The ten with a direction of their own

Heat a crystal of tourmaline and one end acquires a positive charge and the other a negative one. Cool it and the charges reverse. The effect is pyroelectricity, it was known to the ancient world as the ability of the stone to pull ash out of a fire, and it happens because tourmaline has a built-in electric polarisation whose size depends on temperature.

A built-in polarisation is a vector, and a vector is a direction with a magnitude. So a crystal can have one only if it has a direction that its own symmetry does not move.

Ten of the thirty-two classes have such a direction. The rest do not, and for them the whole family of effects is impossible rather than small.

What “polar” means, exactly #

A direction d is left alone by an operation M when Md = d — fixed pointwise, not merely mapped to the same line. A two-fold axis fixes its own axis; a mirror fixes every direction lying in its plane; the inversion fixes nothing at all.

A class is polar when some direction is fixed by every operation at once. That is a much stronger condition than each operation having some fixed direction, and it is why the count is ten and not larger.

Take class 222: three mutually perpendicular two-folds. Each fixes its own axis, and the axis of one is reversed by the other two. No direction survives all three, so 222 is not polar even though every one of its operations has a fixed axis of its own.

Take class mm2: two perpendicular mirrors and the two-fold where they meet. The line of intersection lies in both mirror planes and along the rotation axis, so it is fixed by everything. mm2 is polar, and the 2 at the end of the symbol is pointing at exactly that direction.

The two classes are worth putting side by side, because the arithmetic that separates them is the same in both and the answers are opposite. Averaging a general vector over 222 sends every component to zero: the two-fold about z reverses x and y, the two-fold about x reverses y and z, and between them nothing is left standing. Averaging the same vector over mm2 leaves one component alone. That is the whole of what “polar” decides, and it is one projection run twice.

Two routes, one answer #

The site computes the ten twice, and requires the answers to agree.

As a fixed space. Stack up the conditions Md = d for every M in the group — three linear equations per operation — and solve. The solution space is the set of polar directions, computed exactly by fraction-free elimination over the integers. Its dimension is 0, 1 or 3, and the class is polar when it is not 0.

As a character sum. By Neumann’s principle, the number of independent components of a spontaneous vector is the average of tr M over the group. One sum, no linear algebra, no basis.

The first produces an explicit direction that a figure can draw. The second produces only a number. They share the group and nothing else, and the build fails if they disagree — which is the field’s version of the arrangement running through the whole site, where a pattern’s group is asserted from its point set and the same group’s consequences are asserted from a sum in reciprocal space.

The dimension is one, except once #

For nine of the ten polar classes the fixed space is a line: one independent component, one polar axis, one direction along which the crystal has a head and a tail.

Class 1 is the exception. It has one operation, the identity, which fixes everything, so its fixed space is all of space and its count is three. A triclinic crystal with no symmetry has a polarisation vector that can point anywhere and needs three numbers to specify.

Class m sits in between with two: a mirror fixes every direction in its plane, so the polarisation is confined to the plane and needs two components.

So the polar column of the property table reads 3, 2, 1, 1, 1, 1, 1, 1, 1, 1 across the ten, and zero for the other twenty-two. The count is the dimension of the freedom, not the strength of the effect, and class 1 heading the column means only that it constrains nothing.

Polar, pyroelectric, ferroelectric — three words, three claims #

These are routinely used as synonyms and they are three different statements, each strictly stronger than the last.

Polar is a property of the class: some direction is fixed by every operation. Ten classes, decided exactly, no material involved.

Pyroelectric is a property of the material: it has a spontaneous polarisation whose magnitude changes measurably with temperature. Every pyroelectric crystal is in a polar class, and plenty of crystals in polar classes have no useful pyroelectric response.

Ferroelectric is stronger still: the spontaneous polarisation can be reversed by an applied field. That requires a polar class and a structure with an accessible alternative state to switch into, which is a question about energy barriers that symmetry cannot answer at all. Every ferroelectric is pyroelectric, every pyroelectric is polar, and neither implication reverses.

The classification supplies the outermost of the three circles and nothing inside it. That is the whole content of permitted is not present, and this is the cleanest place on the site to see the three layers separately.

Drawn as stereograms the ten look almost alike, and what they have in common is easy to state: in each of them the polar direction is the centre of the circle, the pole that no operation moves. Seven of the ten have a single axis and nothing else — 1, 2, m, mm2, 3, 4 and 6 — and the remaining three, 3m, 4mm and 6mm, add vertical mirrors that contain that axis and therefore leave it alone. A mirror fixes every direction lying in its own plane, so adding one costs nothing as long as the axis lies in it.

Reading the ten off their symbols #

There is a shortcut, and it is a good one because it explains the list rather than merely reproducing it.

A class is polar exactly when its symbol names no operation that moves the primary direction. Look at the ten: 1, 2, m, mm2, 3, 3m, 4, 4mm, 6, 6mm. Every one is either the trivial class, a single rotation axis, a single mirror, or a rotation axis with mirrors containing it.

What is absent from that list is as informative. There is no /m anywhere — a mirror perpendicular to the axis reverses it, so 2/m, 4/m and 6/m are all out. There is no bar — a rotoinversion reverses its own axis, so 1̅, 4̅ and 6̅ are out. And there is no second numeral — a two-fold perpendicular to the main axis reverses it, so 222, 422, 622 and 32 are out.

So the rule reads straight off the notation: a polar class has one axis position filled, and everything else in the symbol is a mirror that contains that axis. That is why the ten are exactly the ten, and it is a nice illustration of what deriving a symbol from the directions buys — the notation carries the answer because it was built to report positions.

The cubic classes are all excluded for a reason that needs no case analysis: a cubic group has axes along more than one direction by construction, and no direction survives axes pointing several ways.

Why the inversion settles it immediately #

Eleven of the twenty-two non-polar classes are ruled out in one line: a centrosymmetric class cannot be polar, because the inversion sends every direction to its negative and only the zero vector is its own negative.

That is one operation doing all the work, and it is worth noticing that it is the same operation whose presence or absence decides what a diffraction pattern can report and whether piezoelectricity is possible. The inversion is the single most consequential element in the whole classification: eleven classes have it, and having it forbids nearly everything interesting.

The remaining eleven non-polar classes are ruled out for varied reasons — 222 by having three mutually perpendicular axes, 432 and 23 by having axes along both the cube edges and the body diagonals, 4̅ because a rotoinversion reverses its own axis. Those cases need the calculation. The centrosymmetric eleven do not.

The oldest recorded observation in the subject #

Theophrastus described lyngourion around 314 BC — a stone that attracts straws and bits of wood when warmed — and the identification with tourmaline is reasonably secure. That makes pyroelectricity one of the oldest recorded solid-state phenomena, older than magnetism as a laboratory subject and very much older than any theory of crystals.

The chain from that observation to this list of ten is worth tracing, because each link is a different kind of argument.

Haüy and others through the eighteenth century established that the effect is directional — the two ends of the crystal behave differently — which is the observation that makes it a vector rather than a scalar. Brewster named it in 1824. Then the symmetry argument arrives: if it is a vector, and the crystal’s own operations must leave it alone, then most crystals cannot have one, and the list of those that can is a consequence of a classification that was being worked out at the same time for entirely different reasons.

That is the pattern the whole field repeats. A phenomenon is observed; someone identifies what kind of tensor it is; and the classification, which was built by people thinking about the shapes of crystals rather than about their properties, turns out to answer a question nobody had put to it.

Neither half is derivable from the other. Symmetry cannot tell anyone that heating a stone will move ash, and no amount of measuring tourmaline produces the number ten.

Where the exactness stops #

Polar is a statement about the class, and a real crystal’s class is an experimental inference. A structure refined in a centrosymmetric space group when the truth is non-centrosymmetric will be reported as forbidding an effect the material has, and that error is common enough to be a standard cautionary tale in crystallography.

Domains cancel. A ferroelectric grown without a poling field is typically a mosaic of domains with polarisations in different permitted senses, and the bulk polarisation of the mosaic is near zero. The class permits the effect, each domain has it, and the sample shows nothing.

And a polar axis has a sense that diffraction cannot report. Friedel’s law means the two ends of a polar crystal give identical patterns, so which end is positive is not determined by an ordinary structure determination — the same gap that leaves absolute configuration undetermined, and closed the same way, by anomalous scattering.

What a polar axis does to the rest of the table #

A polar class is not merely a class that permits one extra property. Having a fixed direction changes the count of every property, because it is a strong statement about how little cancellation the group can do.

Compare 4mm with 4/mmm. They differ by one operation — the mirror perpendicular to c — and everything else follows from it. 4mm has eight operations and 4/mmm sixteen; but the interesting differences are that 4mm permits one polar component and three piezoelectric moduli where 4/mmm permits none of either, while their elastic counts are the same, at six.

One operation removes two properties entirely and leaves a third untouched. The reason is parity: a mirror perpendicular to the axis reverses the axis, so any odd-rank polar tensor with a component along it must vanish, and elasticity is even-rank and does not notice.

That is the general shape of the polar classes’ position in the table. They are the classes that have declined to add the one operation that would kill the odd-rank properties, and they pay nothing for it in the even-rank ones.

The measurement, and why it is done with heat #

Pyroelectricity is an odd way to measure a spontaneous polarisation, and the reason it is the standard way is worth a paragraph, because it explains why the property is named after temperature rather than after the polarisation itself.

A crystal with a built-in polarisation has bound charge on the faces perpendicular to its polar axis. In equilibrium that charge is compensated — by ions from the air, by surface conduction, by whatever is available — so the crystal appears neutral and the polarisation is invisible.

Changing the temperature changes the polarisation faster than the compensation can follow, and the difference appears as a measurable charge. That is what Theophrastus’s stone was doing when it picked up ash: warming it moved the polarisation, the compensating layer lagged, and the exposed charge attracted light objects.

So the effect being measured is a derivative: dP/dT rather than P. The absolute value of the spontaneous polarisation is much harder to get at, and in a ferroelectric it is usually obtained by reversing it with a field and integrating the switching current — which is available only in the subset of polar materials that can be switched.

This is a general feature of measuring symmetry-permitted quantities and not a peculiarity of pyroelectricity. What symmetry permits is a static property of the crystal; what an experiment sees is nearly always a change in that property under some stimulus. The count says how many independent numbers there are; the experiment reaches them through whichever derivative happens to be accessible.

A final observation about the number itself. Ten is small — under a third of the thirty-two — and that scarcity is the reason the polar classes are worth naming at all.

Every material with a spontaneous polarisation, every ferroelectric, every pyroelectric detector and every piezoelectric transducer working by domain alignment is in one of these ten. That is a very short list for a very large body of technology, and it is short for a reason with no physics in it: a direction fixed by every operation of a group is a strong demand, and most groups do not meet it.

Symmetry narrowed a search space by two-thirds before anybody synthesised anything, and that is the most useful thing a prohibition can do.

The ten are exactly the crystallographic subgroups of one continuous group #

There is a characterisation of the list that is shorter than the rule about symbols and explains rather more, and it comes from the same person who generalised Neumann’s principle.

Consider the symmetry of a cone: every rotation about its axis, and every mirror containing that axis. As a group it is written ∞m, it is infinite, and it is the symmetry of any object with one distinguished direction and no distinguished sense of turning about it — a stationary electric field, for one.

A crystal class is polar exactly when it is a subgroup of ∞m. Check the list against it: 1, 2, m, mm2, 3, 3m, 4, 4mm, 6, 6mm — every one of them consists of rotations about a single axis together with mirrors containing that axis, and nothing else. Every class not on the list has an operation ∞m does not: a two-fold across the main axis (222, 422, 32, 622), an inversion or a horizontal mirror (2/m, 4/m, mmm and the rest), a 4̅ or a 6̅, or the several axes of a cubic class.

So the ten are not ten separate facts. They are the answer to which crystal classes fit inside the symmetry of a field, and Neumann’s principle then does the rest: a property with the symmetry of ∞m can occur only in a class whose symmetry contains it, which is the same statement read from the other end.

Curie’s seven limit groups are the continuous groups this belongs to — the cone, the cylinder, the twisted cylinder, the sphere and three more — and each one is the symmetry of a physical influence rather than of a crystal. An electric field is ∞m; a magnetic field is ∞/m, the cylinder with a sense of rotation; a uniaxial stress is ∞/mm. Asking which classes sit inside each of them produces the permitted-property lists directly, with no tensors and no characters, and it produces them for influences the crystal is subjected to as well as for properties it has by itself.

That second half is the part the character sum on this site does not compute. Neumann’s principle is about a crystal alone; Curie’s is about a crystal in a field, and it asks which symmetry the combination has — the intersection of the two groups. It is the version that predicts what happens when a stress is applied to a crystal that had no polarisation, which is what the converse piezoelectric effect is, and it is why the limit groups are worth knowing even though nothing in the thirty-two is continuous.

Where this goes #

Ten classes permit a spontaneous vector; twenty-one lack a centre of symmetry. Those two facts nearly determine a third — which classes permit piezoelectricity, the coupling between a stress and a polarisation — and “nearly” is where the interest is. Twenty of the twenty-one permit it, and the single exception is a class with a great deal of symmetry and no inversion at all.