Illustrated Physics
physics, drawn
Concept

Paramagnetism — where it appears

The weak attraction of a material into a magnetic field because its atoms carry magnetic moments that partly line up with the field. Thermal motion fights the alignment, so the magnetisation falls as the inverse of temperature, which is Curie's law, and vanishes when the field is removed.

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

The field a classical partition function cannot see. The momentum plane of one classical charge, in units of the root-mean-square thermal momentum. With no field the Boltzmann weight is a set of circles about the origin; with a field the same circles sit about p = qA, displaced and otherwise unaltered, because the energy depends on p only through p − qA. Integrating over the whole plane therefore cannot notice the displacement, and the numbers beside the drawing are that integral evaluated at 6 displacements: the largest departure from the zero-field value is 3.3e-16, which is the precision of the arithmetic and not a physical effect. The classical free energy has no B in it, so the classical magnetisation is exactly zero at every field and every temperature — no paramagnetism, no diamagnetism, no ferromagnetism. Every magnetic material is therefore evidence of something classical mechanics does not contain.

The magnetism classical physics forbids

Write down the partition function of any collection of classical charges in a magnetic field, and the field cancels out. Not approximately, not to leading order — the integral is over all of momentum space and the field only shifts where the middle of it is. So classical statistical mechanics predicts no paramagnetism, no diamagnetism and no ferromagnetism, and a compass needle is a quantum instrument.

electromagnetism · Magnetisation A heat capacity that goes up and comes down. The heat capacity per system, in units of Boltzmann's constant, against temperature in units of the level spacing over k: for a system with two levels (red), Schottky's hump x²eˣ/(1 + eˣ)², x = Δ/kT, and for a harmonic oscillator with the same spacing (blue), which climbs to one k and stays there. Both are frozen out when kT is well below the gap. The oscillator then fills its unending stack of levels and reaches equipartition's value. The two-level system peaks at 0.439k when kT = 0.417Δ, and then, with both levels nearly equally occupied and nothing higher to go to, it stops absorbing heat: its capacity falls back towards zero as 1/T².

The heat capacity that goes up and comes down

Equipartition promises every way a system can move a fixed share of heat, and for a harmonic oscillator the promise is kept: once the temperature is high enough, it holds one k per oscillator and goes on holding it. A system with only two levels cannot keep it. It absorbs heat while its upper level is filling, then stops — the two levels equally full and nowhere higher to go — so its heat capacity rises to a peak and falls back to zero. The hump, named after Walter Schottky, is the signature of a degree of freedom with a ceiling, and the area under it counts how many states were there.

thermodynamics · Equipartition

Named alongside it

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

The Boltzmann factorAngular momentumBohr magnetonDebye modelDiamagnetismEntropyEquipartitionFree energyHeat capacityMagnetisationPartition functionPhase space

All concepts