def initialize_displacive(self, Ux, Uy, Uz, H, K, L, Qx, Qy, Qz, indices,
factors, nu, nv, nw, n_atm, p, centering):
n_uvw = nu * nv * nw
coeffs = displacive.coefficients(p)
U_r = displacive.products(Ux, Uy, Uz, p)
Q_k = displacive.products(Qx, Qy, Qz, p)
U_k, i_dft = displacive.transform(U_r, H, K, L, nu, nv, nw, n_atm)
H_nuc, K_nuc, L_nuc, \
cond = space.condition(H, K, L, nu, nv, nw, centering)
F, F_nuc, \
prod, prod_nuc, \
V_k, V_k_nuc, \
even, \
bragg = displacive.structure(U_k, Q_k, coeffs, cond, p, i_dft, factors)
F_orig = np.zeros(indices.size, dtype=complex)
F_nuc_orig = np.zeros(bragg.size, dtype=complex)
prod_orig = np.zeros(indices.size, dtype=complex)
prod_nuc_orig = np.zeros(bragg.size, dtype=complex)
V_k_orig = np.zeros(indices.size, dtype=complex)
V_k_nuc_orig = np.zeros(bragg.size, dtype=complex)
U_k_orig = np.zeros(n_uvw * coeffs.size, dtype=complex)
F_cand = np.zeros(indices.shape, dtype=complex)
F_nuc_cand = np.zeros(bragg.shape, dtype=complex)
prod_cand = np.zeros(indices.shape, dtype=complex)
prod_nuc_cand = np.zeros(bragg.shape, dtype=complex)
V_k_cand = np.zeros(indices.size, dtype=complex)
V_k_nuc_cand = np.zeros(bragg.size, dtype=complex)
U_k_cand = np.zeros(n_uvw * coeffs.size, dtype=complex)
U_r_orig = np.zeros(coeffs.size, dtype=float)
U_r_cand = np.zeros(coeffs.size, dtype=float)
return U_r, U_r_orig, U_r_cand, Q_k, \
U_k, U_k_orig, U_k_cand, \
V_k, V_k_orig, V_k_cand, \
V_k_nuc, V_k_nuc_orig, V_k_nuc_cand, \
F, F_orig, F_cand, \
F_nuc, F_nuc_orig, F_nuc_cand, \
prod, prod_orig, prod_cand, \
prod_nuc, prod_nuc_orig, prod_nuc_cand, \
i_dft, coeffs, H_nuc, K_nuc, L_nuc, cond, even, bragg
def displacive_parameters(self, p, centering):
coeffs = displacive.coefficients(p)
start = (np.cumsum(displacive.number(np.arange(p + 1))) -
displacive.number(np.arange(p + 1)))[::2]
end = np.cumsum(displacive.number(np.arange(p + 1)))[::2]
even = []
for k in range(len(end)):
even += range(start[k], end[k])
even = np.array(even)
nuclear = ['P', 'I', 'F', 'R', 'C', 'A', 'B']
cntr = np.argwhere([x in centering for x in nuclear])[0][0]
cntr += 1
return coeffs, even, cntr
def test_coefficients(self):
p = 5
coeffs = displacive.coefficients(p)
numbers = displacive.number(np.arange(p + 1))
self.assertEqual(coeffs.size, numbers.sum())
self.assertEqual(coeffs[0], 1)
even = np.isreal(coeffs)
odd = ~np.isreal(coeffs)
end = np.cumsum(numbers)
start = end - numbers
self.assertTrue(even[start[0]:end[0]].all())
self.assertTrue(odd[start[1]:end[1]].all())
self.assertTrue(even[start[2]:end[2]].all())
self.assertTrue(odd[start[3]:end[3]].all())
self.assertTrue(even[start[4]:end[4]].all())
self.assertTrue(odd[start[5]:end[5]].all())
self.assertAlmostEqual(coeffs[0], 1)
self.assertAlmostEqual(coeffs[1], 1j)
self.assertAlmostEqual(coeffs[2], 1j)
self.assertAlmostEqual(coeffs[3], 1j)
self.assertAlmostEqual(coeffs[4], -0.5)
self.assertAlmostEqual(coeffs[5], -1)
self.assertAlmostEqual(coeffs[6], -0.5)
self.assertAlmostEqual(coeffs[7], -1)
self.assertAlmostEqual(coeffs[8], -1)
self.assertAlmostEqual(coeffs[9], -0.5)
def test_displacive(self):
a, b, c, alpha, beta, gamma = 5, 6, 7, np.pi / 2, np.pi / 3, np.pi / 4
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
h_range, nh = [-1, 1], 5
k_range, nk = [0, 2], 11
l_range, nl = [-1, 0], 5
nu, nv, nw, n_atm = 2, 5, 4, 2
u = np.array([0.2, 0.1])
v = np.array([0.3, 0.4])
w = np.array([0.4, 0.5])
atm = np.array(['Fe', 'Mn'])
occupancy = np.array([0.75, 0.5])
U11 = np.array([0.5, 0.3])
U22 = np.array([0.6, 0.4])
U33 = np.array([0.4, 0.6])
U23 = np.array([0.05, -0.03])
U13 = np.array([-0.04, 0.02])
U12 = np.array([0.03, -0.02])
twins = np.eye(3).reshape(1, 3, 3)
variants = np.array([1.0])
W = np.eye(3)
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
U = np.row_stack((U11, U22, U33, U23, U13, U12))
Ux, Uy, Uz = displacive.expansion(nu, nv, nw, n_atm, value=U)
index_parameters = space.mapping(h_range, k_range, l_range, nh, nk, nl,
nu, nv, nw)
h, k, l, H, K, L, indices, inverses, operators = index_parameters
Qh, Qk, Ql = crystal.vector(h, k, l, B)
Qx, Qy, Qz = crystal.transform(Qh, Qk, Ql, R)
Qx_norm, Qy_norm, Qz_norm, Q = space.unit(Qx, Qy, Qz)
ux, uy, uz = crystal.transform(u, v, w, A)
ix, iy, iz = space.cell(nu, nv, nw, A)
rx, ry, rz, atms = space.real(ux, uy, uz, ix, iy, iz, atm)
phase_factor = scattering.phase(Qx, Qy, Qz, ux, uy, uz)
scattering_length = scattering.length(atm, Q.size)
p = 3
coeffs = displacive.coefficients(p)
H_nuc, K_nuc, L_nuc, cond = space.condition(H,
K,
L,
nu,
nv,
nw,
centering='P')
U_r = displacive.products(Ux, Uy, Uz, p)
Q_k = displacive.products(Qx, Qy, Qz, p)
U_k, i_dft = displacive.transform(U_r, H, K, L, nu, nv, nw, n_atm)
factors = space.prefactors(scattering_length, phase_factor, occupancy)
I_ref = displacive.intensity(U_k, Q_k, coeffs, cond, p, i_dft, factors)
reduced_params = space.reduced(h_range, k_range, l_range, nh, nk, nl,
nu, nv, nw)
indices, reverses, symops, Nu, Nv, Nw = reduced_params
symop = symmetry.laue_id(symops)
centering = 1
even, odd = displacive.indices(p)
I = monocrystal.displacive(U_r, coeffs, occupancy, ux, uy, uz, atm,
h_range, k_range, l_range, indices, symop,
W, B, R, twins, variants, nh, nk, nl, nu,
nv, nw, Nu, Nv, Nw, p, even, centering)
np.testing.assert_array_almost_equal(I, I_ref)
def test_structure(self):
a, b, c, alpha, beta, gamma = 5, 6, 7, np.pi / 2, np.pi / 3, np.pi / 4
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
h_range, nh = [-1, 1], 5
k_range, nk = [0, 2], 11
l_range, nl = [-1, 0], 5
nu, nv, nw, n_atm = 2, 5, 4, 2
u = np.array([0.2, 0.1])
v = np.array([0.3, 0.4])
w = np.array([0.4, 0.5])
atm = np.array(['Fe', 'Mn'])
occupancy = np.array([0.75, 0.5])
U11 = np.array([0.5, 0.3])
U22 = np.array([0.6, 0.4])
U33 = np.array([0.4, 0.6])
U23 = np.array([0.05, -0.03])
U13 = np.array([-0.04, 0.02])
U12 = np.array([0.03, -0.02])
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
U = np.row_stack((U11, U22, U33, U23, U13, U12))
Ux, Uy, Uz = displacive.expansion(nu, nv, nw, n_atm, value=U)
index_parameters = space.mapping(h_range, k_range, l_range, nh, nk, nl,
nu, nv, nw)
h, k, l, H, K, L, indices, inverses, operators = index_parameters
Qh, Qk, Ql = crystal.vector(h, k, l, B)
Qx, Qy, Qz = crystal.transform(Qh, Qk, Ql, R)
Qx_norm, Qy_norm, Qz_norm, Q = space.unit(Qx, Qy, Qz)
ux, uy, uz = crystal.transform(u, v, w, A)
ix, iy, iz = space.cell(nu, nv, nw, A)
rx, ry, rz, atms = space.real(ux, uy, uz, ix, iy, iz, atm)
phase_factor = scattering.phase(Qx, Qy, Qz, ux, uy, uz)
scattering_length = scattering.length(atm, Q.size)
p = 3
coeffs = displacive.coefficients(p)
H_nuc, K_nuc, L_nuc, cond = space.condition(H, K, L, nu, nv, nw)
U_r = displacive.products(Ux, Uy, Uz, p)
Q_k = displacive.products(Qx, Qy, Qz, p)
U_k, i_dft = displacive.transform(U_r, H, K, L, nu, nv, nw, n_atm)
factors = space.prefactors(scattering_length, phase_factor, occupancy)
F, F_nuc, \
prod, prod_nuc, \
V_k, V_k_nuc, \
even, bragg = displacive.structure(U_k, Q_k, coeffs, cond,
p, i_dft, factors)
n_hkl = Q.size
n_xyz = nu * nv * nw * n_atm
m = np.arange(n_xyz)
n = np.mod(m, n_atm)
rx_m = rx[m]
ry_m = ry[m]
rz_m = rz[m]
bc = scattering.length(atm, 1)
U_r = U_r.reshape(coeffs.shape[0], n_xyz)
Q_k = Q_k.reshape(coeffs.shape[0], n_hkl)
U_m = U_r[:, m]
c_n = occupancy[n]
b_n = bc[n]
exp_iQ_dot_U_m = np.dot(coeffs * U_m.T, Q_k).T
prod_ref = c_n*b_n*exp_iQ_dot_U_m*np.exp(1j*(Qx[:,np.newaxis]*rx_m+\
Qy[:,np.newaxis]*ry_m+\
Qz[:,np.newaxis]*rz_m))
F_ref = prod_ref.sum(axis=1)
prod_ref = prod_ref.reshape(n_hkl, nu * nv * nw,
n_atm).sum(axis=1).flatten()
np.testing.assert_array_almost_equal(F, F_ref)
np.testing.assert_array_almost_equal(prod, prod_ref)
cos_iQ_dot_U_m = np.dot((coeffs * U_m.T)[:, even], Q_k[even, :]).T
prod_nuc_ref = c_n*b_n*cos_iQ_dot_U_m*\
np.exp(1j*(Qx[:,np.newaxis]*rx_m+\
Qy[:,np.newaxis]*ry_m+\
Qz[:,np.newaxis]*rz_m))
F_nuc_ref = prod_nuc_ref.sum(axis=1)[cond]
prod_nuc_ref = prod_nuc_ref.reshape(n_hkl, nu * nv * nw,
n_atm).sum(axis=1)[cond].flatten()
np.testing.assert_array_almost_equal(F_nuc, F_nuc_ref)
np.testing.assert_array_almost_equal(prod_nuc, prod_nuc_ref)
factors = (c_n * b_n * cos_iQ_dot_U_m).flatten()
F_nuc_ref = space.bragg(Qx, Qy, Qz, rx, ry, rz, factors, cond)
np.testing.assert_array_almost_equal(F_nuc, F_nuc_ref)
def test_intensity(self):
a, b, c, alpha, beta, gamma = 5, 6, 7, np.pi / 2, np.pi / 3, np.pi / 4
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
h_range, nh = [-1, 1], 5
k_range, nk = [0, 2], 11
l_range, nl = [-1, 0], 5
nu, nv, nw, n_atm = 2, 5, 4, 2
u = np.array([0.2, 0.1])
v = np.array([0.3, 0.4])
w = np.array([0.4, 0.5])
atm = np.array(['Fe', 'Mn'])
occupancy = np.array([0.75, 0.5])
U11 = np.array([0.5, 0.3])
U22 = np.array([0.6, 0.4])
U33 = np.array([0.4, 0.6])
U23 = np.array([0.05, -0.03])
U13 = np.array([-0.04, 0.02])
U12 = np.array([0.03, -0.02])
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
U = np.row_stack((U11, U22, U33, U23, U13, U12))
Ux, Uy, Uz = displacive.expansion(nu, nv, nw, n_atm, value=U)
index_parameters = space.mapping(h_range, k_range, l_range, nh, nk, nl,
nu, nv, nw)
h, k, l, H, K, L, indices, inverses, operators = index_parameters
Qh, Qk, Ql = crystal.vector(h, k, l, B)
Qx, Qy, Qz = crystal.transform(Qh, Qk, Ql, R)
Qx_norm, Qy_norm, Qz_norm, Q = space.unit(Qx, Qy, Qz)
ux, uy, uz = crystal.transform(u, v, w, A)
ix, iy, iz = space.cell(nu, nv, nw, A)
rx, ry, rz, atms = space.real(ux, uy, uz, ix, iy, iz, atm)
phase_factor = scattering.phase(Qx, Qy, Qz, ux, uy, uz)
scattering_length = scattering.length(atm, Q.size)
p = 3
coeffs = displacive.coefficients(p)
H_nuc, K_nuc, L_nuc, cond = space.condition(H,
K,
L,
nu,
nv,
nw,
centering='P')
U_r = displacive.products(Ux, Uy, Uz, p)
Q_k = displacive.products(Qx, Qy, Qz, p)
U_k, i_dft = displacive.transform(U_r, H, K, L, nu, nv, nw, n_atm)
factors = space.prefactors(scattering_length, phase_factor, occupancy)
# I = displacive.intensity(U_k, Q_k, coeffs, cond, p, i_dft, factors)
I, F_nuc = displacive.intensity(U_k,
Q_k,
coeffs,
cond,
p,
i_dft,
factors,
subtract=False)
n_hkl = Q.size
n_xyz = nu * nv * nw * n_atm
i, j = np.triu_indices(n_xyz, 1)
k, l = np.mod(i, n_atm), np.mod(j, n_atm)
m = np.arange(n_xyz)
n = np.mod(m, n_atm)
rx_ij = rx[j] - rx[i]
ry_ij = ry[j] - ry[i]
rz_ij = rz[j] - rz[i]
bc = scattering.length(atm, 1)
U_r = U_r.reshape(coeffs.shape[0], n_xyz)
Q_k = Q_k.reshape(coeffs.shape[0], n_hkl)
U_i, U_j, U_m = U_r[:, i], U_r[:, j], U_r[:, m]
c_k, c_l, c_n = occupancy[k], occupancy[l], occupancy[n]
b_k, b_l, b_n = bc[k], bc[l], bc[n]
exp_iQ_dot_U_m = np.dot(coeffs * U_m.T, Q_k).T
exp_iQ_dot_U_i = np.dot(coeffs * U_i.T, Q_k).T
exp_iQ_dot_U_j = np.dot(coeffs * U_j.T, Q_k).T
I_ref = ((c_n**2*(b_n*b_n.conj()).real*\
(exp_iQ_dot_U_m*exp_iQ_dot_U_m.conj()).real).sum(axis=1)\
+ 2*(c_k*c_l*(b_k*b_l.conj()).real*
((exp_iQ_dot_U_i*exp_iQ_dot_U_j.conj()*\
np.cos(Qx[:,np.newaxis]*rx_ij+\
Qy[:,np.newaxis]*ry_ij+\
Qz[:,np.newaxis]*rz_ij)).real+
(exp_iQ_dot_U_i*exp_iQ_dot_U_j.conj()*\
np.sin(Qx[:,np.newaxis]*rx_ij+\
Qy[:,np.newaxis]*ry_ij+\
Qz[:,np.newaxis]*rz_ij)).imag)).sum(axis=1))/n_xyz
np.testing.assert_array_almost_equal(I, I_ref)