Thirty-five classes a squeeze can magnetise

Which magnetism a class permits counted two properties among the hundred and twenty-two magnetic point groups: a spontaneous magnetisation, permitted in thirty-one, and the linear magnetoelectric effect, permitted in fifty-eight. It named a third and left it uncounted. Piezomagnetism is a magnetisation proportional to a mechanical stress, the magnetic analogue of the charge a piezoelectric crystal develops when it is squeezed, and it is the property in which the difference between a magnet and an antiferromagnet becomes something a laboratory can push on.

The count comes out at sixty-six. That number has been in the literature since the magnetic groups were first worked through, and it is reproduced here without being supplied. What the count says when it is taken apart is the subject of this essay: which sixty-six, why they include every class the earlier count found, what separates them from the magnetoelectric fifty-eight, and why exactly three of the ninety classes that are not grey forbid it without the reason that forbids it everywhere else.

The tensor, and the one sign time reversal adds #

A stress is a symmetric second-rank tensor and a magnetisation is an axial vector, so the coefficients that link them form an array with one axial index and a symmetric pair: eighteen components before any symmetry acts. Under a spatial operation the array transforms as an axial vector does in its first index, picking up the operation’s determinant, and as a stress does in the other two. Its character at an operation is therefore the determinant times the trace times the character of the symmetric square, which is the piezoelectric character with one determinant added.

Time reversal adds a sign. A magnetisation reverses when time runs backwards, and a stress does not, so the piezomagnetic coefficients are odd under time reversal. In the arithmetic the operation that reverses time set up, a primed operation contributes its character multiplied by minus one, and the count of independent coefficients a magnetic class permits is the average of that signed character over the class. Nothing else changes. The same average that counts elastic constants and piezoelectric moduli in Neumann’s principle, as one sum counts this, and every magnetic class counts its properties by mirrors has already shown that such signed averages are relative invariants of the class, with the sign as their character.

Sixty-six, and what they contain #

The thirty-two grey classes permit nothing. A grey class contains time reversal on its own, which multiplies every time-odd property by minus one while moving nothing, so every time-odd property is its own negative there. That is the arithmetic form of a physical statement worth having in the plainest words: a crystal with no magnetic order cannot be magnetised by a stress, however low its symmetry. The quartz plate that becomes charged when squeezed has no magnetic counterpart, because quartz has no moments for a squeeze to unbalance.

Of the ninety that remain, sixty-six permit piezomagnetism, and every one of the thirty-one ferromagnetic classes is among them. That containment needs no computation. A class that permits a magnetisation permits an axial vector to stand on its own, as the ten polar classes permit a polar one, and multiplying that vector by the trace of the stress, which every class leaves alone, gives a piezomagnetic tensor at once: a crystal magnetised already can have its magnetisation changed by a uniform pressure. So the thirty-one sit inside the sixty-six, and the computation confirms it in every one of them.

The other thirty-five are the interesting ones. In each of them a crystal may have no net moment at all while it rests, and must be allowed to acquire one when it is stressed. Sixteen of the thirty-five are ordinary classes, the symmetry of a magnetic structure in which no operation needs time reversal at all, and nineteen are black-and-white. Every crystal whose magnetic class lies among them and whose moments are ordered is an antiferromagnet, a ferrimagnet that happens to cancel, or something more exotic; what all of them share is that their moments are arranged so that no symmetry forbids a stress from unbalancing them.

The count was made twice. Once by averaging the signed character over each magnetic class in the integer lattice basis the classes are written in, and once by building the eighteen basis tensors in Cartesian coordinates and averaging each of them over the class, with a primed operation entering with a minus sign and every operation acting on the axial index through its determinant. The two agree in all hundred and twenty-two classes. A reader of the projected tensors finds, by elimination, exactly as many independent entries as the count says, which is a third check rather than a restatement of the second.

A centre forbids one of two, and which depends on its prime #

The census figure is arranged by centre because the centre is where the structure is.

The piezomagnetic tensor is even under inversion, since an axial vector and a stress are both left alone by it, and odd under time reversal. The magnetoelectric tensor, which couples an axial magnetisation to a polar electric field, is odd under both. So the inversion on its own reverses the magnetoelectric tensor and does nothing to the piezomagnetic one, and the inversion combined with time reversal does the opposite: it leaves the magnetoelectric tensor alone and reverses the piezomagnetic one. A class containing the plain inversion forbids the magnetoelectric effect, a class containing the primed inversion forbids piezomagnetism, and no centre forbids both.

The figure shows the consequence in counts. Twenty-one classes contain the plain inversion: twenty are piezomagnetic and none is magnetoelectric. Twenty-one contain the primed inversion: eighteen are magnetoelectric and none is piezomagnetic. Forty-eight have no centre: thirty-nine permit both effects, seven permit piezomagnetism alone and one permits only the magnetoelectric effect. The two properties are often discussed together as the two linear couplings an antiferromagnet can have between its magnetisation and something applied to it, and the centre is what decides between them. A crystal that has one of them with a centre in its group cannot have the other.

That split has a practical edge. Chromium oxide, the crystal in which the magnetoelectric effect was first predicted and found, has a magnetic class with a primed centre, so it can never be piezomagnetic. The rutile-type fluorides of cobalt and manganese have a plain centre, so they can never be magnetoelectric, and they were where piezomagnetism was found. Each was the right crystal for one of the two effects because of a single prime.

Three exceptions, and the four-fold operations that make them #

The centre accounts for twenty-one of the twenty-four non-grey classes that forbid piezomagnetism. The other three are the ordinary cubic classes 432, $\bar{4}3m$ and $m\bar{3}m$, the three cubic classes that contain four-fold operations, and in each of them the reason is a single invariant.

In a cubic crystal a piezomagnetic tensor can only have one form, the one in which a shear across each cube axis produces a moment along that axis, with the same coefficient for all three axes because the three-fold rotations along the body diagonals carry the axes into each other. The classes 23 and $m\bar{3}$, whose operations are the three-fold rotations, the two-fold rotations about the cube axes and, in $m\bar{3}$, the inversion, leave that form alone, and each has one piezomagnetic constant. A four-fold rotation about a cube axis does not leave it alone: it exchanges the other two axes with a sign, and the form comes back as its own negative. So 432 permits no piezomagnetism. An improper four-fold operation, a quarter turn followed by the inversion, does the same, because the inversion does nothing to an axial vector or a stress; so $\bar{4}3m$ permits none, and $m\bar{3}m$, which contains both kinds, permits none either.

This is the exception twenty of the twenty-one found for piezoelectricity, arriving from the other side. The piezoelectric tensor has the same shape with a polar vector in place of the axial one, and for a proper rotation the two cannot be told apart. So in 432 the same invariant is reversed by the same four-fold rotations, and 432 is the one class without a centre that cannot be piezoelectric for exactly the reason it cannot be piezomagnetic. In $\bar{4}3m$ the two part company. An improper four-fold operation reverses a polar vector in its inversion part and leaves an axial one alone, so for the polar invariant its two reversals cancel and $\bar{4}3m$ is piezoelectric, which zinc blende is, while for the axial invariant only one reversal acts and $\bar{4}3m$ is not piezomagnetic.

The black-and-white groups then do what time reversal always does to such an argument. In $4'32'$ the four-fold rotations are primed, so they reverse the invariant once for being four-fold and once more for being primed, and the invariant survives: the class permits one constant. The same is true of $\bar{4}'3m'$ and of $4'/m\bar{3}m'$. The two black-and-white groups of $m\bar{3}m$ that contain the primed inversion forbid it for the other reason. So the cubic family contains the whole argument in miniature: an invariant reversed by a geometric operation, restored by attaching time reversal to that operation, and killed outright by a primed centre.

Of the ninety classes that are not grey, piezomagnetism is forbidden in the twenty-one with a primed centre, in the three ordinary cubic classes with four-fold operations, and nowhere else. The magnetoelectric fifty-eight have no description as short. The magnetoelectric effect is forbidden by the plain centre in twenty-one classes and in eleven more for reasons that reduce to no single rule, three of them with a primed centre and eight with none, which is why the two counts are not the mirror images of each other that the centre argument might suggest.

The rutile group, and the moment a shear makes #

The crystals in which piezomagnetism was first found are cobalt and manganese difluoride. Both have the rutile structure, both order antiferromagnetically at low temperature with the moments along the four-fold axis, up on the metal atom at the corner of the cell and down on the one at its centre, and both have the magnetic class $4'/mm'm$: the four-fold axis primed, because a quarter turn about the axis carries the corner site’s environment onto the centre site’s and so carries up onto down.

The class permits two piezomagnetic constants and no magnetisation. The first gives a moment along the four-fold axis in response to a shear in the basal plane, and the second gives moments across the axis in response to the shears that involve it, the same coefficient for both by the four-fold symmetry. The class contains the plain inversion, since the rutile structure is centrosymmetric and the inversion carries each metal site onto a site of the same kind, so the magnetoelectric effect is forbidden in these crystals, as the centre argument requires.

One detail is a matter of description rather than of physics. The census builds this class with its primed mirrors across the edges of the cell, and in that frame the moment along the axis comes from the difference of the two in-plane compressions rather than from a shear. The rutile structure is written with the primed mirrors across the face diagonals, which is the same group turned through forty-five degrees about the axis, and in that frame the same coefficient multiplies the shear. The matrix in the figure is drawn in the frame the structure is written in. The count, two, is the same in both.

What the shear does to the two sublattices #

The symmetry argument says that a shear in the basal plane may produce a moment along the axis. A picture of the structure says why, at least in outline. The two metal sites have environments that are elongated along the two different face diagonals, because the primed four-fold rotation that relates them turns one environment through a quarter turn. A shear in the basal plane lengthens one face diagonal and shortens the other. So it changes the two sites in opposite senses, and the two moments, which cancelled because the sites were equivalent, need no longer cancel.

The figure takes the simplest model consistent with that picture, in which each moment’s size changes in proportion to the strain along its site’s long axis, and draws the result for both domains of the crystal. The model has one free number and does not pretend to the magnitudes; the shear and the response are both drawn far larger than any real crystal shows. What it fixes are the signs, and the signs are the point.

An antiferromagnet has two domains that differ only by the direction of time. How many domains a transition makes is an index, and ordering out of a grey group always contributes a factor of two to it, because time reversal is lost. Reverse every moment in one domain and the other is the result: the same structure, the same magnetic class, the same energy, and no net moment in either. A magnetometer cannot tell them apart, since both read zero. A neutron cannot either, in the ordinary way, because magnetic reflections land where nuclear ones cannot and their intensities depend on the squares of the moments. A shear can. In one domain it produces a moment pointing up the axis and in the other a moment pointing down, because the piezomagnetic coefficient is odd under time reversal and the two domains are each other’s time reversal.

That is what made piezomagnetism an instrument and not only an effect. Cooling a crystal through its ordering temperature under a stress and a magnetic field together favours the domain whose stress-induced moment lies along the field, and so produces a crystal of one domain, which is otherwise hard to arrange in an antiferromagnet. It is also what makes the effect worth having now: a stress that can choose between two time-reversed states can be used to write one of them, in materials whose antiferromagnetic order carries a property, such as an anomalous Hall signal, that depends on which state it is in.

What the count must refuse #

Two claims look reasonable and are refused. The first is that piezomagnetism, like piezoelectricity, needs a crystal without a centre. It does not: twenty of the classes with the plain inversion permit it, the rutile class among them, because the tensor is even under inversion and a centre cannot reverse it. The piezoelectric rule comes from the polar vector in the piezoelectric tensor, and there is none here. A crystallographer who reaches for the familiar rule will rule out exactly the crystals in which the effect was discovered.

The second is that a magnetic class is piezomagnetic when its unprimed operations, taken alone, would permit piezoelectricity, which is how a reader might try to read a black-and-white group as an ordinary group with a decoration. The verdicts disagree in forty-one of the ninety classes. A primed operation is not a missing operation; it is an operation with a sign, and the sign takes part in the average.

Where the count stops #

A permitted coefficient is not a measured one. The thirty-five classes say where a stress may magnetise a crystal with no moment at rest. They say nothing about how large the effect is, and in most materials it is small: the moment a realistic stress produces in an antiferromagnet is a tiny fraction of one sublattice’s moment, which is why it was found only in 1960 and why a measurement of it is a careful one. Permitted is not present applies here with full force.

The magnetic class must be the right one. The count is a statement about the magnetic point group of the ordered state, and an antiferromagnet’s magnetic group depends on the direction its moments point, not only on its chemical structure. The same crystal with its moments turned into the basal plane would have a different magnetic class and could have a different verdict. Deciding the moment direction is a diffraction problem, and where a powder goes blind is the reminder that a powder does not always answer it.

And the counts are linear. A stress can change an antiferromagnet’s moment at second order in classes that forbid the linear effect, and a large enough stress can change the magnetic class altogether. Neither is counted here.

From Voigt to the difluorides #

A magnetisation produced by stress was discussed by Woldemar Voigt in his crystal physics of the early twentieth century, working from the crystal classes alone, and a theory built on the thirty-two classes has no way to exclude it: the tensor’s spatial form is permitted in twenty-nine of the thirty-two, centrosymmetric or not. What excluded it in ordinary crystals was the operation the thirty-two classes do not contain. Once time reversal is admitted, a crystal with no magnetic order has a grey group, and a grey group forbids every time-odd response at once.

The magnetic point groups were worked out in the 1950s, by Shubnikov’s school and by Tavger and Zaitsev among others, and with them the question became which ordered magnets could show the effect. Dzyaloshinskii pointed in the late 1950s to the rutile-type antiferromagnets, and Borovik-Romanov measured piezomagnetism in cobalt and manganese difluoride in 1960. It was among the first properties to be predicted from magnetic symmetry before it was seen, alongside the magnetoelectric effect in chromium oxide, which Dzyaloshinskii predicted in 1959 and Astrov observed in 1960.

The two discoveries, made within a year of each other in neighbouring laboratories, sit on either side of the centre argument. One crystal had a primed centre and the other a plain one, and each showed the effect the other could not have.

Still open: what a second-order squeeze can do #

The thirty-five classes are where a stress can magnetise a crystal linearly. The magnetic classes that forbid the linear effect may still permit a magnetisation quadratic in the stress, whose tensor is an axial vector against the symmetric square of the stress, a space of sixty-three components before symmetry acts. In a class with a primed centre that tensor is reversed too, since a square of an even quantity is still even, so the primed centre should forbid every order of piezomagnetism, while the three ordinary cubic exceptions might admit a quadratic effect where they forbid a linear one. Whether the second-order count follows the same short description, a primed centre and nothing else, or whether new exceptions appear among the classes with high symmetry, is the same signed average with a larger character. It has not been run here.