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Energy Levels of the Single Excited States in the Boron Isoelectronic Sequences.

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Annalen der Phvsik. 7. Folee. Band 46, Heft 2. 1989. S. 105-112
VEB J. A. Barth, LeiDzie
Energy Levels of the Single Excited States in the Boron
Isoelectronic Sequences
By TH. M. EL-SIIERBINI
and H. M. M. MANSOUR
International Centre for Theoretical Physics, Trieste, Italy
and
A. A. FARRAG
and A. A. RAHMAN
Cairo University, Cairo, Egypt
1s22sznd (2D)and
Abstract. Energy levels of the single excited 1s22s2ns (V), l s 2 2 s 2 n p (2P),
(2P),
n = 3-8 states for the boron isoelectronic sequence are calculated using the simple
configuration Hartree-Fock method. Good agreement is obtained between our results and previous
experimental and theoretical data.
ls22s2nf
Energieniveaus der einfach angeregten Zustande der isoelektronischen Boronreihe
I n h e l t s u bersic ht. Die Energieniveaus der einfach angeregten Zustande (1s22s2ns (W), l s 2 2 s 2 n p
(2P),l s 2 2 s 2 n d (2D)und 1s22sznf (2F),n = 3-8) der isoelektrischen Boronreihe wurden unter Benutzung der Hartree-Fockmethode berechnet. Eine gute Ubereinstimmung mit bisherigen experimentellen und theoretischen Resultaten wird erzielt.
1. Introduction
Boron-like ions are observed in many astrophysical objects, such as solar corona [ 11.
The intercombination lines of the low members of the sequence form some of the most
prominent features of quasar spectra.
Energy level calculations for the boron sequence provide useful data which can be
used to interpret and identify the spectra of solar corona.
2. Method of Calculation
The Hartree-Fock programme of Froese Fischer [21 was used in calculating the average
energies of the various configurations and the method of E d l h [3] for extrapolating
level intervals in terms of Slater parameters was applied in order t o obtain the term
energies of the levels. The method of calculation is explained elsewhere [4,51.
3. Results and Discussion
Our results of the excitation energies calculated in cm-l of the various configurations
relative to the ground state lsZ2s22.p configuration are given in Tables 1-4. A comparison
between our calculated energy levels of the boron isoelectronic sequence and the experimental data of Moore [6] and Bromander [7] are given in Tables 5- 14. Table 16 shows
Ann. Physik Leipzig 46 (1989) 2
106
Table 1. Excitation energies of the lsL2s2nslevels of the boron isoelectronic sequence (in cm-l)
(the energies are relative t o the ground state 1d22~P2p
configuration)
ls22s"s ( 2 8 )
ls22s23s( W ) ls22s24s(W)ls22s25ss(2S) ls22s26s(28)ls22s27s( W ) ls22s28s(%')
BI
c I1
K I11
0 IV
FV
Ne VI
Na V t t
Mg V I l I
A1 IX
Si X
38799.1
ll4J54.6
221 753.1
359 553.0
538028.6
727 109.4
956 756.9
1216948.6
1507672.7
1828924.4
52 564.2
155250.9
300132.3'
486 713.3
714783.9
984244.2
1295046.8
1647 173.2
2040623.1
2475407.9
57443.5
170 948.9
331 841.4
539493.2
793647.5
1094 184.8
1441050.6
1834226.3
227371.2
2 759524.9
59 737.0
178 689.3
347 842.3
566 500.3
834381.9
1151867.1
1517 366.9
1932390.8
2 396 431.8
2 909508.6
60997.5
183074.1
357 041.4
682 1 6 . 6
868 164.3
1184868.4
1562 247.5
1990270.3
24G8940.1
2998276.5
GI 764.1
185 797.8
362814.8
592660.5
873231.4
1206 187.6
1590866.2
2 027 245.3
2 515328.2
3 055 135.4
Table 2. Excitation energies of the 1s22ds2nplevels of the boron isoelectronic sequence (in cm-I)
(the energies are relative t o the ground state ls22s22pconfiguration)
BI
c I1
N I11
0 IV
FV
Ne V I
Na V11
Mg V I I I
A1 IX
Si X
ls22s2np (2P)
ls22s23p( 2 P )ls22s24p( 2 P ) ls22s25p(")
ls22sz(ip( 2 P ) ls22s27p(")
ls22s28p(")
46667.5
131466.6
247 136.2
393462.3
570383.7
777881.9
1016956.5
12846 17.7
1583882.6
19133773.7
62005.9
186403.0
.363807.7
593450.1
875022.9
1208385.4
1593474.5
2 030268.8
2518772.4
3059 006.3
55172.3
161211.3
309494.6
499473.1
730938.9
1003801.2
1318021.9
1 673592.3
2070321.2
2508829.9
58 620.9
173744.8
336313.1
545655.4
801 507.9
1103752.7
1462337.9
1847 247.8
2288 488.3
2770080.7
60366.7
180221.0
350320.6
5G9939.3
838 789.8
1156741.4
1523735.9
1939754.5
2404802.1
2918 899.9
61373.1
1840U2.9
358556.2
584278.9
860870.9
1188194.9
15FG189.9
1994835.3
2474135.4
3004 111.3
Table 3. Excitation energies of the ls22sZndlevels of the boron isoelectronir sequence (in cm-l) (the
energies are relative t o the ground state ls22s22pconfiguration)
BI
c I1
N 111
0 IV
F V
Ne V I
~a VII
Mg V I I I
A1 IX
Si X
1s22s23d(")
is22s24d)D"(
ls22s2nd ( 2 0 )
la22s25d( 2 D ) ls22s26d(*0)
ls22s27d(20)
ls22s28d(20)
51 606.5
14373.6
26G 764.4
420323.2
604 398.5
819015.6
1064206.5
1340 006.1
1646452.4
1983587.6
56997.9
165 725.0
316720.7
509414.5
743 610.1
1019 2221.9
1336215.9
1694586.2
2 094 34 5.9
2535521.1
59498.7
175914.7
339 796.1
550464.9
807 GGO.4
11112G4.1
1461224.1
1857157.9
2300174.2.
2 789194.8
60856.2
181432.0
352 268.5
572626.3
842247.7
1160971.1
1528747.7
1945558.4
2411 408.9
2 926 321.2
G1673.8
184747.9
359 756.6
585943.9
863 008.5
1190 813.5
1569 295.8
1998435.1
2478236.0
3008 720.3
62 203.9
186893.9
364599.8
594 550.3
876437.3
1210119.3
1595532.6
2 032 655.6
2521492.7
3 062 065.2
TH. M. EL-SHERBINI
and H. M.M. NANSOUR,
Levels in the Boron Isoeleetronic Sequences
107
T<ible 4. Excitation energies of the ls22s2nf levels of the boron isoelectronic sequence (in cn-')
(t lie energies are relative t o tlic ground state 1s22sL2pconfiguration)
B1
c JI
N tr[
0IV
PV
Ke VI
Nib VII
Mg VIlI
A1 I X
Si X
1s22s24f( 2 P )
lr22P5f (LP)
570i2.9
166409.8
318507.5
512559.7
748 247.G
1025498.5
1344040.1
1704662.6
2105499.9
2648375.9
:,9 541.1
176285.5
340735.6
552087.3
810021.7
1114395.3
1466144.3
18G2247.7
2305710.3
2795563.8
ls~2s2nf(2P)
('P) ls22s27f (2B')
60881.9
181652.1
352817.6
673575.8
843 606.7
7162765.4
1530!387.1
1948249.8
241
7.9
2929932.9
60874.6
164888.4
3eo 103.9
686535.1
863861.1
1191936.0
1570694.3
2000113.7
2480198.2
3010369.1
1822s28f(*F)
-
62215.3
186988.9
36,4833.1
594 945.9
877006.5
1210867.9
169G464.3
2033773.1
2522798.0
3063560.6
Table 5. A comparison between our calculations for the energy levels of B I and the observed levels
by Moore [GI
Level
This work
Ref. [GI
Level
This work
Ref. [6]
ls22s23s( 2 S )
7 ~ 2 2 ~ (2s)
2 4 ~
lS"S259 ( 2 8 )
ls22s26s ( 2 S )
ls22s27a( W )
38799.1
525G4.2
57443.5
59737.0
60997.<5
40 040
ls22s23d (20)
1s22s24d(20)
l,?2s25d ( 2 0 )
ls22s2Gd(20)
ls22s27d ( 2 D )
ls2?s28d ( 2 0 )
51606.5
54 765
59 989
62 481
63 847
64664
65195
55 009
60 146
62 098
64 156
56997.9
59 498.7
60896.2
61673.8
62203.9
Table 6. 9 comparison between our r,%lcuhtionsfor the energy levels of C I1 and the observed
levels by Moore [GI
Level
Tliir work
1 ~ 2 2 ~ ~ (W)
33s
114 754.6
lt55250.9
ls22v24s ( 2 8 )
ls22s'S~(28) 170948.9
ls22s23p( 2 P )
13146.6
161211.3
1s*2,s24p(2P)
178 744.8
ls229'5p ( 2 P )
Ref. [C]
Level
116,537.!1
137 234.4
173348.2
131724.7
162618.7
175387.9
ls22s2Rd(20) 143 737.6
165725.0
1<22s24d(2U)
175914.7
1922s25d ( 2 0 )
181432.1
ls22s26d ( 2 0 )
1 ~ ~ 2 s ~ 4 f ( ~ P166409.8
)
ls22s25f ( 2 F )
176285.5
ls22s2Gf (2F)
181652.1
This work
Ref.
[GI
145549.9
168 123.9
178494.8
184064.9
168 123.9
178966.5
184376.2
a comparison between our results and the available theoretical results of Dankwort and
Trefftz [8]. The cornparison between our results of the energy levels and the observed
for B 1 which decreases with the increase cf Zamong
ones show a tleviation of about
the members of the isoclectronic sequence until i t reaches about 0.01% for Na VII.
A large deviation for I3 I is expected since we use in our calculations the single configuration one electron model. This model is suitable for t h e calculations of the energy levels
Ann. Physik Leipzig 46 (1989) 2
108
Table 7. A comparison between our calculation8 for the energy levels of N I11 and the observed
levels by Moore [6]
Level
This work
Ref. [6]
Level
This work
Ref. [GI
ls22s23s(W)
ls22s24s(2S)
ls22s25s(2S)
1.s22s23p( 2 P )
ls22s24p(2P)
221 753.1
300132.3
331841.4
247 13G.2
309494.6
221302.4
301088.2
333 713.0
245665.7
311691.3
1922823d ( 2 0 )
1s22s24d(2D)
ls22s25d(2D)
1s2Bs24f( 2 F )
ls22s25f ( 2 F )
ls22s26f (2F)
BGG 764.4
316 720.7
339 796.1
318507.5
340735.5
352817.6
267 238.5
317 750.8
341946.2
320287.5
342 752.0
354956.7
Table 8. A comparison between our calculations for the excited levels of 0 I V and the observed ones
by Moore [6] and Bromander [7]
Level
This work Ref.
359553.0
1s22s23s(2S)
1s22s24s(W) 486713.3
539493.2
1s22s25s(W)
ls22s23p (2P) 393462.3
1s22s24p(2P)
499473.1
ls22s25p (2P) 545 655.4
ls22s23d (20)
420323.2
1s22s24d(20) 509414.5
[GI
Ref. [7]
Level
366 714.8 357 614.3
485823.1 485821.7
539368.0 539368.0
390161.1 390248.0
500037.8
546818.7
419533~5 419550.6
510560.0 510574.1
This work Ref. [ G ]
Ref. [7]
ls22s25d(20) 550464.9 552034.0
ls22s26d (20)
572636.3 574373.0
1s22s24f( 2 P ) 512559.7
513 198.3
ls21s25f (2P) 552087.2 550490.0
ls22s2Gf(2F) 573575.8
574 807.7
ls22s27f(2F) 586535.0
58785.0
1s22a28f(2P) 594945.9
597352.0
Table 9. A comparison between our calculations for the energy levels of F V and the observed
levels by Moore [6]
Level
This work
1s22s23s(W) 528028.6
1s22s24s(W)
714783.9
1s22s23p(2P) 570383.7
Ref. [GI
Level
This work
Ref. [6]
524 751.0
7 12 936.0
56 5 367.0
ls22s23d(”)
ls22s24d ( 2 0 )
ls22s25d ( 2 0 )
ls22s2Gd ( 2 0 )
604398.4
743 610.1
807 660.4
842247.7
602 476.0
744010.0
808663.0
843497.0
Table 10. A compnrison between our calculations for the excited levels 1 ~ ~ 2 8 ~1s22s23p
38,
and
ls22s23d for Ne VI and the observed levels by Moore [6]
Level
This work
ls22s23s( W ) 727 109.4
1.F22.s23p(2P) 771881.9
ls22s23d(2D)
819015.6
Ref. [6]
722 610.0
763096.0
816405.0
TH. M. EL-SHERBINI
and H. M. M. MANYOUR,
Levels in the Boron Isoelectronic Sequences
109
Table 11. A comparison between our calculation for the energy levels of Na VII and the observed
levels by Moore [6]
Level
This work
ls22s23s(V)
95675.9
ls22s24a(W)
1296046.8
1 ~ 2 2 ~(2P)
2 3 ~ 1015 956.5
Ref. [6]
Level
This work
Ref. [S]
951347.0
1294914.0
1008418.0
ls22s23d (W)
ls22s24d (2D)
ls22s25d (2D)
ls22s26d(2D )
1s22s27d(2D)
lse2s28d (2D)
1064206.5
1336215.9
1461224.1
1528 747.7
1569295.8
1595532.6
1060580.0
1335809.0
1461518.0
1529 463.0
1570078.0
1596400.0
Table 12. A comparison between our calculations for the excitation energy levels of Mg VIII and
the observed levels by Moore [6]
Level
This work
Ref. [6]
Level
ls22s23s(2S)
1~228~4s
(W)
121648.6
1647 173.2
1210689.0
1647 879.0
1622823d ('20) 1340006.1
ls22s24d(2D) 1694586.2
is22s25d (2D) 1857157.9
This work
Ref. [6]
1335863.0
1693824.0
1858322.0
Table 13. A comparison between our calculations for the energy levels of A1 IX and the observed
levels by Moore [6]
Level
This work
Ref. [6]
Level
This work
ls22s23s(2S)
1507 622.7
1501020.0
1s22s23d(2D)
ls22s24d (2D)
ls22s25d (2D)
1646452.4
2094345.9
2300174.2
Ref. [GI
1642140.0
2094020.0
2301150.0
Table 14. A comparison between our calculations for the energy level ls22s23d of Si X and the
observed level by Moore [6]
Level
This work
Ref. [6]
ls22s23d(2D)
1983587.6
1979260.0
Table 15. A comparison between our calculations of 1s22s23slevel in B I, C I1 and N I11 and the
theoretical results by Dankwort and Trefftz [8]
BI
c I1
N I11
Total energy in a . u.
Level
This work
Ref. [8]
1 9 2 ~ (28)
~ 3 ~ -24.3582846
1 ~ ~ 2 (~'9)~ 3 9 -36.7830916
1 ~ ~ 2 ( 2~8 )2 3 ~ -51.8328158
-24.416934
-36.836916
-51.897251
Ann. Physik Leipzig 46 (1989) 2
110
of the higher members of the isoelectronic sequence, where the excitation energies of the
excitetl electron are well separated from each other and lie high above the ground level.
Table 15 shows a good agreement between our results ant1 the theoretical estimates by
Dankmort and Trefftz [S].
To study the systematic trends in the isoelectronic sequence from the point of view
of the charge expansion theory of Layzer [9] ; me expand the energy in descending powers
of Z (the atomic charge)
where K O is the zeroth order energy. Fig. 1 shows the Z-energy dependence of our calculated Hartree-E'ock cnergies for some states of the I< I sequence. The linear relations
o1)tained between E'/Z2 and 1/25 (see Fig. 1) are indications of electrostatic interactions
existing between the core electrons ant1 the cxcitetl electron in the various members
of the sequence. The deviation from linearity occuring a t H 1 and C I1 might be due t o
the effects of configuration interactions in these light members of the sequence which
were neglected in our caleulations.
In order t o check the hydrogenic hehavionrof the term energies, we plotted in Figs. 2 , 3 ,
1
our calculated energy values for I3 I and 0 1V against - (where 72 is the principal
r$
Si AlMgNaNe
0,
F
0
x /x v///v//v/v
1v
N
111
c
B
/I
/
r
1
E l Z2
(a. u.)
0.01 -
0.02 -
I
0.05
1$2%3dfD) -
0.05
,
I
0.1
0.15
0.2
112
Fig. 1. Hartree-Fock excitation energies (in n.u.) for 1S2W3s (*S),l.s22ss23p
p")(
states of B I sequence as a function of nuclear charge
and ls21s23d ( W )
TN.M. EL-SHERBINIand H. &I.M. MANSOUR,Levels in the Boron Isorlectronic Sequences
111
quantum number). A linear behaviour is obtained for n = 5 up t o 8 (i.e. for highly excited states), antl a deviation from linearity a t lower values of n (see Figs. 2, 3). This
deviation might be attributed t o electron correlation effects with the same parity antl
angular momentum, which are appreciable in the case of low excited states having relatively small energy separations.
Finally we conclude that the simple configuration Hartree-Pock method is suitable
for the calculations of the energy levels of highly excited states (high n) and moderately
ionized atoms (large 2 ) in the sequence. I n order t o improve the results for the low
members of the sequence, work is in progress taking into account configuration interaction effects in the energy level calculations.
I
Oak- 0.05
0.1
30r
n =3-8
0.15
(
Vn'
Fig. 2. Hartree-Fock term energies of B I a s a
function of the principal quantum number
0.05
0.1
035
0.2
l/n2
Fig. 3. Hartree-Fock term energies of 0 IV as a
function of the principal quantum number
A c k n o w l e d g e m e n t s . Two of the authors (Th. M. El-S. and H. M. M. &I.) would
like to thank Professor Abdus Salam, the International Atomic Energy Agency and
UKESCO for hospitality at the International Centre for Theoretical Physics, Trieste.
References
[I] FLOWER,
D.; NUSSBAUMER,
H.: Astron. Astrophys. 26 (1975) 145.
[ 2 ] FROESE
FISCHER,C.: Comput. Phys. Commun. 1 (1969) 151.
[:I] EI)L$N,
B.: Handbuch der Physik, (Ed. S. Flugge), Berlin: Springer-Verlag, 1964, vol. 27.
[4] EL-SHERBINI,TR. M. : Atomkernenergie Kerntechnik 39 (1981) 53.
[5] EL-SIIERBINI,TH. M.; ALLAM,S. H.: Ann. Physik 39 (1982) 107.
AM. Physik Leipzig 46 (1989) 2
112
[6]
[7]
[S]
[9]
MOORE,
C. E.: Atomic energy levels NBS circular, N. 467, Vol. 1 (1949).
BROMANDER,
J.: Archiv fur Physik, 40 (1969) 257.
DANKWORT,
W.; TREFFTZ,
E.: J. Phys. B 13 (1980) 4325.
LAYZER,
D.: Ann. Phys. 8 (1959) 271; Monthly Not. Roy. Astron. SOC.114 (1954) 692.
Bei der Redaktion eingegangen am 24. September 19%
Anschr. d. Verf.: Dr. TH. M. EL-SHERBINI,
Dr. H. M. M. MANSOUR
A. A. Farrag, A. A. Rahman
Physics Department, Faculty of Science
Cairo University, Cairo, Egypt
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