Where a powder goes blind

The neutron pattern of manganese oxide placed its magnetic cell beyond argument in 1949: the reflections sat at half-integers of the chemical cell, and the cell they implied was doubled along every edge. Which way the moments in that cell pointed was another matter. Analyses of the same kind of powder data reached different answers through the 1950s, until W. L. Roth in 1958 put the moments in the sheets perpendicular to a body diagonal. A year later Gen Shirane wrote down why the question had been so hard: a powder pattern can say some things about the direction of the moments and not others, and which things depends on symmetry alone.

Shirane’s theorem is usually quoted in one line: a cubic powder cannot determine the direction of the moments, and a uniaxial one can determine only their angle to the unique axis. The line is right, and it is easy to misapply. Magnetic reflections land where nuclear ones cannot found that an antiferromagnet’s reflections sit on a coset displaced by its propagation vector, and that the lattice’s symmetry permutes the possible vectors in orbits. The symmetry that decides what a powder can read is the one that carries a ring of those reflections onto itself, and for most cubic antiferromagnets that is not the cubic group. This essay computes the theorem, and then computes which group it has to be applied to.

What a neutron sees of a moment #

A neutron carries a magnetic moment and scatters from the magnetic field of the unpaired electrons. The field of a moment is transverse, and the consequence for diffraction is that only the component of each moment perpendicular to the scattering vector contributes. A moment pointing along the scattering vector scatters nothing, one at right angles to it scatters fully, and in between the intensity goes as the square of the perpendicular part.

For a collinear structure, in which every moment is parallel or antiparallel to one direction $\hat m$, the magnetic intensity of a reflection at scattering vector $Q$ is its magnetic structure factor squared times

$$1 - (\hat Q \cdot \hat m)^2 .$$

The structure factor comes from the signs and from the form factor, and the placement of magnetic reflections showed that it decides where they may appear. The factor above is the only place the direction enters. A single crystal measured one reflection at a time sees it reflection by reflection, and with three or more reflections in independent directions the moment’s direction is fixed up to the symmetry that makes it ambiguous in the first place.

What a ring adds up #

A powder is a great many crystallites in every orientation, and it does not see reflections one at a time. It sees a ring: every reflection with the same length of $Q$ lands at the same scattering angle, and a powder pattern records only that angle. What is measured at the ring is the sum of the intensities of all its members.

For a Bravais lattice with one moment per site the structure factor has the same size on every member of a ring, so the sum is that size squared times the average of the direction factor over the ring. Write the average as a tensor. With $A$ the ring average of $\hat Q \hat Q^{\mathsf T}$,

$$\big\langle 1 - (\hat Q\cdot\hat m)^2 \big\rangle = 1 - \hat m^{\mathsf T} A\, \hat m .$$

Everything a powder can say about $\hat m$ is in $A$, a symmetric three-by-three matrix with trace one, since each $\hat Q \hat Q^{\mathsf T}$ has trace one. And $A$ inherits the symmetry of the ring. If an operation $R$ carries the set of reflections in the ring onto itself, then averaging $R\hat Q (R \hat Q)^{\mathsf T}$ over the ring gives the same answer as averaging $\hat Q \hat Q^{\mathsf T}$, so $RAR^{\mathsf T} = A$. That is Neumann’s principle applied not to a crystal’s property but to an average over its reflections, and the conclusion comes out of the same small argument.

One operation is always in the ring’s group, whatever the crystal. The tensor $\hat Q \hat Q^{\mathsf T}$ is unchanged when $\hat Q$ is replaced by $-\hat Q$, so a reflection and its opposite contribute identically, and a ring behaves as though it had a centre of inversion even when the crystal has none. The same fact is why a diffraction pattern reports only eleven classes rather than thirty-two, and it means the cases to consider are the eleven Laue classes, grouped further by what their rotations force on a symmetric tensor. A symmetric tensor is a quadratic form, and a rotation of order three or more about an axis already makes a quadratic form round about that axis. So the eleven collapse to three behaviours: round in every direction, round about one axis, and not round at all. The average of several orientations is more symmetric than any one of them, and here the averaging is over reflections rather than orientations, and for a cubic ring the extra symmetry is total.

Suppose the ring’s group contains the cubic group. Then $A$ commutes with the three-fold rotation that permutes the axes and with the half-turns that change two signs. The half-turns force every off-diagonal entry to zero, the three-fold forces the three diagonal entries to be equal, and a trace of one makes each of them a third. So $A$ is a third of the identity, and the ring’s intensity is two thirds of its maximum whichever way the moments point. No choice of ring escapes, however many members it has or however they are arranged.

The table is the theorem as a list of cases. A ring with tetragonal, hexagonal or trigonal symmetry has a rotation of order three or more about one axis, and that rotation forces the two entries across the axis to be equal while leaving the entry along it free: $A$ is $a$ in two directions and $1 - 2a$ in the third. The intensity then depends on $\hat m$ only through its angle $\theta$ to the unique axis, as $1 - a - (1 - 3a)\cos^2\theta$, and the azimuth of the moment about the axis drops out. An orthorhombic ring has only half-turns, which make $A$ diagonal and nothing more, so all three entries are free and two or three rings with different entries pin the direction down, up to the signs the half-turns cannot tell apart. The three uniaxial rows share their numbers because a general reflection’s ring averages to the same tensor under any rotation of order three or more about the same axis; it is the order of the axis, not the system, that matters.

The same statement drawn as a map makes the loss visible rather than tabulated.

The cubic panel is the strongest negative result in the subject and it deserves to be read as such. Every point of the map is a different hypothesis about which way the moments lie, and every hypothesis predicts the same number to the last digit. No amount of counting statistics helps and no better instrument helps, because the loss is not noise. It is an identity, and the information was destroyed by the addition before any detector recorded it.

The same numbers, one reflection at a time #

What the sum destroys, the individual reflections still carry, and the cleanest way to see that is a ring small enough to write out.

With the moments along a cube axis, two of the three reflections are at full strength and the third, whose scattering vector lies along the moments, is dark. With the moments along a body diagonal, all three are at two thirds. With the moments along a face diagonal, two are at a half and one at full strength. Three different patterns in a single crystal, and one number in a powder. The powder is blind in this case because the ring is cubic, and the ring is cubic because the propagation vector of this structure is the one halving of the body-centred lattice that the whole cubic group keeps.

A single crystal is not automatically better. A cubic crystal cooled through an ordering transition generally forms domains, and if the structure’s own symmetry is lower than cubic, each kind of domain contributes its own reflections, oriented by an operation the magnetic order lost. A crystal whose domains are equally populated can hand back, reflection by reflection, exactly the average a powder takes. How many kinds of domain there are is the size of the propagation vector’s orbit, which is the index of the group the ordering lands in and so is known before any crystal is measured: one for a kept halving, three or four on the face-centred cubic lattice, six for the body-centred structure with the largest orbit. This is the disguise a merohedral twin wears, and removing it takes a field, a stress or a crystal small enough to hold one domain.

Which group the ring has #

Everything above assumed that the ring’s group contains the cubic group whenever the crystal is cubic. That is where the one-line version of the theorem goes wrong.

A magnetic ring is not a ring of the reciprocal lattice. It is a set of reflections of one length on the coset $k + L^*$, and within one domain only the operations that carry that coset onto itself carry the ring onto itself. Those are the operations that fix the propagation vector modulo the reciprocal lattice, the little group of $k$ that the star of a wavevector defined, and by the orbit–stabiliser count its order is forty-eight divided by the size of $k$’s orbit. A kept halving has an orbit of one and the whole cubic group as its little group. An orbit of three leaves sixteen operations, the tetragonal holohedry about one cube axis. An orbit of four leaves twelve, the trigonal holohedry about one body diagonal. The body-centred orbit of six leaves eight, which is orthorhombic.

So the theorem has to be applied with the little group, and the predictions differ from the naive ones on most of the cubic antiferromagnets there are. Rather than trust that argument, the rings can be summed directly: take the actual magnetic reflections of one domain, group them by length, average $\hat Q\hat Q^{\mathsf T}$ over each shell, and read the tensor.

The rows sort themselves exactly as the orbit column predicts, and no row is an exception. The two antiferromagnets whose halvings are kept — every neighbour opposite on the simple cubic lattice, corner against centre on the body-centred one — have isotropic rings in every shell, and a powder of either cannot say anything about the direction of its moments. Every structure whose propagation vector lies in an orbit of three has rings that are uniaxial about one cube axis, and a powder reads the angle between the moments and that axis. The one in an orbit of four has rings uniaxial about a body diagonal. The body-centred structure with an orbit of six has, beyond its innermost ring, rings with three different entries, and a powder reads its direction outright.

The first ring of that last row is uniaxial rather than general, and the reason is worth a sentence. It holds only the two reflections $\pm k$, and two antipodal vectors average to a tensor that singles out their own direction and treats every direction across it alike. The general tensor needs a ring with several independent directions in it, and the outer shells have them. A reading of the direction therefore comes from the outer rings, where the form factor has already weakened the magnetic intensity, which is one reason a determination that is possible in principle can be hard in practice.

Manganese oxide, read correctly #

Manganese oxide is the type II structure on the face-centred cubic lattice, with propagation vector $(\tfrac12,\tfrac12,\tfrac12)$, one of an orbit of four. Its little group is the trigonal holohedry about the body diagonal that its sheets are perpendicular to, and every one of its magnetic rings is uniaxial about that diagonal. So a powder of manganese oxide can read exactly one number about its moments: the angle they make with the body diagonal of their own domain. It cannot read their azimuth about that diagonal.

That is precisely the form Roth’s answer took. Moments lying in the sheets perpendicular to the body diagonal is a statement about the angle to the diagonal — ninety degrees — and nothing more, and it is the most a powder could have said. The earlier disagreement is what the theorem predicts for anyone who treats the structure as cubic in the powder analysis: a cubic ring would have been blind, and a uniaxial ring analysed as though it were cubic carries a real signal that the wrong model cannot fit consistently. The powder domains do not erase the angle. Each domain’s rings are uniaxial about that domain’s own diagonal, the moments make the same angle with it in every domain because the domains are images of one another, and the powder adds up four copies of the same number.

The one-line theorem would have called manganese oxide blind, because its chemistry is cubic, and it would have been wrong about the most famous antiferromagnet in the subject. The correct version is only slightly longer: a powder is blind to the moments’ direction exactly when the little group of the propagation vector is cubic, which on the cubic lattices happens for the two halvings the cubic group keeps and for no others.

What the computation has to refuse #

The claims above are checked by summing reflections rather than by quoting the theorem, and each test is set against a case it would fail on if the argument were wrong.

The last refusal is the one this essay exists for. A first version of the computation averaged every magnetic ring over the full cubic group, reasoning that the crystal was cubic, and it reported every cubic antiferromagnet as blind. The tensor it produced was exactly a third of the identity every time, which is what an identity produces, and nothing in the output looked wrong. Summing the actual reflections of one domain is what exposed it, because the type I and type II rings then came out uniaxial, and a test that the chemistry’s group and the ring’s group agree would have failed on them at once. The second-to-last refusal is its mirror: for the two kept halvings the rings really are isotropic, and a direction offered from their powder patterns is a direction the data cannot contain.

What the powder leaves for other instruments #

The powder reads as much of the moment’s direction as the little group of the propagation vector allows, and no more. What it cannot read is the orbit of $\hat m$ under that group, and those directions are distinguished only by an experiment that does not add the ring up: a single crystal with its domains made unequal, or a beam of neutrons polarised so that the magnetic scattering of each reflection carries a sign rather than only a size. Both are ways of undoing the sum the powder performed.

Everything here was collinear, with one moment direction throughout. A structure whose moments turn from site to site, a spiral or a structure with several propagation vectors at once, has a direction factor that is not a single $\hat m$, and the ring average then constrains a different tensor. And the symmetry doing the work was only the little group of $k$ acting on reflections, not the whole magnetic space group of the structure. Which magnetism a class permits used the hundred and twenty-two magnetic point groups to decide what a crystal may do; the magnetic space groups, 1,651 of them, would decide in the same way which moment arrangements are compatible with each propagation vector, and that classification is not computed here. The halving a lattice will not permit built the thirty-six magnetic lattices those groups stand on.