Crystallography & Symmetry
patterns derived from their groups
Concept

Time reversal — where it appears

The operation that reverses every magnetic moment while moving nothing, written 1′ and of order two. A magnetic moment is a current loop, so running time backwards reverses it, and admitting that as an operation is what turns thirty-two classes into a hundred and twenty-two.

Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

point-groups · Magnetic 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.

point-groups · Magnetic 4mm: a degeneracy tuned into existence, and gone at the next weight. Four invariant operators on one orbit of 8 points under 4mm, differing only in the weight given to a single class of pairs. The first column is not a choice: it is the value that weight has to take for two levels of different symmetry to arrive at the same number, found by sweeping the weight and closing on the crossing, and the two levels there agree to 1.0e-9. The character table predicts levels of sizes 1, 1, 1, 1, 2, 2; the tuned column shows 1, 1, 1, 2, 3 and every other column shows the predicted pattern again. That is the whole of what an accidental degeneracy is — a property of one choice of weights, not of the group — and it is why the weights have to be moved before a degeneracy is called forced. A degeneracy the group requires would be in all four columns, because nothing respecting the symmetry can lift it.

A coincidence the group did not ask for

Two levels sitting at the same value look identical whether symmetry required it or not. The difference is testable: move the numbers the symmetry does not decide and watch what survives, because a degeneracy the group forces cannot be shifted by anything the group leaves alone.

point-groups · Representations Where time reversal does something, and what. Every wavevector of every plane group at which time reversal changes the answer, with the square of each antiunitary operator, the unitary prediction, Herring's corrected prediction and the measured degeneracies. Case (b) is Kramers' theorem in a crystal with no spin, and it happens exactly where a glide's operator squares to −1. Case (c) is a representation being carried to a different one by the antiunitary operator, so the two become one level. Nine of these rows were open before the criterion was built — three the earlier census called unaccounted and six it could not reach at all.

The degeneracy time reversal forces

A crystal with a glide has levels that stick together at the edge of its zone for a reason no character table contains. The operation responsible is antiunitary, it squares to minus one, and Kramers' theorem then applies to a model with no spin anywhere in it — which closes nine rows an earlier census in this collection had to leave open.

space-groups · K symmetry A spin needs two full turns to come back. The number a rotation about a fixed axis multiplies a state by, against the angle turned through. A vector — anything of integer spin — is back where it started after one full turn; a spin-one-half state is multiplied by minus one and needs a second turn. So the operators acting on such a state do not form the rotation group: a full turn is an operation distinct from doing nothing, and the group is twice as large.

Two turns to come back

A rotation through a full turn does nothing to a crystal and multiplies a spin-one-half state by minus one, so the group acting on such a state is not the point group but a group twice its size. Building those eleven double groups from quaternions and averaging a random operator over each gives the degeneracies a spin may have — and shows that the doubling everybody calls Kramers' is time reversal's doing and not the double group's.

point-groups · Representations 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.

lattices · Magnetic Four inversion centres, and a band's charge on each. A square crystal with four atoms in each cell, at the points a quarter of the way along both diagonals, and the four points of the cell that an inversion maps onto themselves: the corner, the midpoints of the two edges, and the centre. None of the atoms sits on one. For each of four versions of the crystal, differing only in which bonds are stronger, the ring marks where the lowest band's charge is centred — computed from the band's Berry phase and predicted from four parities. Each version puts it on a different inversion centre, and in every case on a point with no atom.

Four signs say where the charge sits

In a plane crystal whose only symmetry beyond its lattice is a centre, a band's charge must sit on one of the cell's four inversion centres — and which one is decided by four signs: whether the band is even or odd under inversion at the four wavevectors inversion leaves alone. The centre can be a point no atom occupies, and when the four signs multiply to minus one there is no centre at all, only a charge that winds round the cell.

space-groups · K symmetry 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.

point-groups · Magnetic 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.

point-groups · Magnetic Every crystal class permits the Faraday effect. The thirty-two crystal classes, each with two counts: the number of independent components of the tensor that turns a magnetic field into an optical rotation, and the number of components of the natural gyration tensor. The first is never zero: 5 classes have 1, 11 classes have 2, 11 classes have 3, 3 classes have 5, 2 classes have 9. The second is zero in 17 classes, every one with an inversion centre or a mirror arrangement that cancels it. The Faraday count is the mean of the squared trace over the class, which is never below one because the isotropic part always survives.

A field lets every class rotate light

Seventeen of the thirty-two classes forbid a crystal to rotate the plane of polarisation on its own. Put the same crystals in a magnetic field along the ray and not one of the thirty-two forbids it, the centrosymmetric ones included. The field supplies the one thing natural rotation needs a handed structure for, and time reversal is what tells the two rotations apart.

point-groups · Optical Which Laue groups have a sampling grid that touches no special point. Every Laue group on every Bravais lattice it occurs on, twenty-four pairs, with the smallest grid of wavevectors that keeps the whole group and has no point fixed by any operation but the identity. Such a grid exists for 1̅, 2/m, mmm and 4/m, at exactly as many points as the group has operations. It does not exist, as far as the search reaches, for 4/mmm or for any group with a three-fold axis: 3̅, 3̅m, 6/m, 6/mmm, m3̅ and m3̅m.

A three-fold axis puts a grid point on itself

A grid of wavevectors offset from the zone centre can keep a crystal's whole symmetry and touch no mirror, no axis and no point where time reversal folds the zone onto itself, and then its share of the zone is exactly one over the order of the group, with no boundary excess at all. Four Laue groups have such a grid, as small as the group itself. Every group with a three-fold axis has none, for a reason that fits in a line: the rotoinversion fixes one point of the plane across its axis, and above that point the three-fold rotation fixes the grid.

operations · Fundamental domain The fifty-eight black-and-white classes, sorted by reflection class. Every black-and-white magnetic point group, grouped by the smallest reflection class containing its underlying crystal class, with the degrees of the mirror products that generate its relative invariants for a polar and for an axial time-odd quantity. Every class's two series are exactly those sums of mirror products, and the number of generators is the class's index in its reflection class: one, two or four.

Every magnetic class counts its properties by mirrors

The invariants of every crystal class turned out to be products of mirrors of a reflection class, times that reflection class's own free invariants. A magnetic class asks for polynomials that some of its operations negate, which is harder in principle. It is not harder in fact: all fifty-eight black-and-white classes, for a time-odd quantity that is polar and for one that is axial, are counted by mirror products — provided the reflection class contains the whole class and not only its unprimed part, which for thirty of them it does not.

point-groups · Invariants Every square-symmetric grid of wavevectors has a point on a mirror. One cell of the square reciprocal lattice, centred on the zone centre, with the mirror lines of 4mm through it and three grids the four-fold rotation keeps with an offset off the axis. The quarter turn allows that offset at one place only, the second four-fold centre of the grid's own lattice, ringed. On an aligned grid of odd side the rotation fixes a point; on an aligned grid of even side the offset lies on a diagonal mirror; on a grid turned by 45 degrees it lies on an axial mirror. No grid escapes all three.

The offset a quarter turn allows lies on a mirror

A census of sampling grids found free grids for the tetragonal group 4/m and none for 4/mmm, to a hundred and twenty-eight points, with no reason. The reason is two lines long. A quarter turn lets a symmetric grid be offset to exactly one place off its axis, the second four-fold centre of the grid's own lattice, and a square lattice that a mirror keeps can only be aligned with its mirrors or turned by half a right angle. Either way that centre lies on a mirror line.

operations · Fundamental domain 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.

point-groups · Magnetic 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.

point-groups · Magnetic

Named alongside it

The objects these essays reach for when they reach for this one.

Magnetic point groupCharacterAxial vectorBrillouin zoneDegeneracyDomain stateLaue classWavevectorFrobenius schur indicatorFundamental domainHolohedryIndex two subgroup

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