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Physics Underlying the ATOMDB

Overview | Emission Models | Assumptions | Calculating Emission
Atomic Data | Collisional Excitation | Radiative Recombination | Dielectronic Recombination | Level Populations

A Postscript copy of this is also available.

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 usefulness requires that the database of atomic information, as well as the codes, be robust, documented, and deterministic.

The following processes are important in this calculation:

I.

Continuum emission processes

Free-bound emission (radiative recombination)

2-photon emission

II.

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 $T_e = 10^7$ K for H-like Fe $N_e > 2 \times 10^{27}$ cm$^{-3}$
• At $T_e = 10^5$ 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 data (and/or by computational limits).

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 $N_e$-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

$$\varepsilon = \frac{hc}{\lambda} N_k A_{kj}$$ (1)

$$\frac{dN_k}{dt} = \sum_i N_i R_{ik} - N_k \sum_i R_{ki} + N_k F_k,$$ (2)

$$R_{mn} = R_{ioniz} + R_{recomb} + \sum_{s} N_s q_{s,cx} + A_{rad} +\sum_{s} N_s q_{s,coll},$$ (3)

$$R_{jk} = N_e q_{ioniz} + \overline{S} \beta_{photoioniz} + N... ...{cxioniz} + \overline{J} B_{jk} + N_e q_{e,ex} + N_p q_{p,ex},$$ (4)

$$R_{kj} = N_e \alpha_{rad} + N_e \alpha_{di} + N_e^2 \alpha_{3-body} + N_{\rm H^o} \: q_{cxrecomb} + A_{kj}$$

$$+ N_e q_{e,de-ex} + N_p q_{p,de-ex}$$ (5)

$\varepsilon$ emissivity

$N_k$ level population density

$A_{kj}$ transition probability from upper level $k$ to lower level $j$.

$R_{ioniz}$ 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

$N_s$ 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 $s$

$\overline{J}$ 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 $s$ (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:

1. Collision Strengths : $\Omega(E)$, $\Upsilon(T)$
2. Ionization/Recombination Rates
   • Ionization
   • Auger ionization
   • Recombination
   • Dielectronic recombination
3. Radiative Processes
   • Absorption
   • Emission
   • Photoionization
4. Atomic Energy Levels
5. 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 :
  
  $$\Omega_{ij} = {{4 \pi \omega_i}\over{\lambda^2}} Q(i\rightarrow j)$$
  
  

• Averaging this over a Maxwellian gives the ``collision strength''
  
  $$\Upsilon(T) = \int_0^\infty \Omega_{ij} \exp \Big( - {{E_j}\over{kT}} \Big) d \Big( {{E_j}\over{kT}} \Big)$$
  
  

• 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)

Proton Collisional Excitation

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

Comparing Excitation Rates: He-like and Hydrogenic

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, $1s2s ^3S_1 \rightarrow {1s^2} ^1S_0$.

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 $\log(T) = 7.5 - 9$.

Calculating Radiative Recombination Continuum From a Hot Plasma

When an electron collides with an atom or ion, it may

• excite the atom/ion ( $I + e \rightarrow I^{*} + e$)
• ionize the atom/ion ( $I + e \rightarrow I^{+} + 2e$)
• scatter inelastically ( $I + e \rightarrow I + e + \gamma$)
• recombine radiatively ( $I + e \rightarrow I^- + \gamma$)
• recombine dielectronically ( $I + e \rightarrow I^{-*}$)

This memo will focus on the penultimate process, where an electron collides and recombines with an ion, emitting a photon in the process. The energy of this photon will equal the kinetic energy of the electron plus the binding energy of the newly-recombined electron. Since the kinetic energy of the electron is not quantized, this forms a continuous spectrum with sharp edges at the binding energy of the levels.

The power emitted per keV by this process is (Tucker & Gould 1966):

$${{dE}\over{dt dV d\omega}} = {{dP}\over{dE}} = n_e n_{Z,j+1} E_{\gamma} \sigma^{rec}(E_e) v_e {{f(v) dv}\over{dE_{\gamma}}}$$ (6)

where $n_e$ is the electron density, $n_{Z,j+1}$ is the density of the ion (Z,j+1) (where Z is the atomic number of the ion, and j+1 is the ionization state), $E_{\gamma}$ is the energy of the emitted photon, $\sigma_n^{rec}(E_e)$ is the recombination cross section to level $n$ at the electon energy $E_e$, $v_e$ is the electron velocity and $f(v) dv$ is the number of electrons with velocities in the range $(v,v+dv)$. In most cases, $f(v)$ is the Maxwell-Boltzman distribution,

$$f(v) dv = 4 \pi \Big({{m}\over{2\pi kT}}\Big)^{3/2} v^2 \exp(-{{m v^2}\over{2kT}}) dv$$ (7)

Equation (6) can be simplified considerably. If we define $I_{Z,j,n}$ to be the binding energy for an electron in level $n$ of the ion $(Z,j)$, and use the fact that $E_e = 0.5 m_e v_e^2$, we get that

$${{dv_e}\over{dE_\gamma}} = {{1}\over{m_e v_e}}$$ (8)

Using this result, and substituting the Maxwell-Boltzman equation for $f(v)$, we can rewrite (6) as

$${{dP}\over{dE}} = n_e n_{Z,j+1} 4 \Big({{E_{\gamma} - I_{Z,j... ...Z,j,n}) \exp(-{{E_{\gamma} - I_{Z,j,n}}\over{kT}} ) E_{\gamma}$$ (9)

Equation (9) completely describes the spectrum of the radiative recombination continuum (RRC). However, cross sections for recombination are not generally calculated by the atomic physics community; photoionization cross sections are. As photoionization and recombination are inverse processes,

$$I_{Z,j,n} + \gamma \stackrel{\textstyle \leftarrow}{\rightarrow} I_{Z,j+1} + e^-$$ (10)

they can be related using detailed balancing. Raymond & Smith (1977) state that:

$${{\sigma^{ph}_{Z,j,n} (\nu)}\over{\sigma^{rec}_{Z,j,n}(v_e)}... ... c^2 v_e^2}\over{E_{\gamma}^2}} {{g_{Z,j+1}}\over{g_{Z,j,n}}}.$$ (11)

Here, $\sigma^{ph}_{Z,j,n}(\nu)$ is the photoionization cross section for the ion $(Z,j)$ in state $n$ for a photon with frequency $\nu$, and $\sigma^{rec}_{Z,j+1}(v_e)$ is the recombination cross section for an electron with velocity $v_e$ to combine with an ion $(Z,j+1)$ (assumed to be in the ground state) to create an ion $(Z,j)$ in state $n$. Additionally, $m_e$ is the mass of the electron, $c$ the speed of light, $E_{\gamma}$ is the energy of the photon, and $g_{Z,j+1}$ and $g_{Z,j,n}$ are the statistical weights for the $(Z,j+1)$ ion in its ground state and the $(Z,j)$ ion in state $n$, respectively.

Detailed Balance

Equation (11) can be derived from first principles, as was first shown by Milne (1924). We begin with the assumption that the system is in thermal equilibrium. In this case the emission due to spontaneous and stimulated recombination is balanced by the ionization of the recombined ions (see also Cowen 1980; Shu 1991):

$$n_{Z,j,n} {{4 \pi B_{\nu}(T)}\over{h\nu}} \sigma^{ph}_{Z,j,n... ... \sigma^{rec}_{Z,j+1}(v) + \alpha^{stim} B_{\nu}(T)\Big\} \\$$ (12)

Here, $B_{\nu}(T)$ is the Planck blackbody emission function, and $f(v)$ is the Boltzman equation (7). The $\alpha^{stim}$ term allows for stimulated recombination. Written in expanded form, Equation (12) becomes

$$n_{Z,j,n} \sigma^{ph}(\nu) {{8 \pi h \nu^3/c^2}\over{e^{{h\n... ...}(v) {{2 h \nu^3/c^2}\over{e^{{h\nu}\over{k T}} - 1}} \Big\}.$$ (13)

Cancelling common terms and solving for $\sigma^{ph}(\nu)$, we get

$$\sigma^{ph}(\nu) = {{n_e n_{Z,j+1}}\over{n_{Z,j,n}}} (e^{{h\... ...m}(v) {{2 h \nu^3/c^2}\over{ e^{{h\nu}\over{k T}} - 1}} \Big\}$$ (14)

Equation (14) can be simplified further using the Saha equation

$${{n_{Z,j+1}}\over{n_{Z,j,n}}} = {{2 g_{Z,j+1}}\over{g_{Z,j,n... ...e k T)^{3/2}}\over{h^3 n_e}} \exp( - {{I_{Z,j,n}\over{k T}}}),$$ (15)

which is applicable because of the assumption of thermal equilibrium. Substituting this in gives

$$\sigma^{ph}(\nu) = {{2 g_{Z,j+1}}\over{g_{Z,j,n}}} {{(2 \pi ... ...}(v) {{2 h \nu^3/c^2}\over{ e^{{h\nu}\over{k T}} - 1}} \Big\}.$$ (16)

This can be simplified further by cancelling common factors, and we also use the relationship $h d\nu = m_e v dv$ to get:

$$\sigma^{ph}(\nu) = {{g_{Z,j+1}}\over{g_{Z,j,n}}} {{m_e^2 c^2 v^2}\over{h^2 \nu^2}} \... ...}\over{k T}}) - 1) + \alpha^{stim}(v) \big({{2 h \nu^3}\over{c^2}} \big) \Big\}$$ (17)

$$= {{g_{Z,j+1}}\over{g_{Z,j,n}}} {{m_e^2 c^2 v^2}\over{h^2 \nu^2}} \... ...})\big(\alpha^{stim}(v)({{2 h \nu^3}\over{c^2}}) - \sigma^{rec}(v) \big) \Big\}$$ (18)

But since $\sigma^{rec}(v)$ and $\alpha^{stim}(v)$ are atomic constants, they cannot depend on the temperature of the medium, or on its equilibrium state. Therefore the following relationship must hold:

$$\alpha^{stim}(v) = {{c^2}\over{2 h \nu^3}} \sigma^{rec}(v),$$ (19)

which leads to our desired result, Equation (11).

Spectral Calculation

Radiative recombination gives rise to a continuum of emission, with a minimum energy equal to the binding energy of the ion in its final state. The power emitted per unit energy at an energy $E_{\gamma}$ is given by

$${{dP}\over{dE}}(E_{\gamma}) = n_e n_{Z,j+1} E_{\gamma} \sigm... ...j+1 \rightarrow Z,j,n}(E_e) v_e f(v) {{dv}\over{dE_{\gamma}}},$$ (20)

where $E_e$ is the initial electron energy, and $v_e$ the initial electron velocity. Using Equation (11), we can restate this in terms of the photoionization cross section $\sigma^{ph}_{Z,j,n}$:

$${{dP}\over{dE}}(E_{\gamma}) = {{4 \pi}\over{c^2}} (2 \pi m_e... ...amma} - I_{Z,j,n}}\over{kT}}) \sigma^{ph}_{Z,j,n}(E_{\gamma}).$$ (21)

After substituting in values for the constants, we get:

$${{dP}\over{dE}}(E_{\gamma}) = 1.31 \times 10^8 {\rm erg cm/... ... ({{T}\over{\rm 1 K}})^{-3/2} \sigma^{ph}_{Z,j,n}(E_{\gamma}).$$ (22)

Practical Consideration

Given the photoionization cross section, the ionization energy, and the statistical weights of the levels involved it is trivial to use Equation (22) to calculate the emission due to radiative recombination. However, a number of complications remain:

1. The photoionization cross sections have not been calculated for all levels of all ions.
2. Each ion has an infinite number of bound states which an electron could recombine into; some method of cutting off the level calculation must be done.
3. Calculating the power emitted per energy bin should properly be done as an integral of dP/dE over the bin. However, this requires a substantial amount of computation that will slow down the entire code.

We will consider each of these problems and discuss how they have been addressed.

Cross Sections

The most substantial problem in creating any plasma emission code is the lack of accurate atomic data. Verner & Yakovlev (1995) used a Hartree-Dirac-Slater (HDS) code to calculate the partial photoionization cross section for all subshells $nl$ of all ions with $Z \le 30$. We use these results to calculate recombination to the ground state of each ion. A subsequent paper (Verner et al. 1996) enhanced these results near threshold by including data from the Opacity Project. However, much of the improvement is due to including autoionization resonances. In the context of recombination, these will be considered in this code as dielectronic recombination, not radiative recombination. Therefore, for our purposes the HDS code data which does not include resonances is more appropriate.

Calculating recombination to non-ground states requires data on the cross section of excited ions. In the case of hydrogenic ions, the exact cross section can be calculated (Karzas & Latter 1961; Boardman 1964). However, for more complex ions, only very limited data is available. The most complete set of data is by Clark, Cowan, & Bobrowicz (1986), who calculated the configuration-averaged photoionization cross sections for all subshells between 1s and 5g for He-like through Al-like ions. One restriction on the data, however, is that it is only valid for ions more than three times ionized. As a result of the configuration averaging, however, some important data is not available. For example, a triplet of lines from the $n=2$ level of He-like ions exists that is very strong in many astrophysical environments. It consists of a resonance line (1s2p $^1P_1 \rightarrow$ 1s$^2$ $^1S_1$), a forbidden line (1s2s $^3S_1 \rightarrow$ 1s$^2$ $^1S_1$), and an intercombination line (1s2s $^3P_1 \rightarrow$ 1s$^2$ $^1S_1$). In addition, the $n=2$ level also has the strictly forbidden transition 1s2s $^1S_1 \rightarrow$ 1s$^2$ $^1S_1$ which gives rise to two-photon emission. The Clark et al. data, however, does not distinguish between the triplet and the singlet states, and so will not allow us to calculate the ratio of the forbidden and resonance lines.

Table 1 lists the sources of the cross sections used in the code to date. The most important issue for the code is not the calculation of the continuum emission, which (except in the case of a photoionized plasma) is usually smaller than the bremsstrahlung emission. Rather, recombination can affect level populations and thereby change line ratios.

Table 1: Sources of photoionization cross section data for all levels

Ion

H

He

C

N

O

Ne

Mg

Al

Si

S

Ar

Ca

Fe

Ni

I

H

II

N

H

III

-

N

IV

-

-

V

-

-

C

C

C

C

C

C

C

C

VI

-

-

H

C

C

C

C

C

C

C

C

VII

-

-

N

H

C

C

C

C

C

C

C

VIII

-

-

-

-

H

C

C

C

C

C

C

C

IX

-

-

-

-

N

C

C

C

C

C

C

C

X

-

-

-

-

-

H

C

C

C

C

C

C

XI

-

-

-

-

-

N

C

C

C

C

C

C

XII

-

-

-

-

-

-

H

C

C

C

C

C

XIII

-

-

-

-

-

-

N

H

C

C

C

C

XIV

-

-

-

-

-

-

-

N

H

C

C

C

C

XV

-

-

-

-

-

-

-

-

N

C

C

C

C

XVI

-

-

-

-

-

-

-

-

-

H

C

C

C

C

XVII

-

-

-

-

-

-

-

-

-

N

C

C

C

C

XVIII

-

-

-

-

-

-

-

-

-

-

H

C

C

C

XIX

-

-

-

-

-

-

-

-

-

-

N

C

C

C

XX

-

-

-

-

-

-

-

-

-

-

-

H

C

C

XXI

-

-

-

-

-

-

-

-

-

-

-

N

C

C

XXII

-

-

-

-

-

-

-

-

-

-

-

-

C

C

XXIII

-

-

-

-

-

-

-

-

-

-

-

-

C

C

XXIV

-

-

-

-

-

-

-

-

-

-

-

-

C

C

XXV

-

-

-

-

-

-

-

-

-

-

-

-

C

C

XXVI

-

-

-

-

-

-

-

-

-

-

-

-

H

C

XXVII

-

-

-

-

-

-

-

-

-

-

-

-

N

C

XXVIII

-

-

-

-

-

-

-

-

-

-

-

-

-

H

XXIX

-

-

-

-

-

-

-

-

-

-

-

-

-

N

Sources:

H

Exact hydrogenic solution available

C

From Clark, Cowen & Bobrowicz, 1986, ADNDT, 34, 3, 419

N

None needed; no electrons to photoionize.

Level Cutoffs

Each ion has an infinite number of bound states but also has a finite total recombination rate. Therefore, the cross section for recombination must drop rapidly as a function of $n$. Of course, as stated above very little data exists beyond the ground state recombination rate. Despite this shortage of data on cross sections on excited ions, we can approximate the recombination to excited levels when necessary by using a hydrogenic model (see Péquignot, Petitjean, & Boisson 1991). This is also the method used by SPEX (Mewe & Kaastra 1994). However, We use a different method than theirs to determine the maximum level to consider for recombination.

The maximum radiative RRC occurs near the ionization energy of the level, since that is where the cross section tends to be large and because of the exponential damping term (see Eq. 9). Therefore, when asking if recombination to a given level will be important we can examine only the case of zero-energy electrons, where the emitted photon's energy comes entirely from the binding energy of the level. In this case, we care only about the hydrogenic cross section at threshold. Using Kramer's semiclassical approximation, this is

$$\sigma_n(E_{th}) = 7.91\times10^{-18} {\rm cm}^2 {{n}\over{Z_{\rm eff}^2}}$$ (23)

where Z$_{\rm eff}$ is the is the atomic number $Z$ of the element under consideration minus the number of electrons $j$ screening its charge. When we use this approximation to the cross section, Equation (22) becomes:

$${{dP}\over{dE}}(I_{Z,j,n}) = 1.31 \times 10^8 {\rm erg cm/s/keV } n_e n_{Z,j+1} ({{I_{Z,j,n}... ...}}\over{g_{Z,j+1}}} ({{T}\over{\rm 1 K}})^{-3/2} \sigma^{ph}_{Z,j,n}(I_{Z,j,n})$$ (24)

$$\approx 5.21\times10^{-15} {\rm erg cm^3/s/keV } n_e n_{Z,j+1} ({{Z_{\rm eff}^4}\over{n^3}}) {{1}\over{g_{Z,j+1}}} ({{T}\over{\rm 1 K}})^{-3/2}$$ (25)

where we also used the fact that $g_{Z,j,n} = 2n^2$. As this shows, the importance of higher $n$ states to the radiative recombination continuum drops off as 1/n$^3$.

To decide where to stop calculating the recombination continuum, we compare the RRC emission at threshold to the bremsstrahlung emission at the same energy. If the RRC emission is less than some chosen fraction of of the bremsstrahlung for this level, then we assume that recombination to this and all higher levels of this ion are negligible.

Integrating dP/dE

After the choice of data is made and the maximum $n$ level to calculate recombination to has been chosen, one further issue exists: how to estimate the total emission in a given energy bin. The exact method would be to integrate the emission over the bin:

$$\Lambda_{RRC}(E_{\rm bin}) = \int_{E_0}^{E_1} {{dP}\over{dE}}(E) dE$$ (26)

where $E_0, E_1$ are the minimum and maximum energies for the bin, respectively. However, doing a numerical integration is slow and if $dP/dE$ is nearly constant over the bin energies, unnecessary.

We calculate the emission as follows:

1. Calculate the emission at the bin edges
2. If the two results differ by less than some constant $\epsilon$, average them and multiply by the bin width to get the total emissivity.
3. If the results differ by more than $\epsilon$, then use a numerical integration routine to calculate the emissivity.
4. If a numerical integration is used, the code returns a low-level warning to the user (which may be safely ignored) suggesting that a finer binning should be used to measure the RRC emission accurately.

References

Boardman, W. J. 1964, ApJS, 9, 185

Clark, R. E. H., Cowan, R. D., & Bobrowicz, F. W. 1986, ADNDT, 34, 415

Cowan R. D. 1980, ``The Theory of Atomic Structure and Spectra'' (Berkeley: University of California Press), pp 547-548

Karzas, W. J. & Latter, R. 1961, ApJS, 6, 167

Mewe, R. & Kaastra, J. S. 1994, ``SPEX - Continuum Radiation Processes'' (SRON Report TRPB03, Version 1.0/Revision 2.0; available at http://saturn.sron.ruu.nl/general/projects/spex/trpb03.ps.Z)

Péquignot, D., Petitjean, P., & Boisson, C. 1991, A&A, 251, 680

Raymond, J. C. & Smith, B. W 1977, ApJS, 35, 419

Shu, F. H. 1991, ``The Physics of Astrophysics: Radiation'' (Mill Valley: Univerisity Science Books), pp 75-76

Tucker, W. H. 1975, ``Radiation Processes in Astrophysics'' (Boston: MIT Press)

Tucker, W. H. & Gould, R. J. 1966, ApJ, 144, 244

Verner, D. A. & Yakovlev, D. G. 1995, A&AS, 109, 125

Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996, ApJ, 465, 487

Calculating Dielectronic Satellite Lines from a Hot Plasma

When an electron collides with an atom or ion, it may

• excite the atom/ion ( $I + e \rightarrow I^{*} + e$)
• ionize the atom/ion ( $I + e \rightarrow I^{+} + 2e$)
• scatter inelastically ( $I + e \rightarrow I + e + \gamma$)
• recombine radiatively ( $I + e \rightarrow I^- + \gamma$)
• recombine dielectronically ( $I + e \rightarrow I^{-*}$)

This memo will focus on the final process, where an electron recombines with the ion in an excited state and excites a second electron while doing so. The ion is left in a highly excited state, which may then autoionize (thereby inverting the dielectronic recombination and converting the process into a simple scattering event) or it may radiatively decay when one of the excited electrons radiatively decays, creating a satellite line. A satellite line is so named because it will be at a slightly longer wavelength than the normal transition from an electron in that energy level. The due to the recombined electron which is also in an excited level.

For a simple example, consider the process of O VIII in the ground state, recombining dielectronically. Thus we may have the following series of events:

$$\hbox{O {\sc viii}} 1s ^2S + e \rightarrow \hbox{O {\sc vii}}\ 2s3p ^3P_2$$ (27)

$$\rightarrow \hbox{O {\sc vii}} 1s2s ^3S_1 + \gamma (16.485\AA)$$ (28)

$$\rightarrow \hbox{O {\sc vii}} {1s^2} ^1S_0 + \gamma (22.10 \AA)$$ (29)

In this case, the first photon emitted is a satellite line of O VIII, so-called since the strength of the line depends upon the abundance of the parent ion, although the line itself is a transition of the daughter ion.

Dielectronic recombination is a resonance phenomenon. The recombining electron must have a kinetic energy that equals the sum of the energies of the two excited levels. For a Maxwellian distribution, Bates & Dalgarno (1962) shows that in the ``isolated resonance approximation,'' the dielectronic recombination rate (in units of $cm^3/s$) is

$$\alpha^{DR}(i\rightarrow d) = \Big[{{4\pi}\over{T}} \Big]^{3/2} a_0^3 \exp\Big({{E_c}\over{k_BT}} \Big) V_a(i\rightarrow d)$$ (30)

where $T$ is in rydbergs, $E_c$ is the energy of the doubly-excited state, $a_0$ is the Bohr radius in cm, and $V_a$ is the capture probability in $s^{-1}$. Since dielectronic recombination is the inverse process of autoionization, the capture probability $V_a$ is related to the autoionization probability $A_a$ by the principle of detailed balance. So, what is $V_a$? That is given by

$$V_a = {{g_s}\over{2g_l}}{{A_a A_r}\over{\sum A_a + \sum A_r}}$$ (31)

where $g_s$ is the statistical weight of the doubly-excited state, $g_l$ is the statistical weight of the parent ion in its initial (ground) state, $A_a$ is the autoionization rate from the doubly-excited state and $A_r$ is the radiative rate. The sums in the denomenator are over all possible transitions from the doubly-excited state, not just the one we're interested in.

Satellite Line Emissivity

Now, we need to move from pure theory to something with actual units that can be seen. Most of this discussion is taken from Kato et al. (1997). In that paper, the equation

$$I_S(T,l\rightarrow d) = 3.30\times 10^{-24} \hbox{ph\ cm}^3\... ...}\over{kT}})^{-3/2} {{Q_{d}}\over{g_l}} exp(-E_{c}/kT) n_l n_e$$ (32)

gives the satellite line intensity. The constant $3.30\times10^{-24}$ equal to ${4\pi a_0}^{3/2}/2$; $I_H$ is the Rydberg constant, and $Q_{d} = g_s A_a B_r$, where

$$B_r = {{A_r}\over{\sum A_a + \sum A_r}}$$ (33)

After dielectronic recombination occurs, the ion is in an ``Auger unstable'' state, which can radiatively stabilize to a singly excited state $j$ of the recombined ion (with rate $A_r$), emitting a satellite line photon. Or, it can autoionize to some state of the parent ion (with rate $A_a$), emitting an electron back into the continuum. Of course, there may be multiple ways for the doubly-ionized state to raditively decay or to autoionize, which is where the sums come from in the denominator.

References

Bates, D. R. & Dalgarno, A. 1962, in ``Atomic and Molecular Processes,'', ed. D. R. Bates (New York: Academic Press), p. 258

Kato, T, Safronova, U. I., Shlyaptseva, A. S., Cornille, M., Dubau, J. & Nilsen, J. 1997, ADNDT, 67, 225

Romanik, C 1988, ApJ, 330, 1022

Calculating Level Populations of Ions in a Hot Plasma

Calculating the level population of an ion in collisional ionization equilibrium is conceptually quite simple. The most trivial example is that of a two level ion, with a ground state (1) and an excited state (2). In this case, the level population is calculated by balancing the excitation and de-excitation rates. Assume the collisional excitation rate is $\gamma_{\ensuremath{1 \rightarrow 2}}(T)$cm$^{-3}$s$^{-1}$, the collisional de-excitation rate is $\gamma_{\ensuremath{2 \rightarrow 1}} (T)$cm$^{-3}$s$^{-1}$, and the radiative rate is $A_{\ensuremath{2 \rightarrow 1}}$s$^{-1}$. Then in equilibrium the excitation rate equals the de-excitation rate so:

$$n_e p_1 \gamma_{\ensuremath{1 \rightarrow 2}} = n_e p_2 \gam... ...emath{2 \rightarrow 1}} + p_2 A_{\ensuremath{2 \rightarrow 1}}$$ (34)

where $p_1$ and $p_2$ are the fractional level populations of level 1 and 2, respectively. In the case of a two-level atom, balancing the excitations and de-excitations into the ground state or the excitated state gives identical equations. This is not the case for atoms with more than 2 levels, however, as will be seen.

Continuing, we can solve equation (34) for $p_2/p_1$ we get

$${{p_2}\over{p_1}} = {{n_e \gamma_{\ensuremath{1 \rightarrow ... ...remath{2 \rightarrow 1}} + A_{\ensuremath{2 \rightarrow 1}}}}.$$ (35)

Finally, using the requirement that $p_1 + p_2 = 1$, we get:

$$p_1 = {{n_e \gamma_{\ensuremath{2 \rightarrow 1}} + A_{\ensuremath{2 \r... ...} + \gamma_{\ensuremath{1 \rightarrow 2}}) + A_{\ensuremath{2 \rightarrow 1}}}}$$ (36)

$$p_2 = {{n_e \gamma_{\ensuremath{1 \rightarrow 2}}}\over{n_e (\gamma_{\e... ...} + \gamma_{\ensuremath{1 \rightarrow 2}}) + A_{\ensuremath{2 \rightarrow 1}}}}$$ (37)

We can now consider the slightly more complex example of a three-level atom. In this case, we will have a larger number of excitation and de-excitation rates: $\gamma_{\ensuremath{1 \rightarrow 2}}$, $\gamma_{\ensuremath{1 \rightarrow 3}}$, $\gamma_{\ensuremath{2 \rightarrow 3}}$, $\gamma_{\ensuremath{3 \rightarrow 1}}$, $\gamma_{\ensuremath{3 \rightarrow 2}}$, $\gamma_{\ensuremath{2 \rightarrow 1}}$, $A_{\ensuremath{2 \rightarrow 1}}$, $A_{\ensuremath{3 \rightarrow 1}}$, and $A_{\ensuremath{3 \rightarrow 2}}$. The method remains the same, however: in equilibrium, the excitation and de-excitation rates out of each level are balanced. Therefore, we can write the following equations:

$$p_1 + p_2 + p_3 = 1$$ (38)

$$n_e (p_2 \gamma_{\ensuremath{2 \rightarrow 1}} + p_3 \gamma_{\ens... ...) + p_2 A_{\ensuremath{2 \rightarrow 1}} + p_3 A_{\ensuremath{3 \rightarrow 1}} = n_e p_1 (\gamma_{\ensuremath{1 \rightarrow 2}} + \gamma_{\ensuremath{1 \rightarrow 3}})$$ (39)

$$n_e (p_1 \gamma_{\ensuremath{1 \rightarrow 2}} + p_3 \gamma_{\ensuremath{3 \rightarrow 2}}) + p_3 A_{\ensuremath{3 \rightarrow 2}} = n_e p_2 (\gamma_{\ensuremath{2 \rightarrow 1}} + \gamma_{\ensuremath{2 \rightarrow 3}}) + A_{\ensuremath{2 \rightarrow 1}}$$ (40)

$$n_e (p_1 \gamma_{\ensuremath{1 \rightarrow 3}} + p_2 \gamma_{\ensuremath{2 \rightarrow 3}}) = n_e p_3 (\gamma_{\ensuremath{3 \rightarrow 2}} +\gamma_{\ensurema... ...rrow 1}}) + A_{\ensuremath{3 \rightarrow 1}} + A_{\ensuremath{3 \rightarrow 2}}$$ (41)

We now have four equations but only three variables ($p_1, p_2$, and $p_3$). This system is degenerate, and so one equation must be removed. This must be done with some care, however. Assume we randomly select a level $i$ and remove it from the equations to be solved. If $A_{\ensuremath{i \rightarrow j}} \equiv \gamma_{\ensuremath{i \rightarrow j}} \equiv \gamma_{\ensuremath{j \rightarrow i}} \equiv 0$ for all $j$, then level $i$ is not connected to any other level. Levels of this type must be removed from the level population calculation in any event, but removing it will not remove the degeneracy. The only level which we can guarantee will be connected to others is the ground state level. Therefore, removing the ground state balancing equation when calculating the level population assures that the population equations will be soluble.

Equations (41) can be written in matrix form, as follows:

$$\left[ \begin{array}{ccc} 1 & 1 & 1 \\ n_e \gamma_{\ensure... ...t] = \left[ \begin{array}{c} 1 \ 0 \ 0 \end{array}\right]$$ (42)

Solving for the level population, then, is done by inverting this matrix equation. This method can easily be extended to $n$ levels. Note the density dependence is explicitly included, since it affects the collisional rates but not the radiative rates.

Equation (42) covers the case of an ion in isolation. However, ionization and recombination can affect the results. Ionization to an excited level can occur, but is rare. However, recombination to an excited state can and does regularly happen, via two different processes: radiative recombination and dielectronic recombination. We will consider these separately.

In the case of radiative recombination to an excited level, the rate for recombination to the $n$th level is $\alpha^{RR}_n$, in units of cm$^{3}$/s. The total rate per unit volume is $n_e n_{I^+} \alpha^{RR}_n$, where $n_I$ is the density of the ionized atom.

Dielectronic recombination occurs when an electron recombines and simultaneously excites an electron in the recombined atom, resulting in a doubly-excited state. This can be resolved by auto-ionization or by radiative stabilization. In this latter case, first one electron radiatively transitions to a lower level (creating a satellite line) and then the atom is left in a singly excited state. The rate for such recombinations can be written as $\alpha^{DR}_n n_e n_{I^+}$, in recombinations per second.

The total recombination rate to an excited level $n$ is therefore $n_e n_{I^+} (\alpha^{RR}_n + \alpha^{DR}_n)$. We can now re-write equations 5-8 including this term and get:

$$p_1 + p_2 + p_3 = 1$$ (43)

$$n_e (p_2 \gamma_{\ensuremath{2 \rightarrow 1}} + p_3 \gamma_{\ens... ...remath{3 \rightarrow 1}} + (\alpha^{DR}_1 + \alpha^{RR}_1){{n_{I^+}}\over{n_I}} = n_e p_1 (\gamma_{\ensuremath{1 \rightarrow 2}} + \gamma_{\ensuremath{1 \rightarrow 3}})$$ (44)

$$n_e (p_1 \gamma_{\ensuremath{1 \rightarrow 2}} + p_3 \gamma_{\ens... ...remath{3 \rightarrow 2}} + (\alpha^{DR}_2 + \alpha^{RR}_2){{n_{I^+}}\over{n_I}} = n_e p_2 (\gamma_{\ensuremath{2 \rightarrow 1}} + \gamma_{\ensuremath{2 \rightarrow 3}}) + A_{\ensuremath{2 \rightarrow 1}}$$ (45)

$$n_e (p_1 \gamma_{\ensuremath{1 \rightarrow 3}} + p_2 \gamma_{\ensuremath{2 \rightarrow 3}}) + (\alpha^{DR}_2 + \alpha^{RR}_2){{n_{I^+}}\over{n_I}} = n_e p_3 (\gamma_{\ensuremath{3 \rightarrow 2}} +\gamma_{\ensurema... ...rrow 1}}) + A_{\ensuremath{3 \rightarrow 1}} + A_{\ensuremath{3 \rightarrow 2}}$$ (46)

Since this rate is not proportional to the any of the level populations of the recombined atom, it cannot be in the $N\times N$ matrix, but rather must be on the right hand side of the equation. In addition, all the rates in equation (42) are proportional to the atom density $n_I$, while the recombination rates are proportional to $n_{I^+}$. This requires adding the factor of $n_{I^+}/n_I$ seen above. Therefore in the case of the 3 level atom, we have for the matrix formulation:

$$\left[ \begin{array}{ccc} 1 & 1 & 1 \\ n_e \gamma_{{1}{2}}... ...over{n_I}} (\alpha^{RR}_3 + \alpha^{DR}_3) \end{array}\right]$$ (47)

Note that arrows have been left out of the subscripts; $\gamma_{21}$ should be read as $\gamma_{\ensuremath{2 \rightarrow 1}}$.

Last modified: 28 June 2002

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