BrillouinZoneMeshes

Documentation for BrillouinZoneMeshes.

This package provides general-purpose multi-dimensional meshes for numerical representation of continuous functions and specialized meshes for functions on Brillouin Zones.

Getting started

Setup with:

    DIM = 2;
    # square lattice
    lattice = Matrix([1.0 0; 0 1]');
    # create Brillouin zone
    br = BZMeshes.Cell(lattice=lattice);
    # uniform mesh
    umesh = BZMeshes.UniformBZMesh(cell=br, size=(4, 4));
    # symmetry reduce map
    mm = MeshMaps.MeshMap(umesh);
    # reduced mesh
    rmesh = ReducedBZMesh(umesh, mm);

and here are some examples of usage:

julia> using BrillouinZoneMeshes; DIM = 2; lattice = Matrix([1.0 0; 0 1]'); br = BZMeshes.Cell(lattice=lattice); umesh = BZMeshes.UniformBZMesh(cell=br, size=(4, 4)); mm = MeshMaps.MeshMap(umesh); rmesh = ReducedBZMesh(umesh, mm);

julia> length(umesh), length(rmesh)
(16, 3)

julia> AbstractMeshes.locate(rmesh, [1,1])
3

julia> rmesh[3]
2-element StaticArraysCore.SVector{2, Float64} with indices SOneTo(2):
 0.7853981633974483
 0.7853981633974483

julia> AbstractMeshes.volume(rmesh) / 4π^2
1.0

julia> data = ones(3)
3-element Vector{Float64}:
 1.0
 1.0
 1.0

julia> AbstractMeshes.integrate(data, rmesh)
39.47841760435743

julia> AbstractMeshes.interp(data, rmesh, [0.3,-0.2])
1.0

General

Various mesh grids for different purposes are defined as concrete types derived from AbstractMeshes.AbstractMesh. All of them are supposed to behave as AbstractArray with elements being SVector representing the mesh points in Cartesian coordinates.

In addition to the interface of AbstractMeshes.AbstractMesh, four useful methods are defined: locate, volume, interp, and integrate.

• locate(mesh, x) finds the mesh point nearest to x
• volume(mesh, i) gives the volume represented by mesh point mesh[i]
• interp(data, mesh, x) gives the interpolation of data on mesh at point x
• integrate(data, mesh) compute integration of data on mesh

If it is known that some of the mesh points are guaranteed to have the same data value, it's possible to define a MeshMap to reveal this fact and create a ReducedBZMesh to save storage space.

Brillouin zone

The information of Brillouin zone is stored in Cells.Cell. Including lattice vector, reciprocal lattice vector and their inverse; volume of unit cell and reciprocal unit cell; G vectors for extended Brillouin zone and symmetries.

Uniform Meshes

Uniform meshes are defined as uniformly distributed meshes on a parallellogram area described by an origin and a set of lattice vectors. The simplest one is BaseMesh.UMesh, while BZMeshes.UniformBZMesh containes additional information about the Brillouin zone stored in its cell field.

Uniform meshes are conventionally used in various ab initio calculations. In this package various frequently used meshes, such as Gamma-centered and Monkhorst-Pack meshes, could be generated via BZMeshes.UniformBZMesh with different parameters. The default parameter of the constructor of BZMeshes.UniformBZMesh generates Gamma-centered mesh, while two constructors for M-P mesh, Monkhorst_Pack and DFTK_Monkhorst_Pack, follow conventions from VASP and DFTK respectively.