What a texture permits

A sonar transducer, an ultrasound probe and the buzzer in a greetings card are all the same material, and it is not a crystal. It is a ceramic: a few billion crystallites of lead zirconate titanate, sintered together, each one sitting at whatever orientation it happened to grow in. There is no cell. There is no lattice. Nothing in the specimen is periodic at any distance larger than a grain, and a grain is a fraction of a micron.

Such a thing has no crystal class, so Neumann’s principle — that a property must be invariant under every operation of the symmetry group — appears to have nothing to act on. It has something to act on. What the ceramic has instead of a class is a texture: a distribution of orientations over the grains, and that distribution has a symmetry group of its own. Poling gives it one. The material is heated, a strong field is applied along a chosen direction, the domains inside each grain swing towards it, and what is left when the field is removed is a distribution which is unchanged by every rotation about that direction and by every mirror containing it.

That is Curie’s group ∞m, the group of a cone, and it is one of the seven a uniform field can have. Neumann’s principle applies to it in exactly the form it takes for a crystal class, and it gives the numbers a manufacturer works to: one pyroelectric coefficient, two dielectric constants, three piezoelectric moduli, five elastic ones.

The group of a distribution #

The first thing to be clear about is what the group is a symmetry of, because it is not a symmetry of any arrangement of atoms.

In a crystal, an operation of the point group carries atoms to atoms. Nothing of the sort happens here. A rotation of 11.3 degrees about the poling axis carries the specimen to a different specimen — different grains in different places, a completely different set of atomic positions. What it does not change is the distribution: how much of the material has its crystallographic c axis within a given cone of directions, how much has it elsewhere. That function on the space of orientations is what ∞m leaves alone.

The bridge to a physical property is that a property measured on a specimen large enough to contain many grains is an average over the grains. A stiffness is a stiffness of the whole block. A piezoelectric charge is the charge on the whole electrode. Each is an integral over the orientation distribution of the corresponding quantity in a single grain, and an integral over a distribution invariant under a group is itself invariant under that group. So the average has the symmetry of the distribution even though nothing in the material does, which is the same manoeuvre the symmetry of an average makes for a different kind of average.

This is worth being blunt about, because the calculation below never mentions it again. Every number here is a statement about a property of an aggregate. None of them is a statement about where an atom is.

The sum becomes an integral #

Neumann’s principle as this collection computes it is an average of a character. A property lives in a space of tensor components — eighteen for the piezoelectric moduli, twenty-one for the elastic constants — the group acts on that space, and the components a crystal may actually have are the ones fixed by every operation. The dimension of that invariant subspace is the trace of the projector that averages over the group, and the trace of that projector is the mean of the character.

For a crystal class the mean is a sum over at most forty-eight matrices. For ∞m it is a sum over an uncountable set, and the natural move is to write an integral and evaluate it.

The natural move is unnecessary, and the reason is the whole essay. Take a rotation through an angle θ about the texture axis. Its trace is 1 + 2 cos θ. Every character in this collection’s list is built out of traces of a matrix and its powers and nothing else — the dielectric character is (tr M)² + tr M² halved, the piezoelectric one is tr M times that, the elastic one is a symmetric square of it — so a character restricted to the rotations about one axis is a trigonometric polynomial in θ, and its degree is at most the rank of the tensor.

A trigonometric polynomial written as a₀ + Σ aₘ cos mθ has mean a₀ over the whole circle. So the average over an infinite group of rotations is one coefficient of one expansion, obtained exactly.

The coefficients are obtained by sampling the character at sixty-four equally spaced angles and taking a discrete cosine transform, which recovers a trigonometric polynomial of degree below thirty-two exactly rather than approximately. That they come out integers is asserted rather than hoped for: a coefficient off by a thousandth would mean the character being sampled was not the character it is claimed to be.

Where the finite count stops changing #

The same expansion answers a question the infinite group did not ask, and the answer is sharper than the question.

The average over the n equally spaced angles of a finite rotation group is a₀ + aₙ + a₂ₙ + …, because the mean of cos mθ over n equally spaced angles is 1 when n divides m and 0 otherwise. A rank-r property has no coefficient above index r. So as soon as n exceeds r there is nothing left to add, and the finite group’s count is the infinite group’s count.

Not approximately. Not in the limit. At n = r + 1 and every n above it, the two numbers are equal, and below it they differ.

This makes the infinite group computable by finite means with no limiting process anywhere, and it also states a fact about crystals that has nothing to do with textures. Two classes whose only difference is the order of the principal axis, both orders exceeding the rank of the property, permit the same number of components. That is why the elastic constants of a trigonal crystal in class 3m and a hexagonal one in class 6mm are both six and five respectively rather than being separated by their obvious difference of symmetry: rank four cannot resolve a three-fold axis from a six-fold one except through the one coefficient at m = 3.

Two routes to one number #

An argument this convenient invites the suspicion that it is a rearrangement of an assumption, so it is checked against machinery that shares nothing with it.

Wherever a rung of one of these ladders happens to be one of the thirty-two — and twenty-five of them are — the count reached by adding cosine coefficients is set beside the count reached the ordinary way, by averaging the same character over that class’s own list of integer matrices. One side is a Fourier sum over an infinite group truncated by divisibility; the other is a sum over between one and twenty-four explicit three-by-three matrices of integers. They share the character functions and no intermediate whatever.

Across every property of rank four or less that is a hundred and fifty comparisons, and they agree at all of them. Two of the entries are worth pausing on, because they are the ones a wrong implementation would get wrong: 4/m carries seven elastic constants and 6̅ carries five, so the family that adds only the mirror across the axis is not monotone in n. More symmetry does not mean fewer constants unless one group contains the other, and 6̅ does not contain 4/m nor the reverse.

Told apart at rank six #

Now the question the whole apparatus was built for. If a texture and a crystal give the same count for every property, in what sense is the texture not a crystal?

They do not give the same count for every property, and the place they separate is exactly where the argument says it must be. A crystal class with an axis of order n keeps the coefficients whose index n divides. A texture keeps only m = 0. The two therefore agree on every property of rank below n and disagree on the first property whose rank reaches n.

For a six-fold axis that means rank six, and rank six is a real physical property rather than a polynomial chosen to make a point: the third-order elastic constants, which say how a material’s stiffness itself changes as it is strained. Fifty-six components with no symmetry imposed. A crystal of class 6mm has ten of them independent. A poled ceramic has nine.

So a poled ceramic and a hexagonal crystal in class 6mm are indistinguishable by any measurement of a property of rank four or lower: the same one pyroelectric coefficient, the same two dielectric constants, the same three piezoelectric moduli, the same five elastic ones, the same zero for optical activity. Nothing in the ordinary catalogue of material constants can tell an oriented aggregate of randomly rotated grains from a single crystal of the right class. The tenth third-order elastic constant is the first thing that can, and measuring third-order elastic constants is difficult enough that the distinction is more often argued than observed.

The reverse case is just as sharp and goes the other way. No crystal class matches ∞∞m at rank four, because two dielectric constants collapse to one and five elastic constants collapse to two, and the cubic classes that come closest still permit three. An isotropic aggregate is detectably not a crystal; a fibre-textured one is not.

Why poling is an argument and not a step #

The counts also settle a question that sounds like process engineering and is not.

A ceramic as fired has its grains oriented every way, with no direction preferred and no sense along any direction. Its texture group is ∞∞m, the group of a sphere. Every grain in it is piezoelectric — lead zirconate titanate in its ferroelectric phase is class 4mm or 3m depending on composition, and both permit piezoelectricity — and the block is not piezoelectric at all, because ∞∞m permits zero piezoelectric moduli. The grains’ contributions cancel exactly, and the cancellation is a theorem rather than an accident of averaging.

The intermediate case is the informative one. Aligning the grains along an axis without giving that axis a sense produces the group ∞/mm — the group of a cylinder, which contains the mirror across the axis — and that group still permits no piezoelectricity and no pyroelectricity. Alignment is not enough. What poling supplies is not order but polarity: the removal of the mirror perpendicular to the axis, which is the operation that maps a polar vector to its negative. Nothing else about the process appears in the argument, and nothing else needs to.

That is Curie’s principle read forwards instead of backwards. The usual reading takes a crystal, applies a field, and asks what is left. This reading takes an aggregate, asks which group the processing leaves it in, and reads off what it may then do.

A property is a pile of spherical harmonics #

There is one more thing in the coefficients, and it explains why any of the above is true rather than merely checking that it is.

The characters of the rotation group’s irreducible representations are χ_ℓ(θ) = 1 + 2 cos θ + … + 2 cos ℓθ. Comparing that with a₀ + Σ aₘ cos mθ gives the number of spherical harmonics of each degree ℓ that a property carries, directly, by differencing adjacent coefficients.

The elastic constants are two scalars, two quadrupoles and one hexadecapole: 2·1 + 2·5 + 1·9 = 21. The piezoelectric moduli are two vectors, a quadrupole and an octupole. The third-order elastic constants reach ℓ = 6, and that ℓ = 6 piece is the entire content of the difference between 6mm and ∞m — the sixth harmonic has an m = 6 component which a six-fold axis keeps and a continuous one averages away.

Once the decomposition is in view the rest is bookkeeping. A texture keeps the components with m = 0; an axis of order n keeps those with m divisible by n; the isotropic groups keep ℓ = 0 alone. The bound on ℓ is the bound on the rank, and every count on this page is a consequence of those three sentences. It is the same statement a shell of orbitals splitting into kinds makes about electronic levels, applied to a tensor instead of a wavefunction, and it is one of the places where the arithmetic of this subject is simply the arithmetic of another one renamed.

Where the exactness stops #

Computed here: for each of the seven limiting groups and seven properties, the exact number of independent components; the cosine coefficients of every character on each of the four cosets of the rotations; the finite ladders in five families up to any order; the spherical-harmonic decomposition of every property; and a hundred and fifty comparisons against counts obtained from the thirty-two classes’ own matrices.

A texture group is not a symmetry of a structure, and this is the limit that matters. ∞m is the symmetry of an orientation distribution. It does not relate any atom to any other, it is not a group a diffraction pattern will show, and it is not what the eleven Laue classes would report about a specimen. The moment the question is about scattering rather than about a bulk average, this group has nothing to say.

Perfect texture, and only perfect texture. The distribution assumed here is exactly invariant under every rotation about the axis. A real poled ceramic is not: grains have preferred orientations left over from sintering, and poling reorients domains rather than grains, so the alignment is partial and the invariance is approximate. The consequence is that the components this page reports as zero are small rather than absent, in the same way and for the same reason that near-symmetry is not symmetry. What symmetry decides is which components must vanish for an ideal texture; it does not bound how far a real one departs.

A permission, again. That ∞m permits three piezoelectric moduli says nothing about their size, and a poled ceramic whose grains happened to cancel would have three permitted moduli all equal to zero. This is permitted is not present in its most practical form: the group decides the shape of the tensor and the material decides the numbers in it.

And rank six is where the ladder stops here. The characters above rank six are computable by the same route with no new idea, and are not computed, because no property of rank eight is measured on a ceramic and a count nobody can check is a count worth leaving out.

Who asked it first #

Curie’s 1894 paper introduced the seven groups and the principle about causes and effects; it did not compute a single tensor. Voigt’s Lehrbuch der Kristallphysik of 1910 has the tensor counts for the thirty-two classes and treats the limiting groups as a curiosity at the edge of the subject.

The two were joined by the piezoelectric ceramics of the early 1950s, when barium titanate and then lead zirconate titanate turned the curiosity into the design basis of an industry: a material with no crystal symmetry whatever, whose entire behaviour is decided by the symmetry of a processing step. The systematic treatment of orientation distributions and their symmetry — texture analysis, and the expansion in spherical harmonics that this essay’s coefficients are a one-dimensional shadow of — is Roe’s and Bunge’s, from the 1960s, and its central object is exactly the harmonic decomposition above with the axis’s continuous symmetry not yet imposed.

The historical order is worth noticing because it is the reverse of the logical one. The group came first and was thought marginal; the arithmetic came second and was thought to be about crystals; the material that needed both arrived third and made the marginal case the important one.

Where the ladder goes next #

Back, to the intersection this essay took for granted: what a crystal keeps in a field, where the limiting groups meet the thirty-two and the residual is computed on matrices.

Sideways, to the counts a genuine crystal class permits and how they were reached: twenty of the twenty-one for piezoelectricity, and twenty-one, thirteen, nine, three for the elastic constants whose ladder this essay climbed past.

And outward, to the other property a texture can carry and a random aggregate cannot: optical activity, which the mirrors of ∞m forbid outright and which ∞ and ∞2 permit — so that a material can be built to rotate light by having a texture with a handedness, with no chiral crystal anywhere in it.