Theoretical ab initio Evolution of Satellite Intensity near Threshold for Cu K-shell transitions

Technical Summary: Theoretical Ab Initio Evolution of Satellite Intensity near Threshold for Cu K-shell Transitions

Problem Statement
Theoretical atomic structure calculations for complex systems often rely on the sudden approximation (SA), which assumes that ionization or nuclear decay occurs on a timescale much shorter than electronic relaxation. While the SA provides an excellent description of high-energy processes, its validity is questionable near ionization thresholds. In this low-energy regime, the outgoing electron possesses low kinetic energy and remains in the vicinity of the atomic system for a comparatively long time, violating the fundamental assumptions of the SA. Consequently, the system approaches an adiabatic limit where the electronic structure evolves continuously. Previous experimental observations of this effect, particularly in X-ray absorption spectroscopy (XAS), have been limited by resolution constraints. Furthermore, while theoretical models such as those by Thomas and Roy exist to predict the evolution of shake intensity (satellite lines) from the adiabatic to the sudden regime, there is a need to validate these models against high-resolution emission data and to determine if standard ab initio methods can accurately simulate this evolution without empirical fitting parameters for intensity.

Methodology
The authors employed a comprehensive ab initio approach using state-of-the-art Multiconfiguration Dirac-Fock (MCDF) methods, specifically utilizing the MCDFGME code. The methodology involved several key steps:

1. Atomic Structure Calculations: All Cu K-shell transitions, including radiative and non-radiative rates, were calculated. The atomic wave functions were optimized using the optimized-level method with full relaxation for radiative rates, while non-radiative rates utilized continuum-electron wavefunctions without relaxation.
2. Shake Probability Determination: The probability of shake-up and shake-off processes was calculated theoretically. The total shake probability was derived from the inner product of initial and final state radial wavefunctions. The authors distinguished between single-electron shake probabilities (using a binomial distribution approach) and total shake probabilities, accounting for monopolar excitations.
3. Synthetic Spectrum Generation: A synthetic spectrum for Cu K$\alpha_{1,2}$ transitions was generated by weighting all possible decay paths by their calculated rates and statistical weights.
4. Modeling Energy Evolution: To simulate the evolution of line intensity near the threshold, the authors developed a modulation function. This involved convolving the natural Lorentzian width of ionized levels with a Gaussian profile representing the excitation beam energy distribution. An "Effective Lorentzian Profile" was used to account for the asymmetric nature of the ionization cross-section near the threshold.
5. Incorporation of Resonant Excitations: Recognizing that ionization data alone could not explain experimental results in the near-threshold region, the authors incorporated resonant excitations from Cu(I) and Cu(II) oxide phases (specifically $1s \to 3d$ and $1s \to 4p$ transitions) identified via X-ray Absorption Near-Edge Structure (XANES) data.
6. Fitting and Analysis: Theoretical results were compared with experimental data from Galambosi et al. [8]. A Bayesian analysis using nested sampling (nested fit) was employed to determine the most likely energy shifts and width multipliers for different decay processes (diagram, shake-off, shake-up, and excitation) to align the simulation with experimental observations.

Key Contributions

• Validation of Ab Initio Methods: The study demonstrates that standard state-of-the-art ab initio methods, without adjustable intensity fitting parameters for ionization, can successfully simulate the evolution of satellite intensity near ionization thresholds.
• Quantification of Shake Probabilities: The authors calculated a total shake probability for Cu K-shell ionization of approximately 25.97%, with a breakdown of $\approx$13.08% for shake-off and $\approx$12.89% for shake-up.
• Identification of Threshold Origins: The work identifies that the "below-threshold" satellite intensity observed in experiments does not solely originate from shake processes but is significantly influenced by resonant excitations in Cu(I) ($1s \to 4p$) and Cu(II) ($1s \to 3d$) oxide phases.
• Refinement of Theoretical Models: The paper provides a rigorous test of the Thomas and Roy models. It confirms the functional form of the Thomas model for the adiabatic-to-sudden transition and highlights discrepancies in the Roy model's parameters when fitted to experimental data, suggesting the experimental data contains physical processes (such as solid-state effects) not fully captured by the simple atomic models.

Results

• Agreement with Experiment: The simulated intensity ratios of satellite lines to total K$\alpha_{1,2}$ intensity showed good agreement with experimental data for excitation energies above 9050 eV, where all decay channels are open. The simulated ratio at 10 keV was found to be $0.262 \pm 0.00004$, consistent with the total shake probability calculated in the sudden approximation.
• Threshold Behavior: Below 9050 eV, the ionization-only simulation failed to reproduce the experimental rise in satellite intensity. However, upon including resonant excitations from Cu oxides, the simulation successfully reproduced the experimental data in the 8975–9060 eV range.
• Model Fitting: Fitting the Thomas model to the simulation yielded parameters ($\Delta E$, $r$, $\mu_\infty$) that were in good agreement with both the experimental fit and theoretical calculations. The Roy model fit showed reasonable proximity to theoretical values but significant deviations in binding energy parameters, indicating the limitations of the model in capturing the full complexity of the experimental spectrum.
• Spectral Features: The inclusion of higher-order shake-up transitions (e.g., $3s$, $3p$, $2p$ shake-off and $3d$ shake-up) significantly altered the theoretical satellite line shape compared to previous Dirac-Fock calculations, highlighting the necessity of detailed shake calculations for high-resolution metrology.

Significance
The paper claims that its primary significance lies in demonstrating that standard theoretical calculations can predict the mean evolution of spectral intensity near ionization thresholds without relying on empirical intensity fitting. By successfully separating satellite intensity (originating from shake processes) from direct diagram intensity and identifying the contribution of resonant excitations in oxide phases, the authors provide a more complete physical picture of the near-threshold region. The work underscores the importance of detailed shake-up and shake-off calculations for state-of-the-art X-ray spectroscopy and suggests that discrepancies between theory and experiment in this regime can often be attributed to solid-state effects (such as the presence of oxides) and the specific line shapes used in data analysis, rather than a failure of the underlying atomic physics models.