Fifteen constants an energy would have forbidden

The parts a property splits into left the elastic tensor in five pieces. Its twenty-one components, decomposed under the full rotation group, come apart as two scalars, two five-dimensional quadrupoles and one nine-dimensional part that no lower-rank quantity can see, and a crystal class keeps whichever of those pieces its operations leave alone. The essay’s working rule was that a property with an index symmetry of its own needs an average of its own, and the elastic tensor was the example.

The photoelastic tensor is the example that rule was waiting for. It has four indices, like the elastic tensor. It is symmetric in its first pair and in its second, like the elastic tensor. And it lacks the one symmetry the elastic tensor has beyond those: it does not stay the same when the two pairs are exchanged. That single missing symmetry is worth fifteen components. They form three pieces of their own, and each piece answers a question the elastic count has no way to ask.

Two tensors with the same indices #

When a crystal is strained its refractive index changes. The optics of a crystal is carried by its impermeability, the inverse of the dielectric tensor, whose level surface is the indicatrix that three optical characters drew as a sphere, a spheroid or an ellipsoid. A small strain changes each of the six independent components of the impermeability by an amount proportional to each of the six independent components of the strain. Pockels wrote that proportionality down at the end of the nineteenth century, and the thirty-six coefficients that make it up are the photoelastic constants. They are what an acousto-optic modulator uses, what makes a stressed glass show colours between crossed polarisers, and what any optical fibre wound on a drum is quietly subject to.

Each coefficient has two index pairs, the first naming a component of the impermeability and the second a component of the strain. Both are symmetric second-rank tensors, so the tensor lives in the product of two copies of the six-dimensional space of symmetric tensors, thirty-six dimensions. Nothing about strain or light ties a coefficient to its mirror image across the diagonal: the response of the yz component of the impermeability to an xx strain has no reason to equal the response of the xx component to a yz strain.

The elastic constants live in the same space and use twenty-one of its dimensions, because for them the mirror-image coefficients are equal. That equality is a fact about where elastic constants come from, not about any crystal.

Where the exchange symmetry comes from, and why light does not have it #

A strained crystal stores energy, and to second order in the strain that energy is a quadratic form. The elastic constants are its coefficients, which is to say its second derivatives with respect to two strain components. Second derivatives commute, so the coefficient that couples an xx strain to a yz strain is the same coefficient as the one that couples a yz strain to an xx strain. The twenty-one is the number of independent coefficients a quadratic form in six variables has, and the exchange symmetry is nothing more than the symmetry of mixed partial derivatives.

The photoelastic tensor is not a second derivative of anything. It couples two different quantities, the strain of the lattice and the response of the electrons to light. A Maxwell relation does tie it to something, the electrostriction a light field produces, but nothing ties it to itself. So the thirty-six coefficients are constrained only by the crystal’s symmetry and by the symmetry within each pair, and the fifteen the elastic tensor gives up to its energy are all available. That is the whole of the physics; everything else here is counting.

The fifteen, split three ways #

The space of tensors with symmetric pairs is the tensor square of the six-dimensional space of symmetric second-rank tensors. Like any tensor square it is the sum of a symmetric part and an antisymmetric part. The symmetric part is the elastic tensor’s twenty-one-dimensional space. The antisymmetric part has the remaining fifteen dimensions, and it is the part of a photoelastic tensor that changes sign when the pairs are exchanged: half the difference between a coefficient and its mirror image across the diagonal.

Under the full rotation group, the six-dimensional space of symmetric tensors is itself the sum of a scalar and a five-dimensional quadrupole, the trace and the traceless remainder, and both of them are left alone by inversion. The antisymmetric square of a sum of two pieces is the product of the two pieces plus the antisymmetric square of each. A scalar times the quadrupole is a quadrupole, five dimensions. The antisymmetric square of a scalar is nothing. And the antisymmetric square of the quadrupole is a classical result from the theory of angular momentum: of the twenty-five ways two objects of angular momentum two can be combined, the ten that change sign under exchange carry angular momenta one and three.

$$\Lambda^2(L_0 \oplus L_2) \;=\; L_2 \,\oplus\, L_1 \,\oplus\, L_3, \qquad 15 = 5 + 3 + 7.$$

The piece with angular momentum one is a vector, and because everything here is even under inversion it is an axial vector: the kind of vector a rotation carries and an inversion leaves alone, like an angular velocity. The piece with angular momentum two is a second quadrupole, independent of the two the elastic tensor already has. The piece with angular momentum three is an octupole, seven components, even under inversion. So a crystal class permits exactly as many photoelastic constants beyond its elastic ones as it leaves invariants among an axial vector, a quadrupole and an octupole, and the rest of this essay is what those three numbers turn out to be.

The identity on ten operations #

A decomposition stated about the rotation group is a claim about every rotation, and a crystal class uses only ten kinds of operation, so the claim can be checked exactly where it is used. The figure above does that, with each character computed from the traces of an operation’s first few powers and its determinant, in the integer lattice coordinates the classes are written in. At the identity the columns are only the dimensions. Elsewhere they say something: a two-fold rotation has character four on the photoelastic space and five on the elastic one, so the extra part has character minus one, and the axial vector, quadrupole and octupole contribute minus one, plus one and minus one. The rows of the improper operations repeat those of their proper parts, which is what being even under inversion means, and it is why a photoelastic count can depend only on a crystal’s Laue class.

Ten rows that agree are a proof that the decomposition holds for every crystal class, since a class’s count is its characters averaged over its own operations and every operation it can contain has been checked. The identity is checked again class by class, in all thirty-two, and holds in all of them.

Eleven counts, and where each comes from #

The figure at the head of this essay is the result. The photoelastic counts run thirty-six, twenty, twelve, ten, seven, twelve, eight, eight, six, four and three, from the triclinic Laue class to the full cubic one, which are the counts the tables have printed since the middle of the twentieth century; the computation reproduces all eleven without being told any of them. They were made twice. Once by averaging characters over the thirty-two classes, and once by a route that shares nothing with it: build the thirty-six basis tensors in Cartesian coordinates, average each over a class’s operations index by index, and take the rank of what comes back. The two agree in all thirty-two classes, on the photoelastic count, its elastic half and its antisymmetric half.

The split into four parts is the new reading. The elastic part runs down the sequence twenty-one, thirteen, nine, three computed. The second quadrupole is present wherever a class is not cubic, with five components in the triclinic classes, three in the monoclinic, two in the orthorhombic and one in every uniaxial class, the same count as the anisotropy of the dielectric tensor: every class that lets a crystal be birefringent lets its photoelastic tensor have a quadrupole part its elastic tensor lacks. At the two ends, the triclinic thirty-six is twenty-one plus three plus five plus seven, and the full cubic three is three plus nothing, because none of the three extra pieces survives a group with three four-fold axes. The axial vector and the octupole are where the interest is.

The vector that no turn of the axes removes #

The axial vector part is invariant in exactly five Laue classes: three components in $\bar{1}$, one in 2/m, and one in each of 4/m, $\bar{3}$ and 6/m. An axial vector is the infinitesimal form of a rotation, so an axial vector a group leaves alone is an axis about which the whole group may be turned without being changed, and the count of invariant axial vectors is the number of independent angles through which a crystal’s frame can turn while every operation stays where it was. A triclinic crystal can be turned any way, which is three angles; a monoclinic one about its two-fold axis, which is one; the uniaxial classes with no mirror or two-fold rotation through their axis, about that axis. The other six Laue classes pin every direction down.

That freedom is not new. The axes a class pins down found a monoclinic crystal’s indicatrix with one principal direction fixed by symmetry and two free to turn as the wavelength changes. And a crystal of class 4/m is printed with seven elastic constants, one of which, $C_{16}$, can be removed by turning the axes about the four-fold axis through the right angle: six intrinsic constants and an angle.

The photoelastic tensor of a 4/m crystal has ten constants, and the same turn is available. In the frame the class is written in, the two coefficients coupling the in-plane shear to the difference of the two in-plane axial components, $p_{16}$ and $p_{61}$, are non-zero and unequal, and as the frame turns both swing through a full cycle every quarter turn. Their half-sum crosses zero at a definite angle, about twelve degrees for this tensor, which is the turn that removes the elastic-like coupling exactly as it removes $C_{16}$. Their difference does not swing at all. It is the same number, to the last digit a double-precision calculation carries, at every angle in the figure.

The difference is the axial vector part, the only piece of the tensor that rotations about the axis leave alone, and so the only piece the freedom of the frame cannot spend. A photoelastic tensor of class 4/m has one constant that a turn of the axes can remove and one that no turn can, and the second exists only because the tensor has no energy behind it. In an elastic tensor the axial piece is forbidden by the exchange symmetry. A crystal of class 4/m is not chiral, since it contains an inversion and a mirror, but nothing in its symmetry reverses a rotation about its axis, and this constant is the property in which that shows. The six-fold case below says what it does to light.

The octupole that splits the cubic crystals #

The octupole survives in eight of the eleven Laue classes and is absent from three: 4/mmm, 6/mmm and $m\bar{3}m$, the three that combine an axis of order four or six with two-fold axes perpendicular to it. Between the two cubic Laue classes it is present in one and absent from the other, and that is the reason the photoelastic count separates them when the elastic count cannot.

In $m\bar{3}m$ the photoelastic matrix has three constants: a diagonal component’s response to the same diagonal strain, its response to either of the other two, and a shear’s response to the same shear. The two off-diagonal responses are equal because a four-fold rotation about x exchanges y and z. Remove the four-fold rotations, as 23 and $m\bar{3}$ do, and only the three-fold rotations along the body diagonals remain, carrying x to y to z. So the pattern cycles: the response of xx to a yy strain equals that of yy to zz and of zz to xx, and the response of xx to a zz strain is a second constant, cycling the other way. Four constants rather than three.

The elastic tensor of the same crystal has three, because in an elastic tensor the coefficient for xx against yy must equal the one for yy against xx, which forces the two cycles to agree. The difference between the cubic class 23 and the cubic class 432 is invisible to an elastic measurement of any precision and visible to an optical one. The invariant octupole of $m\bar{3}$ is the polynomial $xyz$ with a gerade object’s parity, the same product that twelve of the thirty-two are free found among the degree-three invariants of $\bar{4}3m$. A four-fold rotation about a cubic axis changes its sign; a two-fold or three-fold one does not.

Sodium chlorate, of class 23, and barium nitrate, of class $m\bar{3}$, are the standard examples of cubic crystals with four photoelastic constants, and the counting result is usually credited to Bhagavantam’s work of the 1940s. What the decomposition adds is that the fourth constant is not an accident of two cubic classes. It is the one cubic survivor of a seven-dimensional piece of the tensor that eight of the eleven Laue classes keep some of.

Six-fold, with and without its mirrors #

The hexagonal Laue classes show all three extra pieces at once, and the pattern of the matrix says which entry belongs to which.

A crystal of class 6/mmm has six photoelastic constants against five elastic ones, and the sixth is the quadrupole: the response of the in-plane impermeability to a strain along the axis need not equal the response of the axial component to a strain in the plane. A crystal of 6/m has eight, and the two more are the axial vector and the octupole. In the matrix they appear as a coupling between the in-plane shear and the difference of the in-plane axial components, with a minus sign on one side of the diagonal and a plus sign on the other, and as a pair of transverse shears coupled antisymmetrically. Both change sign under a mirror containing the six-fold axis, which is why 6/mmm forbids them.

The axial constant has a consequence that can be stated without any matrix. Stretch a crystal of class 6/mmm along any direction in its basal plane and the birefringence the stretch induces in that plane has its principal axes along the stretch and across it, as common sense expects. Stretch a crystal of class 6/m the same way and the principal axes of the induced birefringence are turned away from the stretch by an angle, and the angle is the same for every direction of stretch and has the same sense every time. For the generic tensor the computation draws, it is eighteen and a half degrees at each of twelve directions spaced fifteen degrees apart, to the precision the arithmetic carries. A mirror containing the axis would carry a turn one way into a turn the other way, so a class with such mirrors can only have a turn of zero, and 6/mmm does. The size of the angle is a property of the material, set by how the axial constant compares with the others, and symmetry says nothing about it except that 6/m permits it to be anything.

In 6/m a turn of the frame changes nothing at all: the whole photoelastic tensor, axial part included, is isotropic in the basal plane, as a hexagonal elastic tensor is. The freedom is there and has nothing to remove. So the tables print five elastic constants for both 6/m and 6/mmm, and eight photoelastic constants against six. The elastic tensor cannot see the mirrors; the photoelastic tensor sees them twice.

Two counts that tell the eleven apart #

The eleven Laue classes are what any centrosymmetric property can distinguish, for the same reason the symmetry diffraction adds collapses the thirty-two classes onto eleven: a property even under inversion cannot tell a class from its centrosymmetric closure. The elastic and photoelastic tensors are both even, so each can at best separate the eleven. Neither does.

The elastic count gives eleven classes seven values, merging four pairs: 4/m with $\bar{3}$, 4/mmm with $\bar{3}m$, 6/m with 6/mmm, and the two cubic Laue classes. A fully symmetric fourth-rank tensor merges the same four pairs. The photoelastic count does better, nine values, and merges a different two: mmm with $\bar{3}$ at twelve, and $\bar{3}m$ with 6/m at eight.

The surprise is the last column. A fourth-rank tensor with no index symmetry at all, eighty-one components in the triclinic class, is the largest space of fourth-rank tensors there is, and it separates ten of the eleven. The orthorhombic Laue class mmm and the tetragonal 4/m both permit exactly twenty-one of its components, for reasons that have nothing to do with each other: three perpendicular two-fold axes on one side, one four-fold axis and a mirror on the other. No single count of a fourth-rank tensor of any index symmetry tells all eleven Laue classes apart.

The two counts together do. Place each Laue class at the point whose coordinates are its elastic and photoelastic counts and the eleven points are all different: the pairs the elastic count merges, the photoelastic count separates, and the other way round. Neither tensor is the better instrument. They fail in different places, and a crystal measured both ways has its Laue class fixed by how many constants of each kind turn out to be non-zero, before any value is examined.

That is counting in principle, and permitted is not present applies in full: a permitted constant can be too small to measure, and a crystal whose fourth photoelastic constant happened to equal its third would be counted as $m\bar{3}m$. But it says the two properties are not redundant even in how many numbers each carries.

Made twice, and refused twice #

Every number here was computed rather than quoted, and held to a test that could fail. The character counts and the Cartesian projection share no step and must agree in all thirty-two classes on three ranks at once. The elastic half must reproduce the count the property table already makes by its own character. The pattern reader, which finds a matrix’s independent entries by elimination rather than by matching equal numbers, must find as many letters as the count. And the eleven published counts must come out unprompted.

Two decompositions that look as reasonable as the right one are refused. If the vector in the extra part were polar, as an antisymmetric object might suggest, the identity would fail in twelve classes, because a polar vector is odd under inversion. If the octupole were odd, as the octupole of a cubic polynomial is, it would fail in eighteen, among them $m\bar{3}$, whose fourth constant would disappear. The wrong reasons fail somewhere a check can see.

One count, five properties #

The space the photoelastic tensor lives in is defined by its indices alone, a symmetric pair against a symmetric pair, and any property of that shape has the same counts. The piezo-optic tensor, the change of impermeability per unit stress, is one. The quadratic electro-optic tensor, the change of impermeability per square of an applied field, is another. Electrostriction, the strain per square of a field, is a third. Magnetostriction in a crystal with no magnetic order is a fourth, since the square of an axial vector transforms like the square of a polar one.

So the thirty-six, twenty, twelve and the rest are numbers of the space, and every property in it inherits the axial vector, the second quadrupole and the octupole. A crystal of class 23 has four independent electrostrictive coefficients and four quadratic electro-optic ones for the same reason it has four photoelastic ones, and a 6/m crystal that turns the axes of a stretch-induced birefringence has room to turn those of a field-induced one too. The counts do not say the turns are equal, only that each property has room for one.

Where the count stops #

The tensor counted is Pockels’ tensor, the response to the symmetric strain. A real deformation also rotates the material locally, by the antisymmetric part of the displacement gradient, which the strain discards. In a birefringent crystal a rotation turns the indicatrix and so changes the impermeability seen in the laboratory frame. Nelson and Lax pointed this out in 1970, and the effect was measured in rutile soon after. Their term adds no new constant, being fixed by the refractive indices, but the coefficient an experiment reports for a shear wave in a birefringent crystal is not the Pockels coefficient alone, and nothing here counts it.

The counts are permissions. A class that permits four constants does not promise that the fourth is large or different from the third in any material; the octupole constant of a cubic crystal without four-fold axes, or the axial constant of a 4/m one, can be near zero for reasons symmetry knows nothing about.

And an index symmetry is not the only kind a property can gain. Two classes go dark far from absorption found the second-harmonic tensor acquiring a symmetry far from absorption, where its three frequencies can be exchanged, and losing eight of eighteen components to it. The photoelastic tensor has no analogue: a strain and an impermeability are never interchangeable, at any frequency. How small its fifteen extra components are is a question about materials.

Pockels, Bhagavantam, and the fourth constant #

Friedrich Pockels set out the photoelastic tensor and its forms for the crystal classes in work culminating in his Lehrbuch der Kristalloptik of 1906. The feature that separates his matrices from the elastic ones, coefficients either side of the diagonal that need not agree, is the one a reader trained on elastic tables is most likely to overlook, because every elastic matrix is symmetric and the eye learns to expect it. The method of counting independent constants by averaging a character, the method of Neumann’s principle, as one sum, was developed by Bhagavantam and his school in the 1940s, and John Nye’s Physical Properties of Crystals of 1957 put the resulting forms in front of every student of the subject.

A symmetry is easy to assume where it does not hold when a neighbouring tensor has it. The elastic tensor is symmetric across its diagonal for a reason, and the reason does not transfer. The counting method is indifferent to the habit: it averages whatever character it is given.

Still open: how many constants survive the choice of frame #

The tables print ten photoelastic constants for class 4/m, and the frame turn shows that one can be removed by choosing the axes; in 2/m the same holds, and in the triclinic class three angles are free. So the constants that describe the material, rather than the material in a frame somebody chose, number the printed count minus the invariant axial vectors, but only where the free angle is spent on a constant: in 6/m it removes nothing. Which classes and properties spend their angles, how many intrinsic constants each then has, and whether the two counts here still separate the eleven Laue classes once the angles are taken out, is a table not drawn here. It needs the orbits of each class’s tensor space under the rotations that commute with it.