def _calculate_reciprocal_angles(self):
self.alpha_ = acos(
(cos(self.beta) * cos(self.gamma) - cos(self.alpha)) /
(sin(self.beta) * sin(self.gamma)))
self.beta_ = acos(
(cos(self.alpha) * cos(self.gamma) - cos(self.beta)) /
(sin(self.alpha) * sin(self.gamma)))
self.gamma_ = acos(
(cos(self.alpha) * cos(self.beta) - cos(self.gamma)) /
(sin(self.alpha) * sin(self.beta)))
def testinterplanarangle_triclinic(self):
lat = self.triclinic
s11 = lat.b**2 * lat.c**2 * sin(lat.alpha)**2
s22 = lat.a**2 * lat.c**2 * sin(lat.beta)**2
s33 = lat.a**2 * lat.b**2 * sin(lat.gamma)**2
s12 = lat.a * lat.b * lat.c**2 * (cos(lat.alpha) * cos(lat.beta) -
cos(lat.gamma))
s23 = lat.a**2 * lat.b * lat.c * (cos(lat.beta) * cos(lat.gamma) -
cos(lat.alpha))
s13 = lat.a * lat.b**2 * lat.c * (cos(lat.gamma) * cos(lat.alpha) -
cos(lat.beta))
v = lat.a * lat.b * lat.c * sqrt(1 - cos(lat.alpha) ** 2 - \
cos(lat.beta) ** 2 - \
cos(lat.gamma) ** 2 + \
2 * cos(lat.alpha) * cos(lat.beta) * cos(lat.gamma))
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
d1 * d2 / v ** 2 * (s11 * h1 * h2 + \
s22 * k1 * k2 + \
s33 * l1 * l2 + \
s23 * (k1 * l2 + k2 * l1) + \
s13 * (l1 * h2 + l2 * h1) + \
s12 * (h1 * k2 + h2 * k1))
for plane1 in self.planes:
for plane2 in self.planes:
d1 = calculations.planespacing(plane1, self.triclinic)
d2 = calculations.planespacing(plane2, self.triclinic)
angle = calculations.interplanarangle(plane1, plane2,
self.triclinic)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.triclinic, *args))
self.assertAlmostEqual(angle, expected_angle, 6)
def testinterplanarangle_triclinic(self):
lat = self.triclinic
s11 = lat.b ** 2 * lat.c ** 2 * sin(lat.alpha) ** 2
s22 = lat.a ** 2 * lat.c ** 2 * sin(lat.beta) ** 2
s33 = lat.a ** 2 * lat.b ** 2 * sin(lat.gamma) ** 2
s12 = lat.a * lat.b * lat.c ** 2 * (cos(lat.alpha) * cos(lat.beta) - cos(lat.gamma))
s23 = lat.a ** 2 * lat.b * lat.c * (cos(lat.beta) * cos(lat.gamma) - cos(lat.alpha))
s13 = lat.a * lat.b ** 2 * lat.c * (cos(lat.gamma) * cos(lat.alpha) - cos(lat.beta))
v = lat.a * lat.b * lat.c * sqrt(1 - cos(lat.alpha) ** 2 - \
cos(lat.beta) ** 2 - \
cos(lat.gamma) ** 2 + \
2 * cos(lat.alpha) * cos(lat.beta) * cos(lat.gamma))
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
d1 * d2 / v ** 2 * (s11 * h1 * h2 + \
s22 * k1 * k2 + \
s33 * l1 * l2 + \
s23 * (k1 * l2 + k2 * l1) + \
s13 * (l1 * h2 + l2 * h1) + \
s12 * (h1 * k2 + h2 * k1))
for plane1 in self.planes:
for plane2 in self.planes:
d1 = calculations.planespacing(plane1, self.triclinic)
d2 = calculations.planespacing(plane2, self.triclinic)
angle = calculations.interplanarangle(plane1, plane2, self.triclinic)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.triclinic, *args))
self.assertAlmostEqual(angle, expected_angle, 6)
def zoneaxis_goniometricangles(zoneaxis, unitcell):
"""
Return the goniometric angles (:math:`\\phi` and :math:`\\rho`)
for a zone axis of a given unit cell.
:type zoneaxis: :class:`vectors.Vector3D`
:type unitcell: :class:`unitcell.UnitCell`
:return: (:math:`\\phi`, :math:`\\rho`)
:rtype: :class:`tuple`
**References**
Example 2.20 from Mathematical Crystallography
"""
a = unitcell.cartesianmatrix
# Change basis of zone axis (lattice basis) into the cartesian basis
r_c = a * zoneaxis
# Normalize r_c to have a unit length
r_c.normalize()
# Find phi and rho
rho = acos(r_c[2])
phi = atan2(r_c[0], r_c[1])
return phi, rho
def facepole_goniometricangles(facepole, unitcell):
"""
Return the goniometric angles (:math:`\\phi` and :math:`\\rho`)
for a face pole of a given unit cell.
:type facepole: :class:`vectors.Vector3D`
:type unitcell: :class:`unitcell.UnitCell`
:return: (:math:`\\phi`, :math:`\\rho`)
:rtype: :class:`tuple`
**References**
Example 2.20 from Mathematical Crystallography
"""
b = matrices.inverse(matrices.transpose(unitcell.cartesianmatrix))
# Change basis of face pole (reciprocal basis) into the cartesian basis
s_c = b * facepole
# Normalize r_c to have a unit length
s_c.normalize()
# Find phi and rho
rho = acos(s_c[2])
phi = atan2(s_c[0], s_c[1])
return phi, rho
def testinterplanarangle_cubic(self):
equation = lambda h1, k1, l1, h2, k2, l2: \
(h1 * h2 + k1 * k2 + l1 * l2) / \
sqrt((h1 ** 2 + k1 ** 2 + l1 ** 2) * (h2 ** 2 + k2 ** 2 + l2 ** 2))
for plane1 in self.planes:
for plane2 in self.planes:
angle = calculations.interplanarangle(plane1, plane2, self.cubic)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(*args))
self.assertAlmostEqual(angle, expected_angle)
def testinterplanarangle_tetragonal(self):
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
((h1 * h2 + k1 * k2) / lat.a ** 2 + l1 * l2 / lat.c ** 2) / \
sqrt(((h1 ** 2 + k1 ** 2) / lat.a ** 2 + l1 ** 2 / lat.c ** 2) * \
((h2 ** 2 + k2 ** 2) / lat.a ** 2 + l2 ** 2 / lat.c ** 2))
for plane1 in self.planes:
for plane2 in self.planes:
angle = calculations.interplanarangle(plane1, plane2, self.tetragonal)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.tetragonal, *args))
self.assertAlmostEqual(angle, expected_angle)
def testinterplanarangle_cubic(self):
equation = lambda h1, k1, l1, h2, k2, l2: \
(h1 * h2 + k1 * k2 + l1 * l2) / \
sqrt((h1 ** 2 + k1 ** 2 + l1 ** 2) * (h2 ** 2 + k2 ** 2 + l2 ** 2))
for plane1 in self.planes:
for plane2 in self.planes:
angle = calculations.interplanarangle(plane1, plane2,
self.cubic)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(*args))
self.assertAlmostEqual(angle, expected_angle)
def testinterplanarangle_tetragonal(self):
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
((h1 * h2 + k1 * k2) / lat.a ** 2 + l1 * l2 / lat.c ** 2) / \
sqrt(((h1 ** 2 + k1 ** 2) / lat.a ** 2 + l1 ** 2 / lat.c ** 2) * \
((h2 ** 2 + k2 ** 2) / lat.a ** 2 + l2 ** 2 / lat.c ** 2))
for plane1 in self.planes:
for plane2 in self.planes:
angle = calculations.interplanarangle(plane1, plane2,
self.tetragonal)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.tetragonal, *args))
self.assertAlmostEqual(angle, expected_angle)
def angle(v1, v2):
"""
Return the angle in radians between *v1* and *v2*.
:arg v1: the first vector
:type v1: :class:`vectors.Vector`
:arg v2: the second vector
:type v2: :class:`vectors.Vector`
:return: angle in radians
:rtype: :class:`float`
"""
costheta = directioncosine(v1, v2)
return acos(costheta)
def angle(v1, v2):
"""
Return the angle in radians between *v1* and *v2*.
:arg v1: the first vector
:type v1: :class:`vectors.Vector`
:arg v2: the second vector
:type v2: :class:`vectors.Vector`
:return: angle in radians
:rtype: :class:`float`
"""
costheta = directioncosine(v1, v2)
return acos(costheta)
def testinterplanarangle_hexagonal(self):
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
(h1 * h2 + k1 * k2 + 0.5 * (h1 * k2 + h2 * k1) + \
(3 * lat.a ** 2) / (4 * lat.c ** 2) * l1 * l2) / \
sqrt((h1 ** 2 + k1 ** 2 + h1 * k1 + (3 * lat.a ** 2) / \
(4 * lat.c ** 2) * l1 ** 2) * \
(h2 ** 2 + k2 ** 2 + h2 * k2 + (3 * lat.a ** 2) / \
(4 * lat.c ** 2) * l2 ** 2))
for plane1 in self.planes:
for plane2 in self.planes:
angle = calculations.interplanarangle(plane1, plane2, self.hexagonal)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.hexagonal, *args))
self.assertAlmostEqual(angle, expected_angle)
def testinterplanarangle_hexagonal(self):
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
(h1 * h2 + k1 * k2 + 0.5 * (h1 * k2 + h2 * k1) + \
(3 * lat.a ** 2) / (4 * lat.c ** 2) * l1 * l2) / \
sqrt((h1 ** 2 + k1 ** 2 + h1 * k1 + (3 * lat.a ** 2) / \
(4 * lat.c ** 2) * l1 ** 2) * \
(h2 ** 2 + k2 ** 2 + h2 * k2 + (3 * lat.a ** 2) / \
(4 * lat.c ** 2) * l2 ** 2))
for plane1 in self.planes:
for plane2 in self.planes:
angle = calculations.interplanarangle(plane1, plane2,
self.hexagonal)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.hexagonal, *args))
self.assertAlmostEqual(angle, expected_angle)
def testinterplanarangle_monoclinic(self):
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
d1 * d2 / sin(lat.beta) ** 2 * \
(h1 * h2 / lat.a ** 2 + \
k1 * k2 * sin(lat.beta) ** 2 / lat.b ** 2 + \
l1 * l2 / lat.c ** 2 - \
(l1 * h2 + l2 * h1) * cos(lat.beta) / (lat.a * lat.c))
for plane1 in self.planes:
for plane2 in self.planes:
d1 = calculations.planespacing(plane1, self.monoclinic)
d2 = calculations.planespacing(plane2, self.monoclinic)
angle = calculations.interplanarangle(plane1, plane2, self.monoclinic)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.monoclinic, *args))
self.assertAlmostEqual(angle, expected_angle, 6)
def testinterplanarangle_orthorhombic(self):
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
(h1 * h2 / lat.a ** 2 + k1 * k2 / lat.b ** 2 + l1 * l2 / lat.c ** 2) / \
sqrt((h1 ** 2 / lat.a ** 2 + \
k1 ** 2 / lat.b ** 2 + \
l1 ** 2 / lat.c ** 2) \
* (h2 ** 2 / lat.a ** 2 + \
k2 ** 2 / lat.b ** 2 + \
l2 ** 2 / lat.c ** 2))
for plane1 in self.planes:
for plane2 in self.planes:
angle = calculations.interplanarangle(plane1, plane2, self.orthorhombic)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.orthorhombic, *args))
self.assertAlmostEqual(angle, expected_angle, 6)
def testinterplanarangle_orthorhombic(self):
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
(h1 * h2 / lat.a ** 2 + k1 * k2 / lat.b ** 2 + l1 * l2 / lat.c ** 2) / \
sqrt((h1 ** 2 / lat.a ** 2 + \
k1 ** 2 / lat.b ** 2 + \
l1 ** 2 / lat.c ** 2) \
* (h2 ** 2 / lat.a ** 2 + \
k2 ** 2 / lat.b ** 2 + \
l2 ** 2 / lat.c ** 2))
for plane1 in self.planes:
for plane2 in self.planes:
angle = calculations.interplanarangle(plane1, plane2,
self.orthorhombic)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.orthorhombic, *args))
self.assertAlmostEqual(angle, expected_angle, 6)
def testinterplanarangle_monoclinic(self):
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
d1 * d2 / sin(lat.beta) ** 2 * \
(h1 * h2 / lat.a ** 2 + \
k1 * k2 * sin(lat.beta) ** 2 / lat.b ** 2 + \
l1 * l2 / lat.c ** 2 - \
(l1 * h2 + l2 * h1) * cos(lat.beta) / (lat.a * lat.c))
for plane1 in self.planes:
for plane2 in self.planes:
d1 = calculations.planespacing(plane1, self.monoclinic)
d2 = calculations.planespacing(plane2, self.monoclinic)
angle = calculations.interplanarangle(plane1, plane2,
self.monoclinic)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.monoclinic, *args))
self.assertAlmostEqual(angle, expected_angle, 6)
def bondangle(atom1, atom2, atom3, unitcell):
"""
Calculate the bond angle (in radians) between the vector
from *atom1* to *atom2* and the vector from *atom1* to *atom3*.
:type atom1: :class:`atomsite.AtomSite`
:type atom2: :class:`atomsite.AtomSite`
:type atom3: :class:`atomsite.AtomSite`
:type unitcell: :class:`unitcell.UnitCell`
:rtype: :class:`float`
**References**
Equation 1.13 from Mathematical Crystallography
"""
atom1vector = atom1.position
atom2vector = atom2.position
atom3vector = atom3.position
metricalmatrix = unitcell.metricalmatrix
# Interatom vectors
vector12 = atom2vector - atom1vector
vector13 = atom3vector - atom1vector
# Dot product
dotproduct = vector12 * (metricalmatrix * vector13)
# Bond distance
bonddistance12 = bonddistance(atom1, atom2, unitcell)
bonddistance13 = bonddistance(atom1, atom3, unitcell)
# Cosine
cosine = dotproduct / (bonddistance12 * bonddistance13)
# Angle
angle = acos(cosine)
return angle
def testinterplanarangle_trigonal(self):
lat = self.trigonal
v = lat.a ** 3 * sqrt(1 - 3 * cos(lat.alpha) ** 2 + 2 * cos(lat.alpha) ** 3)
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
(lat.a ** 4 * d1 * d2) / v ** 2 * \
(sin(lat.alpha) ** 2 * (h1 * h2 + k1 * k2 + l1 * l2) + \
(cos(lat.alpha) ** 2 - cos(lat.alpha)) * \
(k1 * l2 + k2 * l1 + l1 * h2 + l2 * h1 + h1 * k2 + h2 * k1))
for plane1 in self.planes:
for plane2 in self.planes:
d1 = calculations.planespacing(plane1, lat)
d2 = calculations.planespacing(plane2, lat)
angle = calculations.interplanarangle(plane1, plane2, self.trigonal)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.trigonal, *args))
self.assertAlmostEqual(angle, expected_angle, 6)
def testinterplanarangle_trigonal(self):
lat = self.trigonal
v = lat.a**3 * sqrt(1 - 3 * cos(lat.alpha)**2 + 2 * cos(lat.alpha)**3)
equation = lambda lat, h1, k1, l1, h2, k2, l2: \
(lat.a ** 4 * d1 * d2) / v ** 2 * \
(sin(lat.alpha) ** 2 * (h1 * h2 + k1 * k2 + l1 * l2) + \
(cos(lat.alpha) ** 2 - cos(lat.alpha)) * \
(k1 * l2 + k2 * l1 + l1 * h2 + l2 * h1 + h1 * k2 + h2 * k1))
for plane1 in self.planes:
for plane2 in self.planes:
d1 = calculations.planespacing(plane1, lat)
d2 = calculations.planespacing(plane2, lat)
angle = calculations.interplanarangle(plane1, plane2,
self.trigonal)
args = copy.deepcopy(plane1)
args.extend(plane2)
expected_angle = acos(equation(self.trigonal, *args))
self.assertAlmostEqual(angle, expected_angle, 6)
def _calculate_reciprocal_angles(self):
self.alpha_ = acos((cos(self.beta) * cos(self.gamma) - cos(self.alpha)) / (sin(self.beta) * sin(self.gamma)))
self.beta_ = acos((cos(self.alpha) * cos(self.gamma) - cos(self.beta)) / (sin(self.alpha) * sin(self.gamma)))
self.gamma_ = acos((cos(self.alpha) * cos(self.beta) - cos(self.gamma)) / (sin(self.alpha) * sin(self.beta)))
def testacos(self):
self.assertEqual(trigo.acos(4), 0)
self.assertEqual(trigo.acos(-4), pi)
self.assertAlmostEqual(trigo.acos(0.5), 60 / 180.0 * pi)
self.assertAlmostEqual(trigo.acos(0.45675), acos(0.45675))
def testacos(self):
self.assertEqual(trigo.acos(4), 0)
self.assertEqual(trigo.acos(-4), pi)
self.assertAlmostEqual(trigo.acos(0.5), 60 / 180.0 * pi)
self.assertAlmostEqual(trigo.acos(0.45675), acos(0.45675))