)a = 0.26 for the density at which hydrogen-like centres turn metallic, against the radius of their bound orbit. Hydrogen, orbit 0.05 nm: 1.2 × 10²³ cm⁻³. Donors in silicon (2.1 nm), germanium (7.1 nm) and gallium arsenide (10.2 nm): 2 × 10¹⁸, 5 × 10¹⁶ and 1.7 × 10¹⁶ cm⁻³.

The same transition happens in a material far easier to study than compressed hydrogen. A phosphorus atom in silicon has one more electron than the silicon it replaces, and that electron is bound to it like the electron in a hydrogen atom — but in an orbit swollen by the silicon’s permittivity, which weakens the attraction, and by the electron’s small effective mass in the crystal. The orbit is about two nanometres across rather than a twentieth of a nanometre. The product doping cannot move treated these donors at modest concentrations, where each holds or releases its electron independently. At high enough concentrations the orbits overlap, and the electrons stop belonging to individual donors.

Nevill Mott argued in 1949 that the transition happens when an electron’s bound orbit is comparable with the distance over which the other electrons screen the donor’s charge — the length that makes a long-range force stop reaching. When screening is strong enough, the potential of each donor can no longer hold a bound state, and the argument gives a criterion with no free parameters beyond the orbit size: the density at the transition, raised to the power one third, times the orbit radius, is about 0.26. Hydrogen, with its orbit of one Bohr radius, turns metallic near 102310^{23} atoms per cubic centimetre. Silicon, with its twenty-fold larger donor orbit, turns metallic at a few times 101810^{18} donors per cubic centimetre — a few donors per million silicon atoms. Measured transitions in doped silicon, germanium and gallium arsenide, and in several other systems, fall on the same line within a factor of two, over twelve orders of magnitude of density.

This is one of the more striking unifications in physics: the doping level at which a semiconductor wafer stops behaving as a semiconductor and the pressure at which hydrogen in a giant planet becomes a metal are the same criterion applied to orbits of different sizes. The ion-sphere estimate used for the other drawings and Mott’s screening criterion are different arguments that agree within a factor of two on where hydrogen’s transition lies, which is about as well as the physics of the transition is known.

A semiconductor that never freezes out

The doped-semiconductor version of the transition has an everyday consequence. At moderate doping, silicon’s donors hold their electrons at low temperature — freeze-out, the reason ordinary silicon devices stop working near the temperature of liquid helium, which is the Saha balance tipping back towards binding as the Boltzmann factor for the donor’s binding wins. Past the Mott density the donors’ orbits overlap, there is no bound state for the electrons to fall back into, and the silicon conducts at any temperature, however cold. Heavily doped silicon is used for exactly that reason wherever a contact or an interconnect must keep conducting at cryogenic temperatures, and for the electrodes of the superconducting and spin-based devices that are operated a fraction of a degree above absolute zero.

The same arithmetic also sets a limit on large atoms in a gas. An atom in a highly excited state is enormous — a micrometre across at principal quantum number a hundred — and a gas at quite modest density crowds it just as a planetary interior crowds a ground-state atom. Such states are destroyed by their neighbours at densities where the ground state is untouched, which is why the series of spectral lines from the highest levels stops earlier in a denser gas: the atom’s infinite series of bound states is cut off from the top.

Where hydrogen is an atom

Where hydrogen is an atom. Density across and temperature up, both logarithmic, with the lines along which hydrogen is half ionised from Saha's equation with the neighbours' lowering of its binding. To the left of the first line the gas is thin enough that the room a free electron gains ionises it; to the right of the second it is dense enough that there is no bound state. Between them, below about 10⁵ K, hydrogen is made of atoms or molecules. The Sun's surface sits in the atomic region near the thermal line, which is why its spectrum has hydrogen lines at all; the deep interior of Jupiter lies past the crowding line, where hydrogen is a fluid of protons and free electrons — a metal — made by pressure, not by heat.
Fig. 5 Density across, temperature up, with the lines along which hydrogen is half ionised: by room at low density and by crowding at high. Between them, below about 10⁵ K, hydrogen is made of atoms or molecules. Marked: a cool nebula, the Sun’s surface, and Jupiter’s interior eight-tenths of the way down, past the crowding line.

Put together, the two effects divide the plane of density and temperature into three regions. To the left of the first line, hydrogen is ionised because a free electron has room to spare; to the right of the second, because an atom has no room at all. Between them, and below about a hundred thousand kelvin, hydrogen is made of atoms — or, at lower temperatures, of molecules. The atomic region is a band, bounded on both sides, and it closes at the top where the heat alone is enough to ionise at any density.

Almost every place where hydrogen is observed as atoms lies inside that band. The Sun’s visible surface sits in it near the thermal line, which is why the Sun’s spectrum has hydrogen lines at all, and why their strength counts room as well as energy. The cool gas between the stars sits far to the left, thin enough that ultraviolet starlight keeps it ionised in some regions and cold enough in others to be atomic. The interiors of Jupiter and Saturn lie beyond the crowding line: below a depth of about a fifth of Jupiter’s radius, hydrogen is a liquid metal, and the planet’s strong magnetic field is generated by currents flowing in it.

Where the estimate stops

A crude lowering. The ion-sphere lowering is the high-density limit of a family of estimates of how neighbours reduce a binding energy, and its prediction of a sharp density at which the ground state vanishes is an artefact of treating the neighbours as a smooth sphere of charge. In reality the transition is smeared, the atom’s ground state is squeezed and distorted gradually, and hydrogen at these densities is not a gas of separate atoms at all but a dense fluid whose properties must be computed from quantum mechanics directly.

Molecules. Below a few thousand kelvin hydrogen is molecular, and the transition is from a molecular fluid to a metallic one, which involves breaking the molecules as well as freeing electrons. Whether that happens in one step or two, and whether it is a sharp phase transition with a latent heat at low temperatures or a smooth crossover at high ones, is what experiments and calculations disagree about.

Saha’s counting. Near the transition the free electrons are degenerate, as the third drawing shows, and Saha’s equation with its classical room is no longer the right description of them. The curves near the high-density end are illustrations of a mechanism, not predictions of a fraction.

What the curves do not show

The curves show equilibrium fractions, and in the laboratory metallic hydrogen has been made only fleetingly, in shock waves that compress it for nanoseconds, or claimed in diamond anvils at pressures of hundreds of gigapascals whose interpretation has been disputed. The curves also say nothing about the solid. At low temperatures and very high pressures hydrogen is predicted to become a solid metal, perhaps a superconductor at high temperature, perhaps metastable when the pressure is released; none of this is in a gas-phase equilibrium, and none of it has been confirmed.

Still open: the transition inside the planets

Where exactly hydrogen metallises in Jupiter and Saturn, whether the transition is sharp or gradual, and how it depends on temperature matter for the planets directly: they decide where the dynamo that makes the magnetic field operates, how heat moves out of the interior, and whether helium, which does not mix well with metallic hydrogen, separates and rains downward. Measurements from the Juno spacecraft of Jupiter’s gravity field suggest that its core is not compact but diluted through a large part of the planet, which has changed the models the transition sits in. Shock experiments disagree with each other by factors of two in the pressure of the transition, and first-principles calculations disagree about the order of the transition. How much room hydrogen needs to remain an atom, it turns out, is not known to better than a factor of two in density.

The habit worth carrying away is to check whether every participant in an equilibrium has the room the equilibrium assumes. Saha’s equation counts the room a freed electron gains and forgets the room a bound one needs, and a gas compressed until the bound state no longer fits ionises whatever its temperature — so heat and pressure are two different ways of ionising matter, one by giving the electron somewhere to go and the other by taking away the place it was.

Part 7 of 7

This essay is one argument about Chemical potential. The others:

Hydrogen comes apart far below its binding temperature. The fraction of hydrogen that is ionised in equilibrium, against temperature on a logarithmic axis, for densities of hydrogen nuclei of 10²³, 10¹⁸, 10¹², 10⁶ per cubic metre, from Saha's equation. Hydrogen's binding energy, 13.6 eV, corresponds to 157,803 K. It is half ionised at 13,980 K at 10²³ per cubic metre, 7,235 K at 10¹⁸ per cubic metre, 4,519 K at 10¹² per cubic metre, 3,270 K at 10⁶ per cubic metre — at a temperature where kT is between an eleventh and a fiftieth of the energy needed to free the electron. The thinner the gas, the cooler the temperature at which it comes apart, because a free electron in a thin gas has vastly more room to be in than a bound one. The electron that leaves because there is room Part 6 — It takes 13.6 electronvolts to pull the electron off a hydrogen atom, the energy of a particle at a temperature of 158,000 kelvin. Hydrogen is half ionised at the surface of a star at 14,000 kelvin, in a thin nebula at 7,000, and in the early universe at under 4,000 — a fortieth of its binding energy. The electron leaves not because it has enough energy but because a free electron has so much more room to be in than a bound one, and at equilibrium room counts as much as energy. Saha's equation is that trade written down, and it turned out to be the key to what stars are made of.

Where this goes next

Hydrogen comes apart far below its binding temperature. The fraction of hydrogen that is ionised in equilibrium, against temperature on a logarithmic axis, for densities of hydrogen nuclei of 10²³, 10¹⁸, 10¹², 10⁶ per cubic metre, from Saha's equation. Hydrogen's binding energy, 13.6 eV, corresponds to 157,803 K. It is half ionised at 13,980 K at 10²³ per cubic metre, 7,235 K at 10¹⁸ per cubic metre, 4,519 K at 10¹² per cubic metre, 3,270 K at 10⁶ per cubic metre — at a temperature where kT is between an eleventh and a fiftieth of the energy needed to free the electron. The thinner the gas, the cooler the temperature at which it comes apart, because a free electron in a thin gas has vastly more room to be in than a bound one. The electron that leaves because there is room It takes 13.6 electronvolts to pull the electron off a hydrogen atom, the energy of a particle at a temperature of 158,000 kelvin. Hydrogen is half ionised at the surface of a star at 14,000 kelvin, in a thin nebula at 7,000, and in the early universe at under 4,000 — a fortieth of its binding energy. The electron leaves not because it has enough energy but because a free electron has so much more room to be in than a bound one, and at equilibrium room counts as much as energy. Saha's equation is that trade written down, and it turned out to be the key to what stars are made of. How much of copper's electron sea a temperature can reach. The probability that a state of a given energy is occupied, in copper, at 4 temperatures, with energy measured in units of the Fermi energy — 7.04 eV here. At absolute zero the curve is a step: every state below the ceiling is full and every state above it is empty. Raising the temperature rounds the step, and rounds it over a range of about kT, which is the whole point — at room temperature kT is 0.0259 eV against a ceiling of 7.04 eV, so the rounding is 1.6 per cent of the way down the sea and everything deeper is untouched. An electron in the deep is not held there by a force; it simply has nowhere to go, because every state it could be promoted to is occupied. at 0 K the step is spread over 0.00 per cent of E_F, at 300 K the step is spread over 1.61 per cent of E_F, at 3000 K the step is spread over 16.13 per cent of E_F, at 20000 K the step is spread over 107.52 per cent of E_F. The pressure that is not a temperature Copper's conduction electrons are at a temperature of eighty thousand kelvin, in a wire that is at room temperature. That is not a figure of speech, it is what the exclusion principle does to a mole of particles, and it explains the largest unexplained number in the theory of metals. Electrons and holes in silicon against the donors added. The concentrations of free electrons and holes in silicon at 300 K against the concentration of donor atoms added, on logarithmic axes, solved from charge neutrality and the product law n·p = nᵢ². Below the intrinsic concentration of 10¹⁰ per cubic centimetre the donors change almost nothing. Above it the electrons follow the donors one for one and the holes fall in exact proportion: a hundred times more donors, a hundred times fewer holes. At 10¹⁶ donors the electrons outnumber the holes by 10¹² to one. The dashed line is √(n·p), which does not move at all across ten decades of doping — checked to a part in a billion. The product doping cannot move Add one phosphorus atom for every five million silicon atoms and the crystal's free electrons multiply a millionfold — and its holes fall a millionfold, although nothing was done to them. Add a drop of acid to water and the hydroxide ions vanish in exactly the proportion the hydrogen ions grow. Both are one law: when a reaction makes two species together, equilibrium fixes the sum of their chemical potentials, and so the product of their concentrations. Every semiconductor device is built on the minority carriers that law leaves behind.

The objects named here

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

Chemical potentialDegenerate matterIonisationMetallic hydrogenMott transitionPressure ionisationSaha equationScreening