Crystallography & Symmetry
patterns derived from their groups
Ladder

Diffraction — the ladder

4 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. the crystal latticewhere it scatterslong in one is short in the otheraxis ratio 1.7

    The reciprocal lattice

    Nobody has seen a space group. Crystals are read from where they scatter, and where they scatter is a second lattice in which long has become short and short has become long.

    rung 1 · diffraction
  2. 115 present6 absentthe glide's signaturecomputed from the atom positions, not from the grouppg

    Systematic absences

    The most informative part of a diffraction pattern is the part that is not there. A glide plane cancels alternate reflections along a row, exactly, and those missing spots are how a symmetry nobody can see is identified.

    rung 2 · diffraction
  3. right amplitudes, right phases4 of 4 peaks on atomsright amplitudes, wrong phases0 of 4 peaks on atomsall amplitudes 1, right phases4 of 4 peaks on atomsthe peaks were located and compared against the atomsp4, 153 reflections

    The phase problem

    A detector records how much light arrives and not when it arrives, so half of every diffraction measurement is thrown away before it is written down. The half that is lost turns out to be the half that carries the structure.

    rung 3 · diffraction
  4. 44484884481284488484spacing, shortest on the left168 reflections26 peaksan accidental overlap at 2 unrelated reflections(-5 0) (-4 -3) (-4 3) (-3 -4)binned by spacing, multiplicities countedsquare lattice

    What a powder pattern loses

    Grind a crystal up and every orientation is present at once, so a two-dimensional pattern of spots collapses onto a single axis. Reflections that had their own places arrive together, and some of the coincidences are exact and have nothing to do with symmetry.

    rung 4 · diffraction

All ladders