Which magnetism a class permits

Neumann’s principle says that a physical property of a crystal must be invariant under every symmetry operation of its class, and the counting form of it on this site is a character sum: average the character of the property’s representation over the group, and the answer is how many independent components the property may have.

That average counts twenty-one elastic constants in a triclinic crystal and three in a cubic one, eighteen piezoelectric moduli in class 1 and none in any centrosymmetric class. It says nothing whatever about magnetism, because a magnetic moment is a quantity ordinary crystallography has no operation to constrain.

Adding time reversal fixes that, and the change to the arithmetic is a single sign. A primed operation contributes its character multiplied by −1 for a property that reverses under time reversal. Everything else — the projector, the average, the integrality check — is unchanged. That one sign produces two counts that the literature records and that this site can now derive rather than quote: thirty-one magnetic classes admit a spontaneous magnetisation, and fifty-eight admit the linear magnetoelectric effect.

Why a magnetisation is an axial vector, and why that matters #

The character to average depends on what kind of object the property is, and magnetisation is the case that exercises both of the site’s distinctions at once.

It is axial: it changes sign under an improper operation as well as transforming under the rotation, because it is a circulation rather than an arrow. So its character is det(M)·tr(M) rather than tr(M) — the same character optical activity needed, one rank down.

And it is time-odd: it reverses under time reversal, because it is a current. So a primed operation’s contribution is negated.

Those two properties are independent and both are needed. A polar vector that is time-odd, or an axial one that is time-even, would give different counts — and both exist among real properties, which is why the character table on this site carries a sign for each rather than one rule.

Thirty-one, and what a ferromagnet has to give up #

The thirty-one classes that permit a magnetisation are the magnetic analogue of the ten polar classes, and the analogy is close enough to be worth drawing out.

A polar class permits a spontaneous polarisation — a time-even polar vector — and there are ten of them: the classes with a unique direction not reversed by any operation. A ferromagnetic class permits a spontaneous magnetisation — a time-odd axial vector — and there are thirty-one, which is more, because the primes give an operation a second way of being compatible with a moment.

The mechanism is worth one sentence, because it is the whole content of the extra count. An operation that would reverse a moment is fatal to ferromagnetism if it is unprimed and harmless if it is primed: the primed version reverses the moment twice — once by the motion and once by time reversal — and so leaves it alone. So m′ permits a magnetisation lying in the mirror plane, where m does not.

That is why 2′, m′, 4′ and their relatives appear so often in the list. Priming an operation is exactly the move that makes it compatible with a moment it would otherwise have forbidden.

Ferromagnet, antiferromagnet, and the word the count does not use #

A caution about vocabulary, because the count above is often quoted with the wrong word attached.

The thirty-one are the classes that permit a net moment — a magnetisation the crystal has as a whole. Most of the materials in them are not ferromagnets in the everyday sense of iron: they are antiferromagnets whose two sublattices very nearly cancel and leave a small remainder, which is called weak ferromagnetism and is the case Dzyaloshinskii and Moriya explained. Haematite and the orthoferrites are the standard examples, and their moments are a thousandth of iron’s.

The symmetry cannot tell the two apart, and should not be asked to. What it says is that a net moment is not forbidden, and whether that moment comes from parallel spins or from a slight failure of antiparallel ones is a question about interactions. The distinction between a ferromagnet and a canted antiferromagnet is invisible to the character sum, which is a limitation and also a strength: the permission holds whatever the mechanism.

There is a nice consequence of that. A material can be predicted to have some moment from its magnetic class alone, before anything is known about why — which is how weak ferromagnetism was first understood, as a symmetry permission looking for a mechanism rather than the other way round.

The magnetoelectric effect, and its fifty-eight #

The second count is the one that makes the magnetic classification worth a physicist’s attention.

The linear magnetoelectric effect is a magnetisation produced by an electric field, or equivalently a polarisation produced by a magnetic field. Its coefficient α relates a polar vector to an axial one, so it transforms as the product of the two characters; and it is time-odd, because one of its two indices is.

Averaging that character over each of the hundred and twenty-two gives fifty-eight classes with a non-zero answer. The number matters because the effect is rare: it requires a class in which both space inversion and time reversal are broken while their product may survive, which is a strong condition, and it is why the effect was predicted from symmetry before it was measured.

The condition is worth stating in the site’s own terms, because it is a statement about two operations rather than about one. An unprimed inversion kills the effect outright, since α relates a polar quantity to an axial one and inversion treats the two differently; an unprimed time reversal — a grey group — kills it too, since α is time-odd. What survives is the classes in which each of those is either absent or primed, so that the two failures cancel. The eleven Laue classes are what a diffraction measurement can see; these fifty-eight are what an electrical measurement can see, and neither set is a subset of the other.

The history here is the tidiest case in the whole subject of a symmetry argument doing work in advance. Landau and Lifshitz observed in the 1950s that the effect was permitted in some magnetic classes; Dzyaloshinskii identified chromium oxide, Cr₂O₃, as a specific candidate from its magnetic class in 1959; Astrov measured it in 1960. Prediction from a character sum, then the measurement, in that order and about a year apart.

What a permission is not #

Everything above is a permission, and the word is doing exact work. This site’s point-groups field has said it before and it needs saying again in a field where the temptation is stronger, because magnetism has magnitudes and they are what anybody actually wants.

A permitted property may be zero. Thirty-one classes may have a magnetisation; most materials in those classes have none, because their moments happen to cancel. The symmetry says the cancellation is not required, not that it fails to occur.

A permitted property has no size. The character sum returns a count of independent components and no magnitudes whatever. Whether a permitted magnetoelectric coefficient is large enough to measure is a question about electronic structure that nothing here touches, and the coefficients that have been measured span several orders of magnitude.

And the classification is of the ordered state. A magnetic point group describes a material below its ordering temperature. Above it, the group is grey, everything time-odd is forbidden, and the same material permits nothing — so a statement like “Cr₂O₃ is magnetoelectric” carries an implied below 307 K, and the transition is where the permission arrives.

What the counts are worth to somebody looking for a material #

The permission is a filter, and a filter is worth a great deal when the search space is large.

Multiferroics — materials with a spontaneous polarisation and a spontaneous magnetisation at once — are the clearest case. A candidate must be in a class permitting both, and the character sum answers that immediately by asking two questions of one group: is the polar count non-zero, and is the magnetisation count non-zero. The classes surviving both are few, and every search for a multiferroic starts by restricting to them.

The same filter runs in reverse for interpretation. A material that shows an effect its assigned magnetic class forbids has been assigned the wrong class — and that is a real diagnostic, because magnetic classes are inferred from neutron data with assumptions in them, while a measured magnetoelectric coefficient is a direct observation. A permission violated is a structure re-examined.

What this cannot do is rank the survivors. Every class in the list is equally permitted and the effects differ by orders of magnitude between materials in the same class, so the list orders nothing. It is a boundary on the search rather than a direction through it, and confusing the two is the standing error this field’s essays keep coming back to.

Two counts, one calculation, checked against the literature #

The site’s standing arrangement is to compute a count two ways and require agreement. Here the second route is the literature rather than a second calculation, and it is worth being clear about what that does and does not establish.

The character sum is run over each of the hundred and twenty-two groups, which were themselves derived from index-two subgroups rather than looked up. The resulting counts — 31 ferromagnetic, 58 magnetoelectric, 0 ferromagnetic grey groups — are then compared against the standard tables. Agreement says the derivation reproduces the accepted answer; it does not independently confirm the accepted answer, and no essay here claims that it does.

What is independent is the internal consistency. The average of a character over a group must be a whole number, since it is the dimension of a space, and the calculation asserts integrality on every one of the 366 sums it performs. A group assembled wrongly, a character written wrongly, a sign applied to the wrong half — any of those would show up as a non-integer, and none does.

Where the exactness stops #

Three limits, and the third is the largest.

The properties covered are the ones whose characters are written down. Magnetisation, magnetoelectric coupling and polarisation are here; piezomagnetism, magnetostriction and the higher-rank magnetic properties are not, though the same machinery would reach them with the same one-sign change. What is claimed is the two counts computed, not a complete magnetic property table.

The moments are classical arrows. Treating a magnetic structure as a set of axial vectors at atomic sites is exactly enough to decide permissions and nothing like enough to compute a magnetic structure. The quantum mechanics that decides which arrangement a material takes is not in any of this.

The counts are of components, not of directions. A class permitting one component of a magnetisation permits it along one specific direction fixed by the symmetry; a class permitting three permits any direction at all. That distinction is in the numbers and not in the word “permits”, and it matters for an experiment: a single-component class has its moment where the symmetry says, so a measurement along the wrong axis finds nothing and means nothing. The same reading applies to a polar class, where the permitted polarisation lies along the unique axis and nowhere else.

And a point group is not a magnetic structure. The classification says what the macroscopic symmetry permits. Two antiferromagnets with the same magnetic point group can have quite different arrangements of moments inside the cell — that is a magnetic space group question, and the 1,651 magnetic space groups are the object that answers it. This site derives neither the 230 nor the 1,651.

What actually measures a magnetic class #

Every count on this page is about a macroscopic property, and it is worth saying what instrument reports one, because two quite different experiments are involved and they answer different questions.

A magnetometer measures the permission directly. A net moment is what a bulk magnetisation measurement finds, so a material in one of the thirty-one may show a signal and a material outside them may not. That is the count used as a filter, and it is a one-bit answer about the whole sample.

Neutron diffraction measures the arrangement. A neutron carries a magnetic moment of its own, so it scatters from the crystal’s moments as well as from its nuclei, and the magnetic scattering produces its own reflections. Where the moments alternate, those reflections sit at positions the nuclear structure does not use — a doubled repeat gives peaks at half-integer indices — so an antiferromagnet announces itself as a set of extra spots appearing below the ordering temperature and vanishing above it.

The two are not interchangeable and their disagreement is informative. A perfectly compensated antiferromagnet gives strong magnetic reflections and no magnetometer signal at all, which is the case the thirty-one exclude and neutrons see plainly. A canted arrangement gives both. And a material whose moments are ordered only over short distances gives diffuse magnetic scattering rather than peaks, and no macroscopic moment, which neither count on this page describes.

One property of the neutron measurement is worth carrying because it constrains what can be seen. Magnetic scattering comes from the electrons’ spin distribution, which is spread over an orbital rather than concentrated at a nucleus, so the magnetic form factor falls away with scattering angle far faster than the nuclear one. Magnetic reflections are therefore a low-angle phenomenon, and a magnetic structure is determined from many fewer reflections than a nuclear one — which is why magnetic structure determination leans so heavily on symmetry arguments of exactly the kind this page computes.

A permission with a ceiling on it #

Symmetry says which components of the magnetoelectric tensor may be non-zero. There is a second constraint of an entirely different kind that bounds how large they may be, and the two together are what a search for a large effect actually has to satisfy.

The argument is thermodynamic. A magnetoelectric material in an electric and a magnetic field has an energy with a cross term in it, and the energy must be bounded below for the material to be stable at all. Working that requirement through gives an inequality: each magnetoelectric coefficient is bounded by the geometric mean of the corresponding dielectric and magnetic susceptibilities.

That bound is due to Brown, Hornreich and Shtrikman and it explains something the permission count cannot. The magnetoelectric effect is small in every material that has it, and the reason is that magnetic susceptibilities are small in the insulators where the effect is permitted — a good ferromagnet has a large susceptibility and is metallic and generally has a magnetic class outside the fifty-eight. So the search is squeezed from two directions at once: symmetry permits the effect in a minority of classes, and stability bounds it by a product of quantities that are rarely both large.

Neither constraint can be derived from the other, and both are exact. That is the same arrangement as the symmetry counts and the stability criteria in elasticity: one says which numbers exist and the other says which values they may take, and a search that uses only the first will produce candidates that no material can realise.

Where the ladder goes next #

The obvious next rung is the one this essay’s last limit names: magnetic space groups, where the primed operations acquire translations and a whole class of them doubles the cell. The physical consequence is the one that motivated neutron diffraction in the first place — an antiferromagnet’s magnetic cell is often twice its chemical one, so its magnetic reflections appear at positions the chemical structure leaves empty, and reading them is how a magnetic structure is determined at all.