def test_Bragg_condition(self):
al = Lattice.from_symbol('Al')
p = HklPlane(0, 0, 2, lattice=al)
lambda_keV = 42
lambda_nm = lambda_keV_to_nm(lambda_keV)
rod = [0.1449, -0.0281, 0.0616]
o = Orientation.from_rodrigues(rod)
(w1, w2) = o.dct_omega_angles(p, lambda_keV, verbose=False)
# test the two solution of the rotating crystal
for omega in (w1, w2):
alpha = o.compute_XG_angle(p, omega, verbose=True)
theta_bragg = p.bragg_angle(lambda_keV)
self.assertAlmostEqual(alpha, 180 / np.pi * (np.pi / 2 - theta_bragg))
def test_dct_omega_angles(self):
# test with a BCC Titanium lattice
lambda_keV = 30
lambda_nm = 1.2398 / lambda_keV
a = 0.3306 # lattice parameter in nm
Ti_bcc = Lattice.cubic(a)
(h, k, l) = (0, 1, 1)
hkl = HklPlane(h, k, l, lattice=Ti_bcc)
o = Orientation.from_euler((103.517, 42.911, 266.452))
theta = hkl.bragg_angle(lambda_keV, verbose=False)
gt = o.orientation_matrix(
) # our B (here called gt) corresponds to g^{-1} in Poulsen 2004
A = h * gt[0, 0] + k * gt[1, 0] + l * gt[2, 0]
B = -h * gt[0, 1] - k * gt[1, 1] - l * gt[2, 1]
C = -2 * a * np.sin(
theta
)**2 / lambda_nm # the minus sign comes from the main equation
Delta = 4 * (A**2 + B**2 - C**2)
self.assertEqual(Delta > 0, True)
t1 = (B - 0.5 * np.sqrt(Delta)) / (A + C)
t2 = (B + 0.5 * np.sqrt(Delta)) / (A + C)
# verifying A cos(w) + B sin(w) = C:'
for t in (t1, t2):
x = A * (1 - t**2) / (1 + t**2) + B * 2 * t / (1 + t**2)
self.assertAlmostEqual(x, C, 2)
# verifying (A + C) * t**2 - 2 * B * t + (C - A) = 0'
for t in (t1, t2):
self.assertAlmostEqual((A + C) * t**2 - 2 * B * t + (C - A), 0.0,
2)
(w1, w2) = o.dct_omega_angles(hkl, lambda_keV, verbose=False)
self.assertAlmostEqual(w1, 196.709, 2)
self.assertAlmostEqual(w2, 28.334, 2)
# test with an FCC Aluminium-Lithium lattice
a = 0.40495 # lattice parameter in nm
Al_fcc = Lattice.face_centered_cubic(a)
hkl = HklPlane(-1, 1, 1, Al_fcc)
o = Orientation.from_rodrigues([0.0499, -0.3048, 0.1040])
w1, w2 = o.dct_omega_angles(hkl, 40, verbose=False)
self.assertAlmostEqual(w1, 109.2, 1)
self.assertAlmostEqual(w2, 296.9, 1)
def test_bragg_angle(self):
l = Lattice.cubic(0.287) # FCC iron
hkl = HklPlane(2, 0, 0, l) # 200 reflection at 8 keV is at 32.7 deg
self.assertAlmostEqual(hkl.bragg_angle(8), 0.5704164)