Illustrated Physics
physics, drawn
Concept

Fermi energy — where it appears

The energy of the highest occupied state in a system of fermions at absolute zero, filled from the bottom one particle per state. For the conduction electrons of a metal it is several electronvolts, far above the thermal energy, which is why only electrons near it take part in heat and conduction.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

quantum · Exclusion The triangle of collisions that survive. Every collision of a particle ε above a filled Fermi sea with a partner from the sea, at zero temperature: the partner's energy across, one product's energy up, both from the Fermi level, the other product's fixed by energy conservation. Without exclusion every point in the plotted square would be an allowed collision. With it, the partner must come from below the Fermi level, and both products must land above it: the partner no deeper than ε below, the first product above zero, the second product above zero too. What survives is the shaded triangle, whose area is half of ε squared, and the collision rate is proportional to it. Halve the particle's excess energy and the triangle shrinks to a quarter; at the Fermi level itself it vanishes, and a particle there cannot scatter at all.

The collisions the exclusion principle forbids

A copper wire holds one free electron per atom, packed a quarter of a nanometre apart and repelling one another with the full Coulomb force. It seems impossible that an electron could travel any distance through that crowd without colliding. It travels micrometres — thousands of atoms — before another electron deflects it. The collisions are not weak. They are forbidden: to collide, both electrons need empty states to move into, and in a sea filled to the brim almost none exist. The same rule makes cold helium-3 thicken as the square of how cold it gets.

quantum · Exclusion The proton fraction a cold star settles at. The fraction of nucleons that are protons in cold matter in beta equilibrium, where the neutron's chemical potential equals the proton's plus the electron's, against density on logarithmic axes, for free neutrons, protons and electrons with no nuclear forces. Below 1.22·10⁷ g/cm³ the electrons cannot pay the 1.29 MeV difference between a neutron and a proton and the matter is all protons. Above it the fraction collapses, to 0.014 per cent at its lowest near 7.9·10¹¹ g/cm³, and at nuclear density, 2.66·10¹⁴ g/cm³, it is 0.49 per cent. At higher densities it climbs slowly towards one ninth, 11 per cent, which it never reaches. The dashed part lies where real matter is made of nuclei rather than free nucleons.

The protons a star cannot afford

A free neutron decays in fifteen minutes into a proton, an electron and an antineutrino. Inside a neutron star it cannot, and the star is made of neutrons because of it. Beta decay is a reaction like any other, and at equilibrium the neutron's chemical potential must equal the proton's plus the electron's. In a crowded star an electron can only be added at the top of a sea tens of megaelectronvolts deep. So the matter settles where protons are rare: half a per cent of the nucleons at nuclear density, if nucleons were free, and never more than one in nine at any density.

thermodynamics · Chemical potential Which atoms in a Fermi sea are allowed to take a photon's kick. A slice through the occupied momentum states of a gas of identical fermions at zero temperature — a filled disc whose radius is the Fermi momentum — with each state marked by whether it can absorb a recoil of 0.8 of the Fermi momentum to the right, the kick a scattered photon gives. A state can scatter only if its kicked momentum lands outside the sea, in an empty state. The kicked sea is the second circle; the atoms in the overlap of the two (grey) land on occupied states and cannot scatter, and only those in the crescent (blue) can. In this slice 50 per cent of the states are allowed; over the whole sphere the allowed share is S(q) = 3q/4 − q³/16 = 0.568. A kick larger than the sea's diameter, twice the Fermi momentum, is never blocked, which is why the effect needs a sea whose Fermi momentum is comparable with the photon's.

The light a full sea will not scatter

An atom that scatters a photon recoils, and the recoil has to put it into a new state of motion. In a cold, dense gas of identical fermions most of the states it could recoil into are already taken, and the exclusion principle forbids the scattering outright. Nothing about the atom has changed — not its electrons, not its transition, not its cross-section — yet the gas becomes dimmer than the same number of atoms scattered one at a time. The effect was predicted in the 1990s and seen in 2021, and it turns transparency into a measurement of how full a quantum gas is.

quantum · Exclusion The cloud that stops shrinking. The rms size of a cloud of 10⁵ ideal atoms in a harmonic trap, in units of the Fermi radius, against temperature in units of the Fermi temperature, for fermions, for bosons and for a classical gas. The classical cloud shrinks as the square root of half the temperature ratio, all the way to nothing. The bosons follow it down to their condensation temperature, 0.52 of the Fermi temperature, and then collapse into the trap's ground state, whose rms size is 0.054 Fermi radii. The fermions part from the classical line near the Fermi temperature and level off at one over the square root of eight, 0.354 of the Fermi radius: at a quarter of the Fermi temperature they are 1.17 times the classical size, and cooling further barely changes them. Mixed together in one trap, as lithium-7 and lithium-6 were in 2001, the two clouds start the same size and end 6.5 times apart in width for ideal atoms of these numbers — further still in volume. The fermions are held up by nothing but the rule that no two of them may share a state.

The cloud that stops shrinking

Two kinds of lithium atom, cooled together in one magnetic trap to a few hundred billionths of a degree, start as clouds of the same size and end very differently. The lithium-7 atoms, which are bosons, shrink with the temperature and then collapse into a condensate. The lithium-6 atoms, which are fermions, stop shrinking at about a quarter of their Fermi temperature and refuse to get any smaller, however much colder they become. Nothing pushes them apart but the rule that no two identical fermions may share a state. The photographs taken in 2001 were the first time Fermi pressure — the pressure that holds up white dwarfs — was seen as a picture.

quantum · Exclusion The middle of an atom gets smaller as it gets heavier. The radius holding half an atom's electrons, in the Thomas–Fermi model, against atomic number: 0.41 Å at Z = 10 and 0.20 Å at Z = 90, shrinking with charge. Marked above are van der Waals radii of the noble gases, which measure where an atom's outside begins and which grow from 1.54 to 2.2 Å over a ninefold increase in electrons. An atom ninety times heavier is about the same size because the crowd inside is drawn in as the crowd outside is added.

The atom that does not grow with its electrons

A uranium atom has ninety-two electrons and hydrogen one, and uranium is not ninety-two times the volume or anything like it. An atom's middle shrinks as Z to the minus a third while its outside hardly moves, so that nearly all atoms are about the same size. A rough equation from 1927, with no shells and no periodic table in it, explains that, and turns out to be exact in the limit nobody can build.

quantum · Atomic structure

Named alongside it

The objects these essays reach for when they reach for this one.

Pauli exclusionAtomic radiusBeta equilibriumBose einstein condensateChemical potentialDegeneracy pressureDegenerate fermi gasDegenerate gasDegenerate matterDensity of statesElectronElectron capture

All concepts