R-matrix calculations for opacities: III. Plasma broadening of autoionizing resonances

A general formulation is employed to study and quantitatively ascertain the effect of plasma broadening of intrinsic autoionizing (AI) resonances in photoionization cross sections. In particular, R-matrix data for iron ions described in the previous paper in the RMOP series (RMOP-II, hereafter RMOP2) are used to demonstrate underlying physical mechanisms due to electron collisions, ion microfields (Stark), thermal Doppler effects, core excitations, and free–free transitions. Breit–Pauli R-matrix cross sections for a large number of bound levels of Fe ions are considered, 454 levels of Fe XVII, 1184 levels of Fe XVIII and 508 levels of Fe XIX. Following a description of theoretical and computational methods, a sample of results is presented to show significant broadening and shifting of AI resonances due to extrinsic plasma broadening as a function of temperature and density. The redistribution of AI resonance strengths broadly preserves their integrated strengths as well as the naturally intrinsic asymmetric shapes of resonance complexes which are broadened, smeared and flattened, eventually dissolving into the bound-free continua.


Introduction
Resonances arise in most atomic interactions.They are especially important in processes such as (e + ion) scattering and photoionization.At the same time, plasma perturbations markedly affect atomic spectra susceptible to varying temperature, density, and other factors.Whereas a vast body of literature exists on line broadening in laboratory and astrophysics plasma environments [1,2,3,4,12,10], there is relatively little work on systematic theoretical treatment of autoionizing resonances that are more readily susceptible to plasma interactions [6,38], though results have been obtained for K-shell spectra (viz.[23]) observed astrophysically [24].Stark broadening and other broadening mechanisms for plasmas have been reviewed from the perspective of individual lines and spectrum [21,8], and in non-local-thermodynamic-equilibrium [11].However, opacity calculations require a statistical treatment such as implemented in the Opacity Project (hereafter OP [29,30,12,31]).
Resonances are ubiquitous in cross sections, measured and calculated in a variety of ways with ever-increasing precision and resolution.State-of-the-art experimental devices such as synchrotron based ion storage rings and narrowband photon sources can now resolve resonances in many atomic systems.Coupled-channel calculations, mainly using the R-Matrix method, have been carried out for nearly all elements and ions up to at least iron under OP [12,13] and, more extensively, the Iron Project (hereafter IP [14]).A prime feature of these calculations is the presence of resonances all throughout the energy ranges of interest.However, resonances are of different types, and exhibit varying shapes, sizes and heights.Their overall resonance strengths may also be computed in analogy with line oscillator strengths for modeling of radiative processes [15].
But the question remains: how are resonance profiles affected by plasma perturbations?To be more precise, how would the intrinsic autoionization shape be modified by extrinsic particle interactions in a given environment?The complexity of the problem becomes evident when one considers that autoionization profiles are inherently asymmetric, described by the Fano formula for isolated resonances in terms of an asymmetry parameter and energy [17].But, any singular expression is insufficient to treat infinite overlapping series of AI resonances which, in fact, range from extremely narrow Rydberg resonances approaching series limits, to huge photoexcitation-of-core resonances that span hundreds of eV in energy and considerably alter the background continuum below core excitation threshold [15,34].Previous works and conventional approach to plasma modeling of resonances, and collisional-radiative models, generally follow the 'isolated resonance approximation', which treats autoionizing resonances as discrete bound levels and entail the calculation of the oscillator strength at a single energy, followed by a perturbative plasma broadening treatment based on independently calculated autoionization and radiative rates (viz.the Cowan code [?]).Although a physical explanation is lacking, arbitrarily increasing line broadening factors of all lines by up to a factor up to ∼100 in atomic structure calculations is found to recover missing solar opacity quantitatively [20].
Ideally, what is needed is a theoretical method that can be translated into a computational algorithm taking into account the variety of resonance shapes and their positions relative to the excited ion core level.Electron-ion interactions in a plasma lead to dominant forms of broadening: Doppler, Stark and electron impact.The Doppler width is approximated by Gaussian that is more narrowly peaked around the line center, and falls off faster, than the other Lorentzian profiles due to Stark and electron impact.The Stark effect due to ions is particularly important for hydrogenic systems when it is linear due to l -degeneracy; a static approximation is sometimes employed since ions move much slower than electrons [1,16].In contrast, the electron impact broadening profile is a Lorentzian with much wider effect on the line wings, and as the electron density and the temperature of the plasma increases, electron collisions become the dominant source of broadening.That would especially be the case for weakly bound electrons in doubly-excited autoionizing states, which would be perturbed more than bound electrons considered in line broadening theories.
In this paper we present a computational methodology that aims to incorporate electron impact broadening in a generally applicable manner suitable for laboratory and astrophysical plasma sources.Without loss of generality, and based on large-scale coupled channel R-matrix calculations ( [5,34]), we consider the photoionization of a complex atomic system, neon-like to fluorine-like iron, Fe xvii −→ Fe xviii , in this study as exemplar of applicability to atomic processes in plasmas.

Theoretical formulation
We first sketch out the theoretical outline for channel coupling that gives rise to resonances and then the resonance broadening modeled after line broadening due to electron impact.

Resonances and channel coupling
Autoionizing resonances manifest themselves via inter-channel coupling in the coupled channel (CC) framework.In the CC approximation the atomic system is represented as the 'target' or the 'core' ion of N-electrons interacting with the (N+1) th electron.The (N+1) th electron may be bound in the electron-ion system, or in the electron-ion continuum depending on its energy to be negative or positive.The total wavefunction, Ψ E , of the (N+1)-electron system in a symmetry Jπ is an expansion over the eigenfunctions of the target ion, χ i in specific state S i L i (J i )π i , coupled with the (N+1) th electron function, θ i : where the i is over the ground and excited states of the target or the core ion.
The (N+1) th electron with energy k 2 i corresponds to a channel labeled . The Φ j s are bound channel functions of the (N+1)-electron system that account for short range correlation not considered in the first term and the orthogonality between the continuum and the bound electron orbitals of the target.
Depending upon the total energy E of the (e + ion) system, and the channel energy k 2 i > 0 or k 2 i < 0, a channel may be open or closed relative to an ion level E i .Interchannel interactions between open and closed channel wavefunctions result in resonances below the excitation threshold at E i .If E < 0 for all channels then the (e + ion) system is in a pure bound state; otherwise we have a free state with an electron in the continuum and some channels open and some closed.Therefore, the CC wavefunction expansion Eq. ( 1) may be used to obtain either (e + ion) collision strengths or bound-bound and bound-free radiative parameters such as oscillator strengths and photoionization cross sections.
With reference to Fig. 1, we have the position of a given resonance ω r corresponding to an excitation threshold E i in terms of its effective quantum number ν i as That yields Typically, there are many excited levels E i included in coupled channel calculations and may number in the hundreds.Infinite series of resonances E i ν nℓ arise and converge on to each level E i .There can be considerable overlap between weakly bound narrow high-ν Rydberg resonances converging on to and immediately below a given threshold, and deeply bound strong and wide resonances with low ν-values belonging to higher levels.A computational algorithm must successively convolve groups of resonances identified with respect to all ion core levels.
Let σ(ω ′ ) be the computed cross section and σ(ω) the convolved cross section such that where the profile factor is (5)

Resonance broadening mechanisms
A general theoretical approximation for scattering of a free electron with an electron in doubly-excited quasi-bound states is necessarily computationally intensive since it needs to be incorporated within a coupled channel framework, and superimposed on ab initio calculations of cross sections.Primary broadening mechanisms such as electron collisions, Stark broadening due to ion microfields, and Doppler broadening due to thermal motions need to considered a priori.We develop a theoretical treatment that accounts for these physical effects independently within a computational viable procedure.
The parameters in the formulation are derived in analogy with line broadening but modified significantly to apply to AI resonances.In the present formulation we associate the energy to the effective quantum number relative to each threshold ω ′ → ν i to write the total width as: pertaining to collisional γ c , Stark γ s , Doppler γ d , and free-free transition γ f widths respectively, with additional parameters as defined below.Without loss of generality we assume a Lorentizan profile factor that describes collisional-ion broadening which dominates in HED plasmas.We assume this approximation to be valid since collisional profile wings extend much wider as x −2 , compared to the shorter range exp(−x 2 ) for Photon Energy (Ry) e+X (z+1)+ X (z+1)+ X z+

Bound Levels
Core Levels Left: Schematic diagram of a coupled channel calculation for photoionization of bound states (solid lines) of an ion X z+ → X z+1 -AI resonances (dashed lines) correspond to Rydberg series converging on to excited levels of the residual ion with E = −(z + 1) 2 /ν 2 ; Right: ion thresholds of convergence E i , E i+1 , E i+2 , E i+3 .... and a Lorentzian profile with lower and upper energy limits (E ℓ , E u ) spanning narrow high-n resonances below E i and broader ones above.
thermal Doppler, and x −5/2 for Stark broadening (viz.[29]).In principle the limits of integration in Eqs.(4)(5)(6) are ∓∞, which are replaced in practical calculations by ∓γ i / √ δ, where δ is chosen to ensure full Lorentzian profile energy range and for accurate normalization.Convolution by evaluation of Eqs.(3)(4)(5)(6) is carried out for each energy ω throughout the tabulated mesh of energies used to delineate all AI resonance structures, for each cross section, and each core ion threshold.

Electron impact broadening
At sufficiently high densities collisional broadening is dominant and and mathematically represented by a Lorentizan function (Eq.5) that correctly approximates the slowly varying behaviour in the line wings.We develop a numerical procedure for convolving cross sections including resonances over a Lorentzian damping width.Given energy dependent cross sections tabulated at sufficiently fine mesh, we first switch the energy variable to the effective quantum number ν = z/ (E), where E = hω.In photoionization, we take ω to be the photon frequency; henceforth we shall also employ ω as the energy variable assuming atomic units h = 1.The ν is more appropriate since for a resonance it is defined relative to the excited core ion level, as illustrated in Fig. 1.
We consider photoionization of an ion of element X with charge z in an initial state by photon of energy hω into the ground or excited level of a residual ion of charge (z +1) It is assumed that unperturbed photoionization cross sections σP I (hω) are theoretically computed with sufficient resolution in energy to delineate autoionization profiles.According to the impact approximation [12] we may then represent the damping profile with a Lorentzian expression In analogy with electron impact damping of bound-bound line transitions, we define E o as the resonance center, γ as the width and x the energy shift (later we shall assume that |E − E o | >> x).We may further express where N e is the electron density and γ is the damping constant which may be written in terms of the electron distribution f (ǫ, T ) at a given temperature T as Given Q D as the electron impact cross section and a Maxwellian distribution we may obtain the thermally averaged damping rate coefficient where Ω(ǫ) is the collision strength.Then . In Eqs.(8-12) the Υ D is a complex quantity.However, for small δω = (ω − ω o ) in the one-perturber approximation ([12] and references therein), we have γ = N e γ and Now we establish a correspondence between γ(ω) and the electron impact rate coefficient Υ according to the relation where Υ(ν) is computed at the resonance energy corresponding to ν = z/ (E), with E in Rydbergs and atomic units a o = h = 1.We now approximate G(z) is an effective Gaunt factor for electron impact excitation of positive ions, empirically determined for line broadening work in OP [12] to be The behavior of G(z) with ion charge z and temperature T has been further studied for electron impact broadening, and we adopt an improved expression ( [29,18,38]) For example, in Table 1 we compare the two expressions and find that they differ significantly for ν < 10, but G(T, z, ν → G(z) as ν → 10, and exceeds marginally for ν > 10 when BPRM resonance structure calculations are truncated.
Here ω g is the ionization energy of the ground state of the photoionizing ion X z+ .Then from Eq. ( 18) we obtain the temperature-density dependent width at each energy Evaluating the constants with T(K) and N e cm( −3 ), we obtain With the transformation of the unbroadened cross section using Eq. ( 18), we obtain the temperature-density-energy dependent functional representing the photoionization cross section broadened by electron impact.This greatly expands the scope of the calculations since Eq.(19) implies that the convolution must be carried out at each energy in the tabulated energy mesh (transposed as E(ω) → ν) of unbroadened function σ(ω), with another tabulation for the Lorentzian profile Eq. ( 8), and for each temperature and electron density.In the next section we describe the procedure developed for such numerical calculations.
Given N core ion levels corresponding to resonance structures, With x ≡ ω ′ − ω, the summation is over all excited thresholds E i included in the N-level CC or RM wavefunction expansion, and corresponding to total damping width γ i due to all broadening processes.The profile φ(ω ′ , ω) is centered at each continuum energy ω, convolved over the variable ω ′ and relative to each excited core ion threshold i .
We employ the following expressions for computations: where T, N e , z, and A are the temperature, electron density, ion charge and atomic weight respectively, and ν i is the effective quantum number relative to each core ion threshold i : i /z 2 is a continuous variable.A factor (n x /n g ) 4 is introduced for γ c to allow for doubly excited AI levels with excited core levels n x relative to the ground configuration n g (e.g. for Fe xviii n x = 3, 4 relative to the ground configuration n g = 2).

Stark broadening
A treatment of the Stark effect for complex systems entails two approaches, one where both electron and ion perturbations are combined (viz.[19,38]), or separately (viz.[12,29]) employed herein.Excited Rydberg levels are nearly hydrogenic and ion perturbations are the main broadening effect, though collisional broadening competes significantly increasing with density as well as ν 4 i (Eq.14).For bound levels in a plasma microfield of strength F, the Stark sub-levels of a level n span a range given by the highest component (n, k max ) with energy (viz.[12,29]) and the lowest component of sub-level ((n + 1), k min ) with energy In deriving occupation probabilities in the Mihalas-Hummer-Däppen equation-ofstate (MHD-EOS) [31] used in OP work [12], a critical field strength F c is calculated when Stark broadening renders these two components equal, and Stark ionization dissolves level n into the continuum.The total Stark width of a given n -complex is ≈ (3F/z)n 2 .Assuming the dominant ion perturbers to be protons and density equal to electrons, N e =N p , and replacing n by the effective quantum number ν i relative to each excited threshold of an ion with charge z, we take F = [(4/3)πa 3  o N e )] 2/3 , as employed in MHD-EOS for Stark broadening in Eq. ( 6) In addition, in employing Eq. ( 6) a Stark ionization parameter ν * s = 1.2 × 10 3 N −2/15 e z 3/5 is introduced such that AI resonances may be considered fully dissolved into the continuum for ν i > ν * s , analogous to but distinct from the Inglis-Teller series limit [32], or the Stark ionization of bound (not AI) energy levels as considered in the MHD-EOS [31].
All calculations are carried out with and without ν * s as shown later in Table 2, and illustrated in the Figs. 3, 4, 5 presented herein (red and blue curves respectively).Results are practically indistinguishable with and without Stark ionization cut-off, and effect on redistribution of differential oscillator strength or opacities.However, ν * s is a parameter that should prove to be useful in further extension of plasma effects including Debye screening, as discussed later.

Thermal Doppler broadening
The Doppler width is: where ω is not the usual line center but taken to be each AI resonance energy.

Free-free transitions broadening
The last term γ f in Eq. ( 6) accounts for freefree transitions among autoionizing levels with ν i , ν ′ i such that The large number of free-free transition probabilities for +ve energy AI levels E i , E ′ i > 0 may be computed using RM or atomic structure codes (viz.[33,37]).Free-free transitions are not considered in the results in Figs. 2 and 3 but included in the results discussed in Table 1, although it is found to be practically negligible.

Computational algorithm
In order to elicit and illustrate important physical features of the formulation, we sketch a few salient features of the mathematical algorithm developed to implement the procedure (numerical details and the computer program will be presented elsewhere).
We have re-defined the Lorentzian profile Eq. ( 5) as in Eq. ( 8), using a damping rate coefficient Eqs.(10)(11)(12)(13) and Maxwellian electron distribution, dependent on electron density and temperature as in Eqs.(17)(18).Numerical evaluation scheme based on this formulation requires several practical considerations to be incorporated in the computational algorithm and computer program.

Profile limits
The limits of integration in Eq. ( 4) are determined by the extent of the Lorentzian factor in Eq. (8).It needs to be ensured that the profile extends into the resonance wings and/or approaches the background continuum without loss of accuracy.Measuring the energy spread relative to the resonance center ω = ω r , we note that according to Eq. ( 13) ω = ω g + E i , with respect to the ionization potential and the target excitation energy E i above the ground state of the residual ion.Then the profile maximum is (Eq.8) We introduce an accuracy parameter δ and choose the profile limits ±ω o such that Then, Or, For small δ, Therefore, |ω − ω| limits the convolution profile such that Whereas Eq.(4) using Eq. ( 5) has an analytical solution in terms of tan −1 (x/γ)/γ evaluated at limiting values of x → ∓γ/ √ δ, its evaluation for practical applications entails piece-wise integration across multiple energy ranges spanning many excited thresholds and different boundary conditions.For example, the total width γ is very large at high densities and the Lorentzian profile may be incomplete above the ionization threshold and therefore not properly normalized.We obtain the necessary redward leftwing correction for partial renormalization as where a is the lower energy range up to the ionization threshold, reaching the maximum value −γ/2 √ δ.The parameter δ is generally chosen to be 10 −2 so that the total profile ranges over 10γ.

Convolution quadrature
The complexity of the problem arises from the following main factors: (i) wide variety of narrow and broad resonances, (ii) overlapping infinite Rydberg series belonging to a large number of excitation thresholds of the target ion, and (iii) Lorentzian profiles that vary at each energy on a mesh that is independent of the tabulated energy mesh for the original cross section.The schematics are described in Fig. 2.
Here the summation is over all excitation thresholds E i included in the CC wavefunction expansion (Eq. 1) and corresponding damping widths γ i .The profile φ(ω ′ , ω) is centered at ω; we define x ≡ ω ′ − ω (note change of order of variables which is immaterial), then This equation requires discrete summation over all target ion thresholds, and piecewise integration over normalized profile at each energy.First, we consider the endpoints with lower energy limit x ℓ ≡ −(ω o − ω) = −γ i / √ δ, and upper limit Let the tabulated energy mesh be ω 1 , ω 2 , .....ω N .Then Assuming the lower limit x ℓ to lie between x 1 < x ℓ < x 2 ; and the upper limit x u between x N −1 < x u < x N , we have + .... + x N x N−1 (.....)dx . (37)

Interpolation and evaluation
Each of the raw originally tabulated unbroadened cross sections σ(ω ′ ) needs to be interpolated on to the resonance profile mesh.A linear interpolation is sufficient for precision since the CC calculations are usually carried out at a fine mesh to resolve most autoionizing resonances up to ν i ≤= ν max = 10 below each target threshold E i .Suppose the transposed energy mesh ω on to the resonance profile is represented by linearly interpolated segments a j + b j x with a j , b j coefficients such that, . Then for all thresholds i, It is understood that the interpolation and summation is carried out with respect to profiles corresponding to all target ion thresholds at E i .Having determined coefficients a j , b j we need to evaluate expressions for each segment as Evaluating separately, For clarity we have avoided the use of double scripts (i, j), one with respect to thresholds E i and the other for interpolation between respective resonance profile segments.But in principle we may represent the final values of the cross sections convolved over all resonances at the transposed energy mesh ω ′ → ω as subsuming all target ion levels (Fig. 1 and Eq. 1) and interpolation into the computational algorithm.Finally, we compute broadened cross sections at the same energy mesh as the unbroadened cross sections σ(ω ′ ) so that there is one-to-one correspondence ω ′ → ω.However, we note that the intermediate energy mesh of the Lorentzian profile is independent, and interpolated in accordance with the damping width Eq. ( 11-12) at each energy.

Computer program
A general program for convolving AI resonances has been written and will be reported elsewhere.Here we note a few of the main features.The primary loops in the program are over electron temperature T e , density N e , and target thresholds E i .The input is the unbroadened CC cross sections tabulated at a sufficiently fine mesh to resolve resonances so that convolution, interpolation and integration do not result in loss of accuracy.The accuracy parameter δ is chosen to be in the range 10 −2 − 10 −6 ; more importantly, it is ensured that the convolved cross sections have converged, physically implying that the resonance wings have merged into the continuum.The CPU time required depends mainly on the density which determines the total width γ; for example, in the reported calculations for Fe xvii at T=2 × 10 6 K it is few minutes for N e = 10 21 cc and ∼3 hours for N e =10 24 cc.
The program is suitable as a module within a post-processing program for CC cross sections with AI resonances for photoionization, electron-ion collisions and recombination, intended for practical application in specified temperature-density range.

Results and discussion
The complexity and magnitude of RMOP computations has been studied using photoionization data for a large number of bound levels of the three Fe ions described in RMOP2.Since AI plasma broadening must be carried at each temperature-density pair, the resulting cross sections constitute a huge amount of data required for opacities calculations in HED plasma sources.In this section we discuss a small sample of results for those Fe ions to illustrate some physical features.

Fe xvii : Temperature-Density dependence
Owing to its closed shell ground configuration and many excited n-complexes of configurations, Ne-like Fe xvii later works.
is of considerable importance in astrophysical and laboratory plasmas, as described in a number of previous works ( [35] and references therein).The Fe xvii BPRM calculations are carried out with 218 fine structure levels dominated by n = 2, 3, 4 levels of the core ion Fe xviii .The computed Fe xvii bound levels (E < 0) are dominated by configurations 1s 2 2s 2 2p 6 ( 1 S 0 ), 1s 2 2s p 2p q nℓ, [SLJ] (p, q = 0 − 2, n ≤ 10, ℓ ≤ 9, J ≤ 12).The core Fe xvii levels included in the CC calculation for the (e + Fe xviii ) →Fe xvii system are:1s 2 2s 2 2p 5 ( 2 P o 1/2,3/2 ), 1s 2 2s 2 2p q , nℓ, [S i L i J i ] (p = 4, 5, n ≤ 4, ℓ ≤ 3).The Rydberg series of AI resonances correspond to (S i L i J i ) nℓ, n ≤ 10, ℓ ≤ 9, with effective quantum number defined as a continuous variable ν i = z/ (E i − E) (E > 0), throughout the energy range up to the highest 218 th Fe xviii core level; the n = 2, 3, 4 core levels range from E=0-90.7 Ry ( [34,35]).The Fe xvii BPRM calculations were carried out resolving the bound-free cross sections at ∼40,000 energies for 454 bound levels with AI resonance structures (in total 587 bound levels are considered, but the higher lying s (Table 1).Rydberg series of AI resonance complexes with ν i ≤ 10 belonging to 217 excited Fe xviii levels E i broaden and shift with increasing density, also resulting in continuum raising and threshold lowering.levels are included to ensure convergence and completeness as discussed in paper P4, and do not significantly contribute to opacities calculations).Given 217 excited core levels of Fe xviii , convolution is carried out at each energy or approximately 10 9 times for each (T,N e ) pair.
Fig. 3 (left) displays detailed results for plasma broadened and unbroadened photoionization cross section of one particular excited level 2s 2 2p 5 [ 2 P o 3/2 ]3p( 3 D 2 ) (left, ionization energy 37.707 Ry) of Fe xvii along isotherm T = 10 6 K at three representative densities (note the ∼10 orders of magnitude variation in resonance heights along the Y-axis).The main feature evident in the figure are as follows.(i) AI resonances begin to show significant broadening and smearing of a multitude of overlapping Rydberg series at N e = 10 21 cc.The narrower high-n l resonances dissolve into the continua but stronger low-n l resonance retain their asymmetric shapes with attenuated heights and widths.(ii) As the density increases by one to two order of magnitude, to N e = 10 22−23 cc, resonance structures not only broaden but their strengths shift and redistributed over a wide range determined by total width γ(ω, ν i , T, N e ) at each energy hω (Eq.6).(iii) Stark ionization cut-off (Table 1) results in step-wise structures that represent the average due to complete dissolution into continua.(iv) The total AI resonance strengths are conserved, and integrated values generally do not deviate by more than 1-2%.For example, the three cases in Fig. 3 (left): unbroadened structure (black), and broadened without (red) and with Stark cut-off (blue), the integrated numerical values are 59.11, 59.96, 59.94 respectively.This is also an important accuracy check on numerical integration and the computational algorithm, as well as the choice of the parameter δ that determines the energy range of the Lorentizan profile at each T and N e ; in the present calculations it varies from δ = 0.01-0.05for N e =10 21−24 cc.The scale of unbroadened AI features is evident upon a comparison on log and linear scales as in Fig. 4 (black curves), considered for two excited Fe xviii levels.The top and bottom panels on left and right exhibit Logσ P I (MB) and σ P I (MB) respectively.Whereas the log-scale in top panels appropriately displays the full extent of AI resonances, it appears with equal weight for both positive values that rise up to 10 6 MB, and for negative values down to 10 −6 MB that are not significant contributors, as shown in the bottom panels on a much smaller linear scale going from zero only up to 2.5 MB.
Attenuation of AI features due to plasma effects are shown in red and blue curves at two different T-D pairs; cross sections on the left are at a lower temperature and more than three times lower electron density than the ones on the right.Consequently, the AI features on the right in Fig. 4 are much more broadened that the ones on the left.Two other noticeable features are the closing of "opacity windows" in the unbroadened cross sections, and shift of AI resonances leading to temperature-density dependent redistribution of differential oscillator strengths and opacity with energy.

Conservation of differential oscillator strength
It is important to ensure the numerical accuracy of AI plasma broadening in temperature-density-energy space.Theoretically and computationally, that implies an investigation of integrated differential oscillator strengths proportional to σ P I for all levels of a given ion for the three forms computed: (i) unbroadened (black curves), (ii) with all plasma broadening effects included as in Eq. ( 6) (red curves), and (iii) as in (ii) but with Stark ionization cut-off that leads to sharp step-wise structures below each  ionization threshold (blue curves).We had quoted these values for one level of Fe xvii above in Fig. 3.
In Fig. 5 we present σ P I for the ground state of Fe xix 2s 2 p 4 3 P 3 (ionization energy 104.956Ry), as well as an excited state 2s2p 4 ( 2 S)3s 1 S e (ionization energy 24.186 Ry).For these two cross sections of Fe xix we find integrated values over the entire energy range shown to be 21.74,22.98 and 22.90 for the unbroadened, broadened, and broadened with Stark ionization cut-off, respectively for the ground state, and 12.48, 13.57 and 13.56 respectively for the excited state (units are in MB-Ry though only the relative values are indicators of accuracy).The numerical agreement between the three sets of values is well within ∼10% indicating conservation of oscillator strength, despite some uncertainty in integration over extensive narrow and broad resonance structures that vary by nearly 20 orders of magnitude in height for σ P I (2s2p 4 ( 2 S)3s 1 S e ), and widely disparate width distribution among Rydberg vs. Seaton PEC resonances described in RMOP2.
Generally, the agreement between the three sets of calculations for each level of each ion at each temperature-density is also an accuracy check of the plasma broadening treatment presented.Since there are hundreds of levels or each ion considered, there is  2).AI resonances in the unbroadened σ P I on the right range over 20 orders of magnitude.
more than 10% difference in integrated cross sections for highly excited levels at very high densities where the total AI width (Eq.6) is very large.However, the highly excited levels are cut-off by the MHD-EOS and not considered in opacity calculations.3.1 × 10 22.5 5.25(0) 2.71(0) 2.53(0) 6.6 10.0 1.13 10 6

Plasma opacity parameters
3.1 × 10 23.5 3.89(0) 2.71(1) 1.18(0) 4.8 5.6 1.06 be accounted for at even higher densities.The aggregate effect of AI broadening for large-scale applications is demonstrated in Table 2 by the ratio R of the Rosseland Mean Opacity (discussed in paper RMOP1) using broadened/unbroadened cross sections for 454 Fe xvii levels with AI resonances (other higher bound levels have negligible resonances) [39,35].For any atom or ion R is highly dependent on T and N e ; for Fe xvii R yields up to 58% enhancement due to plasma broadening with increasing N e along the 2 × 10 6 K isotherm, but decreasing to 6% along the 10 6 K isotherm.Approximately 70,000 free-free transitions among +ve energy levels are included in the calculation of R, but their contribution has no significant broadening effect since they entail very high-lying levels with negligible level populations.However, different plasma environments with intense radiation fields, or a different equationof-state than [31] employed here, may lead to more discernible effect due to free-free transitions.AI broadening in a plasma environment affects each level cross section differently, and hence its contribution to opacities or rate equations for atomic processes in general.A critical (T,N e ) range can therefore be numerically ascertained where redistribution and shifts of atomic resonance strengths would be significant and cross sections should be modified.

Conclusion
The main conclusions are: (I) The method described herein is generally applicable to AI resonances in atomic processes in HED plasmas.(II) The cross sections become energytemperature-density dependent in a The cross sections become energy-temperaturedensity dependent in a critical range leading to broadening, shifting, and dissolving into continua.(III) Among the approximations necessary to generalize the formalism is the assumption that thermal Doppler widths are small compared to collisional and Stark widths as herein, but given the intrinsic asymmetries of AI resonances it may not lead to significant inaccuracies (although that needs to be verified in future works).(IV) The treatment of Stark broadening and ionization cut-off is ad hoc, albeit based on the equation-of-state formulation [31] and consistent with previous works [12].(V) Since it is negligibly small, the free-free contribution is included post-facto in the computation of the ratio R in Table 2 and not in the cross sections and results shown in Figs. 2  and 3, but may be important in special HED environments with intense radiation and should then be incorporated in the main calculations of total AI width (Eq.6).(VI) The predicted redward shift of AI resonances as the plasma density increases should be experimentally verifiable.(VII) Redistribution of AI resonance strengths should particularly manifest itself in rate coefficients for (e + ion) excitation and recombination in plasma models and simulations, and for photoabsorption in opacity calculations, using temperature-dependent Maxwellian, Planck, or other particle distribution functions.(VIII) The treatment of individual contributions to AI broadening may be improved, and the theoretical formulation outlined here is predicated on the assumption that external plasma effects are perturbations subsumed by and overlying the intrinsic autoionization effect.(IX) The computational formalism is designed to be amenable for practical applications and the computational algorithm and a general-purpose program AUTOBRO are optimized for large-scale computations of AI broadened cross sections for atomic processes in HED plasma and astrophysical models.

Fig. 3 (
right) shows similar results to Fig. 3 (left) for another excited Fe xvii level 2s 2 2p 5 [ 2 P o 3/2 ]4d( 1 F o 3 ) (ionization energy 17.626 Ry), along a higher temperature 2×10 6 K isotherm at different intermediate densities.Both Figs. 2 and 3 show a redward shift of low-n resonances and dissolution of high-n resonances.In addition, the background continuum is raised owing to redistribution of resonance strengths, which merge into one across high lying and overlapping thresholds.

4. 2 .
Fe xviii : Scaling and delineation of resonances Next, we employ plasma broadened cross sections for Fe xviii to highlight the scale, shape, scope, width and magnitude of AI resonances.

Table 1 )
. Rydberg series of AI resonance complexes with ν i ≤ 10 belonging to 276 excited Fe xix levels.

Table 2
[36,) pl15]a parameters corresponding to Figs.3.Their physical significance is demonstrated by a representative sample tabulated temperature T(K) and N e .The maximum width γ 10 corresponding to ν i = 10 in Eqs.(3,6)is set by the CC-BPRM calculations which delineate unbroadened AI resonance profiles up to ν ≤ 10, and employ an averaging procedure up to each threshold 10 < ν < ∞ using quantum defect (QED) theory (viz.[36,12,15]andreferencestherein).γc(10) and γ s (10) are the maximum collisional and Stark width components.The Doppler width γ d is much smaller, 1.18 × 10 −3 and 1.67 × 10 −3 Ry at 10 6 K and 2 × 10 6 K respectively, validating its inclusion in Eq. (6) in HED plasma sources but possibly not when γ d is comparable to γ c or γ s .The ν * s and ν D are effective quantum numbers corresponding to Stark ionization cut-off and the Debye radius respectively.We obtain ν D = 2 5 πz 2 λ 2 D 1/4 , where the Debye length λD = (kT /8πN e ) 1/2 .It is seen in Table 2 that ν D > ν *s at the T, N e considered, justifying neglect of plasma screening effects herein, but which may need to

Table 2 .
[39]ma parameters along isotherms in Fig.2 and 3; ν D corresponds to Debye radius; R is the ratio of Fe xvii Rosseland Mean Opacity with and without broadening[39]; γ 10 is the maximum AI resonance width at ν = 10.