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dosrixs

dosrixs is a Python package for computing resonant inelastic X-ray scattering (RIXS) cross sections using projected dnesity of states (DOS) including polarization-dependent matrix elements which is based on the Kramers-Heisenebrg formalism. It is designed to help researchers interpet itinerant and charge-transfer features in experimental RIXS spectra. Our implementation is based on M. Norman et al., Phys. Rev. B 107, 165124 (2023).

Features

  • Compute RIXS spectra

$$ \tilde{\sigma}(E_{\mathrm{in}}, E_{\mathrm{out}}, \epsilon) = \sum_{i,f,\sigma,\sigma',\epsilon^{\prime}} \int dE \rho_{f\sigma'}(E)\rho_{i\sigma}(E + E_{\mathrm{loss}}) \Big [ \frac{\Gamma}{2} \frac{M_{if\sigma\sigma'}(\epsilon,\epsilon')}{(E - E_{\mathrm{out}})^{2} + \Gamma^{2}/4}\Big ].$$

  • Compute X-ray absoprtion spectra (XAS)

$$ \tilde{\sigma}(E_{\mathrm{in}},\epsilon) = \sum_{i,\sigma} \int dE \frac{\Gamma}{2} \frac{\rho_{i\sigma}M_{i\sigma}(\epsilon)}{(E_{\mathrm{in}}-E)^{2}+ \Gamma^{2}/4}.$$

For more details, please see M. Norman et al., Phys. Rev. B 107, 165124 (2023) for more details on the notation.

Modular Design

Our implementation is based on the abstract expansion of objects in spherical harmonic basis (YlmExpansion). This enables the computation of the polarization matrix elements, $M_{if\sigma\sigma'}(\epsilon,\epsilon')$, for arbitrary polarizations, initial/final states, and core states.

Transition amplitudes and Polarizations

Within the dipole-dipole approximation, we need to compute the following matrix elements:

$$ \propto \sum_{c} \Big | \langle f|\epsilon_{\mathrm{out}} |c\rangle\langle c | \epsilon_{\mathrm{in}} | i\rangle \Big |^{2}$$

where:

  • $\epsilon_{\mathrm{in}}$, $\epsilon_{\mathrm{out}}$ are the incoming and outgoing polarization vectors,
  • $|c\rangle$ is a core state,
  • $|i\rangle$, $|f\rangle$ are valence states.

YlmExpansion

The valence and core states, as well as the polarizations are all written as expansions in a spherical harmonic basis as:

$$ |\psi\rangle = \sum_{m,\sigma} c_{m,\sigma}Y_{\ell}^{m} $$

where:

  • $\ell$ is the angular quantum number,
  • $m \in [-\ell, \ell]$ is the magnetic quantum number,
  • $\sigma \in {\uparrow,\downarrow }$ spins,
  • $c_{m,\sigma} \in \mathbb{C}$ are expansion coefficients.

The YlmExpansion object is the foundation of the entire implementation. Internally, we use Python dictionaries as the data storage.

Dipole transitions are Gaunt coefficients

The angular part of the dipole matrix element is calculated using Gaunt coefficients $G(\ell_{1}=1, m_{c}, \ell_{2}=1, m_{q}, \ell_{3}=2, m_{d})$:

$$ \langle Y_{\ell_{c}}^{m_{c}}| \hat{r_{q}} | Y_{\ell_{d}}^{m_{d}} \rangle = \int d\Omega Y_{\ell_{c}}^{m_{c}}(\theta,\phi) Y_{1}^{m_{q}}(\theta,\phi) Y_{\ell_{d}}^{m_{d}}(\theta,\phi) =G(m_{c},m_{q},m_{d})$$

Contributing

We welcome contributions! Please open issues for bugs or feature requests, or submit pull requests.

References

To cite this work, please include a reference to this GitHub repository, and cite the following paper:

M. R. Norman, A. S. Botana, J. Karp, A. Hampel, H. LaBollita, A. J. Millis, G. Fabbris, Y. Shen, and M. P. M. Dean. "Orbital polarization, charge-transfer, and fluorescence in reduced-valence nickelates." Phys. Rev. B 107, 165124 (2023).

License

MIT License