Crystallography & Symmetry
patterns derived from their groups
Concept

Residual entropy — where it appears

The entropy a crystal keeps as its temperature approaches zero, because it has frozen among many arrangements rather than settled into one. It is the logarithm of the number of arrangements, and in ice it is measured by calorimetry.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Arrangements per site, falling towards the exact value. The number of configurations of an L × L torus obeying the ice rule, taken to the power of one over the number of sites. The largest computed here is 4,484,823,396 configurations on a 7 × 7 torus. The values fall towards Lieb's exact 1.54 from above, and every one of them is above Pauling's estimate of 1.5 — which undercounts, because it treats the vertices as independent and they are not.

The arrangements a crystal keeps at absolute zero

Ice has a residual entropy, and the number a calorimeter measures is the logarithm of a count of arrangements. Pauling's one-line estimate of that count is out by two and a half per cent; the exact count in two dimensions is available, falls towards its limit from above, and the whole disorder is invisible to a diffraction experiment, which sees only the average.

aperiodic · Entropy Square ice scatters a pinch at the origin. The intensity scattered by the horizontal arrows of square ice, averaged over 100 configurations on a 32 by 32 torus, over the whole Brillouin zone with the origin at the centre; darker is more intense. Along the horizontal axis through the origin the intensity falls to zero — 2.9e-32 at the smallest wavevector — while along the vertical axis it stays near 1.75, so the two meet at the origin in a pinch.

The ice rule is a conservation law

Two arrows in and two out at every vertex is a statement that nothing flows in or out anywhere. That makes one half of the arrow field vanish identically, in every arrangement and not merely on average, and what is left scatters with a pinch at the origin: an intensity that approaches different values from different directions. Break the rule now and then and the pinch acquires a width Debye and Hückel predicted for a salt solution.

aperiodic · Entropy Every arrangement on a torus 4 across, sorted by defects. The transfer matrix that counts ice arrangements chooses, at each vertex, the one horizontal arrow the rule permits. Enumerating both choices instead and carrying a polynomial that records how many vertices end up with three arrows in or three out gives the number of arrangements at every defect count at once. The first column, drawn solid, is the ice count — 2970 arrangements with no defect at all, which is the number the earlier transfer matrix gives and is checked against it. The second column is empty: no arrangement has exactly one defective vertex, because a defect carries a charge and the charges on a closed surface must cancel. The columns together add to two raised to the number of edges, which is every assignment of arrows whatever.

What a defect costs the count

Each broken vertex relaxes the rule and so adds arrangements — the question left standing was whether each adds a fixed amount or the cloud around it costs some back. The exact count at every defect number at once answers both halves: almost all of the rise is the freedom to choose which vertices break, and with that removed the first defects subtract rather than add.

aperiodic · Entropy One rule, one lattice, two entropies. The number of arrangements per vertex for square ice, counted two ways on the same lattice with the same rule. On a torus the count falls towards Lieb's exact value of 1.5396 from above. Inside a domain wall — every arrow on the top and bottom edges pointing in, every arrow on the left and right pointing out — the count rises towards 3√3/4, which is 1.2990, from below. A residual entropy is supposed to be a bulk quantity that forgets the boundary; these two differ by sixteen per cent and the only difference between them is the boundary.

The count that depends on the edge

A residual entropy is supposed to be a bulk number: so much per vertex, whatever surrounds the lattice. Square ice has two of them. On a torus the count per vertex heads for 1.5396 and inside a domain wall it heads for 1.2990, with the same rule on the same lattice — and the sixteen per cent between them is sitting in the corners.

aperiodic · Entropy A colouring, and the arrows it writes. A proper three-colouring of the cells of a four-by-four torus — no two cells sharing an edge carry the same colour — with an arrow drawn on each shared edge by the difference of the two colours it separates. The difference is one or two modulo three, never nought, so every edge gets a direction. At each corner four cells meet and their four differences go round a cycle and add to nothing modulo three, which forces two of the arrows in and two out. That is the ice rule, arrived at from a colouring with no arrows in its statement.

Three colours on a chessboard

Colour the cells of a board in three colours so that no two sharing an edge agree. The number of ways is the number of ice arrangements on the same board — the same integer, to the last digit, at every even size — so a residual entropy a calorimeter reads is also the answer to a colouring problem with no physics in it at all. At odd sizes the two counts part company, and why they do is a condition on going round.

aperiodic · Entropy

Named alongside it

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

EntropyLocal rulesCountingCensusEnumerationHeight functionTransfer matrixDisorderConfigurational entropyCorrelation functionDefectDiffuse scattering

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