CHINESE JOURNAL OF PHYSICS
VOL. 49, NO. 6
December 2011
Excitation Energies and E1, E2, and M1 Transition Parameters for Ac III ¨ ∗ and Leyla Ozdemir ¨ G¨ uldem Urer Department of Physics, Sakarya University, 54187, Sakarya, Turkey (Received September 27, 2010) We report on a relativistic multiconﬁguration DiracFock calculation including the transverse Breit and quantum electrodynamic contribution on the lowlying level structure of doubly ionized actinium (Z = 89). Excitation energies and the electric dipole (E1) and quadrupole (E2) transition and the magnetic dipole (M1) transition parameters such as the wavelengths, oscillator strengths, and transition rates for the lowlying levels of Ac III are presented. Moreover, the results obtained are compared with a few other works from the available literature. PACS numbers: 31.15.ag, 31.15.V, 32.70.Cs
I. INTRODUCTION
The spectrochemical detection or determination of actinium was reported by Meggers and coworkers [1, 2]. In addition, in their paper, there can be found details about the previous works on actinium. In pitchblende, actinium was found as a third new element after the discovery of radium and polonium. Actinium was separated from the other elements by chemical means and was revealed by its radioactivity. Actinium had never been concentrated in pure form from mineral sources because of its low abundance, radioactive instability, and lack of commercial uses. However, after the discovery of nuclear ﬁssion and the construction of uranium piles, neutrons became available in suﬃcient quantities to produce, by transmutation, ponderable amounts of any desired element. Discussion of the spectroscopy and chemistry for f elements in molecules and solids is necessary for the understanding of the lowlying energies of actinide ions. The 7s valence orbital is more tightly bound than the 5f and 6d orbitals at low stages of ionization for all isotopes of actinium, while the 5f and 6d orbitals are more tightly bound for the highly ionized cases. Competition between the 5f , 6d, and 7s orbitals leads to problems for the calculations, making it diﬃcult to obtain very accurate excitation energies and line strengths for the transitions between the lowlying 5f , 6d, and 7s states [3]. In calculating the level structure of a heavy element, one often faces diﬃculties which either do not occur for lowand mediumZ elements or are much less pronounced in those cases. These diﬃculties are strong relativistic and quantum dynamical (QED) eﬀects; the low lying level structure of most of these elements is determined by a number of overlapping and nearly degenerated conﬁgurations, and further complexity arises from the large number of electrons, which ∗
Electronic address:
[email protected]
http://PSROC.phys.ntu.edu.tw/cjp
1178
c 2011 THE PHYSICAL SOCIETY ⃝ OF THE REPUBLIC OF CHINA
VOL. 49
¨ ¨ ¨ GULDEM URER AND LEYLA OZDEMIR
1179
have to be treated explicitly in any ab initio theory in order to explain the lowlying spectra of such elements. Furthermore the short lifetimes and radioactivity of actinide elements obstruct also experimental work. The investigation of the level structure of f element atoms and ions (lanthanides and actinides), especially for the actinides, is more complex because of the great experimental diﬃculties and the diﬃculties in the assignment and interpretation of the enormous number of levels. The electronic structure of doubly ionized actinium consists of a single nl electron outside of a core with completely ﬁlled n = 1, 2, 3, 4 shells and 5s, 5p, 5d, 6s, and 6p subshells. There is little data on the Ac III structure. For the spectrochemical determination of actinium, the wavelengths and estimated relative intensities of 109 strong lines were presented, and 7 of them were assigned to Ac III by Meggers [1]. Brewer [4] presented the energies of the electronic conﬁgurations of the singly and doubly and triply ionized lanthanides and actinides. Eliav et al. [5] applied the relativistic coupled cluster method to the transition energies (ionization potentials, excitation energies, electron aﬃnities) of lanthanum, actinium and ekaactinium in several ionization states. Both relativistic energyconsistent smallcore ab initio pseudopotential and fully relativistic density functional allelectron calculations were carried out for the ﬁrst to fourth ionization potential, as well as some excitation energies for a whole series of actinide atoms by Liu et al. [6]. A detailed investigation of the radiative parameters for the electric dipole transition in Frlike ions with Z = 89–92 was presented by Bi´emont et al. [7]. Relativistic manybody perturbation theory was applied to study properties such as the excitation energies, polarizabilities, multipole transition rates, and lifetimes of ions along the francium isoelectronic sequence (Z = 89–100) by Safronova et al. [3]. Some data for actinide and its ions are available on the NIST atomic database [8] and the ACTINIDES website [9] TABLE I: Conﬁguration TABLE I: Configuration sets considered for Ac III.sets considered for Ac Levels A B C 7s, 6d As in A A+6s7s2, 6s6d2, For evenparity 6s7p2, 6s5f2 5 2 5 2 5f, 7p A+6p 7s , 6p 6d , As in A For oddparity 6p55f2
III. D As in C A+6s26p57s2, 6s26p57p2, 6s26p55f2
In this work we have calculated the excitation energies for the lowlying levels and radiative transition parameters such as wavelengths, oscillator strengths, and transition rates (probabilities) for the electric dipole and quadrupole and magnetic dipole in doubly ionized actinium using GRASP [10], based on a fully relativistic multiconﬁguration DiracFock (MCDF) method. We have here investigated four various conﬁguration sets, including valence and outer core correlations besides the transverse Breit and quantum electrodynamic (QED) eﬀects. These conﬁguration sets considered the correlations which are given in Table I. We think that these calculation results can provide good information on the low energy levels of Ac III. These are very demanding calculations because of relativity and correlation. It is well known that available transition rate values in doubly ionized actinium are sparse. We hope that our results will be useful for other experimental and computational
1180
EXCITATION ENERGIES AND E1, E2, . . .
VOL. 49
works about Ac III in the future. Moreover, it is mentioned that this work is a part of our continuing works on the level structure of f elements [11–18].
II. CALCULATION METHOD
We will here brieﬂy summarize the calculation method. A detailed description of MCDF theory may be found in [19–22]. In the MCDF method [10], an atomic state can be expanded as a linear combination of conﬁguration state functions (CSFs), ψα (P JM ) =
nc ∑
cr (α) γr P JM ⟩,
(1)
r=1
and is optimized usually on the basis of the manyelectron DiracCoulomb Hamiltonian in form HDC =
N ∑ i=1
Hi +
N ∑ 1 , rij
(2)
i̸=j
where Hi is the oneelectron Dirac Hamiltonian, including its kinetic energy and the interaction with the nucleus, Hi = c
3 ∑
α ⃗ k .⃗ pk + (βi − 1)c2 +
i=1
Z . ri
(3)
In (??), nc is the number of CSF, J and P are the total angular momentum and parity of the system, respectively; γr is a set of quantum numbers to specify CSF additional to JP , and {cr (α)} are the mixing coeﬃcients and denote the representation of the atomic state. The CSF γr P JM ⟩ are constructed from a product of a single electron wave function through a proper angular momentum coupling and antisymmetrization of the basis states. cr (α) and the radial orbitals are optimized simultaneously, based on the expectation values ⟨ψα  HDC ψα ⟩ of one or several atomic states in a self consistent ﬁeld (SCF) procedure. Besides the DiracCoulomb Hamiltonian, the Breit interaction also plays an important role in understanding the electronic structure of heavy atoms. The Breit interaction arises from the relativistic retardation and the currentcurrent interaction of fastmoving charges: ) ( N ∑ α ⃗ i .⃗ αj cos(ωij rij ) ( ⃗ ) ( ⃗ ) cos(ωij rij ) − 1 + α ⃗ i .∇i × α ⃗ j .∇j . (4) Htransverse = − 2r rij ωij ij i̸=j
This Hamiltonian can be also interpreted as the exchange of a single transverse photon. The Breit contributions are calculated in the low frequency limit (ωij → 0) by diagonalizing the DiracCoulombBreit Hamiltonian matrix. Another important contribution is the quantum electrodynamic (QED) contribution. The dominant QED contributions are selfenergy and vacuum polarization, which are also
VOL. 49
¨ ¨ ¨ GULDEM URER AND LEYLA OZDEMIR
1181
included in the computations of the transition energy. The ﬁnitenucleus eﬀect is taken into account by assuming an extended Fermi distribution for the nucleus. Both the Breit and QED contributions are treated as perturbation and are not included directly in the SCF procedure. Orbitals are here ﬁxed, but the mixing coeﬃcients are calculated by diagonalizing the modiﬁed Hamiltonian. The oscillator strength for the transition from ASF (Atomic State Function) Γi to ASF Γj can be written ⟨
⟩ 2 πc
ˆ (L) (5) fi→j = O Γ P J Γ P J .
j j j i i i 2 (2L + 1)ω ˆ (L) is multipole radiation ﬁeld operator of order L. This, in turn, can be expressed where O M in the terms of CSF matrix elements by
⟩ ⟨ ⟨ ⟩ ∑
ˆ (L)
ˆ (L) (6) crΓi crΓj γr Pr Jr O Γi Pi Ji O
Γj Pj Jj =
γ s P s Js , r,s
and this, in turn, as a sum of singleelectron transition integrals using
⟨ ⟩ ⟨ ⟩ ∑
ˆ (L)
ˆ (L) L dab (rs) na κa O nb κb , γr Pr Jr O γs Ps Js =
(7)
a,b
where dL ab are angular coeﬃcients. Thus ) (
⟨ ⟩ ( (2j + 1)ω )1/2 ja L jb
ˆ (L) b ja −1/2 ¯ ab , na κa O nb κb = M (−1) 1/2 0 −1/2 πc
(8)
¯ ab is one of the radiative transition integrals deﬁned by where M { e ¯ + GM ¯ I , for electric multipole transitions M ab ab ¯ Mab = m ¯ , for magnetic multipole transitions M ab [( ) [ ] ( L+1 )1/2 [ ]] + − ¯ e = −i L 1/2 (κa − κb )I + + (L + 1)I − M − (κ − κ )I − LI a b ab L+1 L+1 L−1 L−1 L+1 L {[ ] [ ] } , + − + − I ¯ Mab = −i (κa − κb )IL+1 + (L + 1)IL+1 + (κa − κb )IL−1 − LIL−1 − (2L + 1)JL , ¯ m = −iL+1 (2L+1)1/2 (κa + κb )I + , M ab L [L(L+1)] ∫∞ IL± = ∫ 0 drjL (ωr/c) (Pna κa (r)Qnb κb (r) ± Qna κa (r)Pnb κb (r)) , ∞ JL = 0 drjL (ωr/c) (Pna κa (r)Pnb κb (r) ± Qna κa (r)Qnb κb (r)) . (9) Here G is the gauge parameter; it takes the value [(L + 1)/L]1/2 in the Babushkin gauge.
III. RESULTS AND DISCUSSION
We have here presented a multiconﬁguration DiracFock calculation for the excitation energies of some lowlying levels and wavelengths, weighted oscillator strengths and transition rates (or probabilities) for the electric dipole (E1) and quadrupole (E2), and magnetic
1182
EXCITATION ENERGIES AND E1, E2, . . .
VOL. 49
dipole (M1) transitions between these levels in doubly ionized actinium. The M1 and E2 transitions combine states with same parity, whereas the E1 transitions combine states with diﬀerent parity. The calculation has been performed using the widelyused atomic structure package GRASP [10] based on the multiconﬁguration DiracFock method. The calculation of heavy atoms such as actinide atoms requires considering the Breit and quantum electrodynamic (QED) contributions besides the electron correlations. The MCDF code is known to be ﬂexible with regard to the shell structure and the computation of—quite diﬀerent types of—excitation and decay rates, but often suﬀers from the size of the wave function expansions which are needed to be treated explicitly [23]. Doubly ionized actinium has a rather simple electronic structure, with one electron moving in the resultant ﬁeld of the nucleus and the 86 inner electrons. Outer correlation is expected to be small, but this may not be true for corevalence correlation. In addition relativistic eﬀects must play a role. For heavy ions such as multiply ionized rareearth (lanthanides and actinides), the consideration of both intervalence (valencevalence) and corevalence correlation is essential for atomic structure calculations. For this reason, we have used four diﬀerent conﬁguration sets. We have considered the conﬁgurations including 6d, 7s, 7p, and 5f outside the [Rn] core according to the valence correlation. These conﬁgurations are given in Table I. In this table the conﬁguration sets are denoted by A, B, C, and D. The B, C, and D conﬁguration sets include the excitations from the 6p for oddparity, from the 6s for evenparity and from the 6s and 6p for both even and oddparity, respectively, whereas A includes only valence excitations. Table II displays the excitation energies of seven lowlying levels (3 for odd and 4 for evenparity) for the four conﬁguration sets given in Table I. The calculation of heavy atoms such as actinide atoms requires considering the Breit and quantum electrodynamic (QED) contributions besides the electron correlations. We have taken the conﬁgurations 7s, 6d, 7p, and 5f outside the [Rn] core for considering valence correlation. MCDF method has been widely used for studying the electronic structure of heavy elements such as actinium. This method has been found useful for providing accurate results. The MCDF code is known to be ﬂexible with regard to the shell structure and the computation of—quite diﬀerent types of—excitation and decay rates, but often suﬀers on the size of the wave function expansions which needed to be treated explicitly [23]. In Table II the columns represent the MCDF energies, E0 , MCDF plus Breit, E1 , and plus quantum electrodynamic, E2 , contributions, respectively. Comparison values have been placed into the last column. In addition, only oddparity levels are indicated with the superscript “o ” in the table. The corevalence correlation could be considered by opening the 6s2 and 6p6 shells in the B, C, and D calculations. It is seen that some MCDF energy values, except for the 5f levels, are in agreement when compared with others. Energy values including the Breit and QED contributions obtained from the calculation A for the 6d levels are often good. The levels 5f from the B and C calculation are somewhat good. For the 7p levels, the C calculation results are in agreement with the others. Especially the ﬁrst excited level, the agreement between our results and the values from the DiracCoulomb DiracCoulombBreit CCSD method by Eliav et al. [5] is good. Also, other high excited levels are convenient with those of the allorder SD manybody method by Safronova et al. [3]. For improving the results,
VOL. 49
¨ ¨ ¨ GULDEM URER AND LEYLA OZDEMIR
1183
TABLE II: MCDF energies, E0 , MCDF plus Breit, E1 , and plus quantum electrodynamic, E2 , −1 contributions, (in(incm for Ac AcIII. III. contributions, cm )) for E0 E1 E2 Other Works 0.00 0.00 0.00 0.00a, b, c, d, e A 1519.88 1573.63 1581.21 825 a, 801.0 b, c, 1266.28d, 1169.49d, B 1108.64 1148.63 1160.67 798.48e C 1941.16 2012.66 2022.66 D 457.53 423.59 436.795 6d 2D5/2 A 4064.74 4031.69 4039.28 4041 a, 4203.9 c, 4742.50d, 4589.25d, B 3674.53 3614.44 3626.50 4202.11e C 4419.93 4386.24 4396.27 D 3090.48 2966.98 2980.20 7p 2P1/2 A 28284.56 28249.99 28252.07 29303 a, 29466 b, 29465.9 c B 27020.54 26923.58 26930.02 C 29292.85 29322.47 29324.37 D 28047.03 28199.80 28204.52 5f 2F5/2 A 32926.88 33113.24 33120.83 24018 a, 23454 b, 23454.5 c B 27882.17 27981.72 27993.73 C 33480.04 33646.09 33656.17 D 29194.20 29293.62 29306.87 5f 2F7/2 A 34747.91 34807.32 34814.91 26420 a, 26080.2 c B 30130.42 30105.92 30118.10 C 35265.28 35297.09 35307.16 D 31152.24 31121.66 31134.94 7p 2P3/2 A 35933.73 35802.14 35810.00 37816 a, 38063.0 c B 35101.56 34905.61 34917.92 C 36806.53 36692.45 36702.89 D 35449.09 35475.97 35489.55 a Safronova et al. [3], b Brewer [4], c ACTINIDES web site [9], d Eliav et al. [5] (converted from eV a Safronova et al. [3], b Brewer [4], c ACTINIDES web site [9], d Eliav et al. [5] (converted from eV unit), e Moor [in 5] (converted from eV unit) Levels 7s 2S1/2 6d 2D3/2
it is necessary that there be selected the conﬁgurations including more ﬁlled 6s and 6p orbitals. In this case, the number of possible interacting conﬁgurations to be introduced in the model is rapidly increasing, and the computer limits quickly impose severe restrictions on that approach. The electric dipole transition parameters, such as wavelengths, λ, weighted oscillator strengths, gf value, and weighted transition rates (or probabilities), gA, for Ac III are given in Table III. It is well known that the strongest transition is the electric dipole, E1, transition. For this reason the E1 transitions are labeled as allowed, whereas higher order electric and magnetic transitions are referred to as forbidden. We have obtained eight electric dipole transitions. Our results in this table do not include the transverse Breit and quantum electrodynamic contributions. In this table, the wavelengths for all transitions, except the 6d−5f transitions are seen to have a good agreement when compared with other works. The comparing values from [7] have been converted from log(gf ) values. Also, the
1184
EXCITATION ENERGIES AND E1, E2, . . .
VOL. 49
TABLE III: Wavelengths, λ (in ˚ A), weighted oscillator strengths, gf value, and weighted transition rates, gA (in s−1 ), for E1 transitions in Ac III. In this table, a(±b) denotes a×10±b . Ȝ (Å) Transitions
This Work
E1 Transitions: 7s 2S1/2  7p 2P3/2
6d
2
D3/2  7p
2
P3/2
6d
2
D5/2  7p
2
P3/2
7s
2
S1/2  7p
2
6d
2
D3/2  7p
2
6d
2
D3/2  5f
2
F5/2
6d
2
D5/2  5f
2
F7/2
6d
2
D5/2  5f
2
F5/2
P1/2
P1/2
A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D
2782.900 2848.875 2716.908 2820.946 2905.807 2941.789 2868.175 2857.831 3137.846 3181.973 3087.696 3090.368 3535.496 3700.887 3413.802 3565.438 3736.266 3859.230 3656.081 3624.566 3184.003 3735.032 3170.689 3479.873 3259.116 3779.876 3241.979 3563.569 3464.746 4130.926 3441.143 3830.871
Other Works 2626.44 a, b, c, e
2682.902a, b, c, e
2952.553 a, b, e
3392.78 a, b, c, e
3487.589 a, b, c, e
4413.089 a, b, e
4569.876 a, b, e
5193.21a, c, e, 5193.197b
gfvalue This Other Work Works 2.133 1.787 2.012 1.820 0.149 0.108 0.119 0.153 1.416 1.016 1.160 1.443 0.894 0.763 0.845 0.778 0.740 0.506 0.623 0.759 1.058 0.577 0.934 0.910 1.573 0.943 1.388 1.347 0.069 0.038 0.061 0.058
0.20 b 1.69c 1.6 c 0.88 b 0.129c 0.13 c 0.03 b 1.22c 1.1c 0.21 b 0.675c 0.62c 0.30b 0.591c 0.50c 0.16 b 0.322c 0.69c 0.02 b 0.504c 0.95c 1.38 b 0.0202c 0.042c
gA (s1) This Work
Other Works
2.755(9) 2.203(9) 2.728(9) 2.288(9) 1.770(8) 1.248(8) 1.458(8) 1.872(8) 14.394 10.044 12.180 15.126 4.774(8) 3.718(8) 4.836(8) 4.086(8) 3.540(8) 2.268(8) 3.110(8) 3.856(8) 6.966(8) 2.760(8) 6.198(8) 5.010(8) 9.880(8) 4.400(8) 8.808(8) 7.072(8) 3.84(7) 1.50(7) 3.42(7) 2.64(7)
1.55(8)b 1.63(9)c 1.55(9)c 1.22(8)b 1.19(8) 1.23(8)c 8.23(8) b 9.30(8) c 8.23(8) c 3.59(8) b 3.91(8) c 3.59(8) c 2.78(8) b 3.24(8) c 2.75(8) c 2.35(8) b 1.10(8) c 2.34(8) c 3.02(8) b 1.61(8) c 3.02(8) c 1.03(7) b 4.99(6) c 1.03(6) c
a
NIST Atomic Spectra Database [8] (The values are taken from Meggers et al. [2], at the same time the values can be found in reference [1 and 7] except for the 5193.21 value), b Bi´emont et al. [7] (converted from log(gf ) value), c Safronova et al. [3].
weighted transition rates seem good. Only the weighted transition rates for the 6d − 5f transitions are somewhat poor. In Table IV we have considered the electric quadrupole, E2, and the magnetic dipole M1 transitions between some lowlying levels. In this table, the wavelength values belonging to the B calculation are not poor. There are no oscillator strength values for the E2 and M1 transitions in the literature. The transition rates are also not good except for the 6d − 7s transitions. But it is noted that the transition parameters for the E2 and M1 transitions,
VOL. 49
¨ ¨ ¨ GULDEM URER AND LEYLA OZDEMIR
1185
TABLE IV: Wavelengths, λ (in ˚ A), weighted oscillator strengths, gf value, and transition rates, A (in s−1 ), for E2 and M1 transitions in Ac III. In this table, a(±b) denotes a×10±b . Ȝ M1 transitions in Ac III. In this table, a(±b) denotes a×10 Ȝ (Å) Transitions Other This Works Work E2 Transitions: 6d 2D3/2  7s 2S1/2 A 65794.55 124844a B 90200.20 C 51515.45 D 218564.56 6d 2D5/2  7s 2S1/2 A 24601.79 B 27214.36 C 22624.77 D 32357.34 6d 2D5/2  6d 2D3/2 A 39294.87 23787.4a B 38972.90 C 40342.60 D 37980.09 5f 2F5/2  7p 2P1/2 A 21540.98 B 116058.83 C 23882.34 D 87170.98 7p 2P3/2  7p 2P1/2 A 13073.32 B 12374.67 C 13309.05 D 13509.76 5f 2F7/2  5f 2F5/2 A 54913.99 B 44479.09 C 56014.94 D 51071.69 7p 2P3/2  5f 2F5/2 A 33257.41 B 13851.58 C 30061.71 D 15987.50 7p 2P3/2  5f 2F7/2 A 84329.90 B 20116.10 C 64882.32 D 23272.84 M1 Transitions: 7s 2S1/2  6d 2D3/2 A 65794.55 124844a B 90200.20 C 51515.45 D 218564.5 6d 2D3/2  6d 2D5/2 A 39294.87 B 38972.90 C 40342.60 D 37980.09 7p 2P1/2  7p 2P3/2 A 13073.32 B 12374.67 C 13309.05 D 13509.76 5f 2F5/2  5f 2F7/2 A 54913.99 B 44479.09 Ȝ C 56014.94 D 51071.69 Work Works 5f 2F5/2  7p 2P3/2 A 33257.41 B 13851.58 C 15987.50 D 30061.71 a a Safronova et al. [3]
Safronova et al. [3]
. A (s1)
gfvalue Other Works
This Work 6.203(11) 1.962(11) 1.001(10) 1.556(12) 1.951(9) 1.187(9) 1.968(9) 7.866(10) 9.901(11) 8.506(11) 7.042(11) 1.067(10) 1.201(9) 7.363(12) 6.408(10) 1.441(11) 3.457(8) 4.020(8) 2.372(8) 2.263(8) 7.737(12) 1.096(11) 5.879(12) 5.964(12) 8.713(11) 1.155(9) 8.432(11) 6.291(10) 3.693(11) 2.591(9) 5.817(11) 1.394(9) 1.290(17) 1.447(17) 2.104(17) 4.142(18) 2.482(7) 2.462(7) 2.397(7) 2.547(7) 3.963(7) 4.182(7) 3.864(7) 3.823(7) 2.521(7) 3.108(7) 2.471(7) 2.710(7) 6.751(18) 2.687(15) 5.673(15) 1.049(17)

2.389(5) 4.021(6) 6.294(5) 5.433(8) 3.583(3) 1.781(3) 4.276(3) 8.352(4) 7.128(5) 6.226(5) 4.810(5) 8.228(5) 2.879(3) 6.077(7) 1.249(3) 2.108(6) 3.373(1) 4.378(1) 2.233(1) 2.068(1) 2.139(6) 4.622(6) 1.562(6) 1.906(6) 1.313(4) 1.004(2) 1.555(4) 4.104(3) 8.660(6) 1.067(2) 2.304(5) 4.292(3)











Works 
Other Works
This Work
4.972(12) 2.966(12) 1.322(11) 1.446(13) 1.816(1) 1.773(1) 1.637(1) 1.963(1) 3.866 4.554 3.637 3.493 6.971(2) 1.309(1) 6.568(2) 8.665(2) 1.017(11) 2.335(8) 3.701(8) 1.936(11)
1.053(6) a

3.696(3) a





4.354(13)a



Works 
1186
EXCITATION ENERGIES AND E1, E2, . . .
VOL. 49
even the E1 transitions, in the literature are very fragmentary and scarce. Experimental work for actinides is very diﬃcult because of their radioactivity. Spectroscopic parameters such as oscillator strengths, transition rates (probabilities) are fundamental characteristics of excited states of atoms and ions. Therefore the values presented here should be very useful in the ﬁelds of quantum electronics, atomic and laser spectroscopy, plasma physics, and astrophysics.
IV. CONCLUSION
In summary, in this work, the calculation on the level structure of Ac III (Z = 89) has been performed using the MCDF atomic code, GRASP [10]. The excitation energies of levels and electric dipole and quadrupole, and magnetic dipole transition parameters between these levels have been investigated in the framework of the transverse Breit and quantum electrodynamic (QED) contributions and presented in tables. We think that the results obtained from this work will be useful for the theoretical and experimental studies for the level structure of doubly ionized actinium in the future.
Acknowledgements The authors are very grateful to the anonymous reviewer for stimulating comments and valuable suggestions, which resulted in improving the presentation of the paper.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
W. F. Meggers, Spectrochim. Acta 10, 1957 (1957). W. F. Meggers, M. Fred, and F. S. Tomkins, J. Res. Nat. Bur. Stand. 58, 297 (1957). U. I. Safronova, W. R. Johnson, and M. S. Safronova, Phys. Rev. A 76, 042504 (2007). L. Brewer, J. Opt. Soc. Am. 61, 1666 (1971). E. Eliav, S. Shmulyian, and U. Kaldor, Journal of Chemical Physics 109, 3954 (1998). W. Liu, W. K¨ uchle, and M. Dolg, Phys Rev. A 58, 1103 (1998). E. Bi´emont, V. Fivet, and P. Quinet, J. Phys B: At. Mol. Opt. Phys. 37, 4193 (2004). http://www.nist.gov/pml/data/asd.cfm. J. Blaise and J. Wyart, Selected constants: Energy levels and Atomic Spectra of Actinides, http://www.lac.upsud.fr/Database/Contents.html. K. G. Dyall, I. P. Grant, C. T. Johnson, F. A. Parpia, and E. P. Plummer, Comp. Phys. Commun. 55, 425 (1989). ¨ ¨ L. Ozdemir and G. Urer, Acta Phys. Pol. A 118, 563 (2010). ¨ ¨ G. Urer and L. Ozdemir, The Arabian Journal for Science and Engineering AScience (DOI: 10.1007/S1336901101545). ¨ ¨ G. Urer , L. Ozdemir, and B. Kara¸coban, Balkan Phys. Lett. 18, 383 (2010). ¨ ¨ B. Kara¸coban, L. Ozdemir, and G. Urer, Balkan Phys. Lett. 18, 393 (2010). ¨ B. Kara¸coban and L. Ozdemir, J. Quant. Spectrosc. Radiat. Transfer. 109, 1968 (2008).
VOL. 49
¨ ¨ ¨ GULDEM URER AND LEYLA OZDEMIR
1187
¨ [16] B. Kara¸coban and L. Ozdemir, Acta. Phys. Pol. A 113, 1609 (2008). ¨ [17] B. Kara¸coban and L. Ozdemir, Cent. Eur. J. Phys. 10, 124 (2012). ¨ [18] B. Kara¸coban and L. Ozdemir, The Arabian Journal for Science and Engineering AScience, 36, 635 (2011). [19] I. P. Grant, Advan. Phys. 19, 747 (1970). [20] I. P. Grant, J. Phys. B 7, 1458 (1974). [21] I. P. Grant, Computational Chemistry (Vol 2) ed. S. Wilson (Plenum, New York, USA, 1988). [22] I. P. Grant, Relativistic Quantum Theory of Atoms and Molecules (Springer, New York, USA, 2007). [23] S. Fritzsche, C. Z. Dong, F. Koike, and A. Uvarov, Eur. Phys. J. D 45, 107 (2007).