SymmetryReduceBZ
The primary purpose of SymmetryReduceBZ is to calculate the irreducible
Brillouin zone (IBZ) for crystal structures in 2D or 3D provided the real-space
lattice vectors, atom positions, and atom types. It also contains methods for
making unit cells primitive and lattice reduction. See the User Guide in the
documentation for more details
and usage examples in Python.
Installation
SymmetryReduceBZ is a registered Julia package and can be installed using
Julia's package manager Pkg.
using Pkg
Pkg.add("SymmetryReduceBZ")
Examples
To calculate the irreducible Brillouin zone, provide the lattice and atomic
basis to calc_ibz. The IBZ will be returned as either a convex hull or
intersection of half spaces.
import SymmetryReduceBZ.Lattices: genlat_CUB
import SymmetryReduceBZ.Symmetry: calc_ibz
a = 2.0
real_latvecs = genlat_CUB(a)
atom_types = [0,0]
atom_pos = Array([0 0 0; 0.5 0.5 0.5]')
ibzformat = "convex hull"
coordinates = "Cartesian"
makeprim = false
convention = "ordinary"
ibz = calc_ibz(real_latvecs,atom_types,atom_pos,coordinates,ibzformat,
makeprim,convention)
The arguments for calc_ibz are as follows:
real_latvecs: the real-space lattice vectors as columns of a matrix.atom_types: a vector of atom types as integers.atom_pos: the positions of atoms in the crystal structure as columns of a matrix.coordinates: indicates the atoms are in"lattice"or"Cartesian"coordinates.ibzformat: the format of the irreducible Brillouin zone. Options include"convex hull"and"half-space".convention: the convention used to go between real and reciprocal space. The two conventions are"ordinary"(temporal) frequency and"angular"frequency.makeprim: make the unit cell primitive before calculating the IBZ iftrue.rtol=sqrt(eps(float(maximum(real_latvecs)))): (optional) a relative tolerance for floating-point comparisons.atol=1e-9: (optional) an absolute tolerance for floating-point comparisons.
The vertices of the ibz are accessed with ibz.points[ibz.vertices,:]. The
vertices of the IBZ and ibz.points should be the same. The rows of the array
are the vertices of the IBZ in Cartesian coordinates. Other attributes of the
IBZ are accessible, such as the volume ibz.volume. The faces of the IBZ are
calculated with
import SymmetryReduceBZ.Utilities: get_uniquefacets
indices = get_uniquefacets(ibz)
facets = [ibz.points[ind] for ind=indices]
facets is a list of points at the corners of each facet. The function
get_uniquefacets returns the indices of points that lie on the same facet. The
facets are also available through ibz.simplices as simplices, but often multiple simplices
lie on the same facet. See the documentation for SciPy
for additional attributes of the IBZ.
The function plot_convexhulls is useful for visualizing the Brillouin zone
and irreducible Brillouin zone. The arguments are the same as those from
calc_ibz.
import SymmetryReduceBZ.Plotting: plot_convexhulls
import SymmetryReduceBZ.Lattices: genlat_CUB
a = 2.0
real_latvecs = genlat_CUB(a)
atom_types = [0,0]
atom_pos = Array([0 0 0; 0.5 0.5 0.5]')
coordinates = "Cartesian"
makeprim = false
convention = "ordinary"
ax=plot_convexhulls(real_latvecs,atom_types,atom_pos,coordinates,
makeprim,convention)
The functions plot_2Dconvexhull and plot_3Dconvexhull allow greater customization of
the appearance of the convex hull.
import SymmetryReduceBZ.Symmetry: calc_bz, calc_ibz
import SymmetryReduceBZ.Plotting: plot_2Dconvexhull
using PyPlot
real_latvecs = [1 0; 0 1]
convention="ordinary"
atom_types=[0]
atom_pos = Array([0 0]')
coords = "Cartesian"
ibzformat = "convex hull"
makeprim=false
bz = calc_bz(real_latvecs,atom_types,atom_pos,coords,ibzformat,makeprim,convention)
ibz = calc_ibz(real_latvecs,atom_types,atom_pos,coords,ibzformat,makeprim,convention)
ax = plot_2Dconvexhull(bz,facecolor="deepskyblue",linewidth=3,edgecolor="cyan",alpha=0.2)
ax = plot_2Dconvexhull(ibz,ax;facecolor="coral",linewidth=3,edgecolor="magenta",alpha=0.4)
axis("off")
import SymmetryReduceBZ.Symmetry: calc_bz, calc_ibz
import SymmetryReduceBZ.Plotting: plot_3Dconvexhull
using PyPlot
real_latvecs = [1 0 0; 0 1 0; 0 0 1]
convention="ordinary"
atom_types=[0]
atom_pos = Array([0 0 0]')
coords = "Cartesian"
bzformat = "convex hull"
makeprim=false
bz = calc_bz(real_latvecs,atom_types,atom_pos,coords,bzformat,makeprim,convention)
ibz = calc_ibz(real_latvecs,atom_types,atom_pos,coords,bzformat,makeprim,convention)
fig = figure()
ax = fig.add_subplot(111, projection="3d")
ax = plot_3Dconvexhull(ibz,ax,facecolors="pink",alpha=1,edgecolors="black",linewidths = 1)
ax = plot_3Dconvexhull(bz,ax,facecolors="deepskyblue",edgecolors="white",linewidths=1,alpha=0.2)
axis("off")
