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Published September 19, 2023 | Version 1.2
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Generalised oscillator strength for core-shell electron excitation by fast electrons based on Dirac solutions

Creators

  • 1. EMAT, University of Antwerp & Department of Materials, University of Oxford
  • 2. Rosalind Franklin Institute
  • 3. Bio21 Institute
  • 4. EMAT, University of Antwerp
  • 5. Department of Materials, University of Oxford

Description

The rich information of electron energy-loss spectroscopy (EELS) comes from the complex inelastic scattering process whereby fast electrons transfer energy and momentum to atoms, exciting bound electrons from their ground states to higher unoccupied states. To quantify EELS, the common practice is to compare the cross-sections integrated within an energy window or fit the observed spectrum with theoretical differential cross-sections calculated from a generalized oscillator strength (GOS) database with experimental parameters [1].
 
The previous Hartree-Fock-based [2] or DFT-based [3] GOS was calculated from Schrödinger's solution of atomic orbitals, which does not include the full relativistic effects. Here, we attempt to go beyond the limitations of the Schrödinger solution in the GOS tabulation by including the full relativistic effects using the Dirac equation within the local density approximation using FAC [4], which is particularly important for core-shell electrons of heavy elements with strong spin-orbit coupling. This has been done for all elements in the periodic table (up to Z = 118) for all possible excitation edges using modern computing capabilities and parallelization algorithms. The relativistic effects of fast incoming electrons were included to calculate cross-sections that are specific to the acceleration voltage. We make these tabulated GOS available under an open-source license to the benefit of both academic users as well as allowing integration into commercial solutions.

Database Details:

- Covers all elements (Z: 1-108) and all edges

- Large energy range: 0.01 - 4000 eV

- Large momentum range: from minimum momentum transfer to double Bethe ridge for each edge. Adaptive momentum sampling is developed in such a manner to maximize the physical information for a given finite number of sampling points. For example, for C edge this range is 0.14 -67 Å-1  

- Fine log sampling: 128 points for energy and 256 points for momentum

- Data format: GOSH [3]

 

Calculation Details:

- Single atoms only; solid-state effects are not considered

- Unoccupied states before continuum states of ionization are not considered; no fine structure

- Plane Wave Born Approximation

- Frozen Core Approximation is employed; electrostatic potential remains unchanged for orthogonal states when a core-shell electron is excited

- Self-consistent Dirac–Fock–Slater iteration is used for Dirac calculations; A modified local density approximation is used for the correct asymptotic behavior of the exchange energy; continuum states are normalized against asymptotic form at large distances

- Both large and small component contributions of Dirac solutions are included in GOS

- Final state contributions are included until the contribution of the last states falls below 0.1%. A convergence log is provided for reference.

 

Version 1.1 release note:

- Update to be consistent with GOSH data format [3], all the edges are now within a single hdf5 file. A notable change in particular, the sampling in momentum is in 1/m, instead of previously in 1/Å. Great thanks to Gulio Guzzinati for his suggestions and sending conversion script. 

Version 1.2 release note:

- Add “File Type / File version” information

[1] Verbeeck, J., and S. Van Aert. Ultramicroscopy 101.2-4 (2004): 207-224.

[2] Leapman, R. D., P. Rez, and D. F. Mayers. The Journal of Chemical Physics 72.2 (1980): 1232-1243.

[3] Segger, L, Guzzinati, G, & Kohl, H. Zenodo (2023). doi:10.5281/zenodo.7645765

[4] Gu, M. F. Canadian Journal of Physics 86(5) (2008): 675-689.

Notes

The authors acknowledge financial support from the Research Foundation Flanders (FWO, Belgium) through Project No.G.0502.18N. This project has also received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 770887 PICOMETRICS and No. 823717 ESTEEM3).

Files

calculation_log.zip

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Additional details

Funding

PICOMETRICS – Picometer metrology for light-element nanostructures: making every electron count 770887
European Commission
ESTEEM3 – Enabling Science and Technology through European Electron Microscopy 823717
European Commission