Astrophys Space Sci (2016) 361:229 DOI 10.1007/s10509-016-2795-z
O R I G I N A L A RT I C L E
E1 and E2 transitions for Fe XVI, Co XVII and Ni XVIII G. Çelik1 · S. ¸ Ate¸s1
Received: 21 January 2016 / Accepted: 21 May 2016 © Springer Science+Business Media Dordrecht 2016
Abstract Electric dipole and electric quadrupole transition data for sodium-like iron, cobalt and nickel have been calculated within the weakest bound electron potential model (WBEPM) theory using experimental energy levels and theoretical expectation values of orbital radii corresponding to those energy levels under the assumption of the LS coupling scheme. The results obtained from this study provide theoretical transition probability and oscillator strength data requested in many fields of researches, especially astrophysics. The calculated transition data results have been compared with available data in the literature. The present results are consistent with earlier calculations. Some new electric quadrupole transition probability values not existing in the data bases, especially for iron have been obtained using this method. Keywords Electric dipole transition · Electric quadrupole transition · WBEPM theory · Sodium-like Fe · Co · Ni
1 Introduction The transitions not allowed by the selection rules of the electric dipole transitions, called as forbidden transitions have a small probability of their spontaneous occurrence. The electric quadrupole (E2) transitions are essential for the diagnostics of the astrophysical and laboratory plasmas. A number of such transitions have arisen in the ultraviolet spectrum of the solar corona. Additionally, forbidden lines of low excitation energy belonging to alkali-like ions have
been observed in spectra of many gaseous nebulae (Charro and Martin 2001). Experimental data for forbidden lines are scarce in the literature, since experimental determination of electric quadrupole transition probability is difficult. In this case, accurate theoretical estimates for forbidden transitions gain importance, and are necessary. The energy levels of highly charged sodium like ions are practically free from effects of configuration mixing. For this reason, the spectra of these ions have a simple structure. As a result of this case, they are well suited for a theoretical interpretation of line intensities (Younis et al. 2006). Highly charged sodiumlike ions have been observed in several types of laboratory sources such as high-voltage vacuum spark tokamak and laser-produced plasmas. These ions have prominent emission lines in the UV and XUV spectrum of the sun (Younis et al. 2006). Emission lines arising from outer-shell transitions in these ions are widely detected in astrophysical objects using present ground and space observatories (Liang et al. 2009). The detailed study of these astronomical objects requires the knowledge of electric quadrupole transition probabilities. Additionally, the determination of electric quadrupole oscillator strengths can help in assessing the contribution of electric quadrupole radiation to multiphoton transitions (Charro and Martin 2003a, 2003b). In this article, we present the electric dipole (E1) and electric quadrupole (E2) transition data results belonging to excited levels of sodium-like iron, cobalt and nickel by using the WBEPM theory.
2 Theory
B
G. Çelik
[email protected]
1
Department of Physics, Faculty of Science, Selçuk University, Campus 42049 Konya, Turkey
The basic idea of the WBEPM theory is the separation of electrons in two parts to be the weakest bound electron (WBE) and the non-weakest bound electrons (NWBE). The
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G. Çelik, S. ¸ Ate¸s
WBE which is the most weakly bound electron to the system as compared with the other electrons in a given system is the electron excited or ionized most easily. So, many atomic qualities such as rates of radiative decay, excitation, ionization etc. can be expressed by the behavior of the WBE (Zheng et al. 2000a). In this theory, complex manyelectron system can be simplified as one-electron problem and so can be solved easily (Zheng et al. 2000a). Solving the one-electron Schrödinger equation for WBE, it is obtained the wave function of WBE. The radial wave function of the WBE can be expressed by means of general Laguerre polynomial and the parameters determined from experimental energy and the expectation values of radii for levels as (Zheng et al. 1999, 2000a, 2000b, 2000c, 2002) ∗ 2Z r Z∗ r ∗ 2l ∗ +1 (1) Rn∗ l ∗ (r) = C exp − ∗ r l Ln−l−1 n n∗ ∗
∗
2l +1 2Z r where Ln−l−1 ( n∗ ) is the general Laguerre polynominal, and the normalization constant C is given to be (Zheng et al. 1999, 2000a)
C=
2Z ∗ n∗
l ∗ +3/2
∗ −1/2 2n∗ ∗ Γ n −l +1 (n − l − 1)!
(2)
The energy eigenvalue of the WBE in a.u. is ε=−
Z ∗2 2n∗2
(3)
64π 4 e2 a02 (EJ − EJ )3 SE1 3h(2J + 1)
32π 5 αca04 1.11995 × 1018 SE2 = SE2 . 5 15λ g λ5 g
ni , li |r k |nf , lf
∞ = r k+2 Rni li (r)Rnf lf (r)dr
nf +ni +lf +li
= (−1)
(4)
(5)
Here, e is electron charge (esb), a0 is Bohr radius (cm), h is Planck constant (erg.s). (EJ –EJ ) is the energy difference between relevant levels in Kaysers (cm−1 ), (2J + 1) is the degeneracy of initial level and SE1 is the electric dipole line strength in atomic units of e2 a02 .
(6)
Here, α is the fine structure constant, λ is the wavelength in angstrom units, c is the speed of light (cm/s), a0 is the Bohr radius (cm), g is the degeneracy of the initial level and SE2 is the E2 line strength in atomic units. Line strength is determined according to the coupling schemes in atomic or ionic systems. LS coupling is the dominant coupling scheme in many light atoms. The radial wave functions of weakest bound electron can be produced easily by using Z ∗ , n∗ , l ∗ parameters obtained from expectation values of radii and experimental ionization energy data. The radial transition integral for transition between any two levels is given as (Zheng et al. 2000c)
×
where n∗ is an effective principal quantum number, l ∗ is an effective angular momentum quantum number. The introduction of d effectively modifies the integer quantum numbers n and l into non-integers n∗ = n + d and l ∗ = l + d. The d and Z ∗ parameters are obtained by solving the Eq. (3) and Eq. (4) simultaneously (Zheng et al. 1999, 2000a, 2000b, 2000c, 2002). The transition probabilities designate the statistical probability that an electron will spontaneously drop to a lower electronic state from an upper electronic state. The total electric dipole transition probability (in s−1 ) from γ J M to all M levels of γ J is given (Cowan 1981) AE1 =
AE2 =
0
and the expectation values for radius of the WBE r is 3n∗2 − l ∗ (l ∗ + 1) r = 2Z ∗
The electric quadrupole transition probability AE2 (in s−1 ) for a transition between two states within the LS coupling in the customary manner is given by (Majumder et al. 2004; Çelik et al. 2012a, 2012b)
n∗f
×
×
−
Zi∗ n∗i
2Zf∗
l ∗ f
n∗f
2Zi∗ n∗i
l ∗ i
−l ∗ −l ∗ −k−3 f
i
x
∗ ∗ n∗4 f Γ (nf + lf + 1)
−1/2
4Zf∗3 (nf − lf − 1)!
×
Zf∗
∗ ∗ n∗4 i Γ (ni + li + 1)
4Zi∗3 (ni − li − 1)!
∗ (−1)m2 Zf m1 !m2 ! n∗f
−
−1/2
x nf −lf −1 ni −li −1
×
Zi∗ m1 +m2 n∗i
m1 =0
m2 =0
Zi∗ −m1 −m2 × + ∗ x n∗f ni × Γ lf∗ + li∗ + m1 + m2 + k + 3
S li∗ − lf∗ + k + m2 + 1 × n∗f − lf∗ − 1 − m1 − m3
Zf∗
m3 =0
lf∗ − li∗ + k + m1 + 1 × n∗i − li∗ − 1 − m2 − m3 ∗ li + lf∗ + k + m1 + m2 + m3 + 2 × m3
(7)
E1 and E2 transitions for Fe XVI, Co XVII and Ni XVIII
Where S = min{nf − lf − 1 − m1 , ni − li − 1 − m2 } and k > −lf∗ − li∗ − 3, Γ () is the gamma function. In Eq. (7), k = 1 for electric dipole transition integrals and k = 2 for electric quadrupole transition integrals are put in theirs place.
3 Results and discussion We calculated the transition probabilities and oscillator strengths for E1 transitions and E2 transition probabilities for forbidden transitions of present three iso-electronic sodium-like systems using the WBEPM theory. In the WBEPM theory, the Z ∗ , n∗ and l ∗ parameters are required for the transition probability calculations. These parameters determined using experimental energy values and expectation values of radii are used in Eqs. (3)–(7) to obtain the transition probabilities and oscillator strengths for all these ions. In the calculations, while the experimental energy values are taken from NIST (Kramida et al. 2015), the expectation values of radii are determined using NCA wave functions (Lindgard and Nielsen 1975). The electric dipole transition probability and oscillator strength results for Co XVII and Ni XVIII have been presented in Table 1 and for Fe XVI in Table 2. Calculated oscillator strength results for Co XVII and Ni XVIII have been compared with the accepted values given by NIST, and some theoretical comparisons for Fe XVI are made with the Multi-configurational Hartree–Fock (MCHF) results given by Froese Fischer (2002), the relativistic Breit–Pauli Rmatrix (BPRM) method results, the relativistic coupled cluster method (RCCM) results, the configuration interaction atomic structure calculation results with code SUPERSTRUCTURE (SS) given by Nahar et al. (2009), the relativistic Dirac–Fock results obtained from GRASP2 given by Aggarwal and Keenan (2007) and the accepted values given by NIST (Kramida et al. 2015). According to these comparisons, it can be seen that the average agreements of our E1 transition probability values for Co XVII and Ni XVIII are within ±3 % to the accepted values given by NIST. These agreements for Fe XVI are within ±7 % to MCHF results, ±3–6 % to the BPRM, RCCM, SS results given by Nahar (2009), ±3 % to GRASP2 code results and ±5 % to the accepted values given by NIST. Present E2 transition probability results for Fe XVI, Co XVII and Ni XVIII are displayed in Table 3, Table 4 and Table 5, respectively. Since the observation of E2 transitions are difficult, the experimental results for the transitions considered in this work are not available. However, some comparisons in Table 3 are made with theoretical results such as results of the all-order many-body coupledcluster method using fully relativistic Dirac–Hartree–Fock (DHF) orbitals given by Ray (2002), the relativistic transition probability data given by Fuhr et al. (1988), the
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Multi-configurational Hartree–Fock (MCHF) results given by Froese Fischer (2002), the Breit–Pauli approximation results using the code SUPERSTRUCTURE given by Nahar et al. (2009) and the accepted values given by NIST (Kramida et al. 2015). The NIST data include the HartreeFock self-consistent field method results given by Krueger and Czyzak (1965) and the results obtained using HartreeFock orbital wave-functions of frozen-core type given by Tull et al. (1972). As a result of these comparisons, it can be seen that the average agreement of our E2 transition probability values is within ±2 % to the results given by Ray (2002), the accepted values given by NIST, the data given by Fuhr et al. (1988), within ±8 % to the MCHF results given by Froese Fischer (2002) and within ±7 % to the SUPERSTRUCTURE results given by Nahar et al. (2009). The comparisons in Table 4 and Table 5 are made with the all-order many-body coupled-cluster method results given by Ray (2002) and the accepted values given by NIST (Kramida et al. 2015). The average agreement of our transition probability values is within ±4 % to the results reported by Ray (2002) and within ±2 % to the accepted values given by NIST for Co XVII and these rates is within ±3 % to the Ray’s results and ±2 % to the accepted values given by NIST for Ni XVIII. Thus, when the spectroscopic transition data values in the tables have been compared, it can be said to be in close agreement of all the values with the literature for both E1 transitions and E2 transitions. In the WBEPM theory, the fitting parameters such as energies and orbital radii used in the determination of Z ∗ , n∗ and l ∗ parameters affect precision of wave functions used for the atomic structure calculations. These parameters depend on the quantum defects and the expectation values of radii of states. Some discrepancies in the transition probabilities and oscillator strengths observed in the results of comparisons in tables can be based on the different atomic energies of levels and the precisions in the expectation values of radii of states. More accurate E1 and E2 transition data can be obtained using high precision expectation values of radii corresponding to the levels and energy levels in the WBEPM theory framework.
4 Conclusion Because of the developments in astrophysics and astronomic fields for the estimation of stellar chemical composition, and the difficulties in observation of E2 transitions, the accurate theoretical transition data are requested. In this work, a good agreement is observed for Fe XVI, Co XVII and Ni XVIII ions. There are some values for E2 transitions in Fe XVI which are not available elsewhere in the literature. On the other hand, most of the transition data have already been
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Table 1 Oscillator strengths and transition probabilities (s−1 ) for E1 transitions for Co XVII and Ni XVIII 1.ST Level
2.ND Level
Terms
2s(2)2p(6)(S1)3s(1) 2s(2)2p(6)(S1)3s(1) 2s(2)2p(6)(S1)3s(1) 2s(2)2p(6)(S1)3s(1) 2s(2)2p(6)(S1)3s(1) 2s(2)2p(6)(S1)3s(1)
2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)5p(1) 2s(2)2p(6)(S1)5p(1)
2S 2P
2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1)
2s(2)2p(6)(S1)4s(1) 2s(2)2p(6)(S1)4s(1) 2s(2)2p(6)(S1)5s(1) 2s(2)2p(6)(S1)5s(1)
2P 2S
2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1) 2s(2)2p(6)(S1)3p(1)
2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)4d(1) 2s(2)2p(6)(S1)4d(1) 2s(2)2p(6)(S1)4d(1) 2s(2)2p(6)(S1)5d(1) 2s(2)2p(6)(S1)5d(1) 2s(2)2p(6)(S1)5d(1)
2P 2D
2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1)
2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)5p(1) 2s(2)2p(6)(S1)5p(1) 2s(2)2p(6)(S1)5p(1)
2D 2P
2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1) 2s(2)2p(6)(S1)3d(1)
2s(2)2p(6)(S1)4f(1) 2s(2)2p(6)(S1)4f(1) 2s(2)2p(6)(S1)4f(1) 2s(2)2p(6)(S1)5f(1) 2s(2)2p(6)(S1)5f(1) 2s(2)2p(6)(S1)5f(1)
2D 2F
2s(2)2p(6)(S1)4s(1) 2s(2)2p(6)(S1)4s(1) 2s(2)2p(6)(S1)4s(1) 2s(2)2p(6)(S1)4s(1)
2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)5p(1) 2s(2)2p(6)(S1)5p(1)
2S 2P
2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)4p(1) 2s(2)2p(6)(S1)4p(1)
2s(2)2p(6)(S1)4d(1) 2s(2)2p(6)(S1)4d(1) 2s(2)2p(6)(S1)4d(1) 2s(2)2p(6)(S1)5d(1) 2s(2)2p(6)(S1)5d(1) 2s(2)2p(6)(S1)5d(1)
2P 2D
2s(2)2p(6)(S1)4d(1) 2s(2)2p(6)(S1)4d(1) 2s(2)2p(6)(S1)4d(1)
2s(2)2p(6)(S1)5p(1) 2s(2)2p(6)(S1)5p(1) 2s(2)2p(6)(S1)5p(1)
2D 2P
2s(2)2p(6)(S1)4d(1) 2s(2)2p(6)(S1)4d(1) 2s(2)2p(6)(S1)4d(1)
2s(2)2p(6)(S1)5f(1) 2s(2)2p(6)(S1)5f(1) 2s(2)2p(6)(S1)5f(1)
2D 2F
2S 2P 2S 2P 2S 2P 2S 2P 2S 2P
2P 2S 2P 2S 2P 2S
2P 2D 2P 2D 2P 2D 2P 2D 2P 2D 2P 2D 2P 2D 2P 2D
2D 2P 2D 2P 2D 2P 2D 2P 2D 2P
2D 2F 2D 2F 2D 2F 2D 2F 2D 2F
2S 2P 2S 2P 2S 2P
2P 2D 2P 2D 2P 2D 2P 2D 2P 2D
2D 2P 2D 2P
2D 2F 2D 2F
J1
J2
Co XVII This work A
f
NIST
Ni XVIII This work A
f
NIST
0.5 0.5 0.5 0.5 0.5 0.5
1.5 0.5 1.5 0.5 1.5 0.5
9.003E+09 6.84E+09 2.51E+11 2.43E+11 1.50E+11 1.56E+11
0.284 0.118 0.155 0.0756 0.0490 0.0256
0.262 0.119 0.1459 0.0786 0.0522 0.0245
9.83E+09 7.35E+09 3.65E+11 3.08E+11 1.89E+11 1.97E+11
0.251 0.113 0.160 0.0784 0.0502 0.0263
0.253 0.114 0.1498 0.08151 0.05052 0.02535
1.5 0.5 1.5 0.5
0.5 0.5 0.5 0.5
2.43E+11 1.17E+11 1.10E+11 5.69E+10
0.0589 0.0548 0.0117 0.0119
0.0920 0.01788 0.01176 0.0122
2.99E+11 1.51E+11 1.32E+11 6.87E+10
0.0583 0.0570 0.0114 0.0116
0.08982 0.01742 – –
1.5 1.5 0.5 1.5 1.5 0.5 1.5 1.5 0.5
1.5 2.5 1.5 1.5 2.5 1.5 1.5 2.5 1.5
2.76E+09 1.70E+10 1.69E+10 9.50E+10 5.72E+11 4.93E+11 5.30E+10 3.19E+11 2.74E+11
2.58E−02 2.35E−02 2.79E−01 0.0344 0.311 0.348 0.0103 0.0926 0.104
2.6E−02 2.37E−02 2.78E−01 0.0326 0.2932 0.3112 0.00988 0.0900 0.1027
2.92E+09 1.81E+09 1.83E+10 1.21E+11 7.27E+11 5.17E+11 6.70E+10 4.03E+11 3.48E+11
2.45E−02 2.23E−01 2.66E−01 0.0357 0.322 0.298 0.0105 0.0948 0.107
2.46E−02 2.25E−01 2.67E−01 0.0337 0.3010 0.3166 0.0104 0.09408 0.1046
1.5 1.5 2.5 1.5 1.5 2.5
1.5 0.5 1.5 1.5 0.5 1.5
9.35E+09 9.81E+10 8.48E+10 3.53E+10 3.68E+10 3.20E+10
0.00635 0.0337 0.0385 9.92E−04 0.00518 0.00601
0.00620 0.0303 0.0368 0.0010 0.00508 0.00601
1.13E+10 1.19E+11 1.03E+11 4.25E+09 4.44E+10 3.85E+10
0.00606 0.0324 0.0369 9.47E−04 0.00498 0.00575
0.00590 0.0299 0.0345 9.6E−04 0.0047 0.0058
1.5 2.5 2.5 1.5 2.5 2.5
2.5 2.5 3.5 2.5 2.5 3.5
1.21E+12 8.38E+10 1.26E+12 4.51E+11 3.20E+10 4.81E+11
0.942 0.0437 0.899 0.174 0.00824 0.165
0.930 0.0443 0.90 0.1696 0.00825 0.1625
1.52E+12 1.05E+11 1.62E+12 5.64E+11 4.01E+10 6.01E+11
0.944 0.0438 0.901 0.173 0.00824 0.165
0.93 0.04421 0.89 0.1692 0.00803 0.1619
0.5 0.5 0.5 0.5
1.5 0.5 1.5 0.5
2.01E+09 1.56E+09 5.08E+10 4.97E+10
0.374 0.171 0.168 0.0832
0.37 0.16 0.169 0.0853
2.24E+01 1.69E+01 6.57E+02 6.36E+02
0.362 0.164 0.177 0.0863
0.342 0.16 0.177 0.0883
1.5 1.5 0.5 1.5 1.5 0.5
1.5 2.5 1.5 1.5 2.5 1.5
5.99E+08 3.72E+09 3.65E+09 1.46E+10 8.69E+10 8.22E+10
0.0415 0.378 0.442 0.0273 0.243 0.300
0.039 0.35 0.41 0.0280 0.257 0.289
6.44E+08 4.02E+09 4.02E+09 2.06E+10 1.11E+11 8.95E+10
0.0395 0.361 0.425 0.0312 0.252 0.264
0.036 0.33 0.39 0.029 0.27 0.30
1.5 1.5 2.5
1.5 0.5 1.5
4.29E+09 4.47E+10 3.50E+10
0.0140 0.0741 0.0851
0.014 0.0713 0.0854
4.79E+09 4.62E+10 4.68E+10
0.0123 0.0603 0.0806
0.013 0.069 0.081
1.5 2.5 2.5
2.5 2.5 3.5
1.98E+11 1.40E+10 2.13E+11
0.728 0.0346 0.698
0.699 0.0345 0.72
2.50E+11 1.77E+10 2.67E+11
0.735 0.0349 0.70
0.73 0.035 0.71
E1 and E2 transitions for Fe XVI, Co XVII and Ni XVIII
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Table 2 Electric dipole transition probabilities (×109 s−1 ) for Fe XVI 1.ST Level
2.ND Level
Terms
Ji
Jf
This work
MCHF
NIST
RCCM
GRASP2
BPRM
SS
1s(2)2s(2)2p(6)(S1)3s(1) 1s(2)2s(2)2p(6)(S1)3s(1) 1s(2)2s(2)2p(6)(S1)3s(1) 1s(2)2s(2)2p(6)(S1)3s(1) 1s(2)2s(2)2p(6)(S1)3s(1) 1s(2)2s(2)2p(6)(S1)3s(1)
1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)5p(1) 1s(2)2s(2)2p(6)(S1)5p(1)
2S 2P
0.5 0.5 0.5 0.5 0.5 0.5
1.5 0.5 0.5 1.5 0.5 1.5
7.94 6.33 189 194 10400 106
7.374 6.335 206.7 195.6 12080 121.4
7.47 6.38 198 186 12000 110
7.693 6.134 207 195 9233 88.3
8.087 6.463 196.9 185.8 11370 108.5
7.78 6.20 205 193 11200 107
7.86 6.29 196 188 10400 102
1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1)
1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)4d(1) 1s(2)2s(2)2p(6)(S1)4d(1) 1s(2)2s(2)2p(6)(S1)4d(1) 1s(2)2s(2)2p(6)(S1)5d(1) 1s(2)2s(2)2p(6)(S1)5d(1)
2P 2D
1.5 1.5 0.5 1.5 1.5 0.5 1.5 0.5
1.5 2.5 1.5 1.5 2.5 1.5 2.5 1.5
2.64 15.9 15.5 73.7 443 310 249 185
2.583 15.91 15.09 70.60 422.3 349.2 244 191.8
2.65 16.3 15.6 69.7 416 341 250 210
2.546 15.69 14.95 70.53 419.9 344.9 213.6 177.5
2.659 16.35 15.60 69.80 416.2 341.5 243.9 202.0
2.59 15.9 15.1 70.2 419 345 232 194
2.63 16.2 15.4 69.7 415 341 241 193
1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1) 1s(2)2s(2)2p(6)(S1)3p(1)
1s(2)2s(2)2p(6)(S1)4s(1) 1s(2)2s(2)2p(6)(S1)4s(1) 1s(2)2s(2)2p(6)(S1)5s(1) 1s(2)2s(2)2p(6)(S1)5s(1)
2P 2S
1.5 0.5 1.5 0.5
0.5 0.5 0.5 0.5
197 101.1 90 43.4
224 109.6 102.3 45.35
218 105 92 47.2
225.9 109.3 94.75 46.05
217.3 105.2 96.76 47.04
224 109 96.6 47.1
216 103 92.5 41.8
1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1)
1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)5p(1) 1s(2)2s(2)2p(6)(S1)5p(1) 1s(2)2s(2)2p(6)(S1)5p(1)
2D 2P
1.5 1.5 2.5 1.5 1.5 2.5
1.5 0.5 1.5 1.5 0.5 1.5
7.69 80.1 69.6 2.88 29.9 26.1
7.642 79.9 69.56 3.235 32.82 29.70
7.4 77.2 67.0 2.90 28.0 26.0
7.72 80.5 69.95 2.472 25.65 22.38
7.366 76.7 66.62 2.905 30.20 26.25
7.64 80.2 70.1 2.81 30.6 25.6
7.28 75.8 66.6 2.65 27.4 24.9
1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1) 1s(2)2s(2)2p(6)(S1)3d(1)
1s(2)2s(2)2p(6)(S1)4f(1) 1s(2)2s(2)2p(6)(S1)4f(1) 1s(2)2s(2)2p(6)(S1)4f(1) 1s(2)2s(2)2p(6)(S1)5f(1) 1s(2)2s(2)2p(6)(S1)5f(1) 1s(2)2s(2)2p(6)(S1)5f(1)
2D 2F
1.5 2.5 2.5 1.5 2.5 2.5
2.5 2.5 3.5 2.5 2.5 3.5
927 65.9 1020 356 25.3 380
881.0 62.80 906.8 372.7 26.53 402
939 66.9 1000 346 25.0 370
914.1 65.17
931.8 66.42
923 65.9
939 67.1
355.9 25.29
347 24.69
332 23.6
346 24.7
1s(2)2s(2)2p(6)(S1)4s(1) 1s(2)2s(2)2p(6)(S1)4s(1) 1s(2)2s(2)2p(6)(S1)4s(1) 1s(2)2s(2)2p(6)(S1)4s(1)
1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)5p(1) 1s(2)2s(2)2p(6)(S1)5p(1)
2S 2P
0.5 0.5 0.5 0.5
1.5 0.5 1.5 0.5
1.83 1.48 35.9 38.0
2.013 1.632 40.83 39.69
1.83 1.46 39.3 39.0
1.817 1.467
1.833 1.475
1.79 1.44
1.82 1.47
32.04
38.62
36.0
36.2
1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1)
1s(2)2s(2)2p(6)(S1)5s(1) 1s(2)2s(2)2p(6)(S1)5s(1) 1s(2)2s(2)2p(6)(S1)6s(1) 1s(2)2s(2)2p(6)(S1)6s(1)
2P 2S
1.5 0.5 1.5 0.5
0.5 0.5 0.5 0.5
66.2 32.0 33.6 16.3
68.14 32.96 38.29 19.19
65.0 33.9 33.4 17.0
66.02 30.03
68.72 33.38
62.5 30.8
66.0 31.6
1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1) 1s(2)2s(2)2p(6)(S1)4p(1)
1s(2)2s(2)2p(6)(S1)4d(1) 1s(2)2s(2)2p(6)(S1)4d(1) 1s(2)2s(2)2p(6)(S1)5d(1) 1s(2)2s(2)2p(6)(S1)5d(1) 1s(2)2s(2)2p(6)(S1)5d(1)
2P 2D
1.5 0.5 1.5 1.5 0.5
2.5 1.5 1.5 2.5 1.5
3.51 3.37 11.1 66.0 53.2
3.541 3.366 11.53 68.56 55.36
3.48 3.33 12 70 57.0
3.421 3.362 10.09 59.64 48.48
3.497 3.351 11.67 69.38 56.20
3.48 3.33 10.7 62.9 52.2
3.53 3.38 11.5 68.3 54.7
1s(2)2s(2)2p(6)(S1)4d(1) 1s(2)2s(2)2p(6)(S1)4d(1) 1s(2)2s(2)2p(6)(S1)4d(1)
1s(2)2s(2)2p(6)(S1)5p(1) 1s(2)2s(2)2p(6)(S1)5p(1) 1s(2)2s(2)2p(6)(S1)5p(1)
2D 2P
1.5 1.5 2.5
1.5 0.5 1.5
3.51 36.4 31.7
3.805 35.53 34.45
3.60 34.0 31.0
31.49
35.90
32.4
35.1
1s(2)2s(2)2p(6)(S1)4d(1) 1s(2)2s(2)2p(6)(S1)4d(1) 1s(2)2s(2)2p(6)(S1)4d(1)
1s(2)2s(2)2p(6)(S1)5f(1) 1s(2)2s(2)2p(6)(S1)5f(1) 1s(2)2s(2)2p(6)(S1)5f(1)
2D 2F
1.5 2.5 2.5
2.5 2.5 3.5
155 11.1 165
119.9 8.59 110.8
160 11.0 160
2S 2P 2S 2P 2S 2P 2S 2P 2S 2P
2P 2D 2P 2D 2P 2D 2P 2D 2P 2D 2P 2D 2P 2D
2P 2S 2P 2S 2P 2S
2D 2P 2D 2P 2D 2P 2D 2P 2D 2P
2D 2F 2D 2F 2D 2F 2D 2F 2D 2F
2S 2P 2S 2P 2S 2P
2P 2S 2P 2S 2P 2S
2P 2D 2P 2D 2P 2D 2P 2D
2D 2P 2D 2P
2D 2F 2D 2F
229
Page 6 of 9
G. Çelik, S. ¸ Ate¸s
Table 3 Electric quadrupole transition probabilities (s−1 ) for Fe XVI 1.ST Level
2.ND Level
Terms J1
Ray (2002) Fuhr et al. (1988) Froese Fischer (2002)
Nahar et al. (2009)
3s(1)
3d(1)
2S 2D
0.5 1.5 6.64E+05
3s(1)
3d(1)
2S 2D
0.5 2.5 6.79E+05
6.7E+05
6.62E+05
6.70E+05
6.61E+05
6.8E+05
6.82E+05
6.80E+05
3s(1)
4d(1)
2S 2D
6.78E+05
3s(1)
4d(1)
2S 2D
0.5 1.5 1.50E+08
1.5E+08
1.37E+08
1.50E+08
1.36E+08
0.5 2.5 1.51E+08
1.44E+08
1.36E+08
1.44E+08
3s(1)
5d(1)
1.36E+08
2S 2D
0.5 1.5 7.66E+07
7.8E+07
7.45E+07
7.80E+07
3s(1)
6.42E+07
5d(1)
2S 2D
0.5 2.5 7.67E+07
7.6E+07
7.44E+07
7.60E+07
6.38E+07
3p(1)
3p(1)
2P 2P
0.5 1.5 2.43E−02
2.43E−02
3p(1)
3p(1)
2P 2P
1.5 0.5 4.85E−02
3p(1)
4p(1)
2P 2P
0.5 1.5 3.58E+07
3.7E+07
3.45E+07
3p(1)
4p(1)
2P 2P
1.5 0.5 6.57E+07
6.7E+07
6.84E+07
3p(1)
4p(1)
2P 2P
1.5 1.5 3.39E+07
3.4E+07
3p(1)
5p(1)
2P 2P
0.5 1.5 1.95E+07
3p(1)
5p(1)
2P 2P
1.5 0.5 3.65E+07
3p(1)
5p(1)
2P 2P
1.5 1.5 1.85E+07
3p(1)
4f(1)
2P 2F
0.5 2.5 2.17E+08
J2
This work
NIST (Kramida et al. 2015) WBEPMT
2.978E−02
2.36−02
3.7E+07
3.578E+07
3.25E+07
6.7E+07
6.957E+07
3.39E+07
3.4E+07
3.471E+07
2.14E+08
2.06E+08
2.14E+08
3p(1)
4f(1)
2P 2F
1.5 2.5 5.87E+07
5.8E+07
5.72E+07
5.8E+07
3p(1)
4f(1)
2P 2F
1.5 3.5 2.48E+08
2.6E+08
2.58E+08
2.6E+08
3p(1)
5f(1)
2P 2F
0.5 2.5 3.61E+07
3.6E+07
3p(1)
5f(1)
2P 2F
1.5 2.5 9.78E+06 1.5 3.5 4.40E+07
3p(1)
5f(1)
2P 2F
3d(1)
4s(1)
2D 2S
1.5 0.5 9.23E+06
3d(1)
4s(1)
2D 2S
2.5 0.5 1.38E+07
3d(1)
5s(1)
2D 2S
1.5 0.5 4.56E+06
3d(1)
5s(1)
2D 2S
2.5 0.5 6.78E+06
3d(1)
3d(1)
2D 2D
1.5 2.5 2.55E−07
3d(1)
3d(1)
2D 2D
2.5 1.5 3.81E−07
3d(1)
4d(1)
2D 2D
1.5 1.5 1.56E+07
1.6E+07
1.54E+07
1.6E+07
1.561E+07
3d(1)
4d(1)
2D 2D
1.5 2.5 4.45E+06
4.5E+06
4.43E+06
4.5E+06
4.468E+06
3d(1)
4d(1)
2D 2D
2.5 1.5 6.66E+06
6.6E+06
6.62E+06
6.6E+06
6.654E+06
3d(1)
4d(1)
2D 2D
2.5 2.5 1.78E+07
1.78E+07
1.75E+07
1.78E+07
1.777E+07
3d(1)
5d(1)
2D 2D
1.5 1.5 7.14E+06
3d(1)
5d(1)
2D 2D
1.5 2.5 2.04E+06
3d(1)
5d(1)
2D 2D
2.5 1.5 3.04E+06
3d(1)
5d(1)
2D 2D
2.5 2.5 8.11E+06
4s(1)
4d(1)
2S 2D
0.5 1.5 8.12E+04
8.18E+04
8.2E+04
4s(1)
4d(1)
2S 2D
0.5 2.5 8.47E+04
8.44E+04
8.4E+04
4s(1)
5d(1)
2S 2D
0.5 1.5 1.39E+07
1.41E+07
1.5E+07
4s(1)
5d(1)
2S 2D
0.5 2.5 1.38E+07
1.41E+07
1.5E+07
4.5E+07
4.33E+07
2.7E−07
calculated by others for the other two ions. But, in our paper we have presented to Fe XVI, Co XVII and Ni XVIII ions for both generating data belonging to these ions and testing the reliability of the WBEPM theory in electric quadrupole transitions of high stripped ions. Moreover, looking at all the tables, it has been seen that the E1 and E2 transition probabilities for Fe XVI, Co XVII and Ni XVIII gradually
3.693E−07
increase with the increase of nuclear charges for each individual transition. In this case, Z nuclear charge in radiative transitions of iso-electronic sequences is significant and important parameter. The reliable of atomic and ionic structure calculations made using the WBEPM theory, a semi-empirical method based on fitting parameters such as energies and orbital
E1 and E2 transitions for Fe XVI, Co XVII and Ni XVIII
Page 7 of 9
229
Table 4 Electric quadrupole transition probabilities (s−1 ) for Co XVII 1.ST Level
2.ND Level
Terms
J1
J2
This work
Ray (2002)
NIST (Kramida et al. 2015)
WBEPMT 3s(1)
3d(1)
2S 2D
0.5
1.5
7.32E+05
7.30E+05
7.40E+05
3s(1)
3d(1)
2S 2D
0.5
2.5
7.69E+05
7.56E+05
7.6E+05
3s(1)
4d(1)
2S 2D
0.5
1.5
2.12E+08
1.93E+08
2.1E+08
3s(1)
4d(1)
2S 2D
0.5
2.5
2.12E+08
1.92E+08
2.05E+08
3s(1)
5d(1)
2S 2D
0.5
1.5
1.06E+08
1.03E+08
3s(1)
5d(1)
2S 2D
0.5
2.5
1.06E+08
1.03E+08
1.0E+08
3p(1)
3p(1)
2P 2P
0.5
1.5
5.14E−02
5.63E−02
5.1E−02
3p(1)
3p(1)
2P 2P
1.5
0.5
1.03E−01
3p(1)
4p(1)
2P 2P
1.5
1.5
4.77E+07
4.78E+07
4.8E+07
3p(1)
4p(1)
2P 2P
1.5
0.5
9.23E+07
1.13E+06
9.4E+07
3p(1)
4p(1)
2P 2P
0.5
1.5
5.08E+07
4.86E+07
5.2E+07
3p(1)
5p(1)
2P 2P
1.5
1.5
2.61E+07
2.58E+07
3p(1)
5p(1)
2P 2P
1.5
0.5
5.14E+07
9.65E+07
3p(1)
5p(1)
2P 2P
0.5
1.5
2.76E+07
2.67E+07
3p(1)
4f(1)
2P 2F
1.5
2.5
8.12E+07
7.92E+07
8.0E+07
3p(1)
4f(1)
2P 2F
1.5
3.5
3.66E+08
3.57E+08
3.61E+08
3p(1)
4f(1)
2P 2F
0.5
2.5
3.01E+08
2.86E+08
2.97E+08
3p(1)
5f(1)
2P 2F
1.5
2.5
1.27E+07
1.28E+07
3p(1)
5f(1)
2P 2F
1.5
3.5
6.07E+07
5.82E+07
3p(1)
5f(1)
2P 2F
0.5
2.5
4.69E+07
4.87E+07
3d(1)
4s(1)
2D 2S
1.5
0.5
1.34E+07
1.32E+07
3d(1)
4s(1)
2D 2S
2.5
0.5
2.01E+07
3d(1)
5s(1)
2D 2S
1.5
0.5
6.72E+06
3d(1)
5s(1)
2D 2S
2.5
0.5
9.97E+06
3d(1)
3d(1)
2D 2D
1.5
2.5
6.78E−07
3d(1)
3d(1)
2D 2D
2.5
1.5
1.01E−06
3d(1)
4d(1)
2D 2D
1.5
1.5
3d(1)
4d(1)
2D 2D
1.5
3d(1)
4d(1)
2D 2D
2.5
3d(1)
4d(1)
2D 2D
3d(1)
5d(1)
3d(1)
5.9E+07 1.3E+07 1.9E+07
6.69E+06 1.07E−06
6.9E−07
2.23E+07
2.21E+07
2.2E+07
2.5
6.38E+06
6.36E+06
6.5E+06
1.5
9.55E+06
9.50E+06
9.7E+06
2.5
2.5
2.55E+07
2.52E+07
2.6E+07
2D 2D
1.5
1.5
1.03E+07
1.03E+07
5d(1)
2D 2D
1.5
2.5
3.01E+06
3.01E+06
3d(1)
5d(1)
2D 2D
2.5
1.5
4.48E+06
4.46E+06
3d(1)
5d(1)
2D 2D
2.5
2.5
1.17E+07
1.18E+07
4s(1)
4d(1)
2S 2D
0.5
1.5
9.11E+04
9.14E+04
9.1E+04
4s(1)
4d(1)
2S 2D
0.5
2.5
9.37E+04
9.47E+04
9.4E+04
4s(1)
5d(1)
2S 2D
0.5
1.5
2.21E+07
2.03E+07
2.16E+07
4s(1)
5d(1)
2S 2D
0.5
2.5
2.23E+07
2.02E+07
2.20E+07
radii, depend on the accuracy of these parameters. Previously, accurate electric dipole and electric quadrupole transition probabilities were calculated using this theory for some multi-electron atoms and ions and it was obtained the good results (Çelik et al. 2012a, 2012b, 2014; Çelik and Ate¸s 2007, 2008; Çelik 2007). In our recent work (Çelik et al. 2015), we have shown that the WBEPM theory produces
reliable the electric dipole transition probabilities and oscillator strengths for Co16+ . One can expect that the same wave functions should be used for electric quadrupole transitions of sodium-like ions. In this work, the electric dipole and quadrupole transition probabilities are obtained using the same WBEPM method. Thus, in this work the accuracy of the transition probabilities for Fe XVI, Co XVII
229
Page 8 of 9
G. Çelik, S. ¸ Ate¸s
Table 5 Electric quadrupole transition probabilities (s−1 ) for Ni XVIII 1.ST Level
2.ND Level
Terms
J1
J2
This work WBEPMT
Ray (2002)
NIST (Kramida et al. 2015)
3s(1) 3s(1) 3s(1) 3s(1) 3s(1) 3s(1) 3p(1) 3p(1) 3p(1) 3p(1) 3p(1) 3p(1) 3p(1) 3p(1) 3p(1) 3p(1) 3p(1) 3p(1) 3p(1) 3p(1) 3d(1) 3d(1) 3d(1) 3d(1) 3d(1) 3d(1) 3d(1) 3d(1) 3d(1) 3d(1) 3d(1) 3d(1) 3d(1) 3d(1) 4s(1) 4s(1) 4s(1) 4s(1)
3d(1) 3d(1) 4d(1) 4d(1) 5d(1) 5d(1) 3p(1) 3p(1) 4p(1) 4p(1) 4p(1) 5p(1) 5p(1) 5p(1) 4f(1) 4f(1) 4f(1) 5f(1) 5f(1) 5f(1) 4s(1) 4s(1) 5s(1) 5s(1) 3d(1) 3d(1) 4d(1) 4d(1) 4d(1) 4d(1) 5d(1) 5d(1) 5d(1) 5d(1) 4d(1) 4d(1) 5d(1) 5d(1)
2S 2D
0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.5 0.5 1.5 1.5 0.5 1.5 1.5 0.5 1.5 1.5 0.5 1.5 1.5 1.5 2.5 1.5 2.5 1.5 2.5 1.5 1.5 2.5 2.5 1.5 1.5 2.5 2.5 0.5 0.5 0.5 0.5
1.5 2.5 1.5 2.5 1.5 2.5 1.5 0.5 1.5 0.5 1.5 1.5 0.5 1.5 2.5 2.5 3.5 2.5 2.5 3.5 0.5 0.5 0.5 0.5 2.5 1.5 1.5 2.5 1.5 2.5 1.5 2.5 1.5 2.5 1.5 2.5 1.5 2.5
8.05E+05 8.30E+05 2.93E+08 2.94E+08 1.45E+08 1.45E+08 1.05E−01 2.10E−01 7.04E+07 1.27E+08 6.58E+07 3.86E+07 7.13E+07 3.63E+07 4.12E+08 1.10E+08 4.97E+08 6.09E+07 1.64E+07 7.36E+07 1.91E+07 2.85E+07 1.00E+07 1.49E+07 1.71E−06 2.48E−06 3.14E+07 8.97E+06 1.34E+07 3.58E+07 1.48E+07 4.26E+06 6.32E+06 1.69E+07 1.02E+05 1.06E+05 3.14E+07 3.15E+07
8.03E+05 8.36E+05 2.67E+08 2.66E+08 1.40E+08 1.40E+08
8.2E+05 8.4E+05 2.9E+08 2.8E+08
6.73E+07 1.33E+08 6.61E+07
7.1E+07 1.3E+08 6.6E+07
3.91E+08 1.08E+08 4.86E+08
4.07E+08 1.1E+08 4.92E+08
2S 2D 2S 2D 2S 2D 2S 2D 2S 2D 2P 2P 2P 2P 2P 2P 2P 2P 2P 2P 2P 2P 2P 2P 2P 2P 2P 2F 2P 2F 2P 2F 2P 2F 2P 2F 2P 2F 2D 2S 2D 2S 2D 2S 2D 2S 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2D 2S 2D 2S 2D 2S 2D 2S 2D
and Ni XVIII is expected to be high. It can be said that the WBEPM theory is also a useful tool for calculating electric quadrupole transition probabilities, particularly for transitions between highly excited states. But, this case can be tested making more electric quadrupole transition calculations with this theory. Acknowledgements The authors gratefully acknowledge the Selçuk University Scientific Research Projects (BAP) Coordinating Office for the support.
1.5E+08 1.1E−01
7.5E+07 1.8E+07 2.7E+07
1.7E−06 3.11E+07 8.95E+06 1.34E+07 3.54E+07
3.2E+07 9.2E+06 1.4E+07 3.6E+07
1.02E+05 1.06E+05 2.85E+07 2.84E+07
1.02E+05 1.06E+05 3.07E+07 3.1E+07
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