Crystallography & Symmetry
patterns derived from their groups
Concept

Magnetic point group — where it appears

A crystal class together with a rule saying which of its operations also reverse every magnetic moment; there are a hundred and twenty-two. Eight of the merges needed to reach that number require a rotation no lattice may have.

Named by 2 essays across one field — 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

Named alongside it

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

Time reversalAntisymmetryCharacterConjugacyFerromagnetismIndexMagnetoelectric effectNeumanns principleShubnikov groups

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