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Configuration interaction effects on the energy levels and oscillator strengths of lowly charged gold ions: Au11+ as an example
ARTICLE IN PRESS
Journal of Quantitative Spectroscopy & Radiative Transfer 102 (2006) 172–180 www.elsevier.com/locate/jqsrt
Configuration interaction effects on the energy levels and oscillator strengths of lowly charged gold ions: Au11þ as an example Jiaolong Zenga,b,, Gang Zhaoa, Jianmin Yuana,b a
National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, People’s Republic of China b Department of Applied Physics, National University of Defense Technology, Changsha 410073, People’s Republic of China Received 28 February 2005; accepted 6 February 2006
Abstract The effects of configuration interaction (CI) on the energy levels and oscillator strengths of Au11þ have been investigated by systematically adding configurations included in calculations. Four sets of calculations, namely, a single configuration Dirac–Fock type, CI with singly, doubly, and triply electron excitations from the ground configuration of Au11þ have been carried out to show the effects. The results show that CI not only affects distribution of the oscillator strengths among one transition array, but also affects the distributions among different transition arrays. For example, the converged oscillator strengths for transition arrays of 5p ! 5d (e.g. 5p6 ! 5p5 5d and 5p5 5d ! 5p4 5d 2 ) are greatly reduced compared with the results of single configuration approximation. On the other hand, for transition arrays of 4f ! 5d (e.g. 5p6 ! 4f 13 5d and 5p5 5d ! 4f 13 5p5 5d 2 ), the converged oscillator strengths are drastically enhanced. The CI effects result in the net transfer of oscillator strengths from transition arrays of 5p ! 5d to 4f ! 5d. r 2006 Elsevier Ltd. All rights reserved. PACS: 32.70.Cs; 32.10.Fn; 32.10.f Keywords: Au11þ ; Configuration interaction effects; Transfer of oscillator strength
1. Introduction During the last two decades, the determination of oscillator strengths has been an active research area because of its considerable interest in astrophysics, plasma physics, laser physics and thermonuclear research (for example, see [1–3]). Yet, many of the modern experimental techniques still encounter difficulties in exact measurement of oscillator strengths. The main reason is due to the complexity of experimental approaches to determine transition probability data [4]. In most practical applications, these atomic data have been obtained by theoretical method. Corresponding author. Tel.: +86 731 4576016; fax: +86 731 4574895.
E-mail address: [email protected] (J. Zeng). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.02.003
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173
Among other practical scientific interests, gold is an element of particular interest for indirectly driven inertial confinement fusion (ICF), where laser radiation heats the inside of a gold hohlraum producing a plasma that emits intense X rays. The X-ray radiation drives the capsule implosion and influences the resulting fusion yield. As a result, much work [5–9] has been devoted to the X-ray emission spectra of highly charged gold ions and gold plasmas. On the other hand, the absorption or radiative opacity of gold plasmas is another important research field. Eidmann et al. [10] experimentally measured the opacity of a gold plasma at a temperature of 22 eV and density of 0:01 g=cm3 in the photon energy range of 50–300 eV. The main ionization stages are Au7þ –Au13þ near such plasma condition. Super-transition-arrays (STA) model [11] has been used to simulate the transmission spectrum. The predicted absorption caused by 5s ! 5p transition is more pronounced than that of 5p ! 5d, which is contrary to the experimental result. To clarify the difference between the theory and measurement, one need to investigate the detailed absorption structures. These structures are caused by transitions from the bound states of gold ions. Therefore it is necessary to study the oscillator strengths of transitions in more detail. This paper considers the energy levels and oscillator strengths of lowly charged gold ions, taking Au11þ as an example. The main motivation is to obtain accurate oscillator strengths in large scale calculations. Because of many electrons of these ions, configuration interaction (CI) plays an important role in the calculation of oscillator strengths. The emphasis is to clarify how different degree of CI influences the values and redistributions of oscillator strengths. To the best of our knowledge, we found no work which reported the oscillator strengths for Au7þ –Au13þ ions. For atomic Au and more lowly charged ions, such as Auþ , Au2þ , and Au3þ , however, a lot of spectrum analyses had been carried out (see, for example [12–14]). 2. Theoretical method The calculations have been carried out using the flexible atomic code (FAC) developed by Gu [15]. A fully relativistic approach based on the Dirac equation is used throughout the entire package. An atomic state is approximated by a linear combination of configuration state functions (CSFs) with same symmetry Fa ðJpÞ ¼
nc X
ai ðaÞfa ðJpÞ,
(1)
i
where nc is the number of CSFs and ai ðaÞ denotes the representation of the atomic state in this basis. The CSFs are anti-symmetrized products of a common set of orthonormal orbitals which are optimized on the basis of the relativistic Hamiltonian. The radial orbitals are derived from a modified Dirac–Fock–Slater iteration on a fictitious mean configuration with fractional occupation numbers, representing the average electron cloud of the configurations included in the calculation. Once the CI wavefunctions have been obtained, the oscillator strengths can be calculated. In this work, we calculate the weighted oscillator strengths in length formalism !2 N X 2DE ! gf ¼ r p jFf i , (2) hFi j 3 p¼1 where DE ¼ E f E i , E i and E f are, respectively, the energies of initial and final states and g is the statistical weight of the lower state, i.e. g ¼ 2J i þ 1, for relativistic wavefunctions. For complex ions, CI plays an important role in the calculation of atomic structure. In the present paper, we study the convergence behavior of the energy levels and oscillator strengths by systematically increasing the electron correlations. In this systematic study, four cases are considered to show the effects of different electron correlations on the calculated energy levels and oscillator strengths. In case A, a single configuration Dirac–Fock (DF) calculation is carried out, in which intra-configuration correlation has been included. In case B, CI among the configurations of singly electron excitation to valence orbitals of 5d, 5f , 6s, 6p, and 6d from the ground configuration ð½Kr4s2 4p6 Þ4d 10 4f 14 5s2 5p6 have been included for both even and odd parity levels. In electron excitations, 4s and 4p electrons are kept full, the electrons in other orbitals can freely be excited to valence orbitals. Explicitly, CI has been included among following configurations: 4d 10 4f 14 5s2 5p5 nl, 4d 10 4f 14 5s5p6 nl, 4d 10 4f 13 5s2 5p6 nl, and 4d 9 4f 14 5s2 5p6 nl (nl ¼ 5d; 5f ; 6s; 6p, and 6d). In case C, in addition to the CI included in case B, configurations of double electron excitations to these valence orbitals
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have also been included. In case D, further electron correlations of triple electron excitations are taken into account. By adding the configurations step by step, we can see how the oscillator strengths converge with the included CI. 3. Results and discussions First, we check the variational behavior of energy levels of Au11þ with the CI. The number of levels included in case D exceeds 20 000, there is no necessity to write down all of them. Table 1 shows the lowest 33 levels which are relevant to our discussion. These levels belong to the ground and the first two excited configurations
Table 1 Lowest 33 energy levels (eV) of Au11þ obtained by using different electron correlations. In the designation of levels, full relativistic subshells have been omitted for simplicity No.
Designation
Jp e
A
B
C
D
1 2
5p6 5p33=2 5d 3=2
0 01
0.00 54.50
0.00 59.34
0.00 60.31
0.00 58.84
3
5p33=2 5d 3=2
11
55.84
60.70
61.64
60.19
4
31
57.57
62.41
63.52
62.04
5
5p33=2 5d 3=2 5p33=2 5d 3=2
21
57.91
62.82
63.72
62.30
6
5p33=2 5d 5=2
41
59.18
63.92
65.17
63.65
7
5p33=2 5d 5=2
21
60.09
64.98
65.80
64.41
8
5p33=2 5d 5=2
31
62.20
67.17
67.96
66.61
9
5p33=2 5d 5=2
11
67.81
72.62
73.23
71.47
10
21
77.27
81.93
82.59
81.01
11
5p31=2 5d 3=2 5p31=2 5d 5=2
21
80.43
85.17
85.50
84.02
12
5p31=2 5d 5=2
31
81.33
86.04
86.54
85.03
13
5p31=2 5d 3=2
11
86.71
90.86
90.00
87.97
14
4f 77=2 5d 3=2
21
97.39
101.83
91.25
89.55
15
4f 77=2 5d 3=2
51
98.35
102.86
92.17
90.47
16
4f 77=2 5d 3=2 4f 77=2 5d 3=2 4f 77=2 5d 5=2 4f 77=2 5d 5=2 4f 77=2 5d 5=2 4f 55=2 5d 3=2 4f 77=2 5d 5=2 4f 77=2 5d 5=2 4f 77=2 5d 5=2 4f 55=2 5d 3=2 4f 55=2 5d 3=2 4f 55=2 5d 5=2 4f 55=2 5d 5=2 4f 55=2 5d 5=2 4f 55=2 5d 5=2 4f 55=2 5d 5=2 4f 55=2 5d 5=2 4f 77=2 5d 5=2
31
99.12
103.70
92.94
91.25
41
99.52
104.11
93.34
91.65
11
101.00
105.45
94.69
93.00
61
101.10
105.57
94.85
93.18
21
101.46
105.99
95.18
93.51
41
101.98
106.48
95.65
93.96
31
102.43
107.01
96.12
94.46
41
102.60
107.17
96.30
94.63
51
102.68
107.27
96.37
94.71
21
102.79
107.35
96.42
94.72
31
103.76
108.37
97.32
95.63
01
103.81
108.18
97.31
95.65
11
104.63
109.35
98.38
96.69
51
105.39
109.87
98.88
97.21
21
106.62
110.32
99.29
97.63
31
105.52
110.98
99.90
98.24
41
105.78
111.19
100.09
98.43
11
106.41
111.12
100.81
98.76
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
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4d 10 4f 14 5s2 5p6 , 4d 10 4f 14 5s2 5p5 5d and 4d 10 4f 13 5s2 5p6 5d. The level designations are selected automatically by the computer program. It names the level according to the term linked with the largest eigenvector component in the matrix. However, it should be noted that there is a strong mixing of some levels. For simplicity, we omit the full relativistic subshells in the designation of levels. From Table 1, one can see that the energies of the excited configurations extend a wide range. For configuration of 5p5 5d, the level energies cover nearly 30 eV, while they cover less than 10 eV for 4f 13 5d. Note also that the number of levels belonging to 5p5 5d is smaller than those of 4f 13 5d, although the level energies of the former configuration span wider than the latter. From the inspection of Table 1, one can see the variational behavior of the energy levels with different degree of CI. With the increasing of configurations included in the calculation (from case A to case D), the relative energies of the low-lying levels (Nos. 2–10) increase with CI, while those of other levels (Nos. 11–33) are greatly reduced with more CI included. In principle, improvement should be feasible with more configurations (e.g. four electron excitations) included in the calculation. However, the effect of additional CI should be small with those included. In fact, there is a good agreement between the results of cases C and D. We believe that the results of case D have been converged. Therefore, the levels shown in Table 1 have been ordered according to the results of case D. Compared with the results of case D, case A predicts a reversed order for level Nos. 30, 31 and 32, 33, case B for Nos. 26, 27 and 32, 33, case C for 26, 27. For high-lying levels, more levels are reversed compared with the converged results. It indicates that CI effects are so strong that the labeling of the levels is not particularly meaningful. Therefore, many of them are just labels to identify the levels. Table 2 gives the weighted oscillator strengths gf for transition arrays of 5p6 ! 5p5 5d and 5p6 ! 4f 13 5d obtained by using CI of cases A, B, C and D. The variational trend of oscillator strengths is complicated with different electron correlations. For transition array of 5p6 ! 5p5 5d, the gf values of the two strong transitions decrease with the increasing of CI. For example, the gf value for 1–13 is 6.555 in single DF calculation, while it decreased to 5.071, 2.534, and 2.666 for CI of cases B, C, and D. This variational trend of oscillator strengths for strong transitions with CI is consistent with 3p ! 3d transitions of transition elements (for example, iron ions [16–18]). However, for 5p6 ! 4f 13 5d, the variational trend is different. The gf values for transitions of 1–28 and 1–33 are 0.434 and 0.695, respectively, in single DF approximation. With the increasing of CI, the gf value of 1–33 increases dramatically to 2.497, 3.297, and 2.980 for cases B, C, and D. The converged value is greater than the result of case A by a factor of 3.2. While for 1–28, the situation is completely on the opposite. The converged value of 0.176 is much smaller than that of case A, the former being just 40% of the latter. The increase or decrease of gf values with the CI are caused by the constructive- or destructive–interference effects in the radial portion of the oscillator strengths. Mixing of basis functions owing to CI effects produces large changes in computing oscillator strengths. For transition of 1–28, the CI effects are destructive, while for 1–33, they are constructive. On the whole, the CI effects result in the net transfer of oscillator strengths from transition array of 5p6 ! 5p5 5d to 5p6 ! 4f 13 5d. The redistribution of gf values between a transition array, or even between different arrays will surely have large effects on many practical applications of radiative atomic data, such as spectra analyses and radiative opacity calculation. The effects of CI on gf values can more obviously and clearly be seen from Figs. 1 and 2, which shows the total absorption cross section for the transition arrays of 5p5 5d ! 5p4 5d 2 and 5p5 5d ! 4f 13 5p5 5d 2 , respectively. In Figs. 1 and 2, plots A, C, and D corresponds to the results of cases A, C, and D. These two
Table 2 Weighted oscillator strengths of Au11þ obtained by using different electron correlations Lower
Upper
A
B
C
D
1 1 1 1 1
9 13 18 28 33
2.055 6.555 0.021 0.434 0.695
1.897 5.071 0.558 0.169 2.497
1.476 2.534 0.041 0.135 3.297
1.611 2.666 0.036 0.176 2.980
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500 400 A 300 200 100
Cross section (Mb)
0 400
C
300 200 100 0 400
D
300 200 100 0 60
70
80
90
Photon energy (eV) Fig. 1. The total absorption cross section caused by the transition array of 5p5 5d ! 5p4 5d 2 .
transition arrays are not possible in case B. The absorption cross section at photon energy hn is obtained by sðhnÞ ¼
X phe2 f 0 SðhnÞ ¼ 109:71f ll 0 SðhnÞ, me c ll ll 0
(3)
where hn is in eV and s in Mb, f ll 0 is the oscillator strengths of transition from level l to l 0 and S is the line profile which is taken to be a Voigt one pffiffiffiffiffiffiffiffi ln 2 Hða; vÞ, (4) SðhnÞ ¼ pGd where Hða; vÞ is the Voigt function Z a þ1 expðx2 Þ Hða; vÞ ¼ dx, p 1 a2 þ ðv xÞ2 a¼
pffiffiffiffiffiffiffiffi ln 2Gl =Gd ,
v¼
pffiffiffiffiffiffiffiffi ln 2ðhn hn0 Þ=Gd ,
(5)
where Gl is the Lorentzian half-width at half-maximum (HWHM) due to electron impact broadening, Gd is the Doppler HWHW. The Lorentzian and Doppler widths are assumed to be caused by the plasma condition under the prototype experiment carried out by Eidmann et al. [10].
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50 40
A
30 20 10
Cross section (Mb)
0 150
C
100 50 0 80
D
60 40 20 0 75
85
95 105 Photon energy (eV)
115
Fig. 2. The total absorption cross section caused by the transition array of 5p5 5d ! 4f 13 5p5 5d 2 .
From the inspection of Figs. 1 and 2, similar conclusion to the transition arrays of 5p6 ! 5p5 5d and 5p6 ! 4f 13 5d discussed above can be drawn for transition arrays of 5p5 5d ! 5p4 5d 2 and 5p5 5d ! 4f 13 5p5 5d 2 . Because of the complexity of the latter two transition arrays, there are far more strong transitions than the former two arrays. Fig. 1 shows that case A predicts the gf values more expanded than cases C and D predict. The convergence behavior can clearly be seen from plot A to D. Fig. 2 shows that the total absorption caused by transition array of 5p5 5d ! 4f 13 5p5 5d 2 . The oscillator strengths predicted by cases C and D are obviously enhanced compared with those of case A. Furthermore, additional relatively weak transitions occur when more CI has been considered. From the discussion mentioned above, CI should be important for all transition arrays of Au11þ . We have also checked the effects of CI on other gold ions near Au11þ , such as Au10þ , Au12þ , etc. For these ions, CI effects also play an important role on the energy levels and oscillator strengths. To save space, we do not discuss them in detail. The weighted oscillator strengths and transition energies for transition arrays of 5p5 5d ! 5p4 5d 2 and 5 5p 5d ! 4f 13 5p5 5d 2 are given in Table 3 for strong transitions whose gf values are greater than unity. For simplicity, only the converged results of case D have been given. The number of lower level refers to the Nos. given in Table 1. The designation and the total angular momentum of the upper level have been given in the second and third columns. The number in the fourth column gives the ordering of the upper level in the total number of energy levels. Unfortunately, we have no experimental or theoretical results to compare with. However, we shall carry out large scale calculations for Au7þ Au13þ to simulate the transmission of a gold plasma [10]. The degree of agreement will indirectly check the accuracy of the present calculations. In conclusion, four sets of calculations (A–D) with different degree of CI have been carried out to show its effects on the energy levels and oscillator strengths of Au11þ . The results show that the radiative atomic data
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Table 3 Weighted oscillator strengths and transition energies (in eV) of Au11þ for transition arrays 5p5 5d ! 5p4 5d 2 and 5p5 5d ! 4f 13 5p5 5d 2 obtained by using full electron correlations. Full relativistic subshells have been omitted for the designation of upper levels Lower
Upper J
2
ð5p1=2 5p33=2 Þ1 ð5d 23=2 Þ2
1
102
2.179
82.417
3
ð5p1=2 5p33=2 Þ1 ð5d 23=2 Þ2
2
109
2.892
83.246
3
1
129
2.023
87.369
0
137
1.101
89.469
2
140
1.474
90.624
3
75
1.476
66.980
2
82
1.231
70.615
3
110
1.209
81.606
4
111
2.834
82.237
4
116
6.440
82.944
4
133
3.069
86.273
3
142
3.344
89.007
3
150
4.609
90.041
4
ð5p1=2 5p33=2 Þ1 ð5d 23=2 Þ0 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ðð5p23=2 Þ2 5d 3=2 Þ7=2 5d 5=2 ðð5p23=2 Þ2 5d 3=2 Þ7=2 5d 7=2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ7=2 5d 5=2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 25=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ð5p1=2 5p33=2 Þ1 ð5d 23=2 Þ2 ðð4f 57=2 5p33=2 Þ4 5d 3=2 Þ11=2 5d 5=2
4
366
1.041
108.470
4
ð4f 57=2 5p33=2 Þ2 ð5d 25=2 Þ4
3
374
1.813
108.850
5
ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2
3
110
6.030
81.342
5
1
139
2.787
88.072
2
140
3.932
88.513
3
142
2.294
88.743
3
375
1.223
108.610
3
81
2.641
68.558
4
83
3.716
69.505
5
136
1.868
85.478
6
ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ðð4f 57=2 5p33=2 Þ2 5d 3=2 Þ3=2 5d 5=2 ð5p23=2 Þ2 ð5d 25=2 Þ4 ð5p23=2 Þ2 ð5d 25=2 Þ4 ð5p1=2 5p33=2 Þ2 ð5d 25=2 Þ4 ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ3=2 5d 5=2
4
144
9.633
87.712
6
ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ5=2 5d 5=2
5
145
12.232
87.771
6
ð5p1=2 5p33=2 Þ2 5d 3=2 Þ1=2 5d 5=2
3
146
7.756
87.788
6
ð4f 55=2 5p33=2 Þ3 ð5d 25=2 Þ2
5
398
1.627
108.290
6
ð4f 55=2 5p33=2 Þ3 ð5d 25=2 Þ2
5
406
2.223
108.690
6
ð4f 77=2 5p33=2 Þ5 ð5d 25=2 Þ4 ðð4f 55=2 5p33=2 Þ2 5d 3=2 Þ3=2 5d 5=2 ð4f 55=2 5p33=2 Þ3 ð5d 25=2 Þ2 ð4f 77=2 5p33=2 Þ2 ð5d 25=2 Þ4 ð4f 77=2 5p33=2 Þ5 ð5d 25=2 Þ4 ð5p23=2 Þ2 ð5d 25=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ1=2 5d 5=2
5
416
1.392
108.970
4
431
3.898
109.540
4
444
1.280
110.000
3
459
1.306
110.510
5
509
1.704
114.150
1
86
1.027
69.565
3
142
3.147
86.634
2
148
6.138
87.359
ð5p1=2 5p33=2 Þ1 ð5d 23=2 Þ2 ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ3=2 5d 5=2 ð4f 55=2 5p33=2 Þ2 ð5d 25=2 Þ2 ð4f 77=2 5p3=2 Þ2 ð5d 25=2 Þ4 ðð4f 55=2 5p33=2 Þ3 5d 3=2 Þ5=2 5d 5=2
3
150
3.301
87.668
1
151
3.017
87.721
3
433
1.325
108.920
3
459
1.015
109.750
2
467
1.934
110.170
ð5p23=2 Þ2 ð5d 25=2 Þ2 ð5p23=2 Þ2 ð5d 25=2 Þ0
3
91
3.249
69.012
2
93
1.788
69.550
3 3 4 4 4 4 4 4 4 4
5 5 5 6 6 6
6 6 6 6 7 7 7 7 7 7 7 7 8 8
No.
gf
Upper designation
DE
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Table 3 (continued ) Lower
Upper J
8
ðð5p1=2 5p33=2 Þ1 5d 3=2 Þ3=2 5d 5=2
2
156
2.612
87.108
8
4
157
10.159
87.277
3
158
8.093
87.472
2
162
2.478
88.186
2
486
1.523
109.120
4
491
2.791
109.380
8
ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ5=2 5d 5=2 ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ5=2 5d 5=2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ0 ðð4f 55=2 5p33=2 Þ2 5d 3=2 Þ1=2 5d 5=2 ð4f 77=2 5p33=2 Þ4 ð5d 5=2 Þ0 ðð4f 55=2 5p33=2 Þ1 5d 3=2 Þ1=2 5d 5=2
3
494
3.161
109.680
9
ð5p3=2 Þ2 ð5d 25=2 Þ4
2
100
3.982
69.095
9
ð5p1=2 5p3=2 Þ1 ð5d 23=2 Þ2 ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ0
2
109
1.204
71.967
2
159
1.908
82.630
9 9 9
ðð5p1=2 5p3=2 Þ1 5d 3=2 Þ5=2 5d 5=2 ðð5p1=2 5p3=2 Þ2 5d 3=2 Þ3=2 5d 5=2
1 2 2
172 206 576
3.712 1.488 1.102
87.871 92.283 110.810
10 10 10
ðð5p1=2 5p3=2 Þ2 5d 3=2 Þ7=2 5d 5=2 ðð5p1=2 5p3=2 Þ2 5d 3=2 Þ3=2 5d 5=2 ð4f 55=2 5p1=2 Þ2 ð5d 25=2 Þ2
1 2 3
169 206 852
4.186 2.763 1.298
76.193 82.736 112.620
10
ð4f 55=2 5p1=2 Þ2 ð5d 25=2 Þ2
2
857
1.664
112.840
11
ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ2
1
166
1.831
72.116
11
ð5p1=2 5p3=2 Þ1 ð5d 25=2 Þ4
3
167
2.913
72.282
11
ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ4 ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ4 ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ2
2
170
2.503
73.333
3
173
5.696
75.900
2
178
1.728
76.697
11 11 11 11 11
5d 3=2 5d 5=2 5d 3=2 5d 5=2 5d 3=2 5d 5=2 5d 3=2 5d 5=2 ðð4f 55=2 5p1=2 Þ2 5d 3=2 Þ3=2 5d 5=2
4 3 2 1 3
390 393 409 415 849
1.026 1.035 2.694 2.790 1.294
87.457 87.573 88.392 88.595 109.470
11
ð4f 55=2 5p1=2 Þ3 ð5d 25=2 Þ2
3
870
1.704
110.610
12
ð5p1=2 5p3=2 Þ1 ð5d 25=2 Þ4 ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ4 ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ2
3
167
1.410
71.275
3
173
5.369
74.893
2
178
5.892
75.691
12 12 12 12 12
5d 3=2 5d 5=2 5d 3=2 5d 5=2 5d 3=2 5d 5=2 5d 3=2 5d 5=2 ð4f 55=2 5p1=2 Þ3 ð5d 25=2 Þ4
3 3 3 4 4
390 393 395 412 864
1.911 1.540 1.789 7.266 2.813
86.451 86.567 86.806 87.508 109.330
12
ðð4f 55=2 5p1=2 Þ2 5d 3=2 Þ1=2 5d 5=2
3
889
2.394
110.350
12
ðð4f 55=2 5p1=2 Þ3 5d 3=2 Þ3=2 5d 5=2
4
1020
2.645
115.780
12
ðð4f 55=2 5p1=2 Þ3 5d 3=2 Þ3=2 5d 5=2
3
1038
1.241
116.440
13 13 13
ðð5p1=2 5p3=2 Þ1 5d 3=2 Þ5=2 5d 5=2 ðð5p1=2 5p3=2 Þ2 5d 3=2 Þ3=2 5d 5=2 5d 23=2
1 2 2
172 206 466
1.553 3.769 3.916
71.373 75.784 86.558
13
5d 23=2
1
470
3.474
86.773
13
5d 23=2 ð4f 55=2 5p1=2 Þ3 ð5d 23=2 Þ2
0
521
1.416
91.043
1
1027
2.267
113.090
8 8 8 8
9
11 11
12 12
13
ð4f 55=2 5p33=2 Þ4 ð5d 25=2 Þ4
No.
gf
Upper designation
DE
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are very sensitive to the CI included in the calculation. CI not only affects the distribution of the oscillator strengths among one particular transition array, but also does among different transition arrays. For 5p6 ! 5p5 5d and 5p5 5d ! 5p4 5d 2 , the oscillator strengths with the largest CI in this work are greatly reduced compared with the results of single configuration approximation. For 5p6 ! 4f 13 5d and 5p5 5d ! 4f 13 5p5 5d 2 , however, they are drastically enhanced. Acknowledgements This work was supported by the National Science Fund for Distinguished Young Scholars under Grant Nos. 10025416, the National Natural Science Foundation of China under Grant Nos. 10204024, 10373014 and 19974075, the National High-Tech ICF Committee in China, and China Research Association of Atomic and Molecular Data. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Martin I, Hernando C, Lavin C, Velasco AM. JQSRT 1999;62:71. Borges FO, Cavalcanti GH, Trigueiros AG, Jupen C. JQSRT 2004;83:751. Peng YL, Xia D, Li JM. JQSRT 2004;87:95. Wiese WL. Phys Scr 1996;65T:188. Wong KL, May MJ, Beiersdorfer P, Fournier KB, Wilson B, Brown GV, et al. Phys Rev Lett 2003;90:235001. May MJ, Fournier KB, Beiersdorfer P, Chen H, Wong KL. Phys Rev E 2003;68:036402. Glenzer SH, Fournier KB, Wilson BG, Springer PT. Phys Rev Lett 2001;87:045002. Foord ME, Glenzer SH, Thoe RS, Wong KL, Fournier KB, Wilson BG, et al. Phys Rev Lett 2000;85:992. Honda K, Mima K, Koike F. Phys Rev E 1997;55:4594. Eidmann K, Bar-Shalom A, Saemann A, Winhart G. Europhys Lett 1998;44:459. Bar-Shalom A, Oreg J. Phys Rev E 1996;54:1850. Wyart J-F, Joshi YN, Raassen AJJ, Uylings PHM, Tchang-Brillet T. Phys Scr 1994;50:672. Wyart J-F, Joshi YN, Tchang-Brillet T, Raassen AJJ. Phys Scr 1996;53:174. Zhang Z, Brage T, Curtis LJ, Lundberg H, Martinson I. J Phys B 2002;35:483. Gu MF. Astrophys J 2003;582:1241. Ramsbottom CA, Noble CJ, Burke VM, Scott MP, Burke PG. J Phys B 2004;37:3609. Griffin DC, Pindzola MS, Badnell NR. Astron Astrophys Suppl 2000;142:317. Zeng JL, Jin FT, Zhao G, Yuan JM. J Phys B 2003;36:3457.
Journal of Quantitative Spectroscopy & Radiative Transfer 102 (2006) 172–180 www.elsevier.com/locate/jqsrt
Configuration interaction effects on the energy levels and oscillator strengths of lowly charged gold ions: Au11þ as an example Jiaolong Zenga,b,, Gang Zhaoa, Jianmin Yuana,b a
National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, People’s Republic of China b Department of Applied Physics, National University of Defense Technology, Changsha 410073, People’s Republic of China Received 28 February 2005; accepted 6 February 2006
Abstract The effects of configuration interaction (CI) on the energy levels and oscillator strengths of Au11þ have been investigated by systematically adding configurations included in calculations. Four sets of calculations, namely, a single configuration Dirac–Fock type, CI with singly, doubly, and triply electron excitations from the ground configuration of Au11þ have been carried out to show the effects. The results show that CI not only affects distribution of the oscillator strengths among one transition array, but also affects the distributions among different transition arrays. For example, the converged oscillator strengths for transition arrays of 5p ! 5d (e.g. 5p6 ! 5p5 5d and 5p5 5d ! 5p4 5d 2 ) are greatly reduced compared with the results of single configuration approximation. On the other hand, for transition arrays of 4f ! 5d (e.g. 5p6 ! 4f 13 5d and 5p5 5d ! 4f 13 5p5 5d 2 ), the converged oscillator strengths are drastically enhanced. The CI effects result in the net transfer of oscillator strengths from transition arrays of 5p ! 5d to 4f ! 5d. r 2006 Elsevier Ltd. All rights reserved. PACS: 32.70.Cs; 32.10.Fn; 32.10.f Keywords: Au11þ ; Configuration interaction effects; Transfer of oscillator strength
1. Introduction During the last two decades, the determination of oscillator strengths has been an active research area because of its considerable interest in astrophysics, plasma physics, laser physics and thermonuclear research (for example, see [1–3]). Yet, many of the modern experimental techniques still encounter difficulties in exact measurement of oscillator strengths. The main reason is due to the complexity of experimental approaches to determine transition probability data [4]. In most practical applications, these atomic data have been obtained by theoretical method. Corresponding author. Tel.: +86 731 4576016; fax: +86 731 4574895.
E-mail address: [email protected] (J. Zeng). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.02.003
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Among other practical scientific interests, gold is an element of particular interest for indirectly driven inertial confinement fusion (ICF), where laser radiation heats the inside of a gold hohlraum producing a plasma that emits intense X rays. The X-ray radiation drives the capsule implosion and influences the resulting fusion yield. As a result, much work [5–9] has been devoted to the X-ray emission spectra of highly charged gold ions and gold plasmas. On the other hand, the absorption or radiative opacity of gold plasmas is another important research field. Eidmann et al. [10] experimentally measured the opacity of a gold plasma at a temperature of 22 eV and density of 0:01 g=cm3 in the photon energy range of 50–300 eV. The main ionization stages are Au7þ –Au13þ near such plasma condition. Super-transition-arrays (STA) model [11] has been used to simulate the transmission spectrum. The predicted absorption caused by 5s ! 5p transition is more pronounced than that of 5p ! 5d, which is contrary to the experimental result. To clarify the difference between the theory and measurement, one need to investigate the detailed absorption structures. These structures are caused by transitions from the bound states of gold ions. Therefore it is necessary to study the oscillator strengths of transitions in more detail. This paper considers the energy levels and oscillator strengths of lowly charged gold ions, taking Au11þ as an example. The main motivation is to obtain accurate oscillator strengths in large scale calculations. Because of many electrons of these ions, configuration interaction (CI) plays an important role in the calculation of oscillator strengths. The emphasis is to clarify how different degree of CI influences the values and redistributions of oscillator strengths. To the best of our knowledge, we found no work which reported the oscillator strengths for Au7þ –Au13þ ions. For atomic Au and more lowly charged ions, such as Auþ , Au2þ , and Au3þ , however, a lot of spectrum analyses had been carried out (see, for example [12–14]). 2. Theoretical method The calculations have been carried out using the flexible atomic code (FAC) developed by Gu [15]. A fully relativistic approach based on the Dirac equation is used throughout the entire package. An atomic state is approximated by a linear combination of configuration state functions (CSFs) with same symmetry Fa ðJpÞ ¼
nc X
ai ðaÞfa ðJpÞ,
(1)
i
where nc is the number of CSFs and ai ðaÞ denotes the representation of the atomic state in this basis. The CSFs are anti-symmetrized products of a common set of orthonormal orbitals which are optimized on the basis of the relativistic Hamiltonian. The radial orbitals are derived from a modified Dirac–Fock–Slater iteration on a fictitious mean configuration with fractional occupation numbers, representing the average electron cloud of the configurations included in the calculation. Once the CI wavefunctions have been obtained, the oscillator strengths can be calculated. In this work, we calculate the weighted oscillator strengths in length formalism !2 N X 2DE ! gf ¼ r p jFf i , (2) hFi j 3 p¼1 where DE ¼ E f E i , E i and E f are, respectively, the energies of initial and final states and g is the statistical weight of the lower state, i.e. g ¼ 2J i þ 1, for relativistic wavefunctions. For complex ions, CI plays an important role in the calculation of atomic structure. In the present paper, we study the convergence behavior of the energy levels and oscillator strengths by systematically increasing the electron correlations. In this systematic study, four cases are considered to show the effects of different electron correlations on the calculated energy levels and oscillator strengths. In case A, a single configuration Dirac–Fock (DF) calculation is carried out, in which intra-configuration correlation has been included. In case B, CI among the configurations of singly electron excitation to valence orbitals of 5d, 5f , 6s, 6p, and 6d from the ground configuration ð½Kr4s2 4p6 Þ4d 10 4f 14 5s2 5p6 have been included for both even and odd parity levels. In electron excitations, 4s and 4p electrons are kept full, the electrons in other orbitals can freely be excited to valence orbitals. Explicitly, CI has been included among following configurations: 4d 10 4f 14 5s2 5p5 nl, 4d 10 4f 14 5s5p6 nl, 4d 10 4f 13 5s2 5p6 nl, and 4d 9 4f 14 5s2 5p6 nl (nl ¼ 5d; 5f ; 6s; 6p, and 6d). In case C, in addition to the CI included in case B, configurations of double electron excitations to these valence orbitals
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have also been included. In case D, further electron correlations of triple electron excitations are taken into account. By adding the configurations step by step, we can see how the oscillator strengths converge with the included CI. 3. Results and discussions First, we check the variational behavior of energy levels of Au11þ with the CI. The number of levels included in case D exceeds 20 000, there is no necessity to write down all of them. Table 1 shows the lowest 33 levels which are relevant to our discussion. These levels belong to the ground and the first two excited configurations
Table 1 Lowest 33 energy levels (eV) of Au11þ obtained by using different electron correlations. In the designation of levels, full relativistic subshells have been omitted for simplicity No.
Designation
Jp e
A
B
C
D
1 2
5p6 5p33=2 5d 3=2
0 01
0.00 54.50
0.00 59.34
0.00 60.31
0.00 58.84
3
5p33=2 5d 3=2
11
55.84
60.70
61.64
60.19
4
31
57.57
62.41
63.52
62.04
5
5p33=2 5d 3=2 5p33=2 5d 3=2
21
57.91
62.82
63.72
62.30
6
5p33=2 5d 5=2
41
59.18
63.92
65.17
63.65
7
5p33=2 5d 5=2
21
60.09
64.98
65.80
64.41
8
5p33=2 5d 5=2
31
62.20
67.17
67.96
66.61
9
5p33=2 5d 5=2
11
67.81
72.62
73.23
71.47
10
21
77.27
81.93
82.59
81.01
11
5p31=2 5d 3=2 5p31=2 5d 5=2
21
80.43
85.17
85.50
84.02
12
5p31=2 5d 5=2
31
81.33
86.04
86.54
85.03
13
5p31=2 5d 3=2
11
86.71
90.86
90.00
87.97
14
4f 77=2 5d 3=2
21
97.39
101.83
91.25
89.55
15
4f 77=2 5d 3=2
51
98.35
102.86
92.17
90.47
16
4f 77=2 5d 3=2 4f 77=2 5d 3=2 4f 77=2 5d 5=2 4f 77=2 5d 5=2 4f 77=2 5d 5=2 4f 55=2 5d 3=2 4f 77=2 5d 5=2 4f 77=2 5d 5=2 4f 77=2 5d 5=2 4f 55=2 5d 3=2 4f 55=2 5d 3=2 4f 55=2 5d 5=2 4f 55=2 5d 5=2 4f 55=2 5d 5=2 4f 55=2 5d 5=2 4f 55=2 5d 5=2 4f 55=2 5d 5=2 4f 77=2 5d 5=2
31
99.12
103.70
92.94
91.25
41
99.52
104.11
93.34
91.65
11
101.00
105.45
94.69
93.00
61
101.10
105.57
94.85
93.18
21
101.46
105.99
95.18
93.51
41
101.98
106.48
95.65
93.96
31
102.43
107.01
96.12
94.46
41
102.60
107.17
96.30
94.63
51
102.68
107.27
96.37
94.71
21
102.79
107.35
96.42
94.72
31
103.76
108.37
97.32
95.63
01
103.81
108.18
97.31
95.65
11
104.63
109.35
98.38
96.69
51
105.39
109.87
98.88
97.21
21
106.62
110.32
99.29
97.63
31
105.52
110.98
99.90
98.24
41
105.78
111.19
100.09
98.43
11
106.41
111.12
100.81
98.76
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
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4d 10 4f 14 5s2 5p6 , 4d 10 4f 14 5s2 5p5 5d and 4d 10 4f 13 5s2 5p6 5d. The level designations are selected automatically by the computer program. It names the level according to the term linked with the largest eigenvector component in the matrix. However, it should be noted that there is a strong mixing of some levels. For simplicity, we omit the full relativistic subshells in the designation of levels. From Table 1, one can see that the energies of the excited configurations extend a wide range. For configuration of 5p5 5d, the level energies cover nearly 30 eV, while they cover less than 10 eV for 4f 13 5d. Note also that the number of levels belonging to 5p5 5d is smaller than those of 4f 13 5d, although the level energies of the former configuration span wider than the latter. From the inspection of Table 1, one can see the variational behavior of the energy levels with different degree of CI. With the increasing of configurations included in the calculation (from case A to case D), the relative energies of the low-lying levels (Nos. 2–10) increase with CI, while those of other levels (Nos. 11–33) are greatly reduced with more CI included. In principle, improvement should be feasible with more configurations (e.g. four electron excitations) included in the calculation. However, the effect of additional CI should be small with those included. In fact, there is a good agreement between the results of cases C and D. We believe that the results of case D have been converged. Therefore, the levels shown in Table 1 have been ordered according to the results of case D. Compared with the results of case D, case A predicts a reversed order for level Nos. 30, 31 and 32, 33, case B for Nos. 26, 27 and 32, 33, case C for 26, 27. For high-lying levels, more levels are reversed compared with the converged results. It indicates that CI effects are so strong that the labeling of the levels is not particularly meaningful. Therefore, many of them are just labels to identify the levels. Table 2 gives the weighted oscillator strengths gf for transition arrays of 5p6 ! 5p5 5d and 5p6 ! 4f 13 5d obtained by using CI of cases A, B, C and D. The variational trend of oscillator strengths is complicated with different electron correlations. For transition array of 5p6 ! 5p5 5d, the gf values of the two strong transitions decrease with the increasing of CI. For example, the gf value for 1–13 is 6.555 in single DF calculation, while it decreased to 5.071, 2.534, and 2.666 for CI of cases B, C, and D. This variational trend of oscillator strengths for strong transitions with CI is consistent with 3p ! 3d transitions of transition elements (for example, iron ions [16–18]). However, for 5p6 ! 4f 13 5d, the variational trend is different. The gf values for transitions of 1–28 and 1–33 are 0.434 and 0.695, respectively, in single DF approximation. With the increasing of CI, the gf value of 1–33 increases dramatically to 2.497, 3.297, and 2.980 for cases B, C, and D. The converged value is greater than the result of case A by a factor of 3.2. While for 1–28, the situation is completely on the opposite. The converged value of 0.176 is much smaller than that of case A, the former being just 40% of the latter. The increase or decrease of gf values with the CI are caused by the constructive- or destructive–interference effects in the radial portion of the oscillator strengths. Mixing of basis functions owing to CI effects produces large changes in computing oscillator strengths. For transition of 1–28, the CI effects are destructive, while for 1–33, they are constructive. On the whole, the CI effects result in the net transfer of oscillator strengths from transition array of 5p6 ! 5p5 5d to 5p6 ! 4f 13 5d. The redistribution of gf values between a transition array, or even between different arrays will surely have large effects on many practical applications of radiative atomic data, such as spectra analyses and radiative opacity calculation. The effects of CI on gf values can more obviously and clearly be seen from Figs. 1 and 2, which shows the total absorption cross section for the transition arrays of 5p5 5d ! 5p4 5d 2 and 5p5 5d ! 4f 13 5p5 5d 2 , respectively. In Figs. 1 and 2, plots A, C, and D corresponds to the results of cases A, C, and D. These two
Table 2 Weighted oscillator strengths of Au11þ obtained by using different electron correlations Lower
Upper
A
B
C
D
1 1 1 1 1
9 13 18 28 33
2.055 6.555 0.021 0.434 0.695
1.897 5.071 0.558 0.169 2.497
1.476 2.534 0.041 0.135 3.297
1.611 2.666 0.036 0.176 2.980
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176
500 400 A 300 200 100
Cross section (Mb)
0 400
C
300 200 100 0 400
D
300 200 100 0 60
70
80
90
Photon energy (eV) Fig. 1. The total absorption cross section caused by the transition array of 5p5 5d ! 5p4 5d 2 .
transition arrays are not possible in case B. The absorption cross section at photon energy hn is obtained by sðhnÞ ¼
X phe2 f 0 SðhnÞ ¼ 109:71f ll 0 SðhnÞ, me c ll ll 0
(3)
where hn is in eV and s in Mb, f ll 0 is the oscillator strengths of transition from level l to l 0 and S is the line profile which is taken to be a Voigt one pffiffiffiffiffiffiffiffi ln 2 Hða; vÞ, (4) SðhnÞ ¼ pGd where Hða; vÞ is the Voigt function Z a þ1 expðx2 Þ Hða; vÞ ¼ dx, p 1 a2 þ ðv xÞ2 a¼
pffiffiffiffiffiffiffiffi ln 2Gl =Gd ,
v¼
pffiffiffiffiffiffiffiffi ln 2ðhn hn0 Þ=Gd ,
(5)
where Gl is the Lorentzian half-width at half-maximum (HWHM) due to electron impact broadening, Gd is the Doppler HWHW. The Lorentzian and Doppler widths are assumed to be caused by the plasma condition under the prototype experiment carried out by Eidmann et al. [10].
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177
50 40
A
30 20 10
Cross section (Mb)
0 150
C
100 50 0 80
D
60 40 20 0 75
85
95 105 Photon energy (eV)
115
Fig. 2. The total absorption cross section caused by the transition array of 5p5 5d ! 4f 13 5p5 5d 2 .
From the inspection of Figs. 1 and 2, similar conclusion to the transition arrays of 5p6 ! 5p5 5d and 5p6 ! 4f 13 5d discussed above can be drawn for transition arrays of 5p5 5d ! 5p4 5d 2 and 5p5 5d ! 4f 13 5p5 5d 2 . Because of the complexity of the latter two transition arrays, there are far more strong transitions than the former two arrays. Fig. 1 shows that case A predicts the gf values more expanded than cases C and D predict. The convergence behavior can clearly be seen from plot A to D. Fig. 2 shows that the total absorption caused by transition array of 5p5 5d ! 4f 13 5p5 5d 2 . The oscillator strengths predicted by cases C and D are obviously enhanced compared with those of case A. Furthermore, additional relatively weak transitions occur when more CI has been considered. From the discussion mentioned above, CI should be important for all transition arrays of Au11þ . We have also checked the effects of CI on other gold ions near Au11þ , such as Au10þ , Au12þ , etc. For these ions, CI effects also play an important role on the energy levels and oscillator strengths. To save space, we do not discuss them in detail. The weighted oscillator strengths and transition energies for transition arrays of 5p5 5d ! 5p4 5d 2 and 5 5p 5d ! 4f 13 5p5 5d 2 are given in Table 3 for strong transitions whose gf values are greater than unity. For simplicity, only the converged results of case D have been given. The number of lower level refers to the Nos. given in Table 1. The designation and the total angular momentum of the upper level have been given in the second and third columns. The number in the fourth column gives the ordering of the upper level in the total number of energy levels. Unfortunately, we have no experimental or theoretical results to compare with. However, we shall carry out large scale calculations for Au7þ Au13þ to simulate the transmission of a gold plasma [10]. The degree of agreement will indirectly check the accuracy of the present calculations. In conclusion, four sets of calculations (A–D) with different degree of CI have been carried out to show its effects on the energy levels and oscillator strengths of Au11þ . The results show that the radiative atomic data
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Table 3 Weighted oscillator strengths and transition energies (in eV) of Au11þ for transition arrays 5p5 5d ! 5p4 5d 2 and 5p5 5d ! 4f 13 5p5 5d 2 obtained by using full electron correlations. Full relativistic subshells have been omitted for the designation of upper levels Lower
Upper J
2
ð5p1=2 5p33=2 Þ1 ð5d 23=2 Þ2
1
102
2.179
82.417
3
ð5p1=2 5p33=2 Þ1 ð5d 23=2 Þ2
2
109
2.892
83.246
3
1
129
2.023
87.369
0
137
1.101
89.469
2
140
1.474
90.624
3
75
1.476
66.980
2
82
1.231
70.615
3
110
1.209
81.606
4
111
2.834
82.237
4
116
6.440
82.944
4
133
3.069
86.273
3
142
3.344
89.007
3
150
4.609
90.041
4
ð5p1=2 5p33=2 Þ1 ð5d 23=2 Þ0 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ðð5p23=2 Þ2 5d 3=2 Þ7=2 5d 5=2 ðð5p23=2 Þ2 5d 3=2 Þ7=2 5d 7=2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ7=2 5d 5=2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 25=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ð5p1=2 5p33=2 Þ1 ð5d 23=2 Þ2 ðð4f 57=2 5p33=2 Þ4 5d 3=2 Þ11=2 5d 5=2
4
366
1.041
108.470
4
ð4f 57=2 5p33=2 Þ2 ð5d 25=2 Þ4
3
374
1.813
108.850
5
ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2
3
110
6.030
81.342
5
1
139
2.787
88.072
2
140
3.932
88.513
3
142
2.294
88.743
3
375
1.223
108.610
3
81
2.641
68.558
4
83
3.716
69.505
5
136
1.868
85.478
6
ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ðð4f 57=2 5p33=2 Þ2 5d 3=2 Þ3=2 5d 5=2 ð5p23=2 Þ2 ð5d 25=2 Þ4 ð5p23=2 Þ2 ð5d 25=2 Þ4 ð5p1=2 5p33=2 Þ2 ð5d 25=2 Þ4 ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ3=2 5d 5=2
4
144
9.633
87.712
6
ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ5=2 5d 5=2
5
145
12.232
87.771
6
ð5p1=2 5p33=2 Þ2 5d 3=2 Þ1=2 5d 5=2
3
146
7.756
87.788
6
ð4f 55=2 5p33=2 Þ3 ð5d 25=2 Þ2
5
398
1.627
108.290
6
ð4f 55=2 5p33=2 Þ3 ð5d 25=2 Þ2
5
406
2.223
108.690
6
ð4f 77=2 5p33=2 Þ5 ð5d 25=2 Þ4 ðð4f 55=2 5p33=2 Þ2 5d 3=2 Þ3=2 5d 5=2 ð4f 55=2 5p33=2 Þ3 ð5d 25=2 Þ2 ð4f 77=2 5p33=2 Þ2 ð5d 25=2 Þ4 ð4f 77=2 5p33=2 Þ5 ð5d 25=2 Þ4 ð5p23=2 Þ2 ð5d 25=2 Þ2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ2 ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ1=2 5d 5=2
5
416
1.392
108.970
4
431
3.898
109.540
4
444
1.280
110.000
3
459
1.306
110.510
5
509
1.704
114.150
1
86
1.027
69.565
3
142
3.147
86.634
2
148
6.138
87.359
ð5p1=2 5p33=2 Þ1 ð5d 23=2 Þ2 ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ3=2 5d 5=2 ð4f 55=2 5p33=2 Þ2 ð5d 25=2 Þ2 ð4f 77=2 5p3=2 Þ2 ð5d 25=2 Þ4 ðð4f 55=2 5p33=2 Þ3 5d 3=2 Þ5=2 5d 5=2
3
150
3.301
87.668
1
151
3.017
87.721
3
433
1.325
108.920
3
459
1.015
109.750
2
467
1.934
110.170
ð5p23=2 Þ2 ð5d 25=2 Þ2 ð5p23=2 Þ2 ð5d 25=2 Þ0
3
91
3.249
69.012
2
93
1.788
69.550
3 3 4 4 4 4 4 4 4 4
5 5 5 6 6 6
6 6 6 6 7 7 7 7 7 7 7 7 8 8
No.
gf
Upper designation
DE
ARTICLE IN PRESS J. Zeng et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 102 (2006) 172–180
179
Table 3 (continued ) Lower
Upper J
8
ðð5p1=2 5p33=2 Þ1 5d 3=2 Þ3=2 5d 5=2
2
156
2.612
87.108
8
4
157
10.159
87.277
3
158
8.093
87.472
2
162
2.478
88.186
2
486
1.523
109.120
4
491
2.791
109.380
8
ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ5=2 5d 5=2 ðð5p1=2 5p33=2 Þ2 5d 3=2 Þ5=2 5d 5=2 ð5p1=2 5p33=2 Þ2 ð5d 23=2 Þ0 ðð4f 55=2 5p33=2 Þ2 5d 3=2 Þ1=2 5d 5=2 ð4f 77=2 5p33=2 Þ4 ð5d 5=2 Þ0 ðð4f 55=2 5p33=2 Þ1 5d 3=2 Þ1=2 5d 5=2
3
494
3.161
109.680
9
ð5p3=2 Þ2 ð5d 25=2 Þ4
2
100
3.982
69.095
9
ð5p1=2 5p3=2 Þ1 ð5d 23=2 Þ2 ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ0
2
109
1.204
71.967
2
159
1.908
82.630
9 9 9
ðð5p1=2 5p3=2 Þ1 5d 3=2 Þ5=2 5d 5=2 ðð5p1=2 5p3=2 Þ2 5d 3=2 Þ3=2 5d 5=2
1 2 2
172 206 576
3.712 1.488 1.102
87.871 92.283 110.810
10 10 10
ðð5p1=2 5p3=2 Þ2 5d 3=2 Þ7=2 5d 5=2 ðð5p1=2 5p3=2 Þ2 5d 3=2 Þ3=2 5d 5=2 ð4f 55=2 5p1=2 Þ2 ð5d 25=2 Þ2
1 2 3
169 206 852
4.186 2.763 1.298
76.193 82.736 112.620
10
ð4f 55=2 5p1=2 Þ2 ð5d 25=2 Þ2
2
857
1.664
112.840
11
ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ2
1
166
1.831
72.116
11
ð5p1=2 5p3=2 Þ1 ð5d 25=2 Þ4
3
167
2.913
72.282
11
ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ4 ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ4 ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ2
2
170
2.503
73.333
3
173
5.696
75.900
2
178
1.728
76.697
11 11 11 11 11
5d 3=2 5d 5=2 5d 3=2 5d 5=2 5d 3=2 5d 5=2 5d 3=2 5d 5=2 ðð4f 55=2 5p1=2 Þ2 5d 3=2 Þ3=2 5d 5=2
4 3 2 1 3
390 393 409 415 849
1.026 1.035 2.694 2.790 1.294
87.457 87.573 88.392 88.595 109.470
11
ð4f 55=2 5p1=2 Þ3 ð5d 25=2 Þ2
3
870
1.704
110.610
12
ð5p1=2 5p3=2 Þ1 ð5d 25=2 Þ4 ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ4 ð5p1=2 5p3=2 Þ2 ð5d 25=2 Þ2
3
167
1.410
71.275
3
173
5.369
74.893
2
178
5.892
75.691
12 12 12 12 12
5d 3=2 5d 5=2 5d 3=2 5d 5=2 5d 3=2 5d 5=2 5d 3=2 5d 5=2 ð4f 55=2 5p1=2 Þ3 ð5d 25=2 Þ4
3 3 3 4 4
390 393 395 412 864
1.911 1.540 1.789 7.266 2.813
86.451 86.567 86.806 87.508 109.330
12
ðð4f 55=2 5p1=2 Þ2 5d 3=2 Þ1=2 5d 5=2
3
889
2.394
110.350
12
ðð4f 55=2 5p1=2 Þ3 5d 3=2 Þ3=2 5d 5=2
4
1020
2.645
115.780
12
ðð4f 55=2 5p1=2 Þ3 5d 3=2 Þ3=2 5d 5=2
3
1038
1.241
116.440
13 13 13
ðð5p1=2 5p3=2 Þ1 5d 3=2 Þ5=2 5d 5=2 ðð5p1=2 5p3=2 Þ2 5d 3=2 Þ3=2 5d 5=2 5d 23=2
1 2 2
172 206 466
1.553 3.769 3.916
71.373 75.784 86.558
13
5d 23=2
1
470
3.474
86.773
13
5d 23=2 ð4f 55=2 5p1=2 Þ3 ð5d 23=2 Þ2
0
521
1.416
91.043
1
1027
2.267
113.090
8 8 8 8
9
11 11
12 12
13
ð4f 55=2 5p33=2 Þ4 ð5d 25=2 Þ4
No.
gf
Upper designation
DE
ARTICLE IN PRESS J. Zeng et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 102 (2006) 172–180
180
are very sensitive to the CI included in the calculation. CI not only affects the distribution of the oscillator strengths among one particular transition array, but also does among different transition arrays. For 5p6 ! 5p5 5d and 5p5 5d ! 5p4 5d 2 , the oscillator strengths with the largest CI in this work are greatly reduced compared with the results of single configuration approximation. For 5p6 ! 4f 13 5d and 5p5 5d ! 4f 13 5p5 5d 2 , however, they are drastically enhanced. Acknowledgements This work was supported by the National Science Fund for Distinguished Young Scholars under Grant Nos. 10025416, the National Natural Science Foundation of China under Grant Nos. 10204024, 10373014 and 19974075, the National High-Tech ICF Committee in China, and China Research Association of Atomic and Molecular Data. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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