.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/example_02_single_atom_RIXS.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_example_02_single_atom_RIXS.py: RIXS calculations for an atomic model ===================================== Here we show how to compute RIXS for a single site atomic model with crystal field and electron-electron interactions. We take the case of Sr\ :sub:`2`\ YIrO\ :sub:`6` from Ref. [1]_ as the material in question. The aim of this example is to illustrate the procedure and to provide what we hope is useful advice. What is written is not meant to be a replacement for reading the docstrings of the functions, which can always be accessed on the `edrixs website `_ or by executing functions with ?? in IPython. .. GENERATED FROM PYTHON SOURCE LINES 15-19 .. code-block:: Python import edrixs import numpy as np import matplotlib.pyplot as plt .. GENERATED FROM PYTHON SOURCE LINES 20-26 Specify active core and valence orbitals ------------------------------------------------------------------------------ Sr\ :sub:`2`\ YIrO\ :sub:`6`\ has a :math:`5d^4` electronic configuration and we want to calculate the :math:`L_3` edge spectrum i.e. resonating with a :math:`2p_{3/2}` core hole. We will start by including only the :math:`t_{2g}` valence orbitals. .. GENERATED FROM PYTHON SOURCE LINES 26-29 .. code-block:: Python shell_name = ('t2g', 'p32') v_noccu = 4 .. GENERATED FROM PYTHON SOURCE LINES 30-38 Slater parameters ------------------------------------------------------------------------------ Here we want to use Hund's interaction :math:`J_H` and spin orbit coupling :math:`\lambda` as adjustable parameters to match experiment. We will take the core hole interaction parameter from the Hartree Fock numbers EDRIXS has in its database. These need to be converted and arranged into the order required by EDRIXS. .. GENERATED FROM PYTHON SOURCE LINES 38-57 .. code-block:: Python Ud = 2 JH = 0.25 lam = 0.42 F0_d, F2_d, F4_d = edrixs.UdJH_to_F0F2F4(Ud, JH) info = edrixs.utils.get_atom_data('Ir', '5d', v_noccu, edge='L3') G1_dp = info['slater_n'][5][1] G3_dp = info['slater_n'][6][1] F0_dp = edrixs.get_F0('dp', G1_dp, G3_dp) F2_dp = info['slater_n'][4][1] slater_i = [F0_d, F2_d, F4_d] # Fk for d slater_n = [ F0_d, F2_d, F4_d, # Fk for d F0_dp, F2_dp, # Fk for dp G1_dp, G3_dp, # Gk for dp 0.0, 0.0 # Fk for p ] slater = [slater_i, slater_n] .. GENERATED FROM PYTHON SOURCE LINES 58-91 Diagonalization ------------------------------------------------------------------------------ The EDRIXS interface separates the physical model from the methods used to compute XAS or RIXS. :func:`~edrixs.models.model_1v1c` returns one-body matrices (:code:`emat_i` and :code:`emat_n`), Coulomb tensors (:code:`umat_i` and :code:`umat_n`), compact Fock-basis specifications (:code:`basis_i` and :code:`basis_n`), and Cartesian dipole matrices (:code:`trans_mat`). The impurity one-body matrix :code:`imp_mat` contains the valence spin--orbit coupling and, below, the crystal field. :code:`i` quantities describe the initial and final states without a core hole, while the :code:`n` quantities describe the intermediate state with a core hole. Both orbital-space Hamiltonians include the core, and basis_i explicitly fixes its occupancy to a full shell. The v_noccu argument counts only valence electrons. :func:`~edrixs.solvers.get_ops` converts these backend-independent ingredients into many-body initial/final and intermediate Hamiltonians, plus dipole operators that map the initial Fock space to the intermediate one. We choose the SciPy backend, which represents these many-body operators as sparse matrices. :func:`~edrixs.solvers.ed` then obtains the retained low-energy eigenpairs used by :func:`~edrixs.solvers.xas` and :func:`~edrixs.solvers.rixs`. Note that the calculation does not know the core hole energy, so we need to adjust the energy at which the resonance will appear by hand. We know empirically that the resonance is at 11215 eV, and there is a further few-eV shift from the Coulomb energy of adding the valence electrons that is not captured by the raw calculation. We fold both into a single offset :code:`off` and tune the small correction (here :math:`\approx6` eV) so the computed spectrum lines up with experiment. In this case we are assuming a perfectly cubic crystal field, which we have already implemented by specifying the use of the :math:`t_{2g}` subshell only, so we do not need to pass an additional :code:`v_cfmat` matrix. .. GENERATED FROM PYTHON SOURCE LINES 91-108 .. code-block:: Python backend = 'scipy' off = 11215 - 6 imp_mat = edrixs.atom_hsoc(shell_name[0], lam) out = edrixs.model_1v1c( shell_name, v_noccu=v_noccu, shell_level=(0, -off), v_othermat=imp_mat, slater=slater, ) emat_i, umat_i, basis_i, emat_n, umat_n, basis_n, trans_mat, shift = out hmat_i, hmat_n, trans_ops = edrixs.get_ops( emat_i, umat_i, basis_i, emat_n, umat_n, basis_n, trans_mat, backend=backend, ) eval_i, evec_i = edrixs.ed(hmat_i, shift=shift, num_evals=2, backend=backend) .. GENERATED FROM PYTHON SOURCE LINES 109-132 Compute XAS ------------------------------------------------------------------------------ To calculate XAS we need to correctly specify the orientation of the x-rays with respect to the sample. By default, the :math:`x, y, z` coordinates of the sample's crystal field, will be aligned with our lab frame, passing :code:`loc_axis` to :code:`model_1v1c` can be used to specify a different convention. The experimental geometry is specified following the angles shown in Figure 1 of Y. Wang et al., `Computer Physics Communications 243, 151-165 (2019) `_. The default setting has x-rays along :math:`z` for :math:`\theta=\pi/2` rad and the x-ray beam along :math:`-x` for :math:`\theta=\phi=0`. Parameter :code:`scatter_axis` can be passed to :code:`xas` to specify a different geometry if desired. Variable :code:`pol_type` specifies a list of different x-ray polarizations to calculate. Here we will use so-called :math:`\pi`-polarization where the x-rays are parallel to the plane spanned by the incident beam and the sample :math:`z`-axis. EDRIXS weights the retained low-energy eigenstates by their Boltzmann factors. The spectral broadening is dominated by the inverse core-hole lifetime :code:`gamma_c`, the Lorentzian half width at half maximum. .. GENERATED FROM PYTHON SOURCE LINES 132-147 .. code-block:: Python ominc = np.linspace(11200, 11230, 50) temperature = 300 # in K thin = 30*np.pi/180 phi = 0 pol_type = [('linear', 0)] gamma_c = info['gamma_c'][0] xas = edrixs.xas( eval_i, evec_i, hmat_n, trans_ops, ominc, gamma_c=gamma_c, thin=thin, phi=phi, pol_type=pol_type, temperature=temperature, backend=backend, ) .. GENERATED FROM PYTHON SOURCE LINES 148-161 Compute RIXS ------------------------------------------------------------------------------ Calculating RIXS is overall similar to XAS, but with a few additional considerations. The spectral width in the energy loss axis of RIXS it not set by the core hole lifetime, but by either the final state lifetime or the experimental resolution and is parameterized by :code:`gamma_f` -- the Lorentzian half width at half maximum. The angle and polarization of the emitted beam must also be specified, so we pass :code:`pol_type_rixs` to the function. Each entry is a 4-tuple that specifies both the incoming and the outgoing x-ray polarization. If, as is common in experiments, the emitted polarization is not resolved, one needs to add both outgoing polarization channels, which is what we do here. .. GENERATED FROM PYTHON SOURCE LINES 161-176 .. code-block:: Python eloss = np.linspace(-.5, 6, 400) pol_type_rixs = [('linear', 0, 'linear', 0), ('linear', 0, 'linear', np.pi/2)] thout = 60*np.pi/180 gamma_f = 0.02 rixs = edrixs.rixs( eval_i, evec_i, hmat_i, hmat_n, trans_ops, ominc, eloss, gamma_c=gamma_c, gamma_f=gamma_f, thin=thin, thout=thout, phi=phi, pol_type=pol_type_rixs, temperature=temperature, backend=backend, ) .. GENERATED FROM PYTHON SOURCE LINES 177-179 The array :code:`xas` will have shape :code:`(len(ominc), len(pol_type))` .. GENERATED FROM PYTHON SOURCE LINES 181-187 Plot XAS and RIXS ------------------------------------------------------------------------------ Let's plot everything. We will use a function so we can reuse the code later. Note that the rixs array :code:`rixs` has shape :code:`(len(ominc), len(eloss), len(pol_type))`. We will use some numpy tricks to sum over the two different emitted polarizations. .. GENERATED FROM PYTHON SOURCE LINES 187-221 .. code-block:: Python fig, axs = plt.subplots(2, 2, figsize=(10, 10)) def plot_it(axs, ominc, xas, eloss, rixscut, rixsmap=None, label=None): axs[0].plot(ominc, xas[:, 0], label=label) axs[0].set_xlabel('Energy (eV)') axs[0].set_ylabel('Intensity') axs[0].set_title('XAS') axs[1].plot(eloss, rixscut, label=f"{label}") axs[1].set_xlabel('Energy loss (eV)') axs[1].set_ylabel('Intensity') axs[1].set_title(f'RIXS at resonance') if rixsmap is not None: art = axs[2].pcolormesh(ominc, eloss, rixsmap.T, shading='auto') plt.colorbar(art, ax=axs[2], label='Intensity') axs[2].set_xlabel('Incident energy (eV)') axs[2].set_ylabel('Energy loss') axs[2].set_title('RIXS map') rixs_pol_sum = rixs.sum(-1) cut_index = np.argmax(rixs_pol_sum[:, eloss < 2].sum(1)) rixscut = rixs_pol_sum[cut_index] plot_it(axs.ravel(), ominc, xas, eloss, rixscut, rixsmap=rixs_pol_sum) axs[0, 1].set_xlim(right=3) axs[1, 0].set_ylim(top=3) axs[1, 1].remove() plt.show() .. image-sg:: /auto_examples/images/sphx_glr_example_02_single_atom_RIXS_001.png :alt: XAS, RIXS at resonance, RIXS map :srcset: /auto_examples/images/sphx_glr_example_02_single_atom_RIXS_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 222-229 Full d shell calculation ------------------------------------------------------------------------------ Some researchers have questioned the appropriateness of only including the :math:`t_{2g}` subshell for iridates [2]_. Let's test this. We specify that the full :math:`d` shell should be used and apply cubic crystal field matrix :code:`v_cfmat`. We shift the energy offset by :math:`\frac{2}{5}10D_q`, which is the amount the crystal field moves the :math:`t_{2g}` subshell. .. GENERATED FROM PYTHON SOURCE LINES 229-267 .. code-block:: Python ten_dq = 3.5 v_cfmat = edrixs.cf_cubic_d(ten_dq) off = 11215 - 6 + ten_dq*2/5 imp_mat = edrixs.atom_hsoc('d', lam) + v_cfmat out = edrixs.model_1v1c( ('d', 'p32'), v_noccu=v_noccu, shell_level=(0, -off), v_othermat=imp_mat, slater=slater, ) emat_i, umat_i, basis_i, emat_n, umat_n, basis_n, trans_mat, shift = out hmat_i, hmat_n, trans_ops = edrixs.get_ops( emat_i, umat_i, basis_i, emat_n, umat_n, basis_n, trans_mat, backend=backend, ) eval_i, evec_i = edrixs.ed(hmat_i, shift=shift, num_evals=2, backend=backend) xas_full_d_shell = edrixs.xas( eval_i, evec_i, hmat_n, trans_ops, ominc, gamma_c=gamma_c, thin=thin, phi=phi, pol_type=pol_type, temperature=temperature, backend=backend, ) rixs_full_d_shell = edrixs.rixs( eval_i, evec_i, hmat_i, hmat_n, trans_ops, np.array([11215]), eloss, gamma_c=gamma_c, gamma_f=gamma_f, thin=thin, thout=thout, phi=phi, pol_type=pol_type_rixs, temperature=temperature, backend=backend, ) fig, axs = plt.subplots(1, 2, figsize=(10, 4)) plot_it(axs, ominc, xas, eloss, rixscut, label='$t_{2g}$ subshell') rixscut = rixs_full_d_shell.sum((0, -1)) plot_it(axs, ominc, xas_full_d_shell, eloss, rixscut, label='$d$ shell') axs[0].legend() axs[1].legend() plt.show() .. image-sg:: /auto_examples/images/sphx_glr_example_02_single_atom_RIXS_002.png :alt: XAS, RIXS at resonance :srcset: /auto_examples/images/sphx_glr_example_02_single_atom_RIXS_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 268-272 As expected, we see the appearance of excitations on the energy scale of :math:`10D_q` in the XAS and RIXS. The low energy manifold is qualitatively, but not quantitatively similar. This makes it clear that the parameterization of Sr\ :sub:`2`\ YIrO\ :sub:`6`\ is dependent on the model. .. GENERATED FROM PYTHON SOURCE LINES 274-281 .. rubric:: Footnotes .. [1] Bo Yuan et al., `Phys. Rev. B 95, 235114 (2017) `_. .. [2] Georgios L. Stamokostas and Gregory A. Fiete `Phys. Rev. B 97, 085150 (2018) `_. .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.959 seconds) .. _sphx_glr_download_auto_examples_example_02_single_atom_RIXS.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: example_02_single_atom_RIXS.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: example_02_single_atom_RIXS.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: example_02_single_atom_RIXS.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_