The seven groups a field can have

Thirty-two and no others is the count of point groups a crystal may have, and every one of them is finite. That finiteness is not an accident of the subject; it is the crystallographic restriction doing its work, because a lattice permits only rotations of order one, two, three, four and six and a group built from those has finitely many elements.

A uniform field has no lattice. An electric field pointing along a direction is unchanged by a rotation about that direction through any angle whatever, and its symmetry group is infinite.

Curie asked in 1894 how many such groups there are, and the answer is seven.

The count is small enough to be suspicious of, and the derivation below is short enough to check. What makes it short is that “having an axis” is a strong assumption: it fixes the rotations completely and leaves only the question of what else there is, and the answer to that turns out to be a list of four.

Two families and a closure #

An object of this kind either has an axis or does not.

With no axis, there are two groups and no argument to make: a sphere, whose symmetry is every rotation and every reflection, and a sphere made of something with a handedness — a solution of a chiral compound, say — whose symmetry is every rotation and no reflection at all.

With an axis, the object has by assumption every rotation about it, and the question is what else it may have. The answer is a list of four:

• σv, a mirror containing the axis;
• σh, the mirror across the axis;
• C2′, a two-fold rotation across the axis;
• i, a centre of inversion.

Sixteen combinations. And they close onto five.

Any two of σv, σh and C2′ produce the third. A mirror containing the axis composed with the mirror across it is a half turn about their line of intersection, which is a two-fold across the axis; the other two products go the same way. So a set with two of them has all three.

And the mirror across the axis brings a centre. σh composed with the half turn in the axis — which every object of this kind has, since it has every rotation about it — is the inversion. So σh and i arrive together and neither can appear alone.

That leaves five closed sets: nothing, {σv}, {σh, i}, {C2′}, and all four. Five groups, and with the two axis-free ones, seven.

That is the frieze argument with a different list of features, and the resemblance is not a coincidence about the number sixteen. Both are the same procedure: take the features an object of a stated kind may have, form every subset, close each one, and count what is left. A classification that can be run this way is a classification whose answer is short.

Why exactly four features and not five. The list of extras is not a choice, and it is worth saying why it closes. A uniaxial object’s symmetry operations either fix the axis pointwise, reverse it, or move it elsewhere — and the third is impossible, since the axis is the object’s own. Of the operations fixing it, the rotations are already assumed. Of the ones reversing it, an operation is determined by its determinant and by whether it fixes a plane: that gives the two-fold across the axis, the mirror across the axis, the mirrors containing it, and the inversion. Four, and there is nothing else for an operation to be.

That is the same style of argument as the four motions: classify by what an operation does to a distinguished object, and the list is short because there are only so many things it can do.

What each group is the symmetry of #

The names are the International Tables’ with an infinity in them, and each is an object.

∞/mm is a cylinder. Every rotation about the axis, the mirror across it, mirrors containing it, two-folds across it, a centre. It is the symmetry of a uniaxial stress — squeezing a body along one direction — and of anything else with an axis and no sense of direction along it.

∞m is a cone. The mirror across the axis is gone, because a cone has a point at one end and a base at the other. It is the symmetry of an electric field, and of a polar vector generally: the arrow has a head.

∞/m is a turning cylinder. The mirrors containing the axis are gone, because reflecting in one of them reverses the turn. What is left is the mirror across the axis and the centre. This is the symmetry of a magnetic field, which is the fact behind everything peculiar about magnetism’s relationship to symmetry — a magnetic field is not an arrow but a circulation, and its group says so.

∞2 is a twisted cylinder, and ∞ is a turning cone.

Where the infinity is, and where it is not #

One thing about the notation is worth clearing up, because the symbols hide it.

The infinity in ∞m is a rotation axis of infinite order, and it is the only infinite thing in the group. There are still finitely many kinds of operation — rotations about the axis, and one kind of mirror — and the closure argument above counted kinds rather than elements. A group with infinitely many elements can have a short and complete description, which is what made the enumeration possible.

Compare that with the plane groups, which are also infinite and are infinite in a different way. A wallpaper group has infinitely many translations and finitely many rotations, and it is classified by the finite part with the translations quotiented out. A limiting group has infinitely many rotations and no translations at all. Neither is finite; both are classified; the thing being counted differs.

Four rules, and the difference between two arrows #

The link between an abstract group and a physical quantity is a rule about how the quantity transforms, and the rules are short enough to write in a row.

A polar vector survives an operation when the operation sends it to itself: M v = v. An axial vector survives when (det M) M a = a — one extra factor, which is −1 for a reflection or an inversion and +1 for a rotation.

That factor is the whole difference, and it accounts for everything odd about the two. A mirror containing an electric field’s direction leaves the field alone; the same mirror reverses a magnetic field along the same direction. A mirror across an electric field reverses it and leaves a magnetic one alone. Neither statement is a convention about how to draw arrows: it is the determinant.

And the rules hold in any basis, which is why this collection can compute the whole of it in a lattice basis without ever choosing a Cartesian frame — the same point a character makes about elastic constants.

Those ten are the polar classes, and this collection has counted them before by a completely different route — averaging a character over the group and asking how many components of a vector survive. Here they arrive by intersecting each class with the cone’s own group and asking which classes lose nothing. Two computations sharing only the thirty-two matrices, required to give the same ten.

A cylinder has a centre and a cone does not, and that single difference decides most of what follows in this field. A crystal in a cone-shaped field can be pyroelectric; a crystal in a cylinder-shaped one cannot, because the centre the cylinder brings survives into the intersection and a centre forbids a polar vector. Curie’s principle turns that observation into a rule, and the rule is the next rung.

Two of the seven have no mirror at all #

The pair worth dwelling on is ∞ and ∞∞, because they are the ones a chemist meets under another name.

∞∞ is the symmetry of a liquid made of a single enantiomer. Every rotation, because a liquid has no direction; no reflection, because a reflection would turn every molecule into its mirror image and produce a different substance. It is the group of a chiral isotropic medium, and its being distinct from ∞∞m is exactly why a sugar solution rotates the plane of polarised light and a solution of a symmetric molecule does not.

∞ is the symmetry of the same liquid in a magnetic field along an axis — or of a cone that is spinning. It has the rotations about the axis and nothing else, and it is the smallest of the seven.

Both are subgroups of everything above them and neither contains a reflection, which is the same distinction the eleven enantiomorphic classes make among the thirty-two. A group with no operation of negative determinant is a group in which a handedness can live.

The fifth quantity, and why a stress is not a vector #

The list of rules has a fifth entry that is neither a scalar nor a vector, and it is the one an experiment most often applies.

A uniaxial stress — a body squeezed along one direction — has an axis but no sense of direction along it: squeezing is the same whichever end is pushed. Its rule is that an operation survives when it maps the axis to plus or minus itself, and the two signs are what separates it from a polar vector. So a stress along an axis has the cylinder’s group, ∞/mm, and it is the most symmetric of the five uniaxial fields.

A second-rank property has the same shape and is met more often. Permittivity, conductivity and thermal expansion are all symmetric second-rank tensors, and each is drawn as an ellipsoid whose axes are the property’s principal values — the optical one is called the indicatrix. That ellipsoid’s own symmetry is a limiting group and not a crystal class: a general one has three unequal axes, one with two equal axes has a cylinder’s group, and a sphere has the isotropic one. A cubic crystal’s indicatrix is a sphere, which is the whole content of the statement that a cubic crystal is optically isotropic — and it is the same fact the figure two sections above measures, that a property of rank r cannot tell an axis of order n from an axis of every order at once once n passes r.

That is why a stress takes less away from a crystal than an electric field of the same direction does. The cylinder contains the cone, so the intersection with a stress contains the intersection with a field, and the residual class is larger. A more symmetric cause leaves more symmetry behind, which sounds obvious and is the whole content of Curie’s principle read one way round.

The seven are not point groups of anything crystalline #

No crystal has a limiting group, and none ever will. The restriction forbids a continuous rotation axis outright — an infinite rotation group applied to a lattice generates infinitely many lattice vectors of the same length, and a lattice has finitely many of each length. The seven live entirely outside the classification of crystals, and they sit above all of it: every one of the thirty-two classes is a subgroup of at least one of the seven, so intersecting a class with a limiting group can only ever land back among the thirty-two.

That containment is what makes the rule usable. If an intersection could leave the classification, the answer to “what does a field leave of this crystal” would be an object with no name and no table of permissions; because it cannot, the answer is always one of thirty-two rows a reader can look up. The limiting groups are the ceiling of the classification and never a part of it.

Their use is in what is done to a crystal. A field applied, a stress imposed, a direction of growth, a beam of light: each is an object with one of these groups, and what the crystal keeps of itself is the intersection of the two — which is Curie’s principle, and the second rung of this ladder.

The answer depends on where the field points #

The intersection rule is stated as though the two groups were fixed objects. They are not: a crystal’s group and a field’s group both sit in space, and the intersection depends on how they are placed relative to each other. That turns a rule into a computation with more than one answer.

Apply an electric field to a crystal of class 4/mmm. Along the four-fold axis, the field’s ∞m and the crystal’s group share the four-fold, the four vertical mirrors and nothing else — the horizontal mirror and the two-folds across the axis are lost, and what remains is 4mm. Along one of the two-fold axes instead, the surviving group is mm2. And along a general direction the intersection is m or the trivial group, depending on whether the direction lies in a mirror.

So one crystal and one kind of field give several different answers, and which one applies is decided by an orbit computation: the distinct results correspond to the distinct classes of direction under the crystal’s own group, which is the same orbit-and-stabiliser arithmetic that sorts points of a cell into Wyckoff positions, applied to directions.

That is why an experiment on a field-induced effect specifies an orientation and not merely a field. The permission changes with the direction, and a measurement made along a general direction is measuring a crystal in a lower symmetry class than the same crystal measured along its axis. The classification supplies the list of cases before the crystal is mounted, which is exactly what makes the experiment designable.

What a field does to a set of domains #

There is a use for the intersection rule that is not about permissions at all, and it is the one that turns a laboratory field into a tool rather than a perturbation.

A crystal that has been through a symmetry-lowering transition is a mosaic of domain states, related by the operations that were lost, and in the absence of any field they are equally favourable — which is why all of them occur. Apply a field and they stop being equivalent.

The bookkeeping is the intersection rule read on the parent. The lost operations form cosets; a field’s own group intersected with the parent’s picks out the operations that survive; and the domain states related by a surviving operation remain equivalent while those related by a lost one do not. The number of distinct states left is the index of what survives, computed exactly as before.

The practical form is poling. A ferroelectric cooled through its transition has domains pointing every permitted way and no net polarisation; an electric field applied along one of the permitted directions makes the domains aligned with it cheaper, they grow at the others’ expense, and the crystal ends up as a single domain. Which directions are available, and how many domains a given field can leave, is decided entirely by the two groups — and how strong a field is needed, and whether the walls can move at all, is not.

That figure is the whole argument of this rung in one row of numbers, and it is worth reading twice. Nothing about the grains changed: the same crystallites, the same class, the same moduli, before and after. What changed is the symmetry of the distribution of their orientations — from a sphere’s worth to a cone’s — and the count of what a measurement can return moved with it. A property permitted by every grain individually can be forbidden by the arrangement, and the arrangement has a group like anything else.

What the seven do not settle #

They are groups of a uniform field. Every claim here assumes the field is the same everywhere, which is what makes the rotation axis continuous. A field with a gradient has a lower symmetry and is not on this list; a field varying in time is a different object again.

The seven are groups of motions, not of anything with units. ∞m is the symmetry of an electric field and equally of a cone made of wood; the group knows nothing about charge. That is the same abstraction every group on this site is: the seventeen are groups of motions of the plane and do not know that a wallpaper is paper. What makes the abstraction useful here is that a physical quantity’s transformation rule is enough to fix its group, which is the four lines above and nothing more.

The list is of symmetries, not of fields. Two quite different physical quantities share ∞m — an electric field and a temperature gradient — and the classification cannot tell them apart, nor is it meant to. What it supplies is the group, and the group is what Curie’s principle takes as input.

And the two families are not on the same footing. The five uniaxial groups form a chain of containments — ∞ inside all the others, ∞/mm containing all of them — while the two axis-free ones sit above everything. A physical situation with more than one axis is not on the list at all: two fields at an angle have as their joint symmetry the intersection of two limiting groups, which is generally finite and generally not a crystal class either. The seven are the groups of one uniform thing, and the moment there are two the arithmetic moves to intersections.

A pseudoscalar is on the list and has no axis. Handedness is a quantity with no direction at all and it is not a scalar: an inversion reverses it. Its group is ∞∞, the sphere without reflections, and it is the reason that list has two entries rather than one. A quantity with no direction is not thereby symmetric under everything.

And nothing here has a magnitude in it. The whole of this essay is a statement about which operations leave something unchanged, and no number describing a field’s strength appears anywhere. That is the standing division on this site between what symmetry decides and what it merely permits, and it matters most here because the temptation to read a permission as a prediction is strongest when the object is something a laboratory can dial up.