1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
|
# creates: periodic-images-1.svg periodic-images-2.svg
from itertools import product
import matplotlib.pyplot as plt
import numpy as np
class CellFigure:
def __init__(self, dim):
"""
Set up a figure for visualizing a cell metric.
"""
self.fig = plt.figure(figsize=(9, 5))
self.ax = self.fig.gca()
self.ax.set_axis_off()
self.ax.autoscale_view(tight=True)
self.ax.set_xlim(-2 * dim, 3 * dim)
self.ax.set_ylim(-dim, 1.5 * dim)
self.ax.set_aspect('equal')
def add_cell(
self,
cell,
offset=[0, 0],
fill_color=None,
atom=None,
radius=0.1,
atom_color='orange',
):
"""
Draw a cell, optionally filled and including an atom.
"""
xvecs = np.array([(0, 0), (0, 1), (1, 1), (1, 0), (0, 0)], dtype=float)
vectors = []
for xv in xvecs:
vectors.append(np.dot(cell, xv + offset))
vectors = np.array(vectors)
from matplotlib.patches import Circle, Polygon
if fill_color is not None:
self.ax.add_patch(
Polygon(vectors, closed=True, color=fill_color, alpha=0.4)
)
for points in vectors:
self.ax.plot(vectors.T[0], vectors.T[1], c='k', ls='-')
if atom:
pos = np.dot(cell, np.array(atom) + offset)
self.ax.add_patch(Circle(pos, radius, color=atom_color))
return pos
def add_vector(self, cell, xvec, xdir, fill_color='red'):
"""
Draw an arrow typically symbolizing a neighbor connection.
"""
from matplotlib.patches import Arrow
pos = np.dot(cell, xvec)
dir = np.dot(cell, xdir)
self.ax.add_patch(Arrow(pos[0], pos[1], dir[0], dir[1], width=0.2))
def annotate_figure(self, text):
"""
Add some annotation to the lower left corner of the plot.
"""
self.ax.text(-2 * dim, 1.35 * dim, text, ha='left', va='center')
# extent of plotted area
dim = 2
# rescaling factor for arrows
rescale = 1.0
# Figure 1
myfig = CellFigure(dim)
prim = np.eye(2)
atompos = [0.5, 0.5]
for i, j in [[-1, 0], [1, 0], [0, -1], [0, 1]]:
myfig.add_vector(prim, atompos, rescale * np.array([i, j]))
for i, j in product(range(-dim, dim + 1), repeat=2):
fill_color = 'blue' if i == 0 and j == 0 else None
myfig.add_cell(prim, [i, j], atom=atompos, fill_color=fill_color)
myfig.annotate_figure('square lattice\n$r_1 = a$, $Z_1=4$')
plt.savefig('periodic-images-1.svg', bbox_inches='tight')
# Figure 2
myfig = CellFigure(dim)
prim = np.array([[2, 0], [0, 0.5]])
for i, j in [[0, -1], [0, 1]]:
myfig.add_vector(prim, atompos, rescale * np.array([i, j]))
for i, j in product(range(-dim, dim + 1), repeat=2):
fill_color = 'blue' if i == 0 and j == 0 else None
myfig.add_cell(prim, [i, j], atom=atompos, fill_color=fill_color)
myfig.annotate_figure(
'rectangular lattice with a 2:1 aspect ratio\n$r_1 = a/2$, $Z_1=2$'
)
plt.savefig('periodic-images-2.svg', bbox_inches='tight')
# Figure 3
myfig = CellFigure(dim)
prim = np.array([[1, 0.5], [0, np.sqrt(3) / 2]])
prim /= np.linalg.det(prim) ** (1.0 / 2)
s = np.sqrt(2.0 / np.sqrt(3))
for i, j in [[-1, 0], [1, 0], [0, -1], [0, 1], [-1, 1], [1, -1]]:
myfig.add_vector(prim, atompos, rescale * np.array([i, j]))
for i, j in product(range(-dim, dim + 1), repeat=2):
fill_color = 'blue' if i == 0 and j == 0 else None
myfig.add_cell(prim, [i, j], atom=atompos, fill_color=fill_color)
myfig.annotate_figure('hexagonal lattice\n$r_1 = %.3f a$, $Z_1=6$' % s)
plt.savefig('periodic-images-3.svg', bbox_inches='tight')
|
