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# creates: supercell-1.svg supercell-2.svg supercell-3.svg
from math import cos, pi, sin
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.mplot3d import Axes3D
def vertices(cell):
"""
Set up vertices for a cell metric.
"""
from scipy.spatial import Voronoi
I = np.indices((3, 3, 3)).reshape((3, 27)) - 1
G = np.dot(cell, I).T
vor = Voronoi(G)
vert1 = []
for vertices, points in zip(vor.ridge_vertices, vor.ridge_points):
if -1 not in vertices and 13 in points:
normal = G[points].sum(0)
normal /= (normal**2).sum() ** 0.5
vert1.append((vor.vertices[vertices], normal))
return vert1
class CellFigure:
def __init__(self, dim, azim, elev):
"""
Set up a figure for visualizing a cell metric.
"""
Axes3D # silence pyflakes
self.fig = plt.figure(figsize=(5, 5))
self.ax = self.fig.add_subplot(projection='3d')
x = sin(azim)
y = cos(azim)
self.view = [x * cos(elev), y * cos(elev), sin(elev)]
self.ax.set_axis_off()
self.ax.autoscale_view(tight=True)
self.ax.set_xlim(0, dim)
self.ax.set_ylim(0, dim)
self.ax.set_zlim(0, dim)
self.ax.view_init(azim=azim / pi * 180, elev=elev / pi * 180)
def add_cell(self, cell):
"""
Draw a cell.
(i.e. edges but no faces)
"""
vert1 = vertices(cell)
shift = -vert1[0][0][0]
for points, normal in vert1:
if np.dot(normal, self.view) < 0:
ls = ':'
else:
ls = '-'
x, y, z = np.concatenate([points + shift, points[:1] + shift]).T
self.ax.plot(x, y, z, c='k', ls=ls)
def add_primitive_cell(self, cell):
"""
Draw a primitive unit cell.
(i.e. a cell with colored faces)
"""
self.add_cell(cell)
uc = prim[0][0]
# plot side faces of unit cell
X, Y = np.meshgrid([0, uc], [0, uc])
Z = np.zeros((2, 2)) + uc
self.ax.plot_surface(
X, Y, Z, color='blue', alpha=0.5, linewidth=0, zorder=1
)
X, Z = np.meshgrid([0, uc], [0, uc])
Y = np.zeros((2, 2)) + uc
self.ax.plot_surface(
X, Y, Z, color='blue', alpha=0.5, linewidth=0, zorder=1
)
Y, Z = np.meshgrid([0, uc], [0, uc])
X = np.zeros((2, 2)) + uc
self.ax.plot_surface(
X, Y, Z, color='blue', alpha=0.5, linewidth=0, zorder=1
)
def add_atom(self, x0, y0, z0, radius=0.06):
"""
Draw an atom.
"""
u = np.linspace(0, 2 * np.pi, 100)
v = np.linspace(0, np.pi, 100)
x = x0 + radius * np.outer(np.cos(u), np.sin(v))
y = y0 + radius * np.outer(np.sin(u), np.sin(v))
z = z0 + radius * np.outer(np.ones(np.size(u)), np.cos(v))
self.ax.plot_surface(
x,
y,
z,
rstride=4,
cstride=4,
color='orange',
linewidth=0.1,
alpha=0.5,
)
def annotate_figure(self, text):
"""
Add some annotation to the lower left corner of the plot.
"""
self.ax.text(1.1, 0, -0.2, text, ha='left', va='center')
# extent of plotted area
dim = 0.82
# view angle
azim = 0.75 * pi / 5
elev = 0.5 * pi / 6
# define unit cell and supercell
prim = 1.0 / 3 * np.eye(3)
supr = np.eye(3)
# Figure 1
myfig = CellFigure(dim, azim, elev)
myfig.add_primitive_cell(prim)
myfig.annotate_figure('primitive unit cell')
plt.savefig('supercell-1.svg', bbox_inches='tight')
# Figure 2
myfig = CellFigure(dim, azim, elev)
myfig.add_primitive_cell(prim)
myfig.add_cell(supr)
myfig.annotate_figure('ideal supercell')
plt.savefig('supercell-2.svg', bbox_inches='tight')
# Figure 3
myfig = CellFigure(dim, azim, elev)
myfig.add_cell(supr)
d = 0.08
myfig.add_atom(0.5, 0.5 - d, 0.5)
myfig.add_atom(0.5, 0.5 + d, 0.5)
myfig.annotate_figure('defect supercell')
plt.savefig('supercell-3.svg', bbox_inches='tight')
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