Plasma Spectral Modeling

Calculating X-ray and UV spectra of hot plasma requires knowledge of the atomic transition rates and energies, as well as a code to evaluate the precise model required. Scientific usefulnessrequires that the database of atomic information, as well as the codes, be robust, .documented, and deterministic

The following processes are important in this calculation

Continuum emission processes

Free-bound emission (radiative recombination)

2-photon emission

Line emission processes

Non-Auger processes

Satellite lines emitted from excited states above the first ionization potential

Impact excitation of inner-shell electrons above the ionization limit
Dielectronic capture

Plasma Emission Models

The appropriate model depends not only on the temperature of the plasma, but also its density. At sufficiently high densities, collisions completely determine the level population. As the density drops, a collisional-radiative model must be used and finally the purely radiative nebular approximation can be used. The breakpoints between these models are discussed below

Local Thermodynamic Equilibrium (LTE)

Level populations determined only by collisional processes

Applies for $N_e > 1.8 \times 10^{14} T_e^{1/2} \Delta E_{ij}^3$

At K for H-like Fe $N_e > 2 \times 10^{27}$ cm $^{-3}$
At K for H-like O $N_e > 1 \times 10^{24}$ cm $^{-3}$

Collisional-Radiative Model (CR Model)

Most general case
Needed for $10^{14}$ cm $^{-3} < N_e < 10^{27}$ cm $^{-3}$
Also needed for complex ions at somewhat lower densities

Coronal and Nebular Models

Applicable for $N_e < 10^{14}$ to $10^{16}$ cm $^{-3}$
Low density approximation
Equilibrium established

Common Simplifying Assumptions

Most of these assumptions break down somewhere in astrophysics

In fact, most of them break down in our own Sun!

The ability to calculate more general cases is limited by availability and accuracy of atomic

The ability to parameterize an astrophysical plasma in full detail is a different, often more difficult, problem

Ionization/recombination may be solved separately from excitation/de-excitation
Either collisional processes dominate or radiative processes dominate
Optical depth effects may be treated in a simple way

ignored

escape probability formalism
Low density
- Ion population mostly in the ground state
  - Coronal approximation (collisionally ionized plasmas)
  - Nebular approximation (photoionized plasmas)
- Rate coefficients are not -sensitive
Time-independent
Maxwellian electrons
Electric and magnetic field effects are ignored
No diffusion

Calculating Line and Continuum Emission

Calculation of Level Populations and Line Intensities

$\begin{displaymath} \varepsilon = \frac{hc}{\lambda} N_k A_{kj} \end{displaymath}$

$\begin{displaymath} \frac{dN_k}{dt} = \sum_i N_i R_{ik} - N_k \sum_i R_{ki} + N_k F_k, \end{displaymath}$

$\begin{displaymath} R_{mn} = R_{ioniz} + R_{recomb} + \sum_{s} N_s q_{s,cx} + A_{rad} +\sum_{s} N_s q_{s,coll}, \end{displaymath}$

$\begin{displaymath} R_{jk} = N_e q_{ioniz} + \overline{S} \beta_{photoioniz} + N... ...{cxioniz} + \overline{J} B_{jk} + N_e q_{e,ex} + N_p q_{p,ex}, \end{displaymath}$

$\displaystyle R_{kj} = N_e \alpha_{rad} + N_e \alpha_{di} + N_e^2 \alpha_{3-body}$	$\textstyle + N_{\rm H^o} \: q_{cxrecomb} + A_{kj}$
	$\textstyle + N_e q_{e,de-ex} + N_p q_{p,de-ex}$		5)

: $\varepsilon$ emissivity
: level population density
: $A_{kj}$ transition probability from upper level to lower level .
: $R_{ioniz}$ sum of photoionization and collisional impact ionization rates
: $R_{recomb}$ sum of radiative, dielectronic, and 3-body recombination rates
: $q_{s,cx}$ individual charge exchange rate coefficient
: population density of the interacting species
: $A_{rad}$ includes stimulated absorption (photo-excitation) as well as spontaneous radiative decay
: $q_{s,coll}$ the collisional rate coefficient for interaction with species
: $\overline{J}$ and $\overline{S}$ radiative source terms for photoexcitation and photoionization, resp.
: $q_{ioniz}$ , $\beta_{photoioniz}$ and $q_{cxioniz}$ collisional, photo-, and charge exchange ionization coefficients, resp.
: $B_{jk}$ photo-excitation probability
: $\alpha_{rad}$ , $\alpha_{di}$ , $\alpha_{3-body}$ , and $q_{cxrecomb}$ radiative, dielectronic, three-body, and charge exchange recombination rate coefficients, resp.
: $q_{s,ex}$ and $q_{s,de-ex}$ collisional excitation and de-excitation rate coefficients, resp. for impact with species (electrons and sometimes protons)

Atomic Database

To calculate all these rates, we need a database of the atomic transitions. This database must include the following parameters:

Collision Strengths : $\Omega(E)$ , $\Upsilon(T)$
Ionization/Recombination Rates
- Ionization
- Auger ionization
- Recombination
- Dielectronic recombination
Radiative Processes
- Absorption
- Emission
- Photoionization
Atomic Energy Levels
References for all of the above

Collisional Excitation

Ions may be excited by collisions with electrons, protons, or other ions. Collisions with electrons are the most common, since they have the highest velocity, but in some cases proton excitation can be important.

Electron Collisional Excitation

Fundamental calculation is the cross section, which becomes a dimensionless quantity :

$\begin{displaymath}\Omega_{ij} = {{4 \pi \omega_i}\over{\lambda^2}} Q(i\rightarrow j)\end{displaymath}$
Averaging this over a Maxwellian gives the ``collision strength''

$\begin{displaymath}\Upsilon(T) = \int_0^\infty \Omega_{ij} \exp \Big( - {{E_j}\over{kT}} \Big) d \Big( {{E_j}\over{kT}} \Big)\end{displaymath}$
High-temperature approximation (see Burgess & Tully 1992, A&A, 254, 436)
- Electric dipole: $\Omega \rightarrow {\rm const} \times ln(E)$
- Multipole : $\Omega \rightarrow {\rm const}$
- Spin-change : $\Omega \rightarrow {\rm const}/E^2$
Threshold effects; R-MATRIX vs DW (from McLaughlin et al, 2001, J. Phys. B. in press)

$\includegraphics[totalheight=3in]{figures/fig1.fe18.1-4.ps}$

$\includegraphics[totalheight=3in]{figures/fig2.fe18.1-4.ps}$

Proton Collisional Excitation

Similar notation
In equilibrium, $1836\times$ slower than electrons
Affects mostly low-lying levels

Comparing Excitation Rates: He-like and Hydrogenic

$\includegraphics[totalheight=3.4in]{figures/O7R_Upsilon.ps}$ $\includegraphics[totalheight=3.4in]{figures/O7F_Upsilon.ps}$

The collison strength for the O VII $1s2p ^1P_1 \rightarrow {1s^{2}} ^1S_0$ (R) line is not strongly affected by resonances. However, the same is not true for the forbidden transition, $1s2s ^3S_1 \rightarrow {1s^2} ^1S_0$ .

$\includegraphics[totalheight=3.4in]{figures/KisVsSamp_4.ps}$ $\includegraphics[totalheight=3.4in]{figures/KisVsSamp_7.ps}$

Collision strengths for hydrogenic iron (Fe XXVI). The Sampson calculations use a non-relativistic distorted wave calculation, while the Kiselius calculations was fully relativistic. Fe XXVI exists in equilibrium between $\log(T) = 7.5 - 9$

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