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Application of the Variational R-matrix Method for the Dirac Equation to the Be Atom

Chris H. Greene, Miguel A. Alarcón

TL;DR

This work implements a non-iterative variational eigenchannel R-matrix method for the Dirac equation and applies it to the photoionization of beryllium. Using a B-spline-based two-electron basis and a streamlined open/closed channel formulation, the study reproduces known non-relativistic results and experimental data while revealing relativistic effects, such as the altered Fano line shape of the $2pnd$ series and a triplet–singlet coupling in the ground-state continuum. The results validate the relativistic R-matrix approach and demonstrate its capability to capture subtle spin-orbit-enabled couplings, paving the way for accurate predictions in heavier systems like radium and guiding future spectroscopic experiments. The methodology provides a robust framework for calculating cross sections and autoionizing resonances in relativistic atomic systems.

Abstract

This paper presents an implementation of the non-iterative eigenchannel R-matrix method for the Dirac equation. It includes a brief introduction, implementation details, and results for the photoionization cross-section of the beryllium atom. Beryllium is a convenient test due to small but significant relativistic effects. The current calculation aligns with other R-matrix calculations and experiments. It observes the change in the Fano line shape of the (2pnd)1P series and, reveals a previously unnoticed coupling between triplet and singlet series observable in the ground state photoionization cross-section.

Application of the Variational R-matrix Method for the Dirac Equation to the Be Atom

TL;DR

This work implements a non-iterative variational eigenchannel R-matrix method for the Dirac equation and applies it to the photoionization of beryllium. Using a B-spline-based two-electron basis and a streamlined open/closed channel formulation, the study reproduces known non-relativistic results and experimental data while revealing relativistic effects, such as the altered Fano line shape of the series and a triplet–singlet coupling in the ground-state continuum. The results validate the relativistic R-matrix approach and demonstrate its capability to capture subtle spin-orbit-enabled couplings, paving the way for accurate predictions in heavier systems like radium and guiding future spectroscopic experiments. The methodology provides a robust framework for calculating cross sections and autoionizing resonances in relativistic atomic systems.

Abstract

This paper presents an implementation of the non-iterative eigenchannel R-matrix method for the Dirac equation. It includes a brief introduction, implementation details, and results for the photoionization cross-section of the beryllium atom. Beryllium is a convenient test due to small but significant relativistic effects. The current calculation aligns with other R-matrix calculations and experiments. It observes the change in the Fano line shape of the (2pnd)1P series and, reveals a previously unnoticed coupling between triplet and singlet series observable in the ground state photoionization cross-section.
Paper Structure (12 sections, 47 equations, 4 figures, 3 tables)

This paper contains 12 sections, 47 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theoretical Background
  3. Two-electron basis
  4. Eigenchannel R-matrix method
  5. The Streamlined Method
  6. Physical Solutions, Dipole moments and Cross Sections
  7. Results for Beryllium
  8. Conclusions
  9. Acknowledgments
  10. Relativistic Quantum Defect Theory
  11. Single Channel Formulation
  12. Multichannel Formulation

Figures (4)

  • Figure 1: Photoionization cross section for the ground state of Be below the $1s^2 2p_{1/2}$ threshold. The relativistic and non-relativistic calculations are shown in the length (solid) and velocity gauges (dashed). Color version online. The dotted line is the data obtained from Kim2000.
  • Figure 2: The photoionization cross-section of the Be ground state between the $1s^2 2p_{1/2}$ and $1s^2 2p_{3/2}$ thresholds showcases the narrow $2p_{3/2} n d_{5/2}$ and broad $2p_{3/2} n s_{1/2}$ autoionizing states. The relativistic results are shown in the length (solid) and velocity (dashed) gauges. The non-relativistic calculations agree exactly with these calculations and for clarity are not shown.
  • Figure 3: The Be ground state photoionization cross section at final state energies just below the $1s^2 3s_{1/2}$ threshold. The relativistic and non-relativistic calculations are shown in the length (solid) and velocity gauges (dashed). Color version online. The points are the data obtained from Kim2000.
  • Figure 4: Photoionization cross section for the ground state of Be below the $1s^2 3d_{3/2}$ threshold. The relativistic and non-relativistic calculations are shown in length (solid) and velocity gauges (dashed), with the points denoting the $LS$ coupled calculation Kim2000. Color version online. Again, the non-relativistic calculation agrees exactly with the relativistic one, and for clarity is not shown.