def test_phase(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
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)
nu, nv, nw = 5, 3, 4
Rx, Ry, Rz = space.cell(nu, nv, nw, A)
atm = np.array(['Fe', 'Co'])
u, v, w = np.array([0, 0.2]), np.array([0, 0.3]), np.array([0, 0.4])
ux, uy, uz = crystal.transform(u, v, w, A)
rx, ry, rz, atms = space.real(ux, uy, uz, Rx, Ry, Rz, atm)
h, k, l = np.array([-7, -3]), np.array([2, 2]), np.array([2, 5])
Qh, Qk, Ql = crystal.vector(h, k, l, B)
Qx, Qy, Qz = crystal.transform(Qh, Qk, Ql, R)
phase_factor = scattering.phase(Qx, Qy, Qz, rx, ry, rz)
np.testing.assert_array_almost_equal(phase_factor, 1 + 0j)
def test_dipole_dipole_interaction_potential(self):
nu, nv, nw, n_atm = 3, 4, 5, 2
n = nu * nv * nw * n_atm
M = 2
Sx = np.random.random((nu, nv, nw, n_atm, M))
Sy = np.random.random((nu, nv, nw, n_atm, M))
Sz = np.random.random((nu, nv, nw, n_atm, M))
u = np.array([0.2, 0.3])
v = np.array([0.5, 0.4])
w = np.array([0.7, 0.2])
atm = np.array(['', ''])
a, b, c, alpha, beta, gamma = 5, 6, 7, np.pi / 2, np.pi / 2, 2 * np.pi / 3
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(*crystal.reciprocal(a, b, c, alpha, beta, gamma))
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
Rx, Ry, Rz = space.cell(nu, nv, nw, A)
ux, uy, uz = crystal.transform(u, v, w, A)
rx, ry, rz, atms = space.real(ux, uy, uz, Rx, Ry, Rz, atm)
i, j = np.triu_indices(n)
Qijm = np.zeros((n, n, 6))
Qijkl = np.zeros((n, n, 3, 3))
Q = interaction.dipole_dipole_matrix(rx, ry, rz, nu, nv, nw, n_atm, A,
B, R)
Qijm[i, j, :] = Q
Qijm[j, i, :] = Qijm[i, j, :]
Qijkl[:, :, 0, 0] = Qijm[:, :, 0]
Qijkl[:, :, 1, 1] = Qijm[:, :, 1]
Qijkl[:, :, 2, 2] = Qijm[:, :, 2]
Qijkl[:, :, 1, 2] = Qijkl[:, :, 2, 1] = Qijm[:, :, 3]
Qijkl[:, :, 0, 2] = Qijkl[:, :, 2, 0] = Qijm[:, :, 4]
Qijkl[:, :, 0, 1] = Qijkl[:, :, 1, 0] = Qijm[:, :, 5]
p0 = simulation.dipole_dipole_interaction_potential(Sx, Sy, Sz, Q)
Sx, Sy, Sz = Sx.reshape(n, M), Sy.reshape(n, M), Sz.reshape(n, M)
S = np.column_stack((Sx, Sy, Sz)).reshape(-1, 3, M)
p = np.einsum('ijkl,jlm->ijklm', Qijkl, S).sum(axis=1)
np.testing.assert_array_almost_equal(p, p0)
def test_metric(self):
a, b, c, alpha, beta, gamma = 5, 6, 7, np.pi / 2, np.pi / 3, np.pi / 4
G = crystal.metric(a, b, c, alpha, beta, gamma)
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
G_ = crystal.metric(a_, b_, c_, alpha_, beta_, gamma_)
np.testing.assert_array_almost_equal(np.eye(3), np.dot(G, G_))
def test_interplanar(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
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
u_, v_, w_ = np.dot(B, [1, 0, 0]), np.dot(B, [0, 1, 0]), np.dot(
B, [0, 0, 1])
h0, k0, l0 = 1, 2, -3
h1, k1, l1 = 2, -1, -4
angle = crystal.interplanar(a, b, c, alpha, beta, gamma, \
h0, k0, l0, h1, k1, l1)
angle_ref = np.arccos(
np.dot(h0 * u_ + k0 * v_ + l0 * w_, h1 * u_ + k1 * v_ + l1 * w_) /
np.linalg.norm(h0 * u_ + k0 * v_ + l0 * w_) /
np.linalg.norm(h1 * u_ + k1 * v_ + l1 * w_))
self.assertAlmostEqual(angle, angle_ref)
h0, k0, l0 = np.array([2, 3]), np.array([3, 2]), np.array([-4, 4])
h1, k1, l1 = np.array([-3, 3]), np.array([2, 2]), np.array([-1, 4])
angle = crystal.interplanar(a, b, c, alpha, beta, gamma, \
h0, k0, l0, h1, k1, l1)
angle_ref = np.zeros(2)
for i in range(2):
dot = np.dot(h0[i] * u_ + k0[i] * v_ + l0[i] * w_,
h1[i] * u_ + k1[i] * v_ + l1[i] * w_)
norm0 = np.linalg.norm(h0[i] * u_ + k0[i] * v_ + l0[i] * w_)
norm1 = np.linalg.norm(h1[i] * u_ + k1[i] * v_ + l1[i] * w_)
if (np.isclose(norm0, 0) or np.isclose(norm1, 0)):
angle_ref[i] = 0
elif np.isclose(dot / norm0 / norm1, 1):
angle_ref[i] = 0
else:
angle_ref[i] = np.arccos(dot / (norm0 * norm1))
np.testing.assert_array_almost_equal(angle, angle_ref)
def crystal_reciprocal_matrices(self, a, b, c, alpha, beta, gamma):
constants = a, b, c, alpha, beta, gamma
inv_constants = crystal.reciprocal(*constants)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
A_ = crystal.cartesian(*inv_constants)
B_ = crystal.cartesian(*constants)
R_ = crystal.cartesian_rotation(*inv_constants)
C_ = crystal.cartesian_moment(*inv_constants)
D_ = crystal.cartesian_displacement(*inv_constants)
return A_, B_, R_, C_, D_
def crystal_matrices(self, a, b, c, alpha, beta, gamma):
constants = a, b, c, alpha, beta, gamma
inv_constants = crystal.reciprocal(*constants)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
A = crystal.cartesian(*constants)
B = crystal.cartesian(*inv_constants)
R = crystal.cartesian_rotation(*constants)
C = crystal.cartesian_moment(*constants)
D = crystal.cartesian_displacement(*constants)
return A, B, R, C, D
def test_cartesian_rotation(self):
a, b, c, alpha, beta, gamma = 5, 6, 7, np.pi / 2, np.pi / 3, np.pi / 4
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
hkl = np.array([-3.3, 1.1, 2.5])
uvw = np.array([4.2, -2.7, 1.8])
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
self.assertAlmostEqual(hkl.dot(uvw), A.dot(uvw).dot(R.dot(B.dot(hkl))))
def test_vector(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
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
h, k, l = -3, 1, 2
Qh, Qk, Ql = crystal.vector(h, k, l, B)
d = crystal.d(a, b, c, alpha, beta, gamma, h, k, l)
Q = np.sqrt(Qh**2 + Qk**2 + Ql**2)
self.assertAlmostEqual(d, 2 * np.pi / Q)
def test_cartesian(self):
a, b, c, alpha, beta, gamma = 5, 6, 7, np.pi / 2, np.pi / 3, np.pi / 4
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
G = crystal.metric(a, b, c, alpha, beta, gamma)
U = scipy.linalg.cholesky(G, lower=False)
np.testing.assert_array_almost_equal(A, U)
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
G_ = crystal.metric(a_, b_, c_, alpha_, beta_, gamma_)
U_ = scipy.linalg.cholesky(G_, lower=False)
np.testing.assert_array_almost_equal(B, U_)
def test_cartesian_displacement(self):
a, b, c, alpha, beta, gamma = 5, 5, 7, np.pi / 2, np.pi / 2, 2 * np.pi / 3
D = crystal.cartesian_displacement(a, b, c, alpha, beta, gamma)
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
U = np.array([[1, 0.1, 0.2], [0.1, 3, 0.3], [0.2, 0.3, 4]])
Up, _ = np.linalg.eig(np.dot(np.dot(D, U), D.T))
Uiso = np.mean(Up).real
Viso = (U[0,0]*(a*a_)**2+U[1,1]*(b*b_)**2+U[2,2]*(c*c_)**2+\
2*U[0,1]*a*a_*b*b_*np.cos(gamma)+\
2*U[0,2]*a*a_*c*c_*np.cos(beta)+\
2*U[1,2]*b*b_*c*c_*np.cos(alpha))/3
self.assertAlmostEqual(Uiso, Viso)
def test_d(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
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
u_, v_, w_ = np.dot(B, [1, 0, 0]), np.dot(B, [0, 1, 0]), np.dot(
B, [0, 0, 1])
h, k, l = 1, 2, -3
d = crystal.d(a, b, c, alpha, beta, gamma, h, k, l)
d_ref = 1 / np.sqrt(
np.dot(h * u_ + k * v_ + l * w_, h * u_ + k * v_ + l * w_))
self.assertAlmostEqual(d, d_ref)
h, k, l = np.array([2, 3]), np.array([3, 2]), np.array([-4, 4])
d = crystal.d(a, b, c, alpha, beta, gamma, h, k, l)
d_ref = np.zeros(2)
for i in range(2):
inv_d = np.sqrt(
np.dot(h[i] * u_ + k[i] * v_ + l[i] * w_,
h[i] * u_ + k[i] * v_ + l[i] * w_))
if np.isclose(inv_d, 0):
d_ref[i] = np.nan
else:
d_ref[i] = 1 / inv_d
np.testing.assert_array_almost_equal(d, d_ref)
def test_reciprocal(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
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
u, v, w = np.dot(A,
[1, 0, 0]), np.dot(A,
[0, 1, 0]), np.dot(A, [0, 0, 1])
V = np.dot(u, np.cross(v, w))
self.assertAlmostEqual(a_, np.linalg.norm(np.cross(v, w) / V))
self.assertAlmostEqual(b_, np.linalg.norm(np.cross(w, u) / V))
self.assertAlmostEqual(c_, np.linalg.norm(np.cross(u, v) / V))
u_, v_, w_ = np.dot(B, [1, 0, 0]), np.dot(B, [0, 1, 0]), np.dot(
B, [0, 0, 1])
self.assertAlmostEqual(
alpha_,
np.arccos(
np.dot(v_, w_) / np.linalg.norm(v_) / np.linalg.norm(w_)))
self.assertAlmostEqual(
beta_,
np.arccos(
np.dot(w_, u_) / np.linalg.norm(w_) / np.linalg.norm(u_)))
self.assertAlmostEqual(
gamma_,
np.arccos(
np.dot(u_, v_) / np.linalg.norm(u_) / np.linalg.norm(v_)))
def reciprocal_lattice_parameters(self, a, b, c, alpha, beta, gamma):
return crystal.reciprocal(a, b, c, alpha, beta, gamma)
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(['Fe3+', 'Mn3+'])
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])
T = space.debye_waller(h_range, k_range, l_range, nh, nk, nl, U11, U22,
U33, U23, U13, U12, a_, b_, c_)
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)
Sx, Sy, Sz = magnetic.spin(nu, nv, nw, n_atm)
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)
form_factor = magnetic.form(Q, atm, g=2)
Sx_k, Sy_k, Sz_k, i_dft = magnetic.transform(Sx, Sy, Sz, H, K, L, nu,
nv, nw, n_atm)
factors = space.prefactors(form_factor, phase_factor, occupancy)
factors *= T
I = magnetic.intensity(Qx_norm, Qy_norm, Qz_norm, \
Sx_k, Sy_k, Sz_k, i_dft, factors)
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]
mf = form_factor.reshape(n_hkl, n_atm)
T = T.reshape(n_hkl, n_atm)
Sx_i, Sx_j, Sx_m = Sx[i], Sx[j], Sx[m]
Sy_i, Sy_j, Sy_m = Sy[i], Sy[j], Sy[m]
Sz_i, Sz_j, Sz_m = Sz[i], Sz[j], Sz[m]
c_k, c_l, c_n = occupancy[k], occupancy[l], occupancy[n]
f_k, f_l, f_n = mf[:, k], mf[:, l], mf[:, n]
T_k, T_l, T_n = T[:, k], T[:, l], T[:, n]
Q_norm_dot_S_i = Qx_norm[:,np.newaxis]*Sx_i\
+ Qy_norm[:,np.newaxis]*Sy_i\
+ Qz_norm[:,np.newaxis]*Sz_i
Q_norm_dot_S_j = Qx_norm[:,np.newaxis]*Sx_j\
+ Qy_norm[:,np.newaxis]*Sy_j\
+ Qz_norm[:,np.newaxis]*Sz_j
Q_norm_dot_S_m = Qx_norm[:,np.newaxis]*Sx_m\
+ Qy_norm[:,np.newaxis]*Sy_m\
+ Qz_norm[:,np.newaxis]*Sz_m
Sx_perp_i = Sx_i - (Q_norm_dot_S_i) * Qx_norm[:, np.newaxis]
Sx_perp_j = Sx_j - (Q_norm_dot_S_j) * Qx_norm[:, np.newaxis]
Sx_perp_m = Sx_m - (Q_norm_dot_S_m) * Qx_norm[:, np.newaxis]
Sy_perp_i = Sy_i - (Q_norm_dot_S_i) * Qy_norm[:, np.newaxis]
Sy_perp_j = Sy_j - (Q_norm_dot_S_j) * Qy_norm[:, np.newaxis]
Sy_perp_m = Sy_m - (Q_norm_dot_S_m) * Qy_norm[:, np.newaxis]
Sz_perp_i = Sz_i - (Q_norm_dot_S_i) * Qz_norm[:, np.newaxis]
Sz_perp_j = Sz_j - (Q_norm_dot_S_j) * Qz_norm[:, np.newaxis]
Sz_perp_m = Sz_m - (Q_norm_dot_S_m) * Qz_norm[:, np.newaxis]
I_ref = ((c_n**2*(f_n*f_n.conj()).real*T_n**2*\
(Sx_perp_m**2+Sy_perp_m**2+Sz_perp_m**2)).sum(axis=1)\
+ 2*(c_k*c_l*(f_k*f_l.conj()).real*T_k*T_l*\
(Sx_perp_i*Sx_perp_j+Sy_perp_i*Sy_perp_j+Sz_perp_i*Sz_perp_j)*\
np.cos(Qx[:,np.newaxis]*rx_ij+\
Qy[:,np.newaxis]*ry_ij+\
Qz[:,np.newaxis]*rz_ij)).sum(axis=1))/n_xyz
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(['Fe3+', 'Mn3+'])
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])
T = space.debye_waller(h_range, k_range, l_range, nh, nk, nl, U11, U22,
U33, U23, U13, U12, a_, b_, c_)
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)
Sx, Sy, Sz = magnetic.spin(nu, nv, nw, n_atm)
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)
form_factor = magnetic.form(Q, atm, g=2)
Sx_k, Sy_k, Sz_k, i_dft = magnetic.transform(Sx, Sy, Sz, H, K, L, nu,
nv, nw, n_atm)
factors = space.prefactors(form_factor, phase_factor, occupancy)
factors *= T
Fx, Fy, Fz, \
prod_x, prod_y, prod_z = magnetic.structure(Qx_norm, Qy_norm, Qz_norm,
Sx_k, Sy_k, Sz_k, 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]
mf = form_factor.reshape(n_hkl, n_atm)
T = T.reshape(n_hkl, n_atm)
Sx_m = Sx[m]
Sy_m = Sy[m]
Sz_m = Sz[m]
c_n = occupancy[n]
f_n = mf[:, n]
T_n = T[:, n]
prod_x_ref = (c_n*f_n*Sx_m*T_n*np.exp(1j*(Qx[:,np.newaxis]*rx_m+\
Qy[:,np.newaxis]*ry_m+\
Qz[:,np.newaxis]*rz_m)))
prod_y_ref = (c_n*f_n*Sy_m*T_n*np.exp(1j*(Qx[:,np.newaxis]*rx_m+\
Qy[:,np.newaxis]*ry_m+\
Qz[:,np.newaxis]*rz_m)))
prod_z_ref = (c_n*f_n*Sz_m*T_n*np.exp(1j*(Qx[:,np.newaxis]*rx_m+\
Qy[:,np.newaxis]*ry_m+\
Qz[:,np.newaxis]*rz_m)))
Fx_ref = prod_x_ref.sum(axis=1)
Fy_ref = prod_y_ref.sum(axis=1)
Fz_ref = prod_z_ref.sum(axis=1)
prod_x_ref = prod_x_ref.reshape(n_hkl, nu * nv * nw, n_atm).sum(axis=1)
prod_y_ref = prod_y_ref.reshape(n_hkl, nu * nv * nw, n_atm).sum(axis=1)
prod_z_ref = prod_z_ref.reshape(n_hkl, nu * nv * nw, n_atm).sum(axis=1)
prod_x_ref = prod_x_ref.flatten()
prod_y_ref = prod_y_ref.flatten()
prod_z_ref = prod_z_ref.flatten()
np.testing.assert_array_almost_equal(Fx, Fx_ref)
np.testing.assert_array_almost_equal(Fy, Fy_ref)
np.testing.assert_array_almost_equal(Fz, Fz_ref)
np.testing.assert_array_almost_equal(prod_x, prod_x_ref)
np.testing.assert_array_almost_equal(prod_y, prod_y_ref)
np.testing.assert_array_almost_equal(prod_z, prod_z_ref)
def test_charge_dipole_matrix(self):
a = b = c = 5.4187
alpha = beta = gamma = np.pi / 2
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
nu, nv, nw, n_atm = 5, 5, 5, 12
n = nu * nv * nw * n_atm
atm = np.array(
['S', 'S', 'S', 'S', 'S', 'S', 'S', 'S', 'Fe', 'Fe', 'Fe', 'Fe'])
x = 0.385
u = np.array([
x, 1 - x, 0.5 - x, 0.5 + x, x, 1 - x, 0.5 + x, 0.5 - x, 0.0, 0.0,
0.5, 0.5
])
v = np.array([
x, 1 - x, 0.5 + x, 0.5 - x, 0.5 - x, 0.5 + x, x, 1 - x, 0.0, 0.5,
0.0, 0.5
])
w = np.array([
x, 1 - x, x, 1 - x, 0.5 + x, 0.5 - x, 0.5 - x, 0.5 + x, 0.0, 0.5,
0.5, 0.0
])
Rx, Ry, Rz = space.cell(nu, nv, nw, A)
ux, uy, uz = crystal.transform(u, v, w, A)
rx, ry, rz, atms = space.real(ux, uy, uz, Rx, Ry, Rz, atm)
z = np.zeros(n)
px = np.zeros(n)
py = np.zeros(n)
pz = np.zeros(n)
i, j = np.triu_indices(n)
Qijk = np.zeros((n, n, 3))
Qijk[i,
j, :] = interaction.charge_dipole_matrix(rx, ry, rz, nu, nv, nw,
n_atm, A, B, R)
Qijk[j, i, :] = Qijk[i, j, :]
z[atms == 'S'] = -1
z[atms == 'Fe'] = +2
px.reshape(nu, nv, nw, n_atm)[:, :, :, [0, 3, 4, 6]] = +np.sqrt(3) / 3
px.reshape(nu, nv, nw, n_atm)[:, :, :, [1, 2, 5, 7]] = -np.sqrt(3) / 3
py.reshape(nu, nv, nw, n_atm)[:, :, :, [0, 2, 5, 6]] = +np.sqrt(3) / 3
py.reshape(nu, nv, nw, n_atm)[:, :, :, [1, 3, 4, 7]] = -np.sqrt(3) / 3
pz.reshape(nu, nv, nw, n_atm)[:, :, :, [0, 2, 4, 7]] = +np.sqrt(3) / 3
pz.reshape(nu, nv, nw, n_atm)[:, :, :, [1, 3, 5, 6]] = -np.sqrt(3) / 3
p = np.column_stack((px, py, pz))
am = np.array([-1.957, -7.458])
ad = np.array([-1.184, -2.898])
bm = 2.632
bd = -2.561
np.testing.assert_array_almost_equal(Qijk, np.swapaxes(Qijk, 0, 1))
E = -np.einsum('i,i->...', z, np.einsum('ijk,jk->i', Qijk, p)) / n
self.assertAlmostEqual(E, -2 * (bm / a**2 + bd / a**3) / a, places=1)
Qij = np.zeros((n, n))
Qij[i, j] = interaction.charge_charge_matrix(rx, ry, rz, nu, nv, nw,
n_atm, A, B, R)
Qij[j, i] = Qij[i, j]
E = np.dot(z, np.dot(Qij, z)) / n
self.assertAlmostEqual(E, (2 * am[0] + am[1]) / a * 2 / 3, places=2)
Qijm = np.zeros((n, n, 6))
Qijkl = np.zeros((n, n, 3, 3))
Qijm[i,
j, :] = interaction.dipole_dipole_matrix(rx, ry, rz, nu, nv, nw,
n_atm, A, B, R)
Qijm[j, i, :] = Qijm[i, j, :]
Qijkl[:, :, 0, 0] = Qijm[:, :, 0]
Qijkl[:, :, 1, 1] = Qijm[:, :, 1]
Qijkl[:, :, 2, 2] = Qijm[:, :, 2]
Qijkl[:, :, 1, 2] = Qijkl[:, :, 2, 1] = Qijm[:, :, 3]
Qijkl[:, :, 0, 2] = Qijkl[:, :, 2, 0] = Qijm[:, :, 4]
Qijkl[:, :, 0, 1] = Qijkl[:, :, 1, 0] = Qijm[:, :, 5]
E = np.einsum('ik,ik->...', p, np.einsum('ijkl,jl->ik', Qijkl, p)) / n
self.assertAlmostEqual(E,
-(2 * ad[0] + ad[1]) / a**3 * 2 / 3,
places=2)
def test_structural(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 = [-5, 5], 11
k_range, nk = [-5, 5], 11
l_range, nl = [-5, 5], 11
nu, nv, nw, n_atm = 5, 5, 5, 3
u = np.array([0.2, 0.1, 0.3])
v = np.array([0.3, 0.4, 0.2])
w = np.array([0.4, 0.5, 0.1])
atm = np.array(['Fe', 'Mn', 'Co'])
occupancy = np.array([0.75, 0.5, 0.6])
U11 = np.array([0.5, 0.3, 0.1])
U22 = np.array([0.6, 0.4, 0.3])
U33 = np.array([0.4, 0.6, 0.2])
U23 = np.array([0.05, -0.03, -0.01])
U13 = np.array([-0.04, 0.02, 0.02])
U12 = np.array([0.03, -0.02, 0.01])
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)
D = crystal.cartesian_displacement(a, b, c, alpha, beta, gamma)
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)
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)
I = monocrystal.structural(occupancy, U11, U22, U33, U23, U13, U12, ux,
uy, uz, atm, h_range, k_range, l_range,
indices, symop, W, B, R, D, twins, variants,
nh, nk, nl, nu, nv, nw, Nu, Nv, Nw)
hmin, hmax = h_range
kmin, kmax = k_range
lmin, lmax = l_range
h, k, l = np.meshgrid(np.arange(hmin, hmax + 1),
np.arange(kmin, kmax + 1),
np.arange(lmin, lmax + 1),
indexing='ij')
h, k, l = h.flatten(), k.flatten(), l.flatten()
Qh, Qk, Ql = crystal.vector(h, k, l, B)
Q = np.sqrt(Qh**2 + Qk**2 + Ql**2)
n_hkl = Q.size
phase_factor = np.exp(2j * np.pi *
(h[:, np.newaxis] * u + k[:, np.newaxis] * v +
l[:, np.newaxis] * w))
scattering_power = scattering.length(atm, n_hkl).reshape(n_hkl, n_atm)
T = np.exp(-2 * np.pi**2 *
(U11 * (h * a_)[:, np.newaxis]**2 + U22 *
(k * b_)[:, np.newaxis]**2 + U33 *
(l * c_)[:, np.newaxis]**2 + U23 *
(k * l * b_ * c_ * 2)[:, np.newaxis] + U13 *
(h * l * a_ * c_ * 2)[:, np.newaxis] + U12 *
(h * k * a_ * b_ * 2)[:, np.newaxis]))
factors = scattering_power * occupancy * T * phase_factor
F = factors.sum(axis=1)
I_ref = np.abs(F)**2 / (nu * nv * nw * n_atm)
np.testing.assert_array_almost_equal(I, I_ref)
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_magnetic(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(['Fe3+', 'Mn3+'])
occupancy = np.array([0.75, 0.5])
g = np.array([2., 2.])
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)
T = space.debye_waller(h_range, k_range, l_range, nh, nk, nl, U11, U22,
U33, U23, U13, U12, a_, b_, c_)
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)
D = crystal.cartesian_displacement(a, b, c, alpha, beta, gamma)
Sx, Sy, Sz = magnetic.spin(nu, nv, nw, n_atm)
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)
form_factor = magnetic.form(Q, atm, g=g)
Sx_k, Sy_k, Sz_k, i_dft = magnetic.transform(Sx, Sy, Sz, H, K, L, nu,
nv, nw, n_atm)
factors = space.prefactors(form_factor, phase_factor, occupancy)
factors *= T
I_ref = magnetic.intensity(Qx_norm, Qy_norm, Qz_norm, Sx_k, Sy_k, Sz_k,
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)
I = monocrystal.magnetic(Sx, Sy, Sz, occupancy, U11, U22, U33, U23,
U13, U12, ux, uy, uz, atm, h_range, k_range,
l_range, indices, symop, W, B, R, D, twins,
variants, nh, nk, nl, nu, nv, nw, Nu, Nv, Nw,
g)
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_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])
T = space.debye_waller(h_range, k_range, l_range, nh, nk, nl, U11, U22,
U33, U23, U13, U12, a_, b_, c_)
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)
A_r = occupational.composition(nu, nv, nw, n_atm, value=occupancy)
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)
A_k, i_dft = occupational.transform(A_r, H, K, L, nu, nv, nw, n_atm)
factors = space.prefactors(scattering_length, phase_factor, occupancy)
factors *= T
F, prod = occupational.structure(A_k, 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)
T = T.reshape(n_hkl, n_atm)
A_m = A_r[m]
c_n = occupancy[n]
b_n = bc[n]
T_n = T[:, n]
prod_ref = (c_n*b_n*A_m*T_n*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)
def test_dipole_dipole_matrix(self):
a = b = c = 4.04
alpha = beta = gamma = np.pi / 2
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
nu, nv, nw, n_atm = 8, 8, 16, 1
n = nu * nv * nw * n_atm
atm = np.array(['Ti'])
u = np.array([0.5])
v = np.array([0.5])
w = np.array([0.5])
Rx, Ry, Rz = space.cell(nu, nv, nw, A)
ux, uy, uz = crystal.transform(u, v, w, A)
rx, ry, rz, atms = space.real(ux, uy, uz, Rx, Ry, Rz, atm)
px = np.zeros(n)
py = np.zeros(n)
pz = np.zeros(n)
i, j = np.triu_indices(n)
Qijm = np.zeros((n, n, 6))
Qijkl = np.zeros((n, n, 3, 3))
Qijm[i,
j, :] = interaction.dipole_dipole_matrix(rx, ry, rz, nu, nv, nw,
n_atm, A, B, R)
Qijm[j, i, :] = Qijm[i, j, :]
Qijkl[:, :, 0, 0] = Qijm[:, :, 0]
Qijkl[:, :, 1, 1] = Qijm[:, :, 1]
Qijkl[:, :, 2, 2] = Qijm[:, :, 2]
Qijkl[:, :, 1, 2] = Qijkl[:, :, 2, 1] = Qijm[:, :, 3]
Qijkl[:, :, 0, 2] = Qijkl[:, :, 2, 0] = Qijm[:, :, 4]
Qijkl[:, :, 0, 1] = Qijkl[:, :, 1, 0] = Qijm[:, :, 5]
np.testing.assert_array_almost_equal(Qijkl, np.swapaxes(Qijkl, 0, 1))
np.testing.assert_array_almost_equal(Qijkl, np.swapaxes(Qijkl, 2, 3))
px.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
py.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
pz.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 1
p = np.column_stack((px, py, pz))
E = np.einsum('ijkl,jl->ik', Qijkl, p) * a**3 / 2
np.testing.assert_array_almost_equal(E[..., 2], -2.09440, 2)
px.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
py.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
pz.reshape(nu, nv, nw, n_atm)[:, :, 0::2, :] = +1
pz.reshape(nu, nv, nw, n_atm)[:, :, 1::2, :] = -1
p = np.column_stack((px, py, pz))
E = np.einsum('ijkl,jl->ik', Qijkl, p)[..., 2] * a**3 / 2
np.testing.assert_array_almost_equal(E[..., 0::2], +4.84372, 2)
np.testing.assert_array_almost_equal(E[..., 1::2], -4.84372, 2)
px.reshape(nu, nv, nw, n_atm)[:, :, 0::2, :] = +1
px.reshape(nu, nv, nw, n_atm)[:, :, 1::2, :] = -1
py.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
pz.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
p = np.column_stack((px, py, pz))
E = np.einsum('ijkl,jl->ik', Qijkl, p)[..., 0] * a**3 / 2
np.testing.assert_array_almost_equal(E[0::2], -2.42186, 2)
np.testing.assert_array_almost_equal(E[1::2], +2.42186, 2)
px.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
py.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
pz.reshape(nu, nv, nw, n_atm)[0::2, 0::2, :, :] = +1
pz.reshape(nu, nv, nw, n_atm)[1::2, 1::2, :, :] = +1
pz.reshape(nu, nv, nw, n_atm)[0::2, 1::2, :, :] = -1
pz.reshape(nu, nv, nw, n_atm)[1::2, 0::2, :, :] = -1
p = np.column_stack((px, py, pz))
E = np.einsum('ijkl,jl->ik', Qijkl, p)[..., 2] * a**3 / 2
E = E.reshape(nu, nv, nw, n_atm)
np.testing.assert_array_almost_equal(E[0::2, 0::2, :, :], -2.67679, 2)
np.testing.assert_array_almost_equal(E[1::2, 1::2, :, :], -2.67679, 2)
np.testing.assert_array_almost_equal(E[0::2, 1::2, :, :], +2.67679, 2)
np.testing.assert_array_almost_equal(E[1::2, 0::2, :, :], +2.67679, 2)
px.reshape(nu, nv, nw, n_atm)[0::2, 0::2, :, :] = +1
px.reshape(nu, nv, nw, n_atm)[1::2, 1::2, :, :] = +1
px.reshape(nu, nv, nw, n_atm)[0::2, 1::2, :, :] = -1
px.reshape(nu, nv, nw, n_atm)[1::2, 0::2, :, :] = -1
py.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
pz.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
p = np.column_stack((px, py, pz))
E = np.einsum('ijkl,jl->ik', Qijkl, p)[..., 0] * a**3 / 2
E = E.reshape(nu, nv, nw, n_atm)
np.testing.assert_array_almost_equal(E[0::2, 0::2, :, :], +1.33839, 2)
np.testing.assert_array_almost_equal(E[1::2, 1::2, :, :], +1.33839, 2)
np.testing.assert_array_almost_equal(E[0::2, 1::2, :, :], -1.33839, 2)
np.testing.assert_array_almost_equal(E[1::2, 0::2, :, :], -1.33839, 2)
px.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
py.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
pz.reshape(nu, nv, nw, n_atm)[0::2, 0::2, 0::2, :] = +1
pz.reshape(nu, nv, nw, n_atm)[1::2, 1::2, 0::2, :] = +1
pz.reshape(nu, nv, nw, n_atm)[0::2, 1::2, 1::2, :] = +1
pz.reshape(nu, nv, nw, n_atm)[1::2, 0::2, 1::2, :] = +1
pz.reshape(nu, nv, nw, n_atm)[1::2, 1::2, 1::2, :] = -1
pz.reshape(nu, nv, nw, n_atm)[1::2, 0::2, 0::2, :] = -1
pz.reshape(nu, nv, nw, n_atm)[0::2, 1::2, 0::2, :] = -1
pz.reshape(nu, nv, nw, n_atm)[0::2, 0::2, 1::2, :] = -1
p = np.column_stack((px, py, pz))
E = np.einsum('ijkl,jl->ik', Qijkl, p) * a**3 / 2
np.testing.assert_array_almost_equal(E, 0, decimal=2)
a = c = 2 * 4.04
b = 4.04
alpha = beta = gamma = np.pi / 2
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
nu, nv, nw, n_atm = 8, 8, 8, 2
n = nu * nv * nw * n_atm
atm = np.array(['Ti', 'Ti'])
u = np.array([0.0, 0.5])
v = np.array([0.0, 0.0])
w = np.array([0.0, 0.5])
Rx, Ry, Rz = space.cell(nu, nv, nw, A)
ux, uy, uz = crystal.transform(u, v, w, A)
rx, ry, rz, atms = space.real(ux, uy, uz, Rx, Ry, Rz, atm)
px = np.zeros(n)
py = np.zeros(n)
pz = np.zeros(n)
i, j = np.triu_indices(n)
Qijm = np.zeros((n, n, 6))
Qijkl = np.zeros((n, n, 3, 3))
Qijm[i,
j, :] = interaction.dipole_dipole_matrix(rx, ry, rz, nu, nv, nw,
n_atm, A, B, R)
Qijm[j, i, :] = Qijm[i, j, :]
Qijkl[:, :, 0, 0] = Qijm[:, :, 0]
Qijkl[:, :, 1, 1] = Qijm[:, :, 1]
Qijkl[:, :, 2, 2] = Qijm[:, :, 2]
Qijkl[:, :, 1, 2] = Qijkl[:, :, 2, 1] = Qijm[:, :, 3]
Qijkl[:, :, 0, 2] = Qijkl[:, :, 2, 0] = Qijm[:, :, 4]
Qijkl[:, :, 0, 1] = Qijkl[:, :, 1, 0] = Qijm[:, :, 5]
px.reshape(nu, nv, nw, n_atm)[0::2, :, 0::2, :] = -np.sqrt(2) / 2
px.reshape(nu, nv, nw, n_atm)[1::2, :, 1::2, :] = -np.sqrt(2) / 2
px.reshape(nu, nv, nw, n_atm)[0::2, :, 1::2, :] = +np.sqrt(2) / 2
px.reshape(nu, nv, nw, n_atm)[1::2, :, 0::2, :] = +np.sqrt(2) / 2
py.reshape(nu, nv, nw, n_atm)[:, :, :, :] = 0
pz.reshape(nu, nv, nw, n_atm)[0::2, :, 0::2, :] = +np.sqrt(2) / 2
pz.reshape(nu, nv, nw, n_atm)[1::2, :, 1::2, :] = +np.sqrt(2) / 2
pz.reshape(nu, nv, nw, n_atm)[0::2, :, 1::2, :] = -np.sqrt(2) / 2
pz.reshape(nu, nv, nw, n_atm)[1::2, :, 0::2, :] = -np.sqrt(2) / 2
p = np.column_stack((px, py, pz))
E = np.einsum('ijkl,jl->ik', Qijkl, p) * a**3 / 16
E = np.sqrt(np.sum(E**2, axis=1))
np.testing.assert_array_almost_equal(E, 2.93226, 1)
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)
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])
T = space.debye_waller(h_range, k_range, l_range, nh, nk, nl, U11, U22,
U33, U23, U13, U12, a_, b_, c_)
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)
A_r = occupational.composition(nu, nv, nw, n_atm, value=occupancy)
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)
A_k, i_dft = occupational.transform(A_r, H, K, L, nu, nv, nw, n_atm)
factors = space.prefactors(scattering_length, phase_factor, occupancy)
factors *= T
I = occupational.intensity(A_k, i_dft, factors)
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)
T = T.reshape(n_hkl, n_atm)
A_i, A_j, A_m = A_r[i], A_r[j], A_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]
T_k, T_l, T_n = T[:, k], T[:, l], T[:, n]
I_ref = ((c_n**2*(b_n*b_n.conj()).real*A_m**2*T_n**2).sum(axis=1)\
+ 2*(c_k*c_l*(b_k*b_l.conj()).real*A_i*A_j*T_k*T_l*\
np.cos(Qx[:,np.newaxis]*rx_ij+\
Qy[:,np.newaxis]*ry_ij+\
Qz[:,np.newaxis]*rz_ij)).sum(axis=1))/n_xyz
np.testing.assert_array_almost_equal(I, I_ref)
def test_charge_charge_matrix(self):
a = b = c = 4.1
alpha = beta = gamma = np.pi / 2
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
nu, nv, nw, n_atm = 4, 4, 4, 2
n = nu * nv * nw * n_atm
atm = np.array(['Cs', 'Cl'])
u = np.array([0.0, 0.5])
v = np.array([0.0, 0.5])
w = np.array([0.0, 0.5])
Rx, Ry, Rz = space.cell(nu, nv, nw, A)
ux, uy, uz = crystal.transform(u, v, w, A)
rx, ry, rz, atms = space.real(ux, uy, uz, Rx, Ry, Rz, atm)
z = np.zeros(n)
i, j = np.triu_indices(n)
Qij = np.zeros((n, n))
Qij[i, j] = interaction.charge_charge_matrix(rx, ry, rz, nu, nv, nw,
n_atm, A, B, R)
Qij[j, i] = Qij[i, j]
z[atms == 'Cs'] = +1
z[atms == 'Cl'] = -1
np.testing.assert_array_almost_equal(Qij, Qij.T)
phi = np.dot(Qij, z) * a * np.sqrt(3) / 2
np.testing.assert_array_almost_equal(phi[atms == 'Cs'], -1.762675, 2)
np.testing.assert_array_almost_equal(phi[atms == 'Cl'], +1.762675, 2)
a = b = c = 5.7
alpha = beta = gamma = np.pi / 2
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
nu, nv, nw, n_atm = 4, 4, 4, 8
n = nu * nv * nw * n_atm
atm = np.array(['Na', 'Na', 'Na', 'Na', 'Cl', 'Cl', 'Cl', 'Cl'])
u = np.array([0.0, 0.0, 0.5, 0.5, 0.5, 0.5, 0.0, 0.0])
v = np.array([0.0, 0.5, 0.0, 0.5, 0.0, 0.5, 0.0, 0.5])
w = np.array([0.0, 0.5, 0.5, 0.0, 0.0, 0.5, 0.5, 0.0])
Rx, Ry, Rz = space.cell(nu, nv, nw, A)
ux, uy, uz = crystal.transform(u, v, w, A)
rx, ry, rz, atms = space.real(ux, uy, uz, Rx, Ry, Rz, atm)
z = np.zeros(n)
i, j = np.triu_indices(n)
Qij = np.zeros((n, n))
Qij[i, j] = interaction.charge_charge_matrix(rx, ry, rz, nu, nv, nw,
n_atm, A, B, R)
Qij[j, i] = Qij[i, j]
z[atms == 'Na'] = +1
z[atms == 'Cl'] = -1
phi = np.dot(Qij, z) * a / 2
np.testing.assert_array_almost_equal(phi[atms == 'Na'], -1.747565, 2)
np.testing.assert_array_almost_equal(phi[atms == 'Cl'], +1.747565, 2)
a = b = 3.819
c = 6.246
alpha = beta = np.pi / 2
gamma = 2 * np.pi / 3
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
nu, nv, nw, n_atm = 6, 6, 6, 4
n = nu * nv * nw * n_atm
atm = np.array(['Zn', 'Zn', 'S', 'S'])
u = np.array([2 / 3, 1 / 3, 2 / 3, 1 / 3])
v = np.array([1 / 3, 2 / 3, 1 / 3, 2 / 3])
w = np.array([0.0, 0.5, 0.625, 0.125])
Rx, Ry, Rz = space.cell(nu, nv, nw, A)
ux, uy, uz = crystal.transform(u, v, w, A)
rx, ry, rz, atms = space.real(ux, uy, uz, Rx, Ry, Rz, atm)
z = np.zeros(n)
i, j = np.triu_indices(n)
Qij = np.zeros((n, n))
Qij[i, j] = interaction.charge_charge_matrix(rx, ry, rz, nu, nv, nw,
n_atm, A, B, R)
Qij[j, i] = Qij[i, j]
z[atms == 'Zn'] = +2
z[atms == 'S'] = -2
phi = np.dot(Qij, z) * c * 0.375
np.testing.assert_array_almost_equal(phi[atms == 'Zn'], -3.28146, 2)
np.testing.assert_array_almost_equal(phi[atms == 'S'], +3.28146, 2)
a = b = c = 5.52
alpha = beta = gamma = np.pi / 2
inv_constants = crystal.reciprocal(a, b, c, alpha, beta, gamma)
a_, b_, c_, alpha_, beta_, gamma_ = inv_constants
A = crystal.cartesian(a, b, c, alpha, beta, gamma)
R = crystal.cartesian_rotation(a, b, c, alpha, beta, gamma)
B = crystal.cartesian(a_, b_, c_, alpha_, beta_, gamma_)
nu, nv, nw, n_atm = 5, 5, 5, 12
n = nu * nv * nw * n_atm
atm = np.array(
['Ca', 'Ca', 'Ca', 'Ca', 'F', 'F', 'F', 'F', 'F', 'F', 'F', 'F'])
u = np.array(
[0, 0, 0.5, 0.5, 0.25, 0.25, 0.25, 0.25, 0.75, 0.75, 0.75, 0.75])
v = np.array(
[0, 0.5, 0, 0.5, 0.75, 0.25, 0.25, 0.75, 0.75, 0.25, 0.25, 0.75])
w = np.array(
[0, 0.5, 0.5, 0, 0.75, 0.75, 0.25, 0.25, 0.25, 0.25, 0.75, 0.75])
Rx, Ry, Rz = space.cell(nu, nv, nw, A)
ux, uy, uz = crystal.transform(u, v, w, A)
rx, ry, rz, atms = space.real(ux, uy, uz, Rx, Ry, Rz, atm)
z = np.zeros(n)
i, j = np.triu_indices(n)
Qij = np.zeros((n, n))
Qij[i, j] = interaction.charge_charge_matrix(rx, ry, rz, nu, nv, nw,
n_atm, A, B, R)
Qij[j, i] = Qij[i, j]
z[atms == 'Ca'] = +2
z[atms == 'F'] = -1
phi = np.dot(Qij, z) * a * 0.25 * np.sqrt(3)
np.testing.assert_array_almost_equal(phi[atms == 'Ca'], -3.276110, 2)
np.testing.assert_array_almost_equal(phi[atms == 'F'], +1.762675, 2)
