A mechanism that is a wave

The framework essays read this collection’s nets as frameworks of rigid bars and asked whether they could move. The rigidity matrix answered it exactly, over the rationals, with no tolerance anywhere — and it produced one result that was recorded honestly and left unexplained.

The kagome framework has one mechanism in a cell of one, two in a cell of two, three in a cell of three, while Maxwell’s count sits at one throughout. The note made at the time was that a mechanism count quoted without its cell is half a result. That is true and it is not an explanation. Here is the other half.

The matrix, with a phase in it #

A periodic framework’s rigidity matrix has one row per bar and one column per joint coordinate; the row for a bar says how the bar’s length changes when its two joints move. A motion of the framework is a vector the matrix sends to zero, and the number of independent ones is the number of columns less the rank.

That machinery asks about motions in which every cell does the same thing. A motion in which the cells do related things — the joints of the cell at R moving by $e^{2\pi i\,\mathbf{k}\cdot\mathbf{R}}$ times the joints of the home cell — is a wave, and the matrix for it is the same one with a phase on the far end of each bar: a bar reaching into the next cell picks up that cell’s factor.

So the rigidity matrix becomes complex and depends on k, and the question “can this framework move” splits into a question at every wavevector.

Where the arithmetic stays exact, and where it stops #

At a wavevector whose components are whole numbers or halves, each factor is ±1. The matrix is rational, the rank comes from the same exact row reduction the framework essays use, and the count of motions is a theorem about that wavevector.

Everywhere else the entries are genuinely complex, and the rank becomes a numerical judgement about how small a pivot may be before it counts as zero. That is a measurement, and the tolerance is reported with it rather than buried — the same division of labour as the levels in the reciprocal-space ladder, where the structure is exact and the values are measured.

The division matters because the interesting statement here is about a line of wavevectors, most of which are not half-integers. The exact computations pin the line at its rational points; the sweep says it is a line; and a reader is told which of the two produced each number.

The identity that closes the loop #

If mechanisms in a cell are waves at the wavevectors that cell can hold, then a supercell’s mechanisms should be the sum over its own wavevectors of the motions found at each. That is a statement with a definite answer and it is checked here rather than assumed:

$$\text{mechanisms of an } N\times N \text{ supercell} \;=\; \sum_{k} \bigl(\text{zero modes at } k\bigr) \; - \; 2$$

with the two subtracted being the rigid translations, which appear once at the centre of the zone and are not motions of anything.

The identity holds on five nets at three cell sizes each. The kagome row runs 1, 4, 7 as the cell grows, which is exactly the sequence the framework essays measured directly by row-reducing a larger and larger matrix — arrived at now from a completely different direction.

What the sum rule is really saying #

The identity above is arithmetic, and it is worth turning round into the statement it makes about descriptions.

A framework on an N × N supercell is the same framework as the one on a single cell. Nothing has been added: the same joints, the same bars, the same geometry, described against a larger unit of repetition. So a property of the framework must not depend on which description is used, and the mechanism count plainly does.

The resolution is that the mechanism count of a described framework is not a property of the framework alone. It is a property of the framework and the set of wavevectors the description can represent, which is exactly the N² characters of the supercell’s translation group. Counting mechanisms in a cell is sampling a function over the zone at the points that cell can hold.

That reading makes the growth unmysterious and it makes the right question available: not “how many mechanisms has this framework”, which has no answer, but “where in the zone are its mechanisms”, which has one and is drawn above. The same substitution — from a number that depends on the description to a function that does not — is what the box essay is about in general.

What a rigid unit mode is #

The motions this counts have a name in mineralogy. A framework silicate is a network of SiO₄ tetrahedra joined at their corners, and the tetrahedra are far stiffer than the corner joints, so the structure’s cheap motions are those in which each tetrahedron rotates as a rigid body and only the joints deform. Those are the rigid unit modes, and they are what a framework of bars and pin joints models.

They matter because they are the motions a crystal actually performs. Quartz’s transition between its two forms is a rigid unit mode frozen in; the negative thermal expansion of some zeolites is rigid unit modes being populated; a framework’s response to pressure is dominated by whichever of them is softest.

And they come with a wavevector. A rigid unit mode at the centre of the zone deforms every cell identically and is a change of the structure’s shape; one at a general wavevector is a modulation, and a structure that condenses into it acquires a superstructure whose cell is set by that wavevector. The modulation essays meet the same object from the diffraction side, where such a wavevector appears as the position of a satellite reflection.

Reading the picture #

The kagome framework’s marks lie along the lines joining the corners of the zone through the edge centres. Three such lines cross the zone, related by the three-fold rotation, and every wavevector on them carries a motion.

The count in a cell of N is then a count of how many points of those lines the cell can hold. A cell of one holds only the centre; a cell of two holds the centre and three edge centres, of which the ones on the lines contribute; a cell of three holds more. The sequence 1, 4, 7 is arithmetic about lattice points on lines rather than a property of the framework.

The two exact points, and what they pin #

A sweep of the zone with a numerical rank is a measurement, and a measurement of a rank is a particularly delicate one: a matrix near a drop in rank has a small singular value, and “small” needs a scale.

The half-integer wavevectors are the antidote. At the centre of the zone, at the three edge centres and at the corner, each factor is ±1, so the matrix is rational and the rank is computed by exact row reduction over ℚ — the same routine the framework essays use, with no tolerance in it at all.

Those exact points sit on the lines the sweep draws, and they pin them. If the numerical sweep had drawn a line that missed the corner, or found a mode at a half-integer wavevector where the exact computation says there is none, the disagreement would be visible immediately. It is a small check and it is the reason the picture can be trusted: a measurement anchored at points where an exact answer is available is a different kind of claim from one that is measured everywhere.

Why Maxwell’s count could not see it #

Maxwell’s count subtracts bars from joint coordinates and is a count for one cell. Doubling the cell doubles the joints and the bars together, so the count is unchanged — while the number of wavevectors the cell can hold has doubled, and if the motions live on a line the number found doubles with it.

That is the precise sense in which the count is half a result. It is right about the balance between constraints and freedoms, which is a local statement; it is silent about where in reciprocal space the freedoms are, which is what decides how the number behaves as the cell grows.

The same gap explains why Maxwell’s count can be wrong in the other direction. The ladder net has a state of self-stress and a mechanism the count cannot see, because the count knows only the difference between the two and not either separately. Rank-counting fixes that; wavevector-counting fixes this one; both are cases of a summary statistic being asked a question it does not contain.

The sharpest case is a framework whose mechanisms are neither a line nor a region but a single point. The looped net has two joints of degree six and six bars — four bars between the two joints and one from each joint back to itself, a bar that a translation carries onto the joint it started at — and its zone carries exactly one wavevector with a motion: the corner, where both components are a half. Maxwell’s number for it is the same whatever cell it is described on. The mechanism count is zero in a cell of one, one in a cell of two, and zero again in a cell of three — not merely growing at a rate the count cannot predict, but not growing at all, because a cell of three holds no wavevector with half-integer components. A summary that returns one number for all three descriptions cannot be describing that.

What condensing into a mode does to a cell #

A framework with a mechanism at a general wavevector has a motion that is not periodic with the cell it was described on. If the structure follows that motion — a phase transition, in the ordinary case — the result is periodic with a larger cell, and how much larger is decided by the wavevector’s denominator.

A mode at the corner of the zone, with coordinates in halves, doubles the cell in both directions. A mode at a wavevector of thirds triples it. A mode at an irrational wavevector produces a structure with no cell at all, which is an incommensurate modulation and needs a fourth dimension to describe.

So the picture above is also a map of the superstructures the framework can adopt, and the lines say that the kagome framework has a continuum of them rather than a handful. A crystal built on that net has, in principle, a soft direction at every point of a line, and which one it chooses is decided by energetics this collection does not compute.

Whether those marks really are a line, rather than a scatter that happens to look like one at the resolution drawn, is a question the sweep can be asked directly: draw it again on a grid four times as dense and see whether the marks fill in along the same directions or spread out around them. They fill in.

A flat level and a line of mechanisms #

The previous essay found the kagome net carrying a state of an entirely different kind: a wave localised on one hexagon, exactly, giving a level that does not move with the wavevector.

The two facts are relatives. Both come from the alternating pattern round a hexagon — signs, in the first case; senses of rotation, in this one — and both die outside the ring by the same cancellation at the triangles. A flat level is a family of localised states; a line of mechanisms is a family of localised motions. In each case the localisation is why the wavevector does not matter, and in each case the count is a count of hexagons rather than of cells.

That two such different questions — where a symmetric operator’s levels sit, whether a framework of bars can move — reduce to the same statement about a cycle in a graph is the strongest evidence this collection has for the thesis of the nets essays: a great deal survives when the distances are thrown away, and what survives is more than anybody would guess.

What is claimed, and what a mineralogist would add #

The claims here are about frameworks of bars and pin joints, which is a mathematical object rather than a mineral. A rigid unit mode analysis of an actual silicate has more in it: the tetrahedra have a size, the bridging angles have preferred values, and the softness of a mode is a number rather than a yes or no.

What transfers is the counting. Whether a mode exists at a wavevector, how the number found depends on the cell, and how a supercell’s count relates to a sweep of the zone — all of that is decided by the connectivity and the geometry of the bars, exactly, and it is what a real analysis starts from before any energy is computed.

What does not transfer is any statement about which mode a crystal chooses, how far it goes, or what temperature it does so at. Those are questions with a free energy in them, and this collection has no free energy anywhere.

The exact statement Maxwell’s count is an approximation to #

The count of joint coordinates less bars is a guess at a difference, and there is an identity that makes it exact. Stating it says precisely what the count can and cannot see, at every wavevector separately.

The rigidity matrix has a null space on one side and a null space on the other. The right null space is the mechanisms — motions of the joints that stretch no bar. The left null space is the states of self-stress — assignments of tension to the bars that exert no net force on any joint. Rank–nullity applied to the same matrix twice gives

$$\text{mechanisms} - \text{self-stresses} = (\text{joint coordinates}) - (\text{bars}),$$

exactly, with no conditions on the framework whatever. That is Calladine’s correction of 1978, and the right-hand side is Maxwell’s count.

So Maxwell’s number is a difference and not a count. A framework with one mechanism and no self-stress and a framework with four mechanisms and three self-stresses have the same Maxwell number, and nothing about that number distinguishes them. The kagome framework is the second kind: its extra mechanisms at general wavevectors are matched, term for term, by states of self-stress it also has.

The identity holds at each wavevector separately once the matrix carries its phases, which is what makes the sweep on this page a measurement of two quantities rather than one. Where a mark appears, the rank has dropped — and the drop shows up in both null spaces at once, so a line of mechanisms in the zone is also a line of self-stresses. That pairing is the reason the sum rule over a supercell’s wavevectors closes: both sides of the identity are being summed over the same set of k.

What a rigid unit mode does to a mineral #

The motions counted here have consequences a mineralogist measures, and two of them are worth naming because they are the reason the counting is done at all.

Displacive phase transitions happen at wavevectors where a mode exists. A framework that can move at some k has a soft direction there, and cooling a crystal whose framework has such a mode very often produces a transition in which that mode freezes in — giving a superstructure whose cell is set by the wavevector, exactly as the section above describes. Which superstructures are available is therefore readable off the picture before any thermodynamics is done, and a transition to a cell the picture does not permit is evidence that the framework model is wrong.

Negative thermal expansion is the other. A framework whose units can rock without stretching has motions that pull the structure inward as their amplitude grows, because a rotation shortens the distance between the far ends of two linked units. Materials with many such modes contract on heating over some range, and the ones that do are the ones the counting here would pick out — corner-linked frameworks with more mechanisms than Maxwell’s number admits.

Neither of those is derived here, and both are the kind of statement the counting makes available rather than establishes. What the arithmetic supplies is the list of wavevectors at which something can happen; which of them a material uses, and at what temperature, is a measurement.

Where this goes #

The rigidity anchor now has the instrument it was missing, and the natural next question is which nets have lines of modes rather than points: a census over the nets, sorted by where in the zone their motions live, would say how common a kagome-like framework is. That is more than one essay’s worth of work, and it is recorded in the plan as such.

The nearer neighbour is the census of nets, which asks a related question about the same objects — how special the nets this collection draws are, among the nets nobody chose — and finds an answer that is uncomfortable for anyone who has been looking only at the pretty ones.