PHYSICAL REVIEW A 76, 022501 共2007兲
Relativistic all-order calculations of In I and Sn II atomic properties
U. I. Safronova*
Physics Department, University of Nevada, Reno, Nevada 89557, USA
M. S. Safronova†
Department of Physics and Astronomy, 217 Sharp Lab, University of Delaware, Newark, Delaware 19716, USA
M. G. Kozlov
Petersburg Nuclear Physics Institute, Gatchina 188300, Russia
共Received 4 May 2007; revised manuscript received 14 June 2007; published 7 August 2007兲
We use all-order relativistic many-body perturbation theory to study 5s2nl configurations of In I and Sn II.
Energies, E1 amplitudes, and hyperfine constants are calculated using all-order method, which accounts for
single and double excitations of the Dirac-Fock wave functions. A comprehensive review of experimental and
theoretical studies of In I and Sn II properties is given. Our results are compared with other studies were
available.
DOI: 10.1103/PhysRevA.76.022501
PACS number共s兲: 31.15.Ar, 31.15.Md, 32.10.Fn, 32.70.Cs
I. INTRODUCTION
In this work, we present a systematic calculation of various In I and Sn II atomic properties and study the importance
of the high-order correlation corrections to those properties
using relativistic all-order method. Previously these atoms
have been studied in a number of experimental and theoretical papers. First theoretical studies were published 30 years
ago by Migdalek 关1兴. They used relativistic semiempirical
method including exchange to calculate the oscillator
strengths in In I for the 5s25p j-5s2ns1/2, 5s26p j-5s2ns1/2,
5s25p j-5s2nd j, and 5s26s1/2-5s2np j transitions.
Later, the oscillator strengths determined from singleconfiguration relativistic Hartree-Fock 共RHF兲 calculations
were reported by Migdalek and Baylis 关2兴 for the lowest
5s25p j-5s26s1/2 and 5s25p j-5s25d j⬘ transitions. A quantum
defect theory was used by Gruzdev and Afanaseva 关3兴 to
calculate oscillator strengths f averaged over j in neutral
indium. Configuration interaction gf values for transitions
between the 5s26s1/2, 5s2nd j, and 5s2np j 共with n = 5 , 6兲 states
were reported for the indium isoelectronic sequence up to
Ba VIII in Ref. 关4兴. A self-consistent-field method was used to
generate one-electron orbitals. The method used in Ref. 关4兴
included relativistic effects albeit in an approximate way, and
the configuration interaction scheme accounts for correlation
effects 关4兴. Hartree-Fock calculations including relativistic
corrections and configuration interaction in an intermediate
coupling scheme were carried out in Ref. 关5兴 to analyze the
spectrum of Sn II. Transition probabilities for 36 lines of Sn II
arising from the 5s2ns, 5s2np, 5s2nd, 5s2nf, and 5s5p2 configurations of Sn II were evaluated in Ref. 关5兴 using the
Cowan code. Radiative transition probabilities and oscillator
strengths for 164 lines arising from the 5s2ns, 5s2np, 5s2nd,
5s2nf, 5s2ng, and 5s5p2 configurations of Sn II were calculated recently by Alonso-Medina et al. in Ref. 关6兴. These
*usafrono@nd.edu; On leave from ISAN, Troitsk, Russia
†
msafrono@udel.edu
1050-2947/2007/76共2兲/022501共10兲
values were obtained in intermediate coupling 共IC兲 using ab
initio relativistic Hartree-Fock 共HFR兲 calculations. The standard method of least square fitting of experimental energy
levels by means of computer codes from Cowan was used 关6兴
to calculate IC transition rates. Recently, energies of the
5s25p j, 5s5p2, 5s26s1/2, 5s25d j, and 5s26p j states in Sn II
were evaluated by Dzuba and Flambaum in Ref. 关7兴 using
many-body perturbation theory 共MBPT兲. It was underlined
that correlations and relativistic corrections were important.
The screening of the Coulomb interaction and hole-particle
interaction was included in all orders of the MBPT 关7兴.
The experimental study of atomic lifetimes in gallium,
indium, and thallium was carried by Andersen and Sørensen
关8兴 using beam-foil technique. Results for the 5s26s and
5s2nd 共n = 5 – 7兲 levels in In I were given in Ref. 关8兴. Lifetimes of the 5s2ns and 5s2nd 共n 艋 20兲 states in indium measured using pulsed laser excitation of an atomic beam were
reported by Jönsson et al. in Ref. 关9兴. Determination of radiative lifetimes of the 5s26p, 5s2ns, and 5s2nd 共n 艋 10兲 levels in In I using a pulsed laser was presented in Refs. 关10,11兴.
The atoms were excited in an atomic beam, with a nitrogenlaser-pumped dye laser. The fluorescence decay from the atoms was observed by a fast photomultiplier 关10,11兴. The
optical emission from a laser produced plasma generated by
1064 nm irradiation of Sn-Pb alloy targets at a flux of 2
⫻ 1010 W cm−2 was recorded and analyzed between 200 and
700 nm 关5兴. Experimental transition probabilities for 36 lines
of Sn II arising from the 5s2ns, 5s2np, 5s2nd, 5s2nf, and
5s5p2 configurations of Sn II were determined by AlonsoMedina et al. in Ref. 关5兴. Lifetime measurements for levels
arising from the 5s25d and 5s24f configurations in Sn II were
presented by Schectman et al. in Ref. 关12兴. These measurements utilized the University of Toledo Heavy Ion Accelerator Beam Foil Facility. The results were discussed in the
context of interpreting vacuum ultraviolet absorption spectra
observed with the Goddard High Resolution Spectrograph on
board the Hubble Space Telescope 关12兴.
Hyperfine structure of the 5s25p j states of In115 and In113
were measured by the magnetic-resonance method 关13,14兴.
022501-1
©2007 The American Physical Society
PHYSICAL REVIEW A 76, 022501 共2007兲
SAFRONOVA, SAFRONOVA, AND KOZLOV
An atomic beam irradiated by a narrow band dye laser was
used in Ref. 关15兴 to observe resonance fluorescence in free
indium atom. From the resonance frequencies, values for the
hyperfine structure of the 5s26s level in In115 and In113 were
derived by Neijzen and Dönszelmann in Ref. 关15兴. The spinforbidden 5s25p 2 P – 5s5p2 4 P In115 transition was analyzed
and absolute wavelengths, hyperfine constants A and B, as
well as improved energy level values were reported by
Karlsson and Litzén in Ref. 关16兴.
A high-resolution study of the ␭ = 451.1 nm transition in
In I using CW dye laser was reported by Zaal et al. in Ref.
关17兴. The blue dye laser setup was tested on the
5s25p3/2-5s26s1/2 ␭ = 451.1 nm transition in natural indium
关17兴. Proposal for laser on self-terminating transition in blue
spectral range on indium atom transition at 451.1 nm was
presented recently by Riyves et al. in Ref. 关18兴. The spectroscopy of dense In vapor was studied recently via resonant
pulsed
laser
excitation
at
␭ = 410.13 nm
共the
5s25p1/2-5s26s1/2 transition兲 关19兴.
In this paper, we conduct both relativistic many-body perturbation theory 共RMBPT兲 and all-order single-double 共SD兲
calculations of In I and Sn II properties. Such calculations
permit one to investigate convergence of perturbation theory
and estimate the uncertainty of theoretical predictions. We
evaluate reduced matrix elements, oscillator strengths, and
transition rates for possible 5s2nl-5s2n⬘l⬘ electric-dipole transitions in In I and Sn II and calculate the lifetimes of the
corresponding levels. Our results are compared with theoretical results from Refs. 关1,3,2,4–6兴 and with measurements
from Refs. 关8,9,11,10兴 in In I and Refs. 关5,12兴 in Sn II. We
also calculate hyperfine constants A for the 5s2np j共n = 5 – 8兲,
5s2ns1/2共n = 6 – 9兲, and 5s2nd j共n = 5 – 8兲 states in 115In using
the relativistic MBPT and SD all-order methods. Where possible, we compare our results with the measurements from
Refs. 关13–15兴.
We
consider
the
three-electron
system
关Ni兴4s24p64d105s2nl in In I and Sn II as a one-electron nl
system with 关Ni兴4s24p64d105s2 core. Recently, the relativistic all-order method was used to evaluate the excitation energies, oscillator strengths, transition rates, and lifetimes in
Ga I 关20兴 as well as in Tl I and Tl-like Pb 关21兴. The 关Ni兴4s2nl
states in Ga I were treated in Ref. 关20兴 as the nl one-electron
system with 关Ni兴4s2 core and 关Xe兴4f 145d106s2nl states in Tl I
and Pb II were evaluated in Ref. 关21兴 as the nl one-electron
system with 关Xe兴4f 145d106s2 core.
To summarize, this work presents both a systematic calculation of various properties of In I and Sn II, and a study of
the importance of the high-order correlation corrections to
these properties. We conclude that all-order SD method, in
general, produce more accurate values than the third-order
MBPT and can be used for the accurate calculation of In and
Sn+ properties. By comparing the all-order and third-order
MBPT results, we were able to study the relative importance
of the correlation corrections for different properties and
single out the cases where the treatment of In as a threeparticle system may be important, i.e., the cases where significant discrepancies between theory and experiment persist
even for the all-order calculations. The development of the
all-order approach that is capable to fully treat In or Sn+ as a
three-particle system is a difficult problem 关22–24兴, and the
initial studies of the applicability of the all-order method to
such systems may be useful. We find that the all-order SD
method works relatively well for In I even without explicit
consideration of the three-particle states. For Sn II, the convergence of MBPT expansion is worse than for In, particularly for d wave, where SD equations diverge. That is caused
by the strong interaction between 5s2nd configurations and
low-lying 5s5p2 configuration, which corresponds to the excitation from the core.
In the next section, we briefly review the RMBPT theory
and all-order SD method for the calculation of atomic properties of the atoms with one unpaired electron. The energies
are given in Table I. Extension of the theory to one-electron
matrix elements is discussed in Sec. III. Our results for E1
transitions are listed in Tables II–V. Calculated and experimental lifetimes for In are given in Table VI. Magnetic hyperfine structure of Ins discussed in Sec. IV and results are
summarized in Table VII.
II. ENERGIES OF In I AND Sn II
We start from the “no-pair” Hamiltonian 关26兴 in the second quantization form
H = H0 + VI ,
共1兲
H0 = 兺 ␧ia†i ai ,
共2兲
VI = 兺 gijkla†i a†j alak ,
共3兲
i
ijkl
where negative energy 共positron兲 states are excluded from
the sums; ␧i are eigenvalues of the one-electron DF equations with a frozen core, and gijkl is the Coulomb two-particle
matrix element.
Considering neutral In as a one-electron system we use
VN−1 DF potential 关Ni兴4s24p64d105s2 to calculate DF orbitals
and energies ␧i. There are a number of advantages associated
with this potential, including a greatly reduced number of the
Goldstone diagrams 关27兴, which leads to important simplifications in calculation. For example, when considering the
total energy of different valence states of a one-electron
atom, that energy can be written as
E = Ev + Ecore,
共4兲
where Ecore is the same for all valence states v. The firstorder correlation correction to valence removal energies vanishes for a VN−1 DF potential and the first nonvanishing corrections appear in the second order 关28兴:
E共2兲
v =兺兺
a
mn
gavmn共gmnav − gmnva兲
␧a + ␧v − ␧n − ␧m
+兺兺
n
ab
gnvba共gabnv − gabvn兲
.
␧a + ␧b − ␧n − ␧v
共5兲
We use indexes a and b to label core states and m and n to
designate any excited states. The second-order Coulomb-
022501-2
PHYSICAL REVIEW A 76, 022501 共2007兲
RELATIVISTIC ALL-ORDER CALCULATIONS OF In…
TABLE I. Valence energies in different approximations for In I and Sn II in cm−1. We calculate zeroth共3兲
order 共DF兲, single-double Coulomb correction ESD, and the part of third-order Eextra
which is not included in
the ESD. Breit corrections B共n兲 are calculated in first and second orders. The sum of these five terms ESD
tot is
共2兲
共3兲
compared with experimental energies ENIST 关25兴, ␦ESD = ESD
−
E
.
The
differences
␦
E
and
␦
E
between
NIST
tot
共3兲
共2兲
共0兲
共2兲
共1兲
共2兲
共3兲
total energies 共E共2兲
tot = E + E + B + B , Etot = Etot + E 兲 and experimental energies ENIST 关25兴 are given for
comparison.
nlj
EDF
ESD
共3兲
Eextra
B共1兲
B共2兲
5p1/2
5p3/2
5d3/2
5d5/2
6s1/2
6p1/2
6p3/2
6d3/2
6d5/2
7s1/2
7p1/2
7p3/2
8s1/2
4f 5/2
4f 7/2
5f 5/2
5f 7/2
7d3/2
7d5/2
8p1/2
8p3/2
8d3/2
8d5/2
9s1/2
−41 507
−39 506
−12 390
−12 374
−20 572
−13 979
−13 719
−6955
−6946
−9867
−7488
−7388
−5816
−6863
−6863
−4393
−4393
−4441
−4436
−4687
−4638
−3078
−3075
−3837
−5554
−5378
−1350
−1337
−2096
−964
−919
−558
−554
−584
−349
−335
−251
−118
−118
−67
−67
−285
−283
−168
−162
−171
−170
−132
913
912
161
160
232
113
112
69
68
72
41
41
32
14
14
8
8
35
35
20
20
21
21
17
105
73
2
1
12
13
10
1
1
4
5
4
2
0
0
0
0
0
0
2
2
0
0
1
In I
−146
−132
−4
−4
−18
−16
−16
−2
−2
−6
−6
−6
−3
0
0
0
0
−1
−1
−3
−3
−1
−1
0
5p1/2
5p3/2
6s1/2
6p1/2
6p3/2
7s1/2
4f 5/2
4f 7/2
7p1/2
7p3/2
8s1/2
5f 5/2
5f 7/2
8p1/2
8p3/2
9s1/2
−111 452
−107 358
−57 995
−44 483
−43 691
−30 735
−27 689
−27 691
−25 253
−24 917
−19 133
−17 759
−17 761
−16 354
−16 179
−13 070
−6848
−6719
−3597
−2244
−2133
−1230
−1189
−1193
−930
−895
−629
−670
−674
−487
−472
−301
1014
1032
467
271
258
172
147
147
115
109
84
85
86
60
57
47
206
146
35
39
28
14
0
0
16
12
7
0
0
8
6
4
Sn II
−233
−216
−44
−38
−37
−17
−1
−1
−16
−15
−9
−1
−1
−8
−8
0
ESD
tot
ENIST
␦E共2兲
␦E共3兲
␦ESD
−46 189
−44 031
−13 581
−13 554
−22 442
−14 833
−14 532
−7445
−7433
−10 381
−7797
−7684
−6036
−6967
−6967
−4452
−4452
−4692
−4685
−4836
−4781
−3229
−3225
−3951
−46 670
−44 457
−13 778
−13 755
−22 297
−14 853
−14 555
−7809
−7697
−10 368
−7809
−7697
−6033
−6963
−6962
−4450
−4450
−4834
−4808
−4843
−4789
−3334
−3315
−3951
−2163
−2040
256
260
−237
−150
−155
375
277
−95
−56
−58
−45
−4
−5
−3
−3
148
129
−28
−28
112
98
−24
547
557
493
493
249
122
115
479
379
64
45
42
27
9
8
5
5
201
181
21
21
144
129
15
481
426
197
201
−145
20
23
364
264
−13
12
13
−3
−4
−5
−2
−2
142
123
7
8
105
90
0
−117 313
−113 115
−61 134
−46 455
−45 575
−31 796
−28 732
−28 738
−26 068
−25 706
−19 680
−18 345
−18 350
−16 781
−16 596
−13 320
−118 017
−113 766
−61 131
−46 523
−45 640
−31 737
−28 731
−28 725
−26 114
−25 751
−19 615
−18 358
−18 352
−16 821
−16 630
−13 337
−2578
−2459
−641
−316
−297
−343
−24
−34
−107
−94
−162
−13
−22
−31
−29
−88
712
736
435
314
300
55
206
196
153
152
26
131
121
102
97
13
704
651
−3
68
65
−59
−1
−13
46
45
−65
13
2
40
34
17
022501-3
PHYSICAL REVIEW A 76, 022501 共2007兲
SAFRONOVA, SAFRONOVA, AND KOZLOV
TABLE II. Wavelengths ␭ 共Å兲, transition rates Ar 共s−1兲, oscillator strengths 共f兲, and line strengths S 共a.u.兲
for transitions in In I calculated in all-order perturbation theory. Numbers in brackets represent powers of 10.
Transition
5p1/2
5p1/2
5p1/2
5p1/2
5p1/2
5p1/2
5p1/2
5p3/2
5p3/2
5p3/2
5p3/2
5p3/2
5p3/2
5p3/2
5p3/2
5p3/2
5p3/2
5p3/2
6p1/2
6p1/2
6p1/2
6p1/2
6p1/2
6p1/2
6p3/2
6p3/2
6p3/2
6p3/2
6p3/2
6p3/2
6p3/2
6p3/2
6p3/2
6p3/2
7p1/2
7p1/2
7p1/2
7p1/2
7p3/2
7p3/2
7p3/2
7p3/2
7p3/2
7p3/2
7p3/2
4f 5/2
4f 5/2
4f 5/2
␭
Ar
f
S
Transition
6s1/2
4153 5.15关7兴 1.33关−1兴 3.64关0兴
5d3/2
3045 1.30关8兴 3.61关−1兴 7.24关0兴
7s1/2
2792 1.37关7兴 1.60关−2兴 2.93关-1兴
6d3/2
2578 3.63关7兴 7.24关−2兴 1.23关0兴
8s1/2
2493 5.81关6兴 5.41关−3兴 8.88关−2兴
7d3/2
2410 1.47关7兴 2.56关−2兴 4.06关−1兴
8d3/2
2329 7.49关6兴 1.22关−2兴 1.87关−1兴
6s1/2
4576 9.05关7兴 1.42关−1兴 8.56关0兴
5d3/2
3266 2.49关7兴 3.98关−2兴 1.71关0兴
5d5/2
3264 1.47关8兴 3.53关−1兴 1.52关1兴
7s1/2
2977 2.31关7兴 1.53关−2兴 6.02关−1兴
6d3/2
2734 6.88关6兴 7.71关−3兴 2.78关−1兴
6d5/2
2734 4.05关7兴 6.81关−2兴 2.45关0兴
8s1/2
2639 9.72关6兴 5.07关−3兴 1.76关-1兴
7d3/2
2547 2.79关6兴 2.71关−3兴 9.08关-2兴
7d5/2
2546 1.63关7兴 2.38关−2兴 7.97关-1兴
8d3/2
2456 1.43关6兴 1.29关−3兴 4.18关−2兴
8d5/2
2456 8.32关6兴 1.13关−2兴 3.65关−1兴
5d3/2 69 156 1.57关5兴 2.25关−1兴 1.03关2兴
7s1/2 22 594 3.48关6兴 2.66关−1兴 3.96关1兴
6d3/2 13 512 7.35关6兴 4.02关−1兴 3.58关1兴
8s1/2 11 463 1.13关6兴 2.22关−2兴 1.68关0兴
7d3/2
9903 3.79关6兴 1.12关−1兴 7.27关0兴
8d3/2
8665 2.19关6兴 4.92关−2兴 2.81关0兴
5d3/2 86 580 1.60关4兴 1.80关−2兴 2.05关1兴
5d5/2 84 890 1.03关5兴 1.66关−1兴 1.86关2兴
7s1/2 24 184 6.36关6兴 2.79关−1兴 8.88关1兴
6d3/2 14 065 1.54关6兴 4.56关−2兴 8.45关0兴
6d5/2 14 043 9.12关6兴 4.04关−1兴 7.48关1兴
8s1/2 11 858 1.96关6兴 2.06关−2兴 3.22关0兴
7d3/2 10 197 7.61关5兴 1.19关-2兴 1.59关0兴
7d5/2 10 190 4.54关6兴 1.06关−1兴 1.42关1兴
8d3/2
8889 4.31关5兴 5.11关−3兴 5.98关-1兴
8d5/2
8886 2.58关6兴 4.58关−2兴 5.36关0兴
6d3/2 229 885 2.00关4兴 3.17关−1兴 4.80关2兴
8s1/2 56 883 8.36关5兴 4.06关−1兴 1.52关2兴
7d3/2 31 928 1.33关6兴 4.05关−1兴 8.51关1兴
8d3/2 21 858 8.39关5兴 1.20关−1兴 1.73关1兴
6d3/2 306 748 1.68关3兴 2.37关−2兴 9.59关1兴
6d5/2 296 736 1.12关4兴 2.22关−1兴 8.67关2兴
8s1/2 60 643 1.53关6兴 4.20关−1兴 3.36关2兴
7d3/2 33 080 2.87关5兴 4.72关−2兴 2.05关1兴
7d5/2 33 003 1.70关6兴 4.16关−1兴 1.81关2兴
8d3/2 22 391 1.75关5兴 1.31关−2兴 3.87关0兴
8d5/2 22 371 1.04关6兴 1.17关−1兴 3.44关1兴
7d3/2 43 066 1.77关5兴 3.28关−2兴 2.79关1兴
7d5/2 42 937 8.24关3兴 2.28关−3兴 1.93关0兴
8d3/2 26 560 7.12关4兴 5.02关-3兴 2.63关0兴
4f 5/2
4f 7/2
4f 7/2
8p1/2
8p1/2
8p3/2
8p3/2
8p3/2
8p3/2
5f 5/2
5f 5/2
5f 7/2
6s1/2
6s1/2
6s1/2
6s1/2
6s1/2
6s1/2
5d3/2
5d3/2
5d3/2
5d3/2
5d3/2
5d3/2
5d5/2
5d5/2
5d5/2
5d5/2
5d5/2
5d5/2
7s1/2
7s1/2
7s1/2
7s1/2
6d3/2
6d3/2
6d3/2
6d3/2
6d5/2
6d5/2
6d5/2
6d5/2
6d5/2
8s1/2
8s1/2
7d3/2
7d5/2
7d5/2
022501-4
8d5/2
7d5/2
8d5/2
7d3/2
8d3/2
7d3/2
7d5/2
8d3/2
8d5/2
8d3/2
8d5/2
8d5/2
6p1/2
6p3/2
7p1/2
7p3/2
8p1/2
8p3/2
7p1/2
7p3/2
4f 5/2
8p1/2
8p3/2
5f 5/2
7p3/2
4f 5/2
4f 7/2
8p3/2
5f 5/2
5f 7/2
7p1/2
7p3/2
8p1/2
8p3/2
4f 5/2
8p1/2
8p3/2
5f 5/2
4f 5/2
4f 7/2
8p3/2
5f 5/2
5f 7/2
8p1/2
8p3/2
5f 5/2
5f 5/2
5f 7/2
␭
Ar
f
S
26 532
42 937
26 532
529 101
61 275
740 741
704 225
63 371
63 211
79 681
79 428
79 428
13 669
13 146
7002
6949
5806
5787
18 116
17 765
15 798
11 816
11 741
11 312
17 838
15 855
15 855
11 773
11 342
11 342
39 370
37 750
18 238
18 060
266 667
39 872
39 032
34 662
274 725
274 725
39 200
34 795
34 795
84 388
80 710
531 915
552 486
552 486
3.32关5兴
1.65关5兴
6.65关4兴
4.67关3兴
3.79关5兴
3.40关2兴
2.38关3兴
8.38关4兴
4.95关5兴
1.22关5兴
5.70关3兴
1.14关5兴
1.43关7兴
1.57关7兴
1.40关6兴
1.96关6兴
4.07关5兴
6.40关5兴
7.58关5兴
6.13关4兴
1.32关7兴
3.07关5兴
2.46关4兴
5.54关6兴
5.67关5兴
9.46关5兴
1.42关7兴
2.29关5兴
3.93关5兴
5.90关6兴
2.42关6兴
2.63关6兴
3.95关5兴
5.16关5兴
1.11关4兴
3.44关5兴
2.82关4兴
1.94关6兴
7.24关2兴
1.09关4兴
2.61关5兴
1.40关5兴
2.10关6兴
6.59关5兴
7.18关5兴
4.89关3兴
3.12关2兴
4.68关3兴
3.51关−2兴
3.42关−2兴
5.26关−3兴
3.92关−1兴
4.27关−1兴
2.80关−2兴
2.66关−1兴
5.05关−2兴
4.44关−1兴
7.75关−2兴
5.39关−3兴
8.09关−2兴
4.02关−1兴
8.13关−1兴
1.03关−2兴
2.84关−2兴
2.06关−3兴
6.42关−3兴
1.86关−2兴
2.90关−3兴
7.43关−1兴
3.21关−3兴
5.09关−4兴
1.59关−1兴
1.80关−2兴
3.56关−2兴
7.13关−1兴
3.17关−3兴
7.59关−3兴
1.52关−1兴
5.62关−1兴
1.12关0兴
1.97关−2兴
5.04关−2兴
1.77关−1兴
4.10关−2兴
6.44关−3兴
5.24关−1兴
8.19关−3兴
1.64关−1兴
4.01关−2兴
2.54关−2兴
5.07关−1兴
7.04关−1兴
1.40关0兴
3.11关−1兴
1.43关−2兴
2.85关−1兴
1.84关1兴
3.86关1兴
3.68关0兴
1.37关3兴
1.72关2兴
2.73关2兴
2.47关3兴
4.21关1兴
3.70关2兴
1.22关2兴
8.46关0兴
1.69关2兴
3.61关1兴
7.03关1兴
4.75关−1兴
1.30关0兴
7.86关−2兴
2.45关−1兴
4.45关0兴
6.78关−1兴
1.55关2兴
5.00关−1兴
7.87关−2兴
2.37关1兴
6.36关0兴
1.12关1兴
2.23关2兴
7.36关−1兴
1.70关0兴
3.40关1兴
1.46关2兴
2.79关2兴
2.37关0兴
6.00关0兴
6.23关2兴
2.15关1兴
3.31关0兴
2.39关2兴
4.45关1兴
8.89关2兴
3.10关1兴
1.74关1兴
3.49关2兴
3.91关2兴
7.45关2兴
2.18关3兴
1.56关2兴
3.11关3兴
PHYSICAL REVIEW A 76, 022501 共2007兲
RELATIVISTIC ALL-ORDER CALCULATIONS OF In…
TABLE III. Wavelengths ␭ 共Å兲, transition rates Ar 共cm−1兲, oscillator strengths 共f兲, and line strengths S 共a.u.兲 for transitions in
Sn II calculated using all-order method.
Transition
5p1/2
5p1/2
5p1/2
5p3/2
5p3/2
5p3/2
6p1/2
6p1/2
6p3/2
6p3/2
7p1/2
7p3/2
6s1/2
6s1/2
6s1/2
6s1/2
6s1/2
6s1/2
7s1/2
7s1/2
7s1/2
7s1/2
8s1/2
8s1/2
6s1/2
7s1/2
8s1/2
6s1/2
7s1/2
8s1/2
7s1/2
8s1/2
7s1/2
8s1/2
8s1/2
8s1/2
6p1/2
6p3/2
7p1/2
7p3/2
8p1/2
8p3/2
7p1/2
7p3/2
8p1/2
8p3/2
8p1/2
8p3/2
␭
Ar
f
S
Lower
Upper
␭SD
1780
1170
1024
1924
1231
1070
6859
3735
7300
3862
15 654
16 592
6813
6428
2852
2823
2255
2245
17 218
16 210
6625
6545
34 495
32 425
3.17关8兴
8.18关7兴
7.29关6兴
5.76关8兴
1.35关8兴
1.73关7兴
3.87关7兴
1.45关7兴
7.38关7兴
2.66关7兴
1.07关7兴
2.04关7兴
5.89关7兴
6.91关7兴
1.76关5兴
1.30关6兴
2.52关5兴
1.38关1兴
1.21关7兴
1.41关7兴
4.04关5兴
9.67关5兴
3.62关6兴
4.26关6兴
1.47关−1兴
1.65关-2兴
1.13关−3兴
1.56关−1兴
1.50关−2兴
1.46关−3兴
2.65关−1兴
3.00关−2兴
2.86关−1兴
2.94关−2兴
3.79关−1兴
4.06关−1兴
4.14关−1兴
8.63关−1兴
2.15关−4兴
3.11关−3兴
1.93关−4兴
2.08关−8兴
5.71关−1兴
1.18关0兴
2.72关−3兴
1.27关−2兴
6.95关−1兴
1.43关0兴
1.70关0兴
1.26关−1兴
7.55关−3兴
3.90关0兴
2.41关−1兴
2.04关−2兴
1.18关1兴
7.34关−1兴
2.71关1兴
1.49关0兴
3.84关1兴
8.72关1兴
1.86关1兴
3.67关1兴
4.05关−3兴
5.78关−2兴
2.86关−3兴
3.08关−7兴
6.69关1兴
1.30关2兴
1.20关−1兴
5.54关−1兴
1.64关2兴
3.16关2兴
5p1/2
5p3/2
5p1/2
5p3/2
5p1/2
5p3/2
5p1/2
5p3/2
5p1/2
5p3/2
5p3/2
5p1/2
5p3/2
5p3/2
5p1/2
5p3/2
5p3/2
5p1/2
5p3/2
5p3/2
6p1/2
6p3/2
6p1/2
6p3/2
6p1/2
6p3/2
6s1/2
6s1/2
6s1/2
6s1/2
6s1/2
6s1/2
6s1/2
6s1/2
7s1/2
7s1/2
8s1/2
8s1/2
9s1/2
9s1/2
5d3/2
5d3/2
5d5/2
6d3/2
6d3/2
6d5/2
7d3/2
7d3/2
7d5/2
8d3/2
8d3/2
8d5/2
7s1/2
7s1/2
8s1/2
8s1/2
9s1/2
9s1/2
6p1/2
6p3/2
7p1/2
7p3/2
8p1/2
8p3/2
4153
4576
2792
2977
2493
2639
2370
2502
3045
3266
3264
2578
2734
2734
2410
2547
2546
2329
2456
2456
22 594
24 184
11 463
11 858
9264
9520
13 669
13 146
7002
6949
5806
5787
Breit contribution B共2兲
is obtained from the E共2兲
expression
v
v
共5兲 by changing gijkl → gijkl + bijkl and keeping only terms that
are linear in bijkl that is a two-particle matrix element of the
Breit interaction 关29兴:
B=−
TABLE IV. Oscillator strengths f and wavelengths ␭ 共Å兲 in In I.
The SD data 共f SD兲 are compared with semiempirical calculations
共f SE兲 from Ref. 关1兴 and experimental data 共f expt兲 from Ref. 关56兴.
再
冎
1
␣
␣1␣2 − 关␣1␣2 − 共␣1r̂12兲共␣2r̂12兲兴 ,
2
r12
共6兲
where ␣1 is the Dirac matrix, r̂12 = r12 / r12, and ␣ is the finestructure constant. The first-order Breit correction is B共1兲
v
= 兺a关bvava − bvaav兴 = −兺abvaav, where direct term vanishes after summing over closed shells.
Even though the number of Goldstone diagrams for the
VN−1 DF potential is much smaller than in general case, the
third-order expression for energy correction still includes 52
terms. The corresponding formula for E共3兲
was presented by
v
Blundell et al. in Ref. 关30兴, where 52 terms were combined
into 12 groups with distinct energy denominators:
共3兲
共3兲
E共3兲
v = EA + ¯ + EL .
共7兲
Expression 共7兲 includes terms with one-, two-, three-, and
four-particle sums over virtual states in addition to sums over
core states.
The all-order SD method was discussed previously in
Refs. 关21,31–36兴. Briefly, we represent the wave function ⌿v
4102
4511
2754
2933
2460
2602
2340
2468
3039
3259
3256
2560
2713
2710
2388
2523
2521
2306
2432
2439
f SD
f SE
0.133
0.142
0.016
0.015
0.0054
0.0051
0.0025
0.0024
0.361
0.040
0.353
0.072
0.0077
0.068
0.026
0.0027
0.024
0.012
0.0013
0.011
0.266
0.279
0.0222
0.0207
0.00729
0.00664
0.402
0.813
0.0103
0.0284
0.00206
0.00642
0.137
0.153
0.0158
0.161
0.00541
0.00539
0.00256
0.00254
0.51
0.056
0.49
0.11
0.011
0.10
0.039
0.0033
0.035
0.017
0.0016
0.016
0.274
0.287
0.233
0.218
0.00764
0.00702
0.467
0.944
0.0110
0.0207
0.00223
0.00704
0.14
0.15
0.017
0.017
0.006
0.006
0.0029
0.0026
0.36
0.06
0.37
0.043
0.006
0.052
0.006
0.0014
0.009
0.0003
0.0013
of the atom with one valence electron as ⌿v ⬵ ⌿SD
v :
TABLE V. Transition probabilities A 共in 107 s−1兲 and wavelengths ␭ 共Å兲 in Sn II. Our SD results are compared with experimental data from Ref. 关5兴.
Lower
Upper
␭SD
␭expt
ASD
Aexpt
6s1/2
6p1/2
6s1/2
6p3/2
6p1/2
6p1/2
7s1/2
6p3/2
8s1/2
8s1/2
6813
6859
6428
3862
3735
6844
6761
6453
3841
3715
5.89
3.87
6.91
2.66
1.45
5.8± 1.1
4.2± 0.1
5.2± 1.0
2.5± 0.5
1.8± 0.3
022501-5
PHYSICAL REVIEW A 76, 022501 共2007兲
SAFRONOVA, SAFRONOVA, AND KOZLOV
TABLE VI. Lifetimes ␶ in ns for the nl levels in indium. The SD
data are compared with experimental results.
Level
␶SD
6s1/2
7s1/2
8s1/2
9s1/2
5d3/2
5d5/2
6d3/2
6d5/2
7d3/2
7d5/2
8d3/2
8d5/2
6p3/2
7.04
21.5
47.7
89.4
6.45
6.78
19.2
20.1
42.0
44.0
75.7
77.2
63.7
␶expt
Level
7.5± 0.7a
19.5± 1.5;c 19.5± 1.5;d 27± 6b
53± 5;c 55± 6b
118± 10;c 104± 12b
6.3± 0.5a
7.6± 0.5a
21± 3a
22± 3;a 18.6± 1.5;c 18.6± 1.5d
50± 5;a 200± 4b
50± 5;a 154± 10;c 147± 10b
317± 22c
300± 60;c 238± 20b
55.0± 4d
6p1/2
7p1/2
7p3/2
8p1/2
8p3/2
4f 5/2
4f 7/2
5f 5/2
5f 7/2
␶SD
69.7
219
192
473
414
70.4
70.4
125
125
a
Reference 关8兴.
Reference 关9兴.
c
Reference 关11兴.
d
Reference 关10兴.
b
冋
†
⌿SD
v = 1 + 兺 ␳maamaa +
+
ma
1
兺 ␳mnabam† a†nabaa
2 mnab
1
␳mvam† av + 兺 ␳mnvaam† a†naaav
兺
2 mna
m⫽
v
册
⌽v ,
共8兲
where ⌽v is the lowest-order atomic wave function, which is
taken to be the frozen-core DF wave function of a state v.
The coupled equations for the single- 共␳mv and ␳ma兲 and
double-excitation coefficients ␳mnva and ␳mnab are obtained
TABLE VII. Hyperfine constants, A 共in MHz兲 for 115In 共I
= 9 / 2, ␮ = 5.5408 关57兴兲. Dirac-Fock 共DF兲 and all-order 共SD兲 calculations are compared to experimental values.
Level
6s1/2
7s1/2
8s1/2
9s1/2
5p1/2
6p1/2
7p1/2
8p1/2
5p3/2
6p3/2
7p3/2
8p3/2
a
DF
Expt.
983.0
1812
1685a
335.6
544.5
153.6
240.8
83.10
128.1
1780
2306
2282b
222.7
263.2
85.15
95.61
41.90
45.97
267.8
262.4
242.2c
35.69
77.82
13.71
30.83
6.767
15.42
Reference 关15兴.
Reference 关13兴.
c
Reference 关14兴.
b
SD
Level
5d3/2
6d3/2
7d3/2
8d3/2
5d5/2
6d5/2
7d5/2
8d5/2
4f 5/2
5f 5/2
4f 7/2
5f 7/2
DF
SD
4.365 −11.48
2.305 −11.20
1.275
−7.692
0.805
−5.385
1.862
47.83
0.981
30.81
0.543
18.95
0.342
12.59
0.0611
0.1871
0.0316
0.1055
0.0339
0.2293
0.0176
0.1658
by substituting the wave function ⌿SD
v into the many-body
Schrödinger equation, with Hamiltonian given by Eqs.
共1兲–共3兲. Note that we again start from VN−1 DF potential. The
coupled equations for the excitation coefficients are solved
iteratively. In the following sections, the resulting excitation
coefficients are used to evaluate hyperfine constants and transition matrix elements.
The valence energy ESD
v is given by
ESD
v = 兺 g̃vavm␳ma +
ma
兺 gabvm˜␳mvab + mna
兺 gvamn˜␳mnva .
mab
共9兲
This expression does not include a certain part of the thirdorder MBPT contribution. This part of the third-order contribution Ev共3兲,extra is given in Ref. 关34兴 and has to be calculated
separately. We use our third-order energy code to separate
out Ev共3兲,extra and add it to the ESD
v . We drop the index v in the
共3兲
SD
E共2兲
,
E
,
and
E
designations
in the text and tables below.
v
v
v
We use B-splines 关37兴 to generate a basis set of DF wave
functions for the calculations of MBPT and all-order expressions. Typically, we use 40 or 50 splines of order k = 7 or 9,
respectively, for each partial wave 共see below for more details兲. Basis orbitals for In I and Sn II are constrained to cavities of radii R = 95 and 85 a.u., respectively. The cavity radii
are chosen large enough to accommodate all orbitals considered in this paper and small enough for 50 splines to approximate inner-shell DF wave functions with good precision.
Results of our all-order SD calculations of energies for the
lowest states of neutral In and In-like Sn ion are given in
SD
also includes the part of the
Table I. Our final answer Etot
共3兲
third-order energies omitted in the SD calculation Eextra
, as
共1兲
well as the first-order Breit correction B and the secondorder Coulomb-Breit B共2兲 correction. Theoretical values are
compared with the recommended values ENIST from the National Institute of Standards and Technology database 关25兴,
SD
␦ESD = Etot
− ENIST. For comparison, we also give the differences between the second-order and third-order MBPT calculations and experimental values in columns labeled ␦E共2兲
and ␦E共3兲. In Sn II the all-order SD equations for d–wave do
not converge and we exclude d orbitals of Sn II from Table I.
The largest correlation contribution to the valence energy
comes from the second-order term E共2兲. As we have discussed above, this term is simple to calculate in comparison
with E共3兲 and ESD terms. Thus, we calculate E共2兲 with better
accuracy than E共3兲 and ESD. To increase the accuracy of the
E共2兲 calculations, we use 50 splines of order k = 9 for each
partial wave and include partial waves up to lmax = 10. Then,
the final value is extrapolated to account for contributions
from higher partial waves 共see, for example, Refs. 关38,39兴兲.
We estimate the numerical uncertainty of E共2兲 caused by incompleteness of the basis set to be approximately 10 cm−1 or
less, depending on the valence state.
Owing to the numerical complexity of the ESD calculation, we use lmax = 6 and 40 splines of order k = 7. As we
noted above, the second-order E共2兲 is included in the ESD
value. Therefore, we use our high-precision calculation of
E共2兲 described above to account for the contributions of the
022501-6
PHYSICAL REVIEW A 76, 022501 共2007兲
RELATIVISTIC ALL-ORDER CALCULATIONS OF In…
higher partial waves by replacing E共2兲关lmax = 6兴 value with the
共2兲
:
final high-precision second-order value Efinal
SD
共2兲
= ESD + Efinal
− E共2兲关lmax = 6兴.
Efinal
The size of this correction varies from ⬃200 cm−1 for the
lowest valence states to ⬃1 – 20 cm−1 for other valence states
considered in this work.
A lower number of partial waves, lmax = 6, is used also in
the third-order calculation. Since the asymptotic l dependence of the second- and third-order energies are similar
共both fall off as l−4兲, we use the second-order remainder to
estimate the numerical uncertainties in the third-order and in
all-order corrections.
In our calculations of the Breit contribution, we use the
whole operator 共6兲 in the first-order correction B共1兲, while the
second-order Coulomb-Breit energies B共2兲 are evaluated using the unretarded Breit operator, also known as Gaunt 关it is
described by the first term in Eq. 共6兲兴. Usually Gaunt part
strongly dominates in the Breit corrections to the valence
energies 关40兴. Table I shows that there is strong cancelation
between first and second order corrections. It is in agreement
with the well known observation that Breit interaction for
valence electrons is screened by the core 关41,42兴.
We have also estimated Lamb shift correction to valence
energies. The vacuum-polarization was calculated in the Uehling approximation. The self-energy contribution is estimated for the s, p1/2, and p3/2 orbitals by interpolating the
values obtained by 关43–45兴 using Coulomb wave functions.
We found, as expected, that Lamb shift correction is very
small 共ELS 艋 3 cm−1 for In I and ELS 艋 10 cm−1 for Sn II兲.
This is well below the accuracy of the present theory, and we
neglect this contribution in Table I.
共2兲
Comparison of the differences ␦E共2兲 = Etot
− ENIST and
共3兲
共3兲
␦E = Etot − ENIST given in Table I shows that convergence of
MBPT series is not very good for both In and Sn+. In particular, the second-order results for d-wave in In and f-wave
in Sn+ are even better than the third-order ones. All-order
results are more accurate than the third-order ones, but the
difference is not very large. For p-waves, SD calculation
without the third-order correction overestimates valence
binding energies and underestimates it when this correction
is included. For the d-wave, both variants lead to underesti共3兲
worsens the
mation of the binding energy and term Eextra
agreement with the experiment.
We conclude that all-order calculation is generally more
accurate than the third-order MBPT calculation. Account of
the missing third-order terms does not lead to improvement
of the accuracy. On the other hand, this term is generally on
the order of our final difference with experiment and can
serve as an estimate of the latter. For most levels, our final
accuracy is better than 1%, but the accuracy for the d-wave
of In is noticeably worse. That can be explained by the existence of the low-lying configuration 5s5p2 which strongly
interacts with configurations 5s2nd. To account for this interaction effectively, one needs to consider In as a three electron
atom 关46兴. The same reason explains mentioned above divergence of the SD equations for the d-wave of Sn II. Interaction
between configurations 5s5p2 and 5s2ns is weaker and SD
equations for s-wave converge for both atoms considered
here. In the opposite parity there is no such a low-lying excitation of the 5s shell, so MBPT works better and no problems with convergence occur.
In order to study the relative role of the valence correlations we have performed the second-order RMBPT calculations of atomic properties of In I and Sn II considering these
atoms as three-electron systems. Corresponding variant of
RMBPT was developed in Ref. 关47–51兴. The energies of the
关He兴2s22p, 关He兴2s2p2, and 关He兴2p3 states of B-like systems
were presented in Ref. 关47兴. The second-order RMBPT was
used by Johnson et al. 关52兴 to calculate 关Ne兴3s23l and
关Ne兴3p23s states in Al I and 关Xe兴4f 145d106s26pl and
关Xe兴4f 145d106s6p2 states in Tl I. Comparing results obtained
for neutral B I, Al I, and Tl I, we find that the discrepancy
between RMBPT and experimental results increases significantly from B I to Tl I. For example, the RMBPT and NIST
values of the ns2np 关 2 P3/2- 2 P1/2兴 splitting in cm−1 for n = 2
are equal to 17 and 15; for n = 3 corresponding values are 123
and 112; finally, for n = 6 we get 6710 and 7793, respectively.
It is evident that for a light system, such as B I, the secondorder three-electron RMBPT treatment works much better
than for a heavy system, such as Tl I. For the latter case it is
more appropriate to consider Tl I as one-electron system with
关Xe兴4f 145d106s2 core but treat correlation more completely.
It was found in Ref. 关21兴 that in such approach the discrepancy between the SD and NIST values of the 6s26p
关 2 P3/2- 2 P1/2兴 splitting is only 41 cm−1 instead of 1083 cm−1
obtained in Ref. 关52兴. Alternatively, one can use CI+ MBPT
method 关23兴, where the discrepancy is 43 cm−1 关46兴.
The
main
difference
between
configurations
关Ni兴4s24p64d105s2nl of In-like ions and 关Ne兴3s2nl configurations of Al-like ions is the necessary size of the model
space for valence electrons. For 5l electrons in In-like ions,
we could not construct sufficiently complete three-electron
model space as we did for 3l electrons. Additionally, in Inlike ions the n = 4 core shell is not filled. Obviously, we cannot expect the same accuracy as in the case of Al-like ions
关50,52兴.
We tried two model spaces to evaluate energies of In-like
ions. Firstly we constructed the model space including 5s,
5p, and 5d electrons, 关spd兴 model space. Secondly, the oddparity model space was 关5s25p + 5p3兴 and even-parity model
space was 关5s25d + 5s5p2兴. We found that in the second case
the RMBPT energies were in better agreement with NIST
data 关25兴 than in the case of more complete 关spd兴 model
space. Theoretical values of the 5s25p 关 2 P3/2- 2 P1/2兴 splitting
were equal to 2669 and 4889 cm−1 in In I and Sn II, respecSD
values from
tively. Comparison of these values with the Etot
Table I 共2158 cm−1 in In I and 4198 cm−1 in Sn II兲 shows that
the one-electron representation with all-order treatment of
correlation correction gives the results that are in substantially better agreement with experiment than the threeelectron model space theory. Because of that, we decided not
include three-electron results in the present paper.
III. ELECTRIC-DIPOLE MATRIX ELEMENTS,
OSCILLATOR STRENGTHS, TRANSITION RATES,
AND LIFETIMES IN In I AND Sn II
The one-body matrix element of the operator Z is given
by 关31兴
022501-7
PHYSICAL REVIEW A 76, 022501 共2007兲
SAFRONOVA, SAFRONOVA, AND KOZLOV
Z wv =
具⌿w兩Z兩⌿v典
冑具⌿v兩⌿v典具⌿w兩⌿w典
共10兲
,
where ⌿v,w are exact wave functions for the many-body “nopair” Hamiltonian H
H兩⌿v典 = E兩⌿v典.
共11兲
In MBPT, we expand the many-electron function ⌿v in powers of VI as
共1兲
共2兲
共3兲
兩⌿v典 = 兩⌿共0兲
v 典 + 兩⌿v 典 + 兩⌿v 典 + 兩⌿v 典 + ¯ .
共12兲
The denominator in Eq. 共10兲 arises from the normalization
condition that starts to contribute in the third order 关53兴. In
the lowest order, we find
共0兲
共0兲
Z共1兲
wv = 具⌿w 兩Z兩⌿v 典 = zwv ,
共13兲
where zwv is the corresponding one-electron matrix element.
Since ⌿w共0兲 is a DF function, we use ZDF designation instead
of Z共1兲 below.
The second-order Coulomb correction to the transition
matrix element in the case of VN−1 DF potential is given by
关54兴
Z共2兲
wv = 兺
na
共gwavn − gwanv兲zna
zan共gwnva − gwnav兲
+兺
.
␧a + ␧v − ␧n − ␧w
na ␧a + ␧w − ␧n − ␧v
共14兲
The second-order Breit corrections are obtained from Eq.
共14兲 by changing gijkl to bijkl 关29兴. The third-order Coulomb
correction is obtained from Eqs. 共10兲 and 共12兲 as
IV. HYPERFINE CONSTANTS FOR INDIUM
共0兲
共2兲
共1兲
共2兲
共0兲
共1兲
Z共3兲
wv = 具⌿w 兩Z兩⌿v 典 + 具⌿w 兩Z兩⌿v 典 + 具⌿w 兩Z兩⌿v 典
Z共1兲
共1兲
共1兲
共1兲
− wv 关具⌿共1兲
v 兩⌿v 典 + 具⌿w 兩⌿w 典兴,
2
共15兲
where the last term arises from the normalization condition.
In Ref. 关53兴, contributions to Z共3兲
wv were presented in a following form:
RPA
+ ZBO + ZSR + Znorm .
Z共3兲
wv = Z
共16兲
The first term here corresponds to the well known random
phase approximation 共RPA兲. Though RPA corresponds to the
summation of certain MBPT terms to all orders, it is possible
to include it here using the procedure described in Ref. 关53兴.
Next term ZBO corresponds to the correction which arise
from substituting DF orbitals with Brueckner ones. The last
two terms in Eq. 共16兲 describe structural radiation ZSR and
normalization Znorm corrections.
In the all-order SD calculation, we substitute the all-order
SD wave function ⌿SD
v into the matrix element expression
given by Eq. 共10兲 关31兴:
ZSD
wv =
zwv + Z共a兲 + ¯ + Z共t兲
冑共1 + Nw兲共1 + Nv兲
,
expression completely incorporates Z共3兲 and certain sets of
MBPT terms are summed to all orders 关31兴. The part of the
fourth-order correction that is not included in the SD matrix
element 共17兲 was recently discussed by 关55兴, but we do not
include it here.
In Tables II and III, we present theoretical transition rates
Ar, oscillator strengths f, and line strengths S for E1 transitions between low-lying states of In I and Sn II, respectively.
These results are obtained by combining all-order E1 ampliSD
tudes 共17兲 in the length gauge and theoretical energies Etot
from Table I using well-known expressions 共see, for example, Ref. 关25兴兲.
Calculation of the transition amplitudes provides another
test of the quality of atomic-structure calculations and another measure of the size of the correlation corrections. In
Tables IV and V, we compare our results with available experimental data. For convenience, we also present theoretical
and experimental wavelengths for all transitions. There is
good agreement with experimental results for the strongest
lines of In. For Sn II, agreement is also good with exception
of the 6s-6p3/2 transition where experimental value is much
smaller than the calculated one. Note that the theory and
experiment are in good agreement for the 6s-6p1/2 transition.
We also use E1 transition rates to calculate the lifetimes
of low-lying levels of In I and Sn II. We compare these lifetimes ␶共SD兲 with available experimental measurements in
Table VI. For 7d j levels, the measurements from Refs.
关8,9,11兴 gave rather different lifetimes. Our calculations support the shorter times obtained in Ref. 关8兴.
共17兲
where zwv is the DF matrix element 共13兲 and the terms Z共k兲,
k = a ¯ t are linear or quadratic function of the excitation
coefficients introduced in Eq. 共8兲. Normalization terms Nv,w
are quadratic functions of the excitation coefficients. This
Calculations of hyperfine constants follow the same pattern as calculations of E1 amplitudes, described in the previous section. The value of the nuclear magnetic moment for
115
In used here is taken from Ref. 关57兴. Hyperfine constants
for another odd isotope 113In can be obtained using the scaling factor 0.99785, which is indistinguishable from unity
within the accuracy of the present theory. In contrast with
dipole amplitudes considered above, the hyperfine structure
is sensitive to the wave function at short distances and to
very different types of correlation corrections.
Table VII shows that SD method significantly improves
DF values of the hyperfine constants of the lowermost levels.
It is rather unusual that correlation correction to the hyperfine structure constant of 5p3/2 level is so small. For other
p3/2 levels, correlation corrections are comparable to the initial DF contribution. This situation is more typical for other
atoms with ns2np3/2 configuration, such as Tl 关46,58兴.
V. CONCLUSION
Summarizing results of the previous sections, we can
make several conclusions. We have seen that all-order SD
calculations, when converge, provide an improvement to the
third-order MBPT calculation. Convergence of the SD equations is hampered by the existence of low-lying excitations
from the uppermost core shell 5s. The lowest such excitation
corresponds to configuration 5s5p2 that has positive parity
022501-8
PHYSICAL REVIEW A 76, 022501 共2007兲
RELATIVISTIC ALL-ORDER CALCULATIONS OF In…
and primarily affects SD equations for the valence d-wave.
Because of that, we were not able to solve these equations
for Sn II. To avoid this problem one has to exclude
5selectrons from the core and consider In I and Sn II as
three-electron systems. However, our attempt to treat In I
and Sn II as three-electron systems within second-order
RMBPT for the valence model space, as suggested in Refs.
关47兴, resulted in rather poor agreement with experimental
spectra. We conclude that valence correlations for atoms in
question cannot be accurately accounted within model space
approach. It would be interesting to perform CI+ MBPT calculations 关23兴, but it goes beyond the scope of the present
paper.
Another interesting observation concerns the addition of
共3兲
to the SD rethe missing part of the third-order term Eextra
sults. It was suggested in Ref. 关34兴 to add this term, so that
all third-order terms are accounted for. For heavy alkali-
metal atoms omission of this term leads to significant discrepancies of the all-order values with experiment. One can
see from Table I, that for atoms considered here this term
does not improve agreement with experimental energies. We
have also found that first- and second-order Breit corrections
tend to cancel each other in agreement with Ref. 关42兴; final
Breit corrections are small and can be neglected within
present accuracy of the theory.
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ACKNOWLEDGMENTS
The work of M.S.S. was supported in part by National
Science Foundation Grant No. PHY-0457078. M.G.K. acknowledges support from Russian Foundation for Basic Research, Grant No. 05-02-16914, grant from Petersburg State
Scientific Center, and thanks University of Delaware for hospitality.
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