[Paper Review] Tables of the partition functions for iron, Fe I -- Fe X

J. Halenka, J. Madej|arXiv (Cornell University)|Apr 23, 2002

Material Properties and Failure Mechanisms3 citations

TL;DR

This paper presents highly accurate, physically motivated tables of the atomic partition functions (APF) for all neutral to highly ionized iron species (Fe I–Fe X), computed using a comprehensive quantum mechanical approach that includes autoionization levels and accounts for plasma effects via electron concentration and ionization energy lowering (LIE). The tables span temperatures from 1,000 K to 1,000,000 K and LIE values from 0.001 to 5.0 eV, offering superior accuracy and completeness over prior approximations, with results available for use in stellar atmosphere modeling and spectral synthesis.

ABSTRACT

We present extensive tables of the Atomic Partition Function for iron ions, Fe I -- Fe X, and discuss details of the computational method. Partition functions are given in wide range of temperatures, 10^3 K < T < 10^6 K, and lowering of ionization energy (0.001 eV < LIE < 5.0 eV). Our APF take into account all energy levels predicted by quantum mechanics, including autoionization levels. The tables can be applied for the computations of model stellar atmospheres and theoretical spectra over all existing spectral and luminosity classes.

Motivation & Objective

• To provide accurate, complete, and physically consistent atomic partition functions (APF) for all iron ions from Fe I to Fe X, covering all astrophysically relevant conditions.
• To address the limitations of prior APF tables, which often omitted high-energy bound levels, autoionization states, or failed to account for plasma effects such as ionization energy lowering (LIE).
• To ensure compatibility with existing stellar atmosphere models and spectral synthesis codes by providing homogeneous, high-resolution data across a wide range of temperature and electron density conditions.
• To improve the accuracy of population and opacity calculations in stellar atmospheres, especially for cool stars (F-type and later) and hot white dwarfs, where iron plays a critical role in electron and ion population balance.
• To extend the methodology used for nickel (Ni I–Ni X) in a prior study to iron, ensuring consistency and reliability in atomic data for astrophysical modeling.

Proposed method

• The APF is computed using a modified partition function formalism that includes all bound energy levels predicted by quantum mechanics, including autoionization states, via the expression $ U^{(r)}(T,N_e) = \sum_{p=1}^{p_{\text{max}}} \sum_{i=1}^{i(p)_{\text{max}}} g^{(r)}_{pi} \exp(-E^{(r)}_{pi}/kT) $, where $ r $ denotes the ionization stage.
• The upper limit of bound levels is determined by the condition $ E^{(r)}_{pi} \leq E^{(r)}_{p\infty} - \Delta E^{(r)} $, where $ \Delta E^{(r)} $ is the lowering of ionization energy (LIE), which accounts for plasma screening effects.
• The method incorporates statistical weights $ g^{(r)}_{pi} = (2J_{pi}+1)G_{pi} $ and uses a hierarchical level sequence approach based on core quantum states $ p $ and valence electron quantum numbers $ nlj $.
• For level sequences with no known levels (group $ \gamma^{(r)} $), the contribution to the APF is estimated using a transfer formula based on the next ionization stage’s partition function and statistical weights.
• The partition function for a given ion is computed as the sum over all level sequences: $ \sum_{p \in \text{terms}} U^{(r)}_p = \sum_{p \in \text{terms}} U^{(r)}_p + \Delta U^{(r)}_p $, where $ \Delta U^{(r)}_p $ accounts for missing levels using a weighted ratio from the next ionization stage.
• The tables are generated at 62 non-equidistant temperature points from 10^3 K to 10^6 K and 9 fixed LIE values (0.001 to 5.0 eV), ensuring broad applicability across stellar atmospheres.

Experimental results

Research questions

• RQ1 How can the atomic partition function for iron ions be computed with full inclusion of autoionization and high-lying bound levels, while accounting for plasma effects?
• RQ2 What is the impact of ionization energy lowering (LIE) on the partition function across a wide range of temperatures and electron densities in stellar atmospheres?
• RQ3 How do the partition functions for Fe I–Fe X compare to those computed from observed levels only, and what is the significance of including unobserved or autoionizing states?
• RQ4 To what extent do the computed APF values diverge as LIE approaches zero, and how does this behavior align with theoretical expectations?
• RQ5 Can a consistent, physically grounded method for APF computation be applied uniformly across multiple ionization stages of iron, ensuring compatibility with existing models and data sets?

Key findings

• The computed APF values for Fe IV show a significant increase—up to a factor of 2–3—compared to those based solely on observed levels, especially at low LIE and high temperatures.
• The inclusion of autoionization levels and unobserved high-lying bound states leads to a strong divergence of APF as LIE → 0, consistent with theoretical expectations in the limit of zero plasma screening.
• The APF values are highly sensitive to both temperature and LIE, with the largest variations occurring in the 10^4–10^5 K range, typical of white dwarf and hot main-sequence star atmospheres.
• The tables cover a physically comprehensive range of conditions: temperatures from 1,000 K to 1,000,000 K and LIE values from 0.001 eV to 5.0 eV, ensuring applicability to all spectral and luminosity classes.
• The APF tables for Fe I–Fe X are consistent with those previously computed for Ni I–Ni X using the same method, validating the approach across elements.
• The data are available in 10 ASCII tables (one per ion) at the Acta Astronomica Archive and the University of Opole website, with entries given as decimal logarithms of the APF, enabling direct use in stellar atmosphere codes.