SCREENING CONSTANT BY UNIT NUCLEAR CHARGE CALCULATIONS OF THE 𝟑𝒔 𝟐 𝟑𝒑 𝟒 , RYDBERG SERIES FROM METASTABLE STATE OF Ca 3+ ION

We report in this paper energy positions of the 3𝑠 2 3𝑝 4 3𝑝 4 1 𝑛𝑑 , 3𝑠 2 3𝑝 4 1 𝑛𝑑 , Rydberg series from metastable state of Ca 3+ ion.. Calculations are performed up to n = 20 using the Screening Constant by Unit Nuclear Charge(SCUNC). The present results compared wellwith the experimental data of Ghassan A Alna’washi which are the only available values.The accurate data presented in this work may be a useful guideline for future experimental and other theoretical studies.


ISSN: 2320-5407
Int. J. Adv. Res. 9(07), 227-236 228 In addition, the determination of the photoionization cross section for multicharged atomic systems allows the development of theoretical models of multiple electron interactions [5,6]. Also, knowledge of the cross section of photoionization leads to accurate values of energies and resonance widths. Thus, many theoretical and experimental methods used in the study of photoionization processes are based on the determination of the cross section of photoionization. However, the complex mathematical formalism of the ab initio methods and the recovery of the peaks of the cross-sections of photoionization from a certain value of the energy of the photons used [2] make it difficult to know the resonance parameters for states of high excitation. Thus, the use of the semi-empirical procedure of the SCUNC method leads to precise results for the energies and resonance widths of the Rydberg series without making use of a complex mathematical formalism.
The Screening Constant by Unit Nuclear Charge (SCUNC) formalism presented by Sakho [7][8] [9] is seen to be a very suitable semi-empirical photoionization method for reproducing excellently high experimental data. The goal of the present study is to enlighten the superimposed ALS lines using the SCUNC method for calculating resonances energies of the 3 2 3 4 1 , 3 2 3 4 1 , 3 2 3 4 0 1 , Rydberg series of Ca 3+ ion The only studies on this atomic system have been made, at the Advanced Light Source (ALS) located in the United States, by Ghassan A. Alna'washi and his group using the technique of collinear beams (merge an ion beam with a photon beam). In this work, we aim to extend the ALS experiments of Ghassan A. Alna'washi [2] along with their DARC calculations to the high lying. Analysis of the present results is achieved in the framework of the standard quantum-defect theory and of the SCUNC-procedure by calculating the effective charge. The present paper is organized as follows. Section 2 presents a brief outline of the theoretical part of the work. In section 3, we present and discuss our results compared with the available experimental results. In section 4, we summarize our study and draw conclusions.

Theory Brief description of the SCUNC formalism
In the framework of the Screening Constant by Unit Nuclear Charge formalism, the total energy of the ( 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 -parameters are screening constants by unit nuclear charge expanded in inverse powers of Z and given by [9]   where are parameters to be evaluated empirically.
For a given Rydberg series originating from a-2S+1 L J state, we obtain using [9] In this equation,  and µ (µ) 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 = 1/2) , I k is the Kshell ionization energy, E n denotes the resonance energy and Z stands for the atomic number. The -parameters are screening constants by unit nuclear charge expanded in inverse powers of Z and given by are screening constants to be evaluated empirically. In Eq.(2), q stands for the number of terms in the expansion of the -parameter. Generally, precise resonance energies are obtained for q 5.
The resonance energy are the in the form [9]   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  the energy limit of the series, Z core represents the electric charge of the core ion, and 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 [9] R n The relationship between Z * and is in the form According to this equation, each Rydberg series must satisfy the following conditions Besides, comparing Eq. (6) and Eq. (8), the effective charge is in the form Besides, the f 2 -parameter in Eq. (3) can be theoretically determined from Eq. (11) We get then f 2 = Z -Z core , where Z core is directly obtain by the photoionization process from an atomic X p+ system X p+ + h X (p+1)+ + e  . We find then Z core = p + 1. So, for the Ca3 + ions, Z core = 4. The remaining f 1 -paramerter is to be evaluated empirically using the experimental data from Advanced Light Source (ALS) experiments of Ghassan A. Alna'washi [2]for a given ( 2S+1 L J ) µl level.
In Tables 3, we show a comparison of the energyresonances (SCUNC) and quantum defect (δ) of the 3 2 3 4 1 2 Rydberg series originating from the metastable state of the Ca 3+ ion. The effective charges of the first eleven resonant states of this Rydberg series are all greater than the charge of the Ca 4+ ion because all quantum defects are positive and verify the SCUNC analysis conditions. We note that quantum defect increase so lightly and monotonously which agrees well with the analysis condition. Our results on the resonance energies of this Rydberg series can therefore be considered to be accurate.

2 2
Rydberg series originating from the metastable state of the Ca 3+ ion. For this series, except for the first two auto-ionizing states which have a quantum defect equal to 0.6, the other resonant states have a quantum defect that is equal to 0.5 or near to this value. Moreover, along the series, the quantum defects are positive because the effective charges of the resonant states are greater than the charge of the Ca 4+ ion. Hereagain, the agreements are seen to bevery good. For Bothexperiment and theory, the energydifferences have neveroverrun 0.001 eV. This justifies to expect our results for the resonance energies of this series up to n = 20 to be accurate.
In Table 5, we compare the present (SCUNC) energyresonances and quantum defect (δ) of the 3s 2 3p 4 S 1 nd D 2 Rydberg series originating from the metastable state of the Ca 3+ ion. For this Rydberg series, the quantum defect decreases when the principal quantum number increases. The maximum decrease is equal to 0.04 and takes place between the second and third resonant states. Also the effective charges of the fourteen resonant states for which we calculated the energies are in agreement with the conditions of SCUNC analysis. This allows us to expect our results to be accurate up to n = 20.
In Table 6, wequote the present SCUNC results for energyresonances and quantum defect (δ) of the first fourteen auto-ionizing states of the Rydberg series originating from the metastable state of the Ca 3+ ion. For the first three levels of this series, the quantum defect behaves in the same way as that obtained from the ALS experiment. The quantum defect of the other resonant states decreases when the principal quantum number increases. The effective charges of this series are greater than the charge of the Ca4 + ion only for the first four levels while the quantum defects of the other levels are negative. This mayexpectourresults for n >9 to beaccurate.
For all the Rydberg seriesinvestigated, the slightdiscrepanciesbetween the presentcalculations and experimentmaybeexplain by the simplicity of the SCUNC formalismwhichdoes not includeexplicitlyanyrelativistic corrections.

Conclusions:-
The Screening Constant by Unit Nuclear Charge (SCUNC) formalism has been applied to the Photoionization of the     Rydberg series from the metastable state of the Ca 3+ ion. The present results (SCUNC) are expressed in eVand compared to the Advanced Light Source (ALS) data of Ghassan A. Alna'washi et al (2010) [2].