by the Screening Constant by Unit Nuclear Charge Method

In the present work, accurate high lying single photoionization resonance energies for Aluminium-like P and magnesium-like P 3+ are reported. Calculations are performed in the framework of the Screening Constant by Unit Nuclear Charge (SCUNC) formalism. The resonance energies and quantum defects obtained compared very well with experimental data of Hernández et al., (2015) along with DARC, Dirac Atomic R-matrix Codes computations of Wang et al., (2016). Analysis of the present results is achieved in the framework of the standard quantum-defect theory and of the SCUNCprocedure based on the calculation of the effective charge. It is demonstrated that the SCUNC-method can be used to assist fruitfully experiments for identifying narrow resonance energies due to overlapping peaks. New precise data for Aluminiumlike P 2+ and magnesium-like P 3+ ions are presented as useful guidelines for investigators focusing their challenge on the Photoionization of aluminum-like P 2+ and magnesium-like P 3+ heavy charged ions in connection with their application in laboratory, astrophysics, and plasma physics. In addition, our predicted data up to n = 30 may be of great importance for the atomic physics community in connection with the determination of accurate abundances for phosphorus in the solar photosphere, in solar twins, in the infrared spectrum of Messier 77 galaxy (NGC1068).


Introduction
Phosphorus is a primary element in the ribonucleic acid (RNA) of all living cells and functions in signal passing for DNA. However, its detection has been difficult in comparison with other basic elements of life, such as carbon, oxygen, etc. Recently, it has been detected in a number of astronomical objects, e.g. in damp galaxies by Molaro et al., 2001 [1] and Welsh et al., 2001 [2]. In addition Caffau and collaborators [3] proposed the possibility that phosphorus could be formed in late stages of stars. Later, Bon-chul et al., 2013 [4] found evidence of phosphorus in supernovae by measuring the infrared spectra in the remnants of Cassiopea A. Extragalactic phosphorus has been observed in the solar photosphere by Caffau et al., 2007 [5], in solar twins by Meléndez et al., 2009 [6], in the infrared spectrum of Messier 77 galaxy (NGC1068) by Oliva et al., 2001 [7] and in globular clusters by Hubrig et al., 2009 [8] thus the determination of observed phosphorus abundances remains an important issue. In a recent past, Hernández et al., 2015 [9] measured the single PI cross sections of Al-like P 2+ and Mg-like P 3+ based on the merged-beams technique [10] and obtained the resonance energies and quantum defects for the assigned Rydberg series. Very recently the theoretical photoionization of the ground and metastable states P 2+ are first time presented in the photon energy range of 30-43.5 eV Momar Talla Gning et al.: Photoionization of Aluminum-Like P 2+ and Magnesium-Like P 3+ by the Screening Constant by Unit Nuclear Charge Method by Wang et al., 2016 [11]. However it should be recalled that, the theoretical PI studies of Al-like P 2+ are really rare, and there is no corresponding theoretical data in the previous reports and the comprehensive databases, such as the Opacity Project TOPbase [12]. Therefore, to benchmark the PI measurement of experiment for Al-like P 2+ [9], the theoretical PI cross sections of Al-like P 2+ are necessary, and the theoretical study can serve as a candidate for the database mentioned above. Moreover, the relative population of ground and metastable states need to be taken into account for determining the absolute PI cross sections. The motivation of this work is to use the screening constant by unit nuclear charge (SCUNC) formalism (Sakho [13][14][15][16]; Ba et al. [17]; Badiane et al. [18]; Khatri et al [19]) to report accurate high lying Photoionization data for aluminum-like P 2+ and magnesium-like P 3+ . The layout of the present paper is as follows. In Section 2, we present a brief outline of the theoretical part of the work. The presentation and the discussion of the results obtained are given in Section 3 where comparisons are made with the available experimental of Hernández et al., 2015 [9] and theoretical of Wang et al., 2016 [11] data. In Section 4 we summarize our study and draw conclusions.

Brief Description of the SCUNC Formalism
In the framework of the screening constant by unit nuclear charge formalism, the total energy of the (Nl,nl'; 2S+1 L π ) excited states is expressed in the form (in Rydbergs).
In this equation, the principal quantum numbers N and n are respectively for the inner and the outer electron of the helium-isoelectronic series. The β-parameters are screening constants by unit nuclear charge expanded in inverse powers of Z and given by ( ) In this equation, ν and µ (µ>ν) denote the principal quantum numbers of the ( 2S+1 L J )nl Rydberg series used in the empirical determination of the f i -screening constants, s represents the spin of the nl-electron (s = ½), E ∞ is the energy value of the series limit, E n denotes the resonance energy and Z stands for the atomic number. Theβ-parameters are screening constants by unit nuclear charge expanded in inverse powers of Z and given by The quantity ( ) is a corrective term introduce to stabilize the resonance energies with increasing the principal quantum number n. In general, resonance energies are analyzed from the standard quantum-defect expansion formula In this equation, R is the Rydberg constant, E ∞ denotes the converging limit, Z core represents the electric charge of the Z core ion, and δ means the quantum defect. In addition, theoretical and measured energy positions can be analyzed by calculating the Z* effective charge in the framework of the SCUNC-procedure The relationship between Z * andδ is in the form Besides, comparing Eq.(5) and Eq. (7), the effective charge is in the form Besides, the f 2 -parameter in eq.(2) can be theoretically determined from eq.(10) by neglecting the corrective term with the condition ( ) We get then f 2 =Z-Z core , where Z core is deduced from the photoionization process of the considered atomic X m+ system, hν+X m+ →X (m+1)+ +e − find then Z core =m+1. As an illustration for P 2+ we have hν+P 2+ →P 3+ +e − from where Z core =3 and for P 3+ we have hν+P 3+ →P 4+ +e − from where Z core =4. So, for the P 2+ ion, f 2 =(15−3)=12.0 and for P 3+ ion, f 2 =(15−4)=11.0. The remaining f 1 -parameter is to be evaluated empirically using the experimental data of Hernández et al., 2015 [9] for a given ( 2S+1 L J )nl level with ν=0 in Eq. (5). The empirical procedure of the determination of the f 1 -screening constant along with the corresponding uncertainty have been explained in details in our previous works (Sakho [13][14][15][16]; Ba et al. [17]; Badiane et al. [18]). The results obtained are quoted in Tables 1-4.

Resonance Energies of the 2p 6 3pnp( 2 P 1/2 ) Rydberg Serie of Magnesium-Like P 3+
Using Eq (5), we obtain the following expressions of the resonance energies for Rydberg series of the ion P 3+ (in Rydberg). i. For the Rydberg serie 2p 6 3p( 2 P 1/2 )np originating from the excited state 2p 6 3s3p( 3 P 0 ) of P 3+ ii. For the Rydberg serie 2p 6 3p( 2 P 1/2 )np originating from the excited state 2p 6 3s3p( 3 P 2 )ofP 3+ iii. For the Rydberg serie 2p 6 3p( 2 P 1/2 )np originating from the ground state 2p 6 3s 2 ( 1 S 0 ) of P 3+ In these expressions, ν denotes the principal quantum numbers of ( 2S+1 L J )nl Rydberg series used in the empirical determination of f 1 screening constant. The principle of determining the screening constant f 1 is described in the appendix.

Results and Discussion
The results obtained in the present work are tabulated in Tables 1-4. Tables 5-11 lists the resonance energies and quantum defects are obtained, where a comparison between the theoretical and experimental data is made. The analysis of the calculated energy values is made on the basis of the general expression of the quantum defect and on the SCUNC analysis conditions (9) recommended by the present formalism. We recall these expressions:  Tables 5-11. These agreements allow one to expect the SCUNC data quoted in Tables 1-4 to be accurate up to n = 30 with a quantum defect almost constant along the series investigated.
In addition the excellent agreement between the experimental measurements along with the R-matrix approach and the SCUNC predictions may demonstrate the accuracy of our results quoted Tables 1-4 where the quantum defect is seen to be quite constant along each series. Is should be mentioned that, the SCUNC conditions analysis (9) are well verified as shown by the data listed in Tables 1-4 for the different Rydberg series studied for aluminum-like P 2+ and magnesium-like P 3+ . It is demonstrated in this work that the SCUNC-method can be used to assist fruitfully experiments for identifying narrow resonance energies due to overlapping peaks. New high lying accurate resonance energies (n = 3-30) are tabulated as benchmarked data for the atomic physics community in connection with the modeling of plasma and astrophysical systems.

Summary and Conclusion
The screening constant by unit nuclear charge (SCUNC) has been applied to the photoionization of the ground and metastable states of aluminum-like P 2+ and magnesium-like P 3+ . Excellent agreements are obtained between the present predictions and previous studies from Advanced Light Source experiments at Lawrence Berkeley National Laboratory of Hernández et al., 2015 [9] and calculations from Dirac R-matrix method of Wang et al., 2016 [11]. The very good results obtained in this work show that the SCUNC-method can be used to assist the sophisticated Rmatrix-method for locating and determining the properties of atomic resonances. Finally, our predicted data up to n=30 may be of great importance for the atomic physics community in connection with the determination of accurate abundances for phosphorus in the solar photosphere, in solar twins, in the infrared spectrum of Messier 77 galaxy (NGC1068).

Appendix Detailed Processes to Evaluate Empirically the Screening Constants f 1
In the framework of the Screening constant by unit nuclear (SCUNC) method, the screening constants f 1 are evaluated from experimental values. They are then determined empirically with a certain absolute error linked to the experimental measurement errors. We move on explaining in detail the principle for determining the absolute values, ∆f 1 .
Within the framework of the SCUNC formalism, the screening constants, f 1 , are presented as f 1 =f 1exp ±∆f 1 . The absolute errors, ∆f 1 are given by Sakho [13].
In general, the experimental resonance energies are expressed in the form E n =E exp ±∆E, with ∆E the absolute error on the resonance energies. The i f ± screening constants are evaluated using the experimental resonance energies for n=ν for the nl 2S+1 L J Rydberg series considered.
As a result, the corrective term in Eq. (5) In the present work, only f 1 is to be evaluated as f 2 =12.0 for P 2+ and f 2 =11.0 for P 3+ . In this case, one equation is required to find the value of f 1 in Eq.(23) using the following relations for n = ν.