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For Review O
nly
Energies, Landé g-Factors, Oscillator Strengths, and Transition Probabilities in Cs-like Pr V
Journal: Canadian Journal of Physics
Manuscript ID cjp-2015-0308.R2
Manuscript Type: Review
Date Submitted by the Author: 03-Aug-2015
Complete List of Authors: Karaçoban Usta, Betül; Sakarya university, Physics Doğan, Sevda; sakarya university,
Keyword: HFR method, Relativistic corrections, Wavelengths, Oscillator strengths, Transition probabilities
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Energies, Landé g-Factors, Oscillator Strengths, and Transition
Probabilities in Cs-like Pr V
Betül Karaçoban Usta* and Sevda Doğan
Physics Department, Sakarya University, 54187, Sakarya, Turkey
Abstract
We have calculated relativistic energies and Landé g-factors for the levels of 5p6nf (n =
4 − 30), 5p6np (n = 6 − 30), 5p6nd (n = 5−30), 5p6ng (n = 5−30) and 5p6ns (n = 6 − 30)
configurations and the transition parameters, such as wavelengths, oscillator strengths, and
transition probabilities (or rates), for the electric dipole (E1) transitions between these
levels in quadruply ionized praseodymium (Pr V, Z = 59) by using the relativistic Hartree-
Fock (HFR) method. We have compared the results with available calculations and
experiments in literature.
Keywords: HFR method; Relativistic corrections; Wavelengths; Oscillator strengths;
Transition probabilities
PACS numbers: 31.15.ag, 31.15.aj, 31.15.V-, 31.30.-i, 32.70.Cs
Introduction
The need for reliable atomic data in the study of astrophysical problems is well-known.
In spectrum synthesis work, particularly as applied to chemically peculiar (CP) stars,
*Corresponding author: Betül Karaçoban Usta (E-mail: [email protected]) Tel.: +90 (264) 2956093; fax: +90 (264) 2955950
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accurate knowledge of transition probabilities and oscillator strengths for heavy and rare-
earth elements is essential to establishing reliable abundances for reliable abundances for
these species [1]. Atomic transition probability is a key parameter for a determination of
the chemical composition of the stars.
Praseodymium is one of the odd-Z lanthanides (Pr, Z = 59). It has one stable isotope
(141Pr) and 14 short-live ones. An accurate determination of Pr abundance in different types
of stars is important in astrophysics in relation with nucleosynthesis, Pr being generated by
both the rapid and slow neutron capture processes [2].
As a member of the cesium (Cs) isoelectronic sequence, the quadruply ionized
praseodymium (Pr V) ion is expected to have a simple electronic structure, with a single
valence electron outside a complete 5p6 subshell. Pr V has ground configuration 5p64f and
excited states of the type 5p6nl. There is substantial spectroscopic literature concerning Pr
V, though less than for the neutral or other ionized species. Kaufman and Sugar [3]
reported twelve spectral lines of quadruply-ionized praseodymium in the region 840 to
2250 Å. Their work is the first example of the structure of the fifth spectrum of a rare earth
[3]. Migdalek and Baylis [4] studied the relative importance of relativistic effects, core
polarization and relaxation in ionization potentials for Cs trough Pr V, and reported
relativistic single-configuration Hartree-Fock oscillator strengths for 6s–6p transitions in
Pr V [5]. The single-configuration relativistic Hartree-Fock ionization potentials were
computed by using the froze-core and relaxed-core approximations with and without
allowance for core polarisation by Migdalek and Bojara [6]. Migdalek and Wyrozumska
[7] calculated oscillator strengths obtained using the relativistic model potential approach
in there different versions: a model potential without valence-core electron exchange but
with core-polarization included (RMP+CP), with semiclassical exchange and core-
polarization (RMP+SCE+CP), and with empirically adjusted exchange and core-
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polarization (RMP+EX+CP) for the 6s–6p, 5d–6p, 4f–5d, 5d–5f, 5d–6f, 6p–6d, and 6p–7d
transition arrays. Energies, transition rates, and electron-dipole-moment enhancement
factors for Pr V were performed using relativistic many-body perturbation theory by
Savukov et al. [8]. Zilitis [9] calculated oscillator strengths by the Dirac-Fock method for
the resonance transitions of Pr V. Glushkov [10] presented oscillator strengths of Cs and
Rb-like ions.
In this work, we have presented the relativistic energies and Landé g-factors for the
levels of 5p6nf (n = 4 − 30), 5p6np (n = 6 − 30), 5p6nd (n = 5−30), 5p6ng (n = 5−30) and
5p6ns (n = 6 − 30) configurations, and the transition parameters, such as the wavelengths,
oscillator strengths, and transition probabilities, for electric dipole (E1) transitions between
these levels in quadruply ionized praseodymium (Pr V). Calculations have been carried out
by the relativistic Hartree-Fock (HFR) method [11]. This method considers the correlation
effects and relativistic corrections. These effects make important contribution to
understanding physical and chemical properties of atoms or ions, especially lanthanides.
The ground-state level of Pr V is [Xe]4f 2 o5/ 2F . We have studied different configuration sets
according to valence excitations and core-valence correlation for correlation effects in Pr
V. In calculations, we have taken into account nf (n = 4 − 30), np (n = 6 − 30), nd (n = 5 −
30), ng (n = 5 − 30), and ns (n = 6 − 30) configurations outside the core [Xe] for the
calculation A, 5p6nf (n = 4 − 10), 5p6np (n = 6 − 10), 5p54f6p, 5p6nd (n = 5−10), 5p6ng (n
= 5−10), 5p6ns (n = 6 − 10), 5p54f5d, 5p56s6p, 5p55d6p, and 5p54f6s configurations
outside the core [Cd] for the calculation B, and 5p6nf (n = 4 − 20), 5p6np (n = 6 − 20),
5p54f2, 5p56p2, 5p54f6p, 5p6nd (n = 5−20), 5p6ng (n = 5−20), 5p6ns (n = 6 − 20), 5p54f5d,
5p56s6p, 5p55d6p, and 5p54f6s configurations outside the core [Cd] for the calculation C.
These configuration sets used in calculations have been denoted by A, B, and C in tables.
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Method of calculation
In this work, the calculations were performed by using the HFR method [11]. We will
here introduce this method briefly.
In HFR method [11], for N electron atom of nuclear charge Z0, the Hamiltonian is
expanded as
2 02 2( ) .i i i i i
i i i j ii ij
ZH r
r rζ
>
= − ∇ − + +∑ ∑ ∑ ∑ l s (1)
in atomic units, with ri the distance of the ith electron from the nucleus and ij i jr = −r r .
2 1( )
2i
VR
r r
αζ
∂ = ∂ is the spin-orbit term, with α being the fine structure constant and V
the mean potential field due to the nucleus and other electrons.
In this method it is calculated single-configuration radial functions for a spherically
symmetrized atom (center-of-gravity energy of the configuration) based on Hartree-Fock
method. The radial wave functions are also used to obtain the total energy of the atom (Eav)
including approximate relativistic and correlation energy corrections. Relativistic terms are
included in the potential function of the differential equation to give approximate
relativistic corrections to the radial functions, as well as improved relativistic energy
corrections in heavy atoms. In addition, a correlation term is included in order to make the
potential function more negative, and thereby help to bind negative ions. Also, Coulomb
integrals Fk and Gk and spin-orbit integrals nlζ are computed with these radial functions.
After radial functions have been obtained based on Hartree-Fock model, the wave function
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JMγ of the M sublevel of a level labeled Jγ is expressed in terms of LS basis states
LSJMα by the formula
LS
JM LSJM LSJ Jα
γ α α γ=∑ . (2)
After the wave functions have been obtained, they are used to calculate the
configuration-interaction Coulomb integrals between each pair of interacting
configurations. Then, it is set up energy matrices for each possible value of J and
dioganalized each matrix to get eigenvalues (energy levels) and eigenvectors
(multiconfiguration, intermediate coupling wave functions in various possible angular-
momentum coupling representations).
If determinant wave functions are used for the atom, the total binding energy is given
by
( )i i ij
k n
i j i
E E E E<
= + +∑ ∑ (3)
where i
kE is the kinetic energy, i
nE is the electron-nuclear Coulomb energy, and ijE is the
Coulomb interaction energy between electrons i and j averaged over all possible magnetic
quantum numbers.
In (1), for brevity, it has been omitted the mass-velocity and Darwin terms. These terms
depend only on ri and have the effect only shifting the absolute energies of a group of
related levels, without affecting the energy differences among these levels. This spin-
orbital term in (1), represents the sum over all electrons of magnetic interaction energy
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between the spin of an electron and its own orbital motion. Unlike the mass-velocity and
Darwin terms, the spin-orbit interaction involves the angular portion of the wave function
through the operators l and s, and has a pronounced effect on energy-level structure. It is
therefore necessary to retain it explicitly in the Hamiltonian. In this method the spin-orbit
contributions are considered as perturbation. This term is related to the separated of level
and J-dependent. All of these relativistic contributions are considered as perturbations with
order α2 to the non-relativistic Hamiltonian.
The Landé g-factor of an atomic level is related to the energy shift of the sublevels
having magnetic number M by
( ) B JE JM Bg Mγγ µ∆ = (4)
where B is the magnetic field intensity and Bµ is the Bohr magneton. The Landé g-factor
of a level, denoted as αJ, belonging to a pure LS-coupling term is given by the formula
( 1) ( 1) ( 1)1 ( 1) .
2 ( 1)LSJ s
J J L L S Sg g
J Jα
+ − + + += + −
+ (5)
This expression is derived from vector coupling formulas by assuming a g value of
unity for a pure orbital angular momentum and writing the g value for a pure electron spin
(S level) as sg [12]. A value of 2 for sg yields the Landé formula. The Landé g-factors for
energy levels are a valuable aid in the analysis of a spectrum.
An electromagnetic transition between two states is characterized by the angular
momentum and the parity of the corresponding photon. If the emitted or absorbed photon
has angular momentum k and parity ( 1)kπ = − then, the transition is an electric multipole
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transition (Ek). However, if the photon has parity 1( 1)kπ += − the transition is a magnetic
multipole transition (Mk).
According to HFR method [11], the total transition probability from a state ''' MJγ to
all states M levels of Jγ is given by
4 2 2 3
064
3 (2 ' 1)
e aA
h J
π σ=
+S (6)
and absorption oscillator strength is given by
2( )
3(2 1)j i
ij
E Ef
J
−=
+S . (7)
where, ( ) /j iE E hcσ = − has units of kaysers (cm−1) and 2
(1) ' 'J Jγ γ=S P is the
electric dipole line strength in atomic units of e2a
20. The strongest transition is electric
dipole (E1) radiation. For this reason, the E1 transitions are understood as being ‘allowed’,
whereas high-order transitions are understood as being ‘forbidden’.
Results and discussion
We have here calculated the relativistic energies and Landé g-factors for the levels of
5p6nf (n = 4 − 30), 5p6np (n = 6 − 30), 5p6nd (n = 5−30), 5p6ng (n = 5−30), and 5p6ns (n =
6 − 30) configurations and the transition parameters (wavelengths, oscillator strengths, and
transition probabilities) for electric dipole (E1) transitions between these levels in Pr V
using HFR [13] code. The configuration sets selected for investigating correlation effects
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are given in Introduction section. Correlation effects in atoms can often be classified as
valence-valence, core-valence and core-core contributions. Generally, these contributions
can be evaluated by multiconfiguration techniques. Only the first two contributions are
usually important, in particular valence-valence correlation although the core-valence
correlation in many electron atoms is important. However, excitations from core to valence
produce too many configurations. We tried to perform the core-valence correlation other
than valence excitations. These type configurations occur resulting in large configuration
state function expansions due to open core and valence subshells and cause the
optimization problems. That is, we have not selected more configurations excited from
core and valence together because of the optimization constraints and computer limits.
Therefore, we have performed three types of calculations for obtaining configuration state
functions (CSFs) according to valence excitations and core-valence correlation.
The results for energy levels and Landé g-factors of Pr V have been reported in Table 1
and Table S1 for low-lying and highly-lying excited levels, respectively. The fitted energy
parameters in Table 2 and Table S2 display the scaling factors (fitted/HFR) belonging to
the calculation A. Table 3 shows wavelengths λ (in Å), oscillator strengths f, and transition
probabilities Aki (in s−1), for 5p66p–5p66s, 5p66p–5p65d, 5p65d–5p64f, 5p67s–5p66p, 5p66d–
5p66p, 5p67d–5p66p 5p65f–5p65d, and 5p66f–5p65d electric dipole (E1) transitions. The
comparing values for these transitions exist in literature. Therefore, it is also made a
comparison with other calculations and experiments in Table 3. In calculations, the data
obtained are too much. For this reason, we have here presented just a part of the results. In
Table S3, we have also reported wavelengths, logarithmic weighted oscillator strengths log
(gf), and weighted transition probabilities gAki, for some atomic data. This table includes
the transition probabilities greater than or equal to 108 s−1. In these tables, the calculations
for the various configuration sets are represented by A, B, and C. References for other
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comparison values are typed below the tables with a superscript lowercase letter. Only
odd-parity states in tables are indicated by the superscript “o” and the number in brackets
represents the power of 10.
We have presented our calculations using the RCN, RCN2, RCG and RCE chain of
programs developed by Cowan [13]. The HFR option of the RCN code was used to derive
initial values of the parameters with appropriate scaling factors in the code RCN2. The
RCE can be used to vary the various radial energy parameters Eav, Fk, Gk, ζ, and Rk to
make a least-squares fit of experimental energy levels by an iterative procedure. The
resulting least-squares-fit parameters can then be used to repeat the RCG calculation with
the improved energy levels and wavefunctions [13]. Transition parameters were calculated
by the RCG code after the fitting of energy parameters. In the calculations, the
Hamiltonian’s eigenvalues were optimized to the observed energy levels via a least-
squares fitting procedure using experimentally determined energy levels, specifically all of
the levels from the NIST compilation [14]. The scaling factors of the Slater parameters (Fk
and Gk) and of configuration interaction integrals (Rk), not optimized in the least-squares
fitting, were chosen equal to 0.75 for calculations A, B, and C, while the spin-orbit
parameters were left at their initial values. This low value of the scaling factors has been
suggested by Cowan for heavy elements [11, 13]. In this work, we have only given valence
excitation levels and E1 transitions between these levels. Therefore, the fitted energy
parameters in Table 2 and Table S2 reported the scaling factors (fitted/HFR) belonging to
the calculation A. In this calculation, there is not the scaling factors Fk(li, li) between
equivalent electrons , Fk(li, lj) and Gk(li, lj) for non-equivalent electrons and configuration
interaction (Rk) radial integrals. The ratio (fitted/HFR) for energy parameters in calculation
A is compared with 1.00 for total binding energy (Eav) and spin-orbit (ζ) in Table 2 and
Table S2. It can be mentioned that the agreement for most of values is good.
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The relativistic energies and Landé g-factors for the levels of 5p6nf (n = 4 − 30), 5p6np
(n = 6 − 30), 5p6nd (n = 5−30), 5p6ng (n = 5−30), and 5p6ns (n = 6 − 30) configurations in
Pr V have been calculated by HFR method. The results obtained have been presented
energies (cm−1) relative to 4f 2 o5 / 2F ground-state level in Table 1 and Table S1. We have
compared our results with previous works [7, 8, 14] in Table 1. Only energy results of the
5p64f (n = 4−6), 5p6np (n = 6, 7), 5p6nd (n = 5−7), and 5p6ns (n = 6, 7) levels are compared
with experimental [14] and theoretical [7, 8] results. Most of our energy results are in good
agreement with others. Moreover, we have calculated [|Ethis work − Eother works|/Eother works] ×
100, the differences in per cent, for the accuracy of our results. In calculations, differences
(%) between our results and other experimental works [14] have been found in the 0.00
range for the energies of 5p64f, 5p66p, 5p65d, and 5p6ns (n = 6, 7) excited levels, except
5p66p 2 o3/ 2P level. When the differences (%) between our results and other theoretical
results [8] (indicated by the superscript b in Table 1) are investigated, the differences in
energies are generally in range of 0.07−1.48, 0.07−2.36 and 0.07−1.48 for calculations A,
B, and C, respectively. For energies of 5p64f 2 o7 / 2F and 5p66p excited levels, there is very
little discrepancies. The differences (%) between our results and other theoretical result [7]
(indicated by the superscript c in Table 1) are in range of 0.00−1.44, 0.00−1.08 and
0.00−1.12 for calculations A, B, and C, respectively. The agreement is somewhat poor for
energies of 5p66d level. The Landé g-factor results are reported for the first time. Moreover
it is well known that Landé g-factors are important in many scientific areas such as
astrophysics. Therefore, new energies and Landé g-factors for 5p6nf (n = 7 − 30), 5p6np (n
= 8 − 30), 5p6nd (n = 8−30), 5p6ng (n = 5−30) and 5p6ns (n = 8 − 30) configurations, not
existing in the data bases for these configurations in Pr V, are presented in Table 1 and
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Table S1. Our new energy levels and Landé g-factors are reliable since the results in
presented Table 1 are in excellent agreements with other works.
We obtained 7 287, 22 637, and 33 618 possible electric dipole transitions between odd-
and even-parity levels in the HFR calculations A, B, and C, respectively. Table 3 shows
the wavelengths λ (in Å), oscillator strengths f, and transition probabilities Aki (in s-1) for
5p66p–5p66s, 5p66p–5p65d, 5p65d–5p64f, 5p67s–5p66p, 5p66d–5p66p, 5p67d–5p66p, 5p65f–
5p65d, and 5p66f–5p65d electric dipole (E1) transitions. We have typed as transition
probabilities (the division of the statistical weight g of the upper level and the weighted
transition probabilities) and oscillator strengths (the division of the statistical weight g of
the lower level and the weighted oscillator strengths) for comparing. The comparing values
for these transitions exist in literature. Therefore, it is also made a comparison with other
works in Table 3. The results obtained are in excellent agreement with those of other works
except some transitions. For some transitions, although the agreement is less in the
oscillator strengths and transition probabilities, it is very good in the wavelengths. The
oscillator strengths computed are compared in Table 3 with other available theoretical
results [5, 7−9] for the 20 transitions of Pr V. As seen from this table, the oscillator
strengths obtained from the calculations A, B, and C are in agreement with other works,
except for 5p65f 2 o5/ 2F – 5p65d 2
3/ 2D , 5p65f 2 o7 / 2F – 5p65d 2
5/ 2D , and 5p66f–5p65d
transitions. We have calculated the mean ratio log gf (this work) / log gf (other works) for the
accuracy of our results. In calculations, the mean ratio between our results and other works
[8] have been found in the values 1.63 (in calculation A), 1.04 (in calculation B), and 0.96
(in calculation C). Also, we have found the values 1.29 (in calculation A), 1.06 (in
calculation B), and 1.01 (in calculation C) for the mean ratio log gf (this work) / log gf [7, c1],
except the transitions 5p65d– 5p64f and 5p66f 2 o5 / 2F – 5p65d 2
3/ 2D . The available theoretical
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transition probabilities are only a work in the literature [8]. Therefore we have compared
for 8 transitions. Generally, the results from the calculations B and C are better when our
results have been compared with other work. The agreement is poor for transition
probability of 5p66p 2 o3/ 2P –5p66s 2
1/ 2S transition. Also, we have found the values 1.65 (in
calculation A), 1.02 (in calculation B), and 0.93 (in calculation C), for the mean ratio gAki
(this work) / gAki [8], except the transition 5p66p 2 o3/ 2P –5p66s 2
1/ 2S . The transition results
obtained from the calculations B and C agree with other works. This calculations include
core correlation (including excitation from 5p shell in core). These results obtained from
the HFR calculations may be improved by adding configurations including excitations
from core (5p6). But this case occurs some program constraints or convergence problems.
Electron correlation effects and relativistic effects play an important role in the spectra
of heavy elements. In the structure calculation and accurate prediction of radiative atomic
properties for heavy atom such as Pr V, complex configuration interaction and relativistic
effects must be considered simultaneously. Migdalek and Bojara [6] demonstrated there
the essential role of relativistic effects and of core polarization in improving the agreement
between theory and experiment, whereas the influence of core relaxation was found to be
much less pronounced. Therefore, the calculations may be improved using the HFR
method modified in order to into account the polarization of the ionic core (CPOL) effects.
Conclusion
It is well known that the correlation, relativistic and radiative effects all play important
roles in fundamental atomic theory. The main purpose of this paper is to perform the HFR
calculations for obtaining description of the Pr V spectrum. Especially, in spectrum
synthesis works, particularly for CP stars, accurate data for transition parameters for
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lanthanides are need to establish reliable abundances for species. We have presented other
our results obtained from this work as supplementary data (Table S1, Table S2, and Table
S3). These energy data and Landé g-factors for Pr V can be useful in investigations of
some radiative properties, and interpretation of many levels of Pr V. We have only given
E1 transitions between the valence excitation levels. A set of oscillator strengths and
transition probabilities is obtained for the first time for these transitions of Pr V. Therefore
we hope that our results obtained using HFR method will be useful for research fields and
technological applications and for interpreting the spectrum of Pr V.
Acknowledgments
The authors are very grateful to the anonymous reviewers for stimulating comments and
valuable suggestions, which is resulted in improving the presentation of the paper.
References
1. D.J. Bord, Astron. Astrophys. Suppl. Ser. 144, 517 (2000). doi:
10.1051/aas:2000226.
2. P. Palmeri, P. Quinet, Y. Frémat, J.-F. Wyart, and E. Biémont, Astrophys. J. Suppl.
Ser. 129, 367 (2000). doi:10.1086/313405.
3. V. Kaufman and J. Sugar, Jour. of Research National B.S.-A. 71(6), 583 (1967).
4. J. Migdalek and W.E. Baylis, Phys. Rev. A, 30, 1603 (1984).
doi:10.1103/PhysRevA.30.1603.
5. J. Migdalek and W.E. Baylis, J. Quant. Spectrosc. Radiat. Transfer, 22, 127 (1979).
doi:10.1016/0022-4073(79)90033-5.
6. J. Migdalek and A. Bojara, J. Phys. B: At. Mol. Phys. 17, 1943 (1984).
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doi:10.1088/0022-3700/17/10/004.
7. J. Migdalek and M. Wyrozumska, J. Quant. Spectrosc. Radiat. Transfer, 37, 581
(1987). doi:10.1016/0022-4073(87)90061-6.
8. I.M. Savukov, W.R. Johnson, U.I. Safronova, and M.S. Safronova, Phys. Rev. A,
67, 042504 (2003). doi:10.1103/PhysRevA.67.042504.
9. V.A. Zilitis, Opt. Spectrosc. 117, 513 (2014). doi:10.1134/S0030400X14100245.
10. A.V. Glushkov, J. Appl. Spectrosc. 56, 5 (1992). doi:10.1007/BF00658239.
11. R.D. Cowan, The Theory of Atomic Structure and Spectra, University of California
Press, 1981.
12. P. Jönsson and S. Gustafsson, Comput. Phys. Commun. 144, 188 (2002). doi:
10.1016/S0010-4655(01)00461-1.
13. http://www.tcd.ie/Physics/People/Cormac.McGuinness/Cowan/
(This webpage serves as a repository for Robert D. Cowan's atomic structure
codes.)
14. A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team (2014). NIST Atomic
Spectra Database (version 5.2), Available: http://physics.nist.gov/asd
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Table 1. Energies, E, and Landé g-factors for low-lying levels in Pr V.
Level E (cm-1) Landé g-factor
Conf. Term
This work Other
works
This work
A B C A B C
5p64f
2 o
5 / 2F 0.00 0.00 0.19 0.00a,b,c
0.857 0.857 0.857
2 o
7 / 2F 3027.400 3027.600 3027.594 3027.4a
1.143 1.143 1.143
2985b
3027.22c
5p65d 2
3/ 2D 115052.300 115052.200 115051.756 115052.3a 0.800 0.800 0.800
113543b
115052.31c
2
5/ 2D 118513.800 118513.900 118514.649 118513.8a 1.200 1.200 1.200
116942b
118513.87c
5p66s
2
1/ 2S 178971.100 178970.700 178979.231 178971.1a
2.002 2.002 2.003
177604b
178971.33c
5p66p
2 o
1/ 2P 223478.100 223478.100 223478.141 223478.1a
0.666 0.666 0.666
220398b
223478.44c
2 o
3/ 2P 230039.500 232039.600 230039.582 230039.5a
1.334 1.334 1.334
226690
b
230039.87c
5p65f
2 o
5 / 2F 290296.500 288629.300 288627.179 286885.21c
0.857 0.857 0.857
2 o
7 / 2F 290533.400 289150.100 289148.365 287062.98c
1.143 1.143 1.143
5p66d 2
5/ 2D 291198.300 290145.405 290266.170 289591b 0.800 0.800 0.792
287050.47c
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Table 1. (continued)
Level E (cm-1) Landé g-factor
Conf. Term
This work Other
works
This work
A B C A B C
2
3/ 2D 292092.500 291022.797 291098.959 290686b
1.200 1.200 1.198
288112.29c
5p67s
2
1/ 2S 304511.500 304511.400 304512.133 304511.5a
2.002 2.002 2.002
304736b
5p67p 2 o
1/ 2P 322422.300 322570.297 322570.267 323243.22c 0.666 0.666 0.659
2 o
3/ 2P 324951.300 324969.100 324969.082 326118.59c 1.334 1.334 1.327
5p65g
2
7 / 2G 351437.900 351489.100 351488.937 − 0.889 0.889 0.889
2
9 / 2G 351442.800 351499.300 351499.115 − 1.111 1.111 1.111
5p67d
2
3/ 2D 356293.100 354047.000 354051.013 353088.53c
0.800 0.800 0.775
2
5/ 2D 356722.600 354501.700 354506.118 353604.29c
1.200 1.200 1.191
5p66f
2 o
5 / 2F 353815.800 354396.900 354395.329 352071.26c
0.857 0.857 0.857
2 o
7 / 2F 353941.100 354620.200 354618.216 352174.85c 1.143 1.143 1.143
5p68s 2
1/ 2S 360476.500 360438.000 360438.391 − 2.002 2.002 2.002
5p68p
2 o
1/ 2P 370452.100 370468.010 370467.994 − 0.666 0.666 0.666
2 o
3/ 2P 371812.900 371830.511 371830.494 − 1.334 1.334 1.334
5p66g
2
7 / 2G 385778.400 385812.200 385812.346 − 0.889 0.889 0.889
2
9 / 2G 385781.100 385817.000 385817.471 − 1.111 1.111 1.111
5p67f
2 o
5 / 2F 387464.900 387606.400 387606.144 − 0.857 0.857 0.857
2 o
7 / 2F 387538.000 387688.500 387688.139 − 1.143 1.143 1.143
5p68d 2
3/ 2D 390069.300 387855.400 387858.682 − 0.800 0.800 0.800
2
5/ 2D 390314.300 388121.000 388124.619 − 1.200 1.200 1.200
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Table 1. (continued)
Level E (cm-1) Landé g-factor
Conf. Term
This work Other
works
This work
A B C A B C
5p69s
2
1/ 2S 391424.900 391399.800 391400.153 − 2.002 2.002 2.002
5p69p
2 o
1/ 2P 397361.300 397364.766 397364.599 − 0.666 0.666 0.666
2 o
3/ 2P 398180.500 398184.479 398184.399 − 1.334 1.334 1.334
5p67g 2
7 / 2G 406481.300 406502.600 406502.875 − 0.889 0.889 0.889
2
9 / 2G 406483.100 406505.100 406505.658 − 1.111 1.111 1.111
5p68f
2 o
5 / 2F 407668.800 407733.200 407733.074 − 0.857 0.857 0.857
2 o
7 / 2F 407715.000 407782.200 407782.172 − 1.143 1.143 1.143
5p69d
2
3/ 2D 410165.100 407983.900 407986.018 − 0.800 0.800 0.800
2
5/ 2D 410318.800 408156.500 408159.701 − 1.200 1.200 1.200
5p610s 2
1/ 2S 410208.200 410189.700 410190.251 − 2.002 2.002 2.002
5p610p 2 o
1/ 2P 414027.000 414027.593 414027.500 − 0.666 0.666 0.666
2 o
3/ 2P 414559.200 414560.199 414560.100 − 1.334 1.334 1.334
5p68g
2
7 / 2G 419907.600 419920.300 419921.260 − 0.889 0.889 0.889
2
9 / 2G 419909.000 419921.900 419922.765 − 1.111 1.111 1.111
5p69f
2 o
5 / 2F 420759.400 420795.600 420795.585 − 0.857 0.857 0.857
2 o
7 / 2F 420790.500 420828.100 420827.984 − 1.143 1.143 1.143
5p610d 2
3/ 2D 423141.500 420975.700 421560.947 − 0.800 0.800 0.800
2
5/ 2D 423244.500 421100.700 421591.298 − 1.200 1.200 1.200
5p69g
2
7 / 2G 429105.900 429103.096 429694.829 − 0.889 0.889 0.890
2
9 / 2G 429106.800 429112.800 430717.294 − 1.111 1.111 1.112
5p610f
2 o
5 / 2F 429732.500 429755.600 429755.391 − 0.857 0.857 0.857
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Table 1. (continued)
Level E (cm-1) Landé g-factor
Conf. Term This work
Other
works
This work
2 o
7 / 2F 429754.500 429778.100 429777.990 − 1.143 1.143 1.143
5p610g
2
7 / 2G 435674.000 435682.100 436580.302 − 0.889 0.889 0.889
2
9 / 2G 435674.500 435683.400 436561.422 − 1.111 1.111 1.112
aNIST Atomic Spectra Database [14]
bSavukov et al. [8]
cMigdalek and Wyrozumska [7, RMP+EX+CP approach]
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Table 2. Energy parameters obtained from the calculation A for low-
lying levels in Pr V.
Conf. Parameter HFR Fitted SF(Fitted/HFR)
5p64f Eav 0 1729.9
ζ4f 923.4642 865.0 0.937
5p65d
Eav 112319.1 117129.20 1.043
ζ5d 1371.381 1384.600 1.010
5p66s Eav 174820.7 178971.10 1.024
5p66p Eav 222011.7 227852.4 1.026
ζ6p 3780.791 4374.30 1.157
5p65f Eav 288431.9 290431.9 1.007
ζ5f 67.7312 67.70 1.000
5p66d Eav 287734.8 291734.8 1.014
ζ6d 357.7022 357.70 1.000
5p67s Eav 301560.4 304511.5 1.010
5p67p Eav 322108.3 324108.3 1.006
ζ7p 1685.971 1686.0 1.000
5p65g Eav 349440.6 351440.6 1.006
ζ5g 1.0872 1.10 1.012
5p67d Eav 352550.8 356550.8 1.011
ζ7d 171.8172 171.80 1.000
5p66f Eav 351887.4 353887.4 1.006
ζ6f 35.7942 35.80 1.000
5p68s Eav 358476.5 360476.5 1.006
5p68p Eav 369359.3 371359.3 1.005
ζ8p 907.1902 907.20 1.000
5p66g Eav 383779.9 385779.9 1.005
ζ6g 0.6422 0.60 0.934
5p67f Eav 385506.7 387506.7 1.005
ζ7f 20.9072 20.90 1.000
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Table 2. (continued)
Conf. Parameter HFR Fitted SF(Fitted/HFR)
5p68d Eav 386216.3 390216.3 1.010
ζ8d 98.0192 98.0 1.000
5p69s Eav 389424.9 391424.9 1.005
5p69p Eav 395907.4 397907.4 1.005
ζ9p 546.0552 546.1 1.000
5p67g Eav 404482.3 406482.3 1.005
ζ7g 0.4192 0.40 0.954
5p68f Eav 405695.2 407695.2 1.005
ζ8f 13.2292 13.20 0.998
5p69d Eav 406257.3 410257.3 1.010
ζ9d 61.4762 61.50 1.000
5p610s Eav 408208.2 410208.2 1.005
5p610p Eav 412381.8 414381.8 1.005
ζ10p 354.7802 354.80 1.000
5p68g Eav 417908.4 419908.4 1.005
ζ8g 0.2732 0.30 1.098
5p69f Eav 418777.2 420777.2 1.005
ζ9f 8.8822 8.90 1.002
5p610d Eav 419203.3 423203.3 1.010
ζ10d 41.1892 41.20 1.000
5p69g Eav 427106.4 429106.4 1.005
ζ9g 0.1892 0.20 1.057
5p610f Eav 427745.1 429745.1 1.005
ζ10f 6.2472 6.30 1.008
5p610g Eav 433674.3 435674.3 1.005
ζ10g 0.1362 0.10 0.734
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Table 3. Wavelengths λ (Å), oscillator strengths f, and transition probabilities Aki (s−1) for electric dipole (E1) transitions in
Pr V. Numbers in brackets represent powers of 10.
Transition λ f Aki
Upper Level Lower Level This w. Other w. This w. Other w. This w. Other w.
5p66p 2 o1/ 2P 5p66s 2
1/ 2S A 2246.84 2246.759a 0.4322 0.308b 5.710(8) 4.07(8)b
B 2246.81 2337b 0.3659 0.338c1 4.834(8)
C 2247.25 0.3551 0.321c2 4.690(8)
0.325c3
0.436d1
0.353d2
0.418e
5p66p 2 o3/ 2P 5p66s 2
1/ 2S A 1958.16 1958.088a 0.9918 0.707b 8.625(8) 6.15(9)b
B 1884.34 2037b 0.8756 0.784c1 8.223(8)
C 1958.47 0.8179 0.746c2 7.113(8)
0.745c3
0.993d1
0.817d2
0.952e
5p66p 2 o1/ 2P 5p65d 2
3/ 2D A 922.29 922.290a 0.1996 0.161b 3.131(9) 2.52(9)b
B 922.29 936b 0.1720 0.181c1 2.698(9)
C 922.29 0.1673 0.174c2 2.623(9)
0.165c3
0.210e
5p66p 2 o3/ 2P 5p65d 2
3/ 2D A 869.66 869.622a 0.0423 0.0306b 3.735(8) 2.70(8)b
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Table 3. (continued)
Transition λ f Aki
Upper Level Lower Level This w. Other w. This w. Other w. This w. Other w.
B 854.79 911b 0.0372 0.220c1 3.400(8)
C 869.66 0.0356 0.0334c2 3.140(8)
0.0311c3
0.0390e
5p66p 2 o3/ 2P 5p65d 2
5/ 2D A 896.65 896.654a 0.2464 0.191b 3.068(9) 2.37(9)b
B 880.86 911b 0.2165 0.220c1 2.790(9)
C 896.66 0.2067 0.205c2 2.573(9)
0.193c3
0.239e
5p65d 23/ 2D 5p64f 2 o
5 / 2F A 869.17 869.170a 0.0611 0.0290b 8.093(8) 3.85(8)b
B 869.17 881b 0.0235 0.0169c1 3.113(8)
C 869.17 0.0196 0.0318c2 2.593(8)
0.0259c3
5p65d 25 / 2D 5p64f 2 o
5 / 2F A 843.78 843.783a 0.0045 0.0021b 4.212(7) 2.01(7)b
B 843.78 863b 0.0018 0.00099c1 1.695(7)
C 843.78 0.0015 0.00236c2 1.427(7)
0.0019c3
5p65d 25 / 2D 5p64f 2 o
7 / 2F A 865.90 865.902a 0.0657 0.0318b 7.795(8) 3.78(8)b
B 865.90 877b 0.0253 0.0175c1 3.003(8)
C 865.90 0.0211 0.0351c2 2.500(8)
0.0285c3
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Table 3. (continued)
Transition λ f Aki
Upper Level Lower Level This w. Other w. This w. Other w. This w. Other w.
5p67s 21/ 2S 5p66p 2 o
1/ 2P A 1234.06 1234.070a 0.2278 − 1.00(9) −
B 1234.06 0.2425 1.06(9)
C 1234.05 0.2456 1.07(9)
5p67s 21/ 2S 5p66p 2 o
3/ 2P A 1342.79 1342.775a 0.2093 − 1.55(9) −
B 1379.85 0.2168 1.52(9)
C 1342.78 0.2258 1.67(9)
5p66d 23/ 2D 5p66p 2 o
1/ 2P A 1476.66 − 1.4733 1.208c1 2.253(9) −
B 1499.99 0.9975 1.185c2 1.479(9)
C 1497.27 1.0885 1.146c3 1.619(9)
5p66d 23/ 2D 5p66p 2 o
3/ 2P A 1635.09 − 0.1331 0.123c1 3.320(8) −
B 1721.00 0.0876 0.119c2 1.972(8)
C 1660.40 0.0989 0.113c3 2.393(8)
5p66d 25 / 2D 5p66p 2 o
3/ 2P A 1611.53 − 1.2150 1.111c1 2.080(9) −
B 1695.40 0.8717 1.080c2 1.349(9)
C 1637.75 0.9121 1.031c3 1.512(9)
5p67d 23/ 2D 5p66p 2 o
1/ 2P A 752.93 − 0.0583 0.0516c1 3.430(8) −
B 765.88 0.0262 0.0391c2 1.488(8)
C 765.85 0.0218 0.0157c3 1.239(8)
5p67d 23/ 2D 5p66p 2 o
3/ 2P A 792.06 − 0.0055 0.00319c1 5.890(7) −
B 819.62 0.0026 0.00222c2 2.570(7)
C 806.38 0.0022 0.00053c3 2.278(7)
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Table 3. (continued)
Transition λ f Aki
Upper Level Lower Level This w. Other w. This w. Other w. This w. Other w.
5p67d 25 / 2D 5p66p 2 o
3/ 2P A 789.37 − 0.0500 0.0336c1 3.570(8) −
B 816.58 0.0237 0.0241c2 1.583(8)
C 803.43 0.0205 0.0070c3 1.412(8)
5p65f 2 o5/ 2F 5p65d 2
3/ 2D A 570.63 − 0.9702 0.667c1 1.325(10) −
B 576.11 0.8489 0.674c2 1.137(10)
C 576.12 0.8369 0.660c3 1.121(10)
5p65f 2 o5/ 2F 5p65d 2
5/ 2D A 582.13 − 0.0453 0.0326c1 8.913(8) −
B 587.84 0.0396 0.0331c2 7.640(8)
C 587.85 0.0390 0.0322c3 7.528(8)
5p65f 2 o7 / 2F 5p65d 2
5/ 2D A 581.33 − 0.9070 0.647c1 1.343(10) −
B 586.04 0.8120 0.660c2 1.183(10)
C 586.05 0.8003 0.641c3 1.166(10)
5p66f 2 o5/ 2F 5p65d 2
3/ 2D A 418.82 − 0.2055 0.108c1 5.208(9) −
B 417.81 0.2116 0.0996c2 5.390(9)
C 417.81 0.2107 0.0831c3 5.368(9)
5p66f 2 o5/ 2F 5p65d 2
5/ 2D A 424.99 − 0.0096 0.00513c1 3.562(8) −
B 423.94 0.0100 0.00461c2 3.718(8)
C 423.94 0.0100 0.00388c3 3.707(8)
5p66f 2 o7 / 2F 5p65d 2
5/ 2D A 424.76 − 0.1930 0.104c1 5.350(9) −
B 423.54 0.1998 0.0932c2 5.573(9)
C 423.54 0.1991 0.0782c3 5.551(9)
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aKaufman and Sugar [3] bSavukov et al. [8] c1,c2,c3Migdalek and Wyrozumska [7] d1,d2 Migdalek and Baylis [5] e Zitilis [9]
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