A squeeze magnetises a cubic crystal across itself

Thirty-five classes a squeeze can magnetise counted the magnetic point groups in which a stress produces a magnetisation in proportion to itself. There were sixty-six. Thirty-one of them are the classes that may already carry a moment at rest, and the other thirty-five may have none and acquire one when squeezed. Among the ninety classes that are not grey, the effect was forbidden in two kinds of place. One was any class holding the inversion combined with time reversal, which reverses the moment and leaves a stress alone. The other was a short list with no prime in it at all: the ordinary cubic classes $432$, $\bar43m$ and $m\bar3m$, whose four-fold operations reverse the one cubic term that every other cubic class keeps.

That essay ended on a question it could pose but not answer. A stress can enter at higher powers, and a coefficient that is forbidden at first order may be permitted at second. The prime on a centre should forbid every power, since inversion leaves any product of stresses alone and the prime reverses the moment regardless. The three cubic classes have no such excuse, and their prohibition rested on a sign that a product of two stresses might undo. Whether the second-order count follows the same short description, or whether new exceptions appear, was the same signed average with a larger character.

This essay takes the average. The count comes out at sixty-nine, the three cubic classes join with one constant each, and nothing else ever joins. The more interesting result is what that one constant does. It makes a squeezed cubic crystal magnetise at right angles to the squeeze, and it vanishes along exactly the cube’s twenty-six rotation axes, with indices that add up to the number a sphere demands.

The tensor at second order #

A magnetisation that depends on the square of a stress has components $M_i = Q_{iabcd}\,\sigma_{ab}\sigma_{cd}$. The stress is symmetric, so it has six independent components, and a product of two of them is unchanged by swapping the factors, so there are twenty-one distinct quadratic terms. Each can feed each of the three components of the moment, which gives sixty-three coefficients before any symmetry acts.

The coefficients transform as an axial vector against the symmetric square of the stress space. Under a spatial operation the moment picks up the operation’s determinant and its trace, and the quadratic terms transform with the character of the symmetric square, which is half the sum of the stress character squared and the stress character of the operation applied twice. The stress character is in turn half the sum of the trace squared and the trace of the square. The whole is a short formula in traces of an operation and of its powers.

Time reversal adds one sign, exactly as it did at first order. A moment reverses when time runs backwards and a stress does not, so a primed operation contributes its character with a minus sign, and the number of independent coefficients a magnetic class permits is the average of the signed character over the class. The operation that reverses time set up that arithmetic and Neumann’s principle, as one sum is the same average without the sign; every magnetic class counts its properties by mirrors counted the time-odd invariants of every black-and-white class by products of mirrors, which is the same signed average written as polynomials. Third order needs only the symmetric cube in place of the symmetric square, a formula with one more term, and it is computed alongside.

The convention is the one that makes the count a count. Stress is taken in Cartesian components with no contraction factors, the moment is axial, and a coefficient is permitted when the group’s average of the signed character is not zero. Nothing in the calculation says how large any permitted coefficient is.

Sixty-nine, and the three that join #

The figure at the head of this essay is the result. Sixty-nine of the hundred and twenty-two classes permit a second-order effect. Sixty-six of them are the classes that already permitted the first-order effect, and the remaining three are $432$, $\bar43m$ and $m\bar3m$ in their ordinary form, each with exactly one independent constant. The thirty-two grey classes permit nothing, as they already permitted no moment at rest in which magnetism a class permits: a grey group contains time reversal on its own, which reverses every moment and touches no stress. And the twenty-one classes with the primed inversion permit nothing at second order either, for the reason given in advance.

The count is made twice, by two routes that share nothing beyond the list of groups. The first is the character average, computed on the integer matrices in which each class is built. The second writes down the sixty-three basis maps that send a stress to one product of its components along one axis, averages each map over the group in Cartesian coordinates, and measures the rank of the averaged maps by evaluating them on thirty random stresses. The two agree in all one hundred and twenty-two classes. The first-order counts from the same routine agree with the earlier census in all one hundred and twenty-two as well, which is the check that the second-order routine is reading the groups the same way.

The prediction for the primed centre held, and the prediction for the cubic classes held in its optimistic form. All three joined, each with one constant, and no fourth class appeared anywhere among the ninety. The short description survives one order up, minus its second clause. At second order, piezomagnetism among the classes that are not grey is forbidden by a primed centre and nothing else.

Why the list stops growing #

The census was also run at third order, and it adds nothing: sixty-nine classes again, with the three cubic ones now carrying four constants each. That is not a coincidence of the numbers, and two arguments close the question for every order at once.

The first says that a class never loses an order it has admitted. The trace of the stress, the pressure, is left alone by every operation and by time reversal. So a permitted coefficient at order $n$, multiplied by the pressure, is a permitted coefficient at order $n + 1$. Down each column of the figure, once a cell is shaded every cell beneath it is shaded too, and the enumeration confirms it in all one hundred and twenty-two classes.

The second says that the classes forbidding the second order forbid every order. They are the grey classes and the classes holding $\bar1'$, and the census found no third kind. In a grey class time reversal alone reverses the moment and leaves every power of the stress unchanged. In a class with $\bar1'$ the spatial part of that operation is the inversion, which also leaves every power of the stress unchanged, since a stress is even under inversion and so is any product of stresses, while the moment, being axial, is even under the inversion and reversed by the prime. Either way the operation sends every coefficient to its own negative, at every order.

Together those settle it. The list at order two is sixty-nine, no class can leave it, and every class outside it is forbidden at every order. No power of a stress, however high, magnetises the remaining fifty-three classes, and the census of the hundred and twenty-two is complete at the second order. That is a stronger statement than a count at third order, which only says the list did not grow once; the two arguments say it cannot grow again.

The cubic classes, side by side #

The cubic family shows the whole pattern in five rows. The ordinary classes $23$ and $m\bar3$ have one constant at first order, the shear across a cube axis giving a moment along it, and four at second. Their four-fold relatives have the black-and-white groups whose four-fold operations are primed, and those keep the first-order constant, because priming the four-fold reverses the sign that the four-fold put on the cubic invariant, and they have three constants at second order. The groups with a primed inversion, $m'\bar3'$ and two of the $m\bar3m$ family, have none at either order. And the three ordinary classes with unprimed four-folds have none at first order and exactly one at second.

The step from four constants in $23$ to one in $432$ is the cost of the four-fold axis at second order, and the step from one to zero is its cost at first. A single four-fold rotation therefore removes all of the first-order effect and three quarters of the second, and leaves exactly one coefficient behind. The rest of this essay is about that coefficient.

A stretch times a shear #

Under the twenty-four rotations of a cube, the group $432$, the six components of a stress fall into three pieces that the rotations never mix. One is the pressure, the sum of the three normal stresses, which every rotation keeps. One is a pair of tetragonal stretches, the differences between normal stresses, which transform among themselves as the two-dimensional representation called $E$. The last is the three shears, which transform as $T_2$, a three-dimensional representation that the cube’s rotations permute with signs. The shell that splits into kinds does the same kind of splitting for the neighbours of an atom, and what a group does to a function says why it is always possible and always unique.

The quadratic terms are products of two pieces. A pressure times anything gives back that thing. A stretch times a stretch gives a pressure-like term and another stretch. A shear times a shear gives a pressure-like term, a stretch-like pair and a shear-like triple. Only one product gives anything new, a stretch times a shear, and it splits into a triple of type $T_2$ and a triple of type $T_1$. Totalled, the twenty-one quadratic terms come to three copies each of the trivial representation, of $E$ and of $T_2$, and one of $T_1$.

A magnetisation, an axial vector, is $T_1$ under these rotations. So the moment can be fed by exactly one combination of products, and that combination is a stretch across one cube axis multiplied by the shear across the same axis:

$$M_x = q\,\sigma_{yz}(\sigma_{yy} - \sigma_{zz}),\qquad M_y = q\,\sigma_{zx}(\sigma_{zz} - \sigma_{xx}),\qquad M_z = q\,\sigma_{xy}(\sigma_{xx} - \sigma_{yy}).$$

The Cartesian projection finds this form without being told it. Averaging the sixty-three basis maps over the ordinary $432$ gives a one-dimensional space, and the map spanning it, scaled to the formula, agrees with it on forty random stresses to within $10^{-16}$. The same form holds for $\bar43m$ and $m\bar3m$, since their extra operations are either a mirror, under which a moment and these products change in the same way, or the inversion, under which neither changes.

The surprising thing 432 shares with itself #

The class $432$ already has a reputation in this subject. It is the one class without a centre that forbids piezoelectricity: a crystal with no inversion, twenty-four operations, and a character sum that cancels to nothing, so that a stress produces no polarisation in it. The reason the two effects fail together is that in a group with no improper operations a polar vector and an axial vector cannot be told apart. Both are $T_1$ under the rotations of the cube, and the piezoelectric tensor and the first-order piezomagnetic tensor differ only in which of the two the moment is. In $432$ the two counts are the same count, and the twenty-four rotations kill both.

So the second-order result transfers too. In $432$ a polarisation quadratic in the stress is permitted, with exactly the same form, $P_x \propto \sigma_{yz}(\sigma_{yy} - \sigma_{zz})$ and its cyclic partners. The class that cannot be made electrically polar by a stress can be made polar by the square of one, in exactly the pattern by which an ordinary cubic magnet can be magnetised. The two calculations are one calculation, and what separates a crystal that answers with a charge from one that answers with a moment is only whether its order is electric or magnetic, which symmetry alone does not say.

A squeeze along every direction #

The formula is easiest to read under a uniaxial stress, the stress a squeeze between two flat plates applies. A stress of size $s$ along a unit direction $n$ has components $s\,n_a n_b$, so the moment it produces at second order is

$$M = q s^2 \big(n_y n_z (n_y^2 - n_z^2),\; n_z n_x (n_z^2 - n_x^2),\; n_x n_y (n_x^2 - n_y^2)\big).$$

This moment is perpendicular to the squeeze, whatever the direction. Its component along $n$ is the sum of three terms that are the same product of the three coordinates times three differences of squares, and the differences add to zero. A cubic crystal of one of these three classes, squeezed along any line, is magnetised across that line and never along it. The figure draws this as a field of arrows on the sphere of directions, each arrow lying tangent to the sphere at the direction it belongs to.

The field vanishes where all three components do. Two zero coordinates make every component zero, which is a cube axis. One zero coordinate leaves one component, which is zero when the other two coordinates are equal in size, a face diagonal. No zero coordinate requires all three to be equal in size, a body diagonal. So the moment vanishes along the six ends of the cube axes, the twelve ends of the face diagonals and the eight ends of the body diagonals: twenty-six directions, which are exactly the cube’s rotation axes, the four-fold, two-fold and three-fold ones. A search over a grid of nearly a hundred and twenty thousand directions finds no other zero, every near-zero sample lying within a hundredth of a radian of one of the twenty-six.

It does not vanish on the mirror planes. The class $m\bar3m$ has nine mirrors, and it is natural to expect a squeeze lying in one to give nothing, since the mirror relates the squeeze to itself. A squeeze along $[112]$ lies in the mirror plane perpendicular to $[1\bar10]$ and produces a moment of about a quarter of the largest one. The mirror does constrain it: the moment points along the mirror’s normal, $[\bar110]$, because a reflection reverses the parts of an axial vector that lie in its plane and keeps the part across it, so the only axial vector a mirror leaves alone is one at right angles to the mirror. What the mirror cannot do is make it zero.

Why some zeros were unavoidable #

There is a second way to see that the field had to vanish somewhere, and it does not use the cube at all. A field of arrows tangent to a sphere, varying continuously, must vanish at some point: the hairy-ball theorem. More precisely, each zero has an index, the number of times the arrows turn round as one walks once round the zero, and the indices of all the zeros add up to two, which is the Euler characteristic of the sphere, the same number that forces twelve pentagons into any closed cage of hexagons.

Here the indices can be read from the figure and are computed by walking round each zero. Round a four-fold axis the arrows circulate, as a cyclone does round its eye, and the index is $+1$. Round a three-fold axis they also circulate, and the index is again $+1$. At a two-fold axis they come in along one line and go out along the other, a saddle, and the index is $-1$. Six four-fold ends and eight three-fold ends give fourteen; twelve two-fold ends take away twelve; the sum is two. The cube’s symmetry decides where the zeros are and the topology of the sphere decides that their indices balance, and the two decisions are compatible because they had to be.

The connection runs in one direction only. Topology requires a tangent field on a sphere to have zeros with index sum two; it does not require twenty-six of them, and a field with only two zeros, at the poles, would satisfy it. What forces the twenty-six is the group, which carries any zero to all its images, and the representation, which makes the moment at a rotation axis lie along that axis while also lying across it.

In one face of the cube #

The cleanest experiment the formula suggests is a squeeze that stays in one cube face. With $n$ in the face perpendicular to $z$, at an angle $\theta$ from the $x$ axis, the first two components vanish identically and the third is $q s^2 \cos\theta \sin\theta (\cos^2\theta - \sin^2\theta)$, which is a quarter of $q s^2 \sin 4\theta$. The moment points along the face normal, at right angles to every squeeze in the face, and as the squeeze turns it changes sign four times in each half revolution: zero along the cube edge, largest at twenty-two and a half degrees, zero again along the face diagonal, largest with the opposite sign at sixty-seven and a half.

Two features of this law are ways of telling a second-order effect from a first-order one in a measurement. Reversing the stress, from compression to tension, changes nothing, since the moment depends on its square. And turning the squeeze by a quarter turn about the face normal, which carries the crystal onto itself, gives the same moment rather than the reversed one, since $\sin 4\theta$ has period ninety degrees. A first-order effect would reverse under a change of sign in the stress.

The largest moment over all directions is not in a face. Searching the whole sphere puts it on a diagonal mirror plane, roughly twenty-five degrees from a cube axis, at about 0.28 in the same units, twelve per cent more than the best squeeze in a face. A sample cut for the purpose would be cut along that direction rather than along any of the directions a crystal is usually cut.

Domains, and what the square cannot do #

A crystal in one of these classes is an antiferromagnet or some other order with no net moment, and it has two domains related by time reversal, the smallest case of the count of domains a transition makes. The coefficient $q$ is odd under time reversal, so the two domains have opposite $q$, and the same squeeze magnetises them in opposite senses. That is the property that made first-order piezomagnetism useful in the rutile difluorides: a stress and a field together favour one domain, and a crystal can be poled into a single one.

The square changes one thing about that. At first order a domain could be selected by the product of the stress and the field, so reversing either reversed the choice. At second order the stress enters squared, so only the field’s sign selects; any squeeze off the rotation axes, compressive or tensile, combined with a field of one sign, favours the same domain. Whether the effect is large enough for that to matter in any real crystal is outside what symmetry can say.

What the count must refuse #

The two refused claims are the two a reader would reach for first. One is that a pressure magnetises these crystals at second order, because a pressure is the simplest stress there is and enters the symmetric square. It does not: every term of the invariant contains a shear, and a pressure has none. The other is the mirror claim above, that a squeeze lying in a mirror plane gives no moment, which is what a first-order intuition about reflections suggests and which the squeeze along $[112]$ refutes.

What the pictures cannot show is a size. The census says that $q$ may be non-zero in sixty-nine classes and must be zero in fifty-three; it says nothing about how large $q$ is in any crystal, and a second-order coefficient is expected to be small beside a first-order one wherever both are permitted. The figures draw the moment in units of $q s^2$, which fixes every direction, every zero and every ratio, and leaves the one number a measurement would need entirely open. Nor does the census name a crystal whose magnetic class is one of the three ordinary cubic ones; whether such a crystal is known, and whether its constant has been measured, is not something a count of characters can settle.

Where this census sits #

The magnetic point groups were enumerated by Tavger and Zaitsev in 1956, and the tables of which magnetic classes permit which property tensors were assembled by Birss in the early 1960s, in a form that covers tensors of high rank. Piezomagnetism itself was predicted by Voigt early in the twentieth century, put on its modern footing by Dzyaloshinskii in 1957 for the difluorides of cobalt and manganese, and measured by Borovik-Romanov in 1960. The second-order count here is a direct extension of that tradition rather than a new kind of argument, and its interest is in what the single cubic constant does geometrically, which a table of nonzero coefficients does not display.

Still open: a strain that varies from place to place #

The census has one more door it did not try. A magnetisation can depend on how a strain changes across a crystal, its gradient, rather than on the strain itself. That coupling has one more polar index, so unlike every power of a uniform stress it is odd under inversion. A class holding $\bar1'$ would then send the coefficient to itself rather than to its negative, so the primed centre, which forbids every power of a uniform stress, should permit the gradient, while the plain centre, which permits every power, should forbid it. Whether the classes a strain gradient can magnetise are exactly the twenty-one that no uniform stress reaches, together with others, and whether the three ordinary cubic classes join it at first order in the gradient, is the same signed average with one more polar index. It has not been computed here.