Crystallography & Symmetry
patterns derived from their groups
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Series — page 4

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
The seven groups a uniform field can have. Five of them have an axis and two do not. A cone has every rotation about its axis and mirrors containing it; a cylinder adds the mirror across the axis and the two-folds that go with it; turning either one destroys the mirrors that would reverse the turn. The sphere and the sphere made of something with a handedness are the two with no axis to speak of. Each drawing is the definition: the group is the set of motions leaving the picture unchanged.

Curie

  1. 1 The seven groups a field can have
  2. 2 What a crystal keeps in a field
  3. 3 What a texture permits
3 essays · point-groups 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.

Disorder

  1. 1 The average scatters sharply and the rest does not
  2. 2 The order a diffuse pattern measures
  3. 3 The threshold a symmetry pins down
3 essays · diffraction Friedel's law, as an equality rather than a resemblance. Each pair of bars is a reflection and its opposite for a structure of three atoms in no particular arrangement. They are the same height, and not approximately: with real scattering factors, negating the indices conjugates the structure factor, and conjugation does not change a modulus. The test behind this figure requires the largest difference over 40 pairs to be below 10⁻⁹ and it comes back exactly zero. This is why a diffraction pattern is centrosymmetric whatever the crystal is, and why the thirty-two classes collapse to eleven before a structure is even proposed.

Friedel's law

  1. 1 The law that hides handedness
  2. 2 One experiment gives the cosine, the other gives the sine
  3. 3 How much of it is the other hand
3 essays · diffraction The seven frieze groups. Every way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.

Friezes

  1. 1 Seven friezes
  2. 2 Why sixteen become seven
  3. 3 The friezes inside the seventeen
3 essays · classification p6m: freezing Γ2 leaves p31m. The same crystal three times. On the left, a pattern with the full symmetry of p6m. In the middle, the displacement each atom takes under the order parameter — the arrows are the mode, drawn in the first colour for one set of atoms and the second for the other where there are two. On the right, the atoms moved by a small multiple of those displacements. The group of the right-hand pattern is p31m, of index 2 in the parent, and it was found by the detector from the point set alone. The prediction — which operations carry the displacement field to itself — is made separately, from the mode and not from the points, and the two lists of operations are identical.

Landau

  1. 1 An order parameter is a representation
  2. 2 Which way the order parameter points
  3. 3 The cell a zone-boundary mode doubles
3 essays · space-groups 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.

Magnetic

  1. 1 The operation that reverses time
  2. 2 Which magnetism a class permits
  3. 3 The halving a lattice will not permit
3 essays · point-groups Three shapes, and nothing else. The dielectric tensor of a crystal is an ellipsoid, and averaging a generic one over a point group leaves exactly three possibilities: a sphere, where all three principal values agree and the crystal is optically isotropic; a spheroid, where two agree and there is one optic axis; and a general ellipsoid, with two. The counts are 5, 19 and 8 of the thirty-two classes, and they were found by computing the eigenvalues rather than by sorting the classes by system.

Optical

  1. 1 Three optical characters, and the arithmetic that assigns them
  2. 2 Fifteen may rotate light, and eleven are chiral
  3. 3 The axes a class pins down
3 essays · point-groups How often a dot gives the wrong group. Every position on a grid inside the cell, tried as a single-dot motif for each of the seventeen groups. The bar is how often the resulting pattern turned out to have more symmetry than the group it was made with — so the caption would have been wrong and nothing about the picture would have shown it.

Ornament

  1. 1 The groups ornament actually uses
  2. 2 Two colours, and a symmetry that swaps them
  3. 3 The Alhambra question
3 essays · classification Y-pentomino: A B C D E F, with 6 arcs. The boundary of the Y-pentomino cut into six arcs. A runs from one corner to another and D is the same arc traversed backwards, so D is a translate of A and the translation is (3, 1) cells. Each of B, C, E and F is carried onto itself by the half turn about its own midpoint, and those midpoints are the four marked dots — That is Conway's criterion, and a shape meeting it tiles the plane by translations and half turns.

Isohedral

  1. 1 A tiling of the whole plane, decided on one tile's edge
  2. 2 One shape, two kinds of tile
2 essays · classification

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