Quickstart tutorial¶
If you are already familiar with multiplet calculations this page and our examples page are a good part to start. Otherwise, checkout our Pedagogical examples to learn the concepts behind how edrixs works.
Use edrixs as an ED calculator¶
edrixs can be used as a simple ED calculator to get eigenvalues (eigenvectors) of a many-body Hamiltonian with small size dimension (\(< 1,000\)). We will give an example to get eigenvalues for a \(t_{2g}\)-orbital system (\(l_{eff}=1\)). There are 6 orbitals including spin.
Launch your favorite python terminal:
>>> import edrixs
>>> import scipy
>>> norb = 6
>>> noccu = 2
>>> Ud, JH = edrixs.UJ_to_UdJH(4, 1)
>>> F0, F2, F4 = edrixs.UdJH_to_F0F2F4(Ud, JH)
>>> umat = edrixs.get_umat_slater('t2g', F0, F2, F4)
>>> emat = edrixs.atom_hsoc('t2g', 0.2)
>>> basis = edrixs.get_fock_bin_by_N(norb, noccu)
>>> H = edrixs.build_opers(4, umat, basis)
>>> e1, v1 = scipy.linalg.eigh(H)
>>> H += edrixs.build_opers(2, emat, basis)
>>> e2, v2 = scipy.linalg.eigh(H)
>>> print(e1)
[1. 1. 1. 1. 1. 1. 1. 1. 1. 3. 3. 3. 3. 3. 6.]
>>> print(e2)
[0.890519 0.890519 0.890519 0.890519 0.890519 1.1 1.1 1.1
1.183391 3.009481 3.009481 3.009481 3.009481 3.009481 6.016609]
Hello RIXS!¶
This is a “Hello World!” example for RIXS simulations at Ni (\(3d^8\)) \(L_{2/3}\) edges. \(L_3\) means transition from Ni-\(2p_{3/2}\) to Ni-\(3d\), and \(L_2\) means transition from Ni-\(2p_{1/2}\) to Ni-\(3d\).
#!/usr/bin/env python
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
import edrixs
# Setup parameters
# ----------------
# Number of occupancy of 3d shell
noccu = 8
res = edrixs.get_atom_data('Ni', v_name='3d', v_noccu=noccu, edge='L23')
name_i, slat_i = [list(i) for i in zip(*res['slater_i'])]
name_n, slat_n = [list(i) for i in zip(*res['slater_n'])]
# Slater integrals for initial Hamiltonian without core-hole
si = edrixs.rescale(slat_i, ([1, 2], [0.65]*2))
si[0] = edrixs.get_F0('d', si[1], si[2]) # F0_dd
# Slater integrals for intermediate Hamiltonian with core-hole
sn = edrixs.rescale(slat_n, ([1, 2, 4, 5, 6], [0.65, 0.65, 0.95, 0.7, 0.7]))
sn[0] = edrixs.get_F0('d', sn[1], sn[2]) # F0_dd
sn[3] = edrixs.get_F0('dp', sn[5], sn[6]) # F0_dp
slater = (si, sn)
# Spin-orbit coupling strengths
zeta_d_i = res['v_soc_i'][0] # valence 3d electron without core-hole
zeta_d_n = res['v_soc_n'][0] # valence 3d electron with core-hole
# E_{L2} - E_{L3} = 1.5 * zeta_p
zeta_p_n = (res['edge_ene'][0] - res['edge_ene'][1]) / 1.5 # core 2p electron
# Tetragonal crystal field
cf = edrixs.cf_tetragonal_d(ten_dq=1.3, d1=0.05, d3=0.2)
# Level shift of the core shell
off = 857.4
# Life time broadening
gamma_c = res['gamma_c'][1] # core hole
gamma_f = 0.1 # final states
# Incident, scattered, azimuthal angles
# See Figure 1 of Y. Wang et al.,
# `Computer Physics Communications 243, 151-165 (2019)
# <https://doi.org/10.1016/j.cpc.2019.04.018>`_
# for the defintion of the scattering angles.
thin, thout, phi = 15 / 180.0 * np.pi, 75 / 180.0 * np.pi, 0.0
# Polarization types
poltype_xas = [('isotropic', 0.0)] # for XAS
poltype_rixs = [('linear', 0, 'linear', 0), ('linear', 0, 'linear', np.pi/2.0)] # for RIXS
# Energy grid
ominc_xas = np.linspace(off - 10, off + 20, 1000) # for XAS
ominc_rixs_L3 = np.linspace(-5.9 + off, -0.9 + off, 100) # incident energy at L3 edge
ominc_rixs_L2 = np.linspace(10.4 + off, 14.9 + off, 100) # incident energy at L3 edge
eloss = np.linspace(-0.5, 5.0, 1000) # energy loss for RIXS
# Run ED
eval_i, eval_n, trans_op = edrixs.ed_1v1c_py(
('d', 'p'), shell_level=(0.0, -off), v_soc=(zeta_d_i, zeta_d_n),
c_soc=zeta_p_n, v_noccu=noccu, slater=slater, v_cfmat=cf
)
# Run XAS
xas = edrixs.xas_1v1c_py(
eval_i, eval_n, trans_op, ominc_xas, gamma_c=gamma_c, thin=thin, phi=phi,
pol_type=poltype_xas, gs_list=[0, 1, 2], temperature=300
)
# Run RIXS at L3 edge
rixs_L3 = edrixs.rixs_1v1c_py(
eval_i, eval_n, trans_op, ominc_rixs_L3, eloss, gamma_c=gamma_c, gamma_f=gamma_f,
thin=thin, thout=thout, phi=phi, pol_type=poltype_rixs, gs_list=[0, 1, 2],
temperature=300
)
# Run RIXS at L2 edge
rixs_L2 = edrixs.rixs_1v1c_py(
eval_i, eval_n, trans_op, ominc_rixs_L2, eloss, gamma_c=gamma_c, gamma_f=gamma_f,
thin=thin, thout=thout, phi=phi, pol_type=poltype_rixs, gs_list=[0, 1, 2],
temperature=300
)
# Plot
fig = plt.figure(figsize=(16, 14))
mpl.rcParams['font.size'] = 20
ax1 = plt.subplot(2, 2, 1)
plt.grid()
plt.plot(range(len(eval_i)), eval_i - min(eval_i), '-o')
plt.xlabel(r'Multiplets')
plt.ylabel(r'Energy (eV)')
plt.title(r'(a) Energy of multiplets')
ax2 = plt.subplot(2, 2, 2)
plt.grid()
plt.plot(ominc_xas, xas[:, 0], '-')
plt.xlabel(r'Incident Energy (eV)')
plt.ylabel(r'XAS Intensity (a.u.)')
plt.title(r'(b) Isotropic XAS')
ax3 = plt.subplot(2, 2, 3)
plt.imshow(np.sum(rixs_L3, axis=2),
extent=[min(eloss), max(eloss), min(ominc_rixs_L3), max(ominc_rixs_L3)],
origin='lower', aspect='auto', cmap='rainbow', interpolation='gaussian')
plt.xlabel(r'Energy loss (eV)')
plt.ylabel(r'Incident Energy (eV)')
plt.title(r'(c) RIXS map at $L_3$ edge (LH)')
ax4 = plt.subplot(2, 2, 4)
plt.imshow(np.sum(rixs_L2, axis=2),
extent=[min(eloss), max(eloss), min(ominc_rixs_L2), max(ominc_rixs_L2)],
origin='lower', aspect='auto', cmap='rainbow', interpolation='gaussian')
plt.xlabel(r'Energy loss (eV)')
plt.ylabel(r'Incident Energy (eV)')
plt.title(r'(d) RIXS map at $L_2$ edge (LH)')
plt.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9, wspace=0.2, hspace=0.3)
plt.show()
