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Screening-constant-by-unit-nuclear-charge calculations of high lying (2pns)1,3P∘ and (2pnd)1,3P∘ states of the B+ ion
Atomic Data and Nuclear Data Tables 99 (2013) 447–458
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Screening-constant-by-unit-nuclear-charge calculations of high lying (2pns)1,3 P◦ and (2pnd)1,3 P◦ states of the B+ ion I. Sakho a,∗ , B. Diop b , M. Faye b , A. Sène b , M. Guèye b , A.S. Ndao b , M. Biaye c , A. Wagué b a
UFR Sciences and Technologies, Department of Physics, University of Ziguinchor, Ziguinchor, Senegal
b
Department of Physics, Atoms Laser Laboratory, Faculty of Sciences and Technologies, University Cheikh Anta Diop, Dakar, Senegal
c
Department of Physics and Chemistry, Faculty of Sciences and Technologies of Formation and Education, University Cheikh Anta Diop, Dakar, Senegal
article
info
Article history: Received 22 March 2012 Received in revised form 29 May 2012 Accepted 4 June 2012 Available online 12 February 2013 Keywords: Semi-empirical calculations Screening constant Energy positions resonant widths Rydberg states
∗
abstract We report in this paper energy positions of the (2pns)1,3 P◦ and (2pnd)1,3 P◦ Rydberg states (n = 3–60) and resonance widths of the (2pns)1 P◦ and (2pnd)1 P◦ (n = 20) members of these series of the B+ ion. Calculations are performed in the framework of the screening-constant-by-unit-nuclear-charge method. Along all the series investigated, the quantum defect is almost constant up to n = 60. The present results compare very well to available theoretical and experimental literature values up to n = 20. The data presented in this work may be a useful guideline for investigators considering the photoionization spectrum of the B+ ion. © 2013 Elsevier Inc. All rights reserved.
Corresponding author. E-mail address: [email protected] (I. Sakho).
0092-640X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.adt.2012.06.005
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I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
Contents 1. 2.
3. 4.
Introduction........................................................................................................................................................................................................................448 Theory .................................................................................................................................................................................................................................448 2.1. Energy of the 2pns 1,3 P◦ and 2pnd 1,3 P◦ doubly excited states of B+ ................................................................................................................448 2.2. Resonance widths of the 2pns 1 P◦ and 2pnd 1 P◦ doubly excited states of B+ ..................................................................................................449 Results and discussions .....................................................................................................................................................................................................449 Summary ............................................................................................................................................................................................................................451 References...........................................................................................................................................................................................................................451 Explanation of Tables .........................................................................................................................................................................................................452 Table 1. Energy positions (E) of doubly (2pns) 1 P◦ excited states of B+ ..................................................................................................................452 Table 2. Energy positions (E) of doubly (2pnd) 1 P◦ excited states of B+ .................................................................................................................452 Table 3. Resonance widths (Γ ) of doubly (2pns) 1 P◦ excited states of B+ ..............................................................................................................452 Table 4. Resonance widths (Γ ) of doubly (2pnd) 1 P◦ excited states of B+ ..............................................................................................................452 Table 5. Energy (E) of doubly (2pns) 3 P◦ and (2pnd) 3 P◦ excited states of B+ ........................................................................................................452 Table 6. Present energy positions and quantum defects of high lying doubly (2pns) 1 P◦ and (2pnd) 1 P◦ excited states of B+ ..........................452
1. Introduction
2. Theory
As supra-helium atoms, beryllium-like systems are attractive candidates for the study of electronic correlation effects in atoms containing two electrons in each of the first principal shells. In the view of astrophysics, boron is very interesting because its cosmic abundance is explained neither by standard big-bang theory nor by standard stellar nucleosynthesis [1]. A considerable number of investigations in both theory and experiment have been recently made to study the photoionization processes of the singly-ionized boron. Schippers et al. [2] studied the photoionization of the B+ valence shell using a photon-ion merged-beams arrangement at the Advanced Light Source (ALS). Kim and Manson [3] investigated the photoionization of the 1 S ground state of Be-like B+ ion leading to the 2s, 2p, 3s and 3p states of B2+ employing a noniterative eigenchannel R-matrix (NER-M) method. In addition, the multiconfiguration relativistic random-phase approximation (MCRRPA) theory has been used by Hsiao et al. [4] to study five Rydberg series of doubly excited 2pns1,3 P◦ , 2pnd1,3 P◦ , and 2pnd3 D◦ states in the photoionization spectrum of the singly-ionized boron. Very recently Sakho [5] reported accurate energy positions for the 2pns1,3 P◦ and 2pnd1,3 P◦ levels of the beryllium atom up to n = 20 along with resonance widths in the particular case of the 2pns1 P◦ states (n = 3 − 15) using the screening-constantby-unit-nuclear-charge (SCUNC) method. In the present work, the previous study is extended to the photoionization spectrum of the Be-like ion B+ in the framework of the SCUNC formalism. In the photon-energy region between the (1s2 2s) 2 S1/2 and (1s2 2p) 2 P◦1/2 +
2pns1 P◦1 ,
2pns3 P◦1 ,
ionization thresholds of B , five Rydberg series, 2pnd1 P◦1 , 2pnd3 P◦1, and 2pnd3 D◦1 , are the most prominent doublyexcited autoionization resonances in its photoionization spectrum, as revealed in the works of Hsiao et al. [4]. In general, in most of the experimental and theoretical studies of the photoionization spectrum of many electron systems, investigations are based mainly on the measurements or the calculations of the photoionization cross section. The SCUNC method is known to be a very suitable technique of calculation that has recently given accurate results [6–9] from simple semi-empirical formulas without needing to compute the photoionization cross section. The purpose of the present work is to report accurate results for energy positions of the (2pns) 1,3 P◦ and (2pnd) 1,3 P◦ Rydberg states (n = 3–60) along with resonance widths of the (2pns) 1 P◦ and (2pnd) 1 P◦ (n = 3–20) members of the series of the B+ ion. In Section 2 we present the theoretical procedure adopted in this work. In Section 3, we present and discuss the results obtained, and compare them to available literature data.
2.1. Energy of the 2pns 1,3 P◦ and 2pnd 1,3 P◦ doubly excited states of B+ In the framework of the SCUNC formalism, the total energy of the N ℓ, nℓ′ Rydbergs)
2S +1
E N ℓnℓ′ ;2S +1 Lπ
Lπ excited state is expressed in the form (in
2 1 1 − β N ℓnℓ′ ;2S +1 Lπ ; Z . (1) 2 2 N n In this equation, the principal quantum numbers N and n are, respectively, for the inner and outer electrons of the helium isoelectronic series. The β parameters are screening constants by unit nuclear charge expanded in inverse powers of Z and given by = −Z 2
1
+
q β N ℓnℓ′ ;2S +1 Lπ ; Z = fk
k 1
(2)
Z
k=1
where fk = fk N ℓ nℓ′ ;2S +1 Lπ are parameters to be evaluated empirically. Using Eqs. (1) and (2), total energies for the 2pns 1,3 P◦ and 2pnd 1 ,3 ◦ P doubly excited states of B+ (Z = Z0 = 5) are given as follows. • For 2pns 1,3 P◦ levels
E 2pns ;2S +1 P1◦ = E∞ −
+ + +
Z02
f1 (s)
n2
1−
Z0 (n − 1 )
−
f2 (s) Z0
f12 (s) × (n − n0 ) × (n − n′0 ) Z02 (n − n0 + 1) × (n + n′0 − n0 ) × (n + n0 − n′0 − 1)2 f1 (s) × S Z02 (n − s − 1) f12 (s) × (n + n0 − n′0 − 1) × (n + n0 − n′0 − 2) × S Z02 (n + n′0 − n0 + s)3
2
. (3)
1,3 ◦
• For 2pnd P levels Z2 E 2pnd ;2S +1 P1◦ = E∞ − 02 1 − n
+ +
+
f1 (d) Z0 (n − 1 )
−
f2 (d) Z0
f1 (d) × (n − n0 ) × (n − n′0 ) Z02 (n + n′0 − n0 )2 × (n + n0 + n′0 ) f1 (d) × S Z0 (n − 1)
−
f1 (d) × S Z02 (n + n0 + s)
f1 (d) × (n − n0 ) × (n − n′0 ) × S Z03 (n + n0 − n′0 − s)2
2
.
(4)
I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
In these equations, n0 and n′0 (n′0 > n0 ) denote the principal quantum numbers of the 2pns 1 P◦ and 2pnd 1 P◦ levels of B+ used in the empirical determination of the fi (l = s or d)-screening constants in Eqs. (3) and (4) and s represents the spin of the nlelectron (s = 1/2). In addition, E∞ , is the energy value of the series limit and defined in the NIST atomic spectra database [10] as E∞ = I2s + Eexci = 31.1533 eV. I2s denotes the ionization energy of the B+ (1s2 2s2 S0 ) ground state and Eexci represents the excitation energy (averaged over both 2 P◦ states) of the B2+ (2s → 2p) ion. The screening constants in Eqs. (3) and (4) are evaluated using experimental data from work at the ALS by Schippers et al. [2] on B+ which are, for the 2p4s 1 P◦ (n0 = 4) and 2p5s 1 P◦ (n′0 = 5) levels, at 26.923 ± 0.005 and 28.580 ± 0.003 (in eV), respectively, and for the 2p3d 1 P◦ (n0 = 3) and 2p4d 1 P◦ (n′0 = 4) states, at 25.458 ± 0.001 and 27.889 ± 0.001, respectively. The infinite Rydberg, 1Ry = 13.605698 eV, is used for energy conversion. Using the experimental data (with Z0 = 5), we obtain from Eqs. (3) and (4) the empirical values of the screening constants f1 (s) = −0.671225274, f2 (s) = 2.993329406, f1 (d) = 0.109800098, and f2 (d) = 3.004128516. Taking into account the uncertainties in the energy positions, we determine the screening constants as explained previously [9]: ∆f1 (s) = 0.0006, ∆f2 (s) = 0.0011, ∆f1 (d) = 0.0008, and ∆f2 (d) = 0.0006. Finally, the fi -fitting parameters are expressed as follows: f1 (s) = −0.6712 ± 0.0006; f2 (s) = 2.9933 ± 0.0011; f1 (d) = 0.1098 ± 0.0008; and f2 (d) = 3.0041 ± 0.0006. Using the semi-empirical values of the screening constants along with the energy limit from NIST [10], Eqs. (3) and (4) are expressed explicitly as follows: E 2pns ;2S +1 P1◦ = 31.1533 −
25
1+
0.13424
(n − 1) (n − 4) × (n − 5) − 0.59866 + 0.018020377 (n − 3) × (n + 1) × (n − 2)2 2 0.026848 (n − 2) × (n − 3) + + 0.018020377 × × S (n − 1.5) (n + 1.5)3 n2
× 13.605698, (5) 0.02196 25 2S +1 ◦ E 2pnd ; P1 = 31.1533 − 2 1 − n (n − 1) (n − 3) × (n − 4) − 0.60082 + 0.004392 × (n + 1)2 × (n + 7) 0.004392 0.02196 − + (n − 1) (n + 3.5) 2 0.0008784 × (n − 3) × (n − 4) + × S × 13.605698. (6) (n − 1.5)2 In these equations, S = 0 for the singlet 2pns 1 P◦ and 2pnd 1 P◦ states and S = 1 for the triplet 2pns 3 P◦ and 2pnd 3 P◦ states. 2.2. Resonance widths of the 2pns 1 P◦ and 2pnd 1 P◦ doubly excited states of B+ The resonance widths of the 2pns 1 P◦ and 2pnd 1 P◦ doubly excited states of the B+ ion are given by (in Rydberg units)
Γ 2pns ; P1 = 1
+
−
f12
◦
Z02 n2
1−
f 1 ( s) Z0 (n − 1)
−
f 2 ( s) Z0
(s) × (n − n0 ) × (n − n0 ) ′
Z0 (n + n0 )2 × (n + n′0 )2 f12 (s) × (n − n0 ) × (n − n′0 ) Z02 (n + n′0 − n0 ) × (n + n′0 − n0 + 1)
2
,
(7)
Γ 2pnd ;1 P1 =
◦
+
−
Z02 n2
1−
449
f1 (d) Z0 (n − 1)
−
f2 (d) Z0
f12 (d) × (n − n0 ) × (n − n′0 ) Z0 (n + n0 )2 × (n + n′0 )2 f12 (d) × (n − n0 ) × (n − n′0 ) Z02 (n + n′0 − n0 ) × (n + n′0 − n0 + 1)
2
.
(8)
Using the photon-ion merged-beams results of Schippers et al. [2], for the 2p4s 1 P◦ (n0 = 4) and 2p5 1 P◦ (n′0 = 5) levels of B+ (with Z0 = 5) equal to 0.220 ± 0.001 and 0.106 ± 0.007 (in eV), respectively, Eq. (5) gives f1 (s) = −0.80773702 and f2 (s) = 4.760605197. In the same way, from the data of Ref. [2] we obtain for the 2p3d 1 P◦ (n0 = 3) and 2p4d 1 P◦ (n′0 = 4) levels of B+ 0.034 ± 0.002 and 0.016 ± 0.002 (in eV), respectively. Eq. (5) gives f1 (d) = −0.076790706 and f2 (d) = 4.888426766. Taking into account the uncertainties in the resonance widths, we obtain ∆f1 (s) = 0.07, ∆f2 (s) = 0.02, ∆f1 (d) = 0.02, and ∆f2 (d) = 0.02. Then the fi -screening constants are expressed as follows: f1 (s) = −0.81 ± 0.07; f2 (s) = 4.76 ± 0.02; f1 (d) = −0.08 ± 0.02; and f2 (d) = 4.89 ± 0.02. Using these semi-empirical values, Eqs. (6) and (7) are expressed as follows:
25 Γ 2pns ;1 P1◦ = 2
1+
n
×
0.162
(n − 1)
− 0.952 + 0.13122
(n − 4) × (n − 5) (n + 4)2 × (n + 5)2
(n − 4) × (n − 5) 2 × 13.605698, (n + 1) × (n + 2) 25 0.016 Γ 2pnd ;1 P1◦ = 2 1 + − 0.978 + 0.00128 n (n − 1) (n − 3) × (n − 4) × (n + 3)2 × (n + 4)2 (n − 3) × (n − 4) 2 − 0.000256 × × 13.605698. (n + 1) × (n + 2) − 0.026244 ×
(9)
(10)
3. Results and discussions The results obtained in the present paper are listed in Tables 1– 5 compared to available theoretical and experimental literature values. The analysis of the current results is achieved by calculating the quantum defect µ given by
µ=n−2
R E∞ − En
.
(11)
In this equation, R denotes the Rydberg constant equal to 13.605898 eV, E∞ is the limit taken from NIST [10] with E∞ = 31.1533 eV, and En is the energy resonance. In Table 1, the present results for energy positions (E) of the (2pns) 1 P◦ doubly excited states of B+ are compared to the MCRRPA values of Hsiao et al. [4], the NER-M results of Kim and Manson [3], the R-matrix (R-M) calculations of Tullyet al. [11], and to the experimental data of Schippers et al. [2] at the ALS. As shown by Kim and Manson [3] for the (2pns) 1 P◦ excited states of B+ , the (2p3) 1 P◦ state is a true bound state, and not an autoionizing resonance. This explains why the corresponding energy position is not quoted in Table 1. The agreement between the calculations and measurements is seen to be very good. In addition, it is also seen that the energy deviations with respect to the experimental data do not exceed 0.01 eV for up to n = 10. It should be mentioned that the present results agree well with the MCRRPA data [4] up to n = 20. In addition, for all the Rydberg series, the quantum defect is almost constant and an average value of 0.405 is obtained. This may
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I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
allow one to expect our results up to n = 25 to be accurate. This average value is in good agreement with the experimental results of Schippers et al. [2] at 0.408 (2) using for the Rydberg constant the value 13.6057 eV and for the energy limit 31.166 (1), which is less accurate than the more exact spectroscopy value of 31.1533 from NIST [10]. Table 2 shows a comparison between the present calculations for energy positions (E) of doubly excited states (2pnd) 1 P◦ of B+ and some theoretical [3,4,11] and experimental [2] data. Here again, the present data compare very well to the experimental values [2] where the energy deviations evaluated relatively to the experimental data do not exceed 0.008 eV for states up to n = 10. By comparing theoretical calculations, the agreement is seen to be very good. The current results compare well to the MCRRPA data [4] up to n = 20. In addition, for all the (2pnd) 1 P◦ resonances considered, the quantum defect is roughly constant. The average value obtained is −0.0869 and agrees very well with the experimental data of −0.087 (1) where the energy limit at 31.173 (2) is used. In Tables 3 and 4 we present the resonance widths (Γ ) of the (2pns) 1 P◦ and (2pnd) 1 P◦ doubly excited states of B+ obtained in this work compared to the MCRRPA values of Hsiao et al. [4], the NER-M results of Kim and Manson [3], the R-M calculations of Tully et al. [11] and to the ALS experiments of Schippers et al. [2]. Comparison indicates a very good agreement between the present theoretical widths and those of Kim and Manson [3] for both (2pns) 1 P◦ and (2pnd) 1 P◦ levels (n ≤ 12). For comparisons with the other data given in the tables the agreement is only satisfactory. Comparing the present calculations to the MCRRPA data [4], discrepancies appear particular for the (2pnd) 1 P◦ levels when the principal quantum number ranges between 12 and 20. As noted previously [5], if the agreement between theory and experiment is very good for energy positions (as shown in Tables 1–2), then this is not necessarily the same for widths. As a result, further new work by experiment and theory is needed to resolve the discrepancies observed for the high-lying doubly excited states of B+ investigated in this work. For the (2pns) 3 P◦ and (2pnd) 3 P◦ doubly excited states of B+ , calculations are very scarce. Only the results obtained from the MCRRPA computations [4] are found in the literature. Comparison of the data given in Table 5 indicates a very good agreement up to n = 20. Along all the investigated Rydberg series, the quantum defects are almost constant with an average value of 0.493 for the (2pns) 3 P◦ resonances and −0.013 for the (2pnd) 3 P◦ states. In Table 6, we have listed the present results for very high-lying states of the (2pns) 1 P◦ and (2pnd) 1 P◦ Rydberg series up to n = 60. For these resonances, the quantum defects are roughly constant with an average value of 0.482 for the (2pns) 1 P◦ states and −0.135 for the (2pnd) 1 P◦ levels. This leads us to consider these results as accurate even if no literature data are available at this time for direct comparison. Overall, the simplicity of the SCUNC formalism should be noted and it gives very accurate results from simple semi-empirical formulas. In the MCRRPA [4] formalism, calculations are done in the jj coupling scheme, in which couplings between fully relativistic channels are treated. The good agreement between the present SCUNC calculations and the MCRRPA computations [4] up to n = 20 may be explained as follows seeing that the SCUNC formalism is a nonrelativistic one. For Be-like systems, the Hamiltonian can be expressed as H = H0 + W .
orbit corrections (Wsoo ), and spin–spin corrections (Wss ). The nonrelativistic Hamiltonian and the perturbation operators are explicitly the following: H0 =
2
i=1
ri
8
2
WM = −
Wso =
(13)
rij
→ δ(− r i ),
(15)
i =1
4 1
M
∇i · ∇ j ,
(16)
i,j=1 i̸=j
→ − → 4 − Z l i· s i ri3
2c 2 i=1
Wsoo = −
(14)
i=1
4 3π α 2
WD =
i,j=1 i̸=j
4 α2 4 − → p i,
Wkin = −
4 1
2c 2
i,j=1 i̸=j
,
(17)
1 − − → − → − → − → (→ ri − rj ) × pi · ( si + 2 sj ), rij3
(18)
and Wss =
4 1 1
c2
i,j=1 j>i
rij3
− → − → − → − → 3( si · rij )( sj · rij ) − → − → s · s − . i
j
rij2
(19)
In these expressions, α denotes the fine structure constant and M is the nuclear mass of the Be-like system. Using Eq. (8), the corresponding eigenvalue is in the form E = E0 + w
(20)
with
w = ⟨Wkin ⟩ + ⟨WD ⟩ + ⟨WM ⟩ + ⟨Wso ⟩ + ⟨Wsoo ⟩ + ⟨Wss ⟩ .
(21)
In the framework of the SCUNC formalism, for four electron systems, the total energy is given by E 1s2 ; N ℓnℓ′ ;2S +1 Lπ = E 1s2 − Z 2
+
1 n2
1 − β N ℓnℓ′ ;2S +1 Lπ ; Z
2
1 N2
.
(22)
This equation can be written as follows: E 1s2 ; N ℓnℓ′ ;2S +1 Lπ = −
Z2 1
−
Z2 1
−
Z2 N2
−
Z2 n2
1 β N ℓnℓ′ ;2S +1 Lπ ; Z 2 − β N ℓnℓ′ ;2S +1 Lπ ; Z . n2 That means +
E 1s2 ; N ℓnℓ′ ;2S +1 Lπ = E0 + w
(23)
(24)
where
2 2 2 2 E0 = − Z − Z − Z − Z 2 2 1 1 N n (25) w = 1 β N ℓnℓ′ ;2S +1 Lπ ; Z 2 − β N ℓnℓ′ ;2S +1 Lπ ; Z . 2 n Using (21), the last equation of (25) can be rewritten in the form
(12)
In this expression, H0 denotes the nonrelativistic Hamiltonian and W is the sum of the perturbation operators, which includes correction to kinetic energy (Wkin ), the Darwin term (WD ), mass polarization (WM ), spin–orbit corrections (Wso ), spin–other
4 4 1 Z 1 − ∇i2 − + ,
1 n2
β N ℓnℓ′ ;2S +1 Lπ ; Z 2 − β N ℓnℓ′ ;2S +1 Lπ ; Z
= ⟨Wkin ⟩ + ⟨WD ⟩ + ⟨WM ⟩ + ⟨Wso ⟩ + ⟨Wsoo ⟩ + ⟨Wss ⟩ .
(26)
I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
This equation indicates clearly that, in the framework of the SCUNC formalism, all the contributions of the perturbation operators are taken into account by the β screening constant by unit nuclear charge. In the independent particle model, which disregards all the perturbation effects, the total energy is given by E0 . Consequently w = 0. This implies automatically β = 0. The accurate results that result from the SCUNC formalism are then explained by the fact that the β parameters are expanded in inverse powers of Z as shown by Eq. (2) where the fk = fk N ℓ nℓ′ ;2S +1 Lπ screening constants are evaluated empirically using experimental data incorporating all the perturbation effects given by Eq. (21). It should be mentioned also that calculation of the perturbation operator W is a very complex computational problem. The merit of the SCUNC method is the possibility to overcome all these difficulties in a semi-empirical procedure as demonstrated in our recent works [5–9]. 4. Summary The energy positions of the (2pns) 1,3 P◦ and (2pnd) 1,3 P◦ autoionizing states and resonance widths of the (2pns) 1 P◦ excited
451
states of the B+ ion are presented in this paper using the SCUNC method. In general, the present results agree very well with both the noted theoretical and experimental literature data. For n ≥ 21, no theoretical and experimental literature values are available for direct comparison. The good accuracy obtained in this work indicates that the results may be of interest for future experimental and theoretical studies of the photoabsorption spectrum of B+ . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
J.R. King, Astron. J. 122 (2001) 3115. S. Schippers, et al., J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 3371. D.-S. Kim, S.T. Manson, J. Phys. B: At. Mol. Opt. Phys. 37 (2004) 4013. J.-T. Hsiao, H.-T. Shiao, K.-N. Huang, Chin. J. Phys. 47 (2009) 173. I. Sakho, Rad. Phys. Chem. 80 (2011) 1295. I. Sakho, A. Wague, Chin. J. Phys 48 (2010) 1. I. Sakho, A. Konte, A.S. Ndao, M. Biaye, A. Wague, Phys. Scr. 82 (2010) 035301. I. Sakho, Eur. Phys. J. D 59 (2010) 171. I. Sakho, Eur. Phys. J. D 61 (2011) 267. W.C. Martin, et al., NIST Atomic Spectra Data Base, second ed., National Institute of Standards and Technology Gaithersburg, Maryland 20899-3460, USA, 1999, http://physics.nist.gov/cgi-bin/AtData/main_asd. [11] J.A. Tully, M.L. Dourneuf, C.J. Zeippen, Astron. Astrophys. 211 (1989) 485.
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Explanation of Tables Table 1
Table 2 Table 3
Table 4 Table 5
Table 6
Energy positions (E) of doubly (2pns) 1 P◦ excited states of B+ The results are expressed in eV. SCUNC Present results MCRRPA Multiconfiguration relativistic random-phase approximation values of Hsiao et al. [4] NER-M Noniterative eigenchannel R-matrix results of Kim and Manson [3] R-M R-matrix calculations of Tullyet al. [11] ALS Advanced Light Source experiments of Schippers et al. [2] |∆E | Energy differences relative to the experimental data [2]. Energy positions (E) of doubly (2pnd) 1 P◦ excited states of B+ Same as for Table 1. Resonance widths (Γ ) of doubly (2pns) 1 P◦ excited states of B+ The results are expressed in eV. The a(−b) (c) notation means a × 10−b and (c) indicate the uncertainties of the experimental widths given in parentheses. SCUNC Present results MCRRPA Multiconfiguration relativistic random-phase approximation values of Hsiao et al. [4] NER-M Noniterative eigenchannel R-matrix results of Kim and Manson [3] R-M R-matrix calculations of Tully et al. [11] ALS Advanced Light Source experiments of Schippers et al. [2]. Resonance widths (Γ ) of doubly (2pnd) 1 P◦ excited states of B+ Same as for Table 3. Energy (E) of doubly (2pns) 3 P◦ and (2pnd) 3 P◦ excited states of B+ The results are expressed in eV. SCUNC Present results MCRRPA Multiconfiguration relativistic random-phase approximation values of Hsiaoet al. [4]. Present energy positions and quantum defects of high lying doubly (2pns) 1 P◦ and (2pnd) 1 P◦ excited states of B+ The results are expressed in eV. E Energy positions µ Quantum defects.
I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
453
Table 1 Energy positions (E) of doubly (2pns) 1 P◦ excited states of B+ . ns
Theory SCUNC
Experiment
Theory
MCRRPA
NER-M
R-M
ALS
SCUNC
E
|∆E |∗
µ
26.9137 28.5634 29.4084 29.8983 30.2075 30.4151 30.5612 30.6680 30.7484 30.8103 30.8591 30.8982 30.9301 30.9564 30.9783 30.9968 31.0125
26.9405 28.5568 29.3924 29.8787 30.1861 30.3927 30.5382 30.6444 30.7244
26.96 28.59 29.42
26.923 (5) 28.580 (3) 29.420 (3) 29.895 (3) 30.205 (3) 30.409 (7) 30.562 (9)
0.000 1 0.000 1 0.000 1 0.0112 0.0078 0.0098 0.0016
0.413 0.401 0.396 0.394 0.393 0.393 0.393 0.394 0.395 0.396 0.399 0.402 0.402 0.404 0.410 0.411 0.416 0.412 0.423 0.426 0.422
E 4s 5s 6s 7s 8s 9s 10s 11s 12s 13s 14s 15s 16s 17s 18s 19s 20s 21s 22s 23s 24s ... ∞s
26.9229 (50) 28.5799 (30) 29.4201 (20) 29.9062 (14) 30.2129 (11) 30.4187 (8) 30.5636 (7) 30.6695 (5) 30.7492 (5) 30.8107 (4) 30.8591 (3) 30.8979 (3) 30.9296 (3) 30.9557 (2) 30.9774 (2) 30.9958 (2) 31.0114 (2) 31.0249 (1) 31.0364 (1) 31.0465 (1) 31.0554 (1) 31.1533
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Table 2 Energy positions (E) of doubly (2pnd) 1 P◦ excited states of B+ . nd
Theory SCUNC
Experiment MCRRPA
NER-M
R-M
E 3d 4d 5d 6d 7d 8d 9d 10d 11d 12d 13d 14d 15d 16d 17d 18d 19d 20d 21d 22d 23d 24d ... ∞d
25.4578 (10) 27.8889 (10) 29.0443 (8) 29.6804 (6) 30.0670 (5) 30.3194 (4) 30.4931 (3) 30.6177 (3) 30.7101 (2) 30.7805 (2) 30.8354 (2) 30.8790 (1) 30.9142 (1) 30.9430 (1) 30.9669 (1) 30.9870 (1) 31.0040 (1) 31.0185 (1) 31.0310 (1) 31.0418 (1) 31.0513 (1) 31.0596 (1) ... 31.1533
25.5731 27.9453 29.0747 29.6983 30.0784 30.3269 30.4983 30.6215 30.7130 30.7828 30.8372 30.8805 30.9156 30.9443 30.9681 30.9881 31.0051 31.0196
25.4262 27.8610 29.0166 29.6533 30.0408 30.2935 30.4676 30.5924 30.6850 30.7556
25.47 27. 90 29.05 29.68
ALS
SCUNC
E
|∆E |
µ
25.458 (1) 27.889 (1) 29.041 (1) 29.676 (1) 30.064 (2) 30.320 (2) 30.490 (4) 30.610 (1)
0.000 2 0.000 1 0.003 3 0.004 4 0.003 0 0.000 6 0.003 1 0.007 7
−0.091 −0.083 −0.080 −0.080 −0.080 −0.080 −0.080 −0.080 −0.081 −0.082 −0.084 −0.086 −0.087 −0.087 −0.087 −0.090 −0.092 −0.093 −0.095 −0.093 −0.091 −0.100
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455
Table 3 Resonance widths (Γ ) of doubly (2pns) 1 P◦ excited states of B+ . States (2p4s) 1 P◦ (2p5s) 1 P◦ (2p6s) 1 P◦ (2p7s) 1 P◦ (2p8s) 1 P◦ (2p9s) 1 P◦ (2p10s)1 P◦ (2p11s)1 P◦ (2p12s)1 P◦ (2p13s)1 P◦ (2p14s)1 P◦ (2p15s)1 P◦ (2p16s)1 P◦ (2p17s)1 P◦ (2p18s)1 P◦ (2p19s)1 P◦ (2p20s)1 P◦
Theory
Experiment
SCUNC
MCRRPA
NER-M
R-M
ALS
2.212 (−1) (100) 1.066 (−1) (44) 5.969 (−2) (28) 3.685 (−2) (25) 2.437 (−2) (21) 1.696 (−2) (20) 1.230 (−2) (20) 9.207 (−3) (13) 7.081 (−3) (11) 5.569 (−3) (10) 4.465 (−3) (10) 3.638 (−3) (10) 3.008 (−3) (10) 2.518 (−3) (10) 2.131 (−3) (5) 1.822 (−3) (4) 1.571 (−3) (4)
2.663 (−1) 1.188 (−1) 6.343 (−2) 3.788 (−2) 2. 442 (−2) 1.663 (−2) 1.179 (−2) 8.629 (−3) 6.465 (−3) 4.927 (−3) 3.808 (−3) 2. 978 (−3) 2. 356 (−3) 1.884 (−3) 1. 521 (−3) 1. 234 (−3) 9. 993 (−4)
2.453 (−1) 1.153 (−1) 6.350 (−2) 3.840 (−2) 2.490 (−2) 1.710 (−2) 1.220 (−2) 9.000 (−3) 6.900 (−3)
2.41 (−1) 1.10 (−1) 5.90 (−2)
2.20 (−1) (1) 1.06 (−1) (7) 4.80 (−2) (6) 2.90 (−2) (7) 2.00 (−2) (6) 4.00 (−2) (1) 2.00 (−2) (2)
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Table 4 Resonance widths (Γ ) of doubly (2pnd) 1 P◦ excited states of B+ . States (2p3d) 1 P◦ (2p4d) 1 P◦ (2p5d) 1 P◦ (2p6d) 1 P◦ (2p7d) 1 P◦ (2p8d) 1 P◦ (2p9d) 1 P◦ (2p10d)1 P◦ (2p11d)1 P◦ (2p12d)1 P◦ (2p13d)1 P◦ (2p14d)1 P◦ (2p15d)1 P◦ (2p16d)1 P◦ (2p17d)1 P◦ (2p18d)1 P◦ (2p19d)1 P◦ (2p20d)1 P◦
Theory
Experiment
SCUNC
MCRRPA
NER-M
R-M
ALS
3.40 (−2) (20) 1.59 (−2) (20) 9.2 (−3) (15) 6.0 (−3) (11) 4.2 (−3) (10) 3.1 (−3) (10) 2.4 (−3) (10) 1.9 (−3) (5) 1.6 (−3) (4) 1.3 (−3) (3) 1.1 (−3) (3) 9.3 (−4) (2) 8.0 (−4) (2) 7.0 (−4) (2) 6.2 (−4) (2) 5.5 (−4) (2) 4.870 (−4) (1) 4.380 (−4) (1)
5.17 (−2) 2.21 (−2) 1.14 (−2) 6.61 (−3) 4.17 (−3) 2.79 (−3) 1.96 (−3) 1.42 (−3) 1.06 (−3) 8.05 (−4) 6.21 (−4) 4.84 (−4) 3.80 (−4) 2.99 (−4) 2.35 (−4) 1.82 (−4) 1.38 (−4) 9.69 (−5)
3.12 (−2) 1.41 (−2) 9.5 (−3) 6.4 (−3) 4.4 (−3) 3.1 (−3) 2.3 (−3) 1.7 (−3) 1.3 (−3) 1.1 (−3)
3.38 (−2) 1.78 (−2) 5.90 (−2) 2.41 (−1)
3.4 (−2) (2) 1.6 (−2) (2) 1.0 (−2) (3) 8.0 (−3) (3) 8.0 (−3) (4) 1.0 (−3) (6) 5.0 (−3) (8)
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457
Table 5 Energy (E) of doubly (2pns) 3 P◦ and (2pnd) 3 P◦ excited states of B+ . ns
SCUNC
MCRRPA
SCUNC
nd
SCUNC
MCRRPA
4s 5s 6s 7s 8s 9s 10s 11s 12s 13s 14s 15s 16s 17s 18s 19s 20s 21s 22s ... ∞s
26.7126 (53) 28.4836 (32) 29.3673 (22) 29.8739 (16) 30.1915 (12) 30.4038 (9) 30.5528 (7) 30.6613 (6) 30.7429 (5) 30.8057 (4) 30.8551 (3) 30.8947 (3) 30.9269 (3) 30.9534 (2) 30.9755 (2) 30.9942 (2) 31.0100 (2) 31.0236 (1) 31.0354 (1) ... 31.1533
26.7050 28.4707 29.3586 29.8680 30.1874 30.4008 30.5504 30.6593 30.7410 30.8039 30.8533 30.8929 30.9250 30.9515 30.9736 30.9922 31.0080
0.499 0.485 0.480 0.478 0.478 0.479 0.480 0.483 0.485 0.488 0.490 0.493 0.496 0.500 0.504 0.505 0.512 0.516 0.515 ...
3d 4d 5d 6d 7d 8d 9d 10d 11d 12d 13d 14d 15d 16d 17d 18d 19d 20d 21d ... ...
25.1515 (32) 27.7757 (18) 28.9891 (12) 29.6490 (8) 30.0472 (6) 30.3060 (5) 30.4835 (4) 30.6106 (3) 30.7046 (3) 30.7762 (2) 30.8319 (2) 30.8761 (2) 30.9118 (2) 30.9410 (1) 30.9652 (1) 30.9855 (1) 31.0027 (1) 31.0174 (1) 31.0300 (1) ... ...
25.1505 27.7498 28.9706 29.6370 30.0394 30.3008 30.4801 30.6083 30.7032 30.7754 30.8315 30.8761 30.9120 30.9414 30.9658 30.9862 31.0035 31.0183
−0.012 −0.014 −0.015 −0.015 −0.015 −0.014 −0.014 −0.014 −0.013 −0.013 −0.013 −0.012 −0.012 −0.012 −0.011 −0.011 −0.011 −0.010 −0.010
... ...
... ...
...
µ
SCUNC
µ
458
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Table 6 Present energy positions and quantum defects of high lying doubly (2pns) 1 P◦ and (2pnd) 1 P◦ excited states of B+ . ns
E
µ
nd
E
µ
25s 26s 27s 28s 29s 30s 31s 32s 33s 34s 35s 36s 37s 38s 39s 40s 41s 42s 43s 44s 45s 46s 47s 48s 49s 50s 51s 52s 53s 54s 55s 56s 57s 58s 59s 60s ... ∞s
31.06317 (10) 31.07006 (9) 31.07619 (9) 31.08167 (8) 31.08658 (7) 31.09101 (7) 31.09501 (7) 31.09863 (6) 31.10193 (6) 31.10494 (5) 31.10769 (5) 31.11021 (5) 31.11253 (5) 31.11467 (4) 31.11664 (4) 31.11847 (4) 31.12016 (4) 31.12173 (4) 31.12319 (3) 31.12456 (3) 31.12583 (3) 31.12702 (3) 31.12813 (3) 31.12918 (3) 31.13016 (3) 31.13108 (2) 31.13195 (2) 31.13277 (2) 31.13354 (2) 31.13427 (2) 31.13496 (2) 31.13562 (2) 31.13623 (2) 31.13682 (2) 31.13738 (2) 31.13791 (2) ... 31.1533
0.427 0.430 0.433 0.430 0.436 0.442 0.444 0.449 0.451 0.453 0.457 0.461 0.464 0.466 0.470 0.471 0.476 0.480 0.486 0.484 0.490 0.493 0.500 0.499 0.504 0.510 0.512 0.513 0.520 0.522 0.526 0.518 0.536 0.534 0.532 0.534
25d 26d 27d 28d 29d 30d 31d 32d 33d 34d 35d 36d 37d 38d 39d 40d 41d 42d 43d 44d 45d 46d 47d 48d 49d 50d 51d 52d 53d 54d 55d 56d 57d 58d 59d 60d ... ∞d
31.06694 (5) 31.07344 (4) 31.07923 (4) 31.08442 (4) 31.08908 (3) 31.09328 (3) 31.09708 (3) 31.10054 (3) 31.10368 (3) 31.10655 (3) 31.10918 (2) 31.11159 (2) 31.11381 (2) 31.11586 (2) 31.11776 (2) 31.11951 (2) 31.12113 (2) 31.12265 (2) 31.12405 (2) 31.12537 (2) 31.12659 (1) 31.12774 (1) 31.12881 (1) 31.12982 (1) 31.13077 (1) 31.13166 (1) 31.13250 (1) 31.13329 (1) 31.13404 (1) 31.13475 (1) 31.13541 (1) 31.13605 (1) 31.13665 (1) 31.13721 (1) 31.13775 (1) 31.13827 (1) ... 31.1533
−0.103 −0.105 −0.106 −0.105 −0.111 −0.112 −0.113 −0.105 −0.117 −0.119 −0.121 −0.122 −0.123 −0.126 −0.132 −0.133 −0.131 −0.138 −0.135 −0.142 −0.139 −0.143 −0.141 −0.144 −0.148 −0.149 −0.151 −0.151 −0.157 −0.165 −0.155 −0.169 −0.172 −0.158 −0.159 −0.174
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Atomic Data and Nuclear Data Tables journal homepage: www.elsevier.com/locate/adt
Screening-constant-by-unit-nuclear-charge calculations of high lying (2pns)1,3 P◦ and (2pnd)1,3 P◦ states of the B+ ion I. Sakho a,∗ , B. Diop b , M. Faye b , A. Sène b , M. Guèye b , A.S. Ndao b , M. Biaye c , A. Wagué b a
UFR Sciences and Technologies, Department of Physics, University of Ziguinchor, Ziguinchor, Senegal
b
Department of Physics, Atoms Laser Laboratory, Faculty of Sciences and Technologies, University Cheikh Anta Diop, Dakar, Senegal
c
Department of Physics and Chemistry, Faculty of Sciences and Technologies of Formation and Education, University Cheikh Anta Diop, Dakar, Senegal
article
info
Article history: Received 22 March 2012 Received in revised form 29 May 2012 Accepted 4 June 2012 Available online 12 February 2013 Keywords: Semi-empirical calculations Screening constant Energy positions resonant widths Rydberg states
∗
abstract We report in this paper energy positions of the (2pns)1,3 P◦ and (2pnd)1,3 P◦ Rydberg states (n = 3–60) and resonance widths of the (2pns)1 P◦ and (2pnd)1 P◦ (n = 20) members of these series of the B+ ion. Calculations are performed in the framework of the screening-constant-by-unit-nuclear-charge method. Along all the series investigated, the quantum defect is almost constant up to n = 60. The present results compare very well to available theoretical and experimental literature values up to n = 20. The data presented in this work may be a useful guideline for investigators considering the photoionization spectrum of the B+ ion. © 2013 Elsevier Inc. All rights reserved.
Corresponding author. E-mail address: [email protected] (I. Sakho).
0092-640X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.adt.2012.06.005
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I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
Contents 1. 2.
3. 4.
Introduction........................................................................................................................................................................................................................448 Theory .................................................................................................................................................................................................................................448 2.1. Energy of the 2pns 1,3 P◦ and 2pnd 1,3 P◦ doubly excited states of B+ ................................................................................................................448 2.2. Resonance widths of the 2pns 1 P◦ and 2pnd 1 P◦ doubly excited states of B+ ..................................................................................................449 Results and discussions .....................................................................................................................................................................................................449 Summary ............................................................................................................................................................................................................................451 References...........................................................................................................................................................................................................................451 Explanation of Tables .........................................................................................................................................................................................................452 Table 1. Energy positions (E) of doubly (2pns) 1 P◦ excited states of B+ ..................................................................................................................452 Table 2. Energy positions (E) of doubly (2pnd) 1 P◦ excited states of B+ .................................................................................................................452 Table 3. Resonance widths (Γ ) of doubly (2pns) 1 P◦ excited states of B+ ..............................................................................................................452 Table 4. Resonance widths (Γ ) of doubly (2pnd) 1 P◦ excited states of B+ ..............................................................................................................452 Table 5. Energy (E) of doubly (2pns) 3 P◦ and (2pnd) 3 P◦ excited states of B+ ........................................................................................................452 Table 6. Present energy positions and quantum defects of high lying doubly (2pns) 1 P◦ and (2pnd) 1 P◦ excited states of B+ ..........................452
1. Introduction
2. Theory
As supra-helium atoms, beryllium-like systems are attractive candidates for the study of electronic correlation effects in atoms containing two electrons in each of the first principal shells. In the view of astrophysics, boron is very interesting because its cosmic abundance is explained neither by standard big-bang theory nor by standard stellar nucleosynthesis [1]. A considerable number of investigations in both theory and experiment have been recently made to study the photoionization processes of the singly-ionized boron. Schippers et al. [2] studied the photoionization of the B+ valence shell using a photon-ion merged-beams arrangement at the Advanced Light Source (ALS). Kim and Manson [3] investigated the photoionization of the 1 S ground state of Be-like B+ ion leading to the 2s, 2p, 3s and 3p states of B2+ employing a noniterative eigenchannel R-matrix (NER-M) method. In addition, the multiconfiguration relativistic random-phase approximation (MCRRPA) theory has been used by Hsiao et al. [4] to study five Rydberg series of doubly excited 2pns1,3 P◦ , 2pnd1,3 P◦ , and 2pnd3 D◦ states in the photoionization spectrum of the singly-ionized boron. Very recently Sakho [5] reported accurate energy positions for the 2pns1,3 P◦ and 2pnd1,3 P◦ levels of the beryllium atom up to n = 20 along with resonance widths in the particular case of the 2pns1 P◦ states (n = 3 − 15) using the screening-constantby-unit-nuclear-charge (SCUNC) method. In the present work, the previous study is extended to the photoionization spectrum of the Be-like ion B+ in the framework of the SCUNC formalism. In the photon-energy region between the (1s2 2s) 2 S1/2 and (1s2 2p) 2 P◦1/2 +
2pns1 P◦1 ,
2pns3 P◦1 ,
ionization thresholds of B , five Rydberg series, 2pnd1 P◦1 , 2pnd3 P◦1, and 2pnd3 D◦1 , are the most prominent doublyexcited autoionization resonances in its photoionization spectrum, as revealed in the works of Hsiao et al. [4]. In general, in most of the experimental and theoretical studies of the photoionization spectrum of many electron systems, investigations are based mainly on the measurements or the calculations of the photoionization cross section. The SCUNC method is known to be a very suitable technique of calculation that has recently given accurate results [6–9] from simple semi-empirical formulas without needing to compute the photoionization cross section. The purpose of the present work is to report accurate results for energy positions of the (2pns) 1,3 P◦ and (2pnd) 1,3 P◦ Rydberg states (n = 3–60) along with resonance widths of the (2pns) 1 P◦ and (2pnd) 1 P◦ (n = 3–20) members of the series of the B+ ion. In Section 2 we present the theoretical procedure adopted in this work. In Section 3, we present and discuss the results obtained, and compare them to available literature data.
2.1. Energy of the 2pns 1,3 P◦ and 2pnd 1,3 P◦ doubly excited states of B+ In the framework of the SCUNC formalism, the total energy of the N ℓ, nℓ′ Rydbergs)
2S +1
E N ℓnℓ′ ;2S +1 Lπ
Lπ excited state is expressed in the form (in
2 1 1 − β N ℓnℓ′ ;2S +1 Lπ ; Z . (1) 2 2 N n In this equation, the principal quantum numbers N and n are, respectively, for the inner and outer electrons of the helium isoelectronic series. The β parameters are screening constants by unit nuclear charge expanded in inverse powers of Z and given by = −Z 2
1
+
q β N ℓnℓ′ ;2S +1 Lπ ; Z = fk
k 1
(2)
Z
k=1
where fk = fk N ℓ nℓ′ ;2S +1 Lπ are parameters to be evaluated empirically. Using Eqs. (1) and (2), total energies for the 2pns 1,3 P◦ and 2pnd 1 ,3 ◦ P doubly excited states of B+ (Z = Z0 = 5) are given as follows. • For 2pns 1,3 P◦ levels
E 2pns ;2S +1 P1◦ = E∞ −
+ + +
Z02
f1 (s)
n2
1−
Z0 (n − 1 )
−
f2 (s) Z0
f12 (s) × (n − n0 ) × (n − n′0 ) Z02 (n − n0 + 1) × (n + n′0 − n0 ) × (n + n0 − n′0 − 1)2 f1 (s) × S Z02 (n − s − 1) f12 (s) × (n + n0 − n′0 − 1) × (n + n0 − n′0 − 2) × S Z02 (n + n′0 − n0 + s)3
2
. (3)
1,3 ◦
• For 2pnd P levels Z2 E 2pnd ;2S +1 P1◦ = E∞ − 02 1 − n
+ +
+
f1 (d) Z0 (n − 1 )
−
f2 (d) Z0
f1 (d) × (n − n0 ) × (n − n′0 ) Z02 (n + n′0 − n0 )2 × (n + n0 + n′0 ) f1 (d) × S Z0 (n − 1)
−
f1 (d) × S Z02 (n + n0 + s)
f1 (d) × (n − n0 ) × (n − n′0 ) × S Z03 (n + n0 − n′0 − s)2
2
.
(4)
I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
In these equations, n0 and n′0 (n′0 > n0 ) denote the principal quantum numbers of the 2pns 1 P◦ and 2pnd 1 P◦ levels of B+ used in the empirical determination of the fi (l = s or d)-screening constants in Eqs. (3) and (4) and s represents the spin of the nlelectron (s = 1/2). In addition, E∞ , is the energy value of the series limit and defined in the NIST atomic spectra database [10] as E∞ = I2s + Eexci = 31.1533 eV. I2s denotes the ionization energy of the B+ (1s2 2s2 S0 ) ground state and Eexci represents the excitation energy (averaged over both 2 P◦ states) of the B2+ (2s → 2p) ion. The screening constants in Eqs. (3) and (4) are evaluated using experimental data from work at the ALS by Schippers et al. [2] on B+ which are, for the 2p4s 1 P◦ (n0 = 4) and 2p5s 1 P◦ (n′0 = 5) levels, at 26.923 ± 0.005 and 28.580 ± 0.003 (in eV), respectively, and for the 2p3d 1 P◦ (n0 = 3) and 2p4d 1 P◦ (n′0 = 4) states, at 25.458 ± 0.001 and 27.889 ± 0.001, respectively. The infinite Rydberg, 1Ry = 13.605698 eV, is used for energy conversion. Using the experimental data (with Z0 = 5), we obtain from Eqs. (3) and (4) the empirical values of the screening constants f1 (s) = −0.671225274, f2 (s) = 2.993329406, f1 (d) = 0.109800098, and f2 (d) = 3.004128516. Taking into account the uncertainties in the energy positions, we determine the screening constants as explained previously [9]: ∆f1 (s) = 0.0006, ∆f2 (s) = 0.0011, ∆f1 (d) = 0.0008, and ∆f2 (d) = 0.0006. Finally, the fi -fitting parameters are expressed as follows: f1 (s) = −0.6712 ± 0.0006; f2 (s) = 2.9933 ± 0.0011; f1 (d) = 0.1098 ± 0.0008; and f2 (d) = 3.0041 ± 0.0006. Using the semi-empirical values of the screening constants along with the energy limit from NIST [10], Eqs. (3) and (4) are expressed explicitly as follows: E 2pns ;2S +1 P1◦ = 31.1533 −
25
1+
0.13424
(n − 1) (n − 4) × (n − 5) − 0.59866 + 0.018020377 (n − 3) × (n + 1) × (n − 2)2 2 0.026848 (n − 2) × (n − 3) + + 0.018020377 × × S (n − 1.5) (n + 1.5)3 n2
× 13.605698, (5) 0.02196 25 2S +1 ◦ E 2pnd ; P1 = 31.1533 − 2 1 − n (n − 1) (n − 3) × (n − 4) − 0.60082 + 0.004392 × (n + 1)2 × (n + 7) 0.004392 0.02196 − + (n − 1) (n + 3.5) 2 0.0008784 × (n − 3) × (n − 4) + × S × 13.605698. (6) (n − 1.5)2 In these equations, S = 0 for the singlet 2pns 1 P◦ and 2pnd 1 P◦ states and S = 1 for the triplet 2pns 3 P◦ and 2pnd 3 P◦ states. 2.2. Resonance widths of the 2pns 1 P◦ and 2pnd 1 P◦ doubly excited states of B+ The resonance widths of the 2pns 1 P◦ and 2pnd 1 P◦ doubly excited states of the B+ ion are given by (in Rydberg units)
Γ 2pns ; P1 = 1
+
−
f12
◦
Z02 n2
1−
f 1 ( s) Z0 (n − 1)
−
f 2 ( s) Z0
(s) × (n − n0 ) × (n − n0 ) ′
Z0 (n + n0 )2 × (n + n′0 )2 f12 (s) × (n − n0 ) × (n − n′0 ) Z02 (n + n′0 − n0 ) × (n + n′0 − n0 + 1)
2
,
(7)
Γ 2pnd ;1 P1 =
◦
+
−
Z02 n2
1−
449
f1 (d) Z0 (n − 1)
−
f2 (d) Z0
f12 (d) × (n − n0 ) × (n − n′0 ) Z0 (n + n0 )2 × (n + n′0 )2 f12 (d) × (n − n0 ) × (n − n′0 ) Z02 (n + n′0 − n0 ) × (n + n′0 − n0 + 1)
2
.
(8)
Using the photon-ion merged-beams results of Schippers et al. [2], for the 2p4s 1 P◦ (n0 = 4) and 2p5 1 P◦ (n′0 = 5) levels of B+ (with Z0 = 5) equal to 0.220 ± 0.001 and 0.106 ± 0.007 (in eV), respectively, Eq. (5) gives f1 (s) = −0.80773702 and f2 (s) = 4.760605197. In the same way, from the data of Ref. [2] we obtain for the 2p3d 1 P◦ (n0 = 3) and 2p4d 1 P◦ (n′0 = 4) levels of B+ 0.034 ± 0.002 and 0.016 ± 0.002 (in eV), respectively. Eq. (5) gives f1 (d) = −0.076790706 and f2 (d) = 4.888426766. Taking into account the uncertainties in the resonance widths, we obtain ∆f1 (s) = 0.07, ∆f2 (s) = 0.02, ∆f1 (d) = 0.02, and ∆f2 (d) = 0.02. Then the fi -screening constants are expressed as follows: f1 (s) = −0.81 ± 0.07; f2 (s) = 4.76 ± 0.02; f1 (d) = −0.08 ± 0.02; and f2 (d) = 4.89 ± 0.02. Using these semi-empirical values, Eqs. (6) and (7) are expressed as follows:
25 Γ 2pns ;1 P1◦ = 2
1+
n
×
0.162
(n − 1)
− 0.952 + 0.13122
(n − 4) × (n − 5) (n + 4)2 × (n + 5)2
(n − 4) × (n − 5) 2 × 13.605698, (n + 1) × (n + 2) 25 0.016 Γ 2pnd ;1 P1◦ = 2 1 + − 0.978 + 0.00128 n (n − 1) (n − 3) × (n − 4) × (n + 3)2 × (n + 4)2 (n − 3) × (n − 4) 2 − 0.000256 × × 13.605698. (n + 1) × (n + 2) − 0.026244 ×
(9)
(10)
3. Results and discussions The results obtained in the present paper are listed in Tables 1– 5 compared to available theoretical and experimental literature values. The analysis of the current results is achieved by calculating the quantum defect µ given by
µ=n−2
R E∞ − En
.
(11)
In this equation, R denotes the Rydberg constant equal to 13.605898 eV, E∞ is the limit taken from NIST [10] with E∞ = 31.1533 eV, and En is the energy resonance. In Table 1, the present results for energy positions (E) of the (2pns) 1 P◦ doubly excited states of B+ are compared to the MCRRPA values of Hsiao et al. [4], the NER-M results of Kim and Manson [3], the R-matrix (R-M) calculations of Tullyet al. [11], and to the experimental data of Schippers et al. [2] at the ALS. As shown by Kim and Manson [3] for the (2pns) 1 P◦ excited states of B+ , the (2p3) 1 P◦ state is a true bound state, and not an autoionizing resonance. This explains why the corresponding energy position is not quoted in Table 1. The agreement between the calculations and measurements is seen to be very good. In addition, it is also seen that the energy deviations with respect to the experimental data do not exceed 0.01 eV for up to n = 10. It should be mentioned that the present results agree well with the MCRRPA data [4] up to n = 20. In addition, for all the Rydberg series, the quantum defect is almost constant and an average value of 0.405 is obtained. This may
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allow one to expect our results up to n = 25 to be accurate. This average value is in good agreement with the experimental results of Schippers et al. [2] at 0.408 (2) using for the Rydberg constant the value 13.6057 eV and for the energy limit 31.166 (1), which is less accurate than the more exact spectroscopy value of 31.1533 from NIST [10]. Table 2 shows a comparison between the present calculations for energy positions (E) of doubly excited states (2pnd) 1 P◦ of B+ and some theoretical [3,4,11] and experimental [2] data. Here again, the present data compare very well to the experimental values [2] where the energy deviations evaluated relatively to the experimental data do not exceed 0.008 eV for states up to n = 10. By comparing theoretical calculations, the agreement is seen to be very good. The current results compare well to the MCRRPA data [4] up to n = 20. In addition, for all the (2pnd) 1 P◦ resonances considered, the quantum defect is roughly constant. The average value obtained is −0.0869 and agrees very well with the experimental data of −0.087 (1) where the energy limit at 31.173 (2) is used. In Tables 3 and 4 we present the resonance widths (Γ ) of the (2pns) 1 P◦ and (2pnd) 1 P◦ doubly excited states of B+ obtained in this work compared to the MCRRPA values of Hsiao et al. [4], the NER-M results of Kim and Manson [3], the R-M calculations of Tully et al. [11] and to the ALS experiments of Schippers et al. [2]. Comparison indicates a very good agreement between the present theoretical widths and those of Kim and Manson [3] for both (2pns) 1 P◦ and (2pnd) 1 P◦ levels (n ≤ 12). For comparisons with the other data given in the tables the agreement is only satisfactory. Comparing the present calculations to the MCRRPA data [4], discrepancies appear particular for the (2pnd) 1 P◦ levels when the principal quantum number ranges between 12 and 20. As noted previously [5], if the agreement between theory and experiment is very good for energy positions (as shown in Tables 1–2), then this is not necessarily the same for widths. As a result, further new work by experiment and theory is needed to resolve the discrepancies observed for the high-lying doubly excited states of B+ investigated in this work. For the (2pns) 3 P◦ and (2pnd) 3 P◦ doubly excited states of B+ , calculations are very scarce. Only the results obtained from the MCRRPA computations [4] are found in the literature. Comparison of the data given in Table 5 indicates a very good agreement up to n = 20. Along all the investigated Rydberg series, the quantum defects are almost constant with an average value of 0.493 for the (2pns) 3 P◦ resonances and −0.013 for the (2pnd) 3 P◦ states. In Table 6, we have listed the present results for very high-lying states of the (2pns) 1 P◦ and (2pnd) 1 P◦ Rydberg series up to n = 60. For these resonances, the quantum defects are roughly constant with an average value of 0.482 for the (2pns) 1 P◦ states and −0.135 for the (2pnd) 1 P◦ levels. This leads us to consider these results as accurate even if no literature data are available at this time for direct comparison. Overall, the simplicity of the SCUNC formalism should be noted and it gives very accurate results from simple semi-empirical formulas. In the MCRRPA [4] formalism, calculations are done in the jj coupling scheme, in which couplings between fully relativistic channels are treated. The good agreement between the present SCUNC calculations and the MCRRPA computations [4] up to n = 20 may be explained as follows seeing that the SCUNC formalism is a nonrelativistic one. For Be-like systems, the Hamiltonian can be expressed as H = H0 + W .
orbit corrections (Wsoo ), and spin–spin corrections (Wss ). The nonrelativistic Hamiltonian and the perturbation operators are explicitly the following: H0 =
2
i=1
ri
8
2
WM = −
Wso =
(13)
rij
→ δ(− r i ),
(15)
i =1
4 1
M
∇i · ∇ j ,
(16)
i,j=1 i̸=j
→ − → 4 − Z l i· s i ri3
2c 2 i=1
Wsoo = −
(14)
i=1
4 3π α 2
WD =
i,j=1 i̸=j
4 α2 4 − → p i,
Wkin = −
4 1
2c 2
i,j=1 i̸=j
,
(17)
1 − − → − → − → − → (→ ri − rj ) × pi · ( si + 2 sj ), rij3
(18)
and Wss =
4 1 1
c2
i,j=1 j>i
rij3
− → − → − → − → 3( si · rij )( sj · rij ) − → − → s · s − . i
j
rij2
(19)
In these expressions, α denotes the fine structure constant and M is the nuclear mass of the Be-like system. Using Eq. (8), the corresponding eigenvalue is in the form E = E0 + w
(20)
with
w = ⟨Wkin ⟩ + ⟨WD ⟩ + ⟨WM ⟩ + ⟨Wso ⟩ + ⟨Wsoo ⟩ + ⟨Wss ⟩ .
(21)
In the framework of the SCUNC formalism, for four electron systems, the total energy is given by E 1s2 ; N ℓnℓ′ ;2S +1 Lπ = E 1s2 − Z 2
+
1 n2
1 − β N ℓnℓ′ ;2S +1 Lπ ; Z
2
1 N2
.
(22)
This equation can be written as follows: E 1s2 ; N ℓnℓ′ ;2S +1 Lπ = −
Z2 1
−
Z2 1
−
Z2 N2
−
Z2 n2
1 β N ℓnℓ′ ;2S +1 Lπ ; Z 2 − β N ℓnℓ′ ;2S +1 Lπ ; Z . n2 That means +
E 1s2 ; N ℓnℓ′ ;2S +1 Lπ = E0 + w
(23)
(24)
where
2 2 2 2 E0 = − Z − Z − Z − Z 2 2 1 1 N n (25) w = 1 β N ℓnℓ′ ;2S +1 Lπ ; Z 2 − β N ℓnℓ′ ;2S +1 Lπ ; Z . 2 n Using (21), the last equation of (25) can be rewritten in the form
(12)
In this expression, H0 denotes the nonrelativistic Hamiltonian and W is the sum of the perturbation operators, which includes correction to kinetic energy (Wkin ), the Darwin term (WD ), mass polarization (WM ), spin–orbit corrections (Wso ), spin–other
4 4 1 Z 1 − ∇i2 − + ,
1 n2
β N ℓnℓ′ ;2S +1 Lπ ; Z 2 − β N ℓnℓ′ ;2S +1 Lπ ; Z
= ⟨Wkin ⟩ + ⟨WD ⟩ + ⟨WM ⟩ + ⟨Wso ⟩ + ⟨Wsoo ⟩ + ⟨Wss ⟩ .
(26)
I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
This equation indicates clearly that, in the framework of the SCUNC formalism, all the contributions of the perturbation operators are taken into account by the β screening constant by unit nuclear charge. In the independent particle model, which disregards all the perturbation effects, the total energy is given by E0 . Consequently w = 0. This implies automatically β = 0. The accurate results that result from the SCUNC formalism are then explained by the fact that the β parameters are expanded in inverse powers of Z as shown by Eq. (2) where the fk = fk N ℓ nℓ′ ;2S +1 Lπ screening constants are evaluated empirically using experimental data incorporating all the perturbation effects given by Eq. (21). It should be mentioned also that calculation of the perturbation operator W is a very complex computational problem. The merit of the SCUNC method is the possibility to overcome all these difficulties in a semi-empirical procedure as demonstrated in our recent works [5–9]. 4. Summary The energy positions of the (2pns) 1,3 P◦ and (2pnd) 1,3 P◦ autoionizing states and resonance widths of the (2pns) 1 P◦ excited
451
states of the B+ ion are presented in this paper using the SCUNC method. In general, the present results agree very well with both the noted theoretical and experimental literature data. For n ≥ 21, no theoretical and experimental literature values are available for direct comparison. The good accuracy obtained in this work indicates that the results may be of interest for future experimental and theoretical studies of the photoabsorption spectrum of B+ . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
J.R. King, Astron. J. 122 (2001) 3115. S. Schippers, et al., J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 3371. D.-S. Kim, S.T. Manson, J. Phys. B: At. Mol. Opt. Phys. 37 (2004) 4013. J.-T. Hsiao, H.-T. Shiao, K.-N. Huang, Chin. J. Phys. 47 (2009) 173. I. Sakho, Rad. Phys. Chem. 80 (2011) 1295. I. Sakho, A. Wague, Chin. J. Phys 48 (2010) 1. I. Sakho, A. Konte, A.S. Ndao, M. Biaye, A. Wague, Phys. Scr. 82 (2010) 035301. I. Sakho, Eur. Phys. J. D 59 (2010) 171. I. Sakho, Eur. Phys. J. D 61 (2011) 267. W.C. Martin, et al., NIST Atomic Spectra Data Base, second ed., National Institute of Standards and Technology Gaithersburg, Maryland 20899-3460, USA, 1999, http://physics.nist.gov/cgi-bin/AtData/main_asd. [11] J.A. Tully, M.L. Dourneuf, C.J. Zeippen, Astron. Astrophys. 211 (1989) 485.
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Explanation of Tables Table 1
Table 2 Table 3
Table 4 Table 5
Table 6
Energy positions (E) of doubly (2pns) 1 P◦ excited states of B+ The results are expressed in eV. SCUNC Present results MCRRPA Multiconfiguration relativistic random-phase approximation values of Hsiao et al. [4] NER-M Noniterative eigenchannel R-matrix results of Kim and Manson [3] R-M R-matrix calculations of Tullyet al. [11] ALS Advanced Light Source experiments of Schippers et al. [2] |∆E | Energy differences relative to the experimental data [2]. Energy positions (E) of doubly (2pnd) 1 P◦ excited states of B+ Same as for Table 1. Resonance widths (Γ ) of doubly (2pns) 1 P◦ excited states of B+ The results are expressed in eV. The a(−b) (c) notation means a × 10−b and (c) indicate the uncertainties of the experimental widths given in parentheses. SCUNC Present results MCRRPA Multiconfiguration relativistic random-phase approximation values of Hsiao et al. [4] NER-M Noniterative eigenchannel R-matrix results of Kim and Manson [3] R-M R-matrix calculations of Tully et al. [11] ALS Advanced Light Source experiments of Schippers et al. [2]. Resonance widths (Γ ) of doubly (2pnd) 1 P◦ excited states of B+ Same as for Table 3. Energy (E) of doubly (2pns) 3 P◦ and (2pnd) 3 P◦ excited states of B+ The results are expressed in eV. SCUNC Present results MCRRPA Multiconfiguration relativistic random-phase approximation values of Hsiaoet al. [4]. Present energy positions and quantum defects of high lying doubly (2pns) 1 P◦ and (2pnd) 1 P◦ excited states of B+ The results are expressed in eV. E Energy positions µ Quantum defects.
I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
453
Table 1 Energy positions (E) of doubly (2pns) 1 P◦ excited states of B+ . ns
Theory SCUNC
Experiment
Theory
MCRRPA
NER-M
R-M
ALS
SCUNC
E
|∆E |∗
µ
26.9137 28.5634 29.4084 29.8983 30.2075 30.4151 30.5612 30.6680 30.7484 30.8103 30.8591 30.8982 30.9301 30.9564 30.9783 30.9968 31.0125
26.9405 28.5568 29.3924 29.8787 30.1861 30.3927 30.5382 30.6444 30.7244
26.96 28.59 29.42
26.923 (5) 28.580 (3) 29.420 (3) 29.895 (3) 30.205 (3) 30.409 (7) 30.562 (9)
0.000 1 0.000 1 0.000 1 0.0112 0.0078 0.0098 0.0016
0.413 0.401 0.396 0.394 0.393 0.393 0.393 0.394 0.395 0.396 0.399 0.402 0.402 0.404 0.410 0.411 0.416 0.412 0.423 0.426 0.422
E 4s 5s 6s 7s 8s 9s 10s 11s 12s 13s 14s 15s 16s 17s 18s 19s 20s 21s 22s 23s 24s ... ∞s
26.9229 (50) 28.5799 (30) 29.4201 (20) 29.9062 (14) 30.2129 (11) 30.4187 (8) 30.5636 (7) 30.6695 (5) 30.7492 (5) 30.8107 (4) 30.8591 (3) 30.8979 (3) 30.9296 (3) 30.9557 (2) 30.9774 (2) 30.9958 (2) 31.0114 (2) 31.0249 (1) 31.0364 (1) 31.0465 (1) 31.0554 (1) 31.1533
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I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
Table 2 Energy positions (E) of doubly (2pnd) 1 P◦ excited states of B+ . nd
Theory SCUNC
Experiment MCRRPA
NER-M
R-M
E 3d 4d 5d 6d 7d 8d 9d 10d 11d 12d 13d 14d 15d 16d 17d 18d 19d 20d 21d 22d 23d 24d ... ∞d
25.4578 (10) 27.8889 (10) 29.0443 (8) 29.6804 (6) 30.0670 (5) 30.3194 (4) 30.4931 (3) 30.6177 (3) 30.7101 (2) 30.7805 (2) 30.8354 (2) 30.8790 (1) 30.9142 (1) 30.9430 (1) 30.9669 (1) 30.9870 (1) 31.0040 (1) 31.0185 (1) 31.0310 (1) 31.0418 (1) 31.0513 (1) 31.0596 (1) ... 31.1533
25.5731 27.9453 29.0747 29.6983 30.0784 30.3269 30.4983 30.6215 30.7130 30.7828 30.8372 30.8805 30.9156 30.9443 30.9681 30.9881 31.0051 31.0196
25.4262 27.8610 29.0166 29.6533 30.0408 30.2935 30.4676 30.5924 30.6850 30.7556
25.47 27. 90 29.05 29.68
ALS
SCUNC
E
|∆E |
µ
25.458 (1) 27.889 (1) 29.041 (1) 29.676 (1) 30.064 (2) 30.320 (2) 30.490 (4) 30.610 (1)
0.000 2 0.000 1 0.003 3 0.004 4 0.003 0 0.000 6 0.003 1 0.007 7
−0.091 −0.083 −0.080 −0.080 −0.080 −0.080 −0.080 −0.080 −0.081 −0.082 −0.084 −0.086 −0.087 −0.087 −0.087 −0.090 −0.092 −0.093 −0.095 −0.093 −0.091 −0.100
I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
455
Table 3 Resonance widths (Γ ) of doubly (2pns) 1 P◦ excited states of B+ . States (2p4s) 1 P◦ (2p5s) 1 P◦ (2p6s) 1 P◦ (2p7s) 1 P◦ (2p8s) 1 P◦ (2p9s) 1 P◦ (2p10s)1 P◦ (2p11s)1 P◦ (2p12s)1 P◦ (2p13s)1 P◦ (2p14s)1 P◦ (2p15s)1 P◦ (2p16s)1 P◦ (2p17s)1 P◦ (2p18s)1 P◦ (2p19s)1 P◦ (2p20s)1 P◦
Theory
Experiment
SCUNC
MCRRPA
NER-M
R-M
ALS
2.212 (−1) (100) 1.066 (−1) (44) 5.969 (−2) (28) 3.685 (−2) (25) 2.437 (−2) (21) 1.696 (−2) (20) 1.230 (−2) (20) 9.207 (−3) (13) 7.081 (−3) (11) 5.569 (−3) (10) 4.465 (−3) (10) 3.638 (−3) (10) 3.008 (−3) (10) 2.518 (−3) (10) 2.131 (−3) (5) 1.822 (−3) (4) 1.571 (−3) (4)
2.663 (−1) 1.188 (−1) 6.343 (−2) 3.788 (−2) 2. 442 (−2) 1.663 (−2) 1.179 (−2) 8.629 (−3) 6.465 (−3) 4.927 (−3) 3.808 (−3) 2. 978 (−3) 2. 356 (−3) 1.884 (−3) 1. 521 (−3) 1. 234 (−3) 9. 993 (−4)
2.453 (−1) 1.153 (−1) 6.350 (−2) 3.840 (−2) 2.490 (−2) 1.710 (−2) 1.220 (−2) 9.000 (−3) 6.900 (−3)
2.41 (−1) 1.10 (−1) 5.90 (−2)
2.20 (−1) (1) 1.06 (−1) (7) 4.80 (−2) (6) 2.90 (−2) (7) 2.00 (−2) (6) 4.00 (−2) (1) 2.00 (−2) (2)
456
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Table 4 Resonance widths (Γ ) of doubly (2pnd) 1 P◦ excited states of B+ . States (2p3d) 1 P◦ (2p4d) 1 P◦ (2p5d) 1 P◦ (2p6d) 1 P◦ (2p7d) 1 P◦ (2p8d) 1 P◦ (2p9d) 1 P◦ (2p10d)1 P◦ (2p11d)1 P◦ (2p12d)1 P◦ (2p13d)1 P◦ (2p14d)1 P◦ (2p15d)1 P◦ (2p16d)1 P◦ (2p17d)1 P◦ (2p18d)1 P◦ (2p19d)1 P◦ (2p20d)1 P◦
Theory
Experiment
SCUNC
MCRRPA
NER-M
R-M
ALS
3.40 (−2) (20) 1.59 (−2) (20) 9.2 (−3) (15) 6.0 (−3) (11) 4.2 (−3) (10) 3.1 (−3) (10) 2.4 (−3) (10) 1.9 (−3) (5) 1.6 (−3) (4) 1.3 (−3) (3) 1.1 (−3) (3) 9.3 (−4) (2) 8.0 (−4) (2) 7.0 (−4) (2) 6.2 (−4) (2) 5.5 (−4) (2) 4.870 (−4) (1) 4.380 (−4) (1)
5.17 (−2) 2.21 (−2) 1.14 (−2) 6.61 (−3) 4.17 (−3) 2.79 (−3) 1.96 (−3) 1.42 (−3) 1.06 (−3) 8.05 (−4) 6.21 (−4) 4.84 (−4) 3.80 (−4) 2.99 (−4) 2.35 (−4) 1.82 (−4) 1.38 (−4) 9.69 (−5)
3.12 (−2) 1.41 (−2) 9.5 (−3) 6.4 (−3) 4.4 (−3) 3.1 (−3) 2.3 (−3) 1.7 (−3) 1.3 (−3) 1.1 (−3)
3.38 (−2) 1.78 (−2) 5.90 (−2) 2.41 (−1)
3.4 (−2) (2) 1.6 (−2) (2) 1.0 (−2) (3) 8.0 (−3) (3) 8.0 (−3) (4) 1.0 (−3) (6) 5.0 (−3) (8)
I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
457
Table 5 Energy (E) of doubly (2pns) 3 P◦ and (2pnd) 3 P◦ excited states of B+ . ns
SCUNC
MCRRPA
SCUNC
nd
SCUNC
MCRRPA
4s 5s 6s 7s 8s 9s 10s 11s 12s 13s 14s 15s 16s 17s 18s 19s 20s 21s 22s ... ∞s
26.7126 (53) 28.4836 (32) 29.3673 (22) 29.8739 (16) 30.1915 (12) 30.4038 (9) 30.5528 (7) 30.6613 (6) 30.7429 (5) 30.8057 (4) 30.8551 (3) 30.8947 (3) 30.9269 (3) 30.9534 (2) 30.9755 (2) 30.9942 (2) 31.0100 (2) 31.0236 (1) 31.0354 (1) ... 31.1533
26.7050 28.4707 29.3586 29.8680 30.1874 30.4008 30.5504 30.6593 30.7410 30.8039 30.8533 30.8929 30.9250 30.9515 30.9736 30.9922 31.0080
0.499 0.485 0.480 0.478 0.478 0.479 0.480 0.483 0.485 0.488 0.490 0.493 0.496 0.500 0.504 0.505 0.512 0.516 0.515 ...
3d 4d 5d 6d 7d 8d 9d 10d 11d 12d 13d 14d 15d 16d 17d 18d 19d 20d 21d ... ...
25.1515 (32) 27.7757 (18) 28.9891 (12) 29.6490 (8) 30.0472 (6) 30.3060 (5) 30.4835 (4) 30.6106 (3) 30.7046 (3) 30.7762 (2) 30.8319 (2) 30.8761 (2) 30.9118 (2) 30.9410 (1) 30.9652 (1) 30.9855 (1) 31.0027 (1) 31.0174 (1) 31.0300 (1) ... ...
25.1505 27.7498 28.9706 29.6370 30.0394 30.3008 30.4801 30.6083 30.7032 30.7754 30.8315 30.8761 30.9120 30.9414 30.9658 30.9862 31.0035 31.0183
−0.012 −0.014 −0.015 −0.015 −0.015 −0.014 −0.014 −0.014 −0.013 −0.013 −0.013 −0.012 −0.012 −0.012 −0.011 −0.011 −0.011 −0.010 −0.010
... ...
... ...
...
µ
SCUNC
µ
458
I. Sakho et al. / Atomic Data and Nuclear Data Tables 99 (2013) 447–458
Table 6 Present energy positions and quantum defects of high lying doubly (2pns) 1 P◦ and (2pnd) 1 P◦ excited states of B+ . ns
E
µ
nd
E
µ
25s 26s 27s 28s 29s 30s 31s 32s 33s 34s 35s 36s 37s 38s 39s 40s 41s 42s 43s 44s 45s 46s 47s 48s 49s 50s 51s 52s 53s 54s 55s 56s 57s 58s 59s 60s ... ∞s
31.06317 (10) 31.07006 (9) 31.07619 (9) 31.08167 (8) 31.08658 (7) 31.09101 (7) 31.09501 (7) 31.09863 (6) 31.10193 (6) 31.10494 (5) 31.10769 (5) 31.11021 (5) 31.11253 (5) 31.11467 (4) 31.11664 (4) 31.11847 (4) 31.12016 (4) 31.12173 (4) 31.12319 (3) 31.12456 (3) 31.12583 (3) 31.12702 (3) 31.12813 (3) 31.12918 (3) 31.13016 (3) 31.13108 (2) 31.13195 (2) 31.13277 (2) 31.13354 (2) 31.13427 (2) 31.13496 (2) 31.13562 (2) 31.13623 (2) 31.13682 (2) 31.13738 (2) 31.13791 (2) ... 31.1533
0.427 0.430 0.433 0.430 0.436 0.442 0.444 0.449 0.451 0.453 0.457 0.461 0.464 0.466 0.470 0.471 0.476 0.480 0.486 0.484 0.490 0.493 0.500 0.499 0.504 0.510 0.512 0.513 0.520 0.522 0.526 0.518 0.536 0.534 0.532 0.534
25d 26d 27d 28d 29d 30d 31d 32d 33d 34d 35d 36d 37d 38d 39d 40d 41d 42d 43d 44d 45d 46d 47d 48d 49d 50d 51d 52d 53d 54d 55d 56d 57d 58d 59d 60d ... ∞d
31.06694 (5) 31.07344 (4) 31.07923 (4) 31.08442 (4) 31.08908 (3) 31.09328 (3) 31.09708 (3) 31.10054 (3) 31.10368 (3) 31.10655 (3) 31.10918 (2) 31.11159 (2) 31.11381 (2) 31.11586 (2) 31.11776 (2) 31.11951 (2) 31.12113 (2) 31.12265 (2) 31.12405 (2) 31.12537 (2) 31.12659 (1) 31.12774 (1) 31.12881 (1) 31.12982 (1) 31.13077 (1) 31.13166 (1) 31.13250 (1) 31.13329 (1) 31.13404 (1) 31.13475 (1) 31.13541 (1) 31.13605 (1) 31.13665 (1) 31.13721 (1) 31.13775 (1) 31.13827 (1) ... 31.1533
−0.103 −0.105 −0.106 −0.105 −0.111 −0.112 −0.113 −0.105 −0.117 −0.119 −0.121 −0.122 −0.123 −0.126 −0.132 −0.133 −0.131 −0.138 −0.135 −0.142 −0.139 −0.143 −0.141 −0.144 −0.148 −0.149 −0.151 −0.151 −0.157 −0.165 −0.155 −0.169 −0.172 −0.158 −0.159 −0.174