The parts a property splits into

Neumann’s principle says that a physical property of a crystal must be invariant under the crystal’s own symmetry, and its practical form is a count: how many independent numbers a measurement of that property can return. This collection has computed such counts twice already — as character sums, and again for a class written on a lattice basis.

What has not been said is what those counts are. For a large class of properties they are coefficients of an invariant ring’s series — the same integers the invariants ladder computes for polynomials — and the identification is exact rather than an analogy. For the rest, and the elastic tensor is the standard example, they are not, and knowing which case a property falls into is the useful part.

A symmetric tensor is a polynomial with indices #

A property of rank r is an array with r indices, each running over the directions of space. Fully symmetric means its value does not depend on the order of those indices, so the array is determined by how many times each direction appears — which is to say by a multiset.

A homogeneous polynomial of degree r is determined by exactly the same data: the coefficient of x^a y^b with a + b = r. So the space of fully symmetric rank-r tensors in the plane and the space of forms of degree r are the same space with two notations, and the group acts on them the same way — on the tensor index by index, on the polynomial by substitution.

That makes the count of independent components of such a property the count of invariant forms of that degree, which is the Molien coefficient. The identification costs nothing and explains a good deal: it says why the counts fall as the rotation order rises, why they are zero at odd ranks for centrosymmetric classes, and why the sequence for a class with an n-fold axis has a period related to n.

The count, made a second way #

Saying two spaces are the same is not a computation. To keep the claim honest the count is made again, on the tensor rather than on the polynomial.

An operation acts on a rank-r array one index at a time: each index is multiplied by the matrix, and the result summed. Averaging that action over the group projects onto the components the class permits, and the rank of the projection is the number of independent components. For rank four in the plane that is a 2⁴ = 16-component array, sixteen matrix multiplications per operation, and a rank taken in exact integers.

It agrees with the Molien coefficient at every rank on every class. The two computations share no code: one substitutes linear forms into a polynomial and eliminates; the other multiplies an array by matrices index by index. Agreement is therefore evidence, and the kind that would have caught a transposed matrix or a wrong action convention — which are the two mistakes this sort of code actually makes.

Where it stops: the elastic tensor #

The elastic constants are the standard rank-four property, and they are not a form.

Cᵢⱼₖₗ relates a symmetric stress to a symmetric strain, so it is symmetric within its first pair of indices and within its second, and — because it comes from a second derivative of an energy — symmetric under exchanging the two pairs. That is not the same as being symmetric under every permutation of the four indices. In the plane it leaves six independent components, where a fully symmetric rank-four tensor has five.

The difference is one component, and one component is enough to break the correspondence everywhere. The elastic counts across the ten plane classes come out 6, 6, 4, 4, 4, 3, 2, 2, 2, 2, and the Molien values at degree four are 5, 5, 3, 3, 3, 2, 1, 1, 1, 1. Every entry differs by exactly one, which is the extra piece surviving in each class, and no reading of the polynomial table gives the elastic numbers.

So a property with its own index symmetry needs its own average — the same projector, applied to an array with the right symmetry imposed on it beforehand. That is what the last column of the table is, and it is why this collection’s earlier elastic counts were computed from a character built for the purpose rather than read off a degree.

Which parts, not only how many #

The count of invariant components answers one question. A more informative one is what happens to the rest of the tensor: the components that are not invariant do not vanish, they transform among themselves, and how they do it is what decides which experiments can see them.

The space of tensors decomposes into pieces the group keeps separate — one piece per irreducible representation, appearing with a multiplicity — and the invariant components are the multiplicity of the trivial one. For the strain of a plane crystal, which is a symmetric rank-two tensor with three components, the decomposition under 4mm is: one invariant (the area change), one one-dimensional piece (the difference of the two axial strains), and one more (the shear). Under 6mm the last two join into a single two-dimensional piece, which is why a hexagonal crystal is elastically isotropic in its plane and a square one is not.

That last statement is a consequence of a decomposition rather than of a count, and it is not visible in the table above. It is visible in the strain essays, which need exactly this splitting to say what a transition does to the shape of a cell.

The basis does not know, and the components do #

The counts are basis-free. Molien’s formula uses each operation only through its trace and its determinant, both of which survive a change of basis, so the number of components a class permits is the same computed on a lattice basis or on Cartesian axes.

The components are not basis-free, and the distinction has bitten this collection before. On a hexagonal basis the invariant quadratic form is x² − xy + y², so the “one permitted second-rank component” of a threefold class is the coefficient of that form and not of x² + y². A reader who takes the count from a lattice-basis computation and then interprets the component in Cartesian axes has made an error the arithmetic cannot catch.

The rule that keeps it straight is short: a count may be taken in any basis; a component must be reported in the basis it was computed in. This site’s earlier essay makes the same point for elastic constants of a hexagonal crystal, where the count is right on integer matrices that are not orthogonal, and the constants themselves have to be transported back.

Ranks a crystal actually uses #

The table runs to rank four, and the four ranks below it name properties a laboratory measures.

Rank zero — a scalar, one component, always permitted. Density, heat capacity.

Rank one — a vector, permitted only where the class fixes a direction. In the plane that is 1 and m, which are the polar classes; the count is zero for every other class, and that zero is pyroelectricity forbidden.

Rank two — permittivity, conductivity, thermal expansion. Every class permits at least one, since the metric is always invariant; the classes with a rotation of order three or more permit exactly one, which is the statement that those crystals are isotropic in the plane for every rank-two property. That is a strong and slightly surprising claim — a threefold axis is not obviously enough to force isotropy — and it is the degree-two column of the table saying so.

Rank three — piezoelectricity. Zero for every centrosymmetric class, which in the plane means every class containing the half-turn.

The odd ranks and the centre #

The pattern of zeros in the table is worth one paragraph on its own, because it is the most-used consequence of the whole subject.

A class containing the half-turn — which in the plane is the inversion — sends every vector to its negative. An odd-rank tensor therefore changes sign under it, and a component that changes sign under an operation of the group must be zero. So every odd rank is empty for every centrosymmetric class, in one line, with no computation.

That is the argument behind the rule that a centrosymmetric crystal cannot be piezoelectric, cannot be pyroelectric, and cannot rotate the plane of polarisation — every one of those properties being odd-rank. It is also the reason the counts in the table alternate with such regularity: half the classes kill every odd row outright.

The interesting cases are the classes without the half-turn, where the odd ranks are not all zero and the counts have to be computed. The plane has five such classes, and their odd-rank counts are exactly what the permitted-property tables of this collection tabulate for three dimensions.

Permitted is still not present #

Nothing in this essay says a permitted component is non-zero. The table counts the numbers a measurement may return, and a crystal is free to return zero for any of them.

This collection has an essay about that gap and the point bears restating here because the arithmetic is so clean that it invites over-reading. A count of two independent elastic constants for a hexagonal crystal does not say the two differ. A permitted piezoelectric coefficient does not say a crystal is useful. What the count fixes is the dimension of the space the answer lives in, and the answer’s position in that space is a fact about the material.

The opposite error is rarer and worse: reading a zero as a statement about a material rather than about a class. A zero is a genuine prohibition, and it is the only kind of statement in this subject that a measurement can refute — which is what makes the odd-rank zeros the most useful entries in the table.

What a partition count explains #

For a class with a free invariant ring the count at rank r has a closed form: it is the number of ways to write r as a·d₁ + b·d₂ with a and b non-negative.

For 6mm, with degrees 2 and 6, that is the number of ways to make r from twos and sixes: one way for r = 0, 2, 4, two for r = 6, 8, 10, three for r = 12. Which is the sequence in the figure above, and it means the elastic-type counts of a hexagonal crystal are a partition count and nothing more.

For the four classes with a relation the same list of products over-counts, and the relation says by how much. The correction is the numerator 1 + t^e of the hypersurface series, and its effect on the property table is to remove one component at every rank from the degree of the relation upwards. A reader counting products of generators without knowing about the relation gets the wrong number for a rotation group, and the wrong number is too large.

The same table in three dimensions #

Everything above is a plane argument, and the plane is not where the properties are measured. It is worth setting out what changes and what does not.

The correspondence survives. A fully symmetric rank-r property in three dimensions is a form of degree r in three variables, and its component count is the Molien coefficient of the class acting on those three variables. Nothing in the derivation used the dimension.

The numbers do not. The strain space goes from three components to six, the elastic space from six to twenty-one, and a class such as 4/mmm permits six elastic constants rather than three. The three-dimensional counts are the ones this collection tabulated earlier, and they were computed there by a character sum built for each property kind rather than by an invariant ring.

One thing appears that has no plane analogue. In three dimensions a property can be axial — changing sign under an operation that reverses handedness, rather than under a rotation — and axial properties are counted by a modified character carrying the determinant of each operation. Optical activity is the example, and it is why fifteen classes may rotate light while eleven are chiral: the two counts are made with different characters and do not have to agree.

What a class cannot decide at all #

The last thing worth saying about a count is which questions it is not an answer to.

A property table says how many numbers a measurement returns and says nothing about their values, which is the permitted-is-not-present rule again. It says nothing about which directions those numbers refer to beyond what the class fixes: two crystals of the same class have the same count and can differ by any amount in the components.

And it says nothing about a property whose symmetry is not the crystal’s. A measurement made on a crystal that is twinned, or averaged over domains, or made at a temperature where the structure is not the one the class describes, is a measurement of something with a different symmetry — often a higher one — and the count that applies is the higher symmetry’s. That failure is common and quiet, because the count that comes back is a legitimate count of a class that is not the crystal’s.

The check against it is not arithmetic. It is knowing which structure the number belongs to, which is why every essay in this field ends by naming the object the count was made on.

What the pictures cannot show #

The figures here count and split; they cannot show that a decomposition is the decomposition.

That a space of tensors breaks into pieces the group keeps separate is a theorem about representations, and the pieces are subspaces rather than objects a figure can draw. What is drawn is the arithmetic — multiplicities, sums, ranks — and the geometry of one particular case, the rank-two surface, where the components happen to describe a shape.

Nor is any of this three-dimensional. Every count here is for a plane crystal, where the strain space has three dimensions and the elastic space six. In space those become six and twenty-one, the decomposition acquires pieces with no plane analogue, and the numbers in the three-dimensional table are the ones a laboratory uses. What survives the change of dimension is the argument: a property with the symmetry of a form is counted by an invariant ring, and a property with its own index symmetry is not.

The pieces the elastic tensor breaks into #

The essay’s “which parts, not only how many” points past the counts to a decomposition, and the elastic tensor is where that decomposition is most worth having — because it is the property whose twenty-one numbers are hardest to think about as a whole.

Decompose the space of elastic tensors under the full rotation group rather than under a crystal class. Twenty-one dimensions come apart as

$$21 = 1 + 1 + 5 + 5 + 9,$$

with the pieces being two scalars, two five-dimensional deviatoric parts, and one nine-dimensional part — the harmonic decomposition, and it is unique.

Each piece means something. The two scalars are the isotropic elasticity: an isotropic material has exactly them and nothing else, which is why engineering elasticity has two constants. The two fives measure the anisotropy that a rank-two quantity can see — the directions in which the material is stiffer, in the way an ellipsoid describes. And the nine is the part no lower-rank object detects, the genuinely fourth-order anisotropy.

That gives the class-by-class counts a reading the numbers alone do not have. A cubic crystal has three constants, and the decomposition says which: the two scalars, none of the fives, and one component of the nine. So cubic anisotropy is invisible to any rank-two measurement — it is entirely in the fourth-order piece — which is exactly why a cubic crystal is optically isotropic and elastically not.

The decomposition is under the rotation group and the counts are under a class, so the two are different calculations. What the first supplies is a vocabulary: instead of three independent constants, a cubic crystal has two isotropic ones and one fourth-order anisotropy, and the second phrasing says which measurements can see which.