Crystallography & Symmetry
patterns derived from their groups
Series

Diffraction — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A lattice and its reciprocal. The reciprocal lattice is where a crystal scatters. A long axis in the crystal gives closely spaced spots and a short one gives widely spaced spots, so the diffraction pattern is the structure turned inside out.

    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.

    part 1 · diffraction
  2. What pg scatters. The diffraction pattern computed from the atom positions alone. Where a glide plane is present, alternate reflections along a row cancel exactly — and those missing spots are how the glide is identified in an experiment, since the glide itself is never seen.

    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.

    part 2 · diffraction
  3. The phase problem. The same structure rebuilt from its diffraction three ways: with the true amplitudes and phases, with the phases scrambled, and with the amplitudes discarded but the phases kept. The atoms survive the loss of the amplitudes and do not survive the loss of the phases, which is what an experiment throws away.

    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.

    part 3 · diffraction
  4. 168 reflections, 26 rings, one axis. Every reflection of a square lattice out to indices of 6, drawn as a point of the reciprocal lattice with the rings of equal length it lies on. A single crystal gives each point its own place on a detector; a powder is every orientation at once, so what is measured is the ring rather than the point, and the 168 reflections arrive as 26 lines on the single axis at the right, with height the number sharing each. The rings are found by sorting the lengths and grouping equal ones, and every point is checked to sit on the ring it was given. 1 of the rings are marked: they hold reflections that no operation of the lattice's point group relates, so the collapse has added together two genuinely independent measurements and no experiment can take them apart again.

    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.

    part 4 · diffraction
  5. The h0l layer of P2₁/c. The h0l reflections of P2₁/c out to 4 in each index, with each spot decided by summing the structure factor over the group's operations: 44 survive and 36 vanish identically, whatever the atoms are. The pattern of holes is the condition h0l: l even, read back off the spots rather than imposed on them.

    The reflections that are not there

    A screw axis and a glide plane leave no mark on the intensity of any reflection. What they do is delete some, exactly, for every possible arrangement of atoms — and the pattern of deletions is computed here from the sum a crystallographer writes down, rather than read from a table.

    part 5 · diffraction
  6. What each group extinguishes. The extinction conditions of 9 space groups, each derived by summing the structure factor over that group's own operations and reading the surviving rule off the result: P1 — nothing; P1̅ — nothing; P2 — nothing; Pm — nothing; P2/m — nothing; P222 — nothing; Pmm2 — nothing; P4 — nothing; P23 — nothing. 9 of the 9 extinguish nothing at all, and diffraction alone cannot distinguish those from each other.

    Where the experiment runs out

    Absences narrow the space group down and often not to one. Two groups can extinguish exactly the same reflections and scatter with exactly the same symmetry, and telling them apart needs something the diffraction pattern does not contain.

    part 6 · diffraction
  7. The eleven Laue classes. Adjoining the inversion to each of the thirty-two crystal classes collapses them onto 11 groups. Friedel's law says a diffraction experiment sees the crystal and its inverse alike, so this — and not the crystal class — is what a diffraction pattern's symmetry reports. The highlighted symbol in each row is the class that is already its own Laue class, which is to say the centrosymmetric one.

    The eleven a diffraction pattern reports

    A diffraction experiment cannot tell a crystal from its inverse. So the thirty-two classes collapse to eleven before a single reflection is indexed, and a structure determination begins by answering a different question from the one it was asked.

    part 7 · point-groups
  8. -2 0 -3 is forbidden and is reached 64 ways. A layer of the reciprocal lattice of Cc: pale spots are reflections the group extinguishes, solid ones are allowed. The path shows a detour — the beam diffracts once at -3 -3 -3 and again at 1 3 0, and the two together send it in exactly the direction a single reflection at -2 0 -3 would. That reflection is forbidden, so intensity arrives where the symmetry said none could. There are 64 such routes to this one spot inside this window alone.

    The absence that fills itself in

    A systematic absence is the strongest evidence this subject has: a whole zone of reflections cancelling exactly, for reasons of symmetry rather than of arithmetic accident. The exactness belongs to a model — that the beam scatters once. A beam that has already been diffracted can be diffracted again, and the two events together land where the group said nothing could.

    part 8 · diffraction
  9. One crystal, two rows: one streaks and one does not. Two rows of reflections from the same faulted crystal, at a fault rate of 0.05, each drawn against the same row from a perfect one. The upper row has h − k not divisible by three, so each layer contributes a different cube root of unity and the sequence of layers enters the sum: the sharp peaks collapse into a streak. The lower row has h − k divisible by three, the phase factor is one, every layer scatters in step, and the peaks are exactly as sharp as in the perfect crystal. The sorting is an integer condition — a reflection either can see the stacking or cannot, decided by h − k modulo three — which is the most direct evidence there is that the disorder is in the stacking and not in the layers. The profiles are averaged over 16 independently faulted crystals, because one crystal gives speckle rather than a diffuse profile.

    The streaks a faulted stack makes

    Close packing settles two directions and leaves the third to chance. A crystal that chooses wrongly now and then has a lattice in the plane of its layers and none across them — and its diffraction pattern says so, with some rows of spots as sharp as ever and others smeared into streaks, sorted by an integer condition.

    part 9 · diffraction

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