The staircase that never reaches the floor

Beside the first two laws, Nernst’s statement of the third is a sentence about entropy at zero temperature, and read on its own it sounds like bookkeeping — a convention fixing an additive constant that nothing measurable depends on. Its content is entirely about cooling — about what a heat engine can and cannot reach read from the other end, and the way to see that is to stop reading it and draw it.

The quarter is the whole essay. Each stage takes the same fraction, not the same amount, so the steps shrink in exact proportion to what is left — and a sequence like that has infinitely many terms before it reaches zero.

The contrast worth holding on to is with almost every other limit in the subject. A vacuum pump removes a fixed fraction of the gas per stroke too, and yet a vacuum system reaches its base pressure in minutes, because the fraction is small and the target is a pressure that outgassing sets rather than nothing. Here the target really is nothing, and there is no outgassing to stop short at. What stops a real machine turns out to be something else entirely, and it arrives long before the arithmetic runs out.

Why the ratio is fixed #

The refrigerant is a mole of magnetic moments that do not talk to each other, which is the same independence a gas of molecules has. In a field B at temperature T each has two states split by 2μB, and the whole of its thermodynamics is a function of one variable, x = μB/kT:

$$\frac{S}{Nk} = \ln(2\cosh x) - x\tanh x.$$

That is ln 2 when the field is irrelevant — both states equally likely, one bit per spin — and zero when the field dominates, because every spin is then in the same state and there is nothing left to be uncertain about — entropy as a count, run down to one.

The crucial property is not the shape of that function. It is that its argument is a ratio. Nothing in the expression depends on B and T separately, so any two states with the same B/T have the same entropy — and a move at constant entropy is a move at constant B/T.

So the ratio is not a property of the machine or of how carefully it is run. It is a property of the refrigerant’s own thermodynamics, and it is the same at the first stage and the thousandth.

It also explains why the technique is worth using at all. The vertical step — the isothermal magnetisation — dumps heat into a bath at the starting temperature, which is a bath that already exists. Everything gained afterwards is gained without any colder reservoir anywhere, which is the property no ordinary refrigerator has: a compression cycle needs somewhere to reject heat that is colder than the thing it is cooling from, and this one does not, because the rejection happens before the cooling rather than after it.

The arithmetic of never arriving #

The geometric sequence deserves its numbers written out, because “infinitely many steps” is a statement people hear as “hopelessly many” and it is nothing of the kind.

Reaching a stated temperature $T$ from a start at $T_0$ takes

$$n = \frac{\ln(T_0/T)}{\ln(B_{\text{high}}/B_{\text{low}})}$$

stages, which is a logarithm of the factor wanted divided by a logarithm of the field ratio available. At a field ratio of four, getting from a kelvin to a millikelvin is five stages; to a microkelvin, ten; to a nanokelvin, fifteen. Every further factor of a thousand costs the same five, for ever, and the cost of the last factor of a thousand is identical to the cost of the first. Nothing gets harder as the temperature falls, in this model, which is exactly why the model cannot be the whole story.

So the unattainability is a strange kind of limit. It forbids nothing anybody wants: no experiment has ever needed absolute zero, and every finite temperature is a small number of stages away. What it forbids is the end of the sequence — and the reason that matters is not practical but structural, because a machine that could reach zero could be run as a Carnot engine against it and turn heat entirely into work.

There is a distinction inside the statement that is worth keeping, because the two halves are not the same theorem. That no finite number of steps reaches zero is what the figures here show, and it follows from the curves meeting. That no finite time reaches zero is a stronger claim, and it needs something the entropy diagram does not contain: how long a stage takes, which depends on how fast heat can be moved and how slowly the field must be changed to stay reversible. Both are true. Only the first is drawn.

The step that is forbidden, and why #

Nothing so far has used the third law. It has used one particular refrigerant whose entropy happens to depend on B/T, and a sceptic is entitled to ask what stops someone finding a substance whose two curves cross the axis at different points. One adiabatic move from the upper curve would then land on the lower one at exactly zero, and the whole argument collapses.

There is a second way to say the same thing, and it makes the geometry obvious. A cooling stage is useful only if the vertical drop it can make at temperature T is bigger than the vertical distance between the two curves at the temperature it is aiming for. As both curves fall toward the same point, that available drop shrinks — and it shrinks in step with the gap it is trying to cross. Two curves converging on one point cannot be joined by a horizontal line that ends on the axis, for the same reason two roads that meet at a junction cannot be joined by a shortcut that arrives before the junction.

The third law, in the form that does this work, says that the entropy change of any isothermal process tends to zero as the temperature does. Raise a field, change a pressure, alter a composition: whatever the process, the entropy it moves shrinks to nothing at the bottom of the scale. That is exactly the statement that the two curves converge, and it is what makes the vertical steps shrink faster than the horizontal ones can exploit them.

The reason they converge is a count. Entropy counts arrangements, and at the bottom of the scale a system has one arrangement available, or a small fixed number of them. A count that has run down to one is a count no process can lower — and it is the same count whether the field is on or off, so the difference between the two curves has to vanish along with the number of arrangements it was made of.

The genuine exception, and why it does not help. Some systems really do keep entropy at zero temperature, in the way a mixture keeps the entropy of its mixing. Ice has a residual 3.4 joules per kelvin per mole, from the number of ways the hydrogens can be arranged consistently with the ice rules, and carbon monoxide keeps about R ln 2 from the two ways a nearly symmetric molecule can lie in a lattice. But a residue that is present in both curves cancels out of the difference, and the difference is what a cooling step uses. A frozen-in degeneracy is a constant, not a resource.

The history the argument needs #

Nernst arrived at this in 1906 from chemistry rather than from cryogenics: he was trying to predict which reactions would go, and the free energy that decides that depends on an entropy difference whose value nobody could pin down. His “heat theorem” was the observation that the entropy difference between reactants and products appeared to vanish as the temperature did, so the free energy and the heat of reaction converge at the bottom of the scale and one measurement fixes both.

The unattainability form came later and from a different direction — it is the thermodynamic consequence of Nernst’s chemical statement, and it is the one that can be tested with a machine rather than with a calorimeter. Adiabatic demagnetisation was proposed independently by Debye and by Giauque in 1926, and Giauque and MacDougall ran it in 1933, taking gadolinium sulphate to 0.25 K. Every stage since has been the same figure with a different salt.

And one thing was settled by the experiment rather than by the argument. Whether a paramagnetic salt’s entropy really is a function of B/T down to millikelvin temperatures is not obvious — it requires the moments to stay independent — and the answer is that it is, until the moments start to feel each other, which is the floor drawn below.

What the same statement does to every engine #

The unattainability argument has a twin that looks quite different.

Carnot’s ceiling is the rung this one grew out of, and it is worth reading from the wrong end. Efficiency is $1 - T_{\text{cold}}/T_{\text{hot}}$, so an engine rejecting heat at absolute zero would have an efficiency of exactly one — heat converted entirely into work, with nothing left over. The second law does not forbid that. It forbids it only if such a reservoir cannot be made, which is precisely what the third law supplies. The two laws lean on each other, and neither is complete without the other.

The same convergence has a third consequence, and it is the one that is measured most often.

The heat capacity has to vanish at the bottom, and the argument is one line: $S(T)$ is the integral of $C/T$ from zero, so a heat capacity that stayed finite would make that integral diverge and the entropy would be minus infinity at the bottom of the scale. Every degree of freedom must therefore freeze out. Every one of them does — hydrogen’s rotations at about 85 K and its vibrations at 6,300 K, each when the spacing of its levels has grown large compared with $kT$.

A metal meets the same requirement with no gap at all, which is the sharper case. Its conduction electrons have states available at every energy, so nothing freezes out in the sense above — and the heat capacity still goes to zero, linearly in $T$, because the number of electrons within $kT$ of the Fermi surface is itself proportional to $T$. Two utterly different mechanisms, one requirement, and the requirement is the third law rather than anything about either material.

What stops a real machine, and it is not the arithmetic #

The staircase drawn above continues for ever, and a real salt does not.

That floor is a property of the salt, and it is why a machine that wants to go lower changes refrigerant rather than repeating a stage. Cerium magnesium nitrate orders in the millikelvin range and was the standard salt for reaching it; going below that means demagnetising nuclear moments instead, whose magnetic moments are about two thousand times smaller and whose ordering temperatures are correspondingly in the microkelvin and nanokelvin range. The same staircase, a different rail.

There is one more scale in a real machine that the model has nothing to say about, and it is often the operative one: the link between the salt and the thing being cooled. Below about a millikelvin the thermal conductance of any solid contact falls steeply — the phonons that carry heat across a boundary have wavelengths comparable with the sample by then, and the boundary resistance grows as the inverse cube of the temperature. A sample can be attached to a refrigerator that is genuinely at a microkelvin and stay at a millikelvin indefinitely, because nothing is getting through.

And the practical limit is usually neither. An engine that has to finish is the nearest relative of this: a demagnetisation stage is a one-shot device: it has a fixed entropy budget, and whatever heat leaks in is charged against it. The hold time at the bottom is the budget divided by the leak, so a machine reaching a millikelvin holds it for hours and a machine reaching a microkelvin holds it for minutes. The temperature is reached; what is scarce is the time.

How much heat a millikelvin will absorb #

The reason a demagnetisation stage is a one-shot device is arithmetic, and it is worth doing, because it explains the whole architecture of a low-temperature laboratory.

The heat a sample can take up at temperature $T$ while its entropy rises by $\Delta S$ is $T\Delta S$, and $T$ is now very small. A mole of spins carrying the full $R\ln 2$ has 5.76 joules per kelvin available, which sounds generous and is worth 5.8 millijoules at a millikelvin and 5.8 microjoules at a microkelvin. Against a heat leak of a microwatt — a realistic figure for a well-made cryostat, and a very small amount of power by any other standard — that is a hold of an hour and a half at the first temperature and six seconds at the second.

Everything peculiar about the field follows from that line. The cooling capacity falls in proportion to the temperature while the leaks do not, so the experiment is arranged around the budget rather than the temperature: the sample is small, every wire into it is thin, and the measurement is designed to be finished before the entropy runs out. And the argument in the previous section — that every further factor of a thousand costs the same five stages — is now visibly not the whole cost, since each of those factors divides the hold time by a thousand as well.

It also gives the honest reason a real machine changes salt rather than repeating a stage. It is not only that the refrigerant orders; it is that a refrigerant already near its ordering temperature has almost no entropy left to spend, so even the stages before the floor return less and less useful capacity as they approach it.

The same effect, four hundred kelvin higher #

The magnetocaloric effect is not a cryogenic curiosity, and looking at it where it is large is a good test of which parts of the argument above were essential.

Gadolinium near room temperature warms by a few kelvin when a field of one tesla is applied and cools by the same amount when it is removed, and refrigerators have been built on that cycle. The mechanism is the one drawn here — aligning moments lowers their entropy, and if there is nowhere for the entropy to go the lattice supplies it and the temperature moves. What is different is that the effect is largest right at the Curie temperature, where the moments are on the edge of ordering themselves.

Which is where the scaling this essay leans on fails. Near a transition the moments are not independent, so the entropy is not a function of $B/T$ alone, and the neat statement that a demagnetisation multiplies the temperature by the field ratio simply does not hold. The paramagnetic salt used at a millikelvin obeys the scaling because it is far above its own ordering temperature; gadolinium at 293 K does not, because it is sitting on it. Both are the same effect and only one of them makes a staircase.

Where the picture stops #

“Constant entropy” is a demand, not a description. Every horizontal move above is a reversible adiabatic. A real demagnetisation is done at a finite rate against eddy currents and a finite thermal link, and any irreversibility puts entropy back in — so the treads are shallower than drawn, by an amount that depends on how slowly the field was lowered.

Underneath all of it is one exponential. The population of an excited state falls as $e^{-E/kT}$, so at a fixed level splitting the excited population dies away exponentially in $1/T$ rather than as a power of it. That is why the entropy of a gapped system falls so steeply at the bottom of the scale, and why two curves taken at two different fields close on each other rather than merely converging slowly.

The refrigerant is assumed to be in equilibrium with itself. A demagnetisation cools the spins, and the spins then have to cool everything else — the lattice they sit in, the container, the sample. Below a kelvin the coupling between a spin system and a lattice becomes slow enough to matter, and the two can sit at genuinely different temperatures for minutes. What the staircase computes is the spin temperature, and it is the lattice temperature that a thermometer eventually reports.

Nothing here measures a temperature. Below a few millikelvin a thermometer is its own research problem: a resistance thermometer self-heats, a gas thermometer has no gas, and what is used instead is usually the very susceptibility this essay has been computing, which is the magnetisation a paramagnet gives — Curie’s law, χ ∝ 1/T, read as a thermometer. That is a pleasing circularity to be aware of rather than to be troubled by: the law is checked against a scale defined some other way at higher temperatures and then extrapolated, and the extrapolation is exactly what fails at the ordering temperature the figure above marks.

And the two-level model is a caricature that happens to be exact for the argument. Real ions have several levels, split by the crystal field before any coil is switched on, and their entropy curves have structure the two-level form does not. None of that changes the conclusion, because the conclusion depends on one property — that the entropy is a function of B/T — and that property follows from the level splittings being proportional to B, which they are.

One last thing about what is being pumped. Entropy is extensive: twice the sample has twice as much of it to remove, and the same number of stages removes the same fraction. So the argument is untouched by the size of the apparatus, which is unusual. Most limits in thermodynamics can be pushed back by building something bigger, and this one cannot be pushed back at all.

Where this ladder goes next #

This rung has treated the third law as a constraint on cooling. The next asks what it constrains at the bottom once the cooling has stopped — what a system at a temperature where its entropy is genuinely near zero actually looks like, and why the residual entropies that do survive are always small integers’ worth of logarithms rather than arbitrary numbers. Ice’s 3.4 J/K/mol is not a measurement of disorder in general; it is a count of hydrogen arrangements, and Pauling’s estimate of that count from the ice rules alone gets it to within one per cent.