IOP PUBLISHING PHYSICA SCRIPTA
Phys. Scr. 79 (2009) 035401 (11pp) doi:10.1088/0031-8949/79/03/035401
A comprehensive set of UV and x-ray
radiative transition rates for Fe XVI
S N Nahar
1
, W Eissner
2
, C Sur
1,3
and A K Pradhan
1
1
Department of Astronomy, The Ohio State University, Columbus, OH 43210, USA
2
Institut für Theoretische Physik, Teilinstitut 1, 70550 Stuttgart, Germany
E-mail: nahar@astronomy.ohio-state.edu, csur@astronomy.ohio-state.edu,
pradhan@astronomy.ohio-state.edu and we@theo1.physik.uni-stuttgart.de
Received 24 July 2008
Accepted for publication 19 November 2008
Published 3 March 2009
Online at stacks.iop.org/PhysScr/79/035401
Abstract
Sodium-like Fe XVI is observed in collisionally ionized plasmas such as stellar coronae and
coronal line regions of active galactic nuclei including black hole-accretion disc environments.
Given its recombination edge from neon-like Fe XVII at ∼25 Å, the Fe XVI bound–bound
transitions lie in the soft x-ray and EUV (extreme ultraviolet) range. We present a
comprehensive set of theoretical transition rates for radiative dipole allowed E1 transitions
including fine structure for levels with n`(SL J ) 6 10, ` 6 9 using the relativistic Breit–Pauli
R-matrix (BPRM) method. In addition, forbidden transitions of electric quadrupole (E2),
electric octupole (E3), magnetic dipole (M1) and magnetic quadrupole (M2) type are
presented for levels up to 5g(SLJ) from relativistic atomic structure calculations in the
Breit–Pauli approximation using code SUPERSTRUCTURE. Some of the computed levels are
autoionizing, and oscillator strengths among those are also provided. BPRM results have been
benchmarked with the relativistic coupled cluster method and the atomic structure Dirac–Fock
code GRASP. Levels computed with the electron collision BPRM codes in bound state mode
were identified with a procedure based on the analysis of quantum defects and asymptotic
wavefunctions. The total number of Fe XVI levels considered is 96, with 822 E1 transitions.
Tabulated values are presented for the oscillator strengths f , line strengths S and Einstein
radiative decay rates A. This extensive dataset should enable spectral modelings up to highly
excited levels, including recombination-cascade matrices.
PACS numbers: 31.10.+z, 31.15.ag, 31.15.aj
1. Introduction
Highly charged iron ions exist in a variety of high-temperature
astrophysical sources emitting or absorbing radiation from
the optical to the x-ray range. The spectral lines of these
ions provide information about physical conditions and
chemical abundances in many sources. Iron, including Fe
XVI, lines have been recently identified in XMM observations
of the narrow-line Seyfert 1 galaxy MKN335, currently in a
historically low state of x-ray flux from black hole accretion
[10]. The analysis and modeling of these spectra require
accurate and extensive radiative data. Coronal iron ions are of
special interest because they provide temperature and density
diagnostics. Their presence implies a high degree of ionization
and could also be a discriminant between photoionized and
3
Present address: Indian Institute of Astrophysics, Banglore, India.
collisionally ionized environments. Spectral lines of Ne-like
Fe XVII are signatures of most coronal plasmas. It follows
that in collisional ionization equilibrium adjacent ionization
stages, Na-like Fe XVI and F-like Fe XVIII, would also be
present; indeed, all three ions are often observed. However,
the energetics of the three ions differ significantly. Whereas
the strongest x-ray lines of Fe XVII and Fe XVIII lie around
15 Å, the Fe XVI lines are found in the softer x-ray region
below the recombination edge
Fe XVII (2p
6 1
S
0
) + e(k
2
→ 0) −→ Fe XVI (n`; SL J ) + hν,
(1)
at 25.3 Å, hence photo-excitations or bound–bound transitions
at λ > 25 Å may be detected. They span a wavelength region
from soft x-rays to the EUV. For example Fe XVI lines from
solar active regions due to transitions among low n(S L J )
levels are observed around 300 Å on the one hand, and from
0031-8949/09/035401+11$30.00 1 © 2009 The Royal Swedish Academy of Sciences Printed in the UK
Phys. Scr. 79 (2009) 035401 S N Nahar et al
Table 1. Binding energies E
c
calculated by BPRM for Fe XVI (with
magnetic interaction between electrons that can be absorbed into ζ )
and comparison with observed energies E
o
from NIST. I
J
is the
positional level index in the given symmetry J π, K or K + L telling
that closed shells 1s
2
or 1s
2
2s
2
2p
6
are screening all spin–orbit
parameters ζ . Case K + L computed with NRANG2=15 basis
functions while RA=4.03.
Level J I
J
E
c
/Ry E
o
/Ry ν
o
K K + L
2p
6
3s
2
S 0.5 1 35.9570 35.9517 35.9611
3p
2
P
o
1.5 1 33.2342 33.2344 33.2442
3p
2
P
o
0.5 1 33.4269 33.4208 33.4351
3d
2
D 2.5 1 29.7436 29.7494 29.7790 2.93 201
3d
2
D 1.5 1 29.7853 29.7767 29.8055 2.93 070
4s
2
S 0.5 2 18.9363 18.9415 18.9410
4p
2
P
o
1.5 2 17.8566 17.8677 17.8665
4p
2
P
o
0.5 2 17.9324 17.9407 17.9394
4d
2
D 2.5 2 16.5695 16.5836 16.5879 3.92 848
4d
2
D 1.5 2 16.5879 16.5965 16.5992 3.92 714
4f
2
F
o
3.5 1 16.0472 16.0524 16.0462
4f
2
F
o
2.5 1 16.0544 16.0572 16.0503
5s
2
S 0.5 3 11.6932 11.6856 11.7032
5p
2
P
o
1.5 3 11.1648 11.1523 11.1641
5p
2
P
o
0.5 3 11.1997 11.1888 11.2004
5d
2
D 2.5 3 10.5448 10.5335 10.5494 4.92 695
5d
2
D 1.5 3 10.5538 10.5403 10.5545 4.92 495
5f
2
F
o
3.5 2 10.2719 10.2671 10.2734
5f
2
F
o
2.5 2 10.2754 10.2695 10.2761
5g
2
G 4.5 1 10.2407 10.2410 10.2379
5g
2
G 3.5 1 10.2428 10.2424 10.2388
6s
2
S 0.5 4 7.92865 7.92884 7.93052
6p
2
P
o
1.5 4 7.62529 7.62755 7.63071
6p
2
P
o
0.5 4 7.64675 7.64840 7.65350
6d
2
D 2.5 4 7.27514 7.28139 7.28653 5.92 734
6d
2
D 1.5 4 7.28069 7.28539 7.29200 5.92 511
6f
2
F
o
3.5 3 7.12545 7.12931 7.13599
6f
2
F
o
2.5 3 7.12755 7.13072 7.13654
7s
2
S 0.5 5 5.73268 5.73174 5.67969
7p
2
P
o
1.5 5 5.54563 5.54512 5.51567
7d
2
D 2.5 5 5.32979 5.33174 5.33523 6.92 697
7d
2
D 1.5 5 5.33325 5.33426 5.33797 6.92 519
7f
2
F
o
3.5 4 5.23597 5.23697 5.24165
7f
2
F
o
2.5 4 5.23727 5.23783 5.24037
8p
2
P
o
1.5 6 4.21396 4.21377 4.17610
8d
2
D 2.5 6 4.07158 4.07258 4.07586 7.92 520
8d
2
D 1.5 6 4.07386 4.07426 4.07768 7.92 343
8f
2
F
o
3.5 5 4.00876 4.00914 4.01390
8f
2
F
o
2.5 5 4.00965 4.00972 4.01481
9p
2
P
o
1.5 7 3.31016 3.31001 3.27395
9d
2
D 2.5 7 3.21126 3.21171 3.20105 8.94 280
9d
2
D 1.5 7 3.21283 3.21289 3.16460 8.99 416
9f
2
F
o
3.5 6 3.16713 3.16723 3.15548
9f
2
F
o
2.5 6 3.16774 3.16764 3.15548
higher levels between 25 Å < λ < 80 Å on the other hand
(e.g. [6, 7]).
Multi-wavelength spectroscopy of ions such as Fe XVI
requires a reasonably complete set of atomic parameters.
Generally, these are obtained from large-scale calculations of
high accuracy, such as those being carried out under the Iron
Project (IP: [11]), and extensions thereof such as the RmaX
Network (viz. www.astronomy.ohio-state/∼nahar). These
calculations require a great deal of effort and computational
resources. The Ohio State University (OSU) group has
been engaged in calculations of such extensive datasets of
Table 2. 1µ
`
= ν
`+1/2
− ν
`−1/2
for lower n from data in table 1.
n` K K + L Obs
3p 0.00801 0.00775 0.00793
4p 0.00801 0.00770 0.00770
3d 0.00208 0.00134 0.00131
4d 0.00218 0.00154 0.00134
4f 0.00090 0.00059 0.00051
i.e. 7.2 mRy 4.7 mRy 4.1 mRy
radiative transition probabilities for a number of iron ions
as well as astrophysical applications and spectral modeling
(www.astronomy.ohio-state/∼pradhan). With particular
relevance to the present work transition rates for all levels
up to high-n(SL J ) are needed in such models. For example,
recently, we have developed a general spectral modeling
(GSM) code for transient, quasi-steady state, and steady-state
laboratory and astrophysical plasmas [ 18]. One of the features
of GSM is the computation of a collision-less transition
matrix that requires a complete set of transition rates in
order to compute recombination-cascade matrices. The first
application of GSM, to interpret hard x-ray spectra from
He-like Ca, Fe and Ni, revealed significant and unexpected
discrepancies with previous work for well-known and widely
used line ratios as diagnostics of high-temperature plasmas
[19, 20]. The new models employed BPRM level-specific
unified recombination rates for n(SL J ) 6 10, including
radiative and dielectronic recombination in an ab initio and
self-consistent manner, and BPRM radiative transition rates
(e.g. [16]). Similarly, we plan to investigate the accuracy and
completeness of spectral models of other important ionization
stages, namely the Na-, Ne- and F-like iron ions.
The National Institute for Standards and Technology
(NIST; (http://physics.nist.gov/PhysRefData/ASD/index.html)
provides the evaluated and compiled table of transitions for Fe
XVI from earlier calculations. Among forbidden transitions,
it includes only E2 transitions by Tull et al [29], who
calculated them in a frozen-core Hartree–Fock approximation.
Later Charro et al [5] calculated radiative decay rates for
E2 transitions of Fe XVI using a relativistic quantum defect
orbital formalism. In the absence of accurate fine structure
transitions, NIST has carried out fine structure splitting of LS
multiplets for some transitions.
This report provides radiative data obtained from the
first R-matrix calculation for Fe XVI-levels. It is part of
a project at OSU for a systematic study of iron and
iron-group atoms and ions by Nahar et al (e.g. Fe V [14,
17], Fe XVII [15], Fe XX [13], Fe XXI [12], Fe XXIV
and Fe XXV [16]). We also benchmark the present results
against other elaborate relativistic calculations using the
multi-configuration Dirac–Fock code (MCDF—GRASP2,
e.g. [21]) and relativistic coupled cluster theory (RCC,
e.g. [27, 28]).
2. Formulation
The relativistic Breit–Pauli R-matrix method with close
coupling (CC) approximation, described in a number of
papers [2, 3, 11, 22, 23, 26], enables calculation of a large
number of fine structure E1 transitions with high accuracy. In
2
Phys. Scr. 79 (2009) 035401 S N Nahar et al
Table 3. Sample table of fine structure energy levels of Fe XVI as sets of L S term components. C
t
is the core configuration, ν is the
effective quantum number.
C
t
(S
t
L
t
π
t
) J
t
nl 2J E/Ry ν SLπ
Nlv = 1,
2
L
e
: S (1)/2
2s
2
2p
6
(
1
S
e
) 0 3s 1 −3.59570E + 01 2.67 2 S e
Nlv(c) = 1:set complete
Nlv = 2,
2
L
o
: P ( 3 1 )/2
2s
2
2p
6
(
1
S
e
) 0 3p 1 −3.34269E + 01 2.77 2 P o
2s
2
2p
6
(
1
S
e
) 0 3p 3 −3.32342E + 01 2.78 2 P o
Nlv(c) = 2:set complete
Nlv = 2,
2
L
e
: D(5 3)/2
2s
2
2p
6
(
1
S
e
) 0 3d 3 −2.97853E + 01 2.93 2 D e
2s
2
2p
6
(
1
S
e
) 0 3d 5 −2.97436E + 01 2.93 2 D e
Nlv(c) = 2:set complete
Table 4. Sample output of fine structure energy levels of Fe XVI.
96 = number of levels, n 6 10, l 6 9
i
e
J i
J
E/Ry Config
2S+1
L
π
J π index
1 0.5e 1 −3.59611E + 01 2p
6
3s
2
S
e
100001
2 0.5e 2 −1.89410E + 01 2p
6
4s
2
S
e
100002
3 0.5e 3 −1.17032E + 01 2p
6
5s
2
S
e
100003
4 0.5e 4 −7.93052E + 00 2p
6
6s
2
S
e
100004
5 0.5e 5 −5.67969E + 00 2p
6
7s
2
S
e
100005
6 0.5e 6 −4.33725E + 00 2s
2
2p
6 1
S
e
8s
2
S
e
100006
7 0.5e 7 −3.39566E + 00 2s
2
2p
6 1
S
e
9s
2
S
e
100007
8 0.5e 8 −2.73047E + 00 2s
2
2p
6
1S
e
10s
2
S
e
100008
9 0.5o 1 −3.34351E + 01 2p
6
3p
2
P
o
110001
10 0.5o 2 −1.79394E + 01 2p
6
4p
2
P
o
110002
11 0.5o 3 −1.12004E + 01 2p
6
5p
2
P
o
110003
12 0.5o 4 −7.65350E + 00 2p
6
6p
2
P
o
110004
13 0.5o 5 −5.55871E + 00 2s
2
2p
6 1
S
e
7p
2
P
o
110005
14 0.5o 6 −4.22249E + 00 2s
2
2p
6 1
S
e
8p
2
P
o
110006
15 0.5o 7 −3.31599E + 00 2s
2
2p
6 1
S
e
9p
2
P
o
110007
16 0.5o 8 −2.67295E + 00 2s
2
2p
6 1
S
e
10p
2
P
o
110008
17 1.5e 1 −2.98055E + 01 2p
6
3d
2
D
e
300001
18 1.5e 2 −1.65992E + 01 2p
6
4d
2
D
e
300002
19 1.5e 3 −1.05545E + 01 2p
6
5d
2
D
e
300003
20 1.5e 4 −7.29200E + 00 2p
6
6d
2
D
e
300004
21 1.5e 5 −5.33797E + 00 2p
6
7d
2
D
e
300005
22 1.5e 6 −4.07768E + 00 2p
6
8d
2
D
e
300006
the CC approximation the wavefunction expansion 9
E
for a
(N + 1)-electron system with total orbital angular momentum
L, spin multiplicity (2S + 1) and total angular momentum
symmetry Jπ, is expanded in terms of the states of the
N -electron target ion as
9
E
(e + ion) = A
X
i
χ
i
(ion)θ
i
+
X
j
c
j
8
j
(e + ion), (2)
where the multi-configuration wavefunction χ
i
describes the
target in specific states of S
i
L
i
π
i
or J
i
π
i
, and θ
i
is the suitably
vector coupled wavefunction of the ‘collisional’ electron in
a channel labelled S
i
L
i
(J
i
)π
i
k
2
i
`
i
(S Lπ or J π), at channel
energies k
2
i
taken from the total energy balance
E = E
target
i
(
+k
2
i
if > 0 i.e. an open channel,
−z
2
/ν
2
i
otherwise (and z = Z − N ) .
(3)
The quantities 8
j
are correlation wavefunctions composed
of (N + 1) electrons orbital functions that (a) compensate
for the orthogonality conditions between the continuum and
the bound orbitals and (b) represent additional short-range
correlation that is often of crucial importance in scattering and
radiative CC calculations for each SLπ.
In the Breit–Pauli approximation relativistic terms
of relative magnitude α
2
, in the fine structure constant
α ≡ (e
2
/
¯
hc) ≈ 1/137.036, are added to the non-relativistic
Hamiltonian of say N electrons in the field of a nucleus with
electric charge number Z :
H
BP
N
= H
NR
N
+ H
mass
N
+ H
d
N
+ H
so
N
+
1
2
N
X
i6= j
g
i j
(so + so
0
) + g
i j
(ss
0
)
+g
i j
(css
0
) + g
i j
(d) + g
i j
(oo
0
)
, (4)
where H
NR
N
is the non-relativistic Hamiltonian
H
NR
N
=
N
X
i=1
−∇
2
i
−
2Z
r
i
+
N
X
j>i
2
r
i j
, (5)
assuming H := H/Ry and r := r/a
0
, i.e. scaling with the
3
Phys. Scr. 79 (2009) 035401 S N Nahar et al
Table 5. Sample set of f -values, line strengths S and coefficients A for E1 dipole allowed and intercombination transitions in Fe XVI.
26 11
I
i
I
k
λ/Å E
i
/Ry E
k
/Ry f S A
ki
∗ s
2 0 2 1 8 8 64 = gi Pi gf Pf Ni Nf NN
1 1 360.75 −3.5961E + 01 −3.3435E + 01 −1.210E − 01 2.873E − 01 6.199E + 09
1 2 50.56 −3.5961E + 01 −1.7939E + 01 −7.854E − 02 2.615E − 02 2.049E + 11
1 3 36.80 −3.5961E + 01 −1.1200E + 01 −2.276E − 02 5.516E − 03 1.121E + 11
1 4 32.19 −3.5961E + 01 −7.6535E + 00 −1.173E − 02 2.486E − 03 7.549E + 10
1 5 29.97 −3.5961E + 01 −5.5587E + 00 −6.351E − 03 1.253E − 03 4.716E + 10
1 6 28.71 −3.5961E + 01 −4.2225E + 00 −3.817E − 03 7.216E − 04 3.088E + 10
1 7 27.91 −3.5961E + 01 −3.3160E + 00 −2.487E − 03 4.572E − 04 2.129E + 10
1 8 27.38 −3.5961E + 01 −2.6730E + 00 −1.720E − 03 3.100E − 04 1.531E + 10
2 1 62.87 −1.8941E + 01 −3.3435E + 01 6.452E − 02 2.671E − 02 1.089E + 11
2 2 909.81 −1.8941E + 01 −1.7939E + 01 −1.790E − 01 1.072E + 00 1.442E + 09
2 3 117.73 −1.8941E + 01 −1.1200E + 01 −7.449E − 02 5.774E − 02 3.584E + 10
2 4 80.73 −1.8941E + 01 −7.6535E + 00 −3.257E − 02 1.732E − 02 3.334E + 10
2 5 68.10 −1.8941E + 01 −5.5587E + 00 −1.445E − 02 6.476E − 03 2.078E + 10
2 6 61.91 −1.8941E + 01 −4.2225E + 00 −7.427E − 03 3.028E − 03 1.293E + 10
2 7 58.32 −1.8941E + 01 −3.3160E + 00 −4.329E − 03 1.662E − 03 8.490E + 09
2 8 56.02 −1.8941E + 01 −2.6730E + 00 −2.763E − 03 1.019E − 03 5.874E + 09
3 1 41.93 −1.1703E + 01 −3.3435E + 01 1.241E − 02 3.426E − 03 4.707E + 10
3 2 146.13 −1.1703E + 01 −1.7939E + 01 9.760E − 02 9.391E − 02 3.049E + 10
3 3 1812.38 −1.1703E + 01 −1.1200E + 01 −2.409E − 01 2.875E + 00 4.891E + 08
3 4 225.02 −1.1703E + 01 −7.6535E + 00 −8.802E − 02 1.304E − 01 1.160E + 10
3 5 148.31 −1.1703E + 01 −5.5587E + 00 −2.807E − 02 2.741E − 02 8.511E + 09
3 6 121.82 −1.1703E + 01 −4.2225E + 00 −1.308E − 02 1.049E − 02 5.879E + 09
3 7 108.65 −1.1703E + 01 −3.3160E + 00 −7.344E − 03 5.254E − 03 4.150E + 09
3 8 100.91 −1.1703E + 01 −2.6730E + 00 −4.608E − 03 3.061E − 03 3.018E + 09
4 1 35.73 −7.9305E + 00 −3.3435E + 01 5.561E − 03 1.308E − 03 2.907E + 10
4 2 91.05 −7.9305E + 00 −1.7939E + 01 3.136E − 02 1.880E − 02 2.524E + 10
4 3 278.69 −7.9305E + 00 −1.1200E + 01 1.543E − 01 2.831E − 01 1.325E + 10
.
.
.
.
.
. . . . . . .
hydrogenic ionization energy Ry ≡ (α
2
/2)m
0
c
2
≈ 13.59 eV
and Bohr radius a
0
≡ (
¯
h/m
0
c)/α = 0.529 × 10
−8
cm.
The BPRM method includes the three one-body terms:
mass–velocity correction H
mass
N
, the Darwin term H
d
N
and
ordinary spin–orbit interaction H
so
N
. The mutual spin–orbit
and spin–other–orbit effects (so + so
0
) by core electrons
enter only via (Blume–Watson) screening of spin–orbit
parameters, as (Fe XVII+e) is effectively a one body rather
than an (N + 1) electron system, and all spin–spin interaction
(ss
0
) among electrons drops out for an unpolarized Ne-like
core. Complications by L-shell polarization correlation are
discussed in a later section.
This paper is concerned with solutions of
h9|H|9i = E
min
(6)
when all channels are closed, i.e. all k
2
i
< 0 in equation (3),
which is a Hartree–Fock eigenvalue problem with a discreet
spectrum E , because ν = n − µ forms Rydberg series along
integer n (while the target states χ
i
with energies E
i
are
‘frozen cores’). At eigenvalues E for target and (N + 1)
electrons in a bound state, and with ‘frozen cores’ χ
i
in
the particular case of expansion (2), to which it is moreover
applied as a first step.
The primary quantity for radiative transitions is the line
strength, for E1-type transitions in the length form the square
of a reduced dipole matrix element:
S =
*
9
f
N +1
X
j=1
C
[1]
r
j
9
i
+
2
, (7)
where 9
i
and 9
f
are the initial and final bound wavefunctions
(and r scaled in Bohr units a
0
). Derived from the line strength,
which does not explicitly depend upon the transition energy
E
ji
= E
j
− E
i
, are secondary quantities such as the oscillator
strength f
i j
, which is a pure number going back to classical
physics, and the radiative transition probability or Einstein’s
A-coefficient (E scaled in Ry), an inverse time:
f
i j
=
E
ji
3g
i
S, A
ji
∗ τ
0
= α
3
g
i
g
j
E
2
ji
f
i j
, (8)
where τ
0
≡
¯
h/Ry = 4.8381 × 10
−17
s is the time 2a
0
/(αc) it
takes a hydrogenic ground state electron to cross the Bohr
orbit, and g
i
, g
j
are the statistical weight factors of initial and
final state.
Radiative data for forbidden transitions of electric
quadrupole (E2), octupole (E3) and magnetic dipole
(M1), quadrupole (M2) type are computed using
configuration interaction atomic structure calculations
with code SUPERSTRUCTURE (SS), in the low-Z
Breit–Pauli approximation [8, 9, 15]. SS represents the
nuclear and electron–electron potential by a statistical
Thomas–Fermi–Dirac–Amaldi model potential and includes
relativistic one-body terms as well as two-body FS terms
in the Breit interaction of equation (3), and ignores the last
three two-body terms. Radiative probabilities for ‘forbidden’
excitation or de-excitation via various modes of higher
powers in α and E
i j
can be obtained with the generalized line
strength
S
Xλ
(i j ) =
9
j
O
Xλ
9
i
2
, S( ji) = S(i j), (9)
4
Phys. Scr. 79 (2009) 035401 S N Nahar et al
Table 6. Sample table of dipole allowed same-spin and intercombination E1 transitions in Fe XVI, grouped as fine structure components of
L S multiplets. Calculated energies have been replaced by observed energies.
C
i
–C
k
T
i
–T
k
g
i
: I–g
j
: K E
ik
f S A
(Å) (s
−1
)
2p63s–2p63p 2Se–2Po 2: 1- 2: 1 360.75 1.21E − 01 2.87E − 01 6.20E + 09
2p63s–2p63p 2Se–2Po 2: 1- 4: 1 335.41 2.62E − 01 5.80E − 01 7.78E + 09
L S 2Se–2Po 2- 6 3.83E − 01 8.67E − 01 7.20E + 09
2p63s–2p64p 2Se–2Po 2: 1- 2: 2 50.56 7.85E − 02 2.61E − 02 2.05E + 11
2p63s–2p64p 2Se–2Po 2: 1- 4: 2 50.36 1.47E − 01 4.87E − 02 1.93E + 11
L S 2Se–2Po 2- 6 2.26E − 01 7.48E − 02 1.98E + 11
2p63s–2p65p 2Se–2Po 2: 1- 2: 3 36.80 2.28E − 02 5.52E − 03 1.12E + 11
2p63s–2p65p 2Se–2Po 2: 1- 4: 3 36.75 4.33E − 02 1.05E − 02 1.07E + 11
L S 2Se–2Po 2- 6 6.61E − 02 1.60E − 02 1.08E + 11
2p63s–2p66p 2Se–2Po 2: 1- 2: 4 32.19 1.17E − 02 2.49E − 03 7.55E + 10
2p63s–2p66p 2Se–2Po 2: 1- 4: 4 32.17 2.24E − 02 4.75E − 03 7.23E + 10
L S 2Se–2Po 2- 6 3.41E − 02 7.24E − 03 7.29E + 10
2p63p–2p64s 2Po–2Se 2: 1- 2: 2 62.87 6.45E − 02 2.67E − 02 1.09E + 11
2p63p–2p64s 2Po–2Se 4: 1- 2: 2 63.71 6.83E − 02 5.73E − 02 2.24E + 11
L S 2Po–2Se 6- 2 6.70E − 02 8.40E − 02 3.33E + 11
2p64s–2p64p 2Se–2Po 2: 2- 2: 2 909.81 1.79E − 01 1.07E + 00 1.44E + 09
2p64s–2p64p 2Se–2Po 2: 2- 4: 2 848.09 3.86E − 01 2.15E + 00 1.79E + 09
L S 2Se–2Po 2- 6 5.65E − 01 3.22E + 00 1.68E + 09
2p64s–2p65p 2Se–2Po 2: 2- 2: 3 117.73 7.45E − 02 5.77E − 02 3.58E + 10
2p64s–2p65p 2Se–2Po 2: 2- 4: 3 117.18 1.38E − 01 1.07E − 01 3.36E + 10
L S 2Se–2Po 2- 6 2.13E − 01 1.65E − 01 3.41E + 10
2p64s–2p66p 2Se–2Po 2: 2- 2: 4 80.73 3.26E − 02 1.73E − 02 3.33E + 10
2p64s–2p66p 2Se–2Po 2: 2- 4: 4 80.57 6.19E − 02 3.28E − 02 3.18E + 10
L S 2Se–2Po 2- 6 9.45E − 02 5.01E − 02 3.24E + 10
2p63p–2p65s 2Po–2Se 2: 1- 2: 3 41.93 1.24E − 02 3.43E − 03 4.71E + 10
2p63p–2p65s 2Po–2Se 4: 1- 2: 3 42.30 1.30E − 02 7.22E − 03 9.66E + 10
L S 2Po–2Se 6- 2 1.28E − 02 1.06E − 02 1.44E + 11
2p64p–2p65s 2Po–2Se 2: 2- 2: 3 146.13 9.76E − 02 9.39E − 02 3.05E + 10
2p64p–2p65s 2Po–2Se 4: 2- 2: 3 147.85 1.03E − 01 2.01E − 01 6.29E + 10
L S 2Po–2Se 6- 2 1.01E − 01 2.95E − 01 9.30E + 10
2p65s–2p65p 2Se–2Po 2: 3- 2: 3 1812.38 2.41E − 01 2.87E + 00 4.89E + 08
2p65s–2p65p 2Se–2Po 2: 3- 4: 3 1690.35 5.18E − 01 5.76E + 00 6.04E + 08
L S 2Se–2Po 2- 6 7.59E − 01 8.63E + 00 5.66E + 08
2p65s–2p66p 2Se–2Po 2: 3- 2: 4 225.02 8.80E − 02 1.30E − 01 1.16E + 10
2p65s–2p66p 2Se–2Po 2: 3- 4: 4 223.76 1.63E − 01 2.40E − 01 1.09E + 10
L S 2Se–2Po 2- 6 2.51E − 01 3.70E − 01 1.11E + 10
. . . . . .
where Xλ represents the electric or magnetic-type operator
of multipolarity λ, thus generalizing the electric dipole case
O
E1
= C
[1]
r of equation (7) to O
Eλ
= C
[λ]
r
λ
(e.g. [ 15]). Along
the line of the E1 case, for which A of equation (8) reads
g
j
A
E1
ji
= 2.6773 × 10
9
s
−1
(E
j
− E
i
)
3
S
E1
(i, j) (10)
with S of equation (7), higher order transition modes lead to
probabilities
g
j
A
E2
ji
= 2.6733 × 10
3
s
−1
(E
j
− E
i
)
5
S
E2
(i, j), (11)
g
j
A
M1
ji
= 3.5644 × 10
4
s
−1
(E
j
− E
i
)
3
S
M1
(i, j), (12)
for electric quadrupole (E2) and magnetic dipole (M1)
radiation and
g
j
A
E3
ji
= 1.2050 × 10
−3
s
−1
(E
j
− E
i
)
7
S
E3
(i, j), (13)
g
j
A
M2
ji
= 2.3727 × 10
−2
s
−1
(E
j
− E
i
)
5
S
M2
(i, j), (14)
for electric octopole (E3) and magnetic quadrupole (M2)
radiation. Regarding contributions of BP order to M1 see [9].
The lifetime of a level k can be obtained from transition
probabilities as
τ
k
=
1
P
i
A
ki
, (15)
where the sum runs over all levels i to which k can decay.
3. Calculations
3.1. BPRM calculations for E1 transitions
With the BPRM code, the CC calculations proceed in several
stages. The BPRM package requires orbital wavefunction
input for the target or core eigenstates (here provided by
SS). From it STG1 computes one- and two-electron radial
integrals as specified. STG2 computes angular coefficients
for target and collisional channels in L S-coupling and target
term energies. Stage RECUPD adds the Breit–Pauli algebra,
in intermediate coupling computing target levels and with
the help of their term coupling coefficients the structure of
collisional channel sets of total J π on recoupling the L S
symmetries in a pair-coupling scheme. STGH completes the
inner region (r/a
0
6 RA = 4.0) task computing R-matrices for
the specified collisional symmetries.
5
![Table 7. Comparison of E1 transitions of Fe XVI line strengths SE1 and radiative decay rates AE1: RCC—relativistic coupled cluster, GRASP2—fully relativistic atomic structure [1], BPRM—relativistic Breit–Pauli R-matrix, SS—Breit–Pauli atomic structure.](/figures/table-7-comparison-of-e1-transitions-of-fe-xvi-line-wnt23cs4.webp)
![Table 8. Comparison of present absorption oscillator strengths f for Fe XVI with those from experiment by Buchet et al [4] (a: simulation procedure, b: fit program) and calculation by Weiss [30].](/figures/table-8-comparison-of-present-absorption-oscillator-31qsgio1.webp)
