The strain the atoms do not follow

What symmetry decides about a material shows which components of a property tensor a crystal class allows to be non-zero. This essay is about a displacement rather than a tensor component, and the same machinery decides it — but the group doing the deciding is the site’s, not the crystal’s.

The rule, and the assumption inside it #

An elastic constant is a second derivative of energy with respect to strain, and computing one requires knowing where the atoms go when the crystal is strained. The Cauchy–Born rule answers that in the simplest possible way: apply the deformation to the cell vectors, and move every atom by the same linear map.

It is exact for a lattice with one atom in its cell, and the reason is not mechanics. Every site of a Bravais lattice is a centre of inversion; inversion survives any homogeneous strain, because a linear map commutes with multiplication by minus one; so the force on an atom at such a site is its own negative and is therefore zero. Nothing is left to relax, and the conclusion is reached before a potential has been named.

It is worth being exact about what the rule claims, because it is easy to read as a definition rather than as an assertion. A homogeneous deformation is a linear map applied to space; applied to a crystal it certainly carries the lattice vectors, since those are what define the cell, and the question is what it does to the atoms within the cell. The rule says: the same map. That is a hypothesis about a structure, not a consequence of the deformation being homogeneous, and it can be false while the deformation is perfectly homogeneous — the cell is uniformly strained and the contents are not uniformly carried.

That figure is the argument made falsifiable. A Bravais lattice normally has no internal coordinate at all, so “there is nothing to relax” is true for a trivial reason and proves nothing about the symmetry claim. Describing it on a doubled cell gives the second atom a coordinate it could move along, and it still does not move — which is the statement the inversion argument actually makes.

The classical form of the same observation is worth recording, since it is where this entered crystallography. For a crystal held together by central forces between atoms each of which sits at a centre of symmetry, the elastic constants satisfy relations Cauchy derived in the 1820s — in the cubic case, that two of the three independent constants are equal. Real crystals violate them, and the violation was read for a century as evidence that interatomic forces are not central. Part of it is that; part of it is this, because a structure whose atoms are not at inversion centres breaks the hypothesis before the forces are considered. Those relations are quoted here rather than computed, since the model below has one shear constant and not the three a cubic crystal has.

Where it fails, and what decides #

With more than one atom in a cell the sites need not be inversion centres, and then the affine positions are a guess. The atoms shuffle: the internal coordinate moves to whatever value minimises the energy at fixed cell, and the difference between the two is the internal strain.

Which strains cause a shuffle is not a question about the potential. It is the same question Neumann’s principle answers about tensors, asked of a vector and asked at a site rather than at the crystal.

The site of a honeycomb net has a three-fold axis and three mirrors through it. A strain keeps whichever of those it commutes with, and the shuffle — being a vector at that site — must be invariant under whatever is left. A three-fold axis fixes only the zero vector, so a strain preserving it forbids the shuffle outright. A single mirror fixes a line, so the shuffle is confined to that line. A strain that keeps nothing constrains nothing.

Those three predictions come from the group. The measurements are in the next column, and they agree.

It is worth noticing how weak the requirement is that does all the work. The shuffle is a vector attached to a site, and it must be left alone by every operation that fixes that site and survives the deformation. That is one line of representation theory — the identity representation appearing in the vector representation of the residual group — and it decides, for every structure and every deformation, whether a whole class of corrections is present or absent. No energy is evaluated to find out.

How the relaxation is done is worth a sentence, because a relaxation that stops is not the same as a relaxation that has finished. The internal coordinate is moved along the force with an adaptive step: the step grows while the energy falls and halves when it does not, so the search cannot overshoot into a rising region and stay there. What makes the answer trustworthy is not the method but the check afterwards — the residual force at the reported configuration is measured by central differences and required to be below one part in a million, so a search that merely ran out of step size fails rather than reports.

What it costs to get wrong #

A calculation that assumes the rule holds computes the energy at the affine positions. That is the clamped-ion value, and it is an upper bound rather than an answer.

The direction of the error is fixed and is easy to see: relaxing minimises over the internal coordinate as well as nothing, so it can only lower the energy, and it lowers it more at larger strain, which reduces the curvature. So a clamped constant is too stiff, always, and the excess is the internal strain contribution.

A third is not a correction. Reporting an elastic constant a third too large is reporting a different material, and the sign of the error means a calculation cannot be trusted to bracket the truth from the other side either. This is why relaxed-ion and clamped-ion elastic constants are reported separately in the literature and why the distinction is not a technicality.

The size of the softening in this model is worth reading off, because it is exact rather than approximate. The uniaxial clamped constant measures 0.37500 and the relaxed one 0.25000, a ratio of exactly two to three. That is a property of the honeycomb’s geometry and of nothing else: the internal coordinate couples to the strain through the three bond directions, and the fraction of the stiffness that coupling recovers is fixed by the angles between them. A different net would give a different fraction, and the fact that this one is a clean ratio is a sign the computation is doing arithmetic rather than accumulating error.

The linearity is what makes the correction usable. Because the shuffle is proportional to the strain at small strain, it is a fixed tensor contracted with the strain — the internal strain tensor — with as many independent components as the site symmetry allows, and those components are what a calculation stores. Its symmetry-allowed shape is decided by exactly the same reasoning as any other property tensor’s, so the machinery that twenty of twenty-one applies to piezoelectricity applies here without alteration.

One reason to care about the linearity beyond bookkeeping: it is what lets the correction be eliminated rather than merely estimated. Because the shuffle is linear in the strain, minimising the energy over it is a quadratic minimisation with a closed-form answer, so the relaxed elastic constants are the clamped ones minus a term built from the internal strain tensor and the curvature of the energy in the internal coordinate. That is how a real calculation does it — one extra matrix, computed once — rather than by re-relaxing at every strain, which is what the code here does because the code here is demonstrating the effect rather than computing efficiently.

Two softenings that look the same #

One row of the elastic-constant figure does something different from the others, and separating them is the most useful thing this essay does.

The uniaxial strains soften by a third and the shears soften to nothing. Both appear in a table as a difference between a clamped and a relaxed constant, and they are not the same phenomenon.

A honeycomb cell holds two atoms and three bonds, so four freedoms and three constraints, and one freedom survives that is not a rigid motion of the crystal. It is a mechanism of the bond network — the network is under-constrained — and a mechanism costs no energy, so the shear modulus of a honeycomb of nearest-neighbour springs is zero. A game that decides what counting only bounds is the general machinery for finding such freedoms and for knowing when the count that found this one can be trusted.

So the model is a spring network with no resistance to bending, and a real material of this connectivity is held up by the angles between its bonds rather than by their lengths. The essay’s finding is not that graphene has no shear modulus; it is that a table of softenings cannot distinguish a genuine internal-strain correction from a model that was floppy to begin with, and only counting the constraints tells them apart.

It is worth saying what the rule is for, since the criticism above could be read as a case against it. The Cauchy–Born rule is the bridge between an atomic description and a continuum one: it is what allows a strain energy density — a function of a deformation gradient, the object continuum elasticity is written in — to be defined from a potential between atoms at all. Without it there is no function to differentiate and no elastic constant to compute. So the rule is not an approximation somebody could do without; it is the definition of the quantity, and what this essay measures is the size of the correction that has to be added back once the definition has done its work.

What this says about a structure #

Three things, and the third is the one that generalises past elasticity.

The first is that the presence or absence of internal strain is decided by site symmetry and can be read off before any calculation. If every atom sits at an inversion centre, the rule is exact and clamped constants are the answer. If any does not, there is a correction, and its allowed form is a tensor question with a standard answer.

The second is about which structures are affected. A cell with one atom is safe; a cell with two atoms related by inversion is safe, since inversion then maps each to the other and the pair cannot displace relatively without breaking it; a cell whose atoms sit at general positions is not. So the exposure follows the Wyckoff positions a structure occupies, and a structure description already contains the answer.

The third is that the same argument governs any response, not only elastic ones. An applied electric field, a magnetic field, a temperature change: each is an external influence, each leaves some subgroup of the site symmetry intact, and each therefore permits or forbids an internal displacement by the same one-line criterion. The piezoelectric response of a multi-atom cell has a clamped-ion part and an internal-strain part for exactly this reason, and the plane a deformation leaves alone is the same distinction met in a transformation strain rather than in an elastic one.

The fourth of those is worth naming because it is about the method rather than the physics. A relaxation is a search, and a search that stops is not the same as a search that has found a minimum — so every relaxed configuration here is required to have a residual force below 10⁻⁶, which is a test the answer would fail if the step size had merely become small. Without it, a badly converged relaxation and a genuinely symmetric one report the same thing.

One more limit is worth naming, and it is the one the rule is usually criticised for. Even a Bravais lattice, where the argument above is exact, obeys the Cauchy–Born rule only while the strained lattice remains stable: at large enough strain a phonon at some non-zero wavevector goes soft, the crystal prefers a periodicity the strained cell does not contain, and the atoms leave the affine positions in a way no internal coordinate of that cell can describe. So the rule’s exactness for a Bravais lattice is a statement about small strains and about that cell. The count that promises a mechanism is the same boundary met from the constraint-counting side, where a freedom exists in a larger cell than the one being counted in — and the mechanisms a count cannot see is what it takes to find them.

One practical consequence deserves stating on its own, because it is where a reader is most likely to meet the distinction. A structure refinement reports atomic positions in fractional coordinates, and a fractional coordinate is precisely the internal coordinate discussed here — so a refinement of a strained crystal that holds the fractional coordinates fixed and varies only the cell has assumed the Cauchy–Born rule, whether or not anybody said so. For a structure whose atoms sit at inversion centres that assumption is free. For any other it is a constraint on the fit, and the residual it leaves is the internal strain the refinement was not allowed to find. The site symmetries in the structure’s own description say which case applies, which is the useful form of everything above: the answer is already in the table of positions.

Where this stops #

The model is two-dimensional, harmonic and nearest-neighbour, and none of those is a property of any material. What it demonstrates is the structure of the argument: which strains may move an atom is a group-theoretic question with an answer before any energy is evaluated, and by how much is a number that only a calculation gives.

The strains chosen are five and are not a basis. A full elastic tensor in the plane has three independent components for an isotropic-in-plane material and more for a lower symmetry, and what is computed here is a curvature along each of five deformation paths rather than the tensor itself. That is enough to demonstrate the effect and its sign, and it is not enough to report an elastic tensor — the two uniaxial rows differ in the fifth decimal place, which is the honeycomb’s three-fold symmetry making them equal and the numerical differencing failing to agree exactly.

The angles are absent, and that absence is visible in the results rather than hidden by them. A model with bond-bending terms would have a finite shear modulus and the shear rows would then show internal strain rather than a mechanism — which is the calculation to run next, and which changes the numbers without changing which strains are allowed to shuffle.

The site symmetries used are those of the undeformed structure, and that is a simplification worth naming. A large strain lowers the symmetry of the crystal, so the residual group at a site is smaller than the argument above assumes and more shuffles become allowed than it predicts. At the strains used here — a few per cent — the distinction does not appear, because a symmetry either survives a strain or does not and the surviving ones do so at any amplitude. What changes with amplitude is only whether the shuffle stays linear, and that is measured rather than assumed.

And nothing here is a lattice dynamics. The internal coordinate relaxed at fixed cell is the zone-centre optic mode’s coordinate, so the internal strain correction is the coupling between a strain and an optic phonon, and its size is set by that mode’s frequency. That connection is why a soft mode makes a crystal elastically soft as well, and it is a separate essay.