Crystallography & Symmetry
patterns derived from their groups
Concept

Coupling — where it appears

A term in an energy involving two different quantities, which forces one to become non-zero when the other does. A coupling linear in a polarisation and quadratic in an order parameter is what makes a crystal polar as a side effect.

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

Also named here as secondary order parameter — the same set of essays touches all of them, so they are one junction rather than several.

Which descents change the shape of the cell, and into how many shapes. Each descent the modes produced, with the number of independent strain components the parent class permits and the number the child permits. A transition is ferroelastic exactly when the second is larger — the child leaves alone a distortion the parent moves — and the difference is a spontaneous strain the crystal acquires without being pushed. The count of distinct shapes is the orbit of that strain under the parent, which can be smaller than the number of domains: two domains may differ in something a change of shape cannot show. Every count here is a rank of an averaged set of quadratic forms, computed twice — once by averaging, once from a character.

The strain that arrives with the transition

A crystal that loses symmetry usually changes shape, and whether it does is a subtraction: how many strain components the child permits, minus how many the parent did. The difference is a distortion nobody applied, and it is what makes a domain visible in a microscope.

applied · Domains A mode with no dipole, landing in a phase that may have one. Two marks per row: the first is filled when the mode itself carries a dipole — the displacements, weighted by charge, summing to something other than zero — and the second when the class of the phase it produces permits a polarisation at all. A row with the first empty and the second filled is an improper case: nothing about the transition was about becoming polar, and the phase that results may be polar anyway, so a polarisation appears as a side effect at second order in an order parameter that is about something else. The zone-boundary rows are where these occur; at the zone centre in the plane there are none, because the only two-dimensional order parameters available there are the polarisation itself.

The polarisation nobody asked for

A mode whose displacements cancel exactly can still leave a phase whose class permits a polarisation. The crystal then becomes polar as a side effect of a transition that was about something else — and in the plane, at the zone centre, the arithmetic says this cannot happen at all.

applied · Domains

Named alongside it

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

Order parameterSecondary order parameterAizu criterionDipole momentDomain stateFerroelasticityImproper ferroelectricityIndependent componentsInvariant metricPolar classSpontaneous strainStrain tensor

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