A&A 455, 1157-1160 (2006)

DOI: 10.1051/0004-6361:20064822

**V. Jonauskas ^{1,2} - P. Bogdanovich^{2} - F. P. Keenan^{1} - R. Kisielius^{1,2}
- M. E. Foord^{3} - R. F. Heeter^{3} - S. J. Rose^{4} - G. J. Ferland^{5} - P. H. Norrington^{6} **

1 - Department of Physics and Astronomy,
The Queen's University of Belfast,
Belfast BT7 1NN, Northern Ireland, UK

2 - Institute of Theoretical Physics and Astronomy of
Vilnius University, A. Gostauto 12,
01108 Vilnius, Lithuania

3 - University of California, Lawrence Livermore National Laboratory,
Livermore, CA 94551, USA

4 - Department of Physics, Clarendon Laboratory, Parks Road,
Oxford OX1 3PU, UK

5 - Department of Physics, University of Kentucky, Lexington,
KY 40506, USA

6 - Department of Applied Mathematics and Theoretical Physics,
The Queen's University of Belfast,
Belfast BT7 1NN, Northern Ireland, UK

Received 6 January 2006 / Accepted 17 May 2006

**Abstract**

The Multiconfiguration Dirac-Fock method is
used to calculate the energies of the 407 lowest levels in Fe XXII.
These results are cross-checked using a suite of codes which employ the
configuration interaction method on the basis
set of transformed radial orbitals with variable parameters,
and takes into account relativistic corrections
in the Breit-Pauli approximation.
Transition probabilities, oscillator and line strengths are presented for
electric dipole (E1), electric quadrupole (E2) and magnetic dipole (M1)
transitions among these levels.
The total radiative transition probabilities, as well as
the five largest values from each level
be it of E1, M1, E2, M2, or E3 type, are also provided. Finally,
the results are compared with data compiled by NIST.

**Key words: **atomic data

Physical processes and conditions in astrophysical sources can be
understood through the use of theoretical models to analyse
high resolution spectral observations.
In particular, X-ray spectra from iron L-shell ions are prominent candidates for
astrophysics plasma diagnostics, as their emission lines are observed in the
wavelength range from 6-18 Å, covered
with large effective areas
by X-ray telescopes on board the space
observatories *Chandra* and *XMM-Newton*.
Behar et al. (2001) identified Fe XXII lines in
a spectrum of the Capella binary system obtained by the High Energy
Transmission Grating Spectrometer on board *Chandra*, while
Kinkhabwala et al. (2002) observed Fe XXII
in the Seyfert 2 galaxy NGC 1068. To reliably analyse such spectra
requires an accurate knowledge of
wavelengths and radiative
transition rates, obtained from calculations or experiment.

Numerous observations of boron-like iron have been previously reported in
the X-ray spectra of the Sun
(McKenzie et al. 1985; Phillips et al. 1996; Fawcett et al. 1987; Doschek et al. 1981,1973; Phillips et al. 1982).
Wargelin et al. (1998) have presented an
analysis of the density sensitivity
of Fe XXII L-shell lines in plasma obtained in
the Princeton Large Torus tokamak.
They analysed the intensity ratios of 4d_{5/2} - 2p_{3/2} and
4d_{3/2} - 2p_{1/2} lines using the HULLAC atomic physics package
(Bar-Shalom et al. 2001).
Recently, Chen et al. (2004) used a radiative-collisional model
to predict the density of a plasma produced using
the Electron Beam Ion Trap at Lawrence Livermore National Laboratory.
They showed that the line ratios of the 2p-3d transitions in
boron-like iron are sensitive to the electron density
in the range
cm^{-3}.
Earlier, electron-excitation rate calculations for 45 fine-structure levels, using
the Breit-Pauli formulation of the R-matrix method, were performed by
Zhang & Pradhan (1997). Badnell et al. (2001)
used the AUTOSTRUCTURE code
(Badnell 1997,1986)
to generate radial orbitals and atomic
structure for 204-level close-coupling calculations in Fe XXII.
A large set of energy levels is presented in
the CHIANTI database at
http://wwwsolar.nrl.navy.mil/chianti.html.

The aim of the present paper is extend
our series of calculations for iron ions
(Jonauskas et al. 2004a,b,2005), by providing highly reliable energy levels
and radiative transition rates for Fe XXII up to
the *n*=5 complex.
We note that in photoionized plasmas these high-lying levels will not
in general be populated via electron impact (although collisional
redistribution among the levels may play a role), but rather by a range of
processes including recombination and charge transfer
(Savin 2001).
Transitions in Fe XXII from high-lying
levels having 5d electrons were observed
by Fawcett et al. (1987)
in a solar flare spectrum. Wargelin et al. (1998)
identified transitions from 2s^{2}4d^{1}, 2s^{1}2p^{1}4p^{1},
2s^{1}2p^{1}5p^{1}, 2s^{2}5d^{1}, 2s^{1}2p^{1}5d^{1} and
2s^{2}6d^{1} configurations which are populated in a radiative-collisional
model by electron collisions from metastable state.

In the present paper calculations are performed using the multiconfiguration Dirac-Fock (MCDF) approach. Energies of the lowest 407 levels, plus E1, E2 and M1 type radiative transition probabilities, as well as line and oscillator strengths among the levels of Fe XXII, are provided. The calculated results are compared with data compiled by NIST, as well as results obtained from other codes. The agreement between the length and velocity forms of electric transition operators is checked for the strongest transitions as an additional measure of the accuracy for the data obtained. The total radiative probabilities and five strongest probabilities required for calculating branching ratios of levels are presented as well.

Results were obtained using the multiconfiguration Dirac-Fock (MCDF) method employed in the GRASP code of Grant et al. (1980) and Parpia et al. (1996) (http://www.am.qub.ac.uk/DARC). To cross-check our MCDF data, we used a code developed by Bogdanovich & Karpuskiene (1999), which adopts the configuration interaction (CI) method on the basis of transformed radial orbitals (TROs) with variable parameters, including relativistic effects in the Breit-Pauli approximation ( CITRO).

In the MCDF approach, intermediate coupling wavefunctions
are
constructed by using an expansion of the form:

where configuration state functions (CSFs) are expressed as antisymmetrized products of two-component orbitals

(2) |

Here

The configuration mixing coefficients
are obtained
through diagonalization of the Dirac-Coulomb-Breit
Hamiltonian:

(3) |

where is one-electron Dirac Hamiltonian, is the instantaneous Coulomb repulsion. The effects of the transverse interaction that corresponds to the Breit interaction:

in the low-frequency limit were evaluated in the first order of perturbation expansion. QED corrections, which include vacuum polarization and self-energy (known as the Lamb shift), are considered in the first order of perturbation theory.

One-electron excitations from the 2s and 2p orbitals of the 1s^{2}2s^{2}2p^{1},
1s^{2}2s^{1}2p^{2} and 1s^{2}2p^{3} configurations up to the 8k orbital, as well as two- and three-electron excitations
from orbitals with *n*=2 to all possible
combinations of two or three electrons in the shells up to *n*=3, were employed to generate
one-electron wavefunctions as a basis set for CSFs in the MCDF method.
Additionally, to extend the CI basis and obtain higher accuracy, an additional 14 configurations with electrons in *n*=4 are included, namely:
1s^{2}2s^{1}4s^{2}, 1s^{2}2s^{1}4p^{2}, 1s^{2}2s^{1}4d^{2},
1s^{2}2s^{1}4f^{2}, 1s^{2}2p^{1}4s^{2}, 1s^{2}2p^{1}4p^{2},
1s^{2}2p^{1}4d^{2}, 1s^{2}2p^{1}4f^{2}, 1s^{2}2s^{1}3p^{1}4p^{1},
1s^{2}2s^{1}3p^{1}4f^{1}, 1s^{2}3s^{1}3p^{1}4p^{1},
1s^{2}3s^{1}3p^{1}4f^{1}, 1s^{2}2p^{1}3p^{1}4s^{1},
1s^{2}2p^{1}3p^{1}4d^{1}. The total number of CSFs in our MCDF calculations is 2253.

In the nonrelativistic, multiconfiguration Hartree-Fock approach,
intermediate coupling wavefunctions
being eigenfunctions of Coulomb-Breit-Pauli Hamiltonian are expanded in terms
of CSFs
obtained in the *LSJ*-coupling scheme:

CSFs are formed from a basis of one-electron spin orbitals:

(6) |

The number of CSFs in the intermediate coupling wavefunction expansion (5) is limited. Hence transformed radial orbitals with variable parameters were proposed by Bogdanovich & Karpuskiene (1999) to mimic the correlation effects from omitted CSFs:

Here

(8) |

with variable parameters

The Coulomb-Breit-Pauli Hamiltonian is used for our configuration interaction
calculations which lead to the expansion coefficients
and the corresponding energies for levels.
This Hamiltonian includes spin-orbit
,
spin-other-orbit
and spin-spin
corrections
as well as orbit-orbit corrections
within a shell of equivalent electrons:

= | |||

(9) |

where is expressed in atomic units. Orbit-orbit interactions between shells omitted in our CITRO calculations are usually smaller than within shells.

The frozen core approximation is used to obtain Hartree-Fock radial orbitals with . To include correlation effects from higher CSFs, the obtained basis is supplemented by TROs with principal quantum numbers and orbital quantum numbers . The large number of admixed configurations is reduced, leaving only configurations with significant influence on the energies of the adjusted configurations. For this, the method presented by Bogdanovich et al. (2002) and Bogdanovich & Momkauskaite (2004) is adopted. The total number of CSFs with odd parity is 2115, and the number with even parity is 1987 in our CITRO calculations.

In Table 1 we list energies of the 407 lowest levels of Fe XXII obtained
with the GRASP code. These levels correspond to 1s^{2}2s^{2}2p^{1}, 1s^{2}2s^{1}2p^{2}, 1s^{2}2p^{3}, 1s^{2}2s^{2}3*l*, 1s^{2}2s^{1}2p^{1}3*l*, 1s^{2}2p^{2}3*l*,
1s^{2}2s^{2}4*l*', 1s^{2}2s^{1}2p^{1}4*l*', 1s^{2}2p^{2}4*l*',
1s^{2}2s^{2}5*l*'', 1s^{2}2s^{1}2p^{1}5*l*'',
1s^{2}2s^{2}6*l*''' (*l* = 0,1,2, *l*' = 0,1,2,3, *l*'' = 0,1,2,3,4 and *l*''' =
0,1,2,3,4,5)
configurations. Energy levels are given in cm^{-1} relative to
the ground state 1s^{2}2s^{2}2p^{1} ^{2}P_{0.5}, along with the leading
percentage compositions
(where these exceed 10%) from the expansion
relation Eq. (1) for intermediate coupling wavefunctions.
The expansion coefficients for the intermediate coupling wavefunctions
with CSFs presented in the *LSJ*-coupling scheme (5) are obtained
from the expansion terms of intermediate coupling wavefunctions
with CSFs presented in *jj*-coupling scheme (1).
Diagonalization of Dirac-Coulomb-Breit matrix gives the expansion
coefficients for intermediate coupling wavefunctions
with CSFs constructed from two-component orbitals.
The indices for the levels provided in the first column of Table 1
are used in all subsequent tables.

Some excited levels due to strong intermediate coupling are assigned
to the same CSFs. In Table 2
new *LSJ*-coupling spectroscopic notations are proposed
for the levels which have the same CSFs with primary contributions
to the intermediate coupling wavefunctions.
For this, the technique presented in our earlier paper for Fe XIX
(Jonauskas et al. 2004b) is used.

In Table 3 we compare our calculated energy levels with values
compiled by NIST (National Institute for Standards and Technology:
www.physics.nist.gov) whose data are commonly used as a reference set.
Fairly good agreement is obtained for the MCDF calculations, with the
many energy levels agreeing to better than within 1% with the NIST recommended
values. However, some excited energy levels (indices 29, 55, 59, 69, 74, 174 and 195)
differ by up to 2% from NIST data. As well, our absolute values of energies for two levels (55 and 174) show the largest deviations from the NIST recommended values,
while another five levels (29, 59, 69, 74 and 195) have MCDF absolute
energies lower than those from NIST.
To cross-check our results for these levels, we present data from
CITRO calculations. For both our calculations, the highly excited levels show the similar
discrepancies from NIST values. On the other hand, the MCDF energies of levels 174
(1s^{2}2s^{1}2p^{1} (^{3}P) 4d^{1} ^{2}F_{3.5}) and 195
(1s^{2}2s^{1}2p^{1} (^{1}P) 4d^{1} ^{2}F_{3.5}) show
good agreement with NIST data after their values are interchanged.
This suggests that there are some typographical errors
in the NIST data for these levels. The calculated energies of level 29
(1s^{2} 2s^{1} 2p^{1}(^{3}P)3p^{1} ^{2}P_{0.5})
has similar discrepancies ()
for both calculations presented in
Table 3, but our MCDF result is closer to the NIST
recommended value. As well, our CITRO and MCDF calculations show the same percentage
composition for the intermediate coupling wavefunction of the level.
The energy of the 1s^{2} 2s^{1} 2p^{1} (^{3}P) 3p^{1} ^{2}P_{0.5} level (29) is lower than
that of 1s^{2} 2s^{1} 2p^{1} (^{3}P) 3p^{1} ^{2}P_{1.5} (28) in the
NIST data, while the ordering of both levels in our calculations is opposite to the NIST order.
It suggests that a bigger CI basis may be required for the MCDF calculations to obtain better agreement with the NIST values for the level.
However, the energy for the level obtained by Zhang & Pradhan (1997)
with SUPERSTRUCTURE
(Eissner et al. 1974),
and the value provided by Badnell et al. (2001)
from their AUTOSTRUCTURE
(Badnell 1997,1986)
calculations show similar differences
from NIST, and the same order as in our data.

For another five levels (55, 59, 69 and 74) with the largest discrepancies from NIST values, our MCDF and CITRO energies are in close agreement with the AUTOSTRUCTURE values of Badnell et al. (2001) and, in general, are slightly closer to the NIST data. Furthermore, as will be seen later, some transitions from those levels have largest discrepancies for line strengths compared with the NIST values. Additionally, our calculated lifetimes for the levels are presented in Table 3, to have an additional way to estimate the accuracy of our calculations. Differences between the calculated lifetimes for those levels is less than 10%, which confirm the accuracy of our results.

Energy levels and intermediate coupling wavefunctions
calculated with the configuration interaction method have been employed
to compute matrix elements of transition operators, which
subsequently are adopted for the calculation of transition probabilities, line and
oscillator strengths. The wavelengths and line strengths are presented in Table 4, where our transition data are compared with NIST results.
All line strengths belong to E1 type transitions except one from the first excited level
which corresponds to M1 type transition. E2 type transition from the level is much weaker
(MCDF and NIST line strengths equal to 2.10
10^{-3} and 2.15
10^{-3} correspondingly). The length form is used for electric transitions, as
they are less sensitive to the accuracy of wavefunction
compared with results obtained in the velocity form of the transition
operator. The wavelengths for many transitions agree to better than within 1% with
the NIST data. The largest differences for the wavelengths are obtained for transitions
between energy levels which showed largest discrepancies from NIST energies.
The largest discrepancy for line strengths is observed for the weak transition
(*f* = 3
10^{-5}) from the 2s^{1}2p^{2} ^{2}P_{0.5} level (8)
to 2s^{2}2p^{1} ^{2}P_{1.5} (2). The length and velocity forms differ by a factor of 3.5,
while our calculated MCDF and CITRO line strengths coincide within 20%
for the transition. Furthermore, the contribution of the
transition to the lifetime of the level is very small.
The line strengths for the transitions from
2s^{2}4s^{1} ^{2}S_{0.5} (126) to the first and second levels
of the ground configuration differ by 80% and 29%, respectively,
from the NIST values, but are in the limits of uncertainties provided by NIST.

Additionally, large discrepancies are observed for some transitions from the above mentioned levels which showed the largest deviations for energy levels compared with the NIST values. The line strengths for transitions , , , , , and differ by 29% to up to a factor of 4. The contributions to the lifetimes of those dipole allowed transitions vary from 7% ( ) to 84% ( ) (Table 5). The large discrepancies observed for transitions from levels 174 and 195 can be explained by a typographical error in the NIST data. As in a case of energy levels, our line strengths and wavelengths for those transitions are in good agreement with the NIST line strengths, after the NIST values are interchanged. Finally, our MCDF results for all those transitions with large discrepancies with NIST line strengths are in close agreement with the CITRO ones. It is possible that the determination of all those energy levels is uncertain in the NIST database. Additionally, the fairly good agreement between length and velocity forms for all those transitions indicate that our transition characteristics are quite accurate and reliable.

Transition probabilities, line and oscillator strengths are calculated for radiative E1, E2, E3, M1 and M2 type transitions among the presented 407 energy levels of Fe XXII. We provide the transition wavelengths, probabilities, oscillator strengths and line strengths of E1, E2 and M1 type for Fe XXII in Tables 6-8. Only the strongest transitions are presented here for every type of transition: electric dipole transitions with , electric quadrupole with and magnetic dipole with . The relation between length (Babushkin gauge in the relativistic approach) and velocity (Coulomb gauge) forms are provided for both electric transitions to have an additional indicator of accuracy of our MCDF results. The difference between both forms for many strong E1 type transitions () does not exceed 20%. Only 4 transitions from highly excited levels have differences larger than 20% but less than 40%. On the other hand, the contribution of these transitions to the lifetimes of the levels is much smaller than 1%. Finally, the length and velocity forms of the strongest electric dipole transitions () agree to better than within 10%, with an average deviation of only 1.8% for 457 transitions. For many E2 transitions, the two forms agree to better than within 10%.

In Table 5 we provide the five largest spontaneous radiative transition
probabilities from each level, and the sum of all E1, E2, E3, M1 and M2 radiative
transition probabilities from the corresponding level to all lower levels.
The sums of radiative transition probabilities are important
for branching ratios, while their inverse values are equal to
the lifetimes of the levels. The contribution to the lifetimes of levels
for many forbidden electric quadrupole and
magnetic dipole transitions is negligible and does not exceed 1%.
However, the highly excited
2s^{1}2p^{1} (^{3}P) 3p^{1} ^{4}D_{0.5} (33) and
2p^{2} (^{1}D) 3d^{1} ^{2}G_{0.5} (114)
levels decay primarily through E2 type transitions.
For another two levels, i.e. 2p^{2} (^{3}P) 3d^{1} ^{4}D_{0.5} (97)
and 2p^{2} (^{3}P) 3d^{1} ^{4}F_{0.5} (96), E2 type transitions contribute 48%
and 46%, respectively. As well, the first excited level decays mainly due to magnetic dipole
transition.

In this paper, energy levels, electric dipole, electric quadrupole and magnetic dipole
radiative transition rates, oscillator and
line strengths have been reported for the lowest 407 levels in Fe XXII in the MCDF approximation. The characteristics of the levels in the *LSJ*-coupling scheme have been provided and checked for their completeness. Calculated values have been compared with NIST
recommended values. Breit-Pauli energy levels and electric dipole transition
characteristics in the basis set of transformed radial
orbitals with variable parameters were used to cross-check the
accuracy of our results. Likely typographical errors for Fe XXII have been
found in both the NIST energy levels and
transition characteristics for the highly excited
2s^{1}2p^{1} (^{3}P) 4d^{1} ^{2}F_{3.5} and
2s^{1}2p^{1} (^{1}P) 4d^{1} ^{2}F_{3.5} levels.

The forbidden transitions have been taken into account to obtain
total radiative transition probabilities from levels.
The total radiative probabilities allow estimates of the lifetimes of
the presented states, and are also important for the calculation
of branching ratios. It was found that the highly excited
2s^{1}2p^{1} (^{3}P) 3p^{1} ^{4}D_{0.5}
and 2p^{2} (^{1}D) 3d^{1} ^{2}G_{0.5}
levels mainly decay due to forbidden E2 type transitions.
Their contributions to the lifetimes of
the 2p^{2} (^{3}P) 3d^{1} ^{4}D_{0.5}
and 2p^{2} (^{3}P) 3d^{1} ^{4}F_{0.5} levels
is 48% and 46%, respectively. The first excited level decays mainly through M1 type transition.
For each level, the five strongest transition probabilities are presented.

To conclude, the comparison with the NIST recommended wavelengths and radiative transition data shows that all the presented theoretical results are reliable, and may be successfully used for the interpretation of astronomical and other spectral observations.

FPK and SJR are grateful to AWE Aldermaston for the award of William Penney Fellowships. This work was supported by PPARC and EPSRC, and also by NATO Collaborative Linkage Grant CLG.979443. We are also grateful to the Defence Science and Technology Laboratory (dstl) for support under the Joint Grants Scheme.

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Online Material

**Table 1:**
MCDF results for energy levels of Fe XXII.
The energies of levels are presented relative to the ground energy.
The leading percentage compositions (>10%) of levels are provided in the last column.

**Table 2:**
Suggested change of spectroscopic identifications
of levels to ensure the completeness of spectroscopic dataset.
The indices of levels for which spectroscopic identifications are
changed are presented in the first column. The second column contains index
of level with the same highest contribution of configuration state function
(Table 1) as level from the first column before change.
Indices of levels in the first two columns are taken from Table 1.

**Table 3:**
Comparison of calculated energies
and
for Fe XXII levels with data compiled by NIST (
).
Indices of levels in the first column and CSFs in the second column are
taken from Table 1. Energies are in cm^{-1}.
The lifetimes of levels are presented in the last two columns.

**Table 4:**
Comparison of calculated wavelengths (
,
)
and line strengths (
,
)
for Fe XXII
with values presented by NIST (
,
).
Indices of levels in the first two columns are taken from Table 1.

**Table 5:**
The five major spontaneous radiative transition probabilities *A*^{r} from each level of Fe XXII and the sum of all radiative probabilities
from the corresponding level. Electric dipole, quadrupole and octupole as well as
magnetic dipole and quadrupole transitions are included.
Arrow marks the final level to which radiative transition happens from the level.

Copyright ESO 2006