Crystallography & Symmetry
patterns derived from their groups
Series

Disorder — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The diffuse intensity of an alloy with α₁ = -0.46. The diffuse part of the scattering across the wavevectors an 12 × 12 block can be asked about, one square per wavevector with darkness the intensity. The marked squares are where the average structure scatters — the sharp part, which is what a Bragg reflection is. The diffuse maximum here is at (0.50, 0.50), which is the zone boundary: the alloy is trying to alternate, and a crystal that succeeded would put a sharp reflection exactly there. Summed over every wavevector, the intensity is exactly one per site whatever the correlations are — order moves scattering about, it does not create it.

    The average scatters sharply and the rest does not

    A crystal whose lattice is perfect and whose occupation is not scatters in two parts: the average structure gives Bragg reflections, and the variance is spread over everything between them. The split is exact, the total is one unit per site whatever the disorder does, and an ensemble of n arrangements mislays exactly a fraction 1/n of it.

    part 1 · diffraction
  2. How much one site knows about another, by separation. The Warren–Cowley parameters: the average of the product of the occupations of two sites a given vector apart, over every pair in every configuration. The value at the origin is exactly one — a site always agrees with itself — and it falls away with distance, alternating in sign where the alloy prefers unlike neighbours. These numbers are the whole of what the diffuse scattering measures: its intensity at a wavevector is their Fourier transform, computed here separately and agreeing to the last bits of the arithmetic. Nothing about them requires the crystal to be ordered, and their falling away is what short-range order means.

    The order a diffuse pattern measures

    Where a diffuse maximum sits says what the crystal is trying to become, and its shape is the Fourier transform of how much each site knows about its neighbours. The correlations are a small array of numbers, the intensity is their transform, and neither route to the other loses anything.

    part 2 · diffraction
  3. One crossing or the other, and never both. Two events on a square patch of lattice: a path of occupied sites crossing from left to right, and a path of vacant sites crossing from top to bottom. On the triangular lattice exactly one of them happens in every configuration tested — the claim is combinatorial rather than statistical, so one counterexample would end it. On the square lattice both can fail at once, and do, in more than a quarter of the configurations. That difference is the whole of what follows.

    The threshold a symmetry pins down

    Occupy sites at random and somewhere the occupied ones first join up across the crystal. For almost every lattice that occupancy is known only to a few digits. For the triangular lattice it is exactly a half, and the reason is that on a lattice whose faces are all triangles an occupied path and a vacant path cannot slip past each other — a statement about one configuration at a time, with no probability in it.

    part 3 · diffraction

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