for review only...moreover, transition energies for the transitions 3s 2s 1/2 – 3p 2p 3/2,1/2....
TRANSCRIPT
For Review Only
Accurate study on the properties of spectral lines for Na-likeCr13+
Journal: Canadian Journal of Physics
Manuscript ID cjp-2018-0218.R2
Manuscript Type: Article
Date Submitted by theAuthor: 06-Jul-2018
Complete List of Authors: Singh, A.K.; University of Delhi, Department of PhysicsDimri, Mayank; University of Delhi, Department of Physics andAstrophysics; Deen Dayal Upadhyaya College, University ofDelhi, DST Research LabDawra, Dishu; University of Delhi, Department of Physics andAstrophysicsJha, Alok; University of Delhi, Department of PhysicsMohan, Man; University of Delhi, Department of Physics
Keyword: Energy levels, transitions wavelength, Extreme Ultraviolet, SoftX-ray, lifetime
Is the invited manuscriptfor consideration in a
Special Issue? :Not applicable (regular submission)
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Accurate study on the properties of spectral lines for Na-
like Cr13+
A. K. Singha,b, Mayank Dimria,b, Dishu Dawrab, Alok K. S. Jhac, Man Mohana,b
a Department of Physics, D.D.U. College, University of Delhi, Delhi-110078, India.
b Department of Physics & Astrophysics, University of Delhi, Delhi-110007, India.
cDepartment of Physics, Kirori Mal College, University of Delhi, Delhi-110007, India.
Abstract
An extended calculation of energy levels, radiative rates and lifetimes are reported for sodium like
chromium. Extensive configuration interaction calculations have been performed using general-
purpose relativistic atomic structure package (GRASP). The radiative rates, oscillator strengths and line
strengths are listed for all electric dipole (E1) transitions. However, for magnetic dipole (M1), electric
quadrupole (E2) and magnetic quadrupole (M2) transitions, only radiative rates are listed. The
importance of valence-valence (VV) and core-valence (CV) correlation effects in the calculation of
energy levels have also been shown. To confirm the accuracy of the present results for energy levels
by GRASP, independent calculations have been performed by using Flexible Atomic Code (FAC) and
configuration interaction method (CIV3). The accuracy of the present levels, wavelengths, transition
rates and lifetimes are assessed by comparing them to available experimental and other theoretical
results. We believe that our extensive results may be beneficial in fusion plasma research and
astrophysical investigations and applications.
PACS. 32.70.Cs Oscillator strengths, lifetimes – 32.10.Fn Fine and hyperfine structure
Keywords: Energy levels, transitions wavelength, Extreme Ultraviolet, Soft X-ray, lifetime
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1. Introduction
During the past few years, the spectra of multiply charged ions have become a subject of Astrophysical
interest as satellite-borne telescopes and instruments expand the range of detectable radiation into the
X-ray region. In particular, the studies of stellar and solar coronae have intensified as a result of quality
data acquisitions by the Extreme-Ultraviolet (EUV) Explorer satellite, Chandra X-ray Observatory, and
the Solar Heliospheric Observatory (SOHO) that recorded superb coronal spectra [1,2]. The Ultraviolet
(UV) through EUV and Soft-X-ray emission lines from multiply charged ions are particularly useful
as they provide detailed knowledge of the coronal atmosphere.
Interest in the line emission from highly ionised atoms in Tokamaks derives, in the first instance, from
the effect of impurities on the overall performance of the Tokamak as a fusion device. In this respect,
the impurities can influence the heating rate, the magnetic field topography, the plasma stability and
equilibrium and the energy and particle confinement. Secondly, the impurity ion line emission may be
used as a diagnostic of the temporal and spatial Plasma conditions [3]. Chromium has been a significant
element that plays an important role in Tokamak as it is present as an impurity in stainless steel which
is a structural material of some Tokamaks [3].
Spectra of sodium like ions are often used to diagnose the hot gases found in Tokamaks, laser produced
plasmas and other devices used for controlled fusion research. A knowledge of the wavelengths of these
spectra is very important for detecting the presence of sodium like ions in the plasmas and for
calibrating spectral measurements [4]. Spectra of different sodium like ions including Cr XIV have
been reported by Peacock et al. [3] by presenting transition wavelengths for D1 and D2 lines
corresponding to the transitions 3s 2S1/2 – 3p 2P3/2,1/2.
In the past, several theoretical calculations and experimental measurements have been done on Na-like
ions by employing various experimental techniques and theoretical methods [5,6]. Edlen [6] has
identified the transitions 3s - np (n = 4,5), 3p -4s, 3p - nd (n = 4,5), and 3d - nf (n = 4-6) in vacuum
spark discharges. The 3d-4p lines at ~101 Å were identified by Fawcett et al. [7]. Reader et al. [4] have
made the measurements for the wavelengths of the 3s-3p, 3p-3d, 3d-4f transitions of the sodium like
ions by photographing laser produced plasmas and tokamak plasmas with grazing incidence
spectrographs. The energies of the transitions were also calculated with Dirac Fock Computer codes.
Cohen et al. [8] have presented the energy levels and term splitting for the Na I isoelectronic sequence
from K X through Mn XV by adopting wavelengths reported by Edlen [6]. Similarly, Fischer et al. [9]
have computed energy levels, lifetimes and transition probabilities for Na-like to Ar-like ions by
applying non-orthogonal spline CI, multiconfiguration Hartree–Fock (MCHF) and multiconfiguration
Dirac–Hartree–Fock (MCDHF) methods. Further, Johnson et al. [10] have presented energy intervals
and transition rates for sodium and lithium like ions with nuclear charge ranging from 3 to 100 using
third-order many-body perturbation theory. Moreover, transition energies for the transitions 3s 2S1/2 –
3p 2P3/2,1/2. have been reported by Kim et al. [11], using Dirac Fock method. Also, Douglas et al. [12]
have used rapid relativistic atomic structure approach to obtain transition energies and electric dipole
oscillator strengths for 71 Na-like ions with 22 ≤ Z ≤ 92. Furthermore, Verner et al. [13] have listed
wavelengths, statistical weights and oscillator strengths for 2249 spectral lines at wavelengths greater
than 228 Å arising from the ground states of various ions including Cr XIV. Also, a comprehensive
study of transition energies of D lines in Na-like ions was presented by Gillaspy et al. [14].
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The main difficulty associated with theoretical studies is electron-electron correlation effects;
moreover, the relativistic corrections have to be taken in to account. The sodium like chromium has a
complete core (1s22s22p6) plus one valence electron, which is the simplest system for studying the
different correlation effects, such as the valence-valence (VV), core-valence (CV), and core-core (CC)
correlations. The energy levels are practically free from effects of configuration mixing and therefore
they are well suited for a theoretical interpretation of line intensities and for diagnostic purposes [15].
Although a lot of work has been done on Na-like ions, only a few works have reported the detailed
structure calculations for Na-like chromium. Therefore, the aim of this work is to perform fully
relativistic calculations in order to upgrade the database for transition energies and radiative properties
such as oscillator strengths, line strengths, wavelengths and transition decay probabilities for E1, E2,
M1 and M2 EUV and SXR transitions for Cr XIV. In the present calculations, we have adopted the
relativistic multi-configuration Dirac–Fock (MCDF) approach. QED corrections due to vacuum
polarization and self-energy effects and Breit correction due to the exchange of virtual photons between
two electrons are fully considered. Further, to test the accuracy of our results for energy levels,
analogous calculations have been performed with CIV3 and Flexible Atomic Code (FAC).
In this work, we have studied with two different configuration sets for MCDF calculations according
to valence-valence correlation (VV) and core-valence correlation (CV) within the framework of
configuration interaction expansion. In our calculations, we have taken in to account the configurations
of 2p6nl with 3 ≤ n ≤ 8 and 0 ≤ l ≤ 4 including one electron excitations from valence to other high
subshells for valence-valence correlation, 2p53l nl' with 3 ≤ n ≤ 5, 2p54l 4l’ including one electron
excitations from 2p subshell to other high subshells and 2s2p6 3l nl' with 3 ≤ n ≤ 5, 2s2p6 4l 4l' including
one electron excitations from 2s subshell to other high subshells for CV correlation. A comparison has
been made wherever possible, and good agreement was achieved. We have also provided a detailed
comparison of our theoretical results with the data available from the NIST database (National Institute
of Standards and Technology, website at http://www.nist.gov/pml/data/asd.cfm). We have theoretically
identified EUV and soft X-ray transitions. An outline of the computational methods is presented in
Section 2 and results are discussed in Sections 3.
2. Theoretical Methods
2.1 Multi Configuration Dirac-Fock Method
To perform the large-scale calculations, fully relativistic MCDF method, revised by Norrington [16],
formerly developed by Grant et al. [17] is applied, which has also been successfully applied in our
previous work [18-22]. QED corrections due to self-energy and vacuum polarization effects and Breit
corrections due to the exchange of virtual photons as a first order perturbation theory have also been
considered. Since the elaborate depiction of this method has been presented elsewhere [16, 22-26], so
only a brief outline is discussed here. The Dirac-Coulomb Hamiltonian in MCDF approach for an N-
electron atom or ion can be written as follows
HDC = ∑Hi
N
i=1
+ ∑ ∑1
|ri − rj| (1)
N
j=i+1
N−1
i=1
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where ��, the one-electron Hamiltonian is given by
Hi = cα i. p i + βmc2 + Vnuc (2)
In equation (2), the first two terms signify kinetic energy of an electron and the last term represents the
Coulomb potential of the nucleus. 𝛼 and 𝛽 are 4x4 Dirac matrices and c is the speed of light.
The N electron wave function constructed from central-field Dirac orbitals is given by
ϕ𝑛𝑘𝑚 =1
𝑟 (
𝑃𝑛𝑘(𝑟) 𝜒𝑘𝑚 (Ɵ, ϕ,σ)
−𝑖 𝑄𝑛𝑘(𝑟) 𝜒−𝑘𝑚 (Ɵ, ϕ, σ)) (3)
where k is the Dirac angular quantum number, k =± (j+1/2) for l = j ± 1/2, so j = k - 1/2, m is the
projection of the angular momentum j, and 𝑃𝑛𝑘 and 𝑄𝑛𝑘 are large and small components of one
electron radial functions. The spin angular momentum 𝜒𝑘𝑚 (Ɵ, ϕ) is a 2 component function defined
by
𝜒𝑘𝑚 (Ɵ, ϕ) = ∑ ⟨𝑙𝑚 − σ 1
2 σ|𝑙
1
2 𝑗𝑚 ⟩ 𝑌𝑙
𝑚−σσ=±
1
2
(Ɵ, ϕ) ϕσ (4)
An atomic state function (ASF) for N electron system constructed by the linear combination of n
electronic configuration state functions (CSFs) is represented by
|ψα(PJM)⟩ = ∑Ci
n
i=1
(α)|γi(PJM)⟩ (5)
where 𝐶𝑖(𝛼) are the expansion mixing coefficients for each CSF and satisfy the relation
(Ci(α))ϯCj(α) = δij (6)
such that ASFs satisfy the orthonormality condition. 𝛼 represents the orbital occupation numbers,
coupling, etc. and γi(PJM) are CSFs which specify a particular state with a given parity and angular
momentum (J,M).
In equation (6), the basis wave-functions are enlarged by considering the important correlations and
relativistic effects. CSFs of particular parity P and symmetry have been generated by taking appropriate
excitations from reference configurations to higher shells.
By taking expectation value of the Dirac–Hamiltonian, we get the energy of N-electron system as
EαPJM = ⟨ψ
α(PJM)|HDC|ψ
α(PJM)⟩
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= ∑Ci∗(α)
ij
Ci(α) ⟨γi(PJM)|HDC|γ
j(PJM)⟩
= (CαDC)ϯ HDC Cα
DC (7)
The elements of Dirac Hamiltonian matrix 𝐻𝐷𝐶 are given by
HrsDC = ⟨γr(PJM)|HDC|γs(PJM)⟩ (8)
Using the condition of normalization
(HDC − EαDC I) Cα
DC = 0 (9)
where I is the (n x n) unit matrix. Thus the predicted atomic energy level EαPJM
can be taken to be
eigenvalues of HDC.
2.2 Configuration Interaction Method
The general configuration interaction program CIV3 of Hibbert [27,28] has been employed to execute
the present calculations. The configuration interaction (CI) atomic state functions (ASFs) are
represented as
Ψi(J) = ∑aij
M
j=1
ϕj(αjLjSjJ) (10)
where {ϕj} denotes a set of single-configuration wave functions, (αj) defines the coupling of the orbital
Lj and spin Sj angular momenta to give the total angular momentum J. The mixing coefficients aij are
obtained by diagonalising the Breit-Pauli Hamiltonian matrix relative to the basis {ϕj}.
The radial functions Pnl(r) are expressed as linear combination of normalized slater-type orbitals in
the form
Pnl(r) = ∑Cjnl
k
j=1
χjnl (r) (11)
where {Cjnl } are the Clementi type [29] coefficients and
χjnl (r) = (2ξjnl)
Ijnl+12
[(2Ijnl)!]12
rIjnl exp(−ξjnl r) (12)
with the integer Ijnl ≥ l+1.
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The parameters Cjnl and ξjnl are determined variationally, while the parameters Ijnl, being integers are
kept fixed in the optimization process. The radial functions are chosen to satisfy the orthonormality
condition
∫ Pnl(r)
∞
0
Pn´l(r)dr = 𝛅nn´ ; l + 1 < n´ ≤ n (13)
The wave functions given by equation (10) are used to calculate the excitation energies of the fine
structure levels.
Substituting the CI expansion of the ASF (10) in a variational principle where the energy of the state
is minimized subject to the normalization condition ⟨Ψ|Ψ⟩ = 1 yields
∑aj
j
( Hij − E δij) = 0 (14)
So that, for a particular Jπ symmetry, the optimum choice of the expansion coefficients { ai } is
identified as the set of eigenvector components of the diagonalized Hamiltonian matrix with typical
element Hij = ⟨Ψi|H|Ψj⟩, for a specific eigenvalue Ei.
The ordered eigenvalues (Ei) of the Hamiltonian matrix are upper bounds to the similarly ordered
energy levels
Ei ≥ Eiexact. (15)
It enables us to optimize different radial functions on different eigenvalues and represent the wave
functions of a variety of states simultaneously to an equivalent degree of accuracy.
In Configuration Interaction calculations using CIV3, we have used 15 orthonormal one-electron
orbitals: 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f and 5g. The 1s, 2s and 2p are chosen to be
Hartree-Fock orbitals of the ground state 1s22s22p63s (2S1/2) of Cr XIV ion, given by Clementi and
Roetti [29]. The spectroscopic orbitals 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f and 5g are optimized on the
excited states 2p63p, 2p63d, 2p6 4s, 2p6 4p, 2p6 4d, 2p64f, 2p6 5s, 2p6 5p, 2p6 5d, 2p6 5f and 2p65g
whereas 3s orbital is optimized on the ground state 2p63s. Moreover, to increase the flexibility of the
radial functions of 3d, 4d, 4f, 5d, 5f and 5g, we have added extra Slater Type Orbital (STO) basis
functions allowing k > n-l in equation (11), so that some of the coefficients 𝐶𝑗𝑛𝑙 could vary freely
while still satisfying orthonormality conditions on 𝑃𝑛𝑙 . The optimized radial function parameters are
listed in Table I. The process of optimizing the radial functions is summarized in Table II, giving a
good representation of valence electron in the excited states for the different LS symmetries. The
configurations included in the CI calculations for both even and odd parity are shown in Table III.
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Table I. Radial function parameters for optimized orbitals of Na-like Cr XIV.
Orbitals Expansion coefficients
(Cjnl)
Power of r
(Ijnl)
Exponents
(ξjnl)
3s 0.19668 1 18.42588
-0.88361 2 7.78541
1.39173 3 5.30220
3p 0.52363 2 9.73570
-1.12187 3 4.86867
3d 0.06275 3 10.63170
0.96139 3 4.87476
4s 0.11751 1 18.86704
-0.80998 2 6.55187
2.00701 3 4.84096
-1.89831 4 3.74207
4p 0.36579 2 9.42200
-2.28188 3 3.82846
2.64893 4 3.68743
4d 0.58736 3 5.07973
0.25990 3 4.99393
-1.29886 4 3.32750
4f 0.85909 4 3.53210
0.14100 4 3.44427
5s 0.08012 1 19.17750
-0.78108 2 5.69738
2.18443 3 4.59778
-4.15885 4 3.28502
3.31909 5 2.95012
5p 0.26629 2 9.28260
-3.84184 3 3.07938
8.02510 4 3.07534
-4.92790 5 2.93884
5d 0.02862 3 9.88394
1.94514 3 3.44615
-5.50036 4 3.25329
4.02585 4 2.53807
5f 1.79671 4 3.01650
0.00764 4 6.93361
0.02264 5 3.96775
-2.25835 5 2.77972
5g 0.99986 5 2.80077
0.00022 5 1.59679
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Table II. Method of Determining the Radial Functions.
Orbital Process of Optimization
1s,2s,2p.............. Hartree-Fock orbitals of 2p63s 2S of Cr XIV (Clementi & Roetti [29])
Orbital Eigenvalues Minimized Configurations
3s....................... 2p63s 2S 2p63s
3p....................... 2p63p 2Po 2p63p
3d....................... 2p63d 2D 2p63d
4s....................... 2p64s 2S 2p63s, 2p64s
4p....................... 2p64p 2Po 2p63p, 2p64p
4d....................... 2p64d 2D 2p63d, 2p64d
4f....................... 2p64f 2Fo 2p64f
5s....................... 2p65s 2S 2p63s, 2p64s, 2p65s
5p....................... 2p65p 2Po 2p63p, 2p64p, 2p65p
5d....................... 2p65d 2D 2p63d, 2p64d, 2p65d
5f....................... 2p65f 2Fo 2p64f, 2p65f
5g....................... 2p65g 2G 2p65g
Table III. Configurations Used. All possible coupling of angular momenta of the orbitals of the
following configurations are included for each LSJ symmetry.
Even Parity
[1s22s22p6] 3s, 3d, 4s, 4d, 5s, 5d, 5g
[1s22s22p5] 3s3p, 3p3d, 3s4p, 3s4f, 3p4s, 3p4d, 3d4p, 3d4f, 4s4p, 4s4f, 4p4d, 4d4f, 3s5p, 3s5f,
3p5s, 3p5d, 3p5g, 3d5p, 3d5f
[1s22s 2p6] 3s3d, 3s2, 3p2, 3d2, 3s4s, 3s4d, 3p4p, 3p4f, 3d4s, 3d4d, 4s4d, 4p4f, 4s2, 4p2, 4d2, 4f2,
3s5s, 3s5d, 3s5g, 3p5p, 3p5f, 3d5s, 3d5d, 3d5g
Odd parity
[1s22s22p6] 3p, 4p, 4f, 5p, 5f
[1s22s22p5] 3s3d, 3s2, 3p2, 3d2, 3s4s, 3s4d, 3p4p, 3p4f, 3d4s, 3d4d, 4s4d, 4p4f, 4s2, 4p2, 4d2, 4f2,
3s5s, 3s5d, 3s5g, 3p5p, 3p5f, 3d5s, 3d5d, 3d5g
[1s22s 2p6] 3s3p, 3p3d, 3s4p, 3s4f, 3p4s, 3p4d, 3d4p, 3d4f, 4s4p, 4s4f, 4p4d, 4d4f, 3s5p, 3s5f,
3p5s, 3p5d, 3p5g, 3d5p, 3d5f
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As this method incorporates two spin independent non-fine structure terms, the Darwin and relativistic
mass correction terms which only affect the overall energy of each LS coupled term and three spin-
dependent fine-structure terms being one body spin-orbit (SO), two body spin-other-orbit (SOO) and
spin-spin (SS) interactions [30], which split each LS state into separate J-dependent levels. However,
in our work, the calculation of SOO and SS matrix elements proved very time consuming due to
extensively employed CSF sets. It has been found [31] that a very little accuracy is lost by neglecting
the SS term and replacing SO terms by a modified spin-orbit operator
H′so = ∝2
2 Z ∑
ξ(l)
ri3
N
i=1
(𝐥𝐢. 𝐬𝐢) (16)
where the parameters {ξ(l)} depend only on the l-value of the interacting electrons in the Breit –Pauli
Hamiltonian matrix element. A reliable approximation to the Breit-Pauli matrix elements was achieved
by taking ξ(s)=0.0, ξ(p)=0.9704 and ξ(d)=0.497.
2.3 FAC Calculations
To illustrate the accuracy of the calculated results from MCDF, parallel calculations have been carried
out using fully relativistic configuration interaction method by using Flexible Atomic Code of Gu [32],
based on self-consistent Dirac-Fock-Slater iteration performed on a selected fictitious mean
configuration in order to derive the local central potential [32,33]. In Flexible Atomic Code (FAC), the
orbitals are optimized self consistently and the average energy of a fictitious mean configuration with
orbital occupation numbers is minimized. We have performed larger calculations up to 3075 fine
structure levels belonging to (2*8) n*1 and (2*7)3*2, 4*2, 3*1 4*1, 3*1 5*1 configurations have been
performed where 3 ≤ n ≤ 20 and n*q represents all possible distributions (without restriction on orbital
angular momentum) of q electrons among the shells specified by their principal quantum numbers.
These results are listed in Table 2.
3. Results and Discussion
3.1 Energy Levels
In the present work, we have performed two sets of calculations using GRASP code. Extensive
configuration interaction (CI) has been incorporated in GRASP and for the optimisation of the orbitals
the option of ‘extended average level’ (EAL), in which a weighted (proportional to 2j+1) trace of the
Hamiltonian matrix is minimized, has been adopted. We have gradually increased the number of
configurations to perform GRASP calculations for up to 2727 levels. In the VV model (MCDF1), we
have included 48 levels generated from the configurations 2p6nl with 3 ≤ n ≤ 8 and l ≤ 4. The CV model
(MCDF2) involves additional configurations, 2p53l nl' with 3 ≤ n ≤ 5, 2p54l 4l', 2s2p6 3l nl' with 3 ≤ n
≤ 5, 2s2p6 4l 4l', obtained by single- and double-excitations from the ground state, generating 2727
levels in total. However, we note that the levels of 2p6nl lie below those of the other configurations.
For this reason, we only list the lowest 21 levels in Table 1, all belonging to 2p6nl. A comparison of
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energies obtained from both the models (MCDF1, MCDF2) is shown in Table 1. Due to the inclusion
of larger CI in MCDF2, one can easily note the convergence of energies for most of the levels. Further,
a point worth mentioning here is that MCDF1 values for the low-lying excited states belonging to the
configurations 1s22s22p63l are in close agreement with NIST values whereas MCDF2 values for higher
excited states belonging to configurations 1s22s22p6nl (n = 4,5) are in close agreement with the NIST.
This is because of the fact that the sodium like chromium system has a complete core 1s22s22p6 plus
one valence electron and hence forms a simpler system for studying the different correlation effects
[15]. In our MCDF1 calculations, the energy levels are practically free from the effects of configuration
mixing and these correlations correct the energies of the low-lying excited levels belonging to
configurations 1s22s22p63l only whereas a deviation from NIST in the energies of higher excited levels
was observed. Therefore, there was a need to expand the configuration interaction set by taking into
account the core-valence correlations as these correlations correct the energies of the high-lying levels.
Also, we have tabulated the energies calculated from Dirac-Coulomb Hamiltonian (without taking into
account Breit and QED corrections) under the column IV and VI in case of MCDF1 and MCDF2
respectively.
0 5 10 15 20
-1
0
1
2
MCDF1
MCDF2
FAC1
CIV3
(E(T
heory
)-E
(NIS
T))
/E(N
IST
)(%
)
E(NIST) (Ryd.)
Figure 1 (online colour). Differences (in Ryd) of various theoretical energies from the NIST compiled values in Cr
XIV.
To assess the effect of further CI on the energy levels, we have adopted the FAC code of Gu
[32], which is also fully relativistic and is available from the website https://wwwamdis.iaea.org/FAC/.
We have performed two sets of calculations with FAC i.e. FAC1 and FAC. FAC1 includes the same
CI as in MCDF2 with additional configurations 2p6nl (6 ≤ n ≤8; 5 ≤ l ≤7) and the results are listed in
column VIII of Table 1, which are confined to the n ≤ 5 levels. For most of the levels, the NIST energies
differ with FAC1 by up to 0.18% and are smaller than 0.13% with those with MCDF2. One can see
that MCDF2 and FAC1 energies are in close agreement with each other. However, smaller
discrepancies between these energies are not because of different levels of CI, but due to
methodological variations. Additionally, the energies obtained from FAC1 are generally lower for most
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of the levels. In FAC, we have performed comparatively larger calculations for up to n = 20 and all
possible values of l generating 3075 levels in total with the same CI as in FAC1 by taking additional
configurations 2p6nl (9 ≤ n ≤20). Since it has not been possible to include higher 2p6nl configurations
in our GRASP calculations due to computational limitation of the code, so in Table 2, the FAC energies
have been listed, for the lowest 396 fine-structure levels, all belonging to 2p6nl configurations with n ≤
20. However, the inclusion of additional CI in the FAC calculations (Table 2) vary the energies up to
0.0002 Ryd (From FAC1) for some of the levels. Therefore, it may be reasonable to say that the
inclusion of CI in our FAC calculations is sufficient to calculate accurate results. Also, there is no
(strong) mixing or ambiguity for the designation of the 2p6nl levels listed in Tables 1 and 2. The
presented data may promote transfer of knowledge between astronomers and experimentalists through
targeted measurements and calculations.
In Table 1, we have also presented our results with CIV3 [27,28]. The fine-tuned [34,35]
excitation energies of the lowest 21 fine-structure levels of Cr XIV are presented under the column
CIV3. The ab-initio calculations (CIV3) have been done with 98 configurations, namely 2p6nl with 3
≤ n ≤ 5 and 0 ≤ l ≤ 4, 2p53l nl' with 3 ≤ n ≤ 5, 2p54l 4l', 2s2p6 3l nl' with 3 ≤ n ≤ 5 and 2s2p6 4l 4l' which
generates 2700 fine-structure levels. For the vast majority of our ab-initio energies, we observe very
satisfactory agreement with NIST.
Comparison of calculated energies by all methods has been made with Fischer et al. [9],
Sampson et al. [12], Johnson et al. [10] and Kim et al. [11]. One can see that our presented results by
all methods are not only in good agreement with all available results but are also very close to NIST
which ensures the reliability of our results and improvement in the energies of these levels due to proper
correlations and optimization. A graphical comparison of the relative differences of the present
MCDF1, MCDF2, FAC1 and CIV3 energies from the NIST is shown in Figure 1. To the best of our
knowledge, the new data presented for energy levels have not been reported elsewhere including NIST.
3.2 Radiative data of EUV and SXR transitions
Apart from energy levels, calculations have been made for absorption oscillator strengths (f-values,
dimensionless), radiative rates (A-values, s−1) and line strengths (S-values, in atomic units, 1a.u. =
6.460 × 10−36 cm2 esu2). Absorption oscillator strength (fij) for a transition i → j is connected to the
radiative rate Aji (in s-1) by the expression given below
fij =mcλji
2
8π2e2
ωj
ωiAji (17)
In the above formula m, c and e are the mass of electron, velocity of light and charge of electron
respectively, and ωj, ωi and λji denotes statistical weights of the upper, lower levels and transition
wavelength in Å respectively. Similarly, f- and A-values are related to S by the following standard
equations –
For the electric dipole (E1) transitions:
Aji =2.0261 × 1018
ωjλji3 Sij and fij =
303.75
λ jiωiSij (18)
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for the magnetic dipole (M1) transitions:
Aji =2.6974 × 1013
ωjλji3 Sij and fij =
4.044 × 10−3
λ jiωiSij (19)
for the electric quadrupole (E2) transitions:
Aji =1.1199 × 1018
ωjλji5 Sij and fij =
167.89
λji3ωi
Sij (20)
and for the magnetic quadrupole (M2) transitions:
Aji =1.4910 × 1013
ωjλji5 Sij and fij =
2.236 × 10−3
λji3ωi
Sij (21)
In Table 3, we have tabulated the transition wavelengths, radiative rates, line oscillator strengths
and line strengths for electric dipole (E1) transitions in Na-like Cr, which have been obtained with
MCDF model. For other types of transitions, namely magnetic dipole (M1), electric quadrupole (E2),
and magnetic quadrupole (M2), only the A-values are listed, because the corresponding results for f-
or S-values can be obtained using equations (10–13) given in [36]. Additionally, we have also listed
the ratio (R) of the velocity (Coulomb gauge) and length (Babushkin gauge) forms which often (but
not necessarily) give an indication of the accuracy. In Table 3, indices i and j are the lower and upper
levels of a transition, λij is the transition wavelength (in Å), AjiE1 is the radiative transition probability
(in s−1), fijE1 is the absorption oscillator strength (dimensionless) and SE1 is the line strength in atomic
unit (a.u.) for the E1 transitions. Also AjiE2, Aji
M1 and AjiM2 are the radiative transition probabilities (in
s−1) for E2, M1 and M2 transitions respectively and R(E1) is the ratio of velocity and length forms of
A- (or f- and S-) values for the E1 transitions. From Table 3, we have identified 90 VUV and 64 SXR
spectral lines in dipole and quadrupole transitions. We have theoretically identified many new Extreme
Ultraviolet (EUV) transitions lying in the range 100-1200 Å along with SXR transitions. To the best of
our knowledge, the data presented for the newly identified lines has not been reported elsewhere
including NIST.
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0 50 100 150 200 250 300 350 400 450
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
2p63d
2D
3/2-2p
64p
2P
o
1/2
2p63d
2D
5/2-2p
64f
2F
o
7/2
2p63d
2D
3/2-2p
64f
2F
o
5/2
((T
heory
)-(N
IST
))/
(NIS
T)%
(NIST) (Å)
Figure 2 (online colour). Percentage differences of the present MCDF wavelengths from the NIST compiled values in Cr XIV.
In Table 4, we compare our calculated transition wavelengths and transition probabilities for
some transitions with the data compiled by NIST and other available references. To illustrate the
comparison, Figure 2 shows the relative differences of MCDF wavelengths from the NIST. Our results
agree within 0.1-0.3% with the values compiled by NIST for most of the transitions while differing up
to 0.5% for some transitions such as 0.48% for 2p63d 2D3/2 - 2p64f 2F5/2o , 2p63d 2D5/2 - 2p64f 2F7/2
o and
0.49% for 2p63d 2D3/2 - 2p64p 2P1/2o . Also, a good agreement was found between our calculated
transition probabilities and the ones given by NIST and other references.
4. Lifetimes
The lifetime τ of a level j can be determined from the inverse of the sum of transition probabilities of
radiative transitions from level i as
τj(s) =1
∑ Aji(s−1)i
(22)
Since this is a measurable quantity, it helps us to check the accuracy of A-values, particularly when a
single type of transition dominates. In Table 1, we have tabulated lifetimes for the lowest 21 fine-
structure levels for Cr XIV calculated by including all possible E1 (electric dipole), M1 (magnetic
dipole), E2 (electric quadrupole) and M2 (magnetic quadrupole) transitions. Comparison of calculated
lifetimes has also been made with Fischer et al. [9], and there is no significant discrepancy for any
level. We have also reported new data of lifetimes for levels 2p65l. We believe that lifetimes reported
by us in the present calculations will be helpful for future comparisons.
5. Conclusion
Motivated by the need of accurate atomic data, in the present work, energy levels and the radiative data
i.e. radiative rates, transition wavelengths, oscillator strengths and line strengths for E1, E2, M1 and
M2 transitions for the lowest 21 levels have been computed by MCDF method. We have included Breit
and QED corrections in our calculations and found that the order of magnitude is equal for QED and
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Breit corrections for high Z ions. Therefore, the importance of QED corrections cannot be avoided for
heavy atoms or ions. We have also reported the energies for 396 fine-structure levels belonging to the
configurations 2p6nl (3 ≤ n ≤ 20; all possible values of l) for Cr XIV computed from FAC method. For
additional accuracy assessments, we have also computed energy levels from CIV3 and our presented
energies are in good agreement with experimentally measured and theoretically calculated energies.
The velocity/length ratio of oscillator strength reaffirms the accuracy of our calculations. In this work,
we have extended the work for Cr XIV and there is no major discrepancy between our calculated and
experimentally observed wavelengths. We have also identified many new EUV and SXR spectral lines
in dipole and quadrupole transitions. Ultimately, we believe that our present work is comprehensive
and may be useful in the diagnosis and classification of EUV and SXR spectral lines. Further, our
presented data may be beneficial in fusion, Astrophysical plasma and plasma modelling.
Acknowledgement
This work was performed under the project No. EMR / 2016 / 001203, sponsored by SERB, the
Department of Science and Technology, Govt. of India, at Deen Dayal Upadhyaya College, University
of Delhi, India. Dr. Avnindra Kumar Singh (PI) is thankful to SERB-DST for providing the financial
support. Mayank Dimri is also thankful to DST for assisting him as a Junior Research Fellow. Prof.
Man Mohan (Co-PI) is also thankful to DST for providing the financial assistance as Co-PI.
References
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B (1987).
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[9] C. F. Fischer, G. I. Tachiev and A. Irimia, At. Data Nucl. Data Tables 92 607 (2006).
[10] W. R. Johnson, Z. W. Liu and J. Sapirstein, At. Data Nucl. Data Tables 64 279 (1996).
[11] Y. K. Kim, D. H. Baik, P. Indelicato and J. P. Desclaux, Phys. Rev. A 44 148 (1991).
[12] D. H. Sampson, H. L. Zhang and C. J. Fontes, At. Data Nucl. Data Tables 44 209 (1990).
[13] D.A. Verner, P.D. Barthel and D. Tyler, Astron. Astrophys. Suppl. Ser. 108 287 (1994).
[14] J. D. Gillaspy, D. Osin, Y. Ralchenko, J. Reader and S. A. Blundell, Phys. Rev. A 87 (2013).
[15] W. O. Younis, S. H. Allam, T. M. El-Sherbini, At. Data Nucl. Data Tables 92 187 (2006).
[16] P. H. Norrington. http://www.am.qub.ac.uk/DARC/ 2009.
[17] I. P. Grant, B. J. McKenzie, P. H. Norrington, D. F. Mayers and N. C. Pyper, Comput. Phys. Commun. 21, 207
(1980).
[18] A. Goyal, N. Singh, S. Aggarwal, A.K. Singh, M. Mohan, Can. J. Phys. 94 712 (2016).
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[20] A. Goyal, I. Khatri, S. Aggarwal, A. K. Singh, M. Mohan, Can. J. Phys. 93 487 (2015).
[21] A. Goyal, I. Khatri, S. Aggarwal. A. K. Singh, M. Mohan, J. Quant. Spectrosc. Radiat. Transfer 161 157 (2015).
[22] A. Goyal, I. Khatri, S. Aggarwal, A. K. Singh, M. Mohan, At. Data Nucl. Data Tables 107 406 (2016).
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[23] P. Jönsson, X. He, C. F. Fischer, I. P. Grant, Comput. Phys. Commun. 177 597 (2007).
[24] F. A. Parpia, C. F. Fischer, I. P. Grant, Comput. Phys. Commun. 94 249 (1996).
[25] J. Olsen, M. R. Godefroid, P. Jönsson, P. A. Malmqvist, C. F. Fischer, Phys. Rev. E 52 4499 (1995).
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[32] M. F. Gu, Can. J. Phys. 86 675 (2008).
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Table 1. Comparison of threshold energies and lifetimes of our calculated lowest 21 fine structure levels of Cr XIV with other references. (aE±b = a× 10±𝑏).
Index Configuration Level
Energies (in Ryd) Lifetimes (in s)
DC1 MCDF1 DC2 MCDF2 FAC1 NIST and
other
references
CIV3
MCDF1 Fischer et al.
(DC+Breit
+QED) (DC+Breit
+QED)
1 2p63s 2S1/2 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 ------ ------
2 2p63p 2P1/2o 2.2190 2.2199 2.2273 2.2281 2.2223 2.2116 2.2116 1.828E-10 1.892E-10
2.2313a
2.2175b
2.2088c
2.2115d
3 2p63p 2P3/2o 2.3477 2.3449 2.3564 2.3532 2.3474 2.3370 2.3374 1.543E-10 1.583E-10
2.3626a
2.3431b
2.3358c
2.3373d
4 2p63d 2D3/2 5.3751 5.3688 5.3782 5.3708 5.3651 5.3567 5.3567 6.435E-11 6.564E-11
5.4099a
5.3661b
5 2p63d 2D5/2 5.3920 5.3834 5.3951 5.3855 5.3800 5.3721 5.3721 6.993E-11 7.163E-11
5.4266a
5.3808b
6 2p64s 2S1/2 13.4493 13.4422 13.4711 13.4649 13.4492 13.4729 13.4729 4.875E-12 4.680E-12
13.5138a
13.4649b
7 2p64p 2P1/2o 14.3208 14.3143 14.3477 14.3423 14.3227 14.3419 14.3419 6.163E-12 5.731E-12
14.3919a
14.3322b
8 2p64p 2P3/2o 14.3707 14.3627 14.3977 14.3905 14.3710 14.3905 14.3905 6.405E-12 5.946E-12
14.4432a
14.3837b
9 2p64d 2D3/2 15.4760 15.4668 15.5061 15.4975 15.4692 15.4965 15.4965 4.306E-12 4.135E-12
15.5589a
15.4862b
10 2p64d 2D5/2 15.4838 15.4737 15.5139 15.5043 15.4762 15.5036 15.5036 4.254E-12 4.132E-12
15.5673a
15.4935b
11 2p64f 2F5/2o 15.9180 15.9070 15.9345 15.9244 15.9224 15.9456 15.9456 1.706E-12 1.727E-12
16.0223a
15.9345b
12 2p64f 2F7/2o 15.9207 15.9096 15.9372 15.9270 15.9249 15.9479 15.9479 1.706E-12 1.730E-12
16.0257a
15.9419b
13 2p65s 2S1/2 19.1278 19.1186 19.1659 19.1588 19.1289 19.1619 19.1619 6.476E-12
19.1390b
14 2p65p 2P1/2o 19.5553 19.5465 19.5955 19.5889 19.5573 19.5858 19.5858 7.740E-12
19.5653b
15 2p65p 2P3/2o 19.5798 19.5703 19.6200 19.6125 19.5809 19.6106 19.6107 8.043E-12
19.5947b
16 2p65d 2D3/2 20.1117 20.1015 20.1549 20.1468 20.1100 20.1457 20.1457 5.523E-12
20.1239b
17 2p65d 2D5/2 20.1158 20.1052 20.1590 20.1504 20.1137 20.1488 20.1488 5.483E-12
20.1313b
18 2p65f 2F5/2o 20.3360 20.3251 20.3715 20.3626 20.3492 20.3695 20.3695 3.161E-12
20.3518b
19 2p65f 2F7/2o 20.3374 20.3264 20.3729 20.3639 20.3504 20.3708 20.3708 3.168E-12
20.3591b
20 2p65g 2G7/2 20.3552 20.3441 20.3904 20.3815 20.3563 20.3928 20.3928 6.062E-12
20.3812b
21 2p65g 2G9/2 20.3560 20.3449 20.3913 20.3823 20.3571 20.3929 20.3929 6.094E-12
20.3812b a - Fisher et al. [9], b- Sampson et al. [12], c- Johnson et al. [10], d- Kim et al. [11] and the values without superscript in column IX is the data from NIST.
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Table 2. Energies (in Ryd) for the 2p6nl (n ≤ 20) fine structure levels of Cr XIV. (aE±b = a× 10±𝑏).
Index Configuration Level FAC NIST
1 2p63s 2S1/2 0.0000 0
2 2p63p 2P1/2o 2.2221 2.2116
3 2p63p 2P3/2o 2.3473 2.3370
4 2p63d 2D3/2 5.3651 5.3567
5 2p63d 2D5/2 5.3800 5.3721
6 2p64s 2S1/2 13.4491 13.4729
7 2p64p 2P1/2o 14.3226 14.3419
8 2p64p 2P3/2o 14.3709 14.3905
9 2p64d 2D3/2 15.4692 15.4965
10 2p64d 2D5/2 15.4762 15.5036
11 2p64f 2F5/2o 15.9224 15.9456
12 2p64f 2F7/2o 15.9249 15.9479
13 2p65s 2S1/2 19.1289 19.1619
14 2p65p 2P1/2o 19.5572 19.5858
15 2p65p 2P3/2o 19.5807 19.6106
16 2p65d 2D3/2 20.1100 20.1457
17 2p65d 2D5/2 20.1137 20.1488
18 2p65f 2F5/2o 20.3492 20.3695
19 2p65f 2F7/2o 20.3505 20.3708
20 2p65g 2G7/2 20.3563 20.3928
21 2p65g 2G9/2 20.3572 20.3929
22 2p66s 2S1/2 22.0853 22.0934
23 2p66p 2P1/2o 22.3259 22.3350
24 2p66p 2P3/2o 22.3395 22.3448
25 2p66d 2D3/2 22.6366 22.6449
26 2p66d 2D5/2 22.6389 22.6469
27 2p66f 2F5/2o 22.7739 22.7735
28 2p66f 2F7/2o 22.7746 22.7749
29 2p66h 2H9/2o 22.7813
30 2p66h 2H11/2o 22.7816
31 2p66g 2G7/2 22.7822
32 2p66g 2G9/2 22.7827
33 2p67s 2S1/2 23.7958 23.8028
34 2p67p 2P1/2o 23.9445 23.9580
35 2p67p 2P3/2o 23.9529 23.9580
36 2p67d 2D3/2 24.1360 24.1402
37 2p67d 2D5/2 24.1375 24.1443
38 2p67f 2F5/2o 24.2194 24.2234
39 2p67f 2F7/2o 24.2199 24.2240
40 2p67h 2H9/2o 24.2266
41 2p67i 2I11/2 24.2268
42 2p67h 2H11/2o 24.2268
43 2p67i 2I13/2 24.2270
44 2p67g 2G7/2 24.2283
45 2p67g 2G9/2 24.2286
46 2p68s 2S1/2 24.8798
47 2p68p 2P1/2o 24.9780 24.9895
48 2p68p 2P3/2o 24.9835 24.9895
49 2p68d 2D3/2 25.1043 25.1100
50 2p68d 2D5/2 25.1053 25.1089
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Table 2 (continued)
Index Configuration Level FAC NIST
51 2p68f 2F5/2o 25.1583
52 2p68f 2F7/2o 25.1586 25.1655
53 2p68h 2H9/2o 25.1648
54 2p68i 2I11/2 25.1648
55 2p68h 2H11/2o 25.1649
56 2p68k 2K13/2o 25.1649
57 2p68i 2I13/2 25.1649
58 2p68k 2K15/2o 25.1650
59 2p68g 2G7/2 25.1668
60 2p68g 2G9/2 25.1670
61 2p69s 2S1/2 25.6097
62 2p69p 2P1/2o 25.6779 25.7057
63 2p69p 2P3/2o 25.6817 25.7057
64 2p69d 2D3/2 25.7655 25.7639
65 2p69d 2D5/2 25.7662 25.7713
66 2p69f 2F5/2o 25.8024
67 2p69f 2F7/2o 25.8026 25.8100
68 2p69i 2I11/2 25.8080
69 2p69h 2H9/2o 25.8080
70 2p69k 2K13/2o 25.8080
71 2p69i 2I13/2 25.8080
72 2p69h 2H11/2o 25.8081
73 2p69k 2K15/2o 25.8081
74 2p69l 2L15/2 25.8081
75 2p69l 2L17/2 25.8081
76 2p69g 2G7/2 25.8099
77 2p69g 2G9/2 25.8101
78 2p610s 2S1/2 26.1246
79 2p610p 2P1/2o 26.1738
80 2p610p 2P3/2o 26.1766
81 2p610d 2D3/2 26.2369
82 2p610d 2D5/2 26.2375 26.2000
83 2p610f 2F5/2o 26.2634
84 2p610f 2F7/2o 26.2636 26.2700
85 2p610i 2I11/2 26.2680
86 2p610k 2K13/2o 26.2681
87 2p610i 2I13/2 26.2681
88 2p610h 2H9/2o 26.2681
89 2p610l 2L15/2 26.2681
90 2p610k 2K15/2o 26.2681
91 2p610l 2L17/2 26.2681
92 2p610m 2M17/2o 26.2681
93 2p610h 2H11/2o 26.2681
94 2p610m 2M19/2o 26.2681
95 2p610g 2G7/2 26.2698
96 2p610g 2G9/2 26.2699
97 2p611s 2S1/2 26.5012
98 2p611p 2P1/2o 26.5379
99 2p611p 2P3/2o 26.5400
100 2p611d 2D3/2 26.5849
101 2p611d 2D5/2 26.5853 26.5700
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Table 2 (continued)
Index Configuration Level FAC NIST
102 2p611f 2F5/2o 26.6047
103 2p611f 2F7/2o 26.6048
104 2p611i 2I11/2 26.6084
105 2p611k 2K13/2o 26.6084
106 2p611i 2I13/2 26.6084
107 2p611l 2L15/2 26.6084
108 2p611k 2K15/2o 26.6084
109 2p611l 2L17/2 26.6085
110 2p611m 2M17/2o 26.6085
111 2p611m 2M19/2o 26.6085
112 2p611n 2N19/2 26.6085
113 2p611h 2H9/2o 26.6085
114 2p611n 2N21/2 26.6085
115 2p611h 2H11/2o 26.6085
116 2p611g 2G7/2 26.6099
117 2p611g 2G9/2 26.6100
118 2p612s 2S1/2 26.7851
119 2p612p 2P1/2o 26.8131
120 2p612p 2P3/2o 26.8147
121 2p612d 2D3/2 26.8491
122 2p612d 2D5/2 26.8494
123 2p612f 2F5/2o 26.8642
124 2p612f 2F7/2o 26.8643
125 2p612i 2I11/2 26.8673
126 2p612k 2K13/2o 26.8673
127 2p612i 2I13/2 26.8673
128 2p612l 2L15/2 26.8673
129 2p612k 2K15/2o 26.8673
130 2p612m 2M17/2o 26.8673
131 2p612l 2L17/2 26.8673
132 2p612m 2M19/2o 26.8673
133 2p612n 2N19/2 26.8673
134 2p612n 2N21/2 26.8674
135 2p612o 2O21/2o 26.8674
136 2p612o 2O23/2o 26.8674
137 2p612h 2H9/2o 26.8674
138 2p612h 2H7/2o 26.8674
139 2p612g 2G9/2 26.8685
140 2p612g 2G7/2 26.8686
141 2p613s 2S1/2 27.0043
142 2p613p 2P1/2o 27.0262
143 2p613p 2P3/2o 27.0275
144 2p613d 2D3/2 27.0543
145 2p613d 2D5/2 27.0546
146 2p613f 2F5/2o 27.0663
147 2p613f 2F7/2o 27.0664
148 2p613i 2I11/2 27.0687
149 2p613k 2K13/2o 27.0688
150 2p613i 2I13/2 27.0688
151 2p613l 2L15/2 27.0688
152 2p613k 2K15/2o 27.0688
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Table 2 (continued)
Index Configuration Level FAC NIST
153 2p613m 2M17/2o 27.0688
154 2p613l 2L17/2 27.0688
155 2p613m 2M19/2o 27.0688
156 2p613n 2N19/2 27.0688
157 2p613n 2N21/2 27.0688
158 2p613o 2O21/2o 27.0688
159 2p613o 2O23/2o 27.0688
160 2p613q 2Q23/2 27.0688
161 2p613q 2Q25/2 27.0688
162 2p613h 2H9/2o 27.0689
163 2p613h 2H11/2o 27.0689
164 2p613g 2G7/2 27.0697
165 2p613g 2G9/2 27.0698
166 2p614s 2S1/2 27.1772
167 2p614p 2P1/2o 27.1946
168 2p614p 2P3/2o 27.1956
169 2p614d 2D3/2 27.2170
170 2p614d 2D5/2 27.2172
171 2p614f 2F5/2o 27.2266
172 2p614f 2F7/2o 27.2267
173 2p614i 2I11/2 27.2286
174 2p614k 2K13/2o 27.2286
175 2p614i 2I13/2 27.2286
176 2p614l 2L15/2 27.2286
177 2p614k 2K15/2o 27.2286
178 2p614m 2M17/2o 27.2286
179 2p614l 2L17/2 27.2286
180 2p614n 2N19/2 27.2286
181 2p614m 2M19/2o 27.2286
182 2p614n 2N21/2 27.2286
183 2p614o 2O21/2o 27.2286
184 2p614o 2O23/2o 27.2286
185 2p614q 2Q23/2 27.2287
186 2p614q 2Q25/2 27.2287
187 2p614r 2Ro25/2 27.2287
188 2p614r 2Ro27/2 27.2287
189 2p614h 2H9/2o 27.2288
190 2p614h 2H11/2o 27.2288
191 2p614g 2G7/2 27.2294
192 2p614g 2G9/2 27.2294
193 2p615s 2S1/2 27.3158
194 2p615p 2P1/2o 27.3300
195 2p615p 2P3/2o 27.3308
196 2p615d 2D3/2 27.3481
197 2p615d 2D5/2 27.3482
198 2p615f 2F5/2o 27.3559
199 2p615f 2F7/2o 27.3560
200 2p615i 2I11/2 27.3576
201 2p615k 2K13/2o 27.3576
202 2p615i 2I13/2 27.3576
203 2p615l 2L15/2 27.3576
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Table 2 (continued)
Index Configuration Level FAC NIST
204 2p615k 2K15/2o 27.3576
205 2p615m 2M17/2o 27.3576
206 2p615n 2N19/2 27.3576
207 2p615l 2L17/2 27.3576
208 2p615m 2M19/2o 27.3576
209 2p615n 2N21/2 27.3576
210 2p615o 2O21/2o 27.3576
211 2p615o 2O23/2o 27.3576
212 2p615q 2Q23/2 27.3576
213 2p615q 2Q25/2 27.3576
214 2p615r 2Ro25/2 27.3576
215 2p615r 2Ro27/2 27.3576
216 2p615t 2T27/2 27.3576
217 2p615t 2T29/2 27.3576
218 2p615h 2H9/2o 27.3577
219 2p615h 2H11/2o 27.3577
220 2p615g 2G7/2 27.3582
221 2p615g 2G9/2 27.3582
222 2p616s 2S1/2 27.4288
223 2p616p 2P1/2o 27.4404
224 2p616p 2P3/2o 27.4410
225 2p616d 2D3/2 27.4552
226 2p616d 2D5/2 27.4554
227 2p616f 2F5/2o 27.4618
228 2p616f 2F7/2o 27.4618
229 2p616i 2I11/2 27.4631
230 2p616k 2K13/2o 27.4631
231 2p616i 2I13/2 27.4631
232 2p616k 2K15/2o 27.4631
233 2p616l 2L15/2 27.4631
234 2p616m 2M17/2o 27.4631
235 2p616n 2N19/2 27.4631
236 2p616l 2L17/2 27.4631
237 2p616m 2M19/2o 27.4631
238 2p616n 2N21/2 27.4631
239 2p616o 2O21/2o 27.4631
240 2p616o 2O23/2o 27.4631
241 2p616q 2Q23/2 27.4631
242 2p616q 2Q25/2 27.4631
243 2p616r 2Ro25/2 27.4631
244 2p616r 2Ro27/2 27.4631
245 2p616t 2T27/2 27.4631
246 2p616u 2Uo29/2 27.4631
247 2p616t 2T29/2 27.4631
248 2p616u 2Uo31/2 27.4631
249 2p616h 2H9/2o 27.4633
250 2p616h 2H11/2o 27.4633
251 2p616g 2G7/2 27.4636
252 2p616g 2G9/2 27.4636
253 2p617s 2S1/2 27.5220
254 2p617p 2P1/2o 27.5316
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Table 2 (continued)
Index Configuration Level FAC NIST
255 2p617p 2P3/2o 27.5322
256 2p617d 2D3/2 27.5440
257 2p617d 2D5/2 27.5441
258 2p617f 2F5/2o 27.5495
259 2p617f 2F7/2o 27.5495
260 2p617i 2I11/2 27.5506
261 2p617k 2K13/2o 27.5506
262 2p617i 2I13/2 27.5506
263 2p617k 2K15/2o 27.5506
264 2p617l 2L15/2 27.5506
265 2p617m 2M17/2o 27.5506
266 2p617n 2N19/2 27.5506
267 2p617o 2O21/2o 27.5506
268 2p617l 2L17/2 27.5506
269 2p617n 2N21/2 27.5506
270 2p617m 2M19/2o 27.5506
271 2p617o 2O23/2o 27.5506
272 2p617q 2Q23/2 27.5506
273 2p617q 2Q25/2 27.5506
274 2p617r 2Ro25/2 27.5506
275 2p617r 2Ro27/2 27.5506
276 2p617t 2T27/2 27.5506
277 2p617t 2T29/2 27.5506
278 2p617u 2Uo29/2 27.5506
279 2p617v 2V31/2 27.5506
280 2p617u 2Uo31/2 27.5506
281 2p617v 2V33/2 27.5506
282 2p617h 2H9/2o 27.5507
283 2p617h 2H11/2o 27.5507
284 2p617g 2G7/2 27.5509
285 2p617g 2G9/2 27.5509
286 2p618s 2S1/2 27.5998
287 2p618p 2P1/2o 27.6079
288 2p618p 2P3/2o 27.6084
289 2p618d 2D3/2 27.6183
290 2p618d 2D5/2 27.6184
291 2p618f 2F5/2o 27.6229
292 2p618f 2F7/2o 27.6230
293 2p618i 2I11/2 27.6239
294 2p618k 2K13/2o 27.6239
295 2p618i 2I13/2 27.6239
296 2p618k 2K15/2o 27.6239
297 2p618l 2L15/2 27.6239
298 2p618n 2N19/2 27.6239
299 2p618m 2M17/2o 27.6239
300 2p618o 2O21/2o 27.6239
301 2p618l 2L17/2 27.6239
302 2p618n 2N21/2 27.6239
303 2p618o 2O23/2o 27.6239
304 2p618m 2M19/2o 27.6239
305 2p618q 2Q23/2 27.6239
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Table 2 (continued)
Index Configuration Level FAC NIST
306 2p618q 2Q25/2 27.6239
307 2p618r 2Ro25/2 27.6239
308 2p618r 2Ro27/2 27.6239
309 2p618t 2T27/2 27.6239
310 2p618t 2T29/2 27.6239
311 2p618u 2Uo29/2 27.6239
312 2p618u 2Uo31/2 27.6239
313 2p618v 2V31/2 27.6239
314 2p618w 2Wo33/2 27.6239
315 2p618v 2V33/2 27.6239
316 2p618w 2Wo35/2 27.6239
317 2p618h 2H9/2o 27.6240
318 2p618h 2H11/2o 27.6240
319 2p618g 2G7/2 27.6241
320 2p618g 2G9/2 27.6241
321 2p619s 2S1/2 27.6655
322 2p619p 2P1/2o 27.6723
323 2p619p 2P3/2o 27.6727
324 2p619d 2D3/2 27.6811
325 2p619d 2D5/2 27.6812
326 2p619f 2F5/2o 27.6851
327 2p619f 2F7/2o 27.6851
328 2p619i 2I11/2 27.6859
329 2p619k 2K13/2o 27.6859
330 2p619i 2I13/2 27.6859
331 2p619k 2K15/2o 27.6859
332 2p619l 2L15/2 27.6859
333 2p619o 2O21/2o 27.6859
334 2p619n 2N19/2 27.6859
335 2p619m 2M17/2o 27.6859
336 2p619l 2L17/2 27.6859
337 2p619q 2Q23/2 27.6859
338 2p619o 2O23/2o 27.6859
339 2p619n 2N21/2 27.6859
340 2p619m 2M19/2o 27.6859
341 2p619q 2Q25/2 27.6859
342 2p619r 2Ro25/2 27.6859
343 2p619r 2Ro27/2 27.6859
344 2p619t 2T27/2 27.6859
345 2p619t 2T29/2 27.6859
346 2p619u 2Uo29/2 27.6859
347 2p619u 2Uo31/2 27.6859
348 2p619v 2V31/2 27.6859
349 2p619v 2V33/2 27.6859
350 2p619w 2Wo33/2 27.6859
351 2p619x 2X35/2 27.6859
352 2p619x 2X37/2 27.6859
353 2p619w 2Wo35/2 27.6859
354 2p619h 2H9/2o 27.6860
355 2p619h 2H11/2o 27.6861
356 2p619g 2G7/2 27.6861
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Table 2 (continued)
Index Configuration Level FAC NIST
357 2p619g 2G9/2 27.6861
358 2p620s 2S1/2 27.7214
359 2p620p 2P1/2o 27.7272
360 2p620p 2P3/2o 27.7276
361 2p620d 2D3/2 27.7348
362 2p620d 2D5/2 27.7348
363 2p620f 2F5/2o 27.7382
364 2p620f 2F7/2o 27.7382
365 2p620i 2I11/2 27.7388
366 2p620k 2K13/2o 27.7389
367 2p620i 2I13/2 27.7389
368 2p620k 2K15/2o 27.7389
369 2p620l 2L15/2 27.7389
370 2p620o 2O21/2o 27.7389
371 2p620n 2N19/2 27.7389
372 2p620m 2M17/2o 27.7389
373 2p620q 2Q23/2 27.7389
374 2p620l 2L17/2 27.7389
375 2p620o 2O23/2o 27.7389
376 2p620q 2Q25/2 27.7389
377 2p620n 2N21/2 27.7389
378 2p620m 2M19/2o 27.7389
379 2p620r 2Ro25/2 27.7389
380 2p620r 2Ro27/2 27.7389
381 2p620t 2T27/2 27.7389
382 2p620t 2T29/2 27.7389
383 2p620u 2Uo29/2 27.7389
384 2p620u 2Uo31/2 27.7389
385 2p620v 2V31/2 27.7389
386 2p620v 2V33/2 27.7389
387 2p620w 2Wo33/2 27.7389
388 2p620w 2Wo35/2 27.7389
389 2p620y 2Yo37/2 27.7389
390 2p620x 2X35/2 27.7389
391 2p620y 2Yo39/2 27.7389
392 2p620x 2X37/2 27.7389
393 2p620g 2G7/2 27.7390
394 2p620g 2G9/2 27.7390
395 2p620h 2H9/2o 27.7390
396 2p620h 2H11/2o 27.7390
Page 24 of 29
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Table 3. Transition wavelengths (λij in Å), radiative rates (Aji in s-1), oscillator strengths (fij, dimensionless), and line strengths (S, in atomic
units) for electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2) and magnetic quadrupole (M2) transitions in Cr XIV. The last
column gives R(E1), the ratio of velocity and length forms of A-values for E1 transitions. (aE±b = a× 𝟏𝟎±𝒃).
i j λij AjiE1 fij
E1 SE1 AjiE2 Aji
M1 AjiM2 R(E1)
1 2 4.105E+02 5.469E+09 1.382E-01 3.734E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00
1 3 3.886E+02 6.484E+09 2.936E-01 7.513E-01 0.000E+00 0.000E+00 4.737E+00 1.0E+00
1 4 1.697E+02 0.000E+00 0.000E+00 0.000E+00 5.435E+05 5.903E-02 0.000E+00 0.0E+00
1 5 1.693E+02 0.000E+00 0.000E+00 0.000E+00 5.520E+05 0.000E+00 0.000E+00 0.0E+00
1 6 6.779E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 3.082E+01 0.000E+00 0.0E+00
1 7 6.366E+01 1.125E+11 6.832E-02 2.864E-02 0.000E+00 0.000E+00 0.000E+00 9.9E-01
1 8 6.345E+01 1.068E+11 1.289E-01 5.385E-02 0.000E+00 0.000E+00 2.930E+03 9.8E-01
1 9 5.892E+01 0.000E+00 0.000E+00 0.000E+00 6.327E+07 1.784E-01 0.000E+00 0.0E+00
1 10 5.889E+01 0.000E+00 0.000E+00 0.000E+00 6.315E+07 0.000E+00 0.000E+00 0.0E+00
1 11 5.729E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.449E-04 0.0E+00
1 13 4.766E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 3.522E+01 0.000E+00 0.0E+00
1 14 4.662E+01 6.549E+10 2.134E-02 6.550E-03 0.000E+00 0.000E+00 0.000E+00 9.8E-01
1 15 4.656E+01 6.282E+10 4.084E-02 1.252E-02 0.000E+00 0.000E+00 3.200E+03 9.8E-01
1 16 4.533E+01 0.000E+00 0.000E+00 0.000E+00 3.627E+07 3.213E-01 0.000E+00 0.0E+00
1 17 4.533E+01 0.000E+00 0.000E+00 0.000E+00 3.627E+07 0.000E+00 0.000E+00 0.0E+00
1 18 4.484E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.566E-04 0.0E+00
2 3 7.289E+03 0.000E+00 0.000E+00 0.000E+00 4.543E-03 2.320E+01 0.000E+00 0.0E+00
2 4 2.894E+02 1.319E+10 3.312E-01 6.310E-01 0.000E+00 0.000E+00 6.943E-01 1.1E+00
2 5 2.881E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 5.067E+00 0.0E+00
2 6 8.120E+01 6.705E+10 6.628E-02 3.544E-02 0.000E+00 0.000E+00 0.000E+00 1.0E+00
2 7 7.535E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 7.719E+00 0.000E+00 0.0E+00
2 8 7.505E+01 0.000E+00 0.000E+00 0.000E+00 1.618E+07 7.841E+02 0.000E+00 0.0E+00
2 9 6.879E+01 1.904E+11 2.701E-01 1.223E-01 0.000E+00 0.000E+00 1.775E+02 9.7E-01
2 10 6.876E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.258E+03 0.0E+00
2 11 6.658E+01 0.000E+00 0.000E+00 0.000E+00 9.973E+07 0.000E+00 0.000E+00 0.0E+00
2 13 5.393E+01 2.964E+10 1.292E-02 4.588E-03 0.000E+00 0.000E+00 0.000E+00 1.0E+00
2 14 5.259E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 8.996E+00 0.000E+00 0.0E+00
2 15 5.252E+01 0.000E+00 0.000E+00 0.000E+00 8.862E+06 5.334E+02 0.000E+00 0.0E+00
2 16 5.096E+01 1.156E+11 9.000E-02 3.020E-02 0.000E+00 0.000E+00 1.963E+02 9.7E-01
2 17 5.095E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.392E+03 0.0E+00
2 18 5.033E+01 0.000E+00 0.000E+00 0.000E+00 1.999E+07 0.000E+00 0.000E+00 0.0E+00
3 4 3.014E+02 2.339E+09 3.185E-02 1.264E-01 0.000E+00 0.000E+00 0.000E+00 1.1E+00
3 5 2.999E+02 1.426E+10 2.885E-01 1.139E+00 0.000E+00 0.000E+00 2.268E+01 1.1E+00
3 6 8.212E+01 1.375E+11 6.952E-02 7.517E-02 0.000E+00 0.000E+00 2.253E+03 1.0E+00
3 7 7.613E+01 0.000E+00 0.000E+00 0.000E+00 3.229E+07 1.833E+03 0.000E+00 0.0E+00
3 8 7.583E+01 0.000E+00 0.000E+00 0.000E+00 1.605E+07 4.779E+01 0.000E+00 0.0E+00
3 9 6.945E+01 3.887E+10 2.810E-02 2.570E-02 0.000E+00 0.000E+00 0.000E+00 9.7E-01
3 10 6.941E+01 2.320E+11 2.514E-01 2.298E-01 0.000E+00 0.000E+00 6.892E+03 9.7E-01
3 11 6.719E+01 0.000E+00 0.000E+00 0.000E+00 2.783E+07 1.439E+00 0.000E+00 0.0E+00
3 12 6.718E+01 0.000E+00 0.000E+00 0.000E+00 1.253E+08 0.000E+00 0.000E+00 0.0E+00
3 13 5.433E+01 6.061E+10 1.341E-02 9.593E-03 0.000E+00 0.000E+00 2.268E+03 1.0E+00
3 14 5.298E+01 0.000E+00 0.000E+00 0.000E+00 1.737E+07 1.315E+03 0.000E+00 0.0E+00
3 15 5.290E+01 0.000E+00 0.000E+00 0.000E+00 8.716E+06 5.496E+01 0.000E+00 0.0E+00
3 16 5.132E+01 2.337E+10 9.225E-03 6.235E-03 0.000E+00 0.000E+00 0.000E+00 9.7E-01
3 17 5.131E+01 1.397E+11 8.270E-02 5.588E-02 0.000E+00 0.000E+00 7.591E+03 9.7E-01
3 18 5.068E+01 0.000E+00 0.000E+00 0.000E+00 5.402E+06 9.745E-01 0.000E+00 0.0E+00
3 19 5.068E+01 0.000E+00 0.000E+00 0.000E+00 2.438E+07 9.745E-01 0.000E+00 0.0E+00
3 20 5.063E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.037E-04 0.0E+00
4 5 6.201E+04 0.000E+00 0.000E+00 0.000E+00 2.193E-08 4.521E-02 0.000E+00 0.0E+00
4 6 1.129E+02 0.000E+00 0.000E+00 0.000E+00 3.869E+06 4.754E-04 0.000E+00 0.0E+00
4 7 1.019E+02 4.896E+10 3.808E-02 5.108E-02 0.000E+00 0.000E+00 2.083E+01 1.1E+00
4 8 1.013E+02 4.746E+09 7.304E-03 9.746E-03 0.000E+00 0.000E+00 0.000E+00 1.1E+00
4 9 9.024E+01 0.000E+00 0.000E+00 0.000E+00 6.925E+06 9.526E+00 0.000E+00 0.0E+00
4 10 9.018E+01 0.000E+00 0.000E+00 0.000E+00 1.979E+06 6.610E+00 0.000E+00 0.0E+00
4 11 8.647E+01 5.472E+11 9.202E-01 1.048E+00 0.000E+00 0.000E+00 1.922E+03 9.9E-01
4 12 8.645E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.462E+03 0.0E+00
4 13 6.628E+01 0.000E+00 0.000E+00 0.000E+00 1.860E+06 4.941E-04 0.000E+00 0.0E+00
4 14 6.428E+01 1.906E+10 5.903E-03 4.996E-03 0.000E+00 0.000E+00 2.038E+01 1.1E+00
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Table 3 (continued)
i j λij AjiE1 fij
E1 SE1 AjiE2 Aji
M1 AjiM2 R(E1)
4 15 6.417E+01 1.851E+09 1.143E-03 9.655E-04 0.000E+00 0.000E+00 0.000E+00 1.1E+00
4 16 6.185E+01 0.000E+00 0.000E+00 0.000E+00 3.258E+06 1.024E+01 0.000E+00 0.0E+00
4 17 6.184E+01 0.000E+00 0.000E+00 0.000E+00 9.329E+05 4.188E+00 0.000E+00 0.0E+00
4 18 6.093E+01 2.051E+11 1.712E-01 1.373E-01 0.000E+00 0.000E+00 1.450E+03 9.9E-01
4 19 6.092E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.858E+03 0.0E+00
4 20 6.085E+01 0.000E+00 0.000E+00 0.000E+00 7.844E+07 0.000E+00 0.000E+00 0.0E+00
5 6 1.131E+02 0.000E+00 0.000E+00 0.000E+00 5.785E+06 0.000E+00 0.000E+00 0.0E+00
5 7 1.020E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.223E+02 0.0E+00
5 8 1.015E+02 4.285E+10 4.411E-02 8.843E-02 0.000E+00 0.000E+00 5.957E+02 1.1E+00
5 9 9.038E+01 0.000E+00 0.000E+00 0.000E+00 2.963E+06 2.640E+01 0.000E+00 0.0E+00
5 10 9.031E+01 0.000E+00 0.000E+00 0.000E+00 7.902E+06 3.167E+01 0.000E+00 0.0E+00
5 11 8.659E+01 3.903E+10 4.388E-02 7.505E-02 0.000E+00 0.000E+00 0.000E+00 9.9E-01
5 12 8.657E+01 5.856E+11 8.773E-01 1.500E+00 0.000E+00 0.000E+00 1.690E+04 9.9E-01
5 13 6.635E+01 0.000E+00 0.000E+00 0.000E+00 2.774E+06 0.000E+00 0.000E+00 0.0E+00
5 14 6.434E+01 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.175E+02 0.0E+00
5 15 6.423E+01 1.670E+10 6.887E-03 8.738E-03 0.000E+00 0.000E+00 5.795E+02 1.1E+00
5 16 6.192E+01 0.000E+00 0.000E+00 0.000E+00 1.391E+06 2.080E+01 0.000E+00 0.0E+00
5 17 6.190E+01 0.000E+00 0.000E+00 0.000E+00 3.718E+06 3.387E+01 0.000E+00 0.0E+00
5 18 6.099E+01 1.460E+10 8.142E-03 9.809E-03 0.000E+00 0.000E+00 0.000E+00 9.9E-01
5 19 6.098E+01 2.193E+11 1.630E-01 1.963E-01 0.000E+00 0.000E+00 1.274E+04 9.9E-01
5 20 6.091E+01 0.000E+00 0.000E+00 0.000E+00 8.716E+06 4.269E-01 0.000E+00 0.0E+00
5 21 6.091E+01 0.000E+00 0.000E+00 0.000E+00 8.711E+07 0.000E+00 0.000E+00 0.0E+00
6 7 1.045E+03 1.223E+09 2.002E-01 1.378E+00 0.000E+00 0.000E+00 0.000E+00 1.0E+00
6 8 9.900E+02 1.443E+09 4.240E-01 2.764E+00 0.000E+00 0.000E+00 1.625E-01 1.0E+00
6 9 4.501E+02 0.000E+00 0.000E+00 0.000E+00 6.469E+04 1.446E-03 0.000E+00 0.0E+00
6 10 4.486E+02 0.000E+00 0.000E+00 0.000E+00 6.589E+04 0.000E+00 0.000E+00 0.0E+00
6 11 3.697E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.289E-08 0.0E+00
6 13 1.605E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 7.187E-01 0.000E+00 0.0E+00
6 14 1.493E+02 2.154E+10 7.198E-02 7.075E-02 0.000E+00 0.000E+00 0.000E+00 9.9E-01
6 15 1.487E+02 2.036E+10 1.350E-01 1.322E-01 0.000E+00 0.000E+00 1.017E+02 9.9E-01
6 16 1.368E+02 0.000E+00 0.000E+00 0.000E+00 6.295E+06 5.534E-05 0.000E+00 0.0E+00
6 17 1.368E+02 0.000E+00 0.000E+00 0.000E+00 6.277E+06 0.000E+00 0.000E+00 0.0E+00
6 18 1.324E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.387E-06 0.0E+00
7 8 1.882E+04 0.000E+00 0.000E+00 0.000E+00 5.416E-04 1.348E+00 0.000E+00 0.0E+00
7 9 7.907E+02 2.773E+09 5.199E-01 2.707E+00 0.000E+00 0.000E+00 1.957E-02 1.0E+00
7 10 7.860E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.436E-01 0.0E+00
7 11 5.721E+02 0.000E+00 0.000E+00 0.000E+00 1.298E+04 0.000E+00 0.000E+00 0.0E+00
7 13 1.897E+02 2.106E+10 1.136E-01 1.419E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00
7 14 1.742E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.008E-01 0.000E+00 0.0E+00
7 15 1.734E+02 0.000E+00 0.000E+00 0.000E+00 2.505E+06 6.765E+01 0.000E+00 0.0E+00
7 16 1.575E+02 3.039E+10 2.259E-01 2.342E-01 0.000E+00 0.000E+00 5.411E+00 9.8E-01
7 17 1.574E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 3.824E+01 0.0E+00
7 18 1.516E+02 0.000E+00 0.000E+00 0.000E+00 1.075E+07 0.000E+00 0.000E+00 0.0E+00
8 9 8.254E+02 4.888E+08 4.992E-02 5.426E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00
8 10 8.202E+02 2.991E+09 4.525E-01 4.888E+00 0.000E+00 0.000E+00 6.360E-01 1.0E+00
8 11 5.901E+02 0.000E+00 0.000E+00 0.000E+00 3.184E+03 2.439E-04 0.000E+00 0.0E+00
8 12 5.891E+02 0.000E+00 0.000E+00 0.000E+00 1.446E+04 0.000E+00 0.000E+00 0.0E+00
8 13 1.916E+02 4.309E+10 1.186E-01 2.992E-01 0.000E+00 0.000E+00 1.296E+02 1.0E+00
8 14 1.758E+02 0.000E+00 0.000E+00 0.000E+00 5.001E+06 1.472E+02 0.000E+00 0.0E+00
8 15 1.750E+02 0.000E+00 0.000E+00 0.000E+00 2.490E+06 1.290E+00 0.000E+00 0.0E+00
8 16 1.588E+02 6.291E+09 2.378E-02 4.973E-02 0.000E+00 0.000E+00 0.000E+00 9.8E-01
8 17 1.587E+02 3.747E+10 2.122E-01 4.434E-01 0.000E+00 0.000E+00 2.129E+02 9.8E-01
8 18 1.528E+02 0.000E+00 0.000E+00 0.000E+00 3.035E+06 1.608E-02 0.000E+00 0.0E+00
8 19 1.528E+02 0.000E+00 0.000E+00 0.000E+00 1.366E+07 0.000E+00 0.000E+00 0.0E+00
8 20 1.524E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.800E-06 0.0E+00
9 10 1.313E+05 0.000E+00 0.000E+00 0.000E+00 9.114E-09 4.770E-03 0.000E+00 0.0E+00
9 11 2.070E+03 1.228E+08 1.183E-01 3.226E+00 0.000E+00 0.000E+00 7.533E-04 1.1E+00
9 12 2.058E+03 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 9.938E-04 0.0E+00
9 13 2.495E+02 0.000E+00 0.000E+00 0.000E+00 1.087E+06 6.936E-05 0.000E+00 0.0E+00
9 14 2.234E+02 2.270E+10 8.491E-02 2.497E-01 0.000E+00 0.000E+00 2.009E+00 1.0E+00
9 15 2.221E+02 2.207E+09 1.632E-02 4.772E-02 0.000E+00 0.000E+00 0.000E+00 1.0E+00
9 16 1.966E+02 0.000E+00 0.000E+00 0.000E+00 1.553E+06 3.969E-01 0.000E+00 0.0E+00
9 17 1.965E+02 0.000E+00 0.000E+00 0.000E+00 4.436E+05 1.175E+00 0.000E+00 0.0E+00
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Table 3 (continued)
i j λij AjiE1 fij
E1 SE1 AjiE2 Aji
M1 AjiM2 R(E1)
9 18 1.876E+02 9.016E+10 7.134E-01 1.762E+00 0.000E+00 0.000E+00 6.734E+01 9.9E-01
9 19 1.875E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 8.618E+01 0.0E+00
9 20 1.868E+02 0.000E+00 0.000E+00 0.000E+00 9.665E+06 0.000E+00 0.000E+00 0.0E+00
10 11 2.103E+03 8.361E+06 5.544E-03 2.303E-01 0.000E+00 0.000E+00 0.000E+00 1.1E+00
10 12 2.091E+03 1.278E+08 1.116E-01 4.609E+00 0.000E+00 0.000E+00 6.324E-03 1.1E+00
10 13 2.500E+02 0.000E+00 0.000E+00 0.000E+00 1.624E+06 0.000E+00 0.000E+00 0.0E+00
10 14 2.238E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.144E+01 0.0E+00
10 15 2.225E+02 1.994E+10 9.859E-02 4.332E-01 0.000E+00 0.000E+00 5.766E+01 1.0E+00
10 16 1.969E+02 0.000E+00 0.000E+00 0.000E+00 6.647E+05 2.986E+00 0.000E+00 0.0E+00
10 17 1.968E+02 0.000E+00 0.000E+00 0.000E+00 1.772E+06 1.307E+00 0.000E+00 0.0E+00
10 18 1.878E+02 6.445E+09 3.409E-02 1.265E-01 0.000E+00 0.000E+00 0.000E+00 9.9E-01
10 19 1.878E+02 9.661E+10 6.810E-01 2.526E+00 0.000E+00 0.000E+00 5.927E+02 9.9E-01
10 20 1.871E+02 0.000E+00 0.000E+00 0.000E+00 1.069E+06 2.291E-02 0.000E+00 0.0E+00
10 21 1.871E+02 0.000E+00 0.000E+00 0.000E+00 1.070E+07 0.000E+00 0.000E+00 0.0E+00
11 12 3.536E+05 0.000E+00 0.000E+00 0.000E+00 1.919E-11 2.614E-04 0.000E+00 0.0E+00
11 13 2.838E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 5.559E-10 0.0E+00
11 14 2.504E+02 0.000E+00 0.000E+00 0.000E+00 4.756E+05 0.000E+00 0.000E+00 0.0E+00
11 15 2.488E+02 0.000E+00 0.000E+00 0.000E+00 6.826E+04 2.364E-05 0.000E+00 0.0E+00
11 16 2.173E+02 3.935E+09 1.856E-02 7.965E-02 0.000E+00 0.000E+00 2.192E+00 1.0E+00
11 17 2.171E+02 1.857E+08 1.312E-03 5.624E-03 0.000E+00 0.000E+00 0.000E+00 1.0E+00
11 18 2.063E+02 0.000E+00 0.000E+00 0.000E+00 9.874E+05 3.872E-01 0.000E+00 0.0E+00
11 19 2.062E+02 0.000E+00 0.000E+00 0.000E+00 1.234E+05 7.478E-02 0.000E+00 0.0E+00
11 20 2.054E+02 1.586E+11 1.337E+00 5.425E+00 0.000E+00 0.000E+00 2.510E+02 1.0E+00
11 21 2.053E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.312E+02 0.0E+00
12 15 2.489E+02 0.000E+00 0.000E+00 0.000E+00 4.092E+05 0.000E+00 0.000E+00 0.0E+00
12 16 2.174E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 3.747E+00 0.0E+00
12 17 2.172E+02 3.724E+09 1.975E-02 1.130E-01 0.000E+00 0.000E+00 1.709E+01 1.0E+00
12 18 2.064E+02 0.000E+00 0.000E+00 0.000E+00 1.645E+05 3.238E-01 0.000E+00 0.0E+00
12 19 2.063E+02 0.000E+00 0.000E+00 0.000E+00 1.028E+06 9.138E-01 0.000E+00 0.0E+00
12 20 2.055E+02 5.869E+09 3.715E-02 2.011E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00
12 21 2.055E+02 1.644E+11 1.300E+00 7.036E+00 0.000E+00 0.000E+00 1.215E+03 1.0E+00
13 14 2.130E+03 3.818E+08 2.596E-01 3.640E+00 0.000E+00 0.000E+00 0.000E+00 1.0E+00
13 15 2.018E+03 4.496E+08 5.489E-01 7.292E+00 0.000E+00 0.000E+00 1.219E-02 1.0E+00
13 16 9.271E+02 0.000E+00 0.000E+00 0.000E+00 1.251E+04 7.986E-05 0.000E+00 0.0E+00
13 17 9.237E+02 0.000E+00 0.000E+00 0.000E+00 1.275E+04 0.000E+00 0.000E+00 0.0E+00
13 18 7.554E+02 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 5.547E-10 0.0E+00
14 15 3.837E+04 0.000E+00 0.000E+00 0.000E+00 1.072E-04 1.591E-01 0.000E+00 0.0E+00
14 16 1.642E+03 8.513E+08 6.881E-01 7.438E+00 0.000E+00 0.000E+00 1.394E-03 1.0E+00
14 17 1.631E+03 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.025E-02 0.0E+00
14 18 1.171E+03 0.000E+00 0.000E+00 0.000E+00 3.440E+03 0.000E+00 0.000E+00 0.0E+00
15 16 1.715E+03 1.497E+08 6.604E-02 1.492E+00 0.000E+00 0.000E+00 0.000E+00 1.0E+00
15 17 1.703E+03 9.178E+08 5.989E-01 1.343E+01 0.000E+00 0.000E+00 4.525E-02 1.0E+00
15 18 1.207E+03 0.000E+00 0.000E+00 0.000E+00 8.438E+02 2.063E-05 0.000E+00 0.0E+00
15 19 1.205E+03 0.000E+00 0.000E+00 0.000E+00 3.832E+03 0.000E+00 0.000E+00 0.0E+00
15 20 1.178E+03 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 2.294E-11 0.0E+00
16 17 2.476E+05 0.000E+00 0.000E+00 0.000E+00 2.854E-09 7.108E-04 0.000E+00 0.0E+00
16 18 4.077E+03 5.652E+07 2.113E-01 1.134E+01 0.000E+00 0.000E+00 8.934E-05 1.0E+00
16 19 4.053E+03 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 1.178E-04 0.0E+00
16 20 3.757E+03 0.000E+00 0.000E+00 0.000E+00 5.970E+00 0.000E+00 0.000E+00 0.0E+00
17 18 4.145E+03 3.840E+06 9.891E-03 8.099E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00
17 19 4.121E+03 5.866E+07 1.991E-01 1.621E+01 0.000E+00 0.000E+00 7.473E-04 1.0E+00
17 20 3.815E+03 0.000E+00 0.000E+00 0.000E+00 6.141E-01 1.040E-07 0.000E+00 0.0E+00
17 21 3.802E+03 0.000E+00 0.000E+00 0.000E+00 6.250E+00 0.000E+00 0.000E+00 0.0E+00
18 19 6.961E+05 0.000E+00 0.000E+00 0.000E+00 6.356E-12 3.427E-05 0.000E+00 0.0E+00
18 20 4.787E+04 2.055E+04 9.413E-03 8.900E+00 0.000E+00 0.000E+00 5.987E-10 1.0E+00
18 21 4.590E+04 0.000E+00 0.000E+00 0.000E+00 0.000E+00 0.000E+00 3.861E-10 0.0E+00
19 20 5.141E+04 6.143E+02 2.434E-04 3.295E-01 0.000E+00 0.000E+00 0.000E+00 1.0E+00
19 21 4.914E+04 1.970E+04 8.915E-03 1.154E+01 0.000E+00 0.000E+00 2.546E-09 1.0E+00
20 21 1.117E+06 0.000E+00 0.000E+00 0.000E+00 1.976E-13 8.610E-06 0.000E+00 0.0E+00
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Table 4. Comparison of computed wavelengths and transition probabilities of Cr XIV with
the data available in NIST and other references. (aE±b = a× 10±𝑏).
i j
λij (in Å) Aji (in s-1)
MCDF NIST and other
references MCDF
NIST and other
references
1 2 4.105E+02 4.120E+02 5.469E+09 5.370E+09
4.121E+02a 5.286E+09d
4.109E+02c 5.139E+09e
4.084E+02d
4.120E+02f
1 3 3.886E+02 3.899E+02 6.484E+09 6.410E+09
3.899E+02a 6.319E+09d
3.889E+02c 6.115E+09e
3.857E+02d
3.898E+02f
1 7 6.366E+01 6.354E+01 1.125E+11 1.130E+11
6.332E+01d 1.222E+11d
1 8 6.345E+01 6.332E+01 1.068E+11 1.070E+11
6.309E+01d 1.170E+11d
1 14 4.662E+01 4.653E+01 6.549E+10 6.700E+10
1 15 4.656E+01 4.647E+01 6.282E+10 6.600E+10
2 4 2.894E+02 2.897E+02 1.319E+10 1.310E+10
2.894E+02c 1.294E+10d
2.867E+02d
2 6 8.120E+01 8.092E+01 6.705E+10 7.000E+10
8.077E+01d 7.099E+10d
6.996E+10e
2 9 6.879E+01 6.859E+01 1.904E+11 1.980E+11
6.837E+01d 1.986E+11d
2 13 5.393E+01 5.376E+01 2.964E+10 3.000E+10
2 16 5.096E+01 5.081E+01 1.156E+11 1.200E+11
3 4 3.014E+02 3.018E+02 2.339E+09 2.300E+09
3.013E+02c 2.292E+09d
2.991E+02d
3 5 2.999E+02 3.003E+02 1.426E+10 1.410E+10
2.999E+02c 1.396E+10d
2.974E+02d
3 6 8.212E+01 8.184E+01 1.375E+11 1.300E+11
8.172E+01d 1.427E+11d
1.435E+11e
3 9 6.945E+01 6.925E+01 3.887E+10 3.800E+10
6.906E+01d 3.984E+10d
3 10 6.941E+01 6.921E+01 2.320E+11 2.310E+11
6.901E+01d 2.390E+11d
3 13 5.433E+01 5.416E+01 6.061E+10 5.900E+10
3 17 5.131E+01 5.116E+01 1.397E+11 1.400E+11
4 7 1.019E+02 1.014E+02 4.896E+10 4.830E+10
1.014E+02b 5.101E+10d
1.015E+02d
4 8 1.013E+02 1.009E+02d 4.746E+09 4.940E+09d
4 11 8.647E+01 8.606E+01 5.472E+11 5.300E+11
8.642E+01c 5.404E+11d
8.587E+01d
4 18 6.093E+01 6.070E+01 2.051E+11 2.050E+11
5 8 1.015E+02 1.011E+02 4.285E+10 4.400E+10
1.011E+02b 4.474E+10d
1.011E+02d
5 11 8.659E+01 8.600E+01d 3.903E+10 3.850E+10d
5 12 8.657E+01 8.616E+01 5.856E+11 5.900E+11
8.652E+01c 5.780E+11d
8.598E+01d
5 15 6.423E+01 6.401E+01 1.670E+10 1.700E+10
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Table 4 (continued)
i j
λij (in Å) Aji (in s-1)
MCDF NIST and other
references MCDF
NIST and other
references
5 19 6.098E+01 6.076E+01 2.193E+11 2.190E+11
6 7 1.045E+03 1.038E+03d 1.223E+09 1.245E+09d
6 8 9.900E+02 9.805E+02d 1.443E+09 1.477E+09d
7 9 7.907E+02 7.809E+02d 2.773E+09 2.860E+09d
8 9 8.254E+02 8.168E+02d 4.888E+08 4.995E+08d
8 10 8.202E+02 8.106E+02d 2.991E+09 3.065E+09d
9 18 1.876E+02 1.870E+02 9.016E+10 9.300E+10
10 19 1.878E+02 1.873E+02 9.661E+10 9.600E+10
11 20 2.054E+02 2.049E+02 1.586E+11
12 21 2.055E+02 2.050E+02 1.644E+11
a- Peacock et al. [3], b-Fawcett et al. [7], c-Reader et al. [4], d- Fischer et al. [9], e- Johnson et al.
[10], f- Verner et al. [13] and the values without superscript in columns IV and VI is the data taken
from NIST.
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