A&A 426, 699-705 (2004)
DOI: 10.1051/0004-6361:20040463
O. Zatsarinny1, - T. W. Gorczyca1 - K.
Korista1 - N. R. Badnell2 - D. W. Savin3
1 - Department of Physics, Western Michigan University,
Kalamazoo, Michigan, 49008-5252, US
2 -
Department of Physics, University of Strathclyde,
Glasgow, G4 0NG, UK
3 -
Columbia Astrophysics Laboratory,
Columbia University, New York, 10027, USA
Received 17 March 2004 / Accepted 28 May 2004
Abstract
Dielectronic recombination (DR) and radiative recombination (RR)
data for neon-like ions forming sodium-like systems
has been calculated as
part of the assembly of a DR database
necessary for modelling of dynamic and/or finite-density plasmas
(Badnell et al. 2003).
Dielectronic recombination coefficients for neon-like ions from
Na+ to Zn20+,
as well as Kr26+, Mo32+, Cd38+, and Xe44+,
are presented and the results discussed.
Key words: atomic data - atomic processes - plasmas
Dielectronic recombination (DR) is an important recombination process for laboratory and astrophysical plasmas. Accurate DR rate coefficients are essential in the determination of the ionization balance and in the interpretation of most types of astrophysical spectra. The DR process has been subject to intense theoretical study, but the existing sophisticated calculations have been carried out only for specific ions, and most available data are based on simplified models. Recent experimental measurements (Savin et al. 1999, 2002) have demonstrated that many of the earlier computations of DR rate coefficients are inaccurate and that systematic computations with improved approximations are needed. A review of the published theoretical DR data for L-shell ions as of late 2001 is given by Savin & Laming (2002).
We have initiated a program to generate a total and final state level-resolved intermediate coupling DR database necessary for spectroscopic modelling of dynamic and/or finite density plasmas (Badnell et al. 2003). To this end, work has been underway in the calculation of DR data for the hydrogen through fluorine-like isoelectronic sequences and beyond. In this paper, we describe calculations and results for dielectronic recombination data for neon-like ions forming sodium-like systems.
Although studies for a few elements of this sequence have been made, there are no systematically determined DR rate coefficients for the entire sequence. The first calculations of DR rate coefficients for neon-like ions were carried out by Jacobs et al. (1977, 1979) for Mg, Si, S, and Fe atoms using a simplified model. In this model, the autoionization rates were obtained from the threshold values of the partial-wave electron-impact excitation cross sections for the corresponding ions by means of the quantum-defect theory relationship between these values. The excitation cross sections were obtained in the distorted-wave approximation in LS coupling, and only dipole 2s-3p, 2p-3s, and 2p-3d transitions were included. In addition, radiative decay of autoionizing states was approximated as the ionic core radiative decay rate. As we will discuss, this simplified model gives fairly accurate high-temperature DR rate coefficients for low-charged Mg2+, but underestimates DR rate coefficients for the heavier Si4+, S6+, and Fe16+ ions by large factors at the peak in the DR rate coefficient.
The first sophisticated study of DR rate coefficients for neon-like ions
was carried out by Chen (1986). The calculations were for seven ions
with atomic number
,
and used the multiconfiguration
Dirac-Fock (MCDF) model to evaluate detailed transition energies and
rates. DR rate coefficients for 10 temperatures in the range
keV were computed. Chen (1986) also studied the dependence of DR rate coefficients from the intermediate states
and
as functions of the principal quantum number n and orbital angular momentum l.
One conclusion of
his work was that relativistic effects
on total DR rate coefficients of neon-like ions are negligible;
however, the LSJ splitting can give rise to a noticeable redistribution in
partial DR rate coefficients.
Calculations of DR rate coefficients for neon-like ions were also
performed by Romanik (1988) in an LS coupling
approximation for ions with low atomic
number. The calculations of Romanik (1988) lie
above those
of Chen (1986) for Si4+ and agree very well for Fe16+. The
close agreement between the results of Chen and Romanik and the large
difference between their data and results of Jacobs et al. (1977, 1979)
strongly suggest inaccuracies in the reported results of Jacobs et al. (1977, 1979). The available DR data in the published literature were
fitted by Mazzotta et al. (1998) for all neon-like ions from Na+to Ni18+. They relied mainly on the results of Chen (1986) for ions
heavier than argon and those of
Romanik (1988) for lighter ions. For Na+, Mazzotta et al. write that
they used the calculations of Shull &
Van Steenberg (1982), and for P5+ and Cl7+ that they adopted the
data of Landini & Monsignori Fossi (1991) multiplied by a factor 4
to take into account the results of Romanik (1988) and Chen (1986) for
adjacent ions.
More recently, Fournier et al. (1997) has reported results for neon-like Ar8+ and Gu (2003b) for neon-like Mg2+, Si4+, S6+, Ar8+, Ca10+, Fe16+ and Ni18+(see also Gu 2003a for Fe16+). We discuss some of these results in Sect. 3. It is important for us to compare with these modern theoretical results and to confirm that various theoretical methods have converged to the same results, as there are no experimental measurements available for this isoelectronic sequence.
In this paper, we apply an intermediate-coupling multiconfiguration Breit-Pauli (MCBP) method, in the isolated resonance approximation, to compute DR rate coefficients for neon-like ions from Na+ to Zn20+, as well as for Kr26+, Mo32+, Cd38+, and Xe44+. The rate coefficients computed here cover a wide range of temperatures and ionic species. The present calculations also attempt to produce final state-resolved DR rate coefficients which are important in modelling plasmas under certain conditions. Total DR rate coefficients, along with radiative recombination (RR) rate coefficients, are presented in compact form using simple fitting formulae. It is impractical to list all level-resolved rate coefficients in a paper publication. As previously discussed (Badnell et al. 2003), this latter data will form part of an Atomic Data and Analysis Structure (ADAS) dataset comprising the adf09 files for each ion, detailing the rate coefficients to each LSJ-resolved final state. This data is available through the ADAS project (Summers 1999) and is also made available online at the Oak Ridge Controlled Fusion Atomic Data Center http://www-cfadc.phy.ornl.gov/data_and_codes. In Sect. 2 we give a brief description of the theory used and the details of our calculations for the neon-like ions. In Sect. 3 we present the results for dielectronic recombination rate coefficients for the aforementioned ions in this sequence. We conclude with a brief summary in Sect. 4.
The theoretical details of our calculations have already been described in detail (Badnell et al. 2003; Zatsarinny et al. 2003). Here we outline only the points specific to the present case. The calculation of the energy dependent DR cross sections were carried out using the code AUTOSTRUCTURE (Badnell 1986; Badnell & Pindzola 1989), which is based on lowest order perturbation theory where both the electron-electron and electron-photon interactions are treated to first order. This independent-processes, isolated-resonance approximation is used to compute configuration-mixed LS and IC energy levels, and radiative and autoionization rates, which are then used to compute Lorentzian DR resonance profiles. This enables the generation of final state level-resolved and total DR rate coefficients. We neglected interference between RR and DR, which we have shown does not affect the dominant rate coefficients (Zatsarinny et al. 2003).
The DR process for neon-like ions can be represented by
Table 1:
Fitting coefficients for dielectronic and radiative
recombination on neon-like ions forming sodium-like systems
for Eqs. (2) and (3), respectively:
c1 and E1 correspond to DR via the
states in Eq. (1),
c2 and E2 correspond to DR via the
states in Eq. (1), and a, b, T0, and T1 correspond
to RR. The coefficients ci and a are in units of 10-11 cm3 s-1, the coefficients
Ei, T0, and T1 are in eV, and b is dimensionless.
The computed DR cross section is a sum of Lorentzian profiles and can
therefore be convoluted analytically with an experimental energy
distribution, in order to compare to measured results, or with a
Maxwellian electron distribution, in order to obtain total DR rate
coefficients. This represents a huge savings in computational effort
over R-matrix calculations since the latter must be performed for an
extremely dense energy mesh in order to fully resolve all resonances
(Gorczyca et al. 2002; Ramirez & Bautista 2002). The total DR rate
coefficients were then fitted as
In order to provide total (DR + RR) recombination rate coefficients,
Table 1 also contains the fitting coefficients for the RR rate
coefficients. These were also obtained using the AUTOSTRUCTURE code
with the same target orbitals and in the same approximation as the DR calculations. RR rate coefficients were fitted with the formula of Verner & Ferland (1996)
AUTOSTRUCTURE is implemented within the ADAS suite of programs as
ADAS701. It produces raw autoionization and radiative rates which must be
post-processed to
obtain the final-state level-resolved and total DR rate coefficients. The
post-processor ADASDR is used to reorganize the resultant data and also to
add in radiative transitions between highly-excited Rydberg states, which are
computed hydrogenically. This post-processor outputs directly the
adf09
file necessary for use by ADAS. The adf09 files generated by our
calculations in the IC configuration mixed approximations are available electronically
(http://www-cfadc.phy.ornl.gov/data_and_codes). This site
has tabulated DR rate coefficients
into final LSJ levels from both the ground and metastable states in a
manner useful to fusion and astrophysical modelers. Separate files adf09
are produced for the
states and the
states, which are thus amenable to selective upgrade.
![]() |
Figure 1:
DR resonance structure for neon-like ions Al3+,
Ar8+, and Fe16+ forming sodium-like ions. The theoretical results have been
convolved with a simulated experimental energy spread represented by a
flattened Maxwellian with
![]() ![]() |
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Examples of the DR electron energy dependent resonance structure in the
neon-like ions is given in Fig. 1
(there are no data from laboratory experiments
to compare with). Here the DR cross sections have been
convoluted with a cooler electron distribution which is characteristic
of recent DR measurements using the heavy-ion Test Storage Ring (see,
for example, Savin et al. 1999, 2002). The cooler electrons employed
in this technique have an anisotropic Maxwellian distribution with
low perpendicular and parallel temperatures of
15 meV and
0.15 meV, which allows for detailed
laboratory studies
of the resonance structure. Figure 1 illustrates three examples of
computed DR rate coefficients for low-, intermediate-, and high-charged
neon-like ions. For low-charged Al3+,
all series with dipole-allowed core transitions
contribute appreciably, and the largest contributions
come from the high-
states. For intermediate-charged Ar8+,
the
series becomes
relatively stronger,
and the high-
contributions are comparable to the
lower-
ones.
For high-charged Fe16+, the
series is now dominant, as are the low-
resonances.
![]() |
Figure 2: Maxwellian-averaged DR and RR rate coefficients for recombination of Ar8+ to Ar7+. Present results for DR (thick solid curve) and RR (thin solid curve) were obtained in the MCBP calculations with the code AUTOSTRUCTURE. Also shown are the MCDF results of Chen (1986, solid circles), which were used for the compiled data of Mazzotta et al. (1998, dotted curve), the FAC results of Gu (2003b, dash-dot-dotted curve), the results of Romanik (1988, dashed curve), and the results of Fournier et al. (1997, solid squares) using the fully relativistic, parametric potential atomic structure code RELAC. |
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In Fig. 2, the Maxwellian DR rate coefficient
for Ar8+ as a function of electron temperature is
compared to other theoretical results.
The MCDF results of Chen (1986), which were used in the critical compilation of Mazzotta
(1998), are lower than ours by 25%; we attribute this difference
to Chen's omission of certain intermediate states, such
as
and
(
).
The results of Romanik (1988) are in somewhat better agreement with ours;
the difference here we attribute to the approximate expressions for
the autoionization and radiative rates used in that earlier work.
Fournier et al. (1997) calculated DR for Ar8+using the fully relativistic, parametric potential atomic structure code
RELAC, with the continuum wave functions computed in the distorted-wave
approximation. Gu (2003b) used a newly developed atomic package FAC,
which uses relativistic multiconfigurational atomic wave
functions. Both the results of Fournier et al. and Gu
are in closer agreement with our results than are previously published
calculations. The calculations of Fournier et al. and Gu both
treat DR in the independent processes,
isolated resonance approximation, as does Chen's MCDF code and
AUTOSTRUCTURE. The agreement of our results with those from
other recent independent calculations is an excellent consistency check
on our applied methodology.
![]() |
Figure 3: Maxwellian-averaged DR and RR rate coefficients for recombination of Fe16+ to Fe15+. Present results for DR (thick solid curve) and RR (thin solid curve) were obtained in the MCBP calculations with the code AUTOSTRUCTURE. Also shown are the MCDF results of Chen (1986, solid circles), which were used for the compiled data of Mazzotta et al. (1998, dotted curve), the FAC results of Gu (2003a, solid squares; 2003b, dash-dot-dotted curve), the results of Romanik (1988, dashed curve), the results of Jacobs et al. (1977, RR+DR, open squares). the RR results of Arnaud & Raymond (1992, open circles). |
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In Fig. 3, we make similar comparisons for Fe16+.
It is seen that the results of Chen (1986), as also used by Mazzotta
(1998), Romanik (1988), and Gu (2003a,b) are all in fairly good
agreement with our results. We expect the differences to be less than for Ar8+because the inaccuracies introduced by either omitted configurations
or approximate rate expressions decrease as the ionic charge increases,
i.e., as the system becomes more hydrogenic.
The results of Jacobs et al. (1977), however, are well below all the
others (factor of 4.0 near the peak rate coefficient).
It is thus clear that the simplified methods used in the earlier
calculations of Jacobs et al. (1977, 1979),
which were discussed in Sect. 1, yielded inaccurate DR rate
coefficients. Figure 3 shows that our computed RR rate coefficient as a function of temperature is in excellent agreement with that of
Arnaud & Raymond (1992).
![]() |
Figure 4: Behavior of DR rate coefficients as a function of atomic number for the neon-like ions discussed in this paper forming sodium-like systems. Not all ions are labeled. |
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In Table 1, we present the fitting parameters for total DR
and RR rate coefficients for each ion from Na+ to Zn20+, and
also for Kr26+, Mo32+, Cd38+ and Xe44+. In
previous work, numerical calculations were performed for only
certain ions in the sequence and the DR rate coefficients for all other
ions were interpolated or extrapolated as a function of atomic number Z.
This is not always a reliable approach, since, as we found in our
study of oxygen-like ions (Zatsarinny et al. 2003),
the rate coefficients for DR via
core excitations
do not scale smoothly with Z. However, in the present case of
neon-like ions, we have only
core excitations,
which do scale smoothly with Z.
Indeed, Fig. 4 clearly illustrates this behavior.
![]() |
Figure 5: Maximum total DR rate coefficients as a function of atomic number Z. The open circles represent our MCBP calculations. Comparison is given with the calculations of Jacobs et al. (1977, 1979, open squares), the Hartree-Fock distorted-wave calculation of Roszman (1979, open diamond), the MCDF calculations of Chen (1986, crosses), the compilation data of Mazzotta et al. (1998, solid squares), and with the relativistic DW calculations of Fournier et al. (1997, solid up triangles) and Gu (2003a, solid down triangles; 2003b, open down triangles . Lines are drawn through our data and those of Mazzotta et al. to guide the eye. |
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A simple way to compare various theoretical DR rate coefficients along the entire neon-like isoelectronic series is to plot the maximum DR rate coefficient vs. the atomic number Z, as is done in Fig. 5. The results from the earlier calculations of Jacobs et al. (1977, 1979), as compiled by Shull & van Steenberg (1982), are obviously inaccurate. But more importantly, we also see considerable inaccuracies in the compiled data of Mazzotta et al. (1998). These data are based on the results of various calculations which used different methods and resorted to different approximations, and erratic behavior as a function of atomic number Zis clearly seen. For instance, the peak DR rate coefficient of Mazzotta et al. (1998) at Z=17 is lower by almost an order of magnitude than the neighboring values at Z=16 and Z=18. Our results, which are in sufficient agreement with those of Chen (1986), show the expected smooth behavior with Z.
In this paper, we have systematically calculated partial and
total DR rate coefficients along the neon-like sequence as part of an
assembly of a DR database necessary for the modelling of finite-density
plasmas (Badnell et al. 2003; Zatsarinny et al. 2003; Colgan et al.
2003). The approximations used for generating
our data have been recently validated by the good agreement
of DR resonance strengths and energies between our theoretical data
and the experimental results for L-shell ions
from the Test Storage Ring in Heidelberg. Comparisons have been made
for systems such as
lithium-like Si and Cu (Kilgus 1992), oxygen-like Fe (Savin et al. 2002),
and fluorine-like Fe and Se (Lampert et al. 1996; Savin et al. 1999),
Since DR via
core excitations of these L-shell ions resembles
DR of neon-like ions, we believe that a similar degree of accuracy
was achieved in our present study as in past studies.
Hence, for the accuracy of our results,
we expect an
variation when using different target
orbitals, an
difference when comparing to other
state-of-the-art theory and an
difference when
comparing with laboratory measurements of
DR.
We have presented selected total DR rate coefficients for some ions of interest, and have made comparisons, where possible, with previous work. We found large discrepancies with previous calculations, including the fits of Mazzotta et al. (1998), which were used in their study of the ionization equilibrium for various atoms. Final-state-resolved rate coefficients have been tabulated, and these data are available from the web site http://www-cfadc.phy.ornl.gov/data_and_codes. Total DR and RR rate coefficients have been fitted by simple analytical formula which will prove of great use to astrophysical and fusion plasma modelers, and are available on our web site http://homepages.wmich.edu/~gorczyca/drdata.
We have calculated our data over a wide temperature range and for a large
number of atomic ions in order to maximize the available information for
plasma spectral modelling work. Our fits are accurate to better than 1%
for all ions in the wide temperature range from 101 to 108 K.
Because there are no
DR resonances for neon-like ions,
the small uncertainties in the
DR resonance energies
(whose energies are significantly larger than these uncertainties), have essentially no
impact on our calculated rate coefficient.
These RR and DR data are suitable for modelling of solar and cosmic plasmas
under conditions of collisional ionization equilibrium, photoionization
equilibrium, and non-equilibrium ionization (e.g., in shocks). Future
papers will present DR data for further isoelectronic sequences as
detailed previously (Badnell et al. 2003).
Acknowledgements
T.W.G., K.T.K., and O.Z. were supported in part by NASA Space Astrophysical Research and Analysis Program grant NAG5-10448. D.W.S. was supported in part by NASA Space Astrophysics Research and Analysis Program grant NAG5-5420 and NASA Solar Physics Research, and Subordinal Program grants NAG5-9581 and NAG5-12798.