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.
11 frameworks, 3 where the count is wrong. Every net in this collection read as a framework of rigid bars and free joints, with the cell free to change shape. Maxwell's count and the number of mechanisms agree on most of them and not on all: a framework with a state of self-stress has a bar the count treats as removing a freedom that the others had already removed, and it has a mechanism the count cannot see. Here that is fes, snb, ring5, where the count says 2, -1, -4 and the rank says 3, 0, 9. The identity Maxwell is always right about — count equals mechanisms minus self-stresses — holds on every row.

Rigidity

  1. 1 The count that promises a mechanism
  2. 2 A fold that keeps its symmetry
  3. 3 A mechanism that is a wave
  4. 4 The mechanisms a count cannot see
  5. 5 The polyhedra that can flex
  6. +2 more
7 essays · applied P2₁/c, in the two diagrams the Tables print. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. The symmetry elements drawn: 2 2₁ screw axes, 2 glide planes, 4 inversion centres. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

Space groups

  1. 1 The step a flat surface has no room for
  2. 2 Forgetting a group in three dimensions
  3. 3 The half of a translation that is not a choice
  4. 4 Sixteen candidates, ten groups
  5. 5 Eleven groups that are their own reflection's rival
  6. +2 more
7 essays · space-groups 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
  6. +2 more
7 essays · operations Sublattices of index n in the plane. For each index up to 12: the number of sublattices found by building every Hermite normal form of that determinant, and the number the Dirichlet series ζ(s)ζ(s−1) predicts — the sum of the divisors in the plane, and a longer sum in space. The two columns are computed by routines that share no code, and the figure does not appear at all if any row disagrees.

Sublattices

  1. 1 How many ways there are to thin a lattice
  2. 2 The sublattices that stay square
  3. 3 The reflections a superlattice adds
  4. 4 Every way down, and no way round
  5. 5 The sublattices that are the same shape
  6. +2 more
7 essays · lattices 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. So are both subperiodic families: the seventy-five rod groups class by class, and the eighty layer groups as seventeen one-sided, seventeen mirrored and the forty-six two-colour classes of the plane group underneath. The two hundred and thirty space groups remain a number from the literature, marked as such wherever it appears: reaching it needs the translation extensions of seventy-three arithmetic classes in full.

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
  5. 5 Eighty is seventeen, seventeen and forty-six
  6. +2 more
7 essays · classification The fourteen Bravais lattices. All fourteen lattices: triclinic P, with 2 symmetries; monoclinic P, with 4 symmetries; monoclinic C, with 4 symmetries; orthorhombic P, with 8 symmetries; orthorhombic C, with 8 symmetries; orthorhombic I, with 8 symmetries; orthorhombic F, with 8 symmetries; tetragonal P, with 16 symmetries; tetragonal I, with 16 symmetries; rhombohedral P, with 12 symmetries; hexagonal P, with 24 symmetries; cubic P, with 48 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything.

The fourteen Bravais lattices

  1. 1 Twenty-five cells, and fourteen lattices
  2. 2 Forty-eight becomes sixteen
  3. 3 A lattice described on somebody else's axes
  4. 4 Thirty-two from fourteen matrices
  5. 5 Seventy-three, without a search
  6. +1 more
6 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
  4. 4 Seventy-three Smith forms
  5. 5 Two groups no single operation reveals
  6. +1 more
6 essays · classification 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
  4. 4 A fibre of achiral crystals can rotate light
  5. 5 A looser fibre, more piezoelectric
  6. +1 more
6 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
  4. 4 A warm plane crystal has no Bragg peaks
  5. 5 A flapping sheet keeps its peaks in the plane
  6. +1 more
6 essays · diffraction The angles between the faces of {102̅}. The form {102̅} of class 3̅m in section, with each face labelled by its indices. Its 6 faces make 15 pairs and only 3 distinct angles, the smallest being 76.43°. Every value is computed from the cell's metric — the one calculation in this family that is not integer arithmetic, because an angle is a real number and a lattice does not constrain it.

Forms

  1. 1 Why a crystal face carries small whole numbers
  2. 2 The angles belong to the substance, the shape to the specimen
  3. 2 A form is an orbit, and whether it closes is an integer question
  4. 3 Five classes grow the same cube
  5. 5 A zone is a vanishing dot product
  6. +1 more
6 essays · applied 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
  4. 4 The two most anomalous wavelengths are the wrong pair
  5. 5 The third wavelength is there to measure the edge
  6. +1 more
6 essays · diffraction Two structures on 8 sites with the same vectors. Two arrangements of 4 atoms on a ring of 8 positions. They are not the same arrangement — no rotation of the ring and no reflection carries one onto the other — and every interatomic vector occurs the same number of times in both. The bars below are the shared vector counts, which is the Patterson function of each: the tall one at the origin is the atom count and carries no information, and everything else is what a diffraction experiment measures. Their diffraction patterns are identical in every intensity, so no measurement of intensities, at any resolution, distinguishes them.

Homometry

  1. 1 Two structures, one Patterson
  2. 2 Where the pairs come from
  3. 3 Two structures on a torus, and one Patterson
  4. 4 When the atoms are not all the same
  5. 5 Symmetry does not rescue a Patterson
  6. +1 more
6 essays · diffraction The hat: eight kites, thirteen sides. The shape a search over the eight-kite polykites returns, drawn on the kite grid it lives in — the Laves tiling [3.4.6.4], in which every hexagon is cut into six kites. The eight kites of the shape are tinted and its outline is drawn heavy. Thirteen sides result, of two lengths only: a half and root three over two, in units of the hexagon's circumradius, with one side of twice the shorter length where two kite edges lie in a line. Its interior angles are 90, 120, 240 and 270 degrees. Nothing about the shape was chosen: it is the one octakite that clears every filter in the search.

Monotile

  1. 1 One tile, and no period
  2. 2 The tile that needs no reflection
  3. 3 The hat and the turtle are one tiling
  4. 4 How much of the hat is a crystal
  5. 5 Aperiodic is two words in space
  6. +1 more
6 essays · aperiodic 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
  4. 4 A field lets every class rotate light
  5. 5 Three places for the two-fold axis
  6. +1 more
6 essays · point-groups 6mm: 6 irreducible characters on 6 classes. The character table of the plane point group 6mm, constructed rather than quoted. The columns are its 6 conjugacy classes, with the number of operations in each; the rows are its 6 irreducible representations, of dimensions 1, 1, 1, 1, 2, 2. The dimensions satisfy 1² + 1² + 1² + 1² + 2² + 2² = 12, which is the order of the group, and that identity together with the count of classes is what proves the list complete. Every entry is an exact element of the twelfth cyclotomic ring; the decimals shown are the numerical value of an integer vector, not a computed approximation.

Representations

  1. 1 What a group does to a function
  2. 2 How large a degeneracy may be
  3. 3 A coincidence the group did not ask for
  4. 4 The shell that splits into kinds
  5. 5 What a group forbids to happen
  6. +1 more
6 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
  6. +1 more
6 essays · diffraction N(z): the fraction of reflections weaker than z. The cumulative distribution of normalised intensities, measured on two structures built from the same atoms — one with an inversion centre, one without — and drawn against the two closed forms, 1 − e^(−z) without a centre and erf(√(z/2)) with one. The curves are furthest apart at small z, which is the useful end: a centrosymmetric structure has far more nearly-absent reflections, because its structure factor is a single real number that can pass through zero rather than a complex one that rarely does.

Statistics

  1. 1 Whether there is a centre is a statistic
  2. 2 The zones that behave as if there were a centre
  3. 3 The average that knows the atoms and not where they are
  4. 4 A twin hides in the statistics
  5. 5 A translation that is nearly there
  6. +1 more
6 essays · diffraction Every way regular polygons can fill a turn. The seventeen multisets of regular polygons whose interior angles add to exactly 360°, listed with the sum that qualifies each of them. They are found by a search over sizes from three upward: the largest polygon that can appear is the forty-two-gon, which needs a triangle and a heptagon beside it, and the search stops there because the smallest interior angle is a third of a turn so at most six polygons can meet. Nothing here is a table looked up — the list is the output of the search, and every count on the page downstream of it is counted from this one.

Tilings

  1. 1 Twenty-one vertices, eleven tilings
  2. 2 Eleven tilings, five groups
  3. 3 Eleven duals, one tile each
  4. 4 Which shapes tile by themselves
  5. 5 Three answers in whole numbers
  6. +1 more
6 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
  4. 4 Where a symmetric motif lifts the group
  5. 5 In space a motif can lift two positions in three
5 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
  4. 4 Where a gradient term survives at the edge of the zone
  5. 5 A screw keeps a line and a glide keeps a plane
5 essays · space-groups

All essays