Properties — the series · Crystallography & Symmetry

A spontaneous vector in 222. Averaging each of the three axes over the 4 operations of 222 leaves 0 independent components. No direction survives, so the class permits no spontaneous polarisation at all — which is a statement about what is forbidden, not about any measurement.

The ten with a direction of their own

A crystal has a spontaneous electric polarisation only if some direction is left completely alone by every one of its symmetry operations. Ten of the thirty-two classes have such a direction, and the same ten are computed here twice by routes that share nothing but the group.

part 1 · point-groups

What the inversion does to three sums in 4/mmm. The 16 operations of 4/mmm come in pairs — every operation together with its own negative, because the class contains the inversion — and each row pairs the two terms they contribute. For an even-rank polar property the pair is two equal bars: the inversion changes an even number of indices and the character cannot see it, so the average is whatever it was before the inversion was added. For an odd-rank polar property and for an axial one the pair is a bar and its reflection, and the sum is exactly zero — which is why piezoelectric moduli and gyration tensor are forbidden here outright rather than merely small. Neither statement is about this class: the cancellation is checked over all 11 centrosymmetric classes and all four odd or axial properties every time this figure is drawn.

Twenty of the twenty-one

Twenty-one crystal classes have no centre of symmetry, and twenty of them permit piezoelectricity. The exception is 432, which has twenty-four operations, no inversion, and a character sum that cancels to nothing — and the reason it fails is not that it has too much symmetry in any ordinary sense.

part 2 · point-groups

May be piezoelectric, against may be optically active. Two questions asked of all thirty-two classes, and the classes where the answers part company. 14 classes are in both lists, 6 in only the first, 1 in only the second and 11 in neither. Both lists are computed from the same character sum with a different tensor, so a class appearing in one and not the other is a statement about which representation survives rather than about anything measured. Every entry is a permission: a class in a column is a class whose symmetry fails to forbid the effect, which is a weaker statement than it is usually read as.

Each permits what the other forbids

432 and 4̅3m are both cubic, both of order twenty-four, both without a centre. One of them can be piezoelectric and the other can be optically active, and it is not the same one — which is as clean a demonstration as the subject offers that "amount of symmetry" is not a quantity.

part 3 · point-groups

Every property count, in every class. For each of these 32 classes, how many independent components a property may have: elastic constants from 21 down to 3, piezoelectric moduli from 18 down to 0, dielectric tensor from 6 down to 1, pyroelectric vector from 3 down to 0, gyration tensor from 6 down to 0. Every number is a character averaged over the class's own operations, computed in the lattice basis with integer arithmetic. A zero means the symmetry forbids the property outright; a positive number means it does not forbid it, which is a weaker statement than it is usually read as.

A filter of great precision and no predictive power

The whole table in one place — thirty-two classes, six properties, 192 exact integers. What it settles, what it merely permits, and why knowing which of the two is happening at any moment is the entire skill of using it.

part 4 · point-groups

Five strains, and which of them move the atoms. A honeycomb of harmonic bonds, strained five ways, with the internal coordinate minimised at fixed cell each time. The shuffle is how far the second atom moves away from where the strain alone would have put it. Which strains produce one is decided before any energy is computed: a strain that leaves the site's three-fold axis intact forces the shuffle to vanish, because the only vector a three-fold rotation of the plane fixes is the zero vector. The prediction and the measurement are in adjacent columns.

The strain the atoms do not follow

The rule that makes an elastic constant computable — deform the cell and move every atom by the same map — is exact for a lattice with one atom in it and wrong for every other, and the reason is a site symmetry rather than a mechanical one. Which strains move the atoms inside the cell is decided by what survives of the site's own group, and a wrong answer here is a constant that is too stiff by a third.

part 5 · point-groups

What Kleinman's rule removes from the second harmonic, class by class. The 20 crystal classes that permit second-harmonic generation, each with its independent susceptibility components drawn as squares: those that survive when the tensor is made totally symmetric, and those Kleinman's rule removes, split into the traceless part of the gyration tensor and a polar vector. In all, 83 components become 48. Two classes, 422 and 622, lose everything; four, 6̅, 6̅2m, 23, 4̅3m, lose nothing.

Two classes go dark far from absorption

The second-harmonic tensor of a crystal has as many components as its piezoelectric tensor, and far from any absorption it gains a symmetry of its own that removes eight of the eighteen. What it removes is not arbitrary: it transforms as a polar vector plus the traceless part of the tensor that carries optical activity. So the classes that lose the most are the polar ones and the handed ones, and two of them, 422 and 622, lose everything.

part 6 · point-groups

How many resonances each crystal class needs for a second-order response. The 32 crystal classes sorted by the number of distinct oscillators a model needs before the class can have any second-order optical response, with the dimensions of the class's invariants among the cubic couplings: fully symmetric (one oscillator), mixed (two), alternating (three). The eleven centrosymmetric classes have none. Eighteen need one; four of those (23, 4̅3m, 6̅, 6̅2m) have no mixed invariant and obey Kleinman's rule at every frequency. 422 and 622 need two. 432 needs three.

A crystal in 422 needs two resonances

Kleinman's rule silences classes 422 and 622 far from absorption, and the obvious guess is that dispersion brings them back near it. In the simplest model of dispersion, one charged oscillator with an anharmonic potential, it does not: they stay dark at every frequency. Two oscillators with different resonances light them, with a response growing as the square of the frequency. 432 needs three, and then only mixes two different frequencies, growing as the sixth power. The number of resonances a class needs is a count of invariants, and it sets how fast the forbidden part can grow.

part 7 · point-groups