The two laws absolute zero will not allow

The melting curve that has to arrive flat read the third law off a phase diagram. The slope of every coexistence line is a difference in entropy over a difference in volume, the entropy difference vanishes at absolute zero, and so every such line must meet the temperature axis lying flat. That argument used one identity, Clausius and Clapeyron’s, and one kind of line.

The same move works on every derivative of the entropy there is, and the results are more surprising than the phase diagram. Thermodynamics has a set of identities, the Maxwell relations, that turn the rate at which entropy changes with pressure, volume or magnetic field into a quantity that can be measured with a ruler, a pressure gauge or a magnetometer. If the entropy at absolute zero is a constant, the same for every pressure and every field, then each of those measurable quantities has to go to zero there as well. Two of the best-known laws in physics fail that test outright. The ideal-gas law and Curie’s law for a paramagnet both give responses that grow without limit as the temperature falls, and the third law says both must break down before absolute zero. It says so without specifying what replaces them.

A ruler that measures an entropy #

The identity at the centre of this essay comes from the Gibbs free energy, $G(T,P)$, whose two partial derivatives are $-S$ and $V$. Because the mixed second derivatives of a smooth function are equal,

$$\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P = -V\alpha ,$$

where $\alpha$ is the volume thermal expansion coefficient. The left side is a statement about disorder: how much the number of available arrangements shrinks when the material is squeezed at fixed temperature. The right side is a statement about a ruler: how much the material grows when it is warmed at fixed pressure. They are the same number.

Nernst’s heat theorem, which is the third law in its original form, says that as the temperature goes to zero the entropy approaches a value that does not depend on pressure, field or any other parameter. A law about spectra identified that value as $k \ln g$, with $g$ the degeneracy of the ground state. For the purposes of this essay only one consequence matters. A limit that does not depend on pressure has a pressure derivative of zero, so the thermal expansion coefficient of every material must vanish at absolute zero. The magnetic version of the same identity, from a free energy whose derivatives are $-S$ and $-M$, reads

$$\left(\frac{\partial S}{\partial B}\right)_T = \left(\frac{\partial M}{\partial T}\right)_B ,$$

and it says that the temperature slope of every magnetisation must vanish too. Neither conclusion needs to know anything about atoms.

The gas law that has to fail #

For an ideal classical gas at constant pressure $V \propto T$, so $\alpha = 1/T$ exactly. Cool it and the expansion coefficient grows in inverse proportion; at a millikelvin it would be a thousand per kelvin. By the Maxwell relation, the entropy’s sensitivity to pressure would then grow in the same way, which the third law forbids. So the ideal-gas law is not merely inaccurate at low temperature. It is inconsistent with thermodynamics there, and something must take its place.

What takes its place is quantum statistics. The figure plots the product $\alpha T$, which for the classical gas is one at every temperature, against the temperature measured in units of the Fermi temperature — the temperature at which the thermal energy equals the energy of the highest filled state in a gas of fermions at that density. The solid curve is the same gas of particles treated as fermions, with the exclusion principle obeyed, computed exactly from the Fermi–Dirac integrals at each temperature.

Far above the Fermi temperature the two curves coincide: the particles are so spread through their available states that the exclusion principle has nothing to exclude. Near the Fermi temperature they part company. At a tenth of it the Fermi gas’s product is 0.048 instead of one, and at a hundredth it is 0.002, falling as $(\pi^2/2)(T/T_F)^2$. So its expansion coefficient itself falls in proportion to the temperature and vanishes at absolute zero, exactly as the Maxwell relation requires.

The mechanism is visible in the physics of the pressure that is not a temperature. A cold Fermi gas has a pressure set by how its particles are stacked into states, not by how fast they are moving. Warming it moves only the particles near the top of the stack, a fraction $T/T_F$ of the total. So the pressure, and with it the volume at fixed pressure, responds to warming only through that thin shell of particles at the Fermi surface, and the response dies away with the shell.

The same would be true, by a different route, of a gas of bosons, which condenses into its ground state instead of stacking. The third law does not care which kind of statistics a gas obeys. It demands only that some statistics other than the classical kind take over before absolute zero.

An entropy that stops being a count #

The classical gas’s expansion coefficient is only the visible symptom. The deeper failure is in its entropy, given by the Sackur–Tetrode formula, which counts each particle’s available positions and momenta in cells of phase space of size $h^3$ and divides by the number of ways of labelling identical particles.

The figure plots both entropies against the same temperature axis. When hot they agree. As the gas cools, the Sackur–Tetrode entropy falls as three halves of the logarithm of the temperature, because each particle’s thermal wavelength grows and the number of cells of phase space it can occupy shrinks. At 0.156 of the Fermi temperature it crosses zero, and below that it is negative.

A negative entropy is not a small error. Entropy is $k$ times the logarithm of a number of arrangements, as entropy is a count establishes, and a negative value means fewer than one arrangement: the formula is counting cells of phase space smaller than a single particle’s share. What has happened physically is that the particles’ wavelengths have grown to overlap, so they can no longer be treated as distinguishable points in separate cells. Once they overlap, whether they are fermions or bosons decides how they are counted, and the Fermi gas’s entropy in the figure is the correct count. It falls linearly with temperature and reaches zero at zero, because a sea of fermions filled to a sharp surface has exactly one arrangement.

Nernst saw this before quantum statistics existed. In 1914 he argued that his heat theorem required gases to depart from the ideal law at low temperature, a departure he called gas degeneracy, although no known gas could be cooled far enough to show it before liquefying. The explanation came a decade later, from Fermi and from Bose and Einstein, and the word survives: a degenerate gas is still the name for one cold enough that the third law has forced its statistics on it. The ordinary gases never reach that point in practice, because they liquefy first, and the correction for them is the first correction to the gas law from the forces between molecules. The gases that do reach it are the electrons in a metal, whose Fermi temperature is tens of thousands of kelvin, and the atoms cooled in magnetic traps.

The magnetisation that has to level off #

Curie’s law says that a paramagnet’s magnetisation in a field $B$ is $M = CB/T$. It describes independent magnetic moments pointing randomly, with a small bias towards the field that grows as the temperature falls, and at room temperature it is excellent. Its temperature slope is $-CB/T^2$. The magnetic Maxwell relation equates that slope to the rate at which the entropy changes with field, and the third law requires that rate to vanish at absolute zero. Curie’s slope diverges instead.

For moments that are genuinely independent, the repair is already inside the theory Curie’s law comes from. The figure draws the exact magnetisation of spin-half moments in a field, the hyperbolic tangent of $\mu B/kT$, beside Curie’s law. At high temperature they agree, to within one per cent above $5.8\,\mu B/k$, because the tangent of a small argument is the argument. Cooled, the true magnetisation bends over and saturates at full alignment, since a moment cannot be more than completely aligned. Its slope, and therefore the field derivative of the entropy, falls exponentially: at a tenth of $\mu B/k$ it is less than a millionth of full alignment per unit of temperature.

So for moments in a field the third law is obeyed as soon as the field has been allowed to do its work. The field splits each moment’s two states by $2\mu B$, the ground state is unique — every moment aligned — and the entropy falls to zero. This is the physics the staircase that never reaches the floor exploited for cooling: magnetise at one temperature, isolate, demagnetise.

Spins that must find an order #

The field was doing essential work in that argument, and without it the argument fails. Remove the field and the independent moments have two states of exactly equal energy each, which is a degeneracy of $2^N$ and a residual entropy of $R\ln 2$ per mole that no amount of cooling removes. That is a violation of the third law, and it means independent moments at zero field cannot exist in nature at arbitrarily low temperature.

The figure shows what happens instead. Any interaction between the moments, however weak, splits the $2^N$-fold degeneracy, and the ground state becomes some particular arrangement: all parallel in a ferromagnet, alternating in an antiferromagnet, or frozen at random in a spin glass. The figure uses the simplest description of the ordering, mean-field theory, for couplings that set the ordering at 1 K, 1 mK and 1 μK. In each case the entropy stays at $R\ln 2$ until the temperature approaches the coupling’s scale and then falls to zero over a factor of a few below it.

The consequence is a statement about every paramagnet. Curie’s law has to fail somewhere, and where it fails is set by the weakest thing that couples the moments to each other. What holds a magnet together is a strong exchange interaction, and iron orders at 1,043 K. Salts designed for magnetic cooling dilute their magnetic ions so that the coupling between them is only the weak dipolar field one moment makes at the next; cerium magnesium nitrate, the classic example, stays paramagnetic to about two millikelvin. Nuclear moments are a thousand times weaker still, and the nuclear spins of copper, cooled in a separate experiment to below a microkelvin, order antiferromagnetically at about 60 nanokelvin. At each stage the third law guaranteed that some order would be found, and the question was only at what temperature. The same argument explains the residual entropy of ice, which the entropy that is still there at zero counts: there the ordering that would remove it is kinetically blocked, so the law’s escape clause, that the system must actually reach its ground state, is doing the work.

A solid that stops growing #

Solids obey the third law without anyone noticing, and the thermal expansion shows how. A perfectly harmonic crystal does not expand at all, a point why heating a perfect spring changes nothing makes in full. The expansion comes from the anharmonic part of the forces between atoms, which makes each vibrational mode’s frequency depend on the volume. Grüneisen packaged that dependence into one number per mode, and the result is a relation between two measured quantities:

$$\alpha = \frac{\gamma\, C_V}{B\, V},$$

with $\alpha$ the volume coefficient, $\gamma$ the Grüneisen parameter, near two for most solids, $B$ the bulk modulus, and $C_V$ and $V$ the heat capacity and volume of a mole. The linear coefficient a length gauge reads is a third of it. Either way, the thermal expansion is proportional to the heat capacity.

The figure applies it to copper, with a Grüneisen parameter of two for both contributions. At room temperature it gives $1.61\times10^{-5}$ per kelvin, within three per cent of the measured value. As the metal cools, the lattice vibrations freeze out in the way how many ways there are to vibrate describes, the lattice heat capacity falls as $T^3$ below about thirty kelvin, and the lattice part of the expansion falls with it. The conduction electrons are a degenerate Fermi gas, and their heat capacity, and so their contribution to the expansion, falls only linearly. Below 3.8 K the electrons dominate. At one kelvin the whole coefficient is $5.1\times10^{-10}$ per kelvin, thirty-two thousand times smaller than at room temperature, and it is still falling.

This is where the Maxwell relation stops being an abstraction. Any solid that expands on warming has an entropy that depends on pressure, and the size of the expansion coefficient is a direct reading of that dependence. The Grüneisen relation is the microscopic reason the reading goes to zero: the expansion is carried by the same vibrations and electrons that carry the heat capacity, and the third law’s staircase already required the heat capacity to vanish. The practical consequence is that a precision instrument cooled to a few kelvin is dimensionally stable in a way no instrument at room temperature can be, which is why cryogenic optical cavities hold the frequencies of the best lasers.

What the argument does not decide #

The Maxwell-relation argument says what must vanish and nothing about how. Three limits are worth stating.

It says nothing about the approach. The expansion coefficient could vanish as $T$, as $T^3$, exponentially, or as some other power, and the third law is satisfied by all of them. The figures supply the approach from particular models: the Fermi gas’s linear law, the Debye crystal’s cube, the gapped paramagnet’s exponential. A material that violated all of them but still went to zero would be no problem for thermodynamics.

It says nothing about the temperature at which the old law fails. The classical gas fails near its Fermi or Bose temperature, which depends on density and mass; the Curie law fails near the ordering temperature, which depends on the coupling. The third law requires only that the failure happen somewhere above zero. For a paramagnet whose coupling is tiny, Curie’s law may hold over every temperature ever reached in a laboratory, and there is no contradiction until the unreached region is entered.

It relies on equilibrium. A glass, a frozen spin arrangement, or ice with its hydrogen disorder can hold entropy at the lowest temperatures reached because it never gets to its ground state in the time available. Its response coefficients can then behave in ways the argument would forbid for an equilibrium system. The third law is a statement about equilibrium states, and the region where equilibrium takes longer than an experiment is the region where it is hardest to test.

Still open: whether a spin liquid escapes the argument #

The ordering argument above assumes the coupled spins find an ordered ground state. Some lattices frustrate that search: on a triangle of antiferromagnetically coupled spins, no arrangement satisfies all three bonds at once, and on certain lattices built from triangles the number of equally good arrangements grows with the size of the system. Classical models of such lattices have a residual entropy, as ice does. Quantum models are expected instead to settle into a quantum spin liquid, a ground state that is unique or nearly so but has no ordered pattern, with its entropy removed by entanglement rather than by order.

Several materials have been proposed as spin liquids, among them herbertsmithite, with copper ions on a kagome lattice, and certain organic salts. The evidence is indirect, and impurities, weak additional couplings and disorder make every candidate contested. The third law says that each of them must lose its entropy somehow as the temperature falls; how a spin liquid does so without ordering, and whether any real material is one, is a question about quantum ground states rather than about thermodynamics, and it is not settled.

The habit worth carrying away is to read every response coefficient as a derivative of the entropy. A Maxwell relation turns a question about disorder into a question about a ruler or a magnetometer, and the third law turns every such question into a requirement that the answer vanish at absolute zero. A law that gives a response growing as the temperature falls — the ideal gas’s expansion, Curie’s magnetisation — is thereby announcing the temperature range in which it will fail, and the physics that replaces it is whatever supplies the order the entropy needs.