GroupTheory`

GTVoronoiCell

GTVoronoiCell[basis,clusterdata,BZpath]

constructs a Voronoi cell to a lattice with basis. clusterdata contains the data for the lattice construction. If a Brillouin zone is considered the path used in electronic structure calculations can be given by BZpath.

Details and Options

Wigner-Seitz cells in real space and Brillouin zones in reciprocal space are special case of a Voronoi cell. The interior points of a Voronoi cell are closer to the inner lattice point definig the cell than to all other lattice points.
Voronoi cells can be constructed by choosing a lattice point, and constructing the planes bisecting the lattice vectors to the nearest lattice points at a right angle. The enclosed volume is the Voronoi cell.
GTVoronoiCell uses a different method to construct the Voronoi cell. All triples of bisecting planes are considered. If they have a common crossing point, this point is one of the vertices of the Voronoi cell. If all vertices are known, the corresponding convex hull is the Voronoi cell.
clusterdata defines the lattice vectors used for the construction of the Voronoi cell.
clusterdata = {cut,smin,smax}
cut is the maximum length of th lattice vectors. lattice vectors between the shell numbers smin and smax are taken into account.
The following options can be given:

GOBZPath	True	Plots the brillouin zone path.
GOOutput	"Voronoi"	Plots the Voronoi cell.
GOVerbose	False	Controls the output of additional information.
VertexLabelStyle	Directive[Black,20,BackgroundYellow]	Style of the k-point labels

See: W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 6.3
Description of the XCrySDen software at http://www.xcrysden.org

Examples

open allclose all

Basic Examples (3)

First load the package:

2-dimensional examples

Only the cell is plotted.

The Brillouin zone together with the path is presented.

3-dimensional examples

bcc

fcc

Options (4)

VertexLabelStyle is a standard option of Mathematica and will not be discussed here.

GOBZPath (1)

The plot of the Brilloiuin zone path can be controlled.

GOOutput (1)

Different modes of output are possible. It is standard th plot the Voronoi cell (GOOutput  "Voronoi")

The Voronoi cell can be plotted as a mesh.

If GOOutput  "Output" is used, the first part of the list gives the vertices of the convex hull. The last two parts of the list are graphic objects representing the coordinate system and the BZ path.

GOVerbose (1)

Addition information about the cluster construction and the calculation of the vertices of the Voronoi cell is given.