The pressure that is not a temperature

The exclusion principle applied to one atom explains the periodic table. Applied to a mole of electrons, it produces something an atom’s worth of counting gives no hint of: a ceiling. Fill the available states from the bottom, one electron per state, and the energy of the last one filled is a definite number — and for the electrons in a metal, that number is enormous.

Everything in this essay is that ratio. The Fermi energy of copper corresponds to a temperature of 81,700 kelvin, and the metal is at 300. The electrons are, in the technical sense, extremely cold — not because they are moving slowly, which they emphatically are not, but because the temperature is nowhere near enough to disturb their arrangement. The floor they are sitting on is the same refusal to be localised that keeps helium liquid, counted over $10^{23}$ particles instead of one.

Where a ceiling comes from #

The box that allows only some energies gives the states. Confine a particle to a region of size $L$ and its allowed momenta are spaced by $h/L$; in three dimensions the allowed states are points on a lattice in momentum space, spaced by $h/L$ in each direction.

In a box those levels go as the square of the quantum number, so the spacing grows with energy rather than crowding together as it does in an atom. But the box here is the whole sample: $L$ is centimetres, the spacing between neighbouring levels is around $10^{-18}$ eV, and nothing could resolve it. That is exactly why the spacing does not matter. What matters is how many states lie below a given energy, and that is a volume in momentum space rather than a list.

Counting them is a volume. The states with energy below $E$ are those inside a sphere of radius $p = \sqrt{2mE}$ in momentum space, and the number of lattice points inside a sphere is its volume divided by the volume per point. Two spin states per point, $N$ electrons to place, and the highest energy filled is

$$E_F = \frac{\hbar^2}{2m}\left(3\pi^2 n\right)^{2/3},$$

with $n$ the electrons per unit volume. Nothing in it but the density and two constants.

That last number is the one to hold on to. The fastest electrons in a piece of copper on a bench are doing half a per cent of the speed of light, and they would be doing it if the copper were cooled to a millikelvin. Their motion has nothing to do with heat.

Each occupied state has its own energy and every one of them rises when the box is squeezed — the allowed momenta are spaced by $h/L$, so shrinking $L$ pushes the whole ladder up. Draw four of them and the argument is already complete; the Fermi sea is the same picture with $10^{23}$ rungs instead of four. Squeezing raises every occupied level, the total energy goes up, and a resistance to being squeezed is a pressure.

It is worth doing the arithmetic that turns a density into a ceiling, because the answer is dominated by one thing. The two-thirds power means that a metal with eight times the electron density has four times the Fermi energy, so the range across ordinary metals is modest — potassium at 2.12 eV, sodium at 3.24, silver at 5.49, copper at 7.04, aluminium at 11.7. All of them are one to two orders of magnitude above the 0.026 eV that room temperature supplies. There is no ordinary metal whose electrons are anything but deeply degenerate, and that is why the effects in this essay are not a special case but the normal condition of every wire and every coin.

The heat capacity that went missing #

Drude’s 1900 theory of metals treated the conduction electrons as a classical gas, and got the electrical conductivity roughly right, the ratio of thermal to electrical conductivity almost exactly right, and one thing catastrophically wrong. A classical gas of $N$ particles has a heat capacity of $\tfrac32 Nk$ — half a $kT$ for every way it can hold energy — and adding that to a metal’s known lattice heat capacity would have made every metal’s specific heat about fifty per cent larger than it is measured to be.

The resolution is in the hero figure and takes one sentence. An electron a long way below the ceiling has nowhere to go: every state it could be promoted to by absorbing $kT$ of energy is already occupied, and the exclusion principle is not a force it can push against. Only the electrons within about $kT$ of the top can absorb anything, that fraction is of order $T/T_F$, and each of them takes up about $kT$. The product is

$$C_{\text{el}} = \frac{\pi^2}{2}\,\frac{T}{T_F}\,Nk,$$

which is not a smaller version of $\tfrac32 Nk$ — it is proportional to $T$ rather than constant, so it is a different law. That is what makes the confirmation sharp: at low temperature the lattice contribution falls as $T^3$ and the electronic one only as $T$, so the electronic term dominates below a few kelvin, and plotting $C/T$ against $T^2$ in a metal gives a straight line whose intercept measures the density of states at the Fermi energy directly.

The same shape of argument governs a molecule’s rotations and vibrations, which freeze out one after another as $kT$ falls below the spacing of their levels — which is why a diatomic gas’s heat capacity falls in steps rather than smoothly. The electron sea is that freezing-out taken to its extreme, with one difference in what the relevant gap is: not the spacing between levels, which is unmeasurably small, but the distance from the ceiling down to the states that are already full. At room temperature ninety-eight per cent of the electrons are frozen out in exactly this sense.

Set that against the Maxwell–Boltzmann distribution of speeds, which is what the electrons were assumed to have. Its whole shape scales with temperature: halve $T$ and the curve narrows and shifts down, and as $T$ goes to zero it collapses onto a spike at the origin. The Fermi distribution does no such thing. Cool a metal to absolute zero and its electrons keep the same enormous spread of speeds they had at room temperature, because that spread was never thermal in the first place.

Two errors of a hundredfold, cancelling #

The heat capacity was Drude’s catastrophe, and the ratio he got right was his triumph, and the two have the same cause — which is the most instructive thing on this page.

The quantity is the ratio of a metal’s thermal conductivity to its electrical conductivity divided by the temperature, and it is remarkable because it comes out very nearly the same for every metal: around $2.4\times10^{-8}$ watt-ohms per kelvin squared, for copper and lead and iron alike. Drude’s account of it is one line. Both conductivities are carried by the same electrons, so the mean free path and the scattering time cancel between them, leaving a ratio built out of the electrons’ heat capacity and their mean square speed and nothing else.

He then put in classical values for both, and got a number about half the measured one — which in 1900 was an extraordinary success, since a factor of two on a quantity nobody could otherwise explain at all is agreement.

It was luck of a very particular kind. The classical heat capacity is about a hundred times too large, as this essay’s figure shows. The classical mean square speed is about a hundred times too small, because the real electrons are moving at the Fermi velocity rather than at a thermal one. The ratio contains the product of the two, and the two errors are both essentially $T/T_F$, in opposite directions. They cancel, almost exactly, leaving a number close to right for reasons that have nothing to do with the derivation.

The correct calculation puts the Fermi values in for both, and the constant that comes out is $\pi^2k^2/3e^2$ — a combination of nothing but fundamental constants, which is why the ratio is the same for every metal. That the classical theory arrived near it is the sort of accident that keeps a wrong theory alive: it is not that Drude’s model had a piece of the truth in it, but that the one quantity he could check was the one insensitive to what was wrong.

The pressure that stays when the heat is gone #

If the electrons have energy at absolute zero, and squeezing them raises it, then they push back. The pressure is

$$P = \frac{2}{5}nE_F,$$

which for copper is 38 gigapascals — 380,000 atmospheres — at zero temperature.

There is a second route to that number which makes its origin clearer. The total kinetic energy of a filled sea is $\tfrac35 N E_F$ — three-fifths rather than a half, because there are more states near the top than near the bottom — and squeezing the box raises every level as the inverse square of its width. Differentiating the total energy with respect to volume gives the pressure, and the two-fifths appears from the two-thirds power in $E_F$ combined with the differentiation. Nothing thermal enters at any point, and there is nowhere in the derivation for a temperature to be inserted.

The obvious question is why a piece of copper does not explode. The answer is that the electrostatic attraction between the electrons and the ion cores is of the same enormous size and pulls the other way, and a metal’s actual bulk modulus is the small difference between two large numbers. The degeneracy pressure is real and it is balanced; what it explains is not why metals hold together but why they are so hard to compress. Squeezing a metal by one per cent means squeezing its electron sea by one per cent, and the sea resists at 38 gigapascals.

The relativistic softening at the far right of the pressure figure is the one place this argument leaves the laboratory. As the density rises, the electrons at the ceiling approach the speed of light, their energy becomes $pc$ rather than $p^2/2m$, and the pressure stops rising as fast — from the $5/3$ power of density to the $4/3$. A support whose pressure grows more slowly than the weight it holds up has a ceiling of its own, and that is where a mass limit for a dead star comes from. The rung below this one hands that argument to the collection that owns stars, and this rung hands it over again; what belongs here is the exponent and the reason it changes.

What the straight line’s slope measures #

The heat capacity being linear in temperature is more than a confirmation. It is an instrument, and it is the standard one for a quantity that is otherwise very hard to get at.

Plotting a metal’s measured heat capacity divided by temperature against the square of the temperature gives a straight line: the lattice contributes the slope, going as $T^3$, and the electrons contribute the intercept. That intercept — the Sommerfeld coefficient — is proportional to the density of states at the Fermi energy, and it is measured on a lump of material in a cryostat with no beam, no probe and no theory of the band structure required.

For copper it is a fraction of a millijoule per mole per kelvin squared, and it agrees with the free-electron estimate to within a factor near one. Where it does not agree, the disagreement is the finding. Some compounds return values a thousand times larger, which by this argument means a density of states at the Fermi surface a thousand times larger, which means carriers with an effective mass of hundreds of electron masses — heavy fermions, so named for exactly this measurement. Nothing is heavy in any mechanical sense; a band has become extremely flat, and a flat band is a large density of states.

The chain of reasoning is worth appreciating for its length and its directness. A thermometer, a heater and a straight line give the number of available states at one particular energy in a solid, and the deviation of that number from a free-electron count is the whole quantitative content of the phrase “the interactions matter here”.

The same sea, made on purpose #

Everything in this essay concerns electrons, which come already degenerate and cannot be persuaded otherwise. It is now possible to build the same arrangement from scratch, with the parameters chosen, and watching it assemble makes the argument concrete in a way a piece of copper does not.

Take a cloud of neutral atoms of a fermionic isotope, trap them with light and magnetic fields, and cool them. The Fermi energy of such a cloud is set by the same expression as copper’s — the density and two constants — and at the densities a trap can hold, which are twenty orders of magnitude below a metal’s, the Fermi temperature is a fraction of a microkelvin rather than eighty thousand kelvin. Cool the cloud below that and the same step function appears.

What can be watched, that a metal will not show, is the crossing. Above the Fermi temperature the cloud behaves classically, its distribution is Maxwellian, and releasing it gives the expansion a classical gas gives. Below it the distribution becomes a rounded step, and the cloud released from the trap expands faster than a classical one at the same temperature would, because it retains the motion the exclusion principle forced on it. That excess expansion is the degeneracy pressure, seen directly, in a system whose temperature can be turned up and down at will.

The same figures apply, with $T/T_F$ pushed to a few per cent rather than the 0.4 per cent of copper at room temperature — which means these gases sit in the interesting middle of the crossover, where the step is visibly rounded, rather than at copper’s extreme where it is a step to four decimal places.

Where the model stops #

The electrons are treated as free. They are not: they move in the periodic potential of the ion cores, which turns the sphere of allowed momenta into a shape with flat faces and corners, and replaces the free-electron mass by an effective mass that can be a tenth of it or ten times it or negative. Band structure is the subject, and the surprising thing is how little of this essay it changes: the linear heat capacity, the density independence of the shape of the distribution, and the existence of a sharp Fermi surface all survive, with the numbers adjusted by factors of order one.

Where the states actually come from is worth a sentence, because the counting above used a box with walls and a metal has none. Bring identical wells together and each level splits into as many levels as there are wells, spread into a band whose width is set by how strongly neighbouring sites are coupled — so the number of states in a band is the number of atoms, which is the number the counting needed. The counting in this essay is right; the states it counts belong to a picture like that one rather than to a box.

The electrons are treated as non-interacting. They repel each other with a Coulomb energy comparable with their kinetic energy, and the fact that the free-electron picture works at all is a substantial piece of theory — Landau’s Fermi-liquid argument, which says that the interactions dress each electron into a quasiparticle with a modified mass and a finite lifetime, while leaving the sharp surface and the linear heat capacity intact.

The counting assumes a definite number of electrons per atom. For the alkali metals that is unambiguous — one loosely bound s electron, and the measured Fermi energies agree with the free-electron formula to a few per cent — and for the transition metals it is not. Copper is quoted here at one electron per atom, which is the standard assignment and which gets the right answer, and the d electrons a little below the Fermi level are ignored; in nickel or iron they are not ignorable at all, and the density of states at the Fermi energy is dominated by them. The valence used is an input, not a result.

And a metal is not the only degenerate thing. The same counting applied to nuclear matter gives a Fermi energy of tens of MeV and explains why a nucleus has a nearly constant density; applied to the electrons in a white dwarf it gives the mass limit; applied to neutrons it gives a stiffer version of the same. Each of those has its own physics on top, and none of them changes the counting.

There is one more limitation that belongs on this list because it is the sharpest test the picture has passed. Everything here is the equilibrium arrangement, and a metal is interesting mostly when a current flows, which is a departure from it. The remarkable fact is how small a departure a current is: passing a substantial current through a copper wire shifts the whole Fermi sphere in momentum space by a few centimetres per second against internal speeds of a million metres per second, a displacement of one part in ten million. Conduction is a very slight asymmetry imposed on an arrangement that is otherwise exactly what it is at absolute zero, which is why the resistivity of a metal is a property of its imperfections rather than of its electrons.

What the pictures cannot show #

Every figure here draws a distribution over energy, and none can show what an individual electron is doing. There is no such thing in this picture: the states are delocalised over the whole sample, an “electron at the Fermi surface” is a state rather than a particle at a place, and the drawing of a filled sea suggests a stack of objects where there is a set of occupation numbers.

The heat-capacity figure also hides the thing that makes the measurement possible. What is plotted is a ratio to the classical value, and at room temperature the electronic contribution is a per cent of a metal’s total heat capacity — undetectable against the lattice. The confirmation comes from the low-temperature regime, where the lattice term has fallen away as $T^3$, and no figure drawn on a linear temperature axis running to 1200 K can show it.

Where the ladder goes next #

This ladder began with no two in the same state and why matter has volume, where the exclusion principle was applied to one atom and to a hand pressing on a table. This rung does the same counting over a mole. The rungs above it: the Fermi surface as a shape rather than a sphere, measured directly by the de Haas–van Alphen effect; screening, where the sea rearranges itself around any inserted charge and cuts off the Coulomb interaction within an ångström; superconductivity, where a weak attraction near the surface pairs the fermions into bosons and the exclusion is evaded rather than broken; and the Fermi liquid proper, which explains why any of this survives the interactions.

The claim to carry forward is about what “cold” means. A temperature is not a statement about how fast things are moving; it is a statement about how the energy is distributed among the states available. Copper’s electrons move at millions of metres a second and are, thermodynamically, almost perfectly cold — and the whole behaviour of a metal follows from the fact that a room-temperature disturbance can reach 1.6 per cent of them.