A strain gradient magnetises what a stress cannot

A squeeze magnetises a cubic crystal across itself finished a list. Among the 122 magnetic point groups, a uniform stress can produce a magnetisation in sixty-six classes in proportion to itself, the census of thirty-five classes a squeeze can magnetise, and in three more, the ordinary cubic classes, in proportion to its square. No higher power adds anything. Sixty-nine classes, and the census stops. The fifty-three it never reaches are the thirty-two grey groups, which contain time reversal as a symmetry and so forbid every moment, and twenty-one classes that hold the inversion combined with time reversal, written $\bar1'$.

The reason for the twenty-one is a single sign. A stress is unchanged when space is inverted through a point, and so is any product of stresses. A magnetisation is an axial vector, also unchanged by inversion, but it reverses under time reversal. An operation that inverts and reverses time therefore sends the coefficient joining the two to minus itself, and a coefficient equal to minus itself is zero. No power of a uniform stress gets round that, since every power is even under inversion.

That essay ended by pointing at the one kind of mechanical stimulus that does get round it. A strain can vary from place to place, and a magnetisation can respond to how fast it varies. A gradient is a derivative with respect to position, which is a polar index, and one more polar index makes the coupling odd under inversion. So the prohibition should move. The primed centre should permit what it forbade before, and the plain centre should forbid what it permitted. This essay counts it.

The coupling and its character #

The coupling is $M_i = q_{ijkl}\,\partial_l \varepsilon_{jk}$. The magnetisation $M$ is an axial vector, odd under time reversal. The strain $\varepsilon_{jk}$ is symmetric in its two indices, six components, and its gradient adds a third index, so the tensor $q$ has three times six times three, fifty-four, components before any symmetry is imposed. The question for each magnetic class is how many of those fifty-four survive.

The count is the same signed average that Neumann’s principle, as one sum sets up, adapted for time reversal as in the operation that reverses time. At a spatial operation $R$ the character of the tensor’s space is the product of four characters: $\det R \cdot \operatorname{tr} R$ for the axial vector, $\tfrac12(\operatorname{tr}^2 R + \operatorname{tr} R^2)$ for the symmetric strain, and $\operatorname{tr} R$ for the gradient’s index. A primed operation contributes the same character with a minus sign, because the property is odd under time reversal. The average over the group is the number of independent constants, the same average every magnetic class counts its properties by mirrors used for other tensors.

That average is checked by doing the same thing a second way. Each of the fifty-four basis tensors is written out in Cartesian components and averaged over the group, each operation acting on all four indices, with the determinant for the axial one and the sign for a primed operation. The averaged tensors span the invariant space, and its dimension is the rank of the list. The two counts agree in all 122 classes.

Sixty-nine, again #

The census comes to sixty-nine classes. The shape of the answer is as simple as it could be, and it is exactly the shape the parity argument predicts. Of the ninety classes that are not grey, forty-eight hold no inversion at all, in either form, and a strain gradient can magnetise every one of them. Twenty-one hold the primed inversion, and a strain gradient can magnetise every one of those too. The remaining twenty-one hold the plain inversion, and a strain gradient magnetises none of them. The thirty-two grey classes forbid it, as they forbid every time-odd property.

So the parity argument is not only necessary but sufficient here. Nothing else in any class kills the coupling. Every rotation, every mirror and every prime other than the one on the centre leaves at least two constants standing, and the tensor is large enough that no combination of them can exhaust it. The prohibition comes from one sign and only from that sign.

Now set this beside the uniform stress. The square of a stress magnetises sixty-nine classes too: the same forty-eight with no centre, and the twenty-one with a plain one. The two lists have the same length and differ in exactly the twenty-one classes whose centre is primed in one and plain in the other. The gradient and the square of the stress are each other’s complements on the centrosymmetric classes, and between them they reach every one of the ninety classes that are not grey. A magnetic crystal that no squeeze can magnetise is one that some bending can, and the other way round.

The equality of the two sixty-nines is not a coincidence of tensor sizes, and it has a reason that needs no tensors. For any centrosymmetric point group, the magnetic groups built on it with the plain inversion and those built on it with the primed inversion come in equal numbers. A magnetic group is a choice of which operations carry a prime, and that choice is a homomorphism from the point group to plus and minus one. Multiplying every such choice by the determinant, which priming every improper operation amounts to, is a bijection that moves the prime onto the centre or off it and changes nothing else. Twenty-one and twenty-one are one list read twice.

Why one more index moves the prohibition #

The figure lays out the whole argument as two signs. Under the plain inversion a magnetisation is unchanged, being axial. A stress, and any product of stresses, is unchanged. A strain gradient reverses, because the derivative it carries is with respect to a position, and a position reverses. Under the primed inversion everything is the same except that the magnetisation also reverses, because time does.

So the coefficient joining a magnetisation to a power of a stress is unchanged by the plain inversion and reversed by the primed one: the primed centre forbids it. The coefficient joining a magnetisation to a strain gradient is reversed by the plain inversion and unchanged by the primed one: the plain centre forbids it. That is all the census found. The interesting content is that nothing else ever forbids it, which is a statement about the rank of the tensor and can only be found by counting.

The magnetoelectric effect, a magnetisation in proportion to an electric field, has exactly the same parity: an axial, time-odd vector against one polar index. So it, too, is forbidden by the plain centre and permitted by the primed one, and which magnetism a class permits counted its classes. There are fifty-eight. That number is the standard one, and reproducing it is the check the census runs first.

The eleven a field cannot reach #

Same parity, but fifty-eight against sixty-nine. Every magnetoelectric class is a gradient class, and eleven gradient classes are not magnetoelectric. They are the ordinary class $\bar43m$, two primed cubic classes, and eight hexagonal ones built on $\bar6$, $\bar62m$, $6'$ and their relatives. In each of them the magnetoelectric tensor, with only nine components, is killed by a combination of rotations and primes that the larger gradient tensor survives.

That is the difference between a parity and a rank. Parity decides which classes can never have a property, and the centre is the whole of it for both tensors. Rank decides whether, among the classes parity allows, the property has enough components left over to survive every remaining operation. The magnetoelectric tensor runs out in eleven classes and the gradient tensor in none. Twenty of the twenty-one is the same lesson about piezoelectricity: of the twenty-one point groups without a centre, the cubic $432$ alone forbids it, by rank and not by parity.

The counts across the sixty-nine are generous. The fewest constants any class has is two, in three classes, and the most is fifty-four, all of them, in the two triclinic classes $1$ and $\bar1'$. Between those the distribution is broad, with most classes between three and fifteen.

The cubic classes #

The earlier essay’s question was specific: whether the three ordinary cubic classes, which no first-order stress could magnetise, join at first order in the gradient. Two of them do. The ordinary $432$ has three gradient constants and $\bar43m$ has two. The ordinary $m\bar3m$ has none, because it holds the plain inversion; so does $m\bar3$. And the ordinary $23$, which a stress already magnetises at first order, has five.

The five can be written out. In $23$ the component $M_x$ can respond to the gradient along $x$ of the strain along $x$; to the gradient along $x$ of the sum of the two transverse strains; to the gradient along $x$ of their difference; to the sum of the gradients of the two shears that involve $x$; and to the difference of those two gradients. Each of the five is repeated for $M_y$ and $M_z$ by cycling the axes, which is what the three-fold rotations of $23$ require. The census checks each form directly, by applying every operation of a class to the tensor and asking whether it comes back unchanged.

$432$ adds the four-fold rotations, which exchange $y$ and $z$ up to a sign, and it keeps the three forms that are symmetric under that exchange. Those are exactly the three a flexoelectric polarisation has in a cubic crystal, the coefficients usually written $\mu_{11}$, $\mu_{12}$ and $\mu_{44}$. That is no accident. Every operation of $432$ is a proper rotation, with determinant one, and under proper rotations an axial vector transforms exactly as a polar one does. So in $432$ a magnetisation is bound by the same rules as a polarisation, and the gradient couples to it through the same three numbers.

$\bar43m$ is where the axial nature shows. Half of its operations are improper, the mirrors and the four-fold rotoinversions, and each of those carries a determinant of minus one that the polar case does not have. A mirror that exchanges $y$ and $z$ therefore keeps exactly the forms that change sign under the exchange, and kills the others. The two constants of $\bar43m$ are precisely the two that $432$ lacks. The two classes split the five forms of $23$ between them, three and two, with nothing shared.

The physical reading of $\bar43m$’s two terms is specific. A bending that compresses a cube along $x$ more on one side than the other, a gradient of $\varepsilon_{xx}$ along $x$, gives no moment. A gradient along $x$ of the difference between the $y$ and $z$ strains does, a strain that makes the cross-section more rectangular as one moves along the bar. So does an imbalance between the two shear gradients that involve $x$. In a zinc-blende-type magnetic ordering, those are the deformations to look for.

The polar analogue is never forbidden #

The comparison that makes the magnetic census informative is the one with flexoelectricity, a polarisation in proportion to a strain gradient. Its tensor has the same fifty-four components and the same index structure, with a polar vector where the magnetisation had an axial one, and it is even under time reversal. Its character at an operation is the same product without the determinant, and its count is the same average without any sign from a prime.

That average is never zero. Every one of the thirty-two ordinary point groups admits flexoelectricity, from three constants in all five cubic classes to fifty-four in the triclinic ones, and so every one of the 122 magnetic classes does too, since a time-even property ignores the primes altogether. The reason is the parity. A polarisation reverses under inversion and so does a gradient, so the inversion sends the coefficient to itself and nothing about a centre can forbid it. The polar coupling is universal, which is why flexoelectricity is described as a property of every dielectric, centrosymmetric ones included.

The magnetic coupling differs from it by one determinant, and that determinant is the whole story of the census. Where every operation is a proper rotation, the two tensors are counted by identical characters and have identical constants: the cubic $432$ has the flexoelectric three, and so does every other proper class, the ordinary $422$ with eight, $622$ with seven, $23$ with five. Where a class has improper operations, the determinant changes which combinations survive, and in the classes with a plain centre it kills them all. The magnetic census is the universal polar one, filtered through the sign of each operation’s determinant and the prime it carries.

What the gradient means in a real crystal #

A uniform strain is a property of a crystal as a whole. A strain gradient exists wherever the strain is not uniform: in a bent plate, at a dislocation, near a free surface, in a thin film clamped to a substrate of a different spacing, and around every inclusion. In a bulk crystal the gradients are usually small, since strain varies over distances much longer than a cell. At the nanometre scale they become large, which is why flexoelectricity, the polar analogue, is a thin-film phenomenon: a gradient of one part in a thousand over ten nanometres is a large gradient.

The census says where to look for the magnetic analogue. The twenty-one classes with a primed centre are the natural place, because in them nothing else mechanical works: no uniform stress at any power magnetises them, and the gradient is the only mechanical handle there is. Antiferromagnets whose magnetic structure combines inversion with time reversal are common, because two sublattices of opposite moment related by a centre give exactly that operation. Lukashev and Sabirianov proposed in 2010 that the antiperovskite antiferromagnet $\mathrm{Mn_3GaN}$ should show the effect. The census does not say how large any constant is, only which classes can have one.

There is a second reading, about domains. How many domains a transition makes counts the orientations a magnetic order can take when it appears. Two domains related by time reversal have opposite magnetic structures, and in a primed-centre class a uniform stress cannot tell them apart at any power. A strain gradient can, though not on its own. The gradient constants change sign between the two domains, since the coupling is odd under time reversal, so the same bending gives the two domains opposite magnetisations. Bending alone favours neither, because an energy needs something else that is odd under time reversal. A magnetic field supplies it. Bending such a crystal in a field gives the two domains different energies, and that is a way to select between two antiferromagnetic domains that no squeeze in any field can separate.

What the census cannot show #

The figures show which classes allow the coupling and how many constants it has. They cannot show the sizes of any of those constants, which depend on the material, and the census is silent about whether the effect is large anywhere. They cannot show the boundary terms either. A gradient coupling in a finite crystal can be partly rewritten, by integrating by parts, as a surface term, and in the polar case there is a long-running argument about which part of the flexoelectric response is a bulk property and which belongs to the surfaces. The same question arises here and is not addressed: the census counts the symmetry-allowed bulk tensor, and how much of it is observable in a given sample shape is a separate matter.

The convention the counts depend on is the form of the coupling: magnetisation in proportion to the gradient of the strain, symmetric in the strain’s two indices and nothing assumed about the gradient index. A different but common convention writes the coupling as an energy, $M \cdot (\partial \varepsilon)$ with a tensor that also mixes in the gradient of the magnetisation. Such an energy has extra symmetries between its indices and fewer independent constants, so its counts would be smaller. The list of classes, which depends only on the centre and on whether the tensor survives at all, would not change.

Who worked out the pieces #

Neumann’s principle is from the nineteenth century, and Birss’s book of 1964 organised the symmetry of property tensors under the magnetic point groups. The magnetoelectric effect was predicted for $\mathrm{Cr_2O_3}$ by Dzyaloshinskii in 1959 and observed by Astrov the next year, and its fifty-eight classes are standard. Flexoelectricity was described by Mashkevich and Tolpygo in the 1950s and by Kogan in 1964. The symmetry of the magnetic analogue was set out by Eliseev and co-workers in 2009, and Lukashev and Sabirianov’s proposal for $\mathrm{Mn_3GaN}$ followed. The count against the squared stress, the bijection that explains why both come to sixty-nine, the eleven gradient classes without a magnetoelectric effect, and the split of the cubic forms between $432$ and $\bar43m$ are what this essay adds.

What the count must refuse #

The census can fail in ways that show. The characters and the Cartesian projection are independent computations and must agree class by class; they do in all 122. The magnetoelectric count has a known answer and must reproduce it. The two sixty-nines must split by the centre exactly as the parity argument requires, and the cubic forms must be kept or killed by the explicit action of each class’s operations rather than inferred from the counts. The claim that some class with a plain centre admits a gradient term is put to the census and refused.

Still open: a gradient of the magnetisation itself #

The gradient here is a gradient of strain, applied from outside. A magnetic crystal has a second gradient of its own: the gradient of the magnetisation, which is non-zero in every domain wall and every spiral. Coupling the two, a strain gradient against a magnetisation gradient, gives terms of the Lifshitz kind that where a gradient term survives at the zone edge counted for order parameters, and they decide whether bending a crystal can twist a collinear magnet into a spiral. Those terms are time-even, since two magnetisations appear, and polar in two indices, so their parity differs from everything counted here. Which classes permit them, and whether the twenty-one primed-centre classes are again special, is the same signed average with a different character. It has not been computed here.