Magnetic — the series · Crystallography & Symmetry

One hundred and twenty-two magnetic point groups. The three kinds, counted. Thirty-two ordinary groups, which contain no primed operation; thirty-two grey groups, which contain time reversal on its own and are the symmetry of anything magnetically disordered; and fifty-eight black-and-white groups, one for each way of splitting a class into a subgroup of index two and its complement. The last number is the one that has to be computed: the index-two subgroups are found by closure inside each class, reduced up to conjugacy, and reduced once more by an equivalence that needs a rotation no lattice may have. 32 + 32 + 58 = 122, and every term is a measurement.

The operation that reverses time

A magnetic moment is a current loop, so running time backwards reverses it and moves nothing. Admitting that as a symmetry operation turns the thirty-two crystal classes into a hundred and twenty-two — and eight of the merges needed to reach that number require a rotation no lattice may have.

part 1 · point-groups

The classes that permit a spontaneous magnetisation. Every magnetic point group permitting a spontaneous magnetisation — 31 of the 122 — with the number of independent components each allows. an axial vector, reversed by time reversal — a ferromagnet has one and nothing else does. The count comes from averaging the character over the group, with a primed operation's contribution multiplied by −1 because the property reverses when time does. That is Neumann's principle with one extra sign in it, and it reproduces the numbers the literature records without being given them.

Which magnetism a class permits

Neumann's principle with one extra sign in it decides which of the hundred and twenty-two magnetic classes may have a spontaneous magnetisation and which may show the magnetoelectric effect. The answers are thirty-one and fifty-eight, and they come out of the same average that counts elastic constants.

part 2 · point-groups

Twenty-two halvings the fourteen lattices permit. Every lattice has exactly seven subgroups of index two, whatever its shape. The third column is how many of the seven the lattice's own group carries onto themselves, and the fourth is how many of those survive as distinct types once a change of basis within the type is allowed to identify them. The running total ends at twenty-two, which with the fourteen grey lattices is the thirty-six magnetic Bravais lattices — and the row that ends at zero is the face-centred cubic lattice.

The halving a lattice will not permit

Admit time reversal and a lattice splits into points that leave the moments alone and points that reverse them. The second set is a coset of a subgroup of index two, and every lattice has exactly seven of those, whatever its shape. What differs is how many of the seven the lattice's own symmetry survives — and the face-centred cubic lattice survives none of them.

part 3 · lattices

Where an antiferromagnet's reflections land. Two cubic antiferromagnets, with their nuclear reflections as dots and their magnetic reflections as triangles, in units of the conventional reciprocal lattice. On the left, a simple cubic lattice with every neighbour opposite: the magnetic reflections sit at half-integer indices, in a layer the nuclear pattern never visits. On the right, a body-centred lattice with the corners up and the centres down: the magnetic reflections sit at integer indices with an odd sum — exactly the reflections the body centring had extinguished, lit up again by the moments. In both, no magnetic reflection falls on a nuclear one.

Magnetic reflections land where nuclear ones cannot

An antiferromagnet scatters neutrons into places its chemical structure leaves dark, and never into a place it lights. The reason is one line of algebra: the magnetic pattern is the nuclear pattern shifted by half a reciprocal-lattice vector. Each of a lattice's seven halvings is one such shift, and how many of them its symmetry keeps decides how many kinds of domain the magnet must form.

part 4 · point-groups

Seven cubic antiferromagnets, and what a powder of each can say. Seven collinear antiferromagnets on the three cubic lattices, each with the size of its propagation vector's orbit, the number of the forty-eight cubic operations that keep the vector, the kind of every magnetic ring out to a fixed radius, summed from the actual magnetic reflections of one domain, and what a powder pattern can therefore read about the moments' direction. The two whose halving the cubic group keeps are blind in every ring. An orbit of three leaves every ring uniaxial about a cube axis, an orbit of four about a body diagonal, and the body-centred orbit of six leaves rings that read the direction. The cubic chemistry is the same in every row; the propagation vector decides.

Where a powder goes blind

A neutron sees only the part of a magnetic moment across its path, so each reflection knows which way the moments point. A powder adds up every reflection of a given length, and for a ring with cubic symmetry that sum comes out the same whichever way they point. But the ring's symmetry is not the chemistry's: it is the symmetry the propagation vector keeps, and most cubic antiferromagnets have lost enough of it to be read.

part 5 · point-groups

The ninety magnetic classes that are not grey, sorted by their centre. The ninety magnetic point groups that are not grey, in three columns by whether they contain the inversion, the inversion combined with time reversal, or no centre at all, each tile shaded by what a stress can do: magnetise a crystal that is already magnetised (31 classes), magnetise one that has no moment at rest (35), or nothing. A dot marks the classes that also permit the magnetoelectric effect. No class with the primed inversion is piezomagnetic and no class with the plain one is magnetoelectric.

Thirty-five classes a squeeze can magnetise

A magnetisation proportional to a stress is permitted in sixty-six of the hundred and twenty-two magnetic classes. Thirty-one are the classes that may be magnetised already; the other thirty-five may have no moment at rest and acquire one when squeezed, and in them a stress tells apart two domains that differ only by the direction of time. Among the ninety classes that are not grey, it is forbidden only by a primed centre and in the three ordinary cubic classes with four-fold operations.

part 6 · point-groups

The first order of stress that can magnetise each magnetic class. The ninety magnetic point groups that are not grey, in four columns by the lowest power of a stress that can give them a magnetisation: 31 have one at rest, 35 acquire one in proportion to the stress, 3 only in proportion to its square, and 21 under no power of it at all. The three that need the square are the ordinary cubic classes 432, 4̅3m and m3̅m, and the twenty-one that no stress magnetises are exactly the classes holding the inversion combined with time reversal.

A squeeze magnetises a cubic crystal across itself

Three ordinary cubic classes forbid a magnetisation proportional to a stress. At the square of the stress they each admit exactly one constant, and nothing else joins, at that order or any higher: sixty-nine classes is the final list. The one constant has a shape no table of coefficients shows. A squeeze along any direction magnetises the crystal at right angles to the squeeze, and along each of the cube's twenty-six rotation axes it magnetises nothing, which is the least a field on a sphere is allowed to vanish.

part 7 · point-groups