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# Plasma Effects on Resonant Phenomena

Anil Pradhan  Department of Astronomy, Chemical Physics Program, The Ohio
State University, Columbus, OH 43210, USA

###### Abstract

The effect of autoionizing resonances in atomic systems and processes is
reviewed. Theoretical framework for treating resonances in the coupled channel
approximation using the R-matrix method, as well as approximations related to
plasma applications are described. The former entails large-scale atomic
computations, and the latter is based on a new method for including
collisional, Stark, thermal and other broadening mechanisms. We focus
particularly on the problem of opacities calculations in high-energy-density
(HED) plasmas such as stellar interiors and inertial confinement fusion
devices. The treatment is generally relevant to radiative and collisional
processes as the cross sections become energy-temperature-density dependent.
While the computational difficulty increases considerably, the reaction rates
are significantly affected. The related issue of the Boltzmann-Saha equation-
of-state and its variants in local-thermodynamic-equilibrium (LTE) is also
explored as the intermediary between atomic data on the one hand and plasma
environments on the other.

PACS: 32.80.-t; 32.80.Fb; 33.60.+q

today

##  1 Introduction

Resonant phenomena are ubiquitous in physics and stem from correlation
effects. In AMO physics autoionizing resonances are due to electron
correlation effects among bound and continuum states. They manifest themselves
prominently in cross sections of various atomic and molecular processes in
laboratory and astrophysical plasma sources. The shapes, magnitudes, and
extent of resonances determines associated rates for spectral formation and
experimental and observed spectra. The well-known Fano profile [2, 3, 4] is
widely used to analyze isolated resonance structures, with parameters that may
be compared with theoretical calculations.

On the other hand, overlapping infinite series of autoionizing resonances
converging on to large numbers of excited levels are also of great importance.
In general, they are not amenable to analytic formulation and require
computationally intensive coupled channel calculations, the most powerful of
which is the R-matrix method by P.G. Burke and collaborators [5].111Following
the original suggestion by U. Fano about the R-matrix theory of nuclear
reactions by A.M. Lane and R.G. Thomas [6], P.G. Burke introduced it for
atomic and molecular processes, as described in [5]. Later, M.J. Seaton and
collaborators adapted the R-matrix method for large-scale calculations
required for the Opacity Project [7]. Whereas the R-matrix method has long
been utilized for a variety of atomic processes and applications, I focus on a
large-scale application to plasma opacities that is of immense importance in
astrophysics and nuclear fusion sources. R-matrix atomic calculations
including resonant phenomena in an ab initio manner, and taking account of
plasma effects, are exceedingly difficult but no longer computationally
intractable for complex atoms and ions [9, 10, 11, 13].

Fano derived his general formula in the early days of experimental and
theoretical autoionization studies, but before the close coupling codes were
developed to delineate resonance profiles precisely (viz.[8]). However, HED
plsama applications require not only a complete description of the bound-free
resonant continua, but also interface with the equation-of-state and resonance
broadening and level-dissolution effects as function of temperature and
density [11, 12]. But in fact, the intrinsically quantum mechanical
autoionization is the dominant broadening mechanism when extrinsic plasma
effects are negligible or small, and autoionization widths are much bigger
than line widths due to radiation damping, and/or Stark, thermal, or
collisional broadening. In particular, the characteristically asymmetric
Seaton resonances due to photoexciation-of-core (PEC) in photoionization cross
sections extend over large energy ranges of up to hundreds of eV, and can only
be described by coupled channel correlations [14, 8, 10]. Furthermore, the
Seaton PECs retain their resonance profiles even as other narrower Rydberg
series of resonances dissolve into the continuum with increasing plasma
densities. For these reasons, R-matrix coupled channel calculations would be
necessary in general for all atomic systems under any plasma conditions for
high precision, and to be preferred over simpler approximations based on the
distorted wave method (DW) or its variants that do not include autoionizing
resonance structures (discussed later).

Opacity determines the light we see or detect. All radiation-matter
interactions need to be considered in order to determine the opacity of a
given plasma source or medium. Primary physical processes contributing to
opacity are:

$\kappa_{ijk}(\nu)=\sum_{k}A_{k}\sum_{j}F_{j}\sum_{i,i^{\prime}}[\kappa_{bb(}(i,i^{\prime};\nu)+\kappa_{bf}(i,\epsilon_{i^{\prime}};\nu)+\kappa_{ff}(\epsilon_{i},\epsilon^{\prime}{{}_{i}^{\prime}};\nu)+\kappa_{sc}(\nu)]\
.$ (1)

The first two are the dominant processes; $bb$ refers to bound-bound and $bf$
to bound-free transitions or photoionization. The other two, free-free
transitions and photon scattering, are generally much smaller and may be
treated by simple approximations (viz. [7, 8]). In Eq. 1, $A_{k}$ is the
abundance of an element $k$, its ionization fraction $F_{j}$ at a given
temperature-density, $i,i^{\prime}$ are initial bound and final
bound/continuum states, and $\epsilon$ represents the free electron energy.

##  2 Coupled channel R-matrix method and atomic-plasma effects

The state-of-the-art R-matrix (RM) method provides a powerful computational
tool to implement the general coupled-channel theoretical framework. An atomic
systems is represented by a N-electron core or target ion wavefunction
$\chi_{i}$ coupled with an $(N+1)^{th}$ free electron wavefunction
$\theta_{i}$ in a bound or continuum state of the (e + ion) system. The total
(e + ion) wavefunction is then a quantum superposition expressed as

$\Psi(E)=\mathcal{A}\sum_{i}(E_{i})\chi_{i}\theta_{i}(k_{i}^{2})+\sum_{j}c_{j}\Phi_{j}(E_{j}).$
(2)

When the free electron kinetic energy $k_{i}^{2}>0$, the first sum on the RHS
of Eq. 2 represents a coupled channel system for electron-ion scattering or
half-scattering photoionization process [4]. Each channel is defined by the
spin-orbital quantum symmetries $(S_{i}L_{i}J_{i})\ \ell_{i}s_{i}[SLJ]$.
Rydberg series of resonances arise from photoexcitation of bound energy levels
into (e + ion) continua comprising of coupled thresholds of the core ion. A
particular type of resonances due to photoexcitation-of-core (PEC) or Seaton
resonances are due to strong dipole transitions in the core ion where the
continuum electron in a Rydberg level remains a ’spectator’ [14, 8, 10]. The
Seaton PEC resonances constitute the detailed balance inverse of the
dielectronic recombination process, wherein an (photo-)excited core ion
undergoes radiative decay by emission of photons redward of the core
transition wavelength. The huge Seaton resonances dominate bound-free opacity
under all plasma conditions.

If the first summation on the RHS of Eq. 2 is neglected then the coupling
effects in the (e + ion) wavefunction are excluded, yielding the distorted
wave (DW) approximation that does not include autoionizing resonances in an ab
initio manner as the RM method. Owing to its simplicity, the DW method has
been employed in existing opacity models. While resonant phenomena are not
included in the DW calculations, that contribution may be included
perturbatively by considering autoionizing resonances on par with lines as
bound-bound transitions and employing line broadening theory for plasma
broadening. However, the detailed auotionization shapes over extended energy
ranges and their precise effect on atomic rates is not taken into account in
DW models.

The Opacity Project (OP [7]) was originally developed to implement the RM
method but was computationally intractable for most complex atoms. In
particular, it was found that inner-shell transitions from closed electronic
shells into outer open subshells implies a large number of channels to be
included in Eq. 2. In recent years, a renewed and extended version of OP has
been initiated [16] and is now in progress for improved opacities with higher
accuracy [9, 10, 11, 13].

For opacity calculations, the transition matrix elements are obtained with
dipole operator ${\bf D}=\sum_{i}{\bf r}_{i}$, where the sum is over all
electrons, which gives the generalized line strength as,

${\bf
S=\left|\left\langle{\mit\Psi}_{f}|\sum_{j=1}^{N}r_{j}|{\mit\Psi}_{i}\right\rangle\right|^{2}}$
(3)

The oscillator strength ($f_{ij}$), radiative decay rate ($A_{ji}$),
photoionization cross section ($\sigma_{PI}$), and mass attenuation
coefficient then can be expressed as follows

$f_{ij}=\left[{E_{ji}\over{3g_{i}}}\right]S,~{}~{}A_{ji}(sec^{-1})=\left[0.8032\times
10^{10}{E_{ji}^{3}\over{3g_{j}}}\right]S,~{}\sigma_{PI}(K\alpha,\nu)=\frac{4\pi^{2}a_{o}^{2}\alpha}{3}\frac{E_{ij}}{g_{k}}S$
(4)

The Breit-Pauli R-matrix (BPRM) method incorporates relativistic effects using
the Breit-Pauli (BP) Hamiltonian for the (e + ion) system in intermediate
coupling, with a pair-coupling scheme
$S_{i}L_{l}(J_{i})l_{i}(K_{i})s_{i}(SLJ\pi)$, whereby states $S_{i}L_{i}$ is
split into fine-structure levels $S_{i}L_{i}J_{i}$, and $SLJ\pi$ is the total
spin-orbital symmetry. Consequently, the number of channels becomes several
times larger than the corresponding $LS$ coupling case. A considerable body of
work with the BPRM codes has been carried out under the follow-on project to
OP, the Iron Project [17]. The IP work is based on BPRM codes and archived in
the large amount of radiative and collisional data in databases NORAD [21] and
OP/IP database Topbase [18].

##  3 Plasma environment and approximations

The practical limitation of the RM method for plasma applications is evident
from Eq. 2. Computational constraints imply that only a finite and usually
small number of excited core ion states and resulting channels may be
explicitly included in the wavefunction expansion. This has implications in
high-energy-density (HED) plasmas such as in stellar interiors or fusion
devices, wherein a large number of excited states exist and differentially
perturbed. The temperature regime may be in excess of $10^{6}-10^{7}$ K, with
electron densities up to $10^{27}$ cm-3. Among the largest R-matrix
calculations carried out thus far are the recent ones for Fe ions Fe xvii , Fe
xviii and Fe xix that constitute $\sim$85% of iron opacity at the boundary
between the solar radiative and convection zones at radius $R_{\odot}$ =
0.713$\pm$0.001, where T=$2\times 10^{6}$K and electron density
$N_{e}=10^{23}$ cm-3 [9]. The number of core levels included in the R-matrix
(e + ion) wavefunction expansion were 218 levels of Fe xviii for the (e+Fe
xviii ) $\rightarrow$ Fe xvii bound and continuum states, 276 levels of Fe xix
for the (e+Fe xix ) $\rightarrow$ Fe xviii , and 99 LS terms of Fe xx for
(e+Fe xx ) $\rightarrow$ Fe xix [10].

###  3.1 Equation-of-state

In addition to the theoretical limit of the RM method, an obvious limit is
imposed by perturbations on atoms by the plasma environment depending on the
specific temperature-density. That manifests itself via the equation-of-state
that determines the atomic ionization state and level populations. The
generally employed approximation is to assume local-thermodynamic-equilibrium
(LTE), as defined by the Saha-Boltzmann equations. From an atomic-plasma
physics point of view, a widely employed formulation (such as in OP) is the
Mihalas-Hummer-Däppen (MHD) equation of state in the so-called ”chemical
picture”[15]. It is based on the concept of occupational probability $w$ of an
excited level being occupied at a given temperature and density such that the
level population is

$N_{ij}=\frac{N_{j}g_{ij}w_{ij}e^{-E_{ij}/kT}}{U_{j}},$ (5)

where $w_{ij}$ are the occupation probabilities of levels $i$ in ionization
state $j$, and $U_{j}$ is the atomic internal partition function. The
occupation probabilities do not have a sharp cut-off, but approach zero for
high-$n$ as they are dissolved due to plasma interactions. The partition
function is re-defined as

$U_{j}=\sum_{i}g_{ij}w_{ij}e^{(-E_{ij}/kT)}.$ (6)

$E_{ij}$ is the excitation energy of level $i$, $g_{ij}$ its statistical
weight and $T$ the temperature. The $w_{ij}$ are obtained by free-energy
minimization, and taking into account Stark ionization due to plasma
microfields [15]. Hence, the exact form of the equation-of-state numerically
determines how many and how much the excited states of an atom contribute to
opacity and radiation transport.

###  3.2 Broadening of autoionizing resonances

Unlike line broadening intrinsic autoionization (AI) decay rates are much
larger relative to radiative rates. Therefore, one expects resonances to
broaden, smear out, and dissolve into the continuum much more than lines when
subjected to extrinsic HED plasma environments. Also, unlike line broadening
for which theoretical formulations are well developed and long employed, there
is no ab initio and general treatment for AI broadening. Even for line
broadening the most elaborate methods are are precise only for hydrogenic and
simple atomic systems and several approximations are necessary to apply those
to complex atoms in realistic sources [7].

Recently, a general theoretical and computational formalism has been
introduced for AI resonance broadening [11]. Analogous to line broadening, the
physical mechanisms considered are: electron collisions (pressure broadening),
ion microfields (Stark broadening), Doppler effect (thermal broadening), and
free-free transitions. It has been shown that extrinsic plasma effects
redistribute and shift resonance strengths, even as the broad intrinsic
asymmetries of resonance profiles are discernible. Furthermore, while the
shapes, magnitudes and extent of resonances are affected, the total integrated
resonance oscillator strengths are conserved and manifest in the bound-free
continua independent of temperature and density. The energy-temperature-
density dependent cross sections would elicit and introduce physical features
in resonant processes in photoionization, (e + ion) excitation and
recombination. The method should be generally applicable to atomic species in
high-energy-density (HED) sources such as fusion plasmas and stellar
interiors.

Whereas the main broadening mechanisms in AI broadening are physically similar
to line broadening, their theoretical and computational treatment is quite
different. Superimposed on intrinsic AI broadening in atomic cross sections
the extent of resonances owing to extrinsic plasma effects renders much of the
line broadening theory inapplicable, particularly for multi-electron systems.
The unbroadened AI resonances themselves vary by orders of magnitude in width,
shapes and heights, and incorporate two types: large features due to
photoexcitation-of-core (PEC) below thresholds corresponding to dipole core
transitions [14], and infinite Rydberg series of resonances converging on to
each excited core level of the (e + ion) system. The generally employed Voigt
line profiles obtained by convolution of a Lorentzian function for radiative
and collisional broadening, and a Gaussian function for Doppler or thermal
broadening, are found to be practically inapplicable for AI broadening.
Numerically, the Voigt kernel is ill-conditioned since the collisional-to-
Doppler width ratio $\Gamma_{c}$ /$\Gamma_{d}$ varies over a far wider range
for resonances than lines and therefore unconstrained a priori.

The physical processes for broadening of AI resonances differ from lines
qualitatively and quantitatively. However, line broadening processes and
formulae may be generalized to develop a theoretical treatment and
computational algorithm outlined herein (details to be presented elsewhere).
The convolved bound-free photoionization cross section of level $i$ may be
written as:

$\sigma_{i}(\omega)=\int\tilde{\sigma}(\omega^{\prime})\phi(\omega^{\prime},\omega)d\omega^{\prime},$
(7)

where $\sigma$ and $\tilde{\sigma}$ are the cross sections with plasma-
broadened and unbroadened AI resonance structures, $\omega$ is the photon
energy (Rydberg atomic units are used throughout), and
$\phi(\omega^{\prime},\omega)$ is the normalized Lorentzian profile factor in
terms of the total width $\Gamma$ due to all AI broadening processes included:

$\phi(\omega^{\prime},\omega)=\frac{\Gamma(\omega)/\pi}{x^{2}+\Gamma^{2}},$
(8)

where $x\equiv\omega-\omega^{\prime}$. The crucial difference with line
broadening is that AI resonances in the (e + ion) system correspond to and are
due to quantum mechanical interference between discretized continua defined by
excited core ion levels in a multitude of channels. The coupled channel (CC)
approximation, such as implemented by the R-matrix (RM) method (viz. [5, 7,
8]), accounts for AI resonances in an (e + ion) system with generally
asymmetric profiles (unlike line profiles that are usually symmetric). Given
$N$ core ion levels corresponding to resonance structures,

$\sigma(\omega)=\sum_{i}^{N}\left[\int\tilde{\sigma}(\omega^{\prime})\left[\frac{\Gamma_{i}(\omega)/\pi}{x^{2}+\Gamma_{i}^{(}\omega)}\right]d\omega^{\prime}\right].$
(9)

With $x\equiv\omega^{\prime}-\omega$, 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 $\Gamma_{i}$ due to all broadening
processes. The profile $\phi(\omega^{\prime},\omega)$ is centered at each
continuum energy $\omega$, convolved over the variable $\omega^{\prime}$ and
relative to each excited core ion threshold $i$ . In the present formulation
we associate the energy to the effective quantum number relative to each
threshold $\omega^{\prime}\rightarrow\nu_{i}$ to write the total width as:

$\displaystyle\Gamma_{i}(\omega,\nu,T,N_{e})$ $\displaystyle=$
$\displaystyle\Gamma_{c}(i,\nu,\nu_{c})+\Gamma_{s}(\nu_{i},\nu_{s}^{*})$
$\displaystyle+$
$\displaystyle\Gamma_{d}(A,\omega)+\Gamma_{f}(f-f;\nu_{i},\nu_{i}^{\prime}),$

pertaining to collisional $\Gamma_{c}$, Stark $\Gamma_{s}$, Doppler
$\Gamma_{d}$, and free-free transition $\Gamma_{f}$ 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 thermal Doppler, and $x^{-5/2}$ for Stark
broadening. In evaluating Eq. (10) from Eq. 9 the limits $\mp\infty$ are
replaced by $\mp\Gamma_{i}/\sqrt{\delta}$; $\delta$ is chosen to ensure the
Lorentzian profile energy range for accurate normalization (see Eq. 15).
Convolution by evaluation of Eqs. (7,9) is carried out for each energy
$\omega$ throughout the tabulated mesh of energies used to delineate all AI
resonance structures, for each cross section, and each core ion threshold. We
employ the following expressions for computations:

$\Gamma_{c}(i,\nu)\ =\
5\left(\frac{\pi}{kT}\right)^{1/2}a_{o}^{3}N_{e}G(T,z,\nu_{i})(\nu_{i}^{4}/z^{2}),$
(11)

where T, $N_{e}$ , $z$, and $A$ are the temperature, electron density, ion
charge and atomic weight respectively, and $\nu_{i}$ is the effective quantum
number relative to each core ion threshold $i$ : $\omega\equiv
E=E_{i}-\nu_{i}^{2}/z^{2}$ is a continuous variable. The Gaunt factor
$G(T,z,\nu_{i})=\sqrt{3}/\pi[1/2+ln(\nu_{i}kT/z)]$. A factor
$(n_{x}/n_{g})^{4}$ is introduced for $\Gamma_{c}$ 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 $n_{g}=2$). A treatment of the Stark effect for complex systems
entails two approaches, one where both electron and ion perturbations are
combined (viz. [20]), or separately (viz. [7, 11]) 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 $\nu_{i}^{4}$ (Eq. 5). The total Stark
width of a given $n$ -complex is $\approx(3F/z)n^{2}$, where F is the plasma
electric microfield. Assuming the dominant ion perturbers to be protons and
density equal to electrons, $N_{e}$ =$N_{p}$, we take $F=[(4/3)\pi
a_{o}^{3}N_{e})]^{2/3}$, as employed in the Mihalas-Hummer-Däppen equation-of-
state formulation [15].

$\Gamma_{s}(\nu_{i},\nu_{s}^{*})=[(4/3)\pi a_{o}^{3}N_{e}]^{2/3}\nu_{i}^{2}.$
(12)

In addition, in employing Eq. (6) a Stark ionization parameter
$\nu_{s}^{*}=1.2\times 10^{3}N_{e}^{-2/15}z^{3/5}$ is introduced such that AI
resonances may be considered fully dissolved into the continuum for
$\nu_{i}>\nu_{s}^{*}$ (analogous to the Inglis-Teller series limit [19, 15]).
Calculations are carried out with and without $\nu_{s}^{*}$ as shown later in
Table 1. The Doppler width is:

$\Gamma_{d}(A,T,\omega)=4.2858\times 10^{-7}\sqrt{(}T/A),$ (13)

where $\omega$ is not the usual line center but taken to be each AI resonance
energy. The last term $\Gamma_{f}$ in Eq. (5) accounts for free-free
transitions among autoionizing levels with $\nu_{i},\nu_{i}^{\prime}$ such
that

$X_{i}+e(E_{i},\nu_{i})\longrightarrow
X_{i}^{\prime}+e^{\prime}(E_{i}^{\prime},\nu_{i}^{\prime}).$ (14)

The large number of free-free transition probabilities for $+ve$ energy AI
levels $E_{i},E_{i}^{\prime}>0$ may be computed using RM or atomic structure
codes.

Whereas Eq.9 has an analytical solution in terms of
$tan^{-1}(x/\Gamma)/\Gamma$ evaluated at limiting values of
$x\rightarrow\mp\Gamma/\sqrt{\delta}$, 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 $\Gamma$ 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 left-wing
correction for partial re-normalization as

$\lim_{a\rightarrow-\Gamma/2\sqrt{\delta}}\int_{a}^{+\Gamma/\sqrt{\delta}}\phi(\omega,\omega^{\prime})d\omega^{\prime}=\left[\frac{1}{4}-\frac{tan^{-1}(\frac{a}{\Gamma/2\sqrt{\delta}})}{\pi}\right],$
(15)

where $a$ is the lower energy range up to the ionization threshold, reaching
the maximum value $-\Gamma/2\sqrt{\delta}$.

##  4 Results and discussion

All atomic cross sections with resonant phenomena are modified by the plasma
environment. An exemplar from large-scale opacity calculations [9, 10, 11, 13]
is presented in Fig. 1. The complexity and magnitude of computations is
demonstrated for the (e + Fe xix ) $\longrightarrow$ Fe xviii system in an
highly excited level $2s^{2}2p^{4}\ [^{3}P^{e}_{0}]\ 5s(^{2}P_{1/2})$ with
ionization energy = 13.79 Ry, compared to the ground level $2s^{2}2p^{5}\
{}^{2}P^{o}_{1/2})$ = 98.9 Ry. We utilize new results from an extensive BPRM
calculation with 276-levels dominated by $n=2,3,4$ levels of the core ion Fe
xix [10], resulting in 1,601 bound levels of Fe xviii with configurations up
to $\ n\leq 10,\ \ell\leq 9,\ J\leq 12$). Rydberg series of AI resonances
correspond to $(S_{i}L_{i}J_{i})\ n\ell,\ n\leq 10,\ell\leq 9$, with effective
quantum number defined as a continuous variable $\nu_{i}=z/\sqrt{(}E_{i}-E)\
(E>0)$, up to the highest $276^{th}$ Fe xix core level. AI resonances are
resolved for all cross sections at $\sim$45,000 photon energies [10, 13].

Figure 1: Plasma broadened photoionization cross sections for
$\hbar\omega+{\rm Fe~{}\sc xviii}\ \rightarrow e~{}+~{}{\rm Fe~{}\sc xix}\ $
of the bound level $2s^{2}2p^{4}\ [^{3}P^{e}_{0}]\ 5s(^{2}P_{1/2})$
(ionization energy 13.79 Ry), along the isotherm $T=2\times 10^{6}$K and
electron densities $N_{e}=10^{20,21,22,23}$/cc: black — unbroadened, red —
broadened, blue — broadened with Stark ionization cut-off $\nu_{s}^{*}$ (Table
1). Rydberg series of AI resonance complexes with $\nu_{i}\leq 10$ belonging
to 276 excited Fe xix levels broaden and shift with increasing density, also
resulting in continuum raising and threshold lowering. The two large features
around 68 RY and 85 Ry are combinations of Seaton PEC and Rydberg series of
resonances.

Fig. 1 displays detailed results for plasma broadened and unbroadened
photoionization cross section of one particular excited level
$2s^{2}2p^{5}[^{2}P^{o}_{3/2}]4d(^{1}F^{o}_{3})$ (ionization energy = 17.626
Ry) of Fe xvii at four densities. The main features are as follows: (I) orders
of magnitude variation in resonance heights and extent. (II) For
$N_{e}>10^{20}$/cc AI resonances begin to exhibit broadening and smearing of
overlapping Rydberg series. $N_{e}=10^{21}$cc. The narrower high-$n$ $l$
resonances dissolve into the continua but stronger low-$n$ $l$ resonance
retain their characteristic asymmetric shapes. With increasing density
$N_{e}=10^{21-23}$cc, resonance structures not only broaden but their
strengths shift and redistributed over a wide range determined by the total
width $\Gamma(\omega,\nu_{i},T,N_{e})$ at each photon energy $\hbar\omega$
(Eq. 9). (III) The averaged step-wise structure due to Stark ionization cut-
off (Table 1) represents complete dissolution into continua. (IV) It is
numerically ascertained that total AI resonance strengths are conserved, and
integrated values generally do not deviate by more than 1-2%. This is also an
important accuracy check on numerical integration and the computational
algorithm, as well as the choice of the parameter $\delta$ that determines the
energy range of the Lorentizan profile at each T and $N_{e}$. In the present
$\delta$ = 0.01 for all $N_{e}$ =$10^{20-23}$/cc.

Plasma effects on AI features Fig. 1 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. The shifts in AI
resonance strengths, akin to line shifts but much more pronouned, is
particularly important since cross sections are integrated over plasma
particle distributions in order to obtain rates for atomic proceses. Also
noteworthy is the height and extent of prominent resonances features dominated
by Seaton PEC resonances. cross sections may range up to 10 orders magnitude
in height and hundreds of eV in energy.

Table 1 gives plasma parameters corresponding to Fe xviii at along two plasma
isotherms and varying densities. The maximum width $\Gamma_{10}$ corresponding
to $\nu_{i}=10$ in Eqs. 3.2,11,12 corresponding to the $\nu$-mesh at which the
unbroadened AI resonance profiles are delineated up to $\nu\leq 10$; an
averaging procedure is employed up to $10<\nu<\infty$ using quantum defect
(QD) theory (viz. [7, 8]). $\Gamma_{c}(10)$ and $\Gamma_{s}(10)$ are the
maximum collisional and Stark width components. The thermal Doppler width
$\Gamma_{d}$ is much smaller, as may be inferred from the fact that the total
width $\Gamma$ (10) $\approx$ $\Gamma_{c}$ \+ $\Gamma_{s}$ . However, in lower
density plasmas $N_{e}<10^{20}$/cc, $\Gamma_{d}$ may be comparable to
$\Gamma_{c}$ or $\Gamma_{s}$.

In Table 1, the $\nu^{*}_{s}$ and $\nu_{D}$ are effective quantum numbers
corresponding to Stark ionization cut-off and the Debye radius respectively.
For HED plasmas with $N_{e}>10^{23}$, one needs to examine if the bound
orbitals are penetrated by the free electrons as the Debye length increases,
and plasma screening effects may need to be considered. We therefore calculate
the corresponding effective quantum number $\nu_{D}=\left[\frac{2}{5}\pi
z^{2}\lambda_{D}^{2}\right]^{1/4}$, where the Debye length
$\lambda_{D}=(kT/8\pi N_{e})^{1/2}$. It is seen in Table 1 that
$\nu_{D}>\nu^{*}_{s}$ for all T, $N_{e}$ considered, justifying neglect of
plasma screening effects herein, but which may need to be accounted for at
even higher densities.

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 redistributed resonance phenomena would be
significant and cross sections should be modified.

Table 1: Plasma parameters along isotherms $10^{6}$K and $2\times 10^{6}$K, and electron densities $N_{e}=10^{20-23}$/cc for the (e + Fe xix ) $\longrightarrow$ Fe xviii system w.r.t Fig. 1. Total AI resonance widths are shown at $\nu\approx 10$, with corresponding collisional widths $\Gamma_{c}$ (10) and Stark width $\Gamma_{s}$ (10). Effective quantum number $\nu_{s}^{\ast}$ refers to Stark ionization, and $\nu_{D}$ corresponds to the Debye radius. AI broadening widths are very weakly dependent on temperature and thermal Doppler widths are negligible in comparison with $\Gamma_{c}$ and $\Gamma_{s}$ . T(K) | $N_{e}(cc)$ | $\Gamma(\nu=10)$ | $\Gamma_{c}(10)$ | $\Gamma_{s}(10)$ | $\nu_{s}^{*}$ | $\nu_{D}$
---|---|---|---|---|---|---
|  | Ry | Ry | Ry |  |
$10^{6}$ | $10^{20}$ | 5.98(-2) | 7.56(-3) | 5.23(-2) | 14.6 | 43.2
$10^{6}$ | $10^{21}$ | 3.18(-1) | 7.56(-2) | 2.43(-2) | 10.8 | 24.3
$10^{6}$ | $10^{22}$ | 1.88(0) | 7.56(-1) | 1.13(0) | 7.93 | 13.7
$10^{6}$ | $10^{23}$ | 1.28(1) | 7.56(0) | 5.22(0) | 5.83 | 7.68
$2\times 10^{6}$ | $10^{20}$ | 5.97(-2) | 7.45(-3) | 5.23(-2) | 14.6 | 51.4
$2\times 10^{6}$ | $10^{21}$ | 3.17(-1) | 7.45(-2) | 2.43(-1) | 10.8 | 28.9
$2\times 10^{6}$ | $10^{22}$ | 1.87(0) | 7.45(-1) | 1.13(0) | 7.93 | 16.2
$2\times 10^{6}$ | $10^{23}$ | 1.27(1) | 7.45(0) | 5.23(0) | 5.83 | 9.13

##  5 Conclusion

Atomic cross sections and rates in HED plasma sources at sufficiently high
densities may be significantly affected by attenuation and broadening of AI
resonant features. Precise evaluation of equation-of-state of the plasma
determines the number of levels in predicting macroscopic properties such as
opacity and radiation transport. Utilizing the R-matrix framwork, a
computationally viable theoretical tratment taking account of plasma effects
is reviewed. The method generalizes the description of AI phenomena of
isolated Fano profiles in plasmas. Analogous to line shapes, atomic cross
sections with resonant features become energy-temperature-density dependent,
leading to broadening, shifting, and dissolving into myriad (e + ion)
continua. However, unlike symmetric line profiles, the intrinsically
asymmetric AI resonance shapes are attenuated over extended energy ranges. The
predicted energy shifts of AI resonances as the plasma density increases
should be experimentally verifiable. Redistribution of AI resonance strengths
should manifest itself in rate coefficients for (e + ion) excitation, (e +
ion) recombination, photoionization, opacities and radiation transport in HED
plasma models, using temperature-dependent Maxwellian, Planck, or other
particle distribution functions. The computational algorithm and a general-
purpose program has been developed for large-scale computations. Finally, the
complex electron correlation and coupling effects may now be computed
precisely for myriad HED plasma applications using the R-matrix method.

Acknowledgments

I would like to thank Sultana Nahar for atomic data for Fe ions used in plasma
AI broadening calculations.

References

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