A six-fold crystal hides its frame until rank six

The tables of physical properties print a number beside every crystal class: so many elastic constants, so many piezoelectric moduli, so many components of the dielectric tensor. Fifteen constants an energy would have forbidden added the photoelastic count to the list and found something the printed numbers hide. A crystal of class $4/m$ is printed with ten photoelastic constants, and one of them can be removed by turning the axes about the four-fold axis through the right angle. Another, the part that has no energy behind it, cannot be removed by any turn. In $6/m$ the same freedom to turn exists and removes nothing at all.

That essay ended on the table it had not drawn: which classes and which properties spend their free angles, how many constants each then really has, and whether the elastic and photoelastic counts still tell the eleven Laue classes apart once the angles are taken out. This essay draws it, for six properties and all thirty-two classes. Three answers come out of it, and none is a fact about any single property.

The first is a rule for when a turn removes anything: an axis of order $n$ is invisible to every property of rank below $n$. The second is that the rule puts the six-fold axis out of reach of every property commonly measured, which is why $6/m$ removed nothing. The third is that a crystal has one frame and not one per property, so a crystal described by several properties spends its angle once, and what is left over is the angle between the frames the different properties would each prefer.

Which turns are free #

A turn of the axes changes every number in a property tensor and nothing about the crystal. Most turns also move the crystal’s symmetry operations, so that the tensor in the new frame is constrained by the class in a different orientation, and the printed pattern of zeros and equalities no longer applies. The turns that do not move them are the ones that commute with every operation of the class. Those are the turns a table could have used and did not, and the constants they can remove are the ones that describe the frame rather than the material.

Near the identity, such turns form a small group, and its dimension is the number of independent angles available. In the two triclinic classes, $1$ and $\bar1$, every operation is the identity or the inversion, both of which commute with every rotation, so there are three free angles. In the three monoclinic classes the free turns are those about the two-fold axis, one angle. In the eight uniaxial classes with no two-fold axis or mirror containing the principal axis, $4$, $\bar4$, $4/m$, $3$, $\bar3$, $6$, $\bar6$ and $6/m$, the free turns are those about the principal axis, one angle again. The other nineteen classes have no free turn: a two-fold axis or mirror through the principal axis is moved by any turn about it, and a cubic or orthorhombic class pins every direction down. The axes a class pins down drew the same division for the indicatrix.

The convention that makes this a count is that a turn is measured as a change of description, never of material, and that every property is written in Cartesian components on the class’s own axes, with an axial tensor picking up the determinant under an improper operation. The printed count is the dimension of the tensors the class leaves alone, the average Neumann’s principle, as one sum computes; the intrinsic count subtracts the number of directions in which the free turns move a generic one of them.

Thirteen classes, six properties #

The figure at the head of this essay is the census. The count is made by taking a random tensor of each property, averaging it over the class to get a generic member of what the class allows, and applying the infinitesimal form of each free turn to it. The number of independent directions that produces is the number of constants the turn can remove. A turn cannot remove more constants than it has angles, and in every one of the hundred and ninety-two cells it removes either all its angles’ worth or none.

The elastic column is the one the elasticity literature already knows. Twenty-one constants in a triclinic crystal come down to eighteen when the frame is chosen to clear three of them. Thirteen in a monoclinic crystal come down to twelve. Seven in the tetragonal classes $4$, $\bar4$ and $4/m$, and seven in the trigonal $3$ and $\bar3$, come down to six. Five in the hexagonal classes stay five. The photoelastic column follows the same pattern one step behind: thirty-six to thirty-three, twenty to nineteen, ten to nine, twelve to eleven, and eight in $6/m$ unchanged.

The lower ranks behave differently. The dielectric tensor, rank two, loses three of its six components in a triclinic crystal, the three angles of its principal axes, and one of four in a monoclinic one. But in every uniaxial class it has two constants and loses neither, since a uniaxial dielectric tensor looks the same from every direction about its axis. The piezoelectric tensor, rank three, loses one in classes $2$, $m$, $\bar4$, $3$ and $\bar6$, and nothing in $4$ or $6$. The pyroelectric vector loses its direction in classes $1$ and $m$, where it can point anywhere in a space of more than one dimension, and keeps its one constant in every class with a polar axis.

What an axis lets through #

The pattern has a single cause, and the figure shows it directly. Turn the frame of a crystal about an axis and every component of a property tensor varies with the angle, but not arbitrarily: the tensor splits into parts that go round once as the frame goes round once, parts that go round twice, and so on, up to parts that go round as many times as the tensor has indices. A rank-four tensor has parts that turn zero, one, two, three and four times; a vector has parts that turn zero and one time. That is what a group does to a function, applied to the group of turns about one axis, and the parts are the harmonics of the angle.

An $n$-fold rotation about the same axis is one of those turns, through a definite angle. A part that goes round $m$ times as the frame goes round once is multiplied by a phase when the frame is turned through an $n$-th of a revolution, and it is left alone only if that phase is one, which is when $m$ is a multiple of $n$. So the class keeps the parts with $m = 0$, $\pm n$, $\pm 2n$ and so on, and discards the rest. Of the parts it keeps, the ones with $m = 0$ do not change at all under a turn of the frame, and the others do. A free turn can spend a constant only on a part that turns, and the first part an $n$-fold axis lets through that turns is the one with $m = n$, which needs $n$ indices.

In the figure, the elastic entries of $4/m$ come back after a quarter turn, because the only turning parts it keeps go round four times. Those of $\bar3$ come back after a third of a turn, because the turning parts go round three times. In $6/m$ the turning parts would have to go round six times, and a rank-four tensor has none. So every entry is flat, and the freedom to turn the frame is real and buys nothing.

The rank at which each axis first shows #

The rule can be tested on tensors with no index symmetry at all, the largest tensor space of each rank, and the figure records the result. For the pure rotation classes $2$, $3$, $4$ and $6$ the lowest rank at which a turn moves anything is two, three, four and six, exactly the order of the axis, for polar and axial tensors alike.

The classes with improper operations shift the count by one in either direction, and the shift has a reason. An inversion multiplies a polar tensor of odd rank by minus one and an axial tensor of even rank likewise, so a centrosymmetric class forbids those tensors entirely and they cannot be the first to show anything. In $\bar1$ the polar vector is forbidden, so the first polar tensor that sees the three free angles has rank two. In $4/m$ the axial tensors of even rank vanish, so the first axial tensor to see the turn has rank five rather than four. A rotoinversion is a rotation combined with the inversion, so it multiplies a part by a phase and a sign: $\bar4$ keeps axial parts that turn twice, which is why its optical activity, a rank-two axial tensor, spends its angle while its dielectric tensor does not. The census of every class follows from the one rule and the one sign.

The six-fold axis is the extreme. The first polar tensor whose components change when a $6/m$ crystal’s frame is turned has rank six, and no axial tensor of rank up to six does. The properties the standard tables list class by class stop at rank four: elasticity, photoelasticity, electrostriction, piezoelectricity with its converse, the dielectric and optical tensors. Nothing in that list can tell one frame about the six-fold axis of a hexagonal crystal from another, which is why the elastic and photoelastic tensors of $6/m$ looked transversely isotropic in the essays that counted them. It is not that the crystal is isotropic in its basal plane. It is that every property of rank below six averages away the only parts that could say otherwise.

There is one rank-six property with a tradition of measurement, and the rule picks it out. The third-order elastic constants, which say how a crystal stiffens as it is strained, form a tensor of rank six symmetric in each of three index pairs and under any exchange of the pairs. Counted the same way, $6/mmm$ has ten of them and $6/m$ twelve, the values in the literature, and in $6/m$ the free turn removes exactly one: twelve printed, eleven intrinsic. The third-order elastic constants are the first property in the usual repertoire in which a six-fold crystal’s frame shows. The same count gives sixteen for $4/m$, of which one is removable as it was at second order, fifty-six for the triclinic classes, of which three are, and six for $m\bar3m$, none.

The frame is spent once #

A crystal has one frame. Each property tensor, taken alone, would choose its own: the elastic tensor of a $4/m$ crystal prefers the frame that clears $C_{16}$, and the photoelastic tensor prefers the one that clears the pair-symmetric half of $p_{16}$ and $p_{61}$. For the generic crystal in the figure, the first is reached by a turn of 37.8 degrees and the second by a turn of 22.5 degrees, 15.2 degrees apart modulo the forty-five degrees after which both repeat. Choosing the elastic frame leaves the photoelastic tensor with all ten of its constants; choosing the photoelastic one leaves the elastic tensor with all seven.

So the count for the two together is not the sum of their separate intrinsic counts. It is the sum of their printed counts less one: sixteen in $4/m$, not fifteen. The census confirms it in every class with one free angle, and in the triclinic classes the two together lose three, not six. The last column of the census figure, all six properties at once, loses one constant per free angle however many of the six could have spent it.

The constant that comes back is the angle between the two preferred frames. Once one tensor has used the turn to clear an entry, the other tensor carries the information the turn removed, in the form of an orientation relative to the first. That is a measurable property of the crystal, and it is the one part of the description that belongs to neither tensor. A reader of a table that prints six intrinsic elastic constants and nine intrinsic photoelastic ones for $4/m$ has been told fifteen numbers about a crystal that needs sixteen.

Where the lattice comes in #

There is a third party to the frame, and it settles the question in practice. The free turns commute with the class, but they do not, in general, carry the lattice to itself: a turn of twenty degrees about the four-fold axis of a tetragonal crystal moves the cell edges. Diffraction fixes the edges, and with them a frame that no property chose. Measured against that frame, every printed constant is a genuine material constant, including the ones a turn could have removed, because each of them then encodes the angle between the property’s own preferred frame and the lattice.

That is why the printed counts are not wrong, and why the intrinsic counts are not a correction to them. The printed count answers the question for a crystal whose lattice is known, which is every crystal a laboratory measures. The intrinsic count answers it for a tensor alone, which is what a comparison between two materials, or a theory that predicts a property without the structure, actually has. The difference between the two counts is the number of angles, and the census says exactly which properties can carry them.

The Laue classes, with the angles spent #

Any property even under inversion can at best separate the eleven Laue classes, for the reason the symmetry diffraction adds gives. The earlier essay found that the printed elastic count gives the eleven Laue classes seven different values and the printed photoelastic count nine, and that the two together give all eleven their own pair. The lead asked whether that survives the removal of the angles. It does, though the two counts change their roles.

With the angles spent, the elastic count gets worse. $4/m$ and $\bar3$ drop from seven to six and land on $4/mmm$ and $\bar3m$, so four Laue classes share the value six and the column takes only six values. The photoelastic count gets better: $\bar3$ drops from twelve to eleven, leaving $mmm$ alone at twelve, and $4/m$ drops from ten to nine, so only $\bar3m$ and $6/m$ still share a value, at eight. The intrinsic photoelastic count separates ten of the eleven, more than any single count of a fourth-rank tensor managed in the printed tables.

Read as a pair, the intrinsic counts still give every Laue class its own point. The four classes the elastic count now merges at six are split by photoelasticity into nine, seven, eleven and eight, and the one pair photoelasticity merges is split by elasticity into six and five. The pair is a stronger instrument than either count alone with or without the angles, and the reason is the same as before: the two tensors fail in different places. Only their sum, the single number for the two together, merges a pair again, $4/mmm$ with $6/m$ at thirteen. That is a coincidence of totals rather than of tensors, and reading the two counts separately avoids it.

What the count must refuse #

Two claims sound natural and are refused. The first is that a class with a free turn always has something the turn can remove. Classes $6$ and $6/m$ are the counterexamples, and they are not borderline: every property in the census, of every rank up to four, is already invariant under the turn, and nothing below rank six could be otherwise. The second is that the intrinsic elastic count separates the Laue classes as well as the printed one. It separates fewer, six values against seven, because the printed count’s distinction between $4/m$ and $4/mmm$ was carried entirely by the constant a turn removes.

The census is checked against itself in other ways. The printed counts reproduce the tables’ values for both fourth-rank tensors, in all eleven Laue classes. The intrinsic elastic counts reproduce the values elasticity texts quote for triclinic, monoclinic and the seven-constant tetragonal and trigonal crystals. And the half-difference of $p_{16}$ and $p_{61}$ in $4/m$, the axial part, stays fixed to the last digit of the arithmetic as the frame turns, which is the earlier essay’s result found again by a different route.

What a count of angles cannot show #

The census counts dimensions, and a dimension says nothing about size; permitted is not present holds here in full. A crystal whose elastic $C_{16}$ is tiny in its lattice frame has a preferred elastic frame barely turned from the lattice, and the angle that comes back when two properties are compared may be too small to measure. Nor does the count say which frame a tensor prefers. It says only that there is one, and that the angle to it is a constant like any other.

Nor do the figures show the parts of a tensor directly. The harmonic decomposition that explains every column of the census is drawn only through its consequences, as entries that repeat after a quarter turn or a third or not at all. A reader who wants the parts themselves would find them in the parts a property splits into, which takes the elastic tensor apart under the full rotation group, and a looser fibre, more piezoelectric, which watches the parts of each degree decay at their own rates.

Who counted what #

That a triclinic elastic tensor has eighteen constants rather than twenty-one once its frame is chosen, and a monoclinic one twelve rather than thirteen, has been part of elasticity since the nineteenth century and is stated in Love’s treatise and in Nye’s tables of 1957; the same reduction for the tetragonal and trigonal classes with seven constants is in the later literature on elastic symmetry classes. What a census over every class and several properties adds is the rule behind the cases, the rank at which each axis first shows, and the observation that the angles belong to the crystal rather than to each tensor separately, which is what becomes of them when two properties are measured at once.

Still open: the magnetic classes, where time decides the turn #

The census ran over the thirty-two ordinary classes, where a free turn is a matter of geometry alone. The magnetic classes add a second way for an operation to act, with or without time reversal, and a time-odd property sees a primed operation with a sign, as the operation that reverses time set up. A turn that commutes with every spatial operation still commutes with every primed one, so the free angles are the same as for the underlying class. What can change is the rank at which they show, because a primed rotation keeps the parts of a time-odd tensor that the plain rotation would discard, those whose phase under the rotation is minus one. A primed six-fold axis should therefore let a time-odd tensor’s three-times-turning parts through, and its frame should show at rank three in a magnetic property where the plain six-fold hides it until rank six. Which of the fifty-eight black-and-white classes bring their free angle down in rank this way, and by how much, is the same computation with one sign added. It has not been run here.