Crystallography & Symmetry
patterns derived from their groups
Concept

Symmetry-adapted basis — where it appears

A basis of functions each transforming as a single irreducible representation. In it an operator commuting with the group is block-diagonal, which is why the decomposition is the first step of any calculation on a symmetric structure.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

p3m1 at (0, 0): the levels the little group requires, and the ones measured. The levels of the p3m1 model at (0, 0), with degenerate ones drawn thick. The little group there has order 6, and its characters predict levels of dimensions 1, 1, 2, 2. The measurement is 1, 2, 2, 1, and the account is "unitary". The values are numerical and the multiplicities are read at a stated gap; the prediction they are compared against is exact.

Where two levels must meet

At most wavevectors nothing of a crystal's symmetry survives, and its levels are as unconstrained as any operator's. At a handful of them a whole point group survives, and where that group has a two-dimensional representation, two levels are obliged to coincide — before anything about the material is known.

space-groups · K symmetry An orbit of 12 points under 6mm, split into 6 kinds. The functions defined on one orbit of 12 points under 6mm form a space of that dimension, and the group acts on it by permuting the points. The character of that action is the cheapest one in the subject — at each operation it is the number of points left where they are — and decomposing it against the table gives the multiplicities on the right. They are whole numbers, they weight the dimensions to 12 again, and the multiplicity of the trivial representation is the number of orbits, which is Burnside's lemma arriving as a special case.

The shell that splits into kinds

The neighbours of an atom carry a space of functions as large as the shell, and the group does not treat that space as one thing. It splits into pieces of a few kinds, in whole numbers, and the count is the cheapest character in the subject: how many neighbours each operation leaves where they are.

point-groups · Representations

Named alongside it

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

Irreducible representationBloch stateBurnside lemmaCharacterCharacter tableDegeneracyLittle groupMultiplicityOrbitPermutation representationWavevector

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