/
csl.py
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/
csl.py
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import sys
import random
from math import degrees, atan, sqrt, pi, ceil, cos, acos, sin, gcd, radians
import numpy as np
from numpy import dot, cross
from numpy.linalg import det, norm, inv
def get_cubic_sigma(uvw, m, n=1):
"""
CSL analytical formula based on the book:
'Interfaces in crystalline materials',
Sutton and Balluffi, clarendon press, 1996.
generates possible sigma values.
arguments:
uvw -- the axis
m,n -- two integers (n by default 1)
"""
u, v, w = uvw
sqsum = u*u + v*v + w*w
sigma = m*m + n*n * sqsum
while sigma != 0 and sigma % 2 == 0:
sigma /= 2
return sigma if sigma > 1 else None
def get_cubic_theta(uvw, m, n=1):
"""
generates possible theta values.
arguments:
uvw -- the axis
m,n -- two integers (n by default 1)
"""
u, v, w = uvw
sqsum = u*u + v*v + w*w
if m > 0:
return 2 * atan(sqrt(sqsum) * n / m)
else:
return pi
def get_theta_m_n_list(uvw, sigma):
"""
Finds integers m and n lists that match the input sigma.
"""
if sigma == 1:
return [(0., 0., 0.)]
thetas = []
max_m = int(ceil(sqrt(4*sigma)))
for m in range(1, max_m):
for n in range(1, max_m):
if gcd(m, n) == 1:
s = get_cubic_sigma(uvw, m, n)
if s == sigma:
theta = (get_cubic_theta(uvw, m, n))
thetas.append((theta, m, n))
thetas.sort(key=lambda x: x[0])
return thetas
def get_sigma_list(uvw, limit):
"""
prints a list of smallest sigmas/angles for a given axis(uvw).
"""
sigmas = []
thetas = []
for i in range(limit):
tt = get_theta_m_n_list(uvw, i)
if len(tt) > 0:
theta, _, _ = tt[0]
sigmas.append(i)
thetas.append(degrees(theta))
return sigmas, thetas
def get_theta_m_n_list(uvw, sigma):
"""
Finds integers m and n lists that match the input sigma.
"""
if sigma == 1:
return [(0., 0., 0.)]
thetas = []
max_m = int(ceil(sqrt(4*sigma)))
for m in range(1, max_m):
for n in range(1, max_m):
if gcd(m, n) == 1:
s = get_cubic_sigma(uvw, m, n)
if s == sigma:
theta = (get_cubic_theta(uvw, m, n))
thetas.append((theta, m, n))
thetas.sort(key=lambda x: x[0])
return thetas
def rot(a, Theta):
"""
produces a rotation matrix.
arguments:
a -- an axis
Theta -- an angle
"""
c = cos(Theta)
s = sin(Theta)
a = a / norm(a)
ax, ay, az = a
return np.array([[c + ax * ax * (1 - c), ax * ay * (1 - c) - az * s,
ax * az * (1 - c) + ay * s],
[ay * ax * (1 - c) + az * s, c + ay * ay * (1 - c),
ay * az * (1 - c) - ax * s],
[az * ax * (1 - c) - ay * s, az * ay * (1 - c) + ax * s,
c + az * az * (1 - c)]])
def create_minimal_cell_method_1(sigma, uvw, R):
"""
finds Minimal cell by means of a numerical search.
(An alternative analytical method can be used too).
arguments:
sigma -- gb sigma
uvw -- rotation axis
R -- rotation matrix
"""
uvw = np.array(uvw)
MiniCell_1 = np.zeros([3, 3])
MiniCell_1[:, 2] = uvw
lim = 20
x = np.arange(-lim, lim + 1, 1)
y = x
z = x
indice = (np.stack(np.meshgrid(x, y, z)).T).reshape(len(x) ** 3, 3)
# remove 0 vectors and uvw from the list
indice_0 = indice[np.where(np.sum(abs(indice), axis=1) != 0)]
condition1 = ((abs(dot(indice_0, uvw) / norm(indice_0, axis=1) /
norm(uvw))).round(7))
indice_0 = indice_0[np.where(condition1 != 1)]
if minicell_search(indice_0, MiniCell_1, R, sigma):
M1, M2 = minicell_search(indice_0, MiniCell_1, R, sigma)
return (M1, M2)
else:
return None
def minicell_search(indices, MiniCell_1, R, sigma):
tol = 0.001
# norm1 = norm(indices, axis=1)
newindices = dot(R, indices.T).T
nn = indices[np.all(abs(np.round(newindices) - newindices) < 1e-6, axis=1)]
TestVecs = nn[np.argsort(norm(nn, axis=1))]
# print(len(indices), len(TestVecs),TestVecs[:20])
Found = False
count = 0
while (not Found) and count < len(TestVecs) - 1:
if 1 - ang(TestVecs[count], MiniCell_1[:, 2]) > tol:
# and (ang(TestVecs[i],uvw) > tol):
MiniCell_1[:, 1] = (TestVecs[count])
count += 1
for j in range(len(TestVecs)):
if (1 - ang(TestVecs[j], MiniCell_1[:, 2]) > tol) and (
1 - ang(TestVecs[j], MiniCell_1[:, 1]) > tol):
if (ang(TestVecs[j],
cross(MiniCell_1[:, 2], MiniCell_1[:, 1])) > tol):
# The condition that the third vector can not be
# normal to any other two.
# and (ang(TestVecs[i],uvw)> tol) and
# (ang(TestVecs[i],MiniCell[:,1])> tol)):
MiniCell_1[:, 0] = (TestVecs[j]).astype(int)
Det1 = abs(round(det(MiniCell_1), 5))
MiniCell_1 = (MiniCell_1).astype(int)
MiniCell_2 = ((np.round(dot(R, MiniCell_1), 7))
.astype(int))
Det2 = abs(round(det(MiniCell_2), 5))
if ((abs(Det1 - sigma)) < tol and
(abs(Det2 - sigma)) < tol):
Found = True
break
if Found:
return MiniCell_1, MiniCell_2
else:
return Found
def create_possible_gb_plane_list(uvw, m=5, n=1, lim=5):
"""
generates GB planes and specifies the character.
arguments:
uvw -- axis of rotation.
m,n -- the two necessary integers
lim -- upper limit for the plane indices
"""
uvw = np.array(uvw)
Theta = get_cubic_theta(uvw, m, n)
Sigma = get_cubic_sigma(uvw, m, n)
R1 = rot(uvw, Theta)
# List and character of possible GB planes:
x = np.arange(-lim, lim + 1, 1)
y = x
z = x
indice = (np.stack(np.meshgrid(x, y, z)).T).reshape(len(x) ** 3, 3)
indice_0 = indice[np.where(np.sum(abs(indice), axis=1) != 0)]
indice_0 = indice_0[np.argsort(norm(indice_0, axis=1))]
# extract the minimal cell:
Min_1, Min_2 = create_minimal_cell_method_1(Sigma, uvw, R1)
V1 = np.zeros([len(indice_0), 3])
V2 = np.zeros([len(indice_0), 3])
GBtype = []
tol = 0.001
# Mirrorplanes cubic symmetry
MP = np.array([[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
[1, 1, 0],
[0, 1, 1],
[1, 0, 1],
], dtype='float')
# Find GB plane coordinates:
for i in range(len(indice_0)):
if common_divisor(indice_0[i])[1] <= 1:
V1[i, :] = (indice_0[i, 0] * Min_1[:, 0] +
indice_0[i, 1] * Min_1[:, 1] +
indice_0[i, 2] * Min_1[:, 2])
V2[i, :] = (indice_0[i, 0] * Min_2[:, 0] +
indice_0[i, 1] * Min_2[:, 1] +
indice_0[i, 2] * Min_2[:, 2])
V1 = (V1[~np.all(V1 == 0, axis=1)]).astype(int)
V2 = (V2[~np.all(V2 == 0, axis=1)]).astype(int)
MeanPlanes = (V1 + V2) / 2
# Check the type of GB plane: Symmetric tilt, tilt, twist
for i in range(len(V1)):
if ang(V1[i], uvw) < tol:
for j in range(len(symmetry_equivalent(MP))):
if 1 - ang(MeanPlanes[i], symmetry_equivalent(MP)[j]) < tol:
GBtype.append('Symmetric Tilt')
break
else:
GBtype.append('Tilt')
elif 1 - ang(V1[i], uvw) < tol:
GBtype.append('Twist')
else:
GBtype.append('Mixed')
return (V1, V2, MeanPlanes, GBtype)
def find_orthogonal_cell(uvw, sigma, theta, R, m, n, GB1,
Min_1, Min_2, tol = 0.001):
"""
finds Orthogonal cells from the CSL minimal cells.
arguments:
basis -- lattice basis
uvw -- rotation axis
m,n -- two necessary integers
GB1 -- input plane orientation
"""
# Inputs
lim = 15
uvw = np.array(uvw)
Theta = theta
Sigma = sigma
#create GB2 from GB1
GB2 = np.round((dot(R, GB1)), 6)
x = np.arange(-lim, lim + 1, 1)
y = x
z = x
indice = (np.stack(np.meshgrid(x, y, z)).T).reshape(len(x) ** 3, 3)
indice_0 = indice[np.where(np.sum(abs(indice), axis=1) != 0)]
indice_0 = indice_0[np.argsort(norm(indice_0, axis=1))]
OrthoCell_1 = np.zeros([3, 3])
OrthoCell_1[:, 0] = np.array(GB1)
OrthoCell_2 = np.zeros([3, 3])
OrthoCell_2[:, 0] = np.array(GB2)
# Find Ortho vectors:
if ang(OrthoCell_1[:, 0], uvw) < tol:
OrthoCell_1[:, 1] = uvw
OrthoCell_2[:, 1] = uvw
else:
for i in range(len(indice_0)):
v1 = (indice_0[i, 0] * Min_1[:, 0] +
indice_0[i, 1] * Min_1[:, 1] +
indice_0[i, 2] * Min_1[:, 2])
v2 = (indice_0[i, 0] * Min_2[:, 0] +
indice_0[i, 1] * Min_2[:, 1] +
indice_0[i, 2] * Min_2[:, 2])
if ang(v1, OrthoCell_1[:, 0]) < tol:
OrthoCell_1[:, 1] = v1
OrthoCell_2[:, 1] = v2
break
OrthoCell_1[:, 2] = np.cross(OrthoCell_1[:, 0], OrthoCell_1[:, 1])
OrthoCell_2[:, 2] = np.cross(OrthoCell_2[:, 0], OrthoCell_2[:, 1])
if (common_divisor(OrthoCell_1[:, 2])[1] ==
common_divisor(OrthoCell_2[:, 2])[1]):
OrthoCell_1[:, 2] = common_divisor(OrthoCell_1[:, 2])[0]
OrthoCell_2[:, 2] = common_divisor(OrthoCell_2[:, 2])[0]
Volume_1 = (round(det(OrthoCell_1), 5))
Volume_2 = (round(det(OrthoCell_2), 5))
if Volume_1 == Volume_2:
return True, OrthoCell_1.astype(float), OrthoCell_2.astype(float)
return False, None, None
def get_tilt_twist_comp(v1, uvw, m, n, tol=0.001):
"""
returns the tilt and twist components of a given GB plane.
arguments:
v1 -- given gb plane
uvw -- axis of rotation
m,n -- the two necessary integers
"""
theta = get_cubic_theta(uvw, m, n)
R = rot(uvw, theta)
v2 = np.round(dot(R, v1), 6).astype(int)
tilt = angv(v1, v2)
twist = 0
gb_type = ""
if abs(tilt - degrees(theta)) < 10e-5:
gb_type = "tilt"
else:
twist = 2 * acos(cos(theta / 2) / cos(radians(tilt / 2)))
#Assign types
MP = np.array([[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
[1, 1, 0],
[0, 1, 1],
[1, 0, 1],
], dtype='float')
return tilt, twist
#Helpers
def common_divisor(a):
"""
returns the common factor of vector a and the reduced vector.
"""
CommFac = []
a = np.array(a)
for i in range(2, 100):
while (a[0] % i == 0 and a[1] % i == 0 and a[2] % i == 0):
a = a / i
CommFac.append(i)
return(a.astype(int), (abs(np.prod(CommFac))))
def integer_array(A, tol=1e-7):
"""
returns True if an array is ineteger.
"""
return np.all(abs(np.round(A) - A) < tol)
def integer_matrix(a):
"""
returns an integer matrix from row vectors.
"""
Found = True
b = np.zeros((3, 3))
a = np.array(a)
for i in range(3):
for j in range(1, 2000):
testV = j * a[i]
if integer_array(testV):
b[i] = testV
break
if all(b[i] == 0):
Found = False
print("Can not make integer matrix!")
return (b) if Found else None
def angv(a, b):
"""
returns the angle between two vectors.
"""
ang = acos(np.round(dot(a, b)/norm(a)/norm(b), 8))
return round(degrees(ang), 7)
def ang(a, b):
"""
returns the cos(angle) between two vectors.
"""
ang = np.round(dot(a, b)/norm(a)/norm(b), 7)
return abs(ang)
def symmetry_equivalent(arr):
"""
returns cubic symmetric eqivalents of the given 2 dimensional vector.
"""
Sym = np.zeros([24, 3, 3])
Sym[0, :] = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
Sym[1, :] = [[1, 0, 0], [0, -1, 0], [0, 0, -1]]
Sym[2, :] = [[-1, 0, 0], [0, 1, 0], [0, 0, -1]]
Sym[3, :] = [[-1, 0, 0], [0, -1, 0], [0, 0, 1]]
Sym[4, :] = [[0, -1, 0], [-1, 0, 0], [0, 0, -1]]
Sym[5, :] = [[0, -1, 0], [1, 0, 0], [0, 0, 1]]
Sym[6, :] = [[0, 1, 0], [-1, 0, 0], [0, 0, 1]]
Sym[7, :] = [[0, 1, 0], [1, 0, 0], [0, 0, -1]]
Sym[8, :] = [[-1, 0, 0], [0, 0, -1], [0, -1, 0]]
Sym[9, :] = [[-1, 0, 0], [0, 0, 1], [0, 1, 0]]
Sym[10, :] = [[1, 0, 0], [0, 0, -1], [0, 1, 0]]
Sym[11, :] = [[1, 0, 0], [0, 0, 1], [0, -1, 0]]
Sym[12, :] = [[0, 1, 0], [0, 0, 1], [1, 0, 0]]
Sym[13, :] = [[0, 1, 0], [0, 0, -1], [-1, 0, 0]]
Sym[14, :] = [[0, -1, 0], [0, 0, 1], [-1, 0, 0]]
Sym[15, :] = [[0, -1, 0], [0, 0, -1], [1, 0, 0]]
Sym[16, :] = [[0, 0, 1], [1, 0, 0], [0, 1, 0]]
Sym[17, :] = [[0, 0, 1], [-1, 0, 0], [0, -1, 0]]
Sym[18, :] = [[0, 0, -1], [1, 0, 0], [0, -1, 0]]
Sym[19, :] = [[0, 0, -1], [-1, 0, 0], [0, 1, 0]]
Sym[20, :] = [[0, 0, -1], [0, -1, 0], [-1, 0, 0]]
Sym[21, :] = [[0, 0, -1], [0, 1, 0], [1, 0, 0]]
Sym[22, :] = [[0, 0, 1], [0, -1, 0], [1, 0, 0]]
Sym[23, :] = [[0, 0, 1], [0, 1, 0], [-1, 0, 0]]
arr = np.atleast_2d(arr)
Result = []
for i in range(len(Sym)):
for j in range(len(arr)):
Result.append(dot(Sym[i, :], arr[j]))
Result = np.array(Result)
return np.unique(Result, axis=0)
# GB building tools
def generate_ortho_unitcell_atoms(ortho, basis):
"""
populates a unitcell from the orthogonal vectors.
"""
Or = ortho.T
Orint = integer_matrix(Or)
LoopBound = np.zeros((3, 2), dtype=float)
transformed = []
CubeCoords = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 0],
[0, 1, 1], [1, 0, 1], [1, 1, 1], [0, 0, 0]],
dtype=float)
for i in range(len(CubeCoords)):
transformed.append(np.dot(Orint.T, CubeCoords[i]))
# Finding bounds for atoms in a CSL unitcell:
LoopBound[0, :] = [min(np.array(transformed)[:, 0]),
max(np.array(transformed)[:, 0])]
LoopBound[1, :] = [min(np.array(transformed)[:, 1]),
max(np.array(transformed)[:, 1])]
LoopBound[2, :] = [min(np.array(transformed)[:, 2]),
max(np.array(transformed)[:, 2])]
# Filling up the unitcell:
Tol = 1
x = np.arange(LoopBound[0, 0] - Tol, LoopBound[0, 1] + Tol + 1, 1)
y = np.arange(LoopBound[1, 0] - Tol, LoopBound[1, 1] + Tol + 1, 1)
z = np.arange(LoopBound[2, 0] - Tol, LoopBound[2, 1] + Tol + 1, 1)
V = len(x) * len(y) * len(z)
indice = (np.stack(np.meshgrid(x, y, z)).T).reshape(V, 3)
Base = basis
Atoms = []
tol = 0.001
if V > 5e6:
print("Warning! It may take a very long time"
"to produce this cell!")
# produce Atoms:
for i in range(V):
for j in range(len(Base)):
Atoms.append(indice[i, 0:3] + Base[j, 0:3])
Atoms = np.array(Atoms)
# Cell conditions
Con1 = dot(Atoms, Or[0]) / norm(Or[0]) + tol
Con2 = dot(Atoms, Or[1]) / norm(Or[1]) + tol
Con3 = dot(Atoms, Or[2]) / norm(Or[2]) + tol
# Application of the conditions:
Atoms = (Atoms[(Con1 >= 0) & (Con1 <= norm(Or[0])) & (Con2 >= 0) &
(Con2 <= norm(Or[1])) &
(Con3 >= 0) & (Con3 <= norm(Or[2]))])
if len(Atoms) == (round(det(Or) * len(Base), 7)).astype(int):
return Atoms
def generate_bicrystal_atoms(ortho1, ortho2, basis):
"""
builds the unitcells for both grains g1 and g2.
"""
Or_1 = ortho1.T
Or_2 = ortho2.T
rot1 = np.array([Or_1[0, :] / norm(Or_1[0, :]),
Or_1[1, :] / norm(Or_1[1, :]),
Or_1[2, :] / norm(Or_1[2, :])])
rot2 = np.array([Or_2[0, :] / norm(Or_2[0, :]),
Or_2[1, :] / norm(Or_2[1, :]),
Or_2[2, :] / norm(Or_2[2, :])])
atoms1 = generate_ortho_unitcell_atoms(ortho1.copy(), basis)
atoms2 = generate_ortho_unitcell_atoms(ortho2.copy(), basis)
atoms1 = dot(rot1, atoms1.T).T
atoms2 = dot(rot2, atoms2.T).T
atoms2[:, 0] = atoms2[:, 0] - norm(Or_2[0, :]) # - tol
return atoms1, atoms2
def expand_super_cell(ortho1, atoms1, atoms2, dim):
"""
expands the smallest CSL unitcell to the given dimensions.
"""
a = norm(ortho1[:, 0])
b = norm(ortho1[:, 1])
c = norm(ortho1[:, 2])
dimX, dimY, dimZ = dim
X = atoms1.copy()
Y = atoms2.copy()
X_new = []
Y_new = []
for i in range(dimX):
for j in range(dimY):
for k in range(dimZ):
Position1 = [i * a, j * b, k * c]
Position2 = [-i * a, j * b, k * c]
for l in range(len(X)):
X_new.append(Position1[0:3] + X[l, 0:3])
for m in range(len(Y)):
Y_new.append(Position2[0:3] + Y[m, 0:3])
atoms1 = np.array(X_new)
atoms2 = np.array(Y_new)
return atoms1, atoms2
def find_overlapping_atoms(ortho1, atoms1, atoms2, dim, overlap=0.0):
"""
finds the overlapping atoms.
"""
periodic_length = norm(ortho1[:, 0]) * dim[0]
periodic_image = atoms2 + [periodic_length * 2, 0, 0]
# select atoms contained in a smaller window around the GB and its
# periodic image
IndX = np.where([(atoms1[:, 0] < 1) |
(atoms1[:, 0] > (periodic_length - 1))])[1]
IndY = np.where([atoms2[:, 0] > -1])[1]
IndY_image = np.where([periodic_image[:, 0] <
(periodic_length + 1)])[1]
X_new = atoms1[IndX]
Y_new = np.concatenate((atoms2[IndY], periodic_image[IndY_image]))
IndY_new = np.concatenate((IndY, IndY_image))
# create a meshgrid search routine
x = np.arange(0, len(X_new), 1)
y = np.arange(0, len(Y_new), 1)
indice = (np.stack(np.meshgrid(x, y)).T).reshape(len(x) * len(y), 2)
norms = norm(X_new[indice[:, 0]] - Y_new[indice[:, 1]], axis=1)
indice_x = indice[norms < overlap][:, 0]
indice_y = indice[norms < overlap][:, 1]
X_del = X_new[indice_x]
Y_del = Y_new[indice_y]
if (len(X_del) != len(Y_del)):
print("Warning! the number of deleted atoms"
"in the two grains are not equal!")
# print(type(IndX), len(IndY), len(IndY_image))
return (X_del, Y_del, IndX[indice_x], IndY_new[indice_y])
def populate_gb(ortho1, ortho2, basis,
lattice_parameter, dim=(1,1,1),
overlap=0.0, rigid=False):
atoms1, atoms2 = generate_bicrystal_atoms(ortho1, ortho2, basis)
if overlap > 0.0:
which_gb = "g1"
elif overlap < 0.0:
which_gb = "g2"
else:
which_gb = "a"
atoms1, atoms2 = expand_super_cell(ortho1, atoms1, atoms2, dim)
xdel, _, x_indice, y_indice = find_overlapping_atoms(ortho1, atoms1, atoms2, dim, overlap=overlap)
if which_gb == "g1":
atoms1 = np.delete(atoms1, x_indice, axis=0)
elif which_gb == "g2":
atoms2 = np.delete(atoms2, y_indice, axis=0)
atoms1 = atoms1*lattice_parameter
atoms2 = atoms2*lattice_parameter
dimx, dimy, dimz = dim
#get box bounds
xlo = -1 * np.round(norm(ortho1[:, 0]) * dimx * lattice_parameter, 8)
xhi = np.round(norm(ortho1[:, 0]) * dimx * lattice_parameter, 8)
LenX = xhi - xlo
ylo = 0.0
yhi = np.round(norm(ortho1[:, 1]) * dimy * lattice_parameter, 8)
LenY = yhi - ylo
zlo = 0.0
zhi = np.round(norm(ortho1[:, 2]) * dimz * lattice_parameter, 8)
LenZ = zhi - zlo
return [[LenX, 0, 0],[0, LenY, 0],[0, 0, LenZ]], atoms1, atoms2