The collisions the exclusion principle forbids

No two in the same state stated the exclusion principle as a counting rule and found in it the reason matter has volume. The pressure that is not a temperature filled a box with electrons and found them moving fast at absolute zero, stacked into states up to a Fermi energy of several electronvolts, and pushing hard enough to hold up a star. The force with no force in it found that the same rule keeps identical fermions apart as if they repelled. This essay asks what the full sea does to motion through it — to collisions, and to everything that depends on them.

The puzzle it answers was sharp in the early days of the theory of metals. The free-electron model of a metal treats its conduction electrons as a gas that does not interact with itself at all, and it works: it explains conductivity, heat capacity, the thermoelectric effects. Yet the electrons are charged, one per atom, a quarter of a nanometre apart, and the Coulomb energy between neighbours is several electronvolts, comparable with their kinetic energy. A gas that dense and that strongly interacting should be all collisions. That it behaves as if there were none is not a property of the Coulomb force, which is as strong as ever. It is a property of the sea.

A sea full to a sharp edge #

Copper’s Fermi energy is seven electronvolts, the energy of a particle at eighty-one thousand kelvin. Room temperature is less than half a per cent of that, and the Fermi–Dirac distribution at that temperature is almost a step: every state more than a few $kT$ below the Fermi level is certainly occupied, every state more than a few $kT$ above it certainly empty, and only a thin shell in between is partly filled. The drawing shows the step at three temperatures, the lowest of them still ten times the relative temperature of copper at room temperature.

That shell is where everything happens. An electron deep in the sea has nowhere to go: every state it could move to by a small change of energy is already taken, and exclusion forbids it. Only electrons within about $kT$ of the edge have both an occupied state to leave and an empty one nearby to go to. The width of the shell, measured as the area under the product of the chance a state is full and the chance it is empty, is exactly $kT$ — the arithmetic that makes the electronic heat capacity of a metal some fifty times smaller than a classical gas’s, because only that fraction of the electrons can take up heat. Collisions are rationed by the same arithmetic, and more severely.

The triangle that survives #

Take an electron with a little energy ε above the Fermi level and let it collide with a partner from the sea. After the collision there are two electrons, which share between them the energy the two had before. Without exclusion, the partner could be any electron in the sea and the two products could land anywhere energy allows; the space of possible collisions is large, of the order of the Fermi energy squared.

Exclusion cuts it down in three ways at once, and the drawing shows each as a line in the plane of the partner’s energy and one product’s energy. The partner must come from below the Fermi level, since only there are there electrons to collide with. The first product must land above the Fermi level, since below it every state is full. And the second product, whose energy is fixed by conservation, must land above the Fermi level too. The three conditions leave a triangle: the partner no more than ε below the edge, the products sharing between them no more than ε above it. Its area is $\varepsilon^2/2$.

So the rate at which a particle collides grows as the square of its excess energy. Halve ε and the rate falls to a quarter; at the Fermi level itself it vanishes. A particle exactly at the edge of a filled sea at absolute zero never collides at all, whatever forces act between it and its neighbours. That is Lev Landau’s argument, and it is the foundation of his theory of Fermi liquids, from 1956: a system of strongly interacting fermions behaves, near its Fermi surface, like a gas of nearly free particles with long lives, because the interactions that would destroy them have almost no states to act through.

Warmth reopens a little #

At a finite temperature the edge is smeared over $kT$, and a few empty states appear just below it and a few occupied ones just above. A particle at the Fermi level can now collide, into the states the warmth has opened. The drawing sums every possible collision numerically — each partner weighted by the chance its state is full, each pair of products by the chance their states are empty — and compares the sum with Landau’s closed form. The rate grows as $\varepsilon^2 + (\pi kT)^2$, with a factor that approaches one when ε is large compared with $kT$ and a half at the Fermi level.

The consequence for a metal is a collision rate between electrons proportional to the square of the temperature. The rate an unblocked gas at the same density would have is of the order of the Fermi energy divided by Planck’s constant, about $10^{16}$ per second for copper. The blocked rate at room temperature is smaller by roughly the square of $kT$ over the Fermi energy, a factor of tens of thousands. Electron–electron scattering does contribute a term to the electrical resistance of metals growing as the square of the temperature, and in very pure metals at a few kelvin, and in some metals whose electrons are unusually heavy, that term is measured. In ordinary metals at room temperature it is swamped by scattering from lattice vibrations, which is the collision that actually limits a copper wire’s conductivity.

One power of temperature for heat, two for collisions #

The shell of width $kT$ explains why the electronic heat capacity of a metal is proportional to the temperature, and the triangle explains why the collision rate is proportional to its square, and the difference of one power is worth seeing clearly. Heating a metal needs an electron to move from an occupied state to an empty one — one electron, taken from the shell, placed in the shell. The number that can take part grows as $kT$. A collision needs three states in the shell at once: the partner’s, and the two products’. With the incoming particle itself near the edge, two of those are free to vary and each is confined to a range of about $kT$, which is where the square comes from.

The same shell underlies the thermoelectric coefficients a metal cannot choose freely, which are fixed by how the shell’s occupation is lopsided about the Fermi level, and the Wiedemann–Franz law, that a metal’s thermal and electrical conductivities stand in a fixed ratio, which holds because the same electrons in the same shell carry both. In every case the metal’s behaviour is set by a sliver of its electrons, and the thickness of that sliver is the temperature. Outside the shell, where the occupation follows the exponential that decides everything, the tail is so thin that its electrons hardly count.

How far an electron gets through the crowd #

The drawing turns the rate into a distance for copper, using the lifetime John Quinn and Richard Ferrell calculated in 1958 for an electron gas of copper’s density. An electron a full electronvolt above the Fermi level — a “hot” electron, of the kind light excites in a metal — survives about twenty femtoseconds and travels a few tens of nanometres before an electron–electron collision removes its energy. One within $kT$ of the edge at room temperature survives picoseconds and travels several micrometres, about fifteen thousand atomic spacings, through a crowd of electrons packed at one per atom.

Hot-electron lifetimes of this size have been measured directly since the 1990s, by striking a metal surface with one ultrashort laser pulse to excite electrons and a second, a few femtoseconds later, to eject them and measure how many are left. The measured lifetimes in the noble metals fall roughly as one over the square of the excess energy, as the triangle says, and are close to the electron-gas values, though the details depend on the metal’s band structure and on its bound d electrons. The hot electrons made by light in a metal nanoparticle, which are being explored for driving chemical reactions and for photodetectors, have a lifetime set by exactly this rationing, and it is why they must be harvested within tens of femtoseconds or not at all.

A liquid that thickens as it cools #

The most striking consequence is in liquid helium-3, whose atoms are fermions. Below about a tenth of a kelvin it is a degenerate Fermi liquid, and its properties follow Landau’s theory closely. Viscosity is the transport of momentum by particles travelling between collisions, and in a gas it grows with the distance they travel; a gas’s viscosity does not care how much gas there is and grows gently with temperature, as the square root, because faster molecules carry momentum faster. In a Fermi liquid the carriers are the quasiparticles near the Fermi surface, their speed is fixed at the Fermi speed, and their mean free path is set by the blocked collision rate — which falls as $T^2$. The viscosity therefore rises as $1/T^2$.

Helium-3 does exactly this. Cooled from a hundred millikelvin to ten, it becomes about a hundred times more viscous; cooled further, towards a few millikelvin, it becomes as viscous as a light oil. Its thermal conductivity follows the same logic and rises as $1/T$. Then, near a thousandth of a kelvin, pairs of helium-3 atoms bind and the liquid becomes superfluid, and the pairing that makes a liquid refuse to slow down takes over from the rationing that made it so thick. Its heavier cousin helium-4, whose atoms are bosons, has no Fermi sea to ration anything and behaves completely differently at the same temperatures: helium-3’s thickening in the cold is a direct, macroscopic reading of the exclusion principle.

The same helium-3 appears in the melting curve that has to arrive flat, where its liquid, being a degenerate Fermi liquid, has less entropy than its disordered solid below a third of a kelvin. The exclusion principle that orders the liquid’s spins at low temperature is the one that rations its collisions.

Nucleons that move as if alone #

The strangest application is inside the atomic nucleus. Protons and neutrons interact through the strong force, whose range is about the spacing between them and whose strength is enormous; a nucleon moving through a nucleus ought to collide within a fraction of its own size. Yet the most successful description of nuclei, the shell model, treats each nucleon as moving independently in an average potential, in orbits that persist long enough to have well-defined energies. Maria Goeppert Mayer and Hans Jensen’s shell model of 1949 worked far better than anyone had a right to expect.

The explanation is the triangle. A nucleus is a degenerate Fermi sea of nucleons, with a Fermi energy of about thirty-five million electronvolts. A nucleon near the top of the sea, in one of the orbits the shell model describes, can scatter only into empty states above the Fermi level, and for low excitation energies there are very few. The strong collisions that would scramble the orbits are mostly forbidden, and nucleons in the outer shells travel several femtometres — the size of a nucleus — between collisions. The shell structure of nuclei, the magic numbers, the ground-state spins: all rest on a mean free path made long by exclusion.

Bosons, which do the opposite #

The rule has a mirror image for bosons, and it is instructive how completely the mirror inverts it. A fermion cannot scatter into an occupied state; each collision’s rate carries a factor of one minus the occupation of every final state. A boson is more likely to scatter into an occupied state; the factor is one plus the occupation. Where fermions ration collisions, bosons encourage them into states that are already crowded.

That encouragement is stimulated emission, the reason a laser’s photons pile into one mode, and it is the reason a cooling gas of bosonic atoms, once a state is heavily occupied, funnels atoms into it faster and faster until a Bose–Einstein condensate forms. It is also the pile-up the exchange symmetry produces with no force at all, seen as a rate rather than as a correlation. The same collision, between the same kind of particles with the same force between them, is suppressed in a Fermi sea and amplified in a Bose gas, and the only difference is the sign in front of the occupation.

A star that conducts heat like copper #

The rationing reaches as far as the stars. A white dwarf’s interior is a degenerate electron sea with a Fermi energy of hundreds of thousands of electronvolts, the sea whose pressure holds the star up, at a temperature of a few million kelvin — a few per cent of the Fermi temperature or less. The electrons that carry heat through it collide rarely with one another, for exactly the reason drawn here, and scatter mainly off the nuclei; they travel far between collisions and conduct heat so well that the interior of a white dwarf is almost the same temperature throughout, like a lump of metal, wrapped in a thin, insulating non-degenerate envelope that controls how fast it cools. The cooling of white dwarfs, used to date the oldest stars in the Galaxy, depends on that envelope precisely because the degenerate interior offers almost no resistance.

Where the triangle stops being the answer #

Momentum as well as energy. The triangle counts energy only, with a constant density of states near the Fermi level. A real collision must also conserve momentum, which restricts the angles; in three dimensions this changes the prefactor but not the square law. In one dimension the momentum constraint changes everything, and interacting fermions in a wire are not a Fermi liquid at all but a different kind of state in which the elementary excitations are collective.

Weak coupling near the edge. Landau’s theory needs the quasiparticles to be well defined — to live longer than their own period — which the $\varepsilon^2$ law guarantees near the Fermi surface. Far from it, or in materials where the interactions are strong enough to change the ground state, the argument fails. A class of metals called strange metals, including the normal state of many high-temperature superconductors, has a resistance growing in proportion to the temperature rather than its square, and a collision rate of about $kT/\hbar$, as if nothing were being rationed at all. Why is one of the central open questions in the physics of solids.

Pairing. An attractive interaction near the Fermi surface, however weak, destroys the Fermi liquid at low enough temperature by binding pairs, as in superconductors and superfluid helium-3. The triangle describes the normal state above that transition.

What the rates do not show #

The rates drawn are for a particle’s energy to be redistributed by colliding with the sea. They say nothing about collisions with the lattice or impurities, which in most metals dominate the electrical resistance by a large factor, and nothing about collisions that transfer momentum without much change of energy. They also do not show the screening that makes the Coulomb force between electrons in a metal short-ranged: the sea rearranges itself round any charge within an ångström, which is part of why the collisions that do occur are gentler than the bare Coulomb force would suggest. Exclusion is the reason there are few collisions; screening is part of the reason each one is mild.

Still open: what the strange metals are doing #

Landau’s picture is so successful that its failures are the interesting cases. In the strange metals the resistance is linear in temperature over enormous ranges, from near absolute zero to hundreds of kelvin, and the scattering rate appears to sit at a universal-looking value set by $kT/\hbar$, the fastest rate at which a system at temperature $T$ can plausibly lose track of its momentum. Whether these metals have quasiparticles at all, what replaces the Fermi surface if not, and why the rate lands at that particular value, are not understood, and the strange metals are among the reasons the high-temperature superconductors that grow out of them are still not explained.

The habit worth carrying away is to ask where a particle can go before asking how hard it is pushed. A collision needs somewhere for its products to land, and in a filled Fermi sea almost every landing place is taken — so the strongest forces in physics can act inside a metal, a nucleus or a liquid of helium-3 while the particles they act on move as if nearly alone.