Modified Atomic Orbital Theory Applied to the Study of High Lying (2pns) 1,3P° and (2pnd) 1,3P° Rydberg Series of B+

Citation: Sow M, Diallo A, Ba AD, Badiane JK, Diallo S, et al. (2017) Modified Atomic Orbital Theory Applied to the Study of High Lying (2pns) 1,3P° and (2pnd) 1,3P° Rydberg Series of B+. Int J At Nucl Phys 2:006 Copyright: © 2017 Sow M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. *Corresponding author: I Sakho, Department of Physics, UFR Sciences and Technologies, University Assane Seck of Ziguinchor, Ziguinchor, Senegal, E-mail: aminafatima_sakho@yahoo.fr VIBGYOR ISSN: 2631-5017


Introduction
Divalence atomic systems such as the Be isoelectronic sequence are an interesting area for the systematic studies of the photoionization process because of their relatively simple quasi-two-electron structure. Many works have shown that relativistic effects play an important role in the photoionization of small atoms, such as Be [1]. Therefore, it is worthwhile to investigate the interplay between electron-correlation and relativistic effects on the photoionization of atoms or ions with a low nuclear charge. Considerable theoretical and experimental efforts have been made recently to investigate the photoionization of the B + ion. On the experimental side, Janniti, et al. [2] investigated the absorption spectrum of B + for photon energies between 400 and 1700 Å by using two-laser produced plasma; whereas Schippers, et al. [3] studied the photoionization of the B + valence shell using a photon-ion merged-beams arrangement at the Advanced Light Source (ALS). On the other hand of theoretical side, Tully, et al. [4] reported energy positions for the 2pns 1 P° (n = 4-6) and 2pnd 1 P° (n = 4-6) levels of B + along with resonance widths respectively by using R-matrix method; Kim and Manson, [5] investigated the photoionization of the 1 S ground state of Be-like B + ion leading to the 2s, 2p, 3s and 3p states employing a Noniterative Eigenchannel R-Matrix (NER-M) method; Hsiao, et al. [6] studied five Rydbergs series of doubly excited 2pns 1,3 P°, 2pnd 1,3 P° and 2pnd 3 D° states in the photoionization spectrum of the singly-ionized boron by using the Multiconfiguration Relativistic Random-Phase Approximation (MCRRPA); Sakho, et al. [7] reported 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) of the B + ion by using the Screening Constant by Unit Nuclear Charge (SCUNC) method. Using the MAOT method [8][9][10][11], Sow, et al. [9] reported accurate energy positions for the 2pns 1,3 P° and 2pnd 1,3 P° levels of the Beryllium atom up to n = 25 along with resonance widths in the particular case of the 2pns 1 P° states (n = 3-25). In this paper, these previous study are extended to the photoionization spectrum of the Be-like B + ion in the framework of the MAOT formalism. The MAOT-method is known to be a very suitable technique of calculation who has given recently accurate results from simple semi-empirical formulas without needing to compute any 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 = 3-30) along with resonance widths of the (2pns) 1 P° and (2pnd) 1 P° (n = 3-25) of the B + ion in the framework of the MAOT formalism.
In Eq. (1), Z stands for the atomic number, σ is the screening constant relative to the electron occupying the νℓ orbital, ν and ℓ denotes respectively the principal quantum number and the orbital quantum number. For an atomic system of several electrons M, the total energy is given by (in Rydberg) With respect to the usual spectroscopic notation σi denotes the screening constant relative to the i-electron and v i represents the principal quantum number of the v i ℓ orbital (v 1 = n = 1 for 1 s, v 2 = n = 2 for 2 s or 2 p and so on). In the photoionisation study, energy resonances are generally measured relatively to the E ∞ converging limit of a given ( 2S+1 L J ) nl-Rydberg series. For these states, the general expression of the energy resonances is given by the formula of Sakho presented previously [11] (in Rydberg units): In this equation m and q (m < q) denote the principal quantum numbers of the ( 2S+1 L J ) nl-Rydberg series of the considered atomic system used in the empirical determination of the σ i ( 2S+1 L J )-screening constants, s represents the spin of the nl-electron (s = ½), E∞ is the energy value of the series limit generally determined from the NIST atomic database [12], E n denotes the corresponding energy resonance, and Z represents the nuclear charge of the considered element. The only problem that one may face by using the MAOT formalism is linked to the determination of the

Energy of the 2pns 1,3 P° and 2pnd 1,3 P° Rydberg series
In the framework of the MAOT formalism, the energy positions are given by (in Rydberg units) using Eq. (3)

• For 2pns 1,3 P o levels
In these equations, m and q (m < q) denote the principal quantum numbers of the 2pns 1 P° and 2pnd 1 P° levels of B + used in the empirical determination of the σ i (l = s or d) -screening constants in equations (4) and (5) and s represents the spin of the nl-electron (s = ½). E ∞ is the energy value of the series limit and defined in NIST [12] E ∞ = 31.1533 eV. The screening constants in equations (4) and (5) are evaluated using experimental data from Advanced Light Source of Schippers, et al. [3] on B + for the 2p4s 1 P° (m = 4) and 2p5s 1 P° (q = 5) levels respectively equal to (in eV) 26.923 ± 0.005 and 28.580 ± 0.003 and that of the 2p3d 1 P° (m = 3) and 2p4d 1 P° (q = 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 these experimental data (with Z = 5), we obtain from equations (4) and (5)

Results and Discussions
The results of the presently calculations for the energy positions of doubly (2pns) 1,3 P° and (2pnd) 1,3 P° excited state of the B + atom are listed in Table 1, Table 2, Table 3,  1 P° excited states of B + . The present MAOT results are compared to the Screening Constant by Unit Nuclear Charge (SCUNC) results of Sakho, et al. [7]; Multiconfiguration Relativistic Random-Phase Approximation (MCRRPA) computations of Hsiao, et al. [6]; Noniterative Eigenchannel R-Matrix (NER-M:) results of Kim and Manson [5]; R-Matrix (R-M) calculations of Tully, et al. [4]; Advanced Light Source (ALS) experiments of Schippers, et al. [3] and to the Vaccum Ultraviolet (VU) measurements of Janniti, et al. [2]. The results are expressed in eV. Besides, it should be mentioned the very good agreement between the present MAOT and the SCUNC [7] predictions up to 2p20d 1 P°. As a result, our listed data for n > 25 are expected to be accurate. Table 3 indicates the present MAOT calculations for the 2pns 3 P° and 2pnd 3 P° doubly excited states compared to the SCUNC results [7] and to the MCRRPA values [6]. For both 2pns 3 P° and 2pnd 3 P° states up to n = 20, the agreements between the calculations are seen to be very good. These agreements allow us to extend the MAOT calculations up to n = 30 and the data are expect them to be accurate. As far as the natural widths of the 2pns 1 P° and 2pns 1 P° Rydberg states are concerned, the results obtained in this work using Eq. (5) and (6) are respectively listed in Table 4 and   [7]; Multiconfiguration Relativistic Random-Phase Approximation (MCRRPA) computations of Hsiao, et al. [6]; Noniterative Eigenchannel R-Matrix (NER-M:) results of Kim and Manson [5]; R-Matrix (R-M) calculations of Tully, et al. [4]; Advanced Light Source (ALS) experiments of Schippers, et al. [3] and to the Vaccum Ultraviolet (VU) measurements of Janniti, et al. [2]. The results are expressed in eV. agreements are obtained between the present MAOT calculations and the quoted literature data, it should mentioned slight discrepancies between the present results and those from the NER-M computations [5]. Let us for instance consider the 10s-state for which ab initio [4], NER-M results of Kim and Manson [5], MCRRPA values of Hsiao, et al. [6], and with the SCUNC results of Sakho, et al. [7]. Here, the agreements between the current MOAT results and all the listed theoretical results are seen to be good up to n = 12. Overall, although good  [7] and to the Multiconfiguration Relativistic Random-Phase Approximation (MCRRPA) computations of Hsiao, et al. [6]. The results are expressed in eV.   ably not well taken into account in the NER-M formalism [5] combined with multichannel quantum-defect theory at the R-matrix surface. In the MAOT formalism, the σ i -screening constants are evaluated using experimental data incorporating all relativistic and electron correlation effects. The excellent agreements between the MAOT calculations and both the MCRRPA [6] computations and the ALS [3] measurements indicate that relativistic and correlation effects are well incorporated in the σ i -screening constants up to high n = 30 levels.

Conclusion
The energy positions of the 2pns 1,3 P° and 2pnd 1,3 P° Rydberg series and widths of the (2pns) 1 P° and (2pnd) 1 P° excited states of the B + ion are presented in this paper using the Modified Atomic Orbital Theory (MAOT). In general, the present results agree very well with both quoted theoretical and experimental literature data. For n ≥ 25, no theoretical and experimental literature values are available for direct comparison. It is shown through this study the simplicity of the formalism in contrast with the ab initio methods cited in the paper. In addition, although the energy resonances of the (2pns) 1,3 P° and (2pnd) 1,3 P° Rydberg states members of the B + ion have been widely calculated by many ab initio methods, the present MAOT calculations have been extended to the high-n Rydberg states and precise data are tabulated within simple analytical formulas up to n = 30. The good accuracy obtained in this work point out that, the Modified Atomic Orbital Theory is suitable for the interpretation of atomic spectra. Extension of the present MAOT formalism to the Be-like (Z > 5) ions is then very challenging.  1 P° excited states of B + . The present MAOT results are compared to the Screening Constant by Unit Nuclear Charge (SCUNC) results of Sakho, et al. [7]; Multiconfiguration Relativistic Random-Phase Approximation (MCRRPA) computations of Hsiao, et al. [6]; Noniterative Eigenchannel R-Matrix (NER-M:) results of Kim and Manson [5]; R-matrix (R-M) calculations of Tully, et al. [4] and to the Advanced Light Source (ALS) experiments of Schippers, et al. [3]. The a (-b) (c) notation means a × 10 −b and (c) indicates the uncertainties of the experimental widths given in parentheses. The results are expressed in eV.