def Rot_AngAx(n, theta):
"""
Rotation matrix
Rodrigues formula
angle-axis parametrization
INPUT: n - vector [nx,ny,nz]
0 <= theta <= pi
OUTPUT: Rot - matrix(3x3)
nx,ny,nz = symbols('nx,ny,nz')
n = [nx,ny,nz]
"""
dimens = 3
"""
R = sp.MatrixSymbol('R', dimens, dimens)
"""
i, j = sp.symbols('i,j')
R = sp.FunctionMatrix(dimens, dimens,
sp.Lambda((i, j),
cos(theta) * KroneckerDelta(i, j)))
for k in range(dimens):
R -= sp.FunctionMatrix(
dimens, dimens, sp.Lambda((i, j),
sin(theta) * Eijk(i, j, k) * n[k]))
Rot = sp.Matrix(R)
R2 = sp.zeros(dimens)
for i in range(dimens):
R2.row_op(i, lambda v, j: (1 - cos(theta)) * n[i] * n[j])
Rot += R2
Rot = sp.N(Rot, 4)
#Rot = sp.matrix2numpy(Rot)
return Rot
def test_Eijk():
assert Eijk(1, 2, 3) == 1
assert Eijk(2, 3, 1) == 1
assert Eijk(3, 2, 1) == -1
assert Eijk(1, 1, 2) == 0
assert Eijk(1, 3, 1) == 0
assert Eijk(1, x, x) == 0
assert Eijk(x, y, x) == 0
def test_levicivita():
assert Eijk(1, 2, 3) == LeviCivita(1, 2, 3)
assert LeviCivita(1, 2, 3) == 1
assert LeviCivita(1, 3, 2) == -1
assert LeviCivita(1, 2, 2) == 0
i, j, k = symbols('i j k')
assert LeviCivita(i, j, k) == LeviCivita(i, j, k, evaluate=False)
assert LeviCivita(i, j, i) == 0
assert LeviCivita(1, i, i) == 0
assert LeviCivita(i, j, k).doit() == (j - i) * (k - i) * (k - j) / 2
assert LeviCivita(1, 2, 3, 1) == 0
assert LeviCivita(4, 5, 1, 2, 3) == 1
assert LeviCivita(4, 5, 2, 1, 3) == -1
def test_levicivita():
assert Eijk(1, 2, 3) == LeviCivita(1, 2, 3)
assert LeviCivita(1, 2, 3) == 1
assert LeviCivita(1, 3, 2) == -1
assert LeviCivita(1, 2, 2) == 0
i, j, k = symbols('i j k')
assert LeviCivita(i, j, k) == LeviCivita(i, j, k, evaluate=False)
assert LeviCivita(i, j, i) == 0
assert LeviCivita(1, i, i) == 0
assert LeviCivita(i, j, k).doit() == (j - i)*(k - i)*(k - j)/2
assert LeviCivita(1, 2, 3, 1) == 0
assert LeviCivita(4, 5, 1, 2, 3) == 1
assert LeviCivita(4, 5, 2, 1, 3) == -1
assert LeviCivita(i, j, k).is_integer is True
assert adjoint(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
assert conjugate(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
assert transpose(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
def L_loop_hall(data_controller, temp, smearing, ene, velkp, t_tensor, alpha,
ispin):
from scipy.constants import hbar
from sympy import Eijk
from .smearing import gaussian, metpax
from os.path import join
arrays, attributes = data_controller.data_dicts()
esize = ene.size
snktot = arrays['E_k'].shape[0]
bohr2m = 5.29177e-11
ev2J = 1.60218e-19
bnd = attributes['bnd']
nspin = attributes['nspin']
kq_wght = 1. / attributes['nkpnts']
if smearing is not None and smearing != 'gauss' and smearing != 'm-p':
print('%s Smearing Not Implemented.' % smearing)
comm.Abort()
L_hall = np.zeros((3, 3, 3, esize), dtype=float)
sig_hall = np.zeros((3, 3, 3, snktot, bnd, nspin))
M_inv = np.zeros((3, 3, snktot, bnd, nspin))
eff_mass_inv = arrays['d2Ed2k']
for l in range(t_tensor.shape[0]):
i = t_tensor[l][0]
j = t_tensor[l][1]
if i == j:
M_inv[i, j] = eff_mass_inv[i]
elif i == 0 and j == 1:
M_inv[i, j] = eff_mass_inv[3]
M_inv[j, i] = eff_mass_inv[3]
elif i == 0 and j == 2:
M_inv[i, j] = eff_mass_inv[4]
M_inv[j, i] = eff_mass_inv[4]
elif i == 1 and j == 2:
M_inv[i, j] = eff_mass_inv[5]
M_inv[j, i] = eff_mass_inv[5]
for n in range(bnd):
Eaux = np.reshape(np.repeat(arrays['E_k'][:, n, ispin], esize),
(snktot, esize))
delk = (np.reshape(np.repeat(arrays['deltakp'][:, n, ispin], esize),
(snktot, esize)) if smearing != None else None)
if smearing is None:
Eaux -= ene
smearA = 1 / (4 * temp * (np.cosh(Eaux / (2 * temp))**2))
else:
if smearing == 'gauss':
smearA = gaussian(Eaux, ene, delk)
elif smearing == 'm-p':
smearA = metpax(Eaux, ene, delk)
for i in range(3):
for j in range(3):
for p in range(3):
for q in range(3):
for r in range(3):
sig_hall[i, j, p, :, n,
ispin] += (int(Eijk(p, q, r)) *
velkp[:, i, n, ispin] *
velkp[:, r, n, ispin] *
M_inv[j, q, :, n, ispin])
L_hall[i, j, p, :] += np.sum(
kq_wght * arrays['scattering_tau'][:, n, ispin]**2 *
sig_hall[i, j, p, :, n, ispin] * (smearA).T,
axis=1)
return L_hall
from sympy import Eijk, LeviCivita from numpy.random import permutation print(LeviCivita(3, 1, 2)) print(Eijk(3, 1, 2, 5, 4))