[Paper Review] Validity of the ICFT R-matrix method: Be-like Al$^{9+}$ a case study

L. Fernández-Menchero, G. Del Zanna+1 more|Strathprints: The University of Strathclyde institutional repository (University of Strathclyde)|Mar 17, 2015

Atomic and Molecular Physics23 references3 citations

TL;DR

This study validates the Intermediate Coupling Frame Transformation (ICFT) R-matrix method by comparing 98-level CI/CC ICFT and Breit–Pauli R-matrix calculations for Be-like Al⁹⁺, showing excellent agreement and confirming the method's robustness. It refutes a recent conjecture questioning ICFT's reliability, demonstrating instead that discrepancies in prior work arose from insufficient close-coupling and configuration interaction expansion convergence, not flaws in ICFT itself.

ABSTRACT

We have carried-out 98-level configuration-interaction / close-coupling (CI/CC) intermediate coupling frame transformation (ICFT) and Breit-Pauli R-matrix calculations for the electron-impact excitation of Be-like Al$^{9+}$. The close agreement that we find between the two sets of effective collision strengths demonstrates the continued robustness of the ICFT method. On the other hand, a comparison of this data with previous 238-level CI/CC ICFT effective collision strengths shows that the results for excitation up to n=4 levels are systematically and increasingly underestimated over a wide range of temperatures by R-matrix calculations whose close-coupling expansion extends only to n=4 (98-levels). Thus, we find to be false a recent conjecture that the ICFT approach may not be completely robust. The conjecture was based upon a comparison of 98-level CI/CC Dirac R-matrix effective collision strengths for Al$^{9+}$ with those from the 238-level CI/CC ICFT R-matrix calculations. The disagreement found recently is due to a lack of convergence of the close-coupling expansion in the 98-level CI/CC Dirac work. The earlier 238-level CI/CC ICFT work has a superior target to the 98-level CI/CC Dirac one and provides more accurate atomic data. Similar considerations need to be made for other Be-like ions and for other sequences.

Motivation & Objective

• To assess the validity of the ICFT R-matrix method for electron-impact excitation in Be-like Al⁹⁺.
• To resolve discrepancies between 98-level Dirac R-matrix and 238-level ICFT R-matrix calculations reported by Aggarwal & Keenan (2015a).
• To determine whether the ICFT method is fundamentally flawed or if differences stem from incomplete convergence in close-coupling and configuration interaction expansions.
• To establish the most accurate and complete atomic data for Be-like ions by evaluating convergence effects in R-matrix calculations.
• To caution against drawing conclusions from comparisons of collision data using different atomic structures and close-coupling expansions.

Proposed method

• Performed 98-level configuration-interaction / close-coupling (CI/CC) intermediate coupling frame transformation (ICFT) R-matrix calculations for electron-impact excitation in Be-like Al⁹⁺.
• Conducted parallel Breit–Pauli R-matrix calculations using the same atomic structure and close-coupling expansion to directly compare ICFT and BP results.
• Compared the new 98-level ICFT results with previously published 238-level CI/CC ICFT R-matrix data and 98-level Dirac R-matrix data from Aggarwal & Keenan (2014c).
• Analyzed the impact of convergence in both configuration interaction and close-coupling expansions on effective collision strengths, particularly for n=4 levels.
• Evaluated the influence of relativistic effects by comparing ICFT and Breit–Pauli treatments using identical atomic structures.
• Assessed the relevance of the Storey et al. (2014) issue on O²⁺ to Al⁹⁺, concluding it is not applicable due to differences in ion charge and R-matrix box size.

Experimental results

Research questions

• RQ1 Are the ICFT and Breit–Pauli R-matrix methods in agreement for electron-impact excitation in Be-like Al⁹⁺ when using the same atomic structure?
• RQ2 Why do 98-level Dirac R-matrix calculations for Al⁹⁺ systematically underestimate effective collision strengths compared to 238-level ICFT R-matrix results?
• RQ3 To what extent do incomplete close-coupling and configuration interaction expansions affect the accuracy of effective collision strengths for higher-lying states (n=4)?
• RQ4 Is the recent conjecture that the ICFT method may not be robust valid, or is it due to convergence issues in prior calculations?
• RQ5 Can the discrepancy between 98-level and 238-level calculations be fully attributed to the size of the close-coupling expansion, or are other factors involved?

Key findings

• The 98-level CI/CC ICFT and Breit–Pauli R-matrix calculations for Al⁹⁺ show excellent agreement, with differences of less than 5% for most transitions, confirming the robustness of the ICFT method.
• The 238-level CI/CC ICFT R-matrix results are more accurate than the 98-level Dirac R-matrix calculations because they include a more complete close-coupling and configuration interaction expansion.
• Effective collision strengths for transitions to n=4 levels are systematically underestimated in 98-level calculations across a wide temperature range, including at peak abundance (1.8×10⁶ K).
• The discrepancy between 98-level Dirac and 238-level ICFT results is not due to flaws in the ICFT method but due to insufficient convergence in the close-coupling and configuration interaction expansions in the 98-level work.
• The issue raised by Storey et al. (2014) regarding O²⁺ does not apply to Al⁹⁺, as it is specific to low-charge ions and small R-matrix box sizes, which were not used in the 238-level ICFT calculations.
• Assigning a single accuracy rating (e.g., 20%) to an entire collision data set is misleading, as accuracy varies significantly with the target state and temperature, especially for higher-lying levels.