Crystallography & Symmetry
patterns derived from their groups
Series

Domains — the series

10 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. m3̅m → 4mm: 6 domain states. The transition from class m3̅m to class 4mm loses 40 of the parent's 48 operations, so the child has index 6 and the crystal comes apart into 6 domain states. Each colour is one state — one coset of 4mm in m3̅m — and each holds the same 8 poles. The lost operations are what carries one state onto another, and they survive in the crystal as the relation between its domains rather than as symmetries of any part of it. The descent changes the crystal system, so the states differ in shape as well as in orientation and the transition is ferroelastic.

    How many domains a transition makes is an index

    Cool a crystal through a symmetry-lowering transition and it has to choose one of several equally good low-symmetry arrangements. Different parts of it choose differently, and the number of available choices is the index of the new group in the old one — a number available before any crystal is grown, and one of the few predictions in this field that is a count rather than a bound.

    part 1 · applied
  2. Everything class m3̅m can descend to. The 25 crystal classes that are subgroups of m3̅m, arranged by order, with the 56 maximal steps between them drawn as edges. The order of each row is printed down the left, so the index of any step is the ratio of the two rows it joins. A symmetry-lowering transition can only be continuous when it goes down one of these edges, and the index on the edge is the number of domain states the transition produces. A descent of several steps is possible but has to happen discontinuously or through the intermediate classes.

    The descent of symmetry is a lattice, not a tree

    Which classes a crystal can fall to when it loses symmetry, drawn as a graph with the index on every edge. It is routinely called a tree and it is not one — a class can be reached from its parent by several different routes of the same total index, and which route a material takes is a physical question the diagram deliberately leaves open.

    part 2 · applied
  3. An antiphase boundary in p4. Where two antiphase states meet. Above the line the species alternate one way and below it the other, so at the boundary two cells of the same species sit next to one another and the ordering is out of step. This is a domain wall with no change of orientation across it: the crystal is not twinned, its lattice is undisturbed, and diffraction sees it only in the width of the superlattice reflections.

    The domains a lost translation makes, which nothing optical can see

    An ordering transition can leave the crystal class untouched and take away translations instead. The domains that result have the same orientation, the same shape and the same optical properties as each other, and where two of them meet the ordering is simply out of step — a boundary with no change of direction across it and no way to find it except by looking at the ordering itself.

    part 2 · applied
  4. How many ways each class can lose symmetry. Every crystal class, with the number of distinct classes it can descend to — 247 parent-and-child pairs in all across the thirty-two, counted up to conjugacy in the parent, which is the equivalence that says two descents differing only by which axis was chosen are one transition. The count rises steeply with the order of the parent, which is why the cubic and hexagonal holohedries dominate the list of materials with rich domain structures.

    Two hundred and forty-seven descents, or two hundred and twelve

    How many distinct ways can a crystal lose symmetry? Counting parent-and-child pairs up to conjugacy in the parent gives 247. The standard enumeration in the ferroics literature gives 212, and the operation that merges the extra thirty-five turns out to be a rotation through forty-five degrees — which no lattice may have, and which no integer matrix in a lattice basis can therefore express.

    part 3 · applied
  5. 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.

    part 5 · applied
  6. 3m → 1: the two directions a wall between domains may take. The difference between the strains of two domains, sampled around a circle of directions: the first colour where that direction is stretched, the second where it is compressed. The two solid lines are the directions where it is neither, and those are the only orientations a straight wall between the two domains can take without straining itself — Sapriel's condition, one dimension down from the planes it is usually written for. There are exactly two, and that is not luck: the two domains are images of one another under the parent group, so their strains have the same area change and their difference changes no area at all. A form that changes no area takes both signs, and its zero set is a pair of directions.

    The walls a strain permits

    Two domains of different shape can only meet along a line neither of them stretches. That condition is a quadratic in a direction, so a pair of domains has exactly two permissible walls — and the reason there are always two rather than sometimes none is that their strains differ by no area at all.

    part 6 · applied
  7. 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.

    part 7 · applied
  8. Every wall is a frieze. Each ferroelastic descent, with the frieze group of each of the two walls its domains permit. A wall is periodic along its length and bounded across it, so its symmetry group is one of the seven — the classification this collection derived early as the same argument on a strip, arriving here as a fact about interfaces. The last column counts the operations in the wall's group that exchange the two domains rather than fixing them: a wall is unchanged by having its sides swapped, so those belong to it, and they are why a wall is often more symmetric than either domain.

    The wall has a group of its own

    A boundary between two domains is periodic along its length and bounded across it, so its symmetry is a frieze. The seven, derived here early on as an exercise on a strip, turn out to be the classification of interfaces.

    part 8 · applied
  9. Three variants, and not one undistorted plane. The three tetragonal variants a cubic parent produces, with the principal stretches of each. Every one of them has the same three numbers in a different order, and the middle one is not one — so none of the three leaves any plane undistorted, and none of them can meet the parent phase across an interface. That is the difficulty the whole of the crystallographic theory of martensite exists to resolve, and it is visible in one column.

    The plane a deformation leaves alone

    Two differently deformed regions can meet across a plane only if that plane is deformed identically from both sides — which forces the two deformations to differ by a rank-one term. Multiplying each side by its own transpose removes the rotation and leaves a condition on a signature: one positive eigenvalue, one negative, one exactly zero. In that form the classical rule that the middle principal stretch must be one is not quoted but derived, and it says that no single variant of a cubic-to-tetragonal transition can meet its parent at all.

    part 9 · applied
  10. Where the laminate's middle eigenvalue crosses zero. The middle eigenvalue of FᵀF − I for the average deformation of a twinned laminate, against the volume fraction of one variant. At both ends the laminate is a single variant and the value is well away from zero; in between it crosses, twice, and each crossing is a volume fraction at which the laminate can meet the parent phase across a plane. The two roots are complementary, which is the same plate with the two variants exchanged.

    The plate that only fits when it is twinned

    No single variant of a cubic-to-tetragonal transition can meet its parent across a plane. A fine mixture of two variants can, because its average deformation carries a free parameter — the volume fraction — and that parameter passes through the compatibility condition twice. Sweeping it gives the two fractions, the two habit planes, and one inequality: the plate exists exactly when the two principal stretches satisfy η₁² + η₃² ≤ 2.

    part 10 · applied

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