A primed axis shows its frame at half the rank

A six-fold crystal hides its frame until rank six asked how many of a property’s printed constants are really an angle. Thirteen crystal classes can be turned about an axis without moving any of their operations, and the turn spends a constant of a property only when the property has a part that turns with the frame. An $n$-fold axis keeps such a part only at rank $n$ or above. So a monoclinic crystal’s free angle shows in its dielectric tensor, a tetragonal crystal’s first in its elastic constants, and a hexagonal crystal’s in nothing anybody measures. The essay ended on a question about magnetic crystals: whether time reversal, which acts on magnetic properties with a sign, can bring the angle down to a lower rank.

It can, and by exactly a factor of two. Of the seventeen black-and-white magnetic classes built on the thirteen with a free turn, nine halve the rank at which the angle shows, from six to three, four to two, or two to one. They are the nine whose highest rotation is combined with time reversal. And no property that is even under time reversal sees any difference at all, in any of the seventeen.

What the sign does #

A magnetic class is an ordinary class in which half the operations are combined with time reversal, written with a prime, as the operation that reverses time set out. A property that is even under time reversal, such as strain or the dielectric tensor, sees a primed operation exactly as it sees the plain one. A property that is odd, such as magnetisation, or any coefficient linking a magnetic quantity to a non-magnetic one, sees it with an extra $-1$. That extra sign is the whole of the computation added here.

The free turns themselves do not change. A turn of the axes commutes with time reversal, since time reversal does nothing to directions, so the turns that commute with every operation of a magnetic class are exactly those that commute with every operation of its ordinary class. What changes is which tensors the class allows, and so which of them a turn can move.

The earlier essay’s argument makes the prediction in one line. Decompose a tensor into parts that turn $m$ times as the frame turns about the axis, each with phase $e^{im\theta}$ under a rotation by θ. An $n$-fold rotation multiplies such a part by $e^{2\pi i m/n}$ and keeps it when that is one, when $m$ is a multiple of $n$. A primed $n$-fold rotation acting on a time-odd tensor multiplies the part by $-e^{2\pi i m/n}$, and keeps it when that is one, when $m \equiv n/2 \pmod n$. A part with $m \neq 0$ is what a turn moves, and a tensor of rank $r$ has parts up to $m = r$. So the first kept part that moves sits at rank $n$ under a plain axis and at rank $n/2$ under a primed one.

The prediction needs the axis’s generating rotation to be primed. A class like $\bar6'$, in which the primes fall on the improper operations and the three-fold rotations stay plain, should keep its ordinary rank, since the proper rotations alone already pin the parts that matter. The census tests both halves.

The census #

The census takes all thirty classes, the thirteen ordinary ones and the seventeen black-and-white ones over them, and for each computes, rank by rank from one to six, the dimension of tensors the class allows and the rank of the free turns’ action on a generic one. It does this for polar and axial tensors, time-even and time-odd, with no index symmetry imposed, so that every part a tensor of that rank can have is present. The first rank at which the action is not zero is the rank at which the frame shows.

The time-even column is the first result, and it is a check rather than a discovery: in all seventeen black-and-white classes, a time-even tensor shows the frame at exactly the rank its ordinary class does. The primes are invisible to it, as they must be.

The time-odd column splits the seventeen cleanly. Nine lower the rank:

• $2'$, $2'/m'$ and $2'/m$, from two to one;
• $4'$, $4'/m'$ and $4'/m$, from four to two;
• $6'$, $6'/m'$ and $6'/m$, from six to three.

In each, the highest rotation about the free axis is primed. The other eight keep their ordinary rank: $\bar1'$, $m'$, $2/m'$, $\bar3'$, $\bar4'$, $4/m'$, $\bar6'$ and $6/m'$. In each of these the primes fall on improper operations, inversion, mirrors or rotoinversions, and the proper rotations about the axis stay plain. No class lowers the rank by more than half, and no three-fold class can lower it at all, because a rotation of odd order cannot be primed: half of a group of rotations of order three is not a subgroup.

One row is worth reading on its own. In the ordinary class $6/m$, axial tensors show no frame at any rank up to six: the mirror across the axis and the six-fold rotation together keep only parts that a turn cannot move, until the census’s range runs out. In $6'/m$ a time-odd axial tensor shows the frame at rank four and a time-odd polar one at rank three. A crystal class whose frame is hidden from every axial property the census can reach acquires, once its six-fold axis is primed, a magnetic property of rank three that sees it.

The census confirms the prediction exactly, including its condition. The rank at which a magnetic crystal’s free angle shows is set by its proper rotations alone, at $n$ if they are plain and at $n/2$ if their generator is primed.

The prime does what an inversion already did #

The halving has a precedent in the ordinary classes, and seeing it makes the magnetic result less surprising and more exact. The earlier census found $\bar4$ showing its frame at rank two and $\bar6$ at rank three, half the order of their axes. A rotoinversion $\bar n$ is a rotation by a turn of $1/n$ combined with inversion, and inversion acts on a polar tensor of rank $r$ by the sign $(-1)^r$. On the parts of odd rank, then, $\bar n$ is a rotation with an extra $-1$, and it keeps the parts with $m \equiv n/2$ exactly as a primed rotation does on a time-odd tensor. The same holds for the mirror across a two-fold axis, which is a half-turn combined with inversion, and the ordinary class $m$ shows its frame at rank one, as $2'$ does for a time-odd vector.

So there is a dictionary. For time-odd tensors, $6'$ behaves as $\bar6$ does for tensors of odd rank, $4'$ as $\bar4$, and $2'$ as $m$. In each pair, a six-fold, four-fold or two-fold operation acts with a sign, and the sign comes from inversion in one and from time reversal in the other. The census’s nine halving classes are the magnetic images of the ordinary classes that were already halved by an inversion, and the eight that do not halve are those in which the prime falls where an inversion already was. In $\bar6'$ the prime sits on the rotoinversion, so the sign time reversal adds meets the sign inversion already supplied, and the census finds the first rank unchanged at three, set by the plain three-fold rotations underneath.

That is also why three-fold classes are untouched by both. A rotation of odd order has no square root of minus one among its powers, and an extra sign on a three-fold operation is the same as combining it with a half-turn that the class does not have. Neither inversion nor time reversal can make a three-fold axis keep a part with $m$ not a multiple of three, and the census finds all three-fold classes, plain or magnetic, at rank three.

The dictionary is not an identity of groups. $6'$ and $\bar6$ are different classes with different property tensors; one is a magnetic class and the other an ordinary one with mirrors. What they share is the action of their six-fold operation on the parts of a tensor, up to a sign, and the rank at which the frame shows depends on nothing else. The axes a class pins down found which directions a class fixes; this is the complementary statement about the direction a class leaves free.

Where the angle shows in a property anybody measures #

The prediction is about generic tensors. The question that matters in a laboratory is which named properties carry an angle, and the census counts three: the magnetisation, an axial vector; the linear magnetoelectric tensor, linking an electric field to a magnetisation; and the piezomagnetic tensor, linking a stress to a magnetisation. All three are time-odd.

The cleanest case is the magnetisation itself. In $2'$, $m'$ and $2'/m'$, the magnetisation lies in the plane across the axis and has two printed components. A turn of the axes about the axis takes one of them to zero. The two numbers are a magnitude and a direction, and the direction is the angle. That was always obvious for a ferromagnet with its moment in a plane, and the census finds it as the rank-one instance of the rule.

In $4'$ and $4'/m'$ the magnetoelectric tensor has two constants and one of them is an angle. A plain four-fold axis would hide the angle from every tensor below rank four; the prime brings it to rank two, the rank of the magnetoelectric tensor. Its two printed components are an amplitude of the response across the axis and the orientation of the response’s own frame in the plane. A crystal in $4'/m$ has no linear magnetoelectric effect at all, and its angle appears instead in the piezomagnetic tensor, four constants of which one is the angle.

In $6'$ and $6'/m'$ the piezomagnetic tensor has two constants and one of them is an angle. The plain six-fold classes hide the frame from the elastic and photoelastic tensors and everything else the tables list, and that was the earlier essay’s title. With the six-fold axis primed, the angle reaches the piezomagnetic tensor at rank three. A hexagonal antiferromagnet in class $6'/m'$ that is piezomagnetic has, intrinsically, one piezomagnetic constant and one angle, where the tables print two constants.

Two classes in the table carry an angle without lowering the rank. $\bar6'$ has a piezomagnetic angle at rank three, which is its ordinary rank, since its proper rotations are three-fold. $\bar4'$ has one at rank three, above its ordinary rank of two. The rule is about the first rank; a class that already shows its frame low can show it again in higher properties, and these do.

The angle against the lattice #

The earlier essay found the practical answer to how a crystal’s frame is fixed, and it holds here unchanged. A turn of the axes that commutes with the class does not in general carry the lattice to itself, so diffraction fixes the cell edges and with them a frame no property chose. Measured against the lattice, every printed constant is a material constant, including the one a turn could remove, because it then measures the angle between the property’s own preferred frame and the cell. In $6'$ that angle is a fact about the magnetic structure: it is where the magnetic order’s own frame sits in the plane, relative to the atoms’ lattice.

What changes in a magnetic crystal is how directly the angle can be reached. In an ordinary hexagonal crystal it is invisible to every property up to rank six, which is beyond anything measured routinely. In a $6'$ crystal it is in the piezomagnetic response, a rank-three tensor that a measurement under uniaxial stress determines. The primes do not create the angle. They lower the rank of the first property in which it is a number rather than a convention.

The angles are also shared, as the earlier essay found for elasticity and photoelasticity together. A crystal has one frame, so a $4'$ crystal whose magnetoelectric tensor carries an angle and whose piezomagnetic tensor carries one too has a single angle between them, not two. Measured together, the two tensors have five intrinsic constants and one angle, where the tables print six. Each tensor has a preferred frame, the one in which its removable constant vanishes, and the two frames need not coincide. The angle between them is then one of the five intrinsic constants, a property of the material that no choice of axes changes, while the angle of either frame to the lattice is the one constant a turn could spend.

What the census rests on #

The counts are dimensions computed in floating point and rounded by a rank with a tolerance. The fixed space of each class is spanned by averaging every basis tensor over the class, with the sign $\det R$ for an axial tensor and an extra $-1$ for a primed operation on a time-odd one. The turns’ action is the derivative of a rotation applied index by index. A generic fixed tensor is an average of a pseudo-random one. A rank that depended on the tolerance would be a defect, and none does: the ranks are small integers separated from zero by orders of magnitude.

Two checks have nothing to do with the magnetic question. The ordinary classes reproduce the earlier essay’s first ranks, polar and axial, class by class. The time-even column of every black-and-white class reproduces its ordinary class’s. The magnetic classes themselves come from the derivation of the fifty-eight black-and-white groups in which magnetism a class permits, and only the seventeen over a class with a free turn enter here. The other forty-one, built on classes with axes in several directions, have no free turn to spend.

The refused claim is the natural one. A turn of the axes commutes with time reversal, so it seems that nothing about time reversal could change which constants a turn spends. The turns are indeed the same. What time reversal changes is which tensors exist in the class, and a time-odd tensor in a class with a primed axis has exactly the parts a turn moves at half the rank.

What a count of angles cannot show #

The census counts dimensions, and the count does not say how large anything is. A $6'$ crystal whose piezomagnetic angle is a degree has a piezomagnetic tensor nearly in the lattice frame, and the angle may be invisible within the error of a measurement. Nor does the census say which magnetic structures realise which classes; that is a question about real materials, answered by neutron diffraction, and which magnetism a class permits is where the classes are matched to kinds of order.

The figures do not show any tensor’s parts directly. The decomposition by $m$ that explains every row of the census is drawn only as a pattern of kept and discarded parts. A reader who wants the parts themselves will find them, for the ordinary rotation group, in the parts a property splits into, and the sign the primes add would change only which of them a class keeps.

Who counted what #

Magnetic point groups and their tensor counts are in Birss’s Symmetry and Magnetism of 1964, and the counts of magnetoelectric and piezomagnetic constants for each class are tabulated there and in later reviews. That some of those counts include an angle, the frame of the tensor relative to the class’s free axis, is the earlier essay’s observation applied to magnetic tensors. The census by rank and the rule that a primed axis halves the rank are computed here.

Still open: whether the moments name the angle #

In a $6'$ crystal the piezomagnetic tensor carries an angle, and the census says only that it exists. A magnetic structure refined from neutron diffraction gives more than a class: it gives the moments on each site and their directions in the plane, and those directions are themselves angles relative to the lattice. The natural conjecture is that the piezomagnetic angle is fixed by them, so that a refined structure predicts the frame of its own piezomagnetic response before any stress is applied. But a tensor’s frame is an average over every site’s contribution, and the moments’ directions need not average to the same angle the response chooses; a model with two kinds of site, each contributing a piezomagnetic term turned with its own moment, would test it. Magnetic reflections land where nuclear ones cannot is where the moments are measured, and the computation that would connect them to the tensor’s frame has not been made here.