What a crystal keeps in a field

Neumann’s principle says a physical property of a crystal must be unchanged by every symmetry the crystal has, and it is the whole of what the point-groups field is built on: symmetry decides which components of a tensor may be non-zero, before anything is measured.

It answers a question about a crystal sitting by itself. Curie’s principle answers the question about a crystal with something done to it, and it is one sentence longer:

the symmetry of an effect contains the intersection of the symmetries of its causes.

The crystal is one cause and the field is another. The crystal’s symmetry is one of the thirty-two; the field’s is one of the seven limiting groups; and the intersection is a subgroup of the class, computable exactly.

The arithmetic, which is four lines #

An operation of the crystal survives the field when it leaves the field alone, and whether it does is decided entirely by how the field’s quantity transforms.

So the intersection is a filter over the class’s own matrices, and it never leaves the world of integers. Nothing here needs the field’s strength, its units or a Cartesian frame — which is why the whole table above is computed in the lattice basis this collection has used since its first essay.

The result is always a subgroup, and that is worth checking rather than assuming: an intersection of two groups is a group, so its order must divide the class’s. The check runs over all thirty-two classes and all three field types and finds no exception, which is the kind of assertion that would catch a sign error in the axial rule immediately.

One detail of the statement is doing work and is easy to skip past. Curie’s principle is about the causes, plural, and the crystal is one of them. The temptation is to read it as a statement about a crystal being acted on by a field — an active thing and a passive one — but the arithmetic is symmetric: the intersection does not know which group came from the crystal and which from the field. That is why the same computation answers “what is left of a crystal in a field” and “what is left of a field in a crystal”, and why the second question, which sounds odd, has a use: a beam of light passing through a crystal is a field whose symmetry has been intersected with the crystal’s.

Three ways a class can respond #

Some classes lose nothing. A class survives an electric field along a direction intact when every one of its operations leaves that direction alone — which is exactly the definition of a polar direction, so the classes that lose nothing to a field are the ten polar classes. This collection counts those ten elsewhere by averaging a character over the group; here they arrive by intersecting with a cone. Two computations, one answer, no shared code.

Some lose almost everything. A cubic class in a field along a body diagonal keeps the three-fold axis about that diagonal and little else; the same class in a field along a general direction keeps the identity alone. The higher the symmetry, the more of it a field destroys, because more operations move the field’s direction.

And what is left decides what may happen next. The residual is a crystal class in its own right, so Neumann’s principle applies to it: the material in the field may have whatever properties the residual class permits, whether or not the original class permitted them.

That last sentence is the whole use of the principle, and it is worth an example.

The three fields, compared #

The table has three columns and the differences between them are the whole of the second rung.

An electric field takes the most away. Its group is the cone, the smallest of the three uniaxial ones with a mirror in it, so its intersection with any class is the smallest. Only the ten polar classes survive intact.

A stress takes the least. Its group is the cylinder, which contains the cone, so its intersection contains the electric field’s for every class at once. Twenty-seven of the thirty-two keep a direction under some stress, because a stress does not care which way along its axis it points.

A magnetic field takes away something different rather than something more. Its group is the turning cylinder — it has the mirror across its axis and none containing it, which is the reverse of the cone. So a class with mirrors containing the axis loses them to a magnetic field and keeps them in an electric one, and a class with the mirror across the axis does the opposite. The two lists of classes that lose nothing are not nested at all; they overlap and neither contains the other.

That last observation is the one that surprises, and the determinant is the whole of it. This collection has met it before from the other side, when time reversal was admitted as an operation and the thirty-two classes became a hundred and twenty-two: a magnetic quantity behaves differently under reflection from an electric one, and every count that involves both has to keep them apart.

Why poling a ceramic works #

A polycrystalline ceramic is a jumble of grains pointing every way. Averaged over the grains it has no direction at all, so its symmetry is one of the two axis-free groups — ∞∞m, a sphere, if the material is not chiral. That group contains a centre of inversion, and a centre forbids a polar vector, so the ceramic has no spontaneous polarisation. It cannot be pyroelectric and it cannot be piezoelectric.

Apply a strong electric field along an axis. The field’s group is ∞m, a cone. The intersection of ∞∞m with ∞m is ∞m — the cone — because the cone is contained in the sphere’s group.

And ∞m has no centre. So the ceramic in the field is permitted a polar vector along the axis, and once the domains have been switched to take it up, the material keeps one after the field is removed. That is what poling a piezoelectric ceramic is, and the argument for why it can work at all is three group intersections with no physics in them.

What the argument does not say is that poling will work. The intersection supplies a permission — a polar vector is now allowed — and whether the material has domains that can be switched, and whether they stay switched, is chemistry and thermodynamics. The principle is a gate, not an engine.

And the same argument runs backwards. A material that is already polar — one of the ten classes — needs no field to have a polarisation, and applying one along its polar axis takes nothing away at all. That is the difference between a ferroelectric single crystal and a poled ceramic: the first is permitted its polarisation by its own class, the second only by the residual class the field created.

Two causes, and the ordering #

The principle is stated about causes in the plural, and with two fields it says something stronger than with one: the effect’s symmetry contains the intersection of all the causes, so adding a second cause can only take symmetry away.

There is a useful corollary in the containments among the seven. A stress along an axis has the cylinder’s group ∞/mm; an electric field along the same axis has the cone’s ∞m; and the cone is a subgroup of the cylinder. So the intersection with a stress always contains the intersection with a field, and

a stress can never break a symmetry that a field along the same direction leaves intact.

That ordering is a fact about the seven and holds for every crystal class at once, which is the kind of statement the limiting groups exist to make.

Counting what a field selects #

There is a second consequence of the residual, and it is a count rather than a permission.

The number of domain states a symmetry-lowering step produces is the index of the new group in the old one — a number available before any crystal is grown. A field applied to a crystal is exactly such a step, so the same arithmetic applies: a class of order |G| left with a residual of order |H| has |G| / |H| distinct orientations of the residual available, and each region of the crystal takes one.

For a class of order 16 reduced to one of order 8, the index is two and there are two states: the field selects between them and the crystal ends up in whichever one the field favours. That is what “poling” means as an operation on domains, and it is why a field can convert a multi-domain crystal into a single-domain one without changing anything about the structure.

And the index is an upper bound on what a field can do. A crystal with an index of one — a class the field leaves whole — has nothing to select between, and a field applied to it changes no orientation whatever. The ten polar classes are precisely the classes an electric field cannot pole, because they are already poled.

A count of states is not a count of components, and the two are easy to run together because a field moves both. The index says how many orientations of the residual are available and therefore how many distinct regions a crystal may break into. The component count says how many independent numbers a measurement of one region can return. A descent of index two may leave the component counts of every property exactly where they were — the two states are the same crystal seen two ways — or it may add a component that was forbidden, which is what happens when the operation lost is the one that was killing it. Which of the two occurred is read off the property table for the residual class and not off the index.

Where it sits beside Neumann #

The two principles are usually taught together and are doing different jobs.

Neumann’s takes a group and a property and returns a count of independent components. It is an average over the group, exact in integers, and this collection computes it for all thirty-two classes and six properties.

Curie’s takes two groups and returns a third. It supplies the input to Neumann’s when the situation involves more than the crystal alone — and every real measurement does, because measuring a crystal means doing something to it.

Put together: a crystal of class 4/mmm has a centre and is permitted no polar vector. Apply a field along its four-fold axis and the residual is 4mm, which has none — and the material in the field is permitted one. Neumann’s principle has not changed; the group it is being applied to has.

Why the answer is always a crystal class #

One feature of the table is worth stating plainly because it is not obvious in advance: every residual is one of the thirty-two.

It could have been otherwise. An intersection of a finite group with an infinite one is a finite group, certainly — but a finite group of isometries need not be crystallographic, and there are infinitely many that are not. What guarantees the answer is that the residual is a subgroup of the class, and a subgroup of a crystallographic point group is crystallographic. So the field can only move a crystal down the list it was already on.

That is the same containment argument the holohedry makes about a lattice and its contents, applied one level out: a crystal never has more point symmetry than its lattice, and a crystal in a field never has more than the crystal.

What the intersection does not decide #

The intersection is not the whole story when the field has a position. Everything above is about point groups — operations that leave a point fixed — because a uniform field has no translations to speak of. A crystal has translations, and a field applied to it does not remove them, so the residual space group is the crystal’s space group with its point group cut down. That step is straightforward and is not made here; what is computed is the point-group half, which is the half every property table is indexed by.

Time reversal is a second kind of operation and it is not covered here. A magnetic field is odd under reversing the direction of time, so the group of a magnetic structure carries operations paired with a time reversal, and the classification of those — the magnetic point groups, of which there are ninety — is where Curie’s principle has to be run for anything magnetically ordered. The intersection rule survives the extension unchanged; what changes is the list of groups being intersected, and the field’s own group has to carry its behaviour under the reversal along with its geometry. This collection names that extension rather than deriving it, and every count above is a count for a structure with no magnetic order in it.

“Contains” is not “equals”, and the gap is real. Curie’s principle gives a lower bound on the effect’s symmetry: the effect has at least the intersection and may have more. A crystal in a field may respond with something more symmetric than the intersection permits — most obviously by not responding at all, which is the most symmetric response available. Reading the intersection as a prediction of the effect’s symmetry is the commonest misuse of the principle and it is a misuse in the same direction as reading a permission as a value.

The direction matters and the table above fixes one. Every residual here is computed for a field along the axis of the class’s own conventional setting, which is the case an experiment usually arranges. Along a general direction the residual is smaller — often the identity alone — and along another symmetry direction it is something else again. The class does not have a residual; it has one per direction.

Nor is it about which effect. The intersection says which class the situation has, and Neumann’s principle then says which properties that class permits — but nothing says which of the permitted effects actually occurs, or how large it is, or whether two of them compete. A residual class of 4mm permits a polarisation along the axis and a piezoelectric response and an optical activity that a higher class forbade; all three become allowed at once, by the same argument, and no ordering among them is implied.

And nothing here is about time. A field switched on has a history, and a crystal that has been in a field and had it removed is not the same object as one that never had it — which is what makes poling permanent and what makes a hysteresis loop a loop. Curie’s principle is a statement about a situation and not about a process, and every essay in this field observes the same limit.

The direction the principle does not run #

Curie’s principle bounds the effect’s symmetry from below, and the standing objection to it is that effects with less symmetry than their causes are everywhere. A rod compressed along its axis buckles to one side; a liquid cooled through a freezing point picks a lattice orientation; a crystal cooled through a transition picks a domain. In each case the cause is more symmetric than what happens.

The objection is answered by being precise about what “the effect” is. The buckled rod is one solution of a symmetric problem, and the reflected buckled rod is another. Both are solutions; the equations do not prefer either. The set of all solutions has the symmetry the cause had, and the symmetry has been broken only by the additional information of which solution was realised — which came from a fluctuation, an imperfection, or a boundary the analysis did not include.

So the principle is a statement about the ensemble. Read that way it survives every apparent counterexample, and it stops being a prediction about an individual sample. A single crystal that has chosen one domain is not a counterexample to Curie any more than a coin that landed heads is a counterexample to the symmetry of the coin.

This collection already computes the ensemble. The number of domain states a transition produces is the index of the residual group in the parent, and those states are exactly the solutions the principle is quantified over. Curie’s principle and the domain count are therefore the same statement seen from two ends: the index counts how many ways the symmetry can be broken, and the principle says that all of them together restore it.

The practical consequence is the one every experimenter meets. A measurement on a multi-domain sample sees the ensemble and returns the parent symmetry, so a genuinely lower-symmetry phase can look like the phase above it. Finding the lower symmetry requires either a single domain — a poled ceramic, a strained thin film, a crystal cooled under a bias — or a technique that resolves domains rather than averaging over them.

The field that reverses with time #

The last caveat above says that nothing on this page is about time, and the omission has a consequence sharp enough to be worth naming, because it decides which of two fields the whole table applies to.

An electric field is unchanged by reversing time, since it is produced by charges sitting still. A magnetic field is reversed by it, since it is produced by charges in motion and reversing time reverses every current.

So the two fields need different limiting groups, and only one of them is on this page. An electric field along an axis has the symmetry of a stationary cone — every rotation about the axis, every mirror containing it, no operation reversing the axis. A magnetic field along the same axis has the symmetry of a spinning cylinder: it keeps rotations about the axis and the plane perpendicular to it, and loses the mirrors that contain the axis, because a mirror reverses a circulation.

The consequence is a different residual and a different list of permitted effects. The intersection computed against an electric field permits a polar vector, which is why poling works. The intersection computed against a magnetic field permits an axial vector, which is a quite different object — it does not change sign under inversion, so a centrosymmetric crystal is free to have one, and the whole argument about centres that governs polarisation does not apply to magnetisation at all.

Doing this properly needs operations of a second kind. Time reversal is a symmetry operation in its own right, and the groups that include it — the magnetic point groups — are the setting in which magnetic effects are constrained. They are built exactly as the antisymmetry groups elsewhere in this collection are: an ordinary group, a subgroup of index two whose operations act unprimed, and the rest acting together with time reversal.

The table above is therefore complete for what it covers and silent past it. Everything computed here is right for an electric field, a stress, and any other cause that is even under time reversal. For a magnetic cause the method is identical and the ingredients are not, and substituting one field for the other in these rows is the commonest way the principle is misapplied.