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from math import *
import numpy as np
# Grid generator for direct and reciprocal space
class UnitCell:
def __init__(self, N, basis):
self.N = N
self.d = basis / N # displacement basis of grid points
# Generate NxNxN grid to represent functions inside the cell
def grid_points(self):
gx = np.zeros((self.N, self.N, self.N))
gy = np.zeros_like(gx)
gz = np.zeros_like(gx)
if self.N % 2 == 0:
ns = range(-self.N // 2, self.N // 2)
else:
ns = range(-(self.N - 1) // 2, (self.N - 1) // 2 + 1)
for i in range(self.N):
for j in range(self.N):
for k in range(self.N):
v = ns[i] * self.d[0] + ns[j] * self.d[1] + ns[k] * self.d[2]
gx[i, j, k] = v[0]
gy[i, j, k] = v[1]
gz[i, j, k] = v[2]
return gx, gy, gz
# Volume of smallest parallelepiped representable on this grid
def differential_volume(self):
dV = np.dot(self.d[0], np.cross(self.d[1], self.d[2]))
return abs(dV)
# Base class for periodic structures, don't use it directly
class Crystal:
def __init__(self, N, basis):
self.a = basis
self.b = self.get_reciprocal_basis()
self.cell_direct = UnitCell(N, self.a)
self.cell_fourier = UnitCell(N, self.b * N)
def get_reciprocal_basis(self):
V = np.dot(self.a[0], np.cross(self.a[1], self.a[2]))
b1 = (2 * pi / V) * np.cross(self.a[1], self.a[2])
b2 = (2 * pi / V) * np.cross(self.a[2], self.a[0])
b3 = (2 * pi / V) * np.cross(self.a[0], self.a[1])
return np.array([b1, b2, b3])
def get_grid_direct(self):
rx, ry, rz = self.cell_direct.grid_points()
dr = self.cell_direct.differential_volume()
return rx, ry, rz, dr
def get_grid_fourier(self):
qx, qy, qz = self.cell_fourier.grid_points()
dq = self.cell_fourier.differential_volume()
return qx, qy, qz, dq
# Convert sequence of letters to coordinates for band structure plots.
# The letters' meanings are stored in a dictionary self.bz_points.
def get_brillouin_zone_path(self, chars):
path = []
for c in chars:
k = self.bz_points[c] # should probably catch KeyError here
p = k[0] * self.b[0] + k[1] * self.b[1] + k[2] * self.b[2]
path.append(p)
return path
# Derived class for simple cubic (sc) crystals
class CubicSimple(Crystal):
def __init__(self, N, a):
a1 = np.array((a, 0, 0))
a2 = np.array((0, a, 0))
a3 = np.array((0, 0, a))
super().__init__(N, np.array([a1, a2, a3]))
# Taken from http://lampx.tugraz.at/~hadley/ss1/bzones/sc.php
bz_points = {
"G" : (0, 0, 0),
"R" : (0.5, 0.5, 0.5),
"X" : (0, 0.5, 0),
"M" : (0.5, 0.5, 0)
}
# Derived class for body-centered cubic (bcc) crystals
class CubicBodyCentered(Crystal):
def __init__(self, N, a):
a1 = np.array((-a/2, a/2, a/2))
a2 = np.array(( a/2, -a/2, a/2))
a3 = np.array(( a/2, a/2, -a/2))
super().__init__(N, np.array([a1, a2, a3]))
# Taken from http://lampx.tugraz.at/~hadley/ss1/bzones/bcc.php
bz_points = {
"G" : ( 0, 0, 0),
"H" : (-0.5, 0.5, 0.5),
"P" : ( 0.25, 0.25, 0.25),
"N" : ( 0, 0.5, 0)
}
# Derived class for face-centered cubic (fcc) crystals
class CubicFaceCentered(Crystal):
def __init__(self, N, a):
a1 = np.array((0, a/2, a/2))
a2 = np.array((a/2, 0, a/2))
a3 = np.array((a/2, a/2, 0))
super().__init__(N, np.array([a1, a2, a3]))
# Taken from http://lampx.tugraz.at/~hadley/ss1/bzones/fcc.php
bz_points = {
"G" : (0, 0, 0),
"X" : (0, 0.5, 0.5),
"L" : (0.5, 0.5, 0.5),
"W" : (0.25, 0.75, 0.5),
"U" : (0.25, 0.625, 0.625),
"K" : (0.375, 0.75, 0.375)
}
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