Crystallography & Symmetry
patterns derived from their groups
Concept

Second harmonic generation — where it appears

The conversion of light of one frequency into light of twice that frequency inside a crystal, through a polarisation quadratic in the field. It needs a crystal without a centre of symmetry, and its tensor has the shape of the piezoelectric one.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

point-groups · Properties 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.

point-groups · Properties

Named alongside it

The objects these essays reach for when they reach for this one.

CharacterGyration tensorIrreducible representationOptical activityPolar vectorProperty tensorPiezoelectricity

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