Dielectronic Recombination of Fe XV forming Fe XIV: Laboratory
  Measurements and Theoretical Calculations
Dielectronic Recombination of Fe XV forming Fe XIV: Laboratory
Measurements and Theoretical Calculations
D. V. Lukić1, M. Schnell2, and D. W. Savin
Columbia Astrophysics Laboratory, Columbia University,
New York, NY 10027, USA
lukic@astro.columbia.edu
C. Brandau3, E. W. Schmidt, S. Böhm4, A. Müller, and S. Schippers
Institut für Atom- und Molekülphysik, Justus-Liebig-Universität, D-35392 Giessen,
Germany
M. Lestinsky, F. Sprenger, and A. Wolf
Max-Planck-Institut für Kernphysik, D-69117 Heidelberg, Germany
Z. Altun
Department of Physics, Marmara University, Istanbul 81040, Turkey
N. R. Badnell
Department of Physics, University of Strathclyde, G4 0NG Scotland, UK
ABSTRACT
We have measured resonance strengths and energies for dielectronic recombi-
nation (DR) of Mg-like Fe XV forming Al-like Fe XIV via N = 3 → N ′ = 3 core
excitations in the electron-ion collision energy range 0–45 eV. All measurements
were carried out using the heavy-ion Test Storage Ring at the Max Planck Insti-
tute for Nuclear Physics in Heidelberg, Germany. We have also carried out new
1On leave from the Institute of Physics, 10001 Belgrade, Serbia
2Present address, Carl Zeiss NTS GmbH, Oberkochen D-73447, Germany
3Present address, Gesellschaft für Schwerionenforschung (GSI), Darmstadt, D-64291, Germany
4Present address, Department of Atomic Physics, Stockholm University, S-106 91 Stockholm, Sweden
http://arxiv.org/abs/0704.0905v1
multiconfiguration Breit-Pauli (MCBP) calculations using the AUTOSTRUC-
TURE code. For electron-ion collision energies . 25 eV we find poor agreement
between our experimental and theoretical resonance energies and strengths. From
25 to 42 eV we find good agreement between the two for resonance energies.
But in this energy range the theoretical resonance strengths are ≈ 31% larger
than the experimental results. This is larger than our estimated total experi-
mental uncertainty in this energy range of ±26% (at a 90% confidence level).
Above 42 eV the difference in the shape between the calculated and measured
3s3p(1P1)nl DR series limit we attribute partly to the nl dependence of the de-
tection probabilities of high Rydberg states in the experiment. We have used our
measurements, supplemented by our AUTOSTRUCTURE calculations, to pro-
duce a Maxwellian-averaged 3 → 3 DR rate coefficient for Fe XV forming Fe XIV.
The resulting rate coefficient is estimated to be accurate to better than ±29% (at
a 90% confidence level) for kBTe ≥ 1 eV. At temperatures of kBTe ≈ 2.5− 15 eV,
where Fe XV is predicted to form in photoionized plasmas, significant discrepan-
cies are found between our experimentally-derived rate coefficient and previously
published theoretical results. Our new MCBP plasma rate coefficient is 19−28%
smaller than our experimental results over this temperature range.
Subject headings: atomic data – atomic processes – plasmas – galaxies: active –
galaxies: nuclei – X-rays: galaxies
1. Introduction
Recent Chandra and XMM Newton X-ray observations of active galactic nuclei (AGNs)
have detected a new absorption feature in the 15-17 Å wavelength range. This has been iden-
tified as an unresolved transition array (UTA) due mainly to 2p− 3d inner shell absorption
in iron ions with an open M-shell (Fe I - Fe XVI). UTAs have been observed in IRAS
13349+2438 (Sako et al. 2001), Mrk-509 (Pounds et al. 2001), NGC 3783 (Blustin et al.
2002; Kaspi et al. 2002; Behar et al. 2003), NGC 5548 (Steenbrugge et al. 2003), MR 2251-
178 (Kaspi et al. 2004), I Zw 1 (Gallo et al. 2004), NGC 4051 (Pounds et al. 2004), and
NGC 985 (Krongold et al. 2005).
Based on atomic structure calculations and photoabsorbtion modeling, Behar et al.
(2001) have shown that the shape, central wavelength, and equivalent width of the UTA
can be used to diagnose the properties of AGN warm absorbers. However, models which fit
well absorption features from second and third row elements cannot reproduce correctly the
observed UTAs due to the fourth row element iron. The models appear to predict too high
an ionization level for iron. Netzer et al. (2003) attributed this discrepancy to an underesti-
mate of the low temperature dielectronic recombination (DR) rate coefficients for Fe M-shell
ions. To investigate this possibility Netzer (2004) and Kraemer et al. (2004) arbitrarily in-
creased the low temperature Fe M-shell DR rate coefficients. Their model results obtained
with the modified DR rate coefficients support the hypothesis of Netzer et al. (2003). New
calculations by Badnell (2006a) using a state-of-the-art theoretical method disscused in § 5
further support the hypotesis of Netzer et al. (2003).
Astrophysical models currently use the DR data for Fe M-shell ions recommended
by Arnaud & Raymond (1992). These data are based on theoretical DR calculations by
Jacobs et al. (1977) and Hahn (1989). The emphasis of this early theoretical work was
on producing data for modeling collisional ionization equilibrium (sometimes also called
coronal equilibrium). Under these conditions an ion forms at a temperature about an
order of magnitude higher than the temperature where it forms in photoionized plasmas
(Kallman & Bautista 2001). The use of the Arnaud & Raymond (1992) recommended DR
data for modeling photoionized plasmas is thus questionable. Benchmarking by experiment
is highly desirable.
Reliable experimentally-derived low temperature DR rate coefficients of M-shell iron
ions are just now becoming available. Until recently, the only published Fe M-shell DR mea-
surements were for Na-like Fe XVI (Linkemann et al. 1995; Müller 1999; here and throughout
we use the convention of identifying the recombination process by the initial charge state
of the ion). The Na-like measurements were followed up with modern theoretical calcula-
tions (Gorczyca & Badnell 1996; Gu 2004; Altun et al. 2007). Additional M-shell experi-
mental work also exists for Na-like Ni XVIII (Fogle et al. 2003) and Ar-like Sc IV and Ti V
(Schippers et al. 1998, 2002). We have undertaken to measure low temperature DR for other
Fe M-shell ions. Our results for Al-like Fe XIV are presented in Schmidt et al. (2006) and
Badnell (2006b). The present paper is a continuation of this research.
DR is a two-step recombination process that begins when a free electron approaches
an ion, collisionally excites a bound electron of the ion and is simultaneously captured into
a Rydberg level n. The electron excitation can be labeled Nlj → N
′l′j′ where N is the
principal quantum number of the core electron, l its orbital angular momentum, and j its
total angular momentum. The intermediate state, formed by simultaneous excitation and
capture, may autoionize. The DR process is complete when the intermediate state emits a
photon which reduces the total energy of the recombined ion to below its ionization limit.
In this paper we present experimental and theoretical results for ∆N=N ′ −N = 0 DR
of Mg-like Fe XV forming Al-like Fe XIV. In specific we have studied 3 → 3 DR via the
resonances:
Fe14+(3s2[1S0]) + e
Fe13+(3s3p[3P o
0,1,2;
1P1]nl)
Fe13+(3s3d[3D1,2,3;
1D2]nl)
Fe13+(3p2[3P0,1,2;
1S0]nl)
Fe13+(3p3d[3Do
1,2,3;
2,3,4;
0,1,2;
; 1Do
; 1F o
Fe13+(3d2[3P0,1,2;
3F2,3,4;
1S0]nl)
Possible contributions due to 3s3p 3P metastable parent ions will be discussed below. Table 1
lists the excitation energies for the relevant Fe XV levels, relative to the ground state, that
have been considered in our theoretical calculations. In our studies we have carried out
measurements for electron-ion center-of-mass collision energies Ecm between 0 and 45 eV.
Our work is motivated by the “formation zone” of Fe M-shell ions in photoionized gas.
This zone may be defined as the temperature range where the fractional abundance of a given
ion is greater than 10% of its peak value (Schippers et al. 2004). We adopt this definition
for this paper. Savin et al. (1997, 1999, 2002a,b, 2006) defined this zone as the temperature
range where the fractional abundance is greater than 10% of the total elemental abundance.
This is narrower than the Schippers et al. (2004) definition. For Fe XV the wider definition
corresponds to a kBTe ≈ 2.5-15 eV (Kallman & Bautista 2001). It should be kept in mind
that this temperature range depends on the accuracy of the underlying atomic data used to
calculate the ionization balance.
The paper is organized as follows: The experimental arrangement for our measure-
ments is described in § 2. Possible contamination of our parent ion beam by metastable
ions is discussed in § 3. Our laboratory results are presented in § 4. In this section the
experimentally-derived DR rate coefficient for a Maxwellian plasma is provided as well.
Theoretical calculations which have been carried out for comparison with our experimental
results are discussed in § 5. Comparison between the experimental and theoretical results is
presented in § 6. A summary of our results is given in § 7.
2. Experimental Technique
DR measurements were carried out at the heavy-ion test storage ring (TSR) of the
Max-Planck Institute for Nuclear Physics (MPI-K) in Heidelberg, Germany. A merged
beams technique was used. A beam of 56Fe14+ with an energy of 156 MeV was provided
by the MPI-K accelerator facility. Ions were injected into the ring and their energy spread
reduced using electron cooling (Kilgus et al. 1990). Typical waiting times after injection
and before measurement were ≈ 1 s. Mean stored ion currents were ≈ 10 µA. Details of
the experimental setup have been given elsewhere (Kilgus et al. 1992; Lampert et al. 1996;
Schippers et al. 1998, 2000, 2001).
Recently a second electron beam has been installed at the TSR (Sprenger et al. 2004;
Kreckel et al. 2005). This allows one to use the first electron beam for continuous cooling
of the stored ions and to use the second electron beam as a target for the stored ions. In
this way a low velocity and spatial spread of the ions can be maintained throughout the
course of a DR measurement. The combination of an electron cooler and an electron target
can be used to scan energy-dependent electron-ion collision cross sections with exceptional
energy resolution. In comparison to the electron cooler, the electron source and the electron
beam are considerably smaller and additional procedures, such as the stabilization of the
beam positions during energy scans and electron beam profile measurements, are required
to control the absolute luminosity product between the ion and electron beam on the same
precise level as reached at the cooler. The target electron beam current was ≈ 3 mA. The
beam was adiabatically expanded from a diameter of 1.6 mm at the cathode to 7.5 mm
in the interaction region using an expansion factor of 22. This was achieved by lowering
the guiding magnetic field from 1.28 T at the cathode to 0.058 T in the interaction region
thus reducing the transverse temperature to approximately 6 meV. The relative electron-
ion collision energy can be precisely controlled and the recombination signal measured as a
function of this energy. We estimate that the uncertainty of our scale for Ecm is . 0.5%.
The electrons are merged and demerged with the ion beam using toroidal magnets.
After demerging, the primary and recombined ion beams pass through two correction dipole
magnets and continue into a bending dipole magnet. Recombined ions are bent less strongly
than the primary ion beam and they are directed onto a particle detector used in single
particle counting mode. Some of the recombined ions can be field-ionized by motional
electric fields between the electron target and the detector and thus are not detected. Here
we assumed a sharp field ionization cutoff and estimated for Fe XV that only electrons
captured into nmax . 80 are detected by our experimental arrangement.
The experimental energy distribution can be described as a flattened Maxwellian dis-
tribution. It is characterized by the transversal and longitudinal temperatures T⊥ and T‖,
respectively. The experimental energy spread depends on the electron-ion collision energy
and can be approximated according to the formula ∆E = ([ln(2)kBT⊥]
2+16 ln(2)EcmkBT‖)
(Pastuszka et al. 1996). For the comparison of our theoretical calculations with our experi-
mental data we convolute the theoretical results described in § 5 with the velocity distribution
function given by Dittner et al. (1986) to simulate the experimental energy spread.
With the new combination of an electron target and an electron cooler we obtain in the
present experiment electron temperatures of kBT⊥ ≈ 6 meV and kBT‖ ≈ 0.05 meV. In order
to verify the absolute calibration of the absolute rate coefficient scale we also performed
a measurement with the electron cooler using the previous standard method (Kilgus et al.
1992, Lampert et al. 1996). We find consistent rate coefficients and spectral shapes, while
the electron temperatures were larger by a factor of about 2 with the electron cooler alone.
Moreover, because of the large density of resonances found in certain regions of the Fe XV
DR spectrum the determination of the background level for the DR signal was considerably
more reliable in the higher resolution electron target data than in the lower resolution cooler
data. Hence, we performed the detailed analysis presented below on the electron target data
only.
Details of the experimental and data reduction procedure are given in Schippers et al.
(2001, 2004) and Savin et al. (2003) and reference therein. The baseline experimental un-
certainty (systematic and statistical) of the DR measurements is estimated to be ±25% at
a 90% confidence level (Lampert et al. 1996). The major sources of uncertainties include
the electron beam density determination, ion current measurements, and corrections for the
merging and demerging of the two beams. Additional uncertainties discussed below result in
a higher total experimental uncertainty as is explained in §§ 3 and 4. Unless stated otherwise
all uncertainties in this paper are cited at an estimated 90% confidence level.
3. Metastable Ions
For Mg-like ions with zero nuclear spin (such as 56Fe), the 1s22s22p63s3p 3P0 level is
forbidden to decay to the ground state via a one-photon transition and the multiphoton
transition rate is negligible. Hence this level can be considered as having a nearly infinite
lifetime (Marques, Parente, & Indelicato 1993; Brage et al. 1998). It is possible that these
metastables are present in the ion beam used for the present measurements.
We estimate that the largest possible metastable 3P0 fraction in our stored beam is 11%.
This assumes that 100% of the initial Fe14+ ions are in 3PJ levels and that the levels are
statistically populated. We expect that the J = 1 and 2 levels will radiatively decay to the
ground state during the ∼ 1 s between injection and measurement. The lifetimes of the 3P1
and 3P2 levels are ∼ 1.4 × 10
−10 s (Marques et al. 1993) and ∼ 0.3 s (Brage et al. 1998),
respectively. These decays leave 1/9th or 11% of the stored ions in the 3P0 level.
Our estimate is only slightly higher than the inferred metastable fraction for the ion
beam used for DR measurements of the analogous Be-like Fe22+ (Savin et al. 2006). The
Be-like system has a metastable 1s22s2p 3P0 state and following the above logic the stored
Be-like ion beam had an estimated maximum 11% 3P0 fraction. Fortunately, for the Be-like
measurements we were able to identify DR resonances due to the 3P0 parent ion and use the
ratio of the experimental to theoretical resonance strengths to infer the 3P0 fraction. There
we determined a metastable fraction of 7% ± 2%. A similar fraction was inferred for DR
measurements with Be-like Ti18+ ions (Schippers et al. 2007).
Using theory as a guide, we have searched our Mg-like data fruitlessly for clearly iden-
tifiable DR resonances due to metastable 3P0 parent ions. First, following our work in the
analogous Be-like Fe22+ with its 2s2p 3P0 → 2p
2 core excitation channel (Savin et al. 2006),
we searched for Fe14+ resonances associated with the relevant 3s3p 3P0 → 3p
2 core exci-
tations. However, most of these yield only very small DR cross sections as they strongly
autoionize into the 3s3p 3PJ=1,2 continuum channels. These are energetically open at Ecm
greater than 0.713 eV and 2.468 eV, respectively (Table 1). Hence, above Ecm ≈ 0.713 eV
there are no predicted significant DR resonances for metastable Fe14+ via 3s3p 3P0 → 3p
core excitations. Below this energy the agreement between theory and experiment is ex-
tremely poor (as can be seen in Fig. 1) and we are unable to assign unambiguously any DR
resonance to either the ground state or metastable parent ion. Second, we searched for reso-
nances associated with 3s3p 3P0 → 3s3p
1P1, 3s3p
3P0 → 3s3p
3P1, and 3s3p
3P0 → 3s3p
core excitation which are energetically possible for capture into the n ≥ 14, 62, and 33 levels,
respectively, and which may contribute to the observed resonance structures. The analogous
2s2p 3P0 → 2s2p
1P1 and 2s2p
3P0 → 2s2p
3P2 core excitations were seen for Be-like Ti
(Schippers et al. 2007). However, again the complexity of the Fe XV DR resonance spec-
trum (cf., Fig. 1) prevented unambiguous identification for DR via any of these three core
excitations. Hence despite these two approaches, we have been unable to directly determine
the metastable fraction of our Fe14+ beam.
Clearly our assumption that the 3PJ levels are statistically populated is questionable.
Ion beam generation using beam foil techniques are known to produce excited levels. The
subsequent cascade relaxation could potentially populated the J levels non-statistically
(Martinson & Gaupp 1974; Quinet et al. 1999). Additionally the magnetic sublevels mJ
can be populated non-statistically (Martinson & Gaupp 1974) which may affect the J lev-
els. However, our argument in the above paragraphs that the 3PJ levels are statistically
populated yields 3P0 fractions of the analgous Be-like Ti
18+ and Fe22+ of 11% while our
measurements found metastable fractions of ∼ 7% for those two beams. From this we con-
clude either (a) that if 100% of the initial ions are in the 3PJ levels, then the J = 0 level is
statistically under-populated or (b) that the fraction of initial ions in the 3PJ levels is less
than 100% by a quantity large enough that any non-statistical populating of the various J
levels still yields only a 7% 3P0 metastable fraction of the ion beam. Thus we believe that
our assumption provides a reasonable upper limit to the metastable fraction of the Fe14+
beam.
Based on our estimates above and the Be-like results we have assumed that 6%±6% of
the Fe14+ ions are in the 3s3p 3P0 metastable state and the remaining fraction in the 3s
2 1S0
ground state. Here, we treat this possible 6% systematic error as a stochastic uncertainty
and add it in quadrature with the 25% uncertainty discussed above.
4. Experimental Results
Our measured 3 → 3 DR resonance spectrum for Fe XV is shown in Figs. 1 - 8. The
data 〈σv〉 represent the summed DR and radiative recombination (RR) cross sections times
the relative velocity convolved with the energy spread of the experiment, i.e., a merged beam
recombination rate coefficient (MBRRC).
The strongest DR resonance series corresponds to 3s2 1S0 → 3s3p
1P1 core excitations.
Other observed features in the DR resonance spectrum are possibly due to double core
excitations discussed in § 1. Trielectronic recombination (TR), as this has been named, has
been observed in Be-like ions (Schnell et al. 2003a,b; Fogle et al. 2005). These ions are the
second row analog to third row Mg-like ions. However in our data unambiguous assignment
of possible candidates for the TR resonances could not be made.
Extracted resonance energies Ei and resonance strengths Si for Ecm ≤ 0.95 eV are listed
in Table 2 along with their fitting errors. These data were derived following the method
outlined in Kilgus et al. (1992). Most of these resonances were not seen in any of the
theoretical calculations for either ground state or metastable Fe14+. Hence their parentage
is uncertain. The implications of this are discussed below.
Difficulties in determining the non-resonant background level of the data contributed
an uncertainty to the extracted DR resonance strengths. For the strongest peaks this was
on the order of ≈ 10% for Ecm . 5 eV and ≈ 3% for Ecm & 5 eV. Taking into account the
25% and 6% uncertainties discussed in §§ 2 and 3, respectively, this results in an estimated
total experimental uncertainty for extracted DR resonance strengths of ±28% below ≈ 5 eV
and ±26% above.
Due to the energy spread of the electron beam, resonances below Ecm . kBT⊥ cannot be
resolved from the near 0 eV RR signal. Here this limit corresponds to ≈ 6 meV. But we can
infer the absence of resonances lying below the lowest resolved resonance at 6.74 meV. For
Ecm . kBT‖, a factor of up to ∼ 2− 3 enhanced MBRRC is observed in merged electron-ion
beam experiments (see e.g., Gwinner et al. 2000; Heerlein et al. 2002). Here this temperature
limit corresponds to Ecm . 0.05 meV. As shown in Fig. 9, at an energy 0.005 meV our
MBRRC is a factor of 2.5 times larger than the fit to our data using the RR cross section
from semi-classical RR theory with quantum mechanical corrections (Schippers et al. 2001)
and the extracted DR resonance strengths and energies. This enhancement is comparable
to that found for systems with no unresolved DR resonances near 0 eV (e.g., Savin et
al. 2003 and Schippers at al. 2004). Hence, we infer that there are no additional significant
unresolved DR resonances below 6.74 meV. Recent possible explanations for the cause of the
enhancement near 0 eV have been given by Hörndl et al. (2005, 2006) and reference therein.
We have generated an experimentally-derived rate coefficient for 3 → 3 DR of Fe XV
forming Fe XIV in a plasma with a Maxwellian electron energy distribution (Fig. 10). For
Ecm ≤ 0.95 eV we have used our extracted resonance strengths listed in Table 2. For
energies Ecm ≥ 0.95 eV we have numerically integrated our MBRRC data after subtracting
out the non-resonant background. The rate coefficient was calculated using the methodology
outlined in Savin (1999) for resonance strengths and in Schippers et al. (2001) for numerical
integration.
In the present experiment only DR involving capture into Rydberg levels with quantum
numbers nmax . 80 contribute to the measured MBRRC. In order to generate a total ∆N=0
plasma rate coefficient we have used AUTOSTRUCTURE calculations (see § 5) to account
for DR into higher n levels. As is discussed in more detail in § 6, between 25-42 eV we find
good agreement between the experimental and AUTOSTRUCTURE resonance energies.
However, the theoretical results lie a factor of 1.31 above the measurement. To account
for DR into n ≥ nmax = 80, above 42 eV we replaced the experimental data with the
AUTOSTRUCTURE results (nmax =1000) reduced by a factor of 1.31. Our resulting rate
coefficient is shown in Fig. 10.
Including the DR contribution due to capture into n > 80 increases our experimentally-
derived DR plasma rate coefficient by < 1% for kBTe < 7 eV, by < 2.5% at 10 eV and by
< 7% at 15 eV. This contribution increases to 20% at 40 eV, rises to 27% at 100 eV and
saturates at ≈ 35% at 1000 eV. Thus we see that accounting for DR into n > nmax = 80
levels has only a small effect at temperatures of kBTe ≈ 2.5-15 eV where Fe XV is predicted
to form in photoionized gas (Kallman & Bautista 2001). Also, any uncertainties in this
theoretical addition, even if relatively large, would still have a rather small effect at these
temperatures on our derived DR total rate coefficient. Hence, we have not included this in
our determination below of the total experimental uncertainty for the experimentally-derived
plasma rate coefficient at kBTe ≥ 1 eV.
The two lowest-energy resonances in the experimental spectrum occur at energies of
6.74 meV and 9.80 meV with resonance strengths of 1.89 × 10−16 cm2 eV and 1.01 ×
10−17 cm2 eV, respectively (see Table 1 and Fig. 9). As already mentioned, the parent-
age for the two lowest energy resonances is uncertain. These resonances dominate the DR
rate coefficient for kBTe < 0.24 eV. The contribution is 50% at 0.24 eV, 16% at 0.5 eV,
6.5% at 1 eV, 2.4% at 2.5 eV, and < 0.31% above 15 eV. At temperatures where Fe XV is
predicted to form in photoionized plasmas, contributions due to these two resonances are
insignificant. Because of this, we do not include the effects of these two resonances when
calculating below the total experimental uncertainty for the experimentally-derived plasma
rate coefficient at kBTe ≥ 1 eV.
An additional source of uncertainty in our results is due to possible contamination of
the Fe XV beam by metastable 3P0 ions. Because we cannot unambiguously identify DR
resonances due to metastable parent ions, we cannot directly subtract out any contributions
they may make to our experimentally-derived rate coefficient. Instead we have used our
AUTOSTRUCTURE calculations for the metastable parent ion as a guide, multiplied them
by 0.06 on the basis of the estimated (6 ± 6)% metastable content. We then integrated
them to produce a Maxwellian rate coefficient and compared the results to our experimental
results, leaving out the two lowest measured resonances at 6.74 and 9.80 meV. As discussed
in the paragraph above, these two resonaces were left out because of the uncertainty in their
parentage and their small to insignificant effects above 1 eV. The metastable theoretical
results are 9.5% of this experimentally-derived rate coefficient at kBTe = 1 eV, 4.9% at
2.5 eV, 2.2% at 5 eV, 1% at 10 eV and < 0.77% above 15 eV.
In reality these are probably lower limits for the unsubtracted metastable contributions
to our experimentally-derived rate coefficient. However, these limits appear to be reasonable
estimates even taking into account the uncertainty in the exact value of the contributions
due to metastable ions. For example, if we assume that we have the estimated maximum
metastable fraction of 11%, then our experimentally-derived rate coefficients would have to
reduced by only 9.0% at 2.5 eV, 4.0% at 5 eV, 1.8% at 10 eV, and less than 1.4% above
15 eV. Alternatively, it is likely that theory underestimates the resonance strength for the
metastable parent ions similar to the case for ground state parent ions (cf., Fig. 1). However,
if the metastable fraction is 6% and the resonance contributions are a factor of 2 higher, then
our experimentally-derived rate coefficients would have to reduced by only 9.8% at 2.5 eV,
4.4% at 5 eV, 2.0% at 10 eV, and less than 1.5% above 15 eV. These are small and not very
significant corrections. We consider it extremely unlikely that we have underestimated by a
factor of nearly 2 both the metastable fraction and the metastable resonance contribution.
Thus we expect contamination due to metastable 3P0 ions to have a small to insignificant
effect on our derived rate coefficient at temperatures where Fe XV is predicted to form in
photoinoized gas.
Taking into account the baseline experimental uncertainty of 25%, the metastable frac-
tion uncertainty of 6%, and the nonresonant background uncertainty of 10%/3%, all dis-
cussed above, as well as the uncertainty due to the possible unsubtracted metastable res-
onances, the estimated uncertainty in the absolute magnitude of our total experimentally-
derived Maxwellian rate coefficient ranges between 26% and 29% for kBTe ≥ 1 eV. Here
we conservatively take the total experimental uncertainty to be ±29%. This uncertainty
increases rapidly below 1 eV due to the ambiguity of the parentage for the two lowest energy
resonances and possible resonance contributions from metastable Fe XV which we have not
been able to subtract out.
We have fitted our experimentally-derived rate coefficient plus the theoretical estimate
for capture into n > 80 using the simple fitting formula
αDR(Te) = T
−Ei/kBTe (2)
where ci is the resonance strength for the ith fitting component and Ei the corresponding
energy parameter. Table 3 lists the best-fit values for the fit parameters. All fits to the total
experimentally-derived Maxwellian-averaged DR rate coefficient show deviations of less than
1.5% for the temperature range 0.001 ≤ kBTe ≤ 10000 eV.
In Table 3, the Experiment (I) column gives a detailed set of fitting parameters where
the first 30 values of ci and their corresponding Ei values are for all the resolved resonances
for Ecm ≤ 0.95 eV given in Table 2. The parentage for these resonances are uncertain, though
the majority are most likely due to ground state and not metastable Fe14+. It is our hope that
future theoretical advances will allow one to determine which resonances are due to ground
state ions and which are due to metastables. Listing the resonances as we have will allow
future researchers to readily exclude those resonances which have been determined to be due
to the metastable parent. The remaining 6 fitting parameters yield the rate coefficient due
to all resonances for Ecm between 0.95 and the 3s3p(
1P1)nl series limit at 43.63 eV. In the
Experiment (II) column of Table 3, the first six sets of ci and Ei give the fitting parameters
for the first six resonances. The remaining sets of fit parameters are due to all resonances
between 0.1 eV and the series limit.
5. Theory
The only published theoretical DR rate coefficient for Fe XV which we are aware of is
the work of Jacobs et al. (1977). Using the work of Hahn (1989), Arnaud and Raymond
(1992) modified the results of Jacobs et al. (1977) to take into account contributions from
2p−3d inner-shell transitions. The resulting rate coefficient of Arnaud and Raymond (1992)
is widely used throughout the astrophysics community.
We have carried out new calculations using a state-of-the-art multiconfiguration Breit-
Pauli (MCBP) theoretical method. Details of the MCBP calculations have been reported in
Badnell et al. (2003). Briefly, the AUTOSTRUCTURE code was used to calculate energy
levels as well as radiative and autoionization rates in the intermediate-coupling approxi-
mation. These must be post-processed to obtain the final state level-resolved and total
dielectronic recombination data. The resonances are calculated in the independent process
and isolated resonance approximation (Seaton & Storey 1976).
The ionic thresholds were shifted to known spectroscopic values for the 3 → 3 transi-
tions. Radiative transitions between autoionizing states were accounted for in the calculation.
The DR cross section was approximated by the sum of Lorentzian profiles for all included
resonances. The AUTOSTRUCTURE calculations were performed with explicit n values up
to 80 in order to compare closely with experiment. The resulting MBRRC is presented for
3 → 3 core excitations in Figs. 1-8.
The theoretical 3 → 3 DR plasma rate coefficient was obtained by convolving calculated
DR cross section times the relative electron-ion velocity with a Maxwellian electron energy
distribution. Cross section calculations were carried out up to nmax = 1000. The resulting
Maxwellian plasma rate coefficient is given in Fig. 10.
We have fit our theoretical 3 → 3 MCBP Maxwellian DR rate coefficients using Eq. 2.
The resulting fit parameters are presented in Table 3. The accuracy of the MCBP fit is
better than 0.5% for the temperature range 0.1 ≤ kBTe ≤ 10000 eV. This lower limit
represents the range over which rate coefficient data were calculated. Data are not presented
below (101z2)/11605 eV, which is estimated to be the lower limit of the reliability for the
calculations (Badnell 2007). Here z = 14 and this limit is 0.17 eV.
6. Discussion
6.1. Resonance Structure
As we have already noted, we find poor agreement between our experimental and the-
oretical resonance energies and strengths for electron-ion collision energies below 25 eV.
Theory does not correctly predict the strength of many DR resonances which are seen in
the measurement. A similar extensive degree of disparity between the theoretical and the
measured resonances was also seen in our recent Fe13+ results (Schmidt et al. 2006; Badnell
2006b).
Some of the weaker peaks in our data below 1 eV may be due to the possible presence
of metastable Fe14+ in our beam. But the estimated small metastable contamination seems
unlikely to be able to account in this range for many of the strong resonances which are not
seen in the present theory. Above ≈ 1 eV, we expect no significant DR resonances due to
metastable Fe14+ (as is discussed in § 3).
In the energy range from 1− 25 eV, the differences between experiment and theory are
extensive. The reader can readily see from Figs. 1-8 that theory does not correctly predict
the strength of many resonances which are observed in the experiment. This conclusion
takes into account the by-eye shifting of the theoretical resonances energies to try to match
up theory with the measured resonances.
Between 25 − 42 eV we find good agreement between the experiment and theory for
resonance energies. The AUTOSTRUCTURE code reproduces well the more regular res-
onance energy structure of high-n Rydberg resonances approaching the 3s3p(1P1)nl series
limit. However the AUTOSTRUCTURE cross section lies ≈ 31% above the measurements.
This discrepancy is larger than the estimated ±26% total experimental uncertainty in this
energy range. A similar discrepancy with theory was found for Fe13+ (Badnell 2006b).
Theory and experiment diverge above 42 eV and approaching the 3s3p(1P1)nl se-
ries limit. We attribute the difference in the shape between the calculated and measured
3s3p(1P1)nl series limit partly to the nl dependence of the field-ionization process in the
experiment. Here we assumed a sharp n cutoff. Schippers et al. (2001) discuss the effects
of a more correct treatment of the field-ionization process in TSR. Their formalism uses the
hydrogenic approximation to take into account the radiative lifetime of the Rydberg level n
into which the initially free electron is captured.
Our theoretical calculations indicate there are no DR resonances due to 2 → 3 or 3 → 4
core excitations below 44 eV, significant or insignificant. The two weak peaks above the
3s3p(1P1)nl series limit at 43.63 eV are attributed to ∆N=1 resonances.
6.2. Rate Coefficients
The recommended rate coefficient of Arnaud & Raymond (1992) is in mixed agreement
with our experimental results (Fig. 10). For temperatures below 90 eV, their rate coefficient
is in poor agreement. At temperatures where Fe XV is predicted to form in photoionzed gas,
their data are a factor of 3 to orders of magnitude smaller than our experimental results. At
temperatures above 90 eV, the Arnaud & Raymond (1992) data are in good agreement with
our combined experimental and theoretical rate coefficient.
As already implied by the work of Netzer et al. (2003) and Kraemer et al. (2004), the
present result shows that the previously available theoretical DR rate coefficients for Fe XV
are much too low at temperatures relevant for photoionized plasmas. Other storage ring
measurements show similar difference with published recommended low temperature DR
rate coefficients for Fe M-shell ions (Müller 1999; Schmidt et al. 2006). The reason for this
discrepancy is primarily because the earlier theoretical calculations were for high temperature
plasmas and did not include the DR channels important for low temperatures plasmas.
At temperatures relevant for the formation of Fe XV in photoionized gas, we find that
the modified Fe XV rate coefficient of Netzer (2004) is up to an order of magnitude smaller
than our experimental results. The modified rate coefficient of Kraemer et al. (2004) is a
factor of over 3 times smaller. These rate coefficients were guesses meant to investigate the
possibility that larger low temperature DR rate coefficients could explain the discrepancy
between AGN observations and models. The initial results were suggestive that this is the
case. Our work confirms that the previously recommended DR data are indeed too low but
additionally shows that the estimates of Netzer et al. (2003) and Kraemer et al. (2004) are
also still too low. A similar conclusion was reached by Schmidt et al. (2006) based on their
measurement for Fe13+. Clearly new AGN modeling studies need to be carried out using our
more accurate DR data (Badnell 2006a).
Our state-of-the-art MCBP calculations are 37% lower than our experimental results at
a temperature of 1 eV. This difference decreases roughly linearly with increasing temperature
to ≈ 25% at 2.5 eV. It is basically constant at ≈ 23% up to 7 eV and then again nearly
monotonically decreases to 19% at 15 eV. As discussed in § 4, a small part of these difference
may be attributed to unsubtracted metastable 3P0 contributions. But these contributions are
< 10% at 2.5 eV, < 5% at 5 eV, < 2.0% at 10 eV, and < 1.4% above 15 eV (hence basically
insignificant). Above 15 eV the difference decreases and at 23 eV and up the agreement is
within . 10% with theory initially smaller than experiment but later greater. Part of the
good agreement at these higher temperatures is due to our use of theory for the unmeasured
DR contribution due to states with n > 80.
7. Summary
We have measured resonance strengths and energies for ∆N=0 DR of Mg-like Fe XV
forming Al-like Fe XIV for center-of-mass collision energies Ecm from 0 to 45 eV and compared
our results with new MCBP calculations. We have generated an experimentally-derived
plasma rate coefficient by convolving the measured MBRRC with a Maxwell-Boltzmann
electron energy distribution. We have supplemented our measured MBRRC with MCBP cal-
culations to account for unmeasured DR into states which are field-ionized before detection.
The resulting plasma recombination rate coefficient has been compared to the recommended
rate coefficient of Arnaud & Raymond (1992) and new calculations using a state-of-the-art
MCBP theoretical method. We have considered the issues of metastable ions in our stored
ion beam, enhanced recombination for collision energies near 0 eV, and field-ionization of
high Rydberg states in the storage ring bending magnets.
As suggested by Netzer et al. (2003) and Kraemer et al. (2004), the present result shows
that the previously available theoretical DR rate coefficients for Fe XV are much too low.
Other storage ring measurements show similar differences with published recommended low
temperature DR rate coefficients for M-shell iron ions (Müller 1999; Schmidt et al. 2006).
We are now in the process of carrying out DR measurements for additional Fe M-shell ions.
As these data become available we recommend that these experimentally-derived DR rate
coefficients be incorporated into AGN spectral models in order to produce more reliable
results.
We gratefully acknowledge the excellent support by the MPI-K accelerator and TSR
crews. CB, DVL, MS, and DWS were supported in part by the NASA Space Astrophysics
Research Analysis program, the NASA Astronomy and Astrophysics Research and Analy-
sis program, and the NASA Solar and Heliosperic Physics program. This work was also
supported in part by the German research-funding agency DFG under contract no. Schi
378/5.
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Table 1. Energy levels for the n = 3 shell of Fe XV relative to the ground state.
Level Energy (eV)a
3s3p(3P o
) 28.9927
3s3p(3P o
) 29.7141
3s3p(3P o
) 31.4697
3s3p(1P o
) 43.6314
3p2(3P0) 68.7522
3p2(1D2) 69.3816
3p2(3P1) 70.0017
3p2(3P2) 72.1344
3p2(1S0) 81.7833
3s3d(3D1) 84.1570
3s3d(3D2) 84.2826
3s3d(3D3) 84.4848
3s3d(1D2) 94.4875
3p3d(3F o
) 115.087
3p3d(3F o
) 116.313
3p3d(3F o
) 117.743
3p3d(1Do
) 117.601
3p3d(3Do
) 121.860
3p3d(3Do
) 123.346
3p3d(3Do
) 123.565
3p3d(3P o
) 121.940
3p3d(3P o
) 123.474
3p3d(3P o
) 123.518
3p3d(1F o
) 131.7351
3p3d(1P o
) 133.2690
3d2(3F2) 169.8994
3d2(3F3) 170.1106
3d2(3F4) 170.3612
3d2(1D2) 173.8992
3d2(1G4) 174.4529
3d2(3P0) 174.2613
Table 1—Continued
Level Energy (eV)a
3d2(3P1) 174.3433
3d2(3P2) 174.5416
3d2(1S0) 184.3712
aRalchenko et al. (2006)
unless otherwise noted.
bChurilov et al. (1989)
Table 2. Measured resonance energies Ei and strengths Si for Fe XV forming Fe XIV via
N = 3 → N ′ = 3 DR for Ecm ≤ 0.95. Fitting errors are presented at a 90% confidence level.
Peak Number Ei (eV) Si (10
−21 cm2 eV)
1 (6.74 ± 0.05)E-3 189430.0 ± 20635.3
2 0.0098 ± 0.0008 10078.0 ± 483.1
3 0.0196 ± 0.0008 613.1 ± 56.8
4 0.0254 ± 0.0003 743.9 ± 51.8
5 0.0444 ± 0.0002 686.3 ± 37.9
6 0.0610 ± 0.0002 2949.3 ± 39.0
7 0.1098 ± 0.0002 805.5 ± 699.5
8 0.1674 ± 0.0014 2424.3 ± 954.1
9 0.1943 ± 0.0018 4408.5 ± 1213.1
10 0.2143 ± 0.0022 4735.5 ± 750.9
11 0.2436 ± 0.0003 4257.6 ± 132.6
12 0.2660 ± 0.0006 4169.1 ± 339.0
13 0.2895 ± 0.0122 213.9 ± 218.4
14 0.3102 ± 0.0074 292.5 ± 188.6
15 0.3346 ± 0.0008 1158.1 ± 118.6
16 0.3596 ± 0.0010 943.5 ± 100.3
17 0.4154 ± 0.0149 193.3 ± 230.2
18 0.4536 ± 0.0005 8013.6 ± 328.0
19 0.4781 ± 0.0072 706.9 ± 310.2
20 0.4988 ± 0.0072 781.3 ± 303.5
21 0.5199 ± 0.0266 216.7 ± 285.6
22 0.5433 ± 0.0290 121.8 ± 270.4
23 0.6164 ± 0.0078 136.2 ± 106.9
24 0.6599 ± 0.0006 1269.1 ± 97.8
25 0.6992 ± 0.0010 3090.3 ± 99.5
26 0.7385 ± 0.0010 2068.5 ± 113.4
27 0.7943 ± 0.0006 1594.4 ± 83.7
28 0.8406 ± 0.0006 1740.6 ± 83.6
29 0.8830 ± 0.0006 2164.2 ± 89.9
30 0.9232 ± 0.0013 1420.7 ± 86.9
Table 3. Fit parameters for the total experimentally-derived DR rate coefficient for Fe XV
forming Fe XIV via N = 3 → N ′ = 3 core excitation channels and including the theoretical
estimate for capture into n > 80 (nmax = 1000). See § 4 for an explanation of the columns labeled
“Experiment (I)” and “Experiment (II)”. Also given are the fit parameters for our calculated
MCBP results (nmax = 1000). The units below are cm
3 s−1 K1.5 for ci and eV for Ei.
Parameter Experiment (I) Experiment (II) MCBP
c1 1.07E-4 1.07E-4 7.07E-4
c2 8.26E-6 8.26E-6 7.18E-3
c3 1.00E-6 1.00E-6 2.67E-2
c4 1.46E-6 1.46E-5 3.15E-2
c5 2.77E-6 2.77E-6 1.62E-1
c6 1.51E-5 1.51E-6 5.37E-4
c7 2.90E-6 3.29E-6 -
c8 2.66E-5 1.63E-4 -
c9 5.62E-5 4.14E-4 -
c10 6.66E-5 2.17E-3 -
c11 6.81E-5 6.40E-3 -
c12 7.28E-5 4.93E-2 -
c13 4.07E-6 1.51E-1 -
c14 5.96E-6 - -
c15 2.54E-5 - -
c16 2.23E-5 - -
c17 5.27E-6 - -
c18 2.40E-4 - -
c19 2.22E-5 - -
c20 2.56E-5 - -
c21 7.40E-6 - -
c23 4.35E-6 - -
c23 5.51E-6 - -
c24 5.50E-5 - -
c25 1.42E-4 - -
c26 1.00E-4 - -
c27 8.32E-5 - -
c28 9.61E-5 - -
c29 1.25E-4 - -
c30 8.61E-5 - -
c31 1.02E-4 - -
Table 3—Continued
Parameter Experiment (I) Experiment (II) MCBP
c32 5.46E-1 - -
c33 2.91E-3 - -
c34 4.83E-3 - -
c35 4.86E-2 - -
c36 1.51E-1 - -
E1 6.74E-3 6.74E-3 4.12E-1
E2 9.80E-3 9.80E-3 2.06E+0
E3 1.97E-2 1.97E-2 1.03E+1
E4 2.54E-2 2.54E-2 2.20E+1
E5 4.45E-2 4.45E-2 4.22E+1
E6 6.10E-2 6.10E-2 3.41E+3
E7 1.10E-1 1.10E-1 -
E8 1.67E-1 1.91E-1 -
E9 1.94E-1 3.33E-1 -
E10 2.14E-1 9.63E-1 -
E11 2.44E-1 2.47E+0 -
E12 2.66E-1 1.08E+1 -
E13 2.90E-1 3.83E+1 -
E14 3.10E-1 - -
E15 3.35E-1 - -
E16 3.60E-1 - -
E17 4.15E-1 - -
E18 4.54E-1 - -
E19 4.78E-1 - -
E20 4.99E-1 - -
E21 5.20E-1 - -
E22 5.43E-1 - -
E23 6.16E-1 - -
E24 6.60E-1 - -
E25 6.99E-1 - -
Table 3—Continued
Parameter Experiment (I) Experiment (II) MCBP
E26 7.39E-1 - -
E27 7.94E-1 - -
E28 8.41E-1 - -
E29 8.83E-1 - -
E30 9.23E-1 - -
E31 1.00E+0 - -
E32 1.16E+0 - -
E33 1.62E+0 - -
E34 3.14E+0 - -
E35 1.08E+1 - -
E36 3.82E+1 - -
Fig. 1.— Fe XV to Fe XIV 3 → 3 DR resonance structure versus center-of-mass energy Ecm
from 0 to 1 eV. The solid curve represents the measured rate coefficient 〈σv〉 which is the summed
DR plus radiative recombination (RR) cross sections times the relative velocity convolved with the
experimental energy spread, i.e., a merged beam recombination rate coefficient (MBRRC). The
dotted curve shows our calculated multiconfiguration Breit-Pauli (MCBP) results (nmax = 80) for
ground state Fe XV (top plot) and 3P0 metastable state Fe XV multiplied by a factor of 0.06
to account for the estimated 6% population in our ion beam (bottom plot). To these results we
have added the convolved, non-resonant RR contribution obtained from semi-classical calculations
(Schippers et al. 2001). The inset shows our results for Ecm from 5× 10
−6 to 1× 10−1 eV.
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
Center of Mass Energy (eV)
 Experiment
 MCBP Theory
2.5 3.0 3.5 4.0 4.5
Center of Mass Energy (eV)
 Experiment
 MCBP Theory
4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0
Center of Mass Energy (eV)
 Experiment
 MCBP Theory
15 16 17 18 19 20 21 22 23 24
Center of Mass Energy (eV)
 Experiment
 MCBP Theory
15 16 17 18 19 20 21 22 23 24
Center of Mass Energy (eV)
 Experiment
 MCBP Theory
Fig. 7.— Same as Fig. 2 but for Ecm from 23 to 36 eV. The dotted curve shows our calculated
MCBP results and the thin solid curve shows our calculated MCBP results reduced by a
factor of 1.31.
Fig. 8.— Same as Fig. 7 but for Ecm from 35 to 45 eV. The weak resonances above 44 eV
are attributed to ∆N=1 DR. These are not included in either our experimentally-derived or
theoretical Maxwellian rate coefficients.
Fig. 9.— Measured and fitted Fe XV to Fe XIV 3 → 3 resonance structure below 0.07 eV.
The experimental MBRRC results are shown by the filled circles. The vertical error bars
show the statistical uncertainty of the data points. The solid curve is the fit to the data
using our calculated RR rate coefficient (dashed curve) and taking into account all resolved
DR resonances. The dotted curves show the fitted DR resonances. At Ecm = 0.005 meV the
difference between the model spectrum α0 and the data is 1 + (∆α/α0) = 2.5.
10-1 100 101 102 103
10-11
10-10
Electron Temperature (eV)
Photoionized
       Zone
Fig. 10.— Maxwellian-averaged 3 → 3 DR rate coefficients for Fe XV forming Fe XIV. The
solid curve represent our experimentally-derived rate coefficient plus the theoretical estimate
for unmeasured contributions due to capture into states with n > 80. The error bars show
our estimated total experimental uncertainty of ±29% (at a 90% confidence level). No error
bars are shown below 1 eV for reasons discussed in § 4. The thin solid curve represents our
experimentally-derived rate coefficient without the two lowest energy resonances included.
The dash-dotted curve represents our experimentally-derived rate coefficient alone (nmax =
80). Also shown is the recommended DR rate coefficient of Arnaud & Raymond (1992; thick
dash-dot-dotted curve) and its modification by Netzer (2004; thin dash-dot-dotted curve).
The filled pentagon at 5.2 eV represents the estimated rate coefficient from Kraemer et al.
(2004). The dashed curve shows our MCBP calculations for nmax = 1000. As a reference
we show the recommended RR rate coefficient of Arnaud & Raymond (1992; dotted curve).
Neither the experimental nor theoretical DR rate coefficients include RR. The horizontal
line shows the temperature range over which Fe XV is predicted to form in photoionized gas
(Kallman & Bautista 2001).
	Introduction
	Experimental Technique
	Metastable Ions
	Experimental Results
	Theory
	Discussion
	Resonance Structure
	Rate Coefficients
	Summary
