The arrangements a crystal keeps at absolute zero

A structure determination reports one arrangement of atoms per cell. Ice cannot be reported that way, and the reason is not a failure of the experiment.

Each oxygen in ordinary ice has four hydrogen bonds. Exactly two of the four protons sit close to it — those are the covalent bonds of the water molecule it belongs to — and every bond carries exactly one proton, close to one end or the other. That is all. Nothing in it says which two, and the number of arrangements consistent with it is enormous.

And the number is measurable, which is what makes it a fact about ice rather than about arithmetic. Boltzmann’s entropy is the logarithm of a count of arrangements; a calorimeter measures entropy; so a count made on paper can be compared with a thermometer.

Pauling’s line #

Linus Pauling wrote the estimate down in 1935 and it takes three sentences.

Ignore the rule. Each of the 2N bonds has two positions for its proton, so there are 2²ᴺ = 4ᴺ arrangements. Now put the rule back one site at a time: of the sixteen ways the four bonds at an oxygen can be occupied, six obey it, so each site survives with probability 6/16. Treat the sites as independent and

$$W \approx 4^N \times (6/16)^N = (3/2)^N.$$

An entropy of R ln(3/2) = 3.37 joules per kelvin per mole. The measured residual entropy of ice, from Giauque and Stout’s calorimetry in 1936, is about 3.4. A count of arrangements agreeing with a thermometer to one per cent, from a calculation that fits in a paragraph.

The agreement hides an assumption doing a great deal of work, and it is exactly the assumption the exact count is able to test: the sites are not independent, since a bond is shared and each site constrains its neighbours.

The same rule where it can be counted exactly #

In two dimensions the rule has a lattice of its own. Square ice puts an arrow on every edge of a square lattice and requires two in and two out at every vertex — the same combinatorial statement with four bonds per site — and on a torus the configurations can be counted exactly.

The count is done twice, by methods sharing nothing.

Brute force. Every edge of the torus is given an arrow and the rule is checked at every vertex. That is two to the power of twice the number of sites, so it stops at three by three, which is 262,144 assignments and 148 legal ones.

A transfer matrix. The state carried from one row to the next is the direction of each vertical edge crossing it, so there are 2^L states; the matrix entry is the number of ways to fill one row consistently with the arrows above and below; and the count on the torus is the trace of the L-th power, because the torus closes.

They agree at every size where both can run, and the second reaches a seven by seven torus: 4,484,823,396 configurations, exactly, on forty-nine sites.

What the exact answer says about the estimate #

Two things, and the second is the one worth carrying away.

The independence assumption undercounts. Correlations between neighbouring vertices leave more arrangements than the estimate allows, not fewer, and the exact answer is 2.6 per cent above it. That is the direction that is not obvious: constraints usually cost, and here the accounting of them was double-counting.

A per-site growth read off a small box is an overestimate. Every finite torus in the ladder above sits above the limit, because a small torus has short loops in it that a large one does not, and the deficit falls slowly. Reporting the L = 3 value of 1.74 as an estimate of the limit would be out by thirteen per cent — worse than Pauling’s line, from a computation a thousand times larger.

Lieb’s exact value for two-dimensional square ice, $(4/3)^{3/2}$, came in 1967 from the Bethe ansatz, and it is one of the few models in this subject with a closed-form answer. Nothing in this collection derives it; it is quoted, and used as the destination the finite counts are measured against.

Why the transfer matrix can be trusted #

The two counting methods are worth comparing as methods, since only one of them survives past nine sites.

Brute force is transparent and hopeless: it visits every assignment of arrows and checks every vertex, so its cost is the number of assignments, and the number of assignments is exponential in twice the area. At L = 3 that is a quarter of a million; at L = 5 it would be a million million million.

The transfer matrix is opaque and cheap. Its cost is the number of states squared times the number of rows — exponential in the width rather than in the area — and the trace at the end is what closes the torus. Everything about the arrangement except the row currently being crossed is forgotten, which is legitimate precisely because the rule is local: what a row does next depends on the arrows crossing it and on nothing else.

The reason to run both is that the second is easy to get wrong in a way that produces a plausible number. An off-by-one in the wrap-around, a state indexed the wrong way up, or a boundary handled as a line rather than a circle each give a count that is too large or too small by a factor nobody would notice. Agreement with an enumeration that reads the rule directly is the only cheap check there is, and it is available exactly where the enumeration can still run.

What a diffraction experiment sees #

Here is where the disorder meets the rest of this collection, and the answer is uncomfortable.

A crystal with this many arrangements still diffracts to sharp spots. Sharpness comes from the lattice, and the lattice is untouched — the oxygens are on it, perfectly, in every arrangement. What the experiment measures is an average over some 10²⁰ cells, and the average of all the arrangements is half a proton at each of the two positions on every bond.

That average is a perfectly good periodic structure. It has a symmetry, and the symmetry is higher than any arrangement’s.

A half-occupied site is a statement about a population, and it is worth being clear that it is not a statement about a proton. No proton is in two places, none is halfway along its bond, and nothing about any individual cell is fractional. What is fractional is the fraction of the crystal’s cells in which that position is taken, and the experiment measures a fraction of cells because the beam illuminates all of them at once. A structure report is a census, and a census of a disordered population returns a number no member of it has.

So the reported structure is a structure no cell has, and the fractional occupancies in it are not a fitting compromise but the correct answer to the question the experiment asks. This is the symmetry of an average in its purest form: there, several orientations of a molecule average to something more symmetric than any of them; here, an unlistable number of arrangements do.

The practical consequence is that the disorder has to be found somewhere other than in the Bragg peaks. It shows in diffuse scattering — intensity between the spots, which is where all the information about correlations goes — and the shape of that diffuse intensity — which is where what a measurement cannot separate is decided — is what distinguishes a structure obeying the ice rule from one whose protons are simply random.

What the count is not #

Three limits, and the first is a boundary this collection keeps deliberately.

No thermodynamics is done here. Entropy appears only as the logarithm of a count of arrangements, with every arrangement weighted equally. No temperature is introduced, no free energy is written down, and nothing is derived about the transition ice does or does not make at low temperature. The calorimetric number is quoted as a measurement made by others.

Equal weighting is a choice. It is the right one for a rule — the rule prefers nothing — and it is not right for a material at a temperature, where arrangements differ slightly in energy and the differences are what the low-temperature behaviour of real ice is about.

And two dimensions is not three. Square ice is the ice rule on a lattice with four bonds per site and it is not ice. Pauling’s estimate has the same form in both, which is why it is a useful comparison; Lieb’s exact value is two-dimensional and has no three-dimensional counterpart in closed form. The best estimates for real ice come from series expansions and simulation, and they sit about 1.4 per cent above Pauling’s number — smaller than the two-dimensional correction, in the direction the two-dimensional calculation predicts.

The entropy per site, and what it means for a mole #

The number 1.5 is small and the consequences are not, which is worth doing arithmetic on once.

A mole of ice has 6 × 10²³ oxygens, so the number of arrangements is (3/2) raised to that power — a number with about a hundred thousand million million million digits. Its logarithm times Boltzmann’s constant is 3.4 joules per kelvin per mole, which is a perfectly ordinary laboratory quantity, and that is the whole of why an unimaginable count is measurable: the logarithm is the observable, and the logarithm of an unimaginable number is small.

The same arithmetic says how much better a rule would have to be to matter. A rule permitting 1.0001 arrangements per site — which sounds like almost no freedom at all — still allows more arrangements of a mole than there are atoms in the observable universe, and it would show up in a calorimeter as 0.0008 joules per kelvin per mole, far below anything measurable. Between a rule that decides the structure and a rule a calorimeter can catch being undecided, there is an enormous range where a structure determination is reporting one arrangement out of many and nothing whatever says so.

Where else the same count turns up #

The six-vertex model is one of the small number of exactly solvable objects in two dimensions, and it is not confined to water.

Spin ice. In pyrochlore magnets, whose magnetic symmetry is a group with time reversal in it, such as holmium titanate, the magnetic moments sit on a lattice of corner-sharing tetrahedra and obey a two-in-two-out rule of exactly the same form. Their residual entropy has been measured and comes out near Pauling’s value — a magnetic system reproducing a number computed for a hydrogen-bond network, because the combinatorics is the same and the physics is not.

Alternating sign matrices. As with the Aztec diamond’s power of two, the configurations of square ice with particular boundary conditions are in bijection with a class of matrices whose count is a product formula proved in the 1990s, which is the same kind of coincidence as the Aztec diamond’s power of two and turned out to be the same subject.

And ferroelectrics. Slater’s model of potassium dihydrogen phosphate is the six-vertex model with energies attached to the six vertices, which is where the model came from historically — the counting problem is the zero-energy case of an interacting one.

The rule as a covering, which is the previous rung #

The ice rule and the dimer rule are the same shape of statement with a different number in it. A dimer covering says one of the bonds at each site is occupied; the ice rule says two of the four are. Both are conditions on a site’s neighbourhood and nothing else, both leave exponentially many arrangements, and both are counted by carrying a boundary from one row to the next.

What differs is the number the growth comes to, and the difference is instructive: 1.3385 per site for dimers on the square lattice, 1.5396 for square ice. The looser rule permits more, as it must — but the two are the same order of magnitude, and neither is anywhere near the 2 that free choice at every site would give. A local rule with any content at all costs most of the freedom and leaves an exponential amount of it, which is the shape of every disorder problem in this subject.

The two rules also differ in how many independent routes their counts have, and that is a fact about the mathematics rather than about the materials. The dimer count on a rectangle has three: an enumeration, Kasteleyn’s determinant, and the product of cosines that diagonalises it. The ice count on a torus has two — an enumeration and a transfer matrix — because no product formula of that kind exists for the six-vertex model on a torus, and Lieb’s closed form is a statement about the infinite lattice rather than a formula for a finite one. Where a third route exists it is worth having; where it does not, the two that remain have to be independent enough to be worth running both, which is why the brute-force check here reads the rule directly at every vertex rather than sharing any machinery with the matrix.

Who counted it #

Bernal and Fowler wrote the rules down in 1933, from the geometry of the water molecule and the shape of the hydrogen bond. Pauling turned them into a count in 1935 and compared it with Giauque and Stout’s calorimetry, which had been published as an anomaly nobody could explain. Lieb solved the two-dimensional case exactly in 1967, and Nagle produced the accurate three-dimensional series estimates in 1966.

The order is worth noticing. The measurement came first and was a puzzle; the count explained it; and the exact solution arrived thirty years later and moved the number by two and a half per cent. A one-line estimate that survives a thirty-year correction of that size is a good estimate, and knowing exactly how good it is takes the harder computation.

It is also a case where the estimate’s error and the experiment’s error were the same size, which made the agreement look better than it was. Pauling’s 3.37 against a calorimetric 3.4 is a match to one per cent, and the exact two-dimensional correction is two and a half per cent — larger than the apparent discrepancy. So the 1935 agreement was partly luck: had the calorimetry been ten times more precise, it would have shown the estimate to be low, and the correction would have been demanded three decades before the mathematics could supply it. That is a common shape and worth recognising rather than admiring. An agreement is evidence in proportion to how precisely both sides are known, and quoting a match without quoting the two uncertainties is a claim with the interesting part removed.

What a residual entropy is not a violation of #

A count of arrangements surviving at absolute zero sounds like a contradiction of a law, and the resolution is worth stating because it says what the measured number is a measurement of.

The third law says that the entropy of a system in internal equilibrium tends to zero as the temperature does. Ice is not in internal equilibrium at low temperature: rearranging the protons means breaking and reforming hydrogen bonds, the rate at which that happens falls off exponentially as the crystal cools, and by a few tens of kelvin it has effectively stopped. The crystal is frozen into whichever arrangement it happened to have.

So the residual entropy is a measurement of a freezing, not of a ground state. What a calorimeter reports is the entropy the crystal failed to shed, and the count on this page is the count of arrangements it was choosing among when the choosing stopped.

There is a genuine ordered arrangement underneath. Ordinary ice has a low-temperature ordered phase in which the protons take one particular arrangement and the entropy really is zero, and it is reachable — the transformation is catalysed by a dopant that lets the protons move at temperatures where they otherwise could not. The ordered phase exists, the third law holds, and the residual entropy is the cost of not getting there.

That is a useful distinction to carry into every other disordered structure. A measured residual entropy says a system fell out of equilibrium and how many arrangements it fell among; it says nothing about whether an ordered arrangement exists, and finding one is a separate experiment.

Where the disorder does show #

The essay says the disorder has to be found somewhere other than in the Bragg peaks, and the somewhere has a shape that is itself a consequence of the rule.

The scattering from a disordered crystal is the sharp part — the Bragg peaks of the average structure — plus a diffuse part carrying everything the average threw away. The diffuse part is the transform of the correlations: how the arrangement at one site is related to the arrangement at another.

For a rule of this kind the correlations have an unusually distinctive form. The ice rule is a conservation law — two in and two out at every site is a statement that something has no source anywhere — and a conserved quantity with no source has correlations that fall off as a power of distance rather than exponentially, with a shape borrowed from electrostatics.

In reciprocal space that produces sharp features at particular points where the diffuse intensity is not smooth: it approaches different values from different directions, giving a bow-tie or pinch shape that no ordinary short-range correlation produces. Those features are the direct signature of the rule, they have been measured in spin ice by neutron scattering, and they are what distinguishes an arrangement obeying a local constraint from one that is merely random.

So the count and the correlations are two different things the same rule produces, and only the second is visible in an experiment. A count says how much freedom the rule leaves; the diffuse pattern says what the rule is.

Where the ladder goes next #

The count is the simplest measure of a disordered structure and the least informative. What it does not say is anything about correlations — which arrangements are near which, how the disorder is spatially organised, and what the diffuse scattering actually looks like. Those are the questions the next rungs of this anchor are for, and they are measurements rather than counts.

The other direction runs back through this collection’s own ground. What diffraction cannot tell apart is the general form of the difficulty here: an experiment measuring an average cannot distinguish arrangements that share one, and no improvement in the instrument changes that. Ice is the case where the number of arrangements sharing the average is not two or three but a number with 10²⁰ digits.