Two turns to come back

How large a degeneracy may be reads the possible degeneracies of a level off the irreducible representations of the crystal classes: a class with no representation above dimension one can hold no degenerate level, a cubic class can hold a triplet, and nothing in a crystal is degenerate five ways because no crystallographic point group has a five-dimensional representation. It ends with a caveat, and the caveat is the subject here.

The caveat is that a state with spin does not transform under the point group. It transforms under a group twice as large, and the extra representations of that larger group are degeneracies the point group’s own list does not contain.

Where the factor of two comes from #

Rotate anything through a full turn and it comes back. That is what “a full turn” means for a crystal, a tensor, an orbital, a pattern on a wall.

It is not what happens to a spin-one-half state. The operator implementing a rotation through angle θ about an axis acts on such a state through the half-angle: it multiplies by cos(θ/2) and a term in sin(θ/2), and at θ = 2π that is −1 rather than +1. A spinor changes sign under a rotation nobody performed. Two full turns are needed to get back.

The consequence is a statement about groups rather than about physics. If the full turn is not the identity operator, then the operators do not form the rotation group — they form a group in which the full turn is a distinct element −1 that squares to +1 and commutes with everything. Every rotation has two operators above it, ±q, and the group is twice the size. That is the double group G*, and it is the group a spin actually lives in.

The clean way to build it is with quaternions. A unit quaternion q gives a rotation, q and −q give the same rotation, and the unit quaternions covering the rotations of a point group form its double group with no further work. Closing a set of quaternion generators under multiplication produces G* directly, and the covering being two-to-one is then something to check rather than to arrange — the trivial group is the case that catches an implementation that forgets it, since the double group of “no rotations at all” still has two elements.

Only the eleven proper classes are here — the ones made of rotations alone. That is not a limitation of the method but a choice of scope: a class containing an inversion or a mirror is a proper class times a two-element group in most cases, and the extra structure is bookkeeping rather than a new phenomenon. The eleven carry the whole of what is interesting.

More classes, and therefore more representations #

The number of irreducible representations of a finite group is the number of its conjugacy classes. That is the one fact this essay borrows without deriving, and everything below is a consequence of it.

Every double group here has more classes than the rotation group under it. 222 has four classes and its double group has five; 432 has five and its double group has eight; the cyclic groups double their class count exactly. So every one of the eleven has representations that do not come from the rotation group, and those are the spinor representations: the ones in which the full turn acts as −1 rather than as +1.

222 is the smallest case worth staring at. Its four elements are the identity and three perpendicular half turns, and every one of its four representations is one-dimensional — the class has no degenerate level at all as far as an ordinary tensor is concerned. Its double group is the quaternion group of order eight, with five classes, so it has one representation left over after the four; the squares of the dimensions must add to eight, so that representation is two-dimensional.

A spin-one-half state at a site of symmetry 222 is therefore doubly degenerate, and nothing in the ordinary point group says so. That is not a subtlety at the edge of the theory. It is the ordinary situation for a transition-metal ion on a low-symmetry site, and reading the degeneracy off the ordinary character table gives the wrong answer.

Splitting a level without a character table #

The degeneracies could be got from the double groups’ character tables. They are got here another way, because the other way is shorter to justify and is the same trick an accidental degeneracy is separated from a forced one by.

Take a level of spin j, which is a space of 2j+1 states carrying the spin-j representation of SU(2). Build the matrices of that representation for every element of G* — the symmetric power of the quaternion’s own two-by-two matrix, which is exact enough that its unitarity is checked to fifteen decimal places rather than assumed. Then take a random Hermitian matrix and average it over the group:

H = Σ_{g ∈ G*} D(g) A D(g)†

The result commutes with every group element by construction, so it is as general a Hamiltonian as the site’s symmetry permits. Its eigenvalue multiplicities are the degeneracies, because a generic invariant operator has no degeneracy beyond the one symmetry forces — which is exactly the argument that separates a forced degeneracy from an accident, run here for the purpose it was built for.

Two things make the numerical arithmetic safe to read as a group-theoretic count. The multiplicities must be whole numbers adding to 2j+1, which a slip would break immediately. And they must not depend on the random matrix the average started from, which is checked across several seeds rather than assumed.

The picture a chemist already draws #

The integer-spin rows are a validation rather than a result, and they are a strong one.

A d level is j = 2, five states. At a site of cubic symmetry it splits into a triplet and a doublet — t₂g and e_g, the diagram in the first chapter of every book on transition-metal chemistry. The computation above produces 3 + 2 with no crystal-field parameter anywhere in it: the only input is which rotations the site has.

An f level is j = 3, seven states, and at the same site it goes to 3 + 3 + 1. Lower the symmetry to tetragonal and the d level goes to 1 + 2 + 1 + 1; lower it to orthorhombic and it goes to five singlets, because 222 has no degeneracy to offer an integer spin at all.

Symmetry gives the pattern and never the size. That division is the one this collection keeps everywhere — what a symmetry argument decides is which quantities may be non-zero, never how large they are — and it is why a crystal-field diagram can be drawn correctly by someone who knows nothing about the ion.

A degeneracy no tensor can have #

There is one entry in the table that ought to stop a reader, and it is in the cubic rows.

How large a degeneracy may be establishes a ceiling: no crystallographic point group has an irreducible representation of dimension above three, so nothing in a crystal is degenerate four ways for reasons of symmetry. The largest degeneracies belong to the cubic classes and they are triplets, which is why a p orbital keeps its three states in a cubic site and why the vibrations of a cubic molecule come in threes.

At a cubic site a level of spin three-halves comes out four-fold degenerate, in one piece.

That is not a violation of the ceiling; it is the ceiling being a statement about the wrong group. The cubic rotation group has twenty-four elements, five classes, and dimensions one, one, two, three and three — the ceiling is three because 1 + 1 + 4 + 9 + 9 = 24 and there is no room for more. Its double group has forty-eight elements and eight classes, so there are three dimensions left to find after the five, and the squares of all eight must add to forty-eight. The three new ones are two, two and four.

So the ceiling for a crystal is three for anything without spin and four with it, and the four is reached only in the cubic classes. The four-fold level has a name in the literature — Γ₈ — and it is the reason a spin-three-halves ion in a cubic site behaves so unlike one in any lower symmetry: there is a single level holding four states, and lowering the symmetry at all breaks it into two Kramers pairs. The table shows exactly that happening: 4 at a cubic site, 2 + 2 at every tetragonal, trigonal, hexagonal and orthorhombic one.

A cubic site is therefore doing something no argument about tensors could have predicted. Every quantity a crystal has that is not a spin — a strain, a polarisation, a set of orbitals, a normal mode — is constrained by a ceiling of three, and the constraint is a genuine one that this collection derives from the class list. A spin ignores it, because a spin is not a tensor, and the ceiling it obeys is a different number computed from a different group.

Why anyone measures this #

The practical setting for all of it is magnetic resonance, and the connection is direct enough to state in a paragraph.

An electron paramagnetic resonance experiment drives transitions between the levels of an unpaired electron’s spin in a magnetic field. What it can see depends on there being levels to split: a Kramers pair is degenerate at zero field and splits linearly once a field is applied, which gives a clean resonance at a field proportional to the frequency. A non-degenerate level has nothing to split, and an even-electron ion at a low-symmetry site frequently has none — the crystal field has already separated everything, the resonance is at a field the spectrometer cannot reach, and the ion is called EPR-silent.

So the parity of the electron count decides whether the technique works at all, and the reason is the table above. An odd number of electrons means a half-integer total spin, means every level is a Kramers pair, means there is always a transition at low field. An even number means integer spin, means the levels may be singlets, means there may be nothing to see. That is not a rule of thumb; it is the third and fourth columns of the splitting table, read as an experimental prediction.

The same arithmetic decides what a four-fold Γ₈ level does under a small distortion, which is the beginning of the Jahn–Teller story — a degenerate level in a site whose symmetry the ion itself can lower, with the lowering paying for itself. This collection has the symmetry half of that in the cubic term that forbids a continuous change; the spin half needs the double group, and needs it for exactly the reason above: the degeneracy being broken is one the ordinary point group cannot see.

Where Kramers’ degeneracy actually comes from #

Now the half-integer rows, and the thing the computation was really for.

Every half-integer entry in the table is a pair or larger. That is Kramers’ theorem: a system with an odd number of electrons has every level at least doubly degenerate, whatever the crystal field, and no arrangement of charges around it can split the last pair. It is the reason an odd-electron ion always has a magnetic resonance to measure and an even-electron one may not.

The usual statement attributes it to the half-integer spin, and that statement is incomplete in a way the computation makes visible. Run the same average without time reversal and several sites split a half-integer level into single levels. At a site whose only rotation is a half turn, j = ½ comes out as 1 + 1. The double group alone does not forbid it: its spinor representations there are one-dimensional, and a one-dimensional representation carries a single level.

What supplies the doubling is time reversal, and it is an antiunitary operation rather than a unitary one: it conjugates as well as rotating. For a half-integer spin it squares to −1, and an antiunitary operator squaring to −1 cannot fix any state up to phase, so every eigenstate has a partner it cannot equal. Adding it to the average — which means averaging over the group extended by that one antiunitary element — turns every one-dimensional spinor level into a pair, and the table fills with even numbers.

This is the same theorem, from the other side, as the degeneracy time reversal forces. That essay finds Θ² = −1 in a model with no spin anywhere in it: the minus sign comes from a glide whose square is a half translation, at a wavevector where a half translation is a sign. Here it comes from the spin. The theorem does not care which, and having the two derivations side by side is what makes that visible: what Kramers’ theorem needs is an antiunitary symmetry squaring to minus one, and half-integer spin is one way to have one rather than the definition of one.

What this does not settle #

The eleven, not the thirty-two. Improper classes — those with mirrors, inversion or rotoinversion — are not built here. Adding them is mechanical, since an improper class is either a proper one with the inversion adjoined or is isomorphic to a proper one, but “mechanical” is not “done” and the count of eleven is what the figures show.

No energies, again. Everything above says how many levels there are and how degenerate each is. Which one lies lowest, and how far apart they sit, depends on the actual charges and orbitals and is not a symmetry question. Permission is not presence applies to a level as much as to a property.

The eleven double groups are not eleven different objects. Read down the column of orders and there are only four abstract groups among them: the cyclic ones of order two, four, six, eight and twelve, the binary dihedral ones covering 222, 32, 422 and 622, and the binary tetrahedral and octahedral groups covering 23 and 432. That is the same collapse the ordinary classes show — thirty-two classes, eighteen abstract groups — arriving one level up, and for the same reason: an abstract group does not know which rotations of space it was built from.

And spin–orbit coupling is assumed, not derived. Treating a state as carrying a definite total angular momentum j is what makes the double group the right object; in a regime where the crystal field is much stronger than the spin–orbit interaction, the orbital and spin parts are better treated separately and the ordinary point group does most of the work with the spin degeneracy tacked on afterwards. Which regime a material is in is a question about magnitudes, and this collection does not compute magnitudes.

What to carry away #

Three things, in the order they matter.

The point group is the wrong group for half the questions asked of it. Anything with an odd number of electrons — most transition-metal ions in most oxidation states, every radical, every doped semiconductor with an unpaired carrier — transforms under the double group, and a degeneracy read off the ordinary character table can simply be wrong. The correction is not small: 222 goes from having no degeneracy at all to having a forced pair.

The double group is not exotic and is not extra physics. It is the group of operators that actually act, and the ordinary point group is its quotient by an element that happens to act trivially on everything without spin. Getting it by closing eleven small sets of quaternions takes a page of code.

And a degeneracy has more than one possible cause, which is the habit this collection keeps insisting on. A pair of levels may be stuck together because the site’s rotations require it, because time reversal requires it, or because two numbers happened to be equal. The three are told apart by removing one cause at a time and seeing whether the pair survives — which is what the two tables above do, and what a coincidence the group did not ask for does for the third.