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 2 essays across one field — 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

Named alongside it

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

Degeneracy pressureDensity of statesElectronEquipartitionExclusionFermi dirac distributionFermi liquidHeat capacityMean free pathMetalPauli exclusionQuantum statistics

All concepts