Crystallography & Symmetry
patterns derived from their groups
Depth

Series — page 3

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.
Neumann's principle for elastic constants in mmm. Each bar is one operation's contribution to the character of the representation the elastic constants live in — the stiffness of the crystal in every direction and shear. The identity contributes the unconstrained count of 21; every other operation of mmm subtracts from it, and the average over all 8 is 9, which is how many independent components the property may have. The sum is exact in integers and is taken in the lattice basis, so no Cartesian frame is chosen anywhere in it.

Neumann's principle

  1. 1 Neumann's principle, as one sum
  2. 2 A character does not know its basis
  3. 3 Twenty-one, thirteen, nine, three
  4. 4 Permitted is not present
  5. 5 The parts a property splits into
5 essays · point-groups The origins of p2 that change nothing. One cell of p2 with its pattern, and every point marked to which the origin may be moved without a single operation of the group changing its translation part. There are 4 of them per cell, and the count does not change when the search grid is refined, so it is a fact about the group rather than about the grid. Two coordinate lists differing by one of these vectors describe the identical arrangement, which is why no structure's coordinates are ever unique.

Normalisers

  1. 1 The same pattern, described twice
  2. 2 One crystal, and sixteen coordinate lists
  3. 3 The normaliser is not a function of the group
  4. 4 The same site under two names
  5. 5 Three of them, and they are equivalent
5 essays · operations A structure, and the vectors between its atoms. On the left, 4 atoms in a cell. On the right, every one of the 16 vectors between them, each drawn from a common origin: 13 distinct positions, with the 4-fold peak at the origin being each atom paired with itself. That right-hand picture is what a Patterson map shows, and it is the thing a diffraction experiment gives without phases. It has more peaks than the structure has atoms — n² against n — which is why interpreting one is hard, and why it is always symmetric about its centre.

The Patterson function

  1. 1 The map that needs no phases
  2. 2 Where symmetry stacks the vectors
  3. 3 Solving from the vector set
  4. 4 Seventeen groups, seven vector sets
  5. 5 A map of the atoms that break the law
5 essays · diffraction A spontaneous vector in 222. Averaging each of the three axes over the 4 operations of 222 leaves 0 independent components. No direction survives, so the class permits no spontaneous polarisation at all — which is a statement about what is forbidden, not about any measurement.

Properties

  1. 1 The ten with a direction of their own
  2. 2 Twenty of the twenty-one
  3. 3 Each permits what the other forbids
  4. 4 A filter of great precision and no predictive power
  5. 5 The strain the atoms do not follow
5 essays · point-groups Everything measurable at λ = 1.54 Å, and it is a finite set. A section through the reciprocal lattice of a cubic cell of 10 Å, with the limiting sphere drawn. Bragg's law reaches a reflection only if its spacing is at least half the wavelength, so the measurable reflections are the lattice points inside a sphere of radius 2/λ and the points outside it are not merely unmeasured — no experiment at this wavelength can reach them. In this section 516 points lie inside; in the full sphere there are 9,092.

Resolution

  1. 1 How many reflections there are
  2. 2 As sharp as the sphere is wide
  3. 3 The unknowns against the observations
  4. 4 Every reflection, several times over
  5. 5 What one turn of the crystal reaches
5 essays · diffraction The subgroups of p4m of index 2. p4m has 7 subgroup(s) of index 2 with cyclic quotient. 3 of them keep every translation and lose operations — the lattice is untouched and the pattern loses a symmetry at every point. 4 keep every operation and lose translations, and each is named beside the basis of the sublattice it keeps, written in the parent's own axes. Each subgroup is the kernel of a homomorphism onto a cyclic group, found by enumeration; each name is found by searching changes of basis and origin until the operation sets match exactly.

Subgroups

  1. 1 Two ways down from a group
  2. 2 The descent with no shortcut
  3. 3 How many subgroups of index three
  4. 4 The quotient each normal subgroup leaves
  5. 5 Going up costs the cell a parameter
5 essays · operations p3, twinned. p3 twinned by a rotation. To the left of the composition line the motif sits where p3 puts it; to the right every copy has been carried over by the twin law, which is one of the 3 operations the hexagonal lattice has and p3 does not. 24 images on the left, 24 on the right, and the lattice runs through the line unbroken — which is exactly why a twinned crystal looks like a single one.

Twinning

  1. 1 A twin is a symmetry the lattice has and the crystal does not
  2. 2 Twenty-five of the thirty-two can twin, and seven cannot
  3. 3 Quartz has exactly three twin laws, and its lattice is why
  4. 3 A merohedral twin moves no spot at all
  5. 5 The index and the angle a twin misses by
5 essays · applied What each group leaves distinct. The number of genuinely different ways of putting 2 species on the cells of a 4 × 4 block, for 9 plane groups. Every row starts from the same 65,536 arrangements; what differs is the group identifying them. Each count is Burnside's average of fixed points, and each was required to divide exactly by its group's order.

Counting

  1. 1 Counting what a group cannot tell apart
  2. 2 Every colour count at once
  3. 3 An orbit is what the invariants cannot tell apart
  4. 4 The table that decides every action
4 essays · operations 8 tiles over 5 colours. Wang tiles: unit squares with a colour on each edge, which may be laid side by side only where the touching edges agree, and which may never be turned or reflected. That last restriction is what makes them a computational object rather than a jigsaw — an edge colour is a symbol passed from one tile to its neighbour, and turning a tile would let a symbol change direction. The set here was generated from a stated seed.

Decidability

  1. 1 Nothing decides whether a set of tiles tiles the plane
  2. 2 Surrounded twice over, and covering nothing
  3. 3 How much room a hard question needs
  4. 4 The argument that closes eleven
4 essays · classification A circuit that closes on the wrong point: (1, 0). A square lattice with one extra half-column, drawn as a graph: the rows above the core have one more site than the rows below, and the core is the site at the end of the extra column. The path is 4 steps east, 4 north, 4 west and 4 south — the same number out as back — and it ends one lattice vector from where it started. Every one of the 12 circuits in the survey that goes round the core fails by that vector, and all 10 that miss it close exactly.

Defects

  1. 1 The circuit that does not close
  2. 2 How many dislocations a lattice has
  3. 3 The defect that needs two laps
  4. 4 The point defect whose charge has no sign
4 essays · applied p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers.

Flat space

  1. 1 The two that fold into a surface
  2. 2 Ten ways for space to be flat
  3. 3 Every net folds onto a torus
  4. 4 Crystallography in a box
4 essays · classification Three lattices at 2 forms each: 6, 14, 12 faces. The shape each cubic lattice predicts, built as the solid bounded by its top 2 forms, with each face's distance from the centre inversely proportional to its interplanar spacing. The three lattices have the same metric and the same list of indices; every difference between these solids comes from which reflections are systematically absent. Pm-3m leads on {100} and comes out with 6 faces; Fm-3m leads on {111} and comes out with 14 faces; Im-3m leads on {110} and comes out with 12 faces. Taking more than the leading form matters only where the extinction correction has moved something: in a cubic metric a form's planes are placed at a distance proportional to the root of the sum of the squares of its indices, which is exactly where the corresponding corner of the cube already is, so an uncorrected second form arrives tangent and cuts nothing off.

Growth

  1. 1 Which faces a crystal shows
  2. 2 The fast faces are the ones that vanish
  3. 3 The step that never runs out
  4. 4 Which faces are flat
4 essays · applied 7 cells explain the lines; one of them is right. A line list from a face-centred cubic cell of 5.64 Å, with a realistic error added, handed to a sweep over every cubic cell between 2 and 12 Å in all three centrings. 7 distinct cells explain every line within the tolerance, and each is a genuine solution rather than a numerical accident. The true cell comes top by de Wolff's figure of merit — the last Q over twice the mean discrepancy times the number of lines the candidate says should have been visible — which punishes a candidate for predicting lines nobody saw. That is the whole of what makes indexing decidable in practice: not the arithmetic, which has many answers, but a criterion for preferring one.

Indexing

  1. 1 Indexing a powder pattern
  2. 2 A cell from a bag of spots
  3. 3 The figure of merit a supercell always beats
  4. 4 Every alias is a supercell
4 essays · applied The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry.

Lengths

  1. 1 How many vectors of each length
  2. 2 The lengths do not name the lattice
  3. 3 The sum that turns a lattice into its dual
  4. 4 The sum whose answer depends on the shape
4 essays · lattices P2₁/c under every cell choice. One group, described in each of the 6 bases that keep its cell the shape its system requires, with the symbol derived from the operations each time. The distinct symbols are P2₁/c, P2₁/a, P2₁/n — 3 names for one group. Nothing about the crystal has changed: the basis changes all have determinant ±1, so the lattice is untouched, and the census of rotations, screws, mirrors and glides is identical in every row, since conjugation cannot turn one kind into another. What changes is which lattice vector a glide's translation is half of, and the glide letter names exactly that.

Settings

  1. 1 One group, three symbols
  2. 2 Two origins for one group
  3. 3 Six ways to name one group
  4. 4 One matrix, four rules
4 essays · space-groups Six classifications, and which are enumerated here. The families of symmetry groups by how many directions they repeat in and how many they live in. The thirty-two crystal classes, the seven friezes and the seventeen plane groups are each built from their own operations and counted. The seventy-five rod groups, the eighty layer groups and the two hundred and thirty space groups are numbers from the literature, marked as such wherever they appear: reaching them needs the translation extensions and their equivalences in full, which is the content of the classification rather than an application of it. The subperiodic cases sit exactly between the two halves, which is why they are so easy to assume are already known.

Subperiodic

  1. 1 A layer is not a wallpaper
  2. 2 What a cleave leaves
  3. 3 Seventy-five ways to be a thread
  4. 4 What a thread scatters
4 essays · classification Why p4 cannot be drawn with dots. The same group applied to a single dot and to a motif with no symmetry of its own. The dot's orbit turns out to have more symmetries than the group it was made with, so a figure drawn that way illustrates a different group from the one in its caption.

Accidental symmetry

  1. 1 The motif must be a comma
  2. 2 The symmetry diffraction adds
  3. 3 Near-symmetry, and the tolerance that is not here
3 essays · classification One lattice, two cells, and the absences the choice creates. The same set of points described on a centred rectangular cell and on its primitive rhombic cell, with what the centred description scatters. Half the reflections vanish, and they vanish because of how the cell was drawn rather than because of anything the crystal does.

Centring

  1. 1 Centring, and why cm is not pm
  2. 2 Centring, counted as a sublattice
  3. 3 Why the bigger cell wins
3 essays · lattices 18 extension classes, 17 groups. Each of the thirteen arithmetic classes with the number of ways translations may be attached to it — its cohomology — the shape of that group, and how many distinct plane groups the classes come to once the changes of basis that are mere relabellings are quotiented out. The two columns differ in exactly one row, 2mmp, where four extension classes are three groups because two of them are the same group with the axes swapped. No lattice is drawn anywhere in this computation.

Cohomology

  1. 1 Seventeen, without a picture
  2. 2 The screw a dimension does not have
  3. 3 The denominator a group actually needs
3 essays · classification Subgroups of index 3, across the seventeen. Every subgroup of index 3 with cyclic quotient in each of the seventeen plane groups, sorted into the two kinds: 4 keep all the translations and lose operations, 22 keep all the operations and lose translations, and the total is 26. The split is decided by whether the homomorphism onto ℤ3 kills the two lattice translations, which is a property of the kernel and not a judgement. Every one of them is found by enumeration inside the finite quotient by 3Λ, and the count for the whole classification is a measurement.

Colour

  1. 1 Three colours, and why most patterns cannot have them
  2. 2 Seventy-four colourings, forty-six groups
  3. 3 What a half-turn does to three colours
3 essays · classification

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