Twenty-one, thirteen, nine, three

Push on a crystal and it deforms. How much, and in which direction, is a linear relation between two symmetric 3 × 3 matrices — stress and strain — and the coefficients are the elastic constants.

A rank-four tensor has 81 components. The symmetry of stress and of strain cuts that to 36, and the fact that elastic energy is a function of state cuts it to 21. Those 21 are what a crystal with no symmetry at all has, and every symmetry operation takes some away.

The seven numbers #

Read down the systems and the count falls: 21 triclinic, 13 monoclinic, 9 orthorhombic, then the tetragonal and trigonal systems which need two numbers each, 5 hexagonal, 3 cubic.

The falls are not even, and the reason is instructive. Going from triclinic to monoclinic adds one two-fold axis and removes eight constants. Going from orthorhombic to tetragonal adds a four-fold and removes two or three. Going from hexagonal to cubic — adding the four three-fold axes on the body diagonals — removes two.

The first operation is worth far more than the last. A two-fold axis in a triclinic crystal has an enormous amount to cancel; a three-fold added to a crystal that already has four-folds and mirrors has almost nothing left to work on. That is a general feature of the averaging sum rather than anything about elasticity: each new operation only removes what the previous ones left.

The split nobody predicts by counting #

Two systems carry two numbers, and this is the part worth the essay.

The tetragonal classes split 7 and 6. Classes 4, 4̅ and 4/m have seven elastic constants; classes 422, 4mm, 4̅2m and 4/mmm have six.

The trigonal classes split the same way. 3 and 3̅ have seven; 32, 3m and 3̅m have six.

Neither split follows the order of the group. 4̅ has four operations and seven constants; 422 has eight operations and six. But 3̅ has six operations and seven constants, while 32 also has six operations and six constants. Two classes of identical size, differing by one elastic constant.

What separates them is not how many operations there are but whether any of them acts in the basal plane. A group with only a principal axis leaves the plane perpendicular to it under-constrained: there is a component coupling shear in that plane to shear out of it, and nothing in the group forces it to vanish. Add a single two-fold lying in the plane, or a mirror containing the axis, and that component is reversed by it and must be zero.

The seventh constant is that one. In Voigt’s notation it is c₁₆ for the tetragonal classes and c₁₄ for the trigonal ones, and its presence is the difference between a crystal whose elastic behaviour is symmetric about its axis and one whose is not.

The same split, in a different property, in different places #

If the seventh constant were a peculiarity of elasticity the split would be a curiosity. It is not: every property splits its system somewhere, and the splits are in different places for different properties, which is the strongest evidence that “amount of symmetry” is not a single quantity.

Elasticity splits the tetragonal system 7/6 and the trigonal 7/6, and nowhere else. The dielectric tensor splits nothing — every tetragonal, trigonal and hexagonal class has two independent dielectric constants, and every orthorhombic class has three — because a symmetric rank-2 tensor is too small an object for the differences between classes in a system to reach it.

The piezoelectric moduli split everything. Within the tetragonal system the counts are 4, 4, 0, 1, 3, 2, 0 — seven classes, five distinct answers, and two of them zero. Piezoelectricity is a rank-three polar property, which is odd-rank, so the inversion kills it outright and every other improper operation constrains it heavily.

The contrast between the two columns is worth drawing rather than describing, because it is the clearest evidence available that the classes inside a system are genuinely different objects. Run the same seven groups against the elastic character and they give two answers; run them against the piezoelectric character and they give five, from eighteen down to nothing at all. Nothing about the groups changed between the two runs. What changed is how many indices the property carries and whether that number is odd, and a property with an odd number of indices notices an operation that reverses a direction where an even one cannot.

Two properties give two orderings, and there is no reason to stop at two. Adding the dielectric tensor and the gyration tensor to the same seven classes gives four columns, and the four disagree in four different ways: the dielectric column is flat, the elastic column takes two values, the piezoelectric column five and the optical one three. Set side by side they are the whole argument of this section in a form short enough to check by eye.

That is the practical reason the whole table is computed rather than a rule of thumb applied. There is no ordering of the thirty-two classes by symmetry that predicts all six columns, and any intuition that says a bigger group means fewer constants is right on average and wrong in specific cases that matter.

The cubic three, and why it is not two #

Cubic crystals have three: c₁₁, c₁₂ and c₄₄ in the usual notation — a stiffness along an axis, a coupling between perpendicular axes, and a shear.

An isotropic material has two, which is why elasticity in an engineering course has two constants and not three. So a cubic crystal is not isotropic, and the gap between three and two is a real anisotropy that the cubic symmetry does not remove.

The quantity measuring it is the Zener ratio, 2c₄₄/(c₁₁ − c₁₂), which is one exactly when the third constant is redundant. It is about 1.2 for aluminium, 3.2 for copper and 8.7 for lithium — so a cubic crystal can be strongly anisotropic while having the most symmetric lattice there is.

Symmetry gives an upper bound on how few numbers are needed, and does not make a material behave simply. Isotropy is not a crystal class; there is no group in the thirty-two whose elastic count is two, because full rotational symmetry is not crystallographic.

Where the numbers came from historically #

The counts are Voigt’s, worked out by hand class by class in the Lehrbuch der Kristallphysik of 1910, and the notation crystal physics still uses — c₁₁ through c₆₆ on a 6 × 6 matrix, with pairs of tensor indices collapsed to one — is his.

Working them out by hand is genuinely laborious: for each class, write the transformation law for a rank-four tensor, apply each operation, and collect the linear relations among the 21 components. Voigt did it for all thirty-two, and the results have stood.

The character sum was not available to him in this form — group characters as a tool arrived with Frobenius in the 1890s and reached physics through Wigner in the 1930s — and the striking thing is how much shorter it makes the answer. The 192 numbers this field computes take a few milliseconds and are the same 192 numbers that took a chapter.

That is not a criticism of the chapter. Voigt’s tables give the shape — which component is zero and which pairs are equal — and the character sum gives only the count. The shape needs the longer calculation, and this site does it too, in a Cartesian frame, as the second of two routes that must agree.

Reading the fall as a sequence #

Laid out as a sequence the seven numbers say something the table does not.

Each system is the previous one with symmetry added, and the reductions shrink: eight, four, two or three, one, two. Adding the first two-fold axis to a triclinic crystal removes more than the four three-fold axes that make a hexagonal crystal cubic.

The reason is the averaging sum again. When a group is small, most of the twenty-one components are unconstrained and a new operation finds plenty to cancel; when a group is already large, the surviving components are the ones that have survived everything so far, and a new operation is unlikely to be the one that catches them.

The corollary is worth stating for anybody choosing a material. If elastic simplicity is what is wanted, the step from no symmetry to a single axis buys most of it, and the step from hexagonal to cubic buys almost nothing. And even the cubic three is one more than an isotropic solid needs.

Where the exactness stops #

These are counts of independent constants, not of large ones. A hexagonal crystal has five and graphite’s span four orders of magnitude; a cubic crystal has three and copper’s give a Zener ratio of three. Symmetry decides how many numbers there are and says nothing about their sizes.

The intrinsic symmetries are an input. That the elastic tensor is symmetric under exchanging its first index pair with its second comes from elastic energy being a state function, which is thermodynamics. Hand the calculation a different intrinsic symmetry and it computes a different, equally correct, answer to a different question.

And these are the constants of a perfect single crystal. A polycrystal, a composite or a crystal with a texture has an effective elasticity that is an average over orientations, and averaging is not a symmetry operation. A randomly oriented polycrystal of a cubic material behaves isotropically with two constants, and neither of them is one of the three.

Why the count is a class function and not a lattice one #

One thing about this table is easy to misread: the counts belong to the crystal class, not to the lattice or to the space group.

Two crystals with the same point group have the same number of elastic constants whatever their Bravais lattice — a primitive cubic and a face-centred cubic crystal in class m3̅m both have three, because centring adds translations and elasticity is a point property that translations do not touch. And two crystals with the same space group obviously agree, since the space group determines the class.

Going the other way, the screws and glides are invisible here. P2₁/c and P2/m are different space groups with the same point group 2/m, so both have thirteen elastic constants. A systematic absence tells a crystallographer which of the two the crystal is and tells them nothing new about its elasticity.

That is worth saying because it cuts both ways as a check. An elasticity measurement constrains the class, so it can rule out a proposed structure whose class is wrong and can never distinguish two structures sharing one. And since diffraction reports only the Laue class, which is coarser still, the two experiments answer overlapping but different questions: diffraction gives eleven possibilities, elasticity gives whichever of the seven-or-nine counts the material shows, and neither alone gives the class.

What the numbers are for #

Three uses, and they are why anybody computes this rather than measuring 21 numbers and seeing which come out zero.

It says how many measurements are needed. An experimentalist determining the elasticity of an orthorhombic crystal needs nine independent measurements, not 21, and knows in advance which orientations will give them.

It is a consistency check on a measurement. A tetragonal crystal reported with a non-zero c₁₄ has been mis-indexed, mis-oriented or is not tetragonal. The symmetry constraint is exact and a measurement violating it is wrong, not surprising.

And it constrains what a model may contain. An interatomic potential fitted to a cubic material must reproduce three constants, and a potential producing four independent values has a bug rather than a discovery.

Voigt notation, and the compression that makes the table readable #

A word about the notation, because the counts are almost always quoted in it and it is a genuine piece of engineering rather than an abbreviation.

The elastic tensor has four indices, each running 1 to 3, so writing it out means eighty-one numbers arranged in a shape nothing prints well. Voigt’s compression uses the symmetry of stress and strain: a symmetric pair of indices takes only six distinct values, so 11 → 1, 22 → 2, 33 → 3, 23 → 4, 13 → 5, 12 → 6. Two index pairs become two indices from 1 to 6, and the tensor becomes a 6 × 6 matrix.

The exchange symmetry then makes that matrix symmetric, so it has 21 independent entries — which is where the number at the head of this essay comes from, and why it is 21 rather than anything else. A symmetric 6 × 6 matrix has 6·7/2 = 21 entries above and on its diagonal.

Every count in the elastic column is therefore a count of surviving entries in a 6 × 6 table, and the shapes crystal physics prints — the patterns of dots and lines and linked circles showing which entries are equal and which vanish — are pictures of that table. This site computes the dimension rather than the pattern, and the two agree by construction because the dimension is the number of independent entries the pattern leaves.

The compression has one trap worth knowing, and it is the reason strain and stress are written with different factors of two in some conventions: the map from tensor to Voigt indices is not an isometry, and getting the factors wrong produces an elastic matrix that is not symmetric. That is a bookkeeping hazard rather than a symmetry one, and no count in this field depends on it.

One last thing the sequence is good for: it is a sanity check on a model rather than on a measurement.

An interatomic potential, a density-functional calculation or a machine-learned force field, asked for the elastic tensor of a cubic crystal, must return three independent numbers. If it returns four, something in the calculation has broken the symmetry — an under-converged k-point mesh, a supercell that is not commensurate with the symmetry, a relaxation that drifted. The check costs nothing and catches a class of error that otherwise shows up much later as a result that is slightly wrong everywhere.

Symmetry constraints are the cheapest test a computational result can be put through, and they are exact.

The other constraint, which is not symmetry at all #

Symmetry says which constants may be non-zero and says nothing whatever about their values. There is a second constraint that does the opposite, it is exact, and the pair of them together is what a computed elastic tensor should be checked against.

An elastic solid stores energy when it is deformed, and the energy is a quadratic form in the strain with the elastic tensor as its matrix. A material that is stable stores positive energy under every deformation, so that matrix must be positive definite — which is a condition on the numbers rather than on which of them survive.

Written out for the cubic case the condition is three inequalities: c₁₁ > |c₁₂|, c₁₁ + 2c₁₂ > 0, and c₄₄ > 0. These are the Born stability criteria, and they are as sharp as the symmetry counts and of an entirely different kind. Symmetry says a cubic crystal has three constants; stability says which triples of values a real cubic crystal may have; and neither statement can be derived from the other.

The pair is what makes a computed result testable. A calculation returning four independent constants for a cubic material has a bug, by the symmetry count. A calculation returning three that violate the inequalities has produced a structure that is not at a minimum of its energy — usually because the geometry was not relaxed, or because the structure is genuinely unstable and the calculation has found a saddle. Those are different diagnoses from the same numbers, and both are available before any comparison with an experiment.

What happens to the count in a polycrystal #

The remark that averaging over orientations is not simple deserves its own paragraph, because the effective elasticity of an aggregate is what most measurements of most materials are actually of.

A polycrystal with randomly oriented grains is isotropic on average, so it has two effective constants whatever the single crystal has. The difficulty is that the average is not the average of the constants. Assume uniform strain across the grains and the effective moduli come out as one average — the Voigt average; assume uniform stress and they come out as another — the Reuss average. The two disagree, and neither is right, because a real aggregate has neither uniform strain nor uniform stress.

What is exact is that the truth lies between them. The Voigt average is an upper bound on the effective moduli and the Reuss average a lower one, and the two together bracket the answer with no further assumptions. The Hill average, the mean of the two bounds, is what is usually quoted and is a convenience rather than a result.

The width of the bracket is worth noticing, because it is a measurement of anisotropy that needs no single-crystal orientation. The two bounds coincide exactly when the single crystal is elastically isotropic — a cubic material whose Zener ratio is one — and they separate as the anisotropy grows. So a polycrystal’s two effective constants carry a trace of the three the crystal had, and how much of a trace is set by the same ratio that says whether the third constant was needed at all.

And the counts are permissions like every other entry in this table: thirteen constants is not thirteen large ones, and the arithmetic that produced them is one character averaged over a finite group with nothing physical in it anywhere.

Where this goes #

The counts are exact, they are old, and they are routinely over-read. The next essay is about the over-reading: a class permitting a property does not make the material have it, and the distinction between permitted and present is the one thing a symmetry argument most often gets credit for settling and does not. What the count does not say is the honest boundary of this whole field.