Magnetic reflections land where nuclear ones cannot

In 1949 Clifford Shull and Samuel Smart put a powder of manganese oxide in a neutron beam, cooled it below 120 kelvin, and watched reflections appear that no X-ray pattern of the same powder had ever shown. They sat at indices that were half-integers of the chemical cell — positions where, as far as the atoms were concerned, there was nothing to scatter from. That pattern was the first direct observation of an antiferromagnet, the arrangement Louis Néel had proposed on thermodynamic grounds in the 1930s, in which neighbouring moments point opposite ways and the crystal as a whole carries none.

The striking thing is not that the new reflections appeared but where. None of them fell on a reflection the atoms already made. The halving a lattice will not permit described an antiferromagnet as a lattice split into two cosets of a subgroup of index two, the translations that keep each moment and the ones that reverse it. This essay follows that split into reciprocal space, where it becomes a statement about positions: the magnetic reflections are the nuclear reflections moved by half a reciprocal-lattice vector, and a half-vector never lands on a whole one. The same shift, counted, turns out to be the seven halvings again, and the lattice’s own symmetry acting on those seven decides something a neutron experimenter meets on the first day, which is how many kinds of domain the magnet breaks into.

The shift, in one line #

Take a lattice of sites, each carrying a moment of fixed size along a fixed direction, and let the sign of the moment be the only thing that varies. A halving is a homomorphism from the lattice onto the two-element group, and every such homomorphism can be written as a phase. There is a vector $k$ with $2k$ in the reciprocal lattice for which the sign on the site at $r$ is

$$s(r) = e^{2\pi i\, k\cdot r} = \pm 1 .$$

The vector $k$ is the propagation vector of the magnetic structure: the wavevector of the one plane wave of signs the moments follow. Because $2k$ is a reciprocal-lattice vector, $2k\cdot r$ is an integer for every lattice vector and the phase really is plus or minus one; because $k$ itself is not, some sites get minus.

Now write down what the two kinds of scattering see. The nuclear amplitude at a point $h$ of reciprocal space sums a unit phase over the sites, and the magnetic amplitude sums the same phase weighted by the sign:

$$F_{\text{nuc}}(h) = \sum_r e^{2\pi i\, h\cdot r}, \qquad F_{\text{mag}}(h) = \sum_r s(r)\, e^{2\pi i\, h\cdot r} = \sum_r e^{2\pi i\, (h+k)\cdot r} = F_{\text{nuc}}(h+k).$$

That is the whole theorem. The magnetic amplitude at $h$ is the nuclear amplitude at $h + k$, so the magnetic pattern is the nuclear pattern translated by $-k$, which is the same set as the nuclear pattern translated by $k$ since $2k$ is a reciprocal-lattice vector. The nuclear reflections of a Bravais lattice sit on its reciprocal lattice. The magnetic ones sit on the coset of the reciprocal lattice that contains $k$, and a coset other than the lattice itself shares no point with it.

The two panels are the two ways the shift can look, and the difference between them is only a matter of which cell the indices are written in. On the simple cubic lattice with every nearest neighbour reversed, the propagation vector is $(\tfrac12, \tfrac12, \tfrac12)$ and the magnetic reflections carry three half-integer indices. Half-integer indices are what Shull and Smart saw, though the manganese ions in their oxide sit on a face-centred lattice rather than a simple one; the same propagation vector does the same thing there, as the section on that lattice below shows. On the body-centred lattice with corners up and centres down, the propagation vector is $(1, 0, 0)$ in conventional units, which looks like a whole reciprocal-lattice vector and is not one: the body-centred lattice’s reciprocal lattice holds only the points with an even index sum, which is why a body-centred lattice extinguishes every reflection whose sum is odd. The magnetic reflections sit precisely on those extinguished points. Centring put a hole at every odd-sum position, and the antiferromagnet fills every one of them, and nothing else.

An alloy whose two species cancel #

The right-hand panel has a familiar shape. A body-centred lattice whose corners hold one kind of atom and whose centres hold another is the caesium chloride structure, and the reflections a superlattice adds worked out what ordering two species does to a pattern: the fundamental reflections carry the sum of the two scattering powers and the new superlattice reflections carry their difference. For an alloy of two neighbouring elements that difference is small and the superlattice reflections are faint.

An antiferromagnet is that alloy taken to its limit, with the two species chosen equal and opposite. As far as the magnetic scattering is concerned, the corner sites carry $+\mu$ and the centre sites $-\mu$. The superlattice reflections carry the difference, $2\mu$ per pair, and they are as strong as they could be. The fundamental reflections carry the sum, which is zero. So the magnetic contribution does not merely add a new set of spots beside the old ones; it is entirely absent from the old ones, and that absence is exact rather than a matter of the moments being small. The one-line theorem above is the general form of this cancellation, for any halving of any lattice, and the alloy is the case a reader can check with a pencil.

That is also why the discovery needed neutrons. Time reversal, the operation that turns each moment round and moves nothing, is invisible to an X-ray: it scatters from electron density, and the electron density of a manganese ion with its moment up is the same as one with its moment down, so it sees a crystal with every site identical and sees nothing new below the ordering temperature. A neutron carries its own magnetic moment and interacts with the ion’s, so it sees the signs. The atoms and the moments occupy the same sites and produce two superimposed patterns, and the theorem says the two can never be confused, because they do not share a single position.

A caution the one-line argument hides: a real magnetic amplitude also carries the magnetic form factor, which falls with the length of $h$ as the unpaired electrons are spread over the ion, and a factor for the direction of the moment relative to $h$, which the next essay makes its whole subject. Both multiply the amplitude and neither can move a reflection. Where a magnetic reflection may appear is decided by the signs alone, and it is decided before anything is known about how strong the reflection will be.

Seven halvings are seven places to look #

The propagation vector was introduced as a device for writing a sign, and it is worth seeing that it is exactly as much information as a halving. A vector $k$ with $2k$ in the reciprocal lattice matters only up to adding a reciprocal-lattice vector, since that leaves every phase $e^{2\pi i k\cdot r}$ unchanged. So the propagation vectors live in the half-lattice modulo the lattice, a group with two choices in each of three directions: eight points, of which the origin gives every site the same sign and is not a halving at all. That leaves seven, and these are the seven index-two subgroups the halving essay counted from the other side, as homomorphisms decided by where they send a basis. Seven halvings in real space are seven propagation vectors in reciprocal space, and each of them is a different place in the pattern where magnetic reflections could appear.

The eight points are not new either. They are the wavevectors that equal their own negatives modulo the reciprocal lattice, the points inversion and time reversal leave alone, and in the plane the same count gives four of them, three besides the zone centre — the four points whose signs locate a band’s charge. In space there are eight, and the seven that are not the centre are the full list of places a commensurate, collinear, two-sublattice antiferromagnet can put its reflections. Anything else — a propagation vector at a third of a reciprocal vector, or at an irrational fraction of one — is a magnetic cell three times larger or no cell at all, and is outside what a halving can describe.

The lattice’s point symmetry acts on the seven exactly as it acts on any set of wavevectors: an operation carries $k$ to its image, and the star of a wavevector is the orbit it traces. Written as three bits — the value of the halving on each primitive basis vector — the seven fall into orbits that differ from one lattice to the next.

The simple cubic panel is readable at a glance once the bits are translated. The orbit whose members have a single one are the three propagation vectors $(\tfrac12,0,0)$, $(0,\tfrac12,0)$ and $(0,0,\tfrac12)$, alternating sheets perpendicular to one cube axis. The orbit with two ones are the face diagonals, alternating rows. The lone survivor with three ones is $(\tfrac12,\tfrac12,\tfrac12)$, every neighbour opposite, the only one of the seven that treats the three axes alike. The body-centred lattice’s orbit of six is the centres of the six pairs of opposite faces of its Brillouin zone, and its survivor is $(1,0,0)$, the corner-against-centre structure of the figure before. The face-centred lattice keeps none, which is the halving essay’s result for fcc arriving again from the reciprocal side.

An orbit is a set of domains #

Orbit sizes are not bookkeeping. They are physical, because a crystal does not order all at once.

Cool a paramagnet through its ordering temperature and each region of it picks a propagation vector. If the lattice’s symmetry carries the chosen $k$ onto a different $k'$, then the structure with $k'$ is the image of the structure with $k$ under an operation of the paramagnet, so it has exactly the same energy and is exactly as likely. Different regions pick differently. The result is a crystal made of k-domains, one kind for each member of the orbit, and each kind lights its own set of magnetic reflections. By the orbit–stabiliser count, the number of kinds is the index of the stabiliser of $k$ in the holohedry, which is the orbit’s size: that essay’s rule that a transition makes as many domains as the index of the group it lands in, applied to the one piece of symmetry a propagation vector can break.

A halving kept whole is the opposite case. Its orbit has one member, the structure has only one propagation vector available to it, and the crystal forms no k-domains at all. (It may still form domains of other kinds — the two structures related by reversing every moment are always both present, and are distinguished only by a phase no intensity can read.) Keeping a halving, which the earlier essay needed in order to call a coloured lattice a Bravais lattice of its type — the same ceiling the holohedry sets on any structure built on a lattice — is here the statement that an antiferromagnet with that propagation vector orders as a single domain of propagation.

The table rewards reading down its columns rather than across its rows. Every orbit partition sums to seven, as it must. The lattices with the least symmetry keep the most, because an operation that cannot move a halving cannot put it in an orbit with another: the triclinic lattice’s only operations are the identity and inversion, and inversion fixes every propagation vector since $k$ and $-k$ differ by a reciprocal-lattice vector. The same is true of every operation of the primitive monoclinic and orthorhombic lattices, which modulo two are all the identity. So on those three lattices an antiferromagnet on any halving orders as a single domain of propagation.

The body-centred cubic lattice has the largest orbit in the table, six, which makes it the lattice on which an antiferromagnet with a propagation vector at an edge centre shatters into the most k-domains. The face-centred orthorhombic lattice keeps three halvings though its cubic parent kept none. Lowering the symmetry splits the fcc orbit of three into three orbits of one, because the three cube axes become inequivalent; this is the reciprocal-space face of the fact that a crystal which distorts as it orders can remove its own domains.

The lattice where every antiferromagnet has domains #

The face-centred cubic row deserves its own figure because it is where the argument meets materials. Manganese oxide, nickel oxide, cobalt oxide and iron oxide all have rock-salt structures whose metal ions sit on a face-centred cubic lattice, and every one of them is an antiferromagnet. The table says none of them can order as a single domain of propagation.

The two orderings are the two orbits. Type I alternates sheets perpendicular to a cube axis. Its propagation vector $(0,0,1)$ is one of three, the X points of the face-centred cubic zone, and like the body-centred case it looks like a whole reciprocal vector and sits on a reflection centring had extinguished: face centring keeps only reflections with indices all even or all odd, and type I lights exactly the mixed ones. A type I crystal has three kinds of k-domain, one for each axis. Type II alternates sheets perpendicular to a body diagonal. Its propagation vector $(\tfrac12,\tfrac12,\tfrac12)$ is one of four, the L points, and a type II crystal has four kinds of domain, one for each body diagonal. Manganese oxide is type II, which is why Shull and Smart’s reflections were at half-integers; the doubled cell they reported is the cell a single L-point halving produces.

Each kind of domain alone would give a pattern with lower symmetry than the cubic chemistry, since one body diagonal has been singled out. A crystal with all four kinds present in equal amounts gives a pattern that looks cubic again, the four sets of magnetic reflections superimposed. That is the same disguise a merohedral twin wears, where domains related by a lost symmetry sum to a pattern with the symmetry restored. Separating them is an experimental problem with its own methods — applying a field or a stress to favour one domain, or measuring a crystal small enough to hold only one — and the table above says in advance which lattices force the problem on the experimenter.

The halving essay drew the same seven fcc halvings from the lattice side, with how many operations each keeps.

Read against the orbit table, the bars are the orbit–stabiliser theorem written out twice. Forty-eight divided by sixteen is three, the size of the type I orbit; forty-eight divided by twelve is four, the size of the type II orbit. The halving essay’s negative result — no face-centred cubic black-and-white Bravais lattice — and the neutron experimenter’s positive one — three or four domains in every face-centred cubic antiferromagnet — are the same computation, read as a count of operations kept in one case and a count of images made in the other.

What the computation has to refuse #

Each claim above is computed from sites and signs rather than from a table of known structures, and each is tested against a case built to break it.

The first line holds a number worth a sentence. The fourteen holohedries between them keep thirty-nine of the ninety-eight halvings, and those thirty-nine are not twenty-two Bravais types until the normaliser has identified the ones that differ only by a relabelling of axes. A halving kept by its lattice’s symmetry and a black-and-white lattice type are different objects, and a computation that confused them would report thirty-nine where the literature reports twenty-two. The refusal that matters most is the third. The structure factors are summed site by site over a doubled cell, with no appeal to the shift theorem, and the test is that no magnetic reflection coincides with a nuclear one in any of the seven cubic structures. A sign error in the moments, or a halving whose $2k$ was not a reciprocal-lattice vector, would put magnetic intensity on a nuclear reflection and fail it at once.

Where the next question is #

Everything here was decided by the signs, and none of it by the direction the moments point. Where a magnetic reflection may appear, how many sets of them a crystal shows and which lattices force domains are all fixed by the propagation vector. The direction enters only through the intensity, as the part of each moment perpendicular to the scattering vector, and that is the one piece of a magnetic structure the positions cannot give.

Whether the intensities can give it is a separate question with a sharper answer than it first appears to have. Where a powder goes blind takes the sum a powder performs over a ring of reflections of one length and finds that for a ring with cubic symmetry it destroys the direction completely, for a uniaxial ring it leaves only the angle to the unique axis, and below that it leaves enough. The orbits of this essay decide which case a given antiferromagnet is in, because the symmetry of a magnetic ring is the symmetry its propagation vector keeps and not the symmetry of the chemistry. It is the question Shull and Smart’s own manganese oxide pattern raised and could not settle, since their reflections placed the doubled cell exactly and said nothing reliable about which way its moments lay.