The electron that leaves because there is room

The reaction that cannot go all the way makes the chemical potential the slope of free energy with particle number, and it shows why no reaction reaches completion: the entropy of mixing has an infinite slope at the pure ends. The product doping cannot move applies the same accounting to a reaction that makes two things at once — an electron and a hole, a hydrogen ion and a hydroxide ion — and finds that equilibrium fixes the product of their concentrations. The most consequential reaction of that kind is the simplest: an atom coming apart into a nucleus and an electron.

$$\mathrm{H} \rightleftharpoons p^+ + e^-$$

In 1920 the Indian physicist Meghnad Saha wrote down the equilibrium of that reaction for a hot gas, treating the three species as ideal gases and asking where the chemical potentials balance. His equation was the first quantitative theory of stellar spectra, and within five years it had been used to show that the stars are made mostly of hydrogen. It rests on one idea, which the drawings take apart: a bound electron is confined to one atom, a free electron has the whole gas, and at equilibrium the room a particle has counts as much as the energy it costs.

Hydrogen comes apart far below its binding temperature #

The equilibrium condition is that the chemical potential of a hydrogen atom equal the sum of the proton’s and the electron’s. For an ideal gas the chemical potential of a species is its energy per particle minus $kT$ times the logarithm of how much room each particle has — the number of quantum states per particle, which is the ratio of a quantum concentration $n_Q$ to the actual concentration. Setting up the balance and solving it gives Saha’s equation, which for hydrogen reads

$$\frac{x^2}{1-x} = \frac{n_Q}{n}\,e^{-\chi/kT}, \qquad n_Q = \left(\frac{2\pi m_e kT}{h^2}\right)^{3/2},$$

where $x$ is the fraction ionised, $n$ the density of hydrogen nuclei and $\chi$ the binding energy. Everything about the atom is in $\chi$; everything about the gas is in $n$.

The drawing solves it at four densities, from the surface of a star to interstellar space, and the curves share a shape and disagree about where. At the density of a stellar atmosphere hydrogen is half ionised at 14,000 kelvin; at the density of interstellar gas, at 3,300 kelvin. None of them is anywhere near 158,000 kelvin, the temperature corresponding to the binding energy. The Boltzmann factor $e^{-\chi/kT}$, the suppression the exponential that decides everything describes, is at 14,000 kelvin about one in a hundred thousand. Something else is overcoming it, and it is the other factor.

The ratio at which hydrogen lets go #

The factor $n_Q/n$ is enormous. $n_Q$ is roughly one over the cube of an electron’s thermal wavelength, the size of the box an electron at that temperature effectively occupies — about $10^{27}$ per cubic metre at 10,000 kelvin — and $n$ is the density of the gas. In a stellar atmosphere the ratio is ten thousand; in interstellar space, ten million million million. That is how many distinct places each free electron has to be, compared with one for an electron bound in its atom.

At half ionisation the left side of Saha’s equation is a half, so the equation says that $\chi/kT$ equals the logarithm of $n_Q/n$, up to a small correction. The drawing confirms it: the binding energy measured in units of $kT$ at half ionisation runs from 11 at stellar densities to 48 in interstellar gas, tracking the logarithm of the room. A binding energy is therefore not a temperature. What a bond can withstand at equilibrium depends on how much more freedom breaking it would buy, and in a thin gas that is a great deal.

The same arithmetic, from the other side, makes the site that fills like an electron level: a site that can hold one particle, in contact with a reservoir, fills according to how far the reservoir’s chemical potential lies above the site’s energy. An atom is a site for its electron, the rest of the gas is the reservoir, and the reservoir’s chemical potential is low when the gas is thin. Ionisation is a site emptying because the reservoir it drains into is vast.

Energy on one side, entropy on the other #

The drawing makes the balance explicit. The horizontal line is the energy cost of ionising an atom, fixed by the atom. The rising curve is what the electron’s freedom is worth, as a free energy: $kT$ times the entropy gained per electron by moving from one atomic state into the whole gas. At low temperature the energy cost wins and atoms stay bound; at high temperature the entropy wins and they come apart; the crossing is close to where the full equation puts half ionisation.

This is the form in which the result generalises. What a system actually minimises at fixed temperature is its free energy, energy minus temperature times entropy, and every equilibrium between bound and free — a molecule dissociating, a vapour over a liquid, a dopant atom giving up its electron in a semiconductor, electron–positron pairs appearing in a gas hot enough to make matter — is the same competition between a binding energy and the logarithm of a ratio of rooms. Saha’s equation for hydrogen and the law of mass action for $np = n_i^2$ in silicon are one equation with different constants.

Why the hydrogen lines are strongest in warm stars #

Saha’s equation was immediately a tool for reading stars. A star’s spectrum is a row of dark absorption lines on a bright background, and each line is a subtraction: atoms in a particular state absorbing at a particular wavelength. In the late nineteenth century astronomers at Harvard had sorted hundreds of thousands of spectra into classes by the strength of their hydrogen lines — A for the strongest, then B, and so on — and then reordered them into a sequence O B A F G K M that nobody could explain.

The visible hydrogen lines, the Balmer series, are absorbed only by hydrogen atoms that are neutral and already in their second energy level. The drawing multiplies the two requirements. The share of neutral atoms excited to the second level rises steeply with temperature, as a Boltzmann factor for 10.2 electronvolts. The share of atoms that are neutral at all falls steeply, as Saha’s equation says. Their product peaks near ten thousand kelvin, and at that peak only about nine atoms in a million are able to absorb a Balmer photon. Cooler stars have neutral hydrogen but almost none of it excited; hotter stars have almost none of it neutral. Sirius, at 9,900 kelvin, has the strongest hydrogen lines of any bright star — and no more hydrogen than the Sun.

In 1925 Cecilia Payne, a doctoral student at Harvard, applied Saha’s equation to the lines of many elements across the spectral sequence and showed that the sequence is one of temperature, with the relative strengths of the lines set by ionisation and excitation rather than by abundance. Correcting the line strengths for those factors, she found that the stars have broadly the same composition, and that hydrogen and helium outnumber everything else by an enormous factor — so different from the composition of the Earth that her thesis, on the advice of the most senior astronomer to read it, described the abundance as almost certainly not real. Within a few years it was accepted. The finding that the universe is mostly hydrogen came out of a chemical-potential calculation.

The universe’s hydrogen holding on #

The early universe is the most extreme case of a thin gas. For every nucleon there are about 1.6 billion photons, and the density of hydrogen, compared with the room available to a free electron at a few thousand kelvin, is tiny. The drawing applies Saha’s equation with that density and finds hydrogen half ionised at 3,734 kelvin, where the binding energy is 42 times $kT$. The typical photon at that temperature has less than a tenth of the energy needed to ionise hydrogen, but there are so many photons per atom that the few in the far tail of the Planck spectrum with enough energy still outnumber the atoms. Hydrogen could not stay bound until the radiation had cooled to a fortieth of its binding energy.

The real history was slower than Saha’s, which is the equation’s lesson about its own limits. Each recombination to the ground state emits a photon energetic enough to ionise another atom, and each recombination through the second level releases a Lyman-alpha photon that can excite another atom so that the next photon ionises it easily. The universe has nowhere for those photons to go, and recombination proceeds only as fast as they can be lost — by the redshift of the expansion or by a slow two-photon decay. Jim Peebles, and independently Yakov Zel’dovich and colleagues, worked this out in 1968. The light of the cosmic microwave background last scattered off free electrons when the universe was at about 2,970 kelvin, some 380,000 years after the Big Bang, and every map of that background is a picture of Saha’s equation lagging.

Ionisation makes a gas soft #

A gas that is partly ionised has the same weakness as a gas that is making pairs. Squeeze it, and some of the work goes into ionising more atoms rather than into raising the temperature, so its pressure rises less steeply than an ordinary gas’s; expand it, and recombination releases heat that props the pressure up. Its adiabatic exponent falls, in the middle of the ionisation zone, well below the 5/3 of a monatomic gas.

That softness has visible consequences. The outer third of the Sun is convective — hot gas rising and cool gas sinking in the granules that cover its surface — largely because the zone where hydrogen is partly ionised lies just below the surface, and a soft gas convects more readily. And the regular pulsation of Cepheid variable stars, the standard candles by which the distances to galaxies were first measured, is driven by the zone where helium loses its second electron. On compression that layer absorbs heat into ionisation and becomes more opaque, trapping the heat that would otherwise escape; on expansion it releases it. The layer works as a valve, pushing on the star in step with its oscillation, and a Cepheid’s period and brightness are set by where in the star that valve sits.

The same equation at room temperature #

The trade between energy and room is not special to stars. A phosphorus atom in silicon holds its fifth electron with an energy of about 45 millielectronvolts — less than twice $kT$ at room temperature — and it lets the electron go into the conduction band, which plays the part of the free gas. The room available there is set by the conduction band’s effective density of states, a few times $10^{25}$ per cubic metre, against a dopant density that is typically a thousand times smaller. Saha’s equation, with those numbers, says the donors are almost all ionised at room temperature, which is why the electron density in doped silicon simply equals the dopant density, the starting point of the product doping cannot move.

Cool the crystal and the balance tips. Below about fifty kelvin the logarithm of the room no longer pays for the binding, and the electrons fall back onto their donors — freeze-out, the reason ordinary silicon electronics stop working near liquid-helium temperature and why cryogenic electronics are built from more heavily doped or different materials. The semiconductor’s freeze-out and the universe’s recombination are one curve drawn at different densities and binding energies.

A thermometer and a barometer made of ions #

Saha’s equation run backwards is how the temperature and pressure of a stellar atmosphere are measured. An element like iron or calcium shows lines from its neutral atoms and from its singly ionised ions in the same spectrum, and the ratio of the two populations depends on the temperature through the Boltzmann factor and on the electron density through the $1/n$ in Saha’s equation. The excitation of lines within one ionisation stage depends on temperature alone. Two measurements, two unknowns: a spectroscopist fits the lines of several stages of several elements at once and reads off both the temperature and the electron pressure of the layer where the lines form.

The dependence on density is what makes giants and dwarfs distinguishable. A giant star and a dwarf with the same surface temperature have atmospheres of very different density — the giant’s is far thinner — so the giant’s atoms are more ionised at the same temperature, and its spectrum shows stronger lines of ions relative to neutral atoms. Astronomers use exactly that difference to sort stars of the same colour into luminosity classes, and through the luminosity to estimate distances. A chemical-potential calculation, applied line by line, measures how far away a star is.

An atom with infinitely many states #

The drawings use a simplification that hides one of the stranger problems in statistical mechanics. The neutral hydrogen atom was given only its ground state. In fact it has infinitely many bound states, crowding towards the ionisation limit, and each contributes a Boltzmann factor to the atom’s partition function. The sum diverges: counting every bound state, a hydrogen atom in a box of any size has an infinite partition function, because the number of highly excited states grows faster than their Boltzmann factors shrink.

The resolution is that the highly excited states are enormous. An atom in its hundredth level is the size of a bacterium, and in any real gas such an atom overlaps its neighbours and is disrupted by their fields long before it can be counted as bound. The states are cut off by the density of the gas itself, and in stellar atmospheres the cut-off falls at levels of a few tens. How best to make that cut-off — sharply, smoothly, with an occupation probability for each level — is a question that stellar-atmosphere modellers still argue about, because the answer changes the populations of the high levels that produce the observed lines. At the densities in the drawings the ground state dominates, and the simplification changes the numbers only slightly.

Where Saha’s equation stops #

Equilibrium. The equation describes a gas in thermal equilibrium with its radiation. The solar corona, heated to millions of kelvin at very low density, is far from it: ionisation there is set by the balance of collision rates and recombination rates, not by chemical potentials, and the corona’s ionisation state is a function of temperature alone, with no density dependence at all. The early universe left equilibrium for the reason described above.

Ideal gases. Electrons, protons and atoms are treated as non-interacting. At high densities the electric fields of neighbouring charges lower the ionisation energy — an effect that, taken to its extreme in the interiors of stars and planets, ionises atoms by pressure alone, with no help from temperature.

Hydrogen only. Real gases have helium, which holds its electrons more tightly and ionises in two stages at higher temperatures, and metals, which ionise easily and supply most of the free electrons in cool stellar atmospheres; the electron pressure used for the Balmer drawing is where those enter.

What the pictures cannot show #

The pictures show fractions, and the gas in each is a violently active thing. At every instant atoms are being ionised by collisions and photons and electrons are being captured, at rates that in a stellar atmosphere run to billions per second per atom. The equilibrium fraction is a statistic of that traffic, stable only because the rates in both directions have adjusted to match. Saha’s equation predicts the statistic without saying anything about the rates, which is its power and the reason it fails whenever the rates are too slow for the traffic to settle.

Still open: where hydrogen stops being an atom #

Saha’s equation assumes a clean distinction between a bound atom and a free electron. In the interiors of Jupiter and Saturn, at millions of atmospheres, hydrogen is compressed until its atoms overlap, and the distinction dissolves: somewhere in the planets’ depths hydrogen becomes a metal, its electrons free to wander through a fluid of protons without any temperature having ionised them. Where that transition happens, whether it is sharp or gradual, and how it depends on temperature have been pursued with diamond-anvil cells and shock compression for decades, with experiments disagreeing about the pressure by factors that matter for models of the planets’ interiors and magnetic fields. It is a question about the same trade as Saha’s — energy against room — in a regime where the room has run out.

The habit worth carrying away is to ask how much room an equilibrium is trading. A binding energy is resisted not by a temperature but by the logarithm of the space the freed particle gains, so a thin enough gas comes apart at a small fraction of its binding energy — which is why the stars’ spectra could be read at all, why the universe stayed opaque until it was cold, and why “hot enough to ionise” always needs “at what density” after it.