# A comprehensive set of UV and x-ray radiative transition rates for Fe XVI

Abstract: Sodium-like Fe XVI is observed in collisionally ionized plasmas such as stellar coronae and coronal line regions of active galactic nuclei including black hole-accretion disc environments. Given its recombination edge from neon-like Fe XVII at ~\!25 A, the Fe XVI bound–bound transitions lie in the soft x-ray and EUV (extreme ultraviolet) range. We present a comprehensive set of theoretical transition rates for radiative dipole allowed E1 transitions including fine structure for levels with nl(SLJ)≤10, l≤9 using the relativistic Breit–Pauli R-matrix (BPRM) method. In addition, forbidden transitions of electric quadrupole (E2), electric octupole (E3), magnetic dipole (M1) and magnetic quadrupole (M2) type are presented for levels up to 5g(SLJ) from relativistic atomic structure calculations in the Breit–Pauli approximation using code SUPERSTRUCTURE. Some of the computed levels are autoionizing, and oscillator strengths among those are also provided. BPRM results have been benchmarked with the relativistic coupled cluster method and the atomic structure Dirac–Fock code GRASP. Levels computed with the electron collision BPRM codes in bound state mode were identified with a procedure based on the analysis of quantum defects and asymptotic wavefunctions. The total number of Fe XVI levels considered is 96, with 822 E1 transitions. Tabulated values are presented for the oscillator strengths f, line strengths S and Einstein radiative decay rates A. This extensive dataset should enable spectral modelings up to highly excited levels, including recombination-cascade matrices.

## Summary (2 min read)

### 1. Introduction

- Highly charged iron ions exist in a variety of high-temperature astrophysical sources emitting or absorbing radiation from the optical to the x-ray range.
- Spectral lines of Ne-like Fe XVII are signatures of most coronal plasmas.
- Generally, these are obtained from large-scale calculations of high accuracy, such as those being carried out under the Iron Project (IP: [11]), and extensions thereof such as the RmaX Network (viz. www.astronomy.ohio-state/∼nahar).
- Similarly, the authors plan to investigate the accuracy and completeness of spectral models of other important ionization stages, namely the Na-, Ne- and F-like iron ions.

### 2. Formulation

- The relativistic Breit–Pauli R-matrix method with close coupling (CC) approximation, described in a number of papers [2, 3, 11, 22, 23, 26], enables calculation of a large number of fine structure E1 transitions with high accuracy.
- (3) The quantities 8 j are correlation wavefunctions composed of (N + 1) electrons orbital functions that (a) compensate for the orthogonality conditions between the continuum and the bound orbitals and (b) represent additional short-range correlation that is often of crucial importance in scattering and radiative CC calculations for each SLπ .
- The mutual spin–orbit and spin–other–orbit effects (so + so′) by core electrons enter only via (Blume–Watson) screening of spin–orbit parameters, as (Fe XVII+e) is effectively a one body rather than an (N + 1) electron system, and all spin–spin interaction (ss′) among electrons drops out for an unpolarized Ne-like core.
- Complications by L-shell polarization correlation are discussed in a later section.

### 3.1. BPRM calculations for E1 transitions

- With the BPRM code, the CC calculations proceed in several stages.
- STG2 computes angular coefficients for target and collisional channels in L S-coupling and target term energies.
- A demanding step in the calculations is the theoretical level identification, since the computed bound levels are initially known only by their negative eigenvalues or effective quantum numbers.
- All transitions are processed for proper energies and transition wavelengths using code PBPRAD.

### 3.2. Atomic structure calculations for forbidden transitions

- Radiative transition probabilities for forbidden transitions in Fe XVI were obtained from configuration mixing atomic structure calculations using SS, which computes multipole transitions in Breit–Pauli approximation.
- The forbidden transitions considered are electric quadrupole (E2) and octupole (E3), and magnetic dipole (M1) and quadrupole (M2).
- The computed data have been processed by replacing the calculated energies by the limited number of observed energies available, using the code PRCSS.

### 4. Results and discussion

- Oscillator strengths ( f -values), line strengths S and radiative transition probabilities A of Fe XVI are presented for allowed electric dipole and intercombination transitions E1, and for forbidden E2, E3, M1 and M2 transitions.
- The large set of atomic parameters should comprise a reasonably complete set for all practical applications.
- The results for energy levels and radiative transition rates are discussed separately.

### 4.1. Energy levels

- Table 1 presents a partial set of binding energies and comparison with available observed energy levels compiled by NIST.
- In their case excited target levels start at about 35 Ry above (Fe XVII) 1S, and from there bound collisional electrons with n > 3 never drop below the target ground state (and a few allowed electrons associated with Rydberg n = 2 remain).
- [K]nl clearly lead to significantly overestimated FS splittings because of neglected magnetic shielding by L-shell electrons.
- This line is followed by energy levels of the same configurations; Nlv(c) at the end specifies total number of calculated J -levels found for the set.
- It may be noted that levels in the table are grouped consistently in energies and effective quantum numbers, also confirming consistent level identification.

### 4.2. Oscillator strengths for allowed E1 transitions

- The 92 bound levels of Fe XVI yield 822 dipole allowed E1 (same-spin multiplet) and intercombination (spin-change multiplet) transitions.
- Whereas they differ only in factors of energies and dynamical constants, the authors provide all three not only for easy usage but also for self-consistency in converting one form to the other.
- The first two columns are level indices Ii and Ik whose identification can be found from the energy table 4; the third column is the transition wavelength λ.
- Calculated energies have been replaced by the observed energies wherever available.

### 4.3. Radiative decay rates for forbidden transitions

- Forbidden transitions are relatively weak and are observed mostly among low-lying levels.
- They provide important diagnostics for ambient conditions in many plasma sources.
- A total of 27083 transitions are obtained among the 230 fine structure levels.
- Comparison of these levels with the measured values (NIST compilation) shows agreement within 1%.
- The parity remains unchanged for the E2 and M1 transitions and hence they are presented together.

### 5. Conclusion

- An extensive set of parameters for radiative transitions is presented for both allowed and forbidden transitions in Fe XVI.
- Very good agreement, within 1%, is found between calculated and measured results.
- One main objective of the present work was to benchmark the BPRM method for radiative decay rates.
- The present results are expected to be accurate and complete enough for most astrophysical and laboratory applications.
- A for forbidden transitions are largely in agreement with those available at NIST and [5].

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