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For Review O

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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

https://mc06.manuscriptcentral.com/cjp-pubs

Canadian Journal of Physics

For Review O

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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)


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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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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Other

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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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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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Transition λ f Aki


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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Transition λ f Aki


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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Transition λ f Aki


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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