Crystallography & Symmetry
patterns derived from their groups
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Magnetic — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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
  2. 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
  3. 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

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