LatticeData

LatticeData[lattice,"property"]

gives the specified property for a lattice.

LatticeData[n]

gives a list of named lattices of dimension n.

Details

  • Lattices can be specified by standard names such as "FaceCenteredCubic" and "CoxeterTodd".
  • LatticeData[] gives a list of classical named lattices.
  • LatticeData[patt] gives a list of all named lattices that match the string pattern patt.
  • LatticeData[{"type",id},] gives data for the lattice of the specified type with identifier id.
  • Type-related lattices include:
  • {"BarnesWall",n} BarnesWall lattice BWn
    {"Bravais",{"system","centering"}}Bravais lattice for the specified crystal system
    {"CoxeterBarnes", {r, n}}CoxeterBarnes lattice
    {"DualRootLattice",group}dual root lattice for the specified Lie group
    {"IntegerLattice",n}integer lattice
    {"KappaLattice",{m,n}}-lattice
    {"LaminatedLattice",{m,n}}laminated lattice
    {"MordellWeil",n}MordellWeil lattice
    {"Niemeier",n}^(th) Niemeier lattice
    {"PerfectLattice",n}perfect lattice
    {"Quebbemann",n}Quebbemann lattice
    {"RootLattice",group}root lattice for the specified Lie group
  • Crystal systems are specified by standard names such as "Trigonal", "Monoclinic", etc. Centering is "FaceCentered", "BodyCentered", or "BaseCentered".
  • Groups can be specified either for example as "A5" or {"A",5}.
  • LatticeData[group,] is equivalent to LatticeData[{"RootLattice",group},].
  • LatticeData["Properties"] gives a list of possible properties for lattices.
  • Lattice points properties include:
  • "Basis"basis vectors
    "Determinant"determinant of Gram matrix
    "Dimension"dimension of lattice
    "Dual"dual lattice
    "GeneratorMatrix"matrix of generators
    "Genus"genus of lattice
    "GlueVectors"glue vectors (when applicable)
    "GramMatrix"Gram matrix
    "Image"configuration of points (when applicable)
    "KissingNumber"kissing number
    "MinimalNorm"minimal norm
    "MinimalVectors"minimal vectors
    "ModularNumber"modular number
    "RadialFunction"lattice points as a function of radius ( series coefficients)
    "ThetaSeriesFunction"pure function for the theta series
  • Lattice-packing-related properties include:
  • "CenterDensity"center density
    "CoveringRadius"covering radius
    "CoxeterNumber"Coxeter number
    "Density"average sphere packing density
    "HermiteInvariant"Hermite invariant
    "PackingRadius"packing radius
    "Thickness"thickness
    "Volume"volume of the fundamental region
  • Other properties include:
  • "QuadraticForm"quadratic form
    "AutomorphismGroupOrder"order of the automorphism group
  • Naming-related properties include:
  • "AlternateNames"alternate English names
    "Name"English name
    "Notation"standard notation for display
    "StandardName"Wolfram Language name
  • LatticeData[lattice,"Classes"] gives a list of the classes in which lattice occurs.
  • LatticeData["class"] gives a list of named lattices in the specified class.
  • LatticeData[lattice,"class"] gives True or False depending on whether lattice is in the specified class.
  • Basic classes of lattices include:
  • "Even"even
    "Extremal"extremal
    "Integral"integral
    "Odd"odd lattice
    "Unimodular"unimodular
  • Negative classes of lattices include:
  • "Nonextremal"not extremal
    "Nonintegral"not integral
    "Nonunimodular"not unimodular
  • Using LatticeData may require internet connectivity.

Examples

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Basic Examples  (3)

The face-centered cubic lattice:

A basis for the face-centered cubic lattice:

Delaunay mesh from points corresponding to minimal vectors of the hexagonal close packing lattice:

Scope  (24)

Names and Classes  (15)

Obtain a list of classical named lattices, including non-lattice packings:

A lattice can be specified by its standard Wolfram Language name:

A BarnesWall lattice is specified by {"BarnesWall",n}:

Bravais lattices can be specified by a crystal system with a lattice centering:

The integer lattice :

Laminated lattices :

MordellWeil lattice :

The root lattices :

The root lattice :

The root lattice :

Short string forms can be used for root lattices:

Find the English name of a lattice:

A list of alternate names can also be found:

Find the list of classes:

Find the list of named lattices belonging to a class:

Test whether a lattice belongs to a class:

Get a list of classes in which a lattice belongs:

A list of lattices that are even and integral:

Properties  (4)

Get a list of possible properties:

Get a list of available properties for a particular lattice:

Certain 3-dimensional lattices have images:

Get notation forms:

Property Values  (5)

A property value can be any valid Wolfram Language expression:

For the property "ThetaSeriesFunction", lattice data is given as pure functions:

A property that is not applicable to a lattice has the value Missing["NotApplicable"]:

A property that is not available for a lattice has the value Missing["NotAvailable"]:

Specify a list of properties for a lattice:

Generalizations & Extensions  (1)

LatticeData works on nonlattice structures such as tetrahedral packings:

Applications  (3)

Number of lattice points in the unit sphere:

The number of lattice points on the sphere as a function of radius:

The number of lattice points in the ball as a function of radius:

Given a 2D lattice basis:

Generate and display some of the elements in the lattice:

Given a 3D lattice basis:

Generate and display some of the elements in the lattice:

Possible Issues  (2)

The generator matrix of a lattice may be given as a SparseArray:

Use Normal to convert to a matrix expression:

Extra coordinates may be used to describe an n-dimensional lattice:

Introduced in 2007
 (6.0)