Atomic Structure Calculations and Study of Plasma Parameters of Al-Like Ions

In the present paper, the spectroscopic properties and plasma characteristics of Al-like ions are investigated in an extensive and detailed manner by adopting the GRASP2K package based on fully relativistic Multi-Configuration Dirac–Hartree–Fock (MCDHF) wave-functions in the active space approximation. We have presented energy levels for Al-like ions for Valence-Valence (VV) and Core-Valence (CV) correlations under the scheme of active space. We have also provided radiative data for E1 transitions for Al-like ions and studied the variation of the transition wavelength and transition probability for electric dipole (E1) Extreme Ultraviolet (EUV) transitions with nuclear charge. Our calculated energy levels and transition wavelengths match well with available theoretical and experimental results. The discrepancies of the GRASP2K code results with CIV3 and RMPBT (Relativistic Many Body Perturbation Theory) results are also discussed. The variations of the line intensity ratio, electron density, plasma frequency and plasma skin depth with plasma temperature and nuclear charge are discussed graphically in detail for optically thin plasma in Local Thermodynamic Equilibrium (LTE). We believe that our obtained results may be beneficial for comparisons and in fusion and astrophysical plasma research.


Introduction
In recent years, medium Z-elements are the most probable candidates for plasma diagnostics and modeling, as spectral lines of higher wavelengths and the atomic spectra of their ions may be employed to compute various parameters, like plasma temperature, electron density, etc. [1][2][3][4].The necessity of highly accurate and consistent plasma parameters required in order to quantify the plasma and for the detailed understanding of the particle confinement, H modes, fusion products, etc., arises from the fact that the accurate and precise measurements of these parameters at superfluous conditions are very cumbersome due to a limited number of experimental techniques.In the past few years, the identification and detailed study and analysis of spectral lines on the experimental side through atomic data have boosted the research progress in plasma physics for highly ionized elements.Therefore, the calculations of various atomic and plasma parameters by implementing several theoretical and experimental methods have become an extensive and active area of research owing to a specific interest in fusion and astrophysical plasma applications, laser physics and nanoplasmonic applications.Since a number of experimental discoveries and examinations on neutral to highly ionized medium to high Z ions have been presented in the literature [5][6][7][8][9][10], in this paper, we have furnished the plasma characteristics with Multi-Configuration Dirac-Hartree-Fock (MCDHF) calculations for highly ionized Al-like ions.
In the last few years, various theoretical and experimental studies of highly ionized Al-like ions have been performed by implementing several types of technology , which has boosted the fame and prominence of Al-like ions.Safronova et al. [38,39] have presented energy levels and radiative data of Al-like ions by making use of Relativistic Many Body Perturbation Theory (RMBPT).Wavelengths and oscillator strengths of Al-like ions from Z = 26 ´31 have been reported by Wei et al. [40] by using the MCDHF method.Gupta et al. [41] have listed the energy levels and radiative data for Al-like Cu by performing CIV3 calculations.Sansonetti [42,43] has compiled the spectroscopic data of strontium ions from singly-ionized to H-like and rubidium ions from neutral to Rb XXXXVII.Dong et al. [44] have calculated lifetimes, excitation energies and radiative rates for Al-like iron group elements by applying the MCDHF method.Atomic binding energies of the lithium to dubnium isoelectronic sequence (including Al-like ions) have been studied by Rodrigues et al. [45] in Dirac-Fock approximation.Santana et al. [46] have performed multi-reference Møller-Plesset perturbation theory calculations to check the lowest 40 levels of Al-like iron group elements.Recently, Argamam et al. [47] have calculated the total energy, spin and charge density of all ions of elements of Z = 1 ´29 under the formulation of Density-Functional Theory (DFT).
Although much work on the theoretical and experimental side of Al-like ions has been published in the literature, there is a scarcity of atomic data of Al-like ions and the elaborate study of plasma parameters using accurate atomic data.In recent years, characterization of plasmas has become an interesting and significant area of research for understanding and analyzing the utilization of multi-lateral spectroscopic sources.Therefore, we have computed energy levels and radiative data for Extreme Ultraviolet (EUV) transitions from ground state and studied graphically the effect of the increase of nuclear charge on transition wavelength and transition probability for EUV transitions for Al-like ions.We have also investigated the effect of nuclear charge on the line intensity ratio, electron density, plasma frequency and skin depth at high values of plasma temperature in optically thin plasma in LTE for Al-like ions.We have also analyzed the effect of electron density and plasma temperature on the plasma parameter and coupling parameters for the diagnosis and characterization of cold, dense and strongly-coupled plasma at high temperature.The present calculations will be beneficial and helpful for the investigation and breakdown of fusion and astrophysical plasma [48][49][50][51] and in nanoplasmonic applications in various fields, such as biomedical, chemical physics, optical physics, etc., [52][53][54][55][56][57][58][59].

Computational Procedure
The description of the theoretical base of the MCDHF method has been explained in an elaborate form by Grant [60][61][62][63].Therefore, we discuss only a brief synopsis here.The Dirac-Coulomb Hamiltonian is given by: where α and β are 4 ˆ4 Dirac spin matrices, c is the speed of light, V N represents the monopole part of the electron-nucleus Coulomb interaction and the last term denotes Coulomb interaction between the electrons.An Atomic State Function (ASF) interpreting fine structure levels for the N electron system is constructed by the linear combination of symmetrically fitted and appropriate Configuration State Functions (CSFs).
|ψ α pPJMqy " In Equation ( 2), φ is CSF, which represents a particular state with a given parity and angular momentum (J, M). δ represents the orbital occupation numbers, population, coupling, etc.The configuration state functions γ i are fabricated by the multiplication of one-electron Dirac orbitals.In Equation (2), C i (δ) are the expansion mixing coefficients for each CSFs, which are attained by means of diagonalization of the Dirac-Coulomb Hamiltonian, given in Equation (1).After this, the radial part of the Dirac orbitals and expansion mixing coefficients are both optimized to self-consistency through the Relativistic Self-Consistent Field (RSCF) method.In the next step, the Breit interaction: due to the exchange of virtual photons between two electrons and QED corrections due to vacuum polarization and self-energy effects are taken into account by performing Relativistic Configuration Interaction (RCI) calculations.

Calculation Procedure
In our MCDHF calculations, we have considered different correlations in a simple and systematic way.To categorize the correlation, atomic electrons may be split into two parts, namely valence electrons and core electrons.Valence-Valence (VV) correlation is the correlation between valence electrons, whereas the correlation between valence electrons and core electrons is defined as Core-Valence (CV) correlation.On the other hand, the correlation between core-electrons is called Core-Core (CC) correlation.In the MCDHF method, these correlations are described by the different constraints included in Equation ( 2), during the fabrication of CSFs.In VV correlation, CSFs are constructed by the excitation of valence electrons to higher shells, keeping core electrons fixed.In CV correlation, CSFs are generated by the single excitation from the core and one or more excitations from the valence.At last, the CC correlation can be considered by allowing doublet, triplet or more excitations from the core.
In the present paper, to construct atomic state functions, we have used the active space approach in the EOL (Define)scheme.For this purpose, we excite electrons from reference configurations to the active set.For the relativistic fine-structure terms belonging to the 3s 2 3p ground state, we perform Single and Double (SD) excitations from the 3s 2 3p set to the active space, and for terms related to 3s3p 2 and 3s 2 3d, we have done SD excitations from the {3s3p 2 , 3s 2 3d} multi-reference set to generate configuration state functions.In the above process, we produce CSFs of definite parity P and total angular momentum J.By applying limitation or boundation to the excitations, the effect of different types of correlations may be included in a proper way.Since the orbitals having the same principle quantum number acquire similar energies, therefore orbitals have been increased in a systematic way in order to handle the convergence of our computations and to enhance the active set layer by layer.To reduce processing time as a result of the large number of orbitals, we optimized the set of orbitals for even and odd parity states separately.Thus, we enhanced the size of the active set as shown below until the convergence is achieved.
AS1 " tn " 3, l " 0 ´2u AS2 " AS1 `tn " 4, l " 0 ´3u AS3 " AS2 `tn " 5, l " 0 ´4u AS4 " AS3 `tn " 6, l " 0 ´4u AS4 " AS3 `tn " 7, l " 0 ´4u In the present work, in order to explain the inner properties, such as fine structure, transitions, etc., we consider VV and CV correlation effects, and during each active set only newly-added orbitals are optimized by keeping previously optimized orbitals frozen to reduce the computational time.We have also included QED and Breit interactions by performing Relativistic Configuration Interaction (RCI) calculations in order to examine and analyze the effect of higher order correlation effects.Ultimately, we enlarge the size of the multi-reference set to {3s 2 3p, 3s3p3d, 3s3p4s} and {3s3p 2 , 3s 2 3d} for odd parity states and even parity states, respectively.In the next step, we have determined energies and rates from optimized orbitals.To compute transition parameters, we have employed the biorthogonal transformation [51] of the ASFs.

Energy Levels and Radiative Data
The accomplishment of our computations relies on judicious selection of the multi-reference set.
In Table 1, we present our fine structure energies of 12 levels belonging to the configurations 3s 2 3p, 3s3p 2 and 3s 2 3d of Cu XVII, Zn XVIII, Ge XX, Rb XXVI and Sr XXVII calculated by the MCDHF method.To assist readers with the identification of energy levels, we list the configurations in LSJ (L is orbital angular momentum, S is spin and J is total angular momentum) under the first column "Configurations" in Table 1.In the next columns of Table 1, we list energy levels for VV and CV correlations with increasing active set.Since for higher principal quantum numbers the number of CSFs increases very rapidly, so it is very cumbersome to get convergence.Therefore, in this paper, we have limited our CV calculations up to n = 6 in which we have an excited electron from core orbital 2p, whereas for VV calculations, it is not very difficult to achieve convergence for higher values of principal quantum number.Furthermore, due to the limitation of our computer calculation, we have done calculations only for VV and CV models.By comparing our VV and CV calculations for n = 6, we predict that the adding of additional CSFs by taking excitation from the core pushed down the ground state, except for 3s3p 2 2 S 1/2 , 3s3p 2 2 P 1/2 , 3s3p 2 2 P 3/2 and 3s 2 3d 2 D 3/2,5/2 .We also predict that the position of 3s3p 2 2 S 1/2 and 3s3p 2 2 P 1/2 changes for Al-like Ge, Rb and Sr.From Table 1, one can see that the maximum difference of our CV calculations with NIST is 3179 cm ´1 for the 3s3p 2 2 S 1/2 level for Sr XXVII.In Table 1, we have also studied the discrepancy of our results calculated by using the GRASP2K code with other results calculated by other codes.Our reported energy levels for Al-like Cu for levels 3s 2 3p 2 P o 3{2 , 3s3p 2 4 P 1/2 , 3s3p 2 4 P 5/2 , 3s3p 2 2 D 3/2 and 3s3p 2 2 D 5/2 are close to NIST, as compared to the results reported by Gupta et al. [41] using the CIV3 code.On the other hand, for other levels, the results by Gupta et al. [41] are close to the NIST results.The maximum discrepancy of our results with Gupta et al. [41] is 2785 cm ´1 for 3s3p 2 2 P 3/2 .This shows that our results are also in good agreement with the results by Gupta et al. [41].Further, energy levels experimentally observed by Sugar et al. [12] for Al-like Zn are close to NIST as compared, theoretically calculated by us using the GRASP2K code, except for 3s3p 2 2 D 3/2 , 3s3p 2 2 D 5/2 , and the maximum percentage difference with Sugar et al. [12] is 2985 cm ´1 for 3s3p 2 2 P 3/2 .Further, results reported by Safronova et al. [38] using RMBPT for Al-like Zn are close to NIST as compared to the results by GRASP2K, except for the levels, and the maximum deviation of our results with Safronova et al. [38] is 3225 cm ´1 for 3s3p 2 2 P 3/2 .The difference between our results and the RMBPT results is the amount of electron correlation.These are the reasons behind the discrepancies of our results calculated by GRASP2K and other codes, such as RMBPT and CIV3.Therefore, it will be very useful for readers to understand the difference between other methods and the currently-developed code GRASP2K.The transition parameters, such as radiative rates (A ji ), oscillator strength (f ij ) and line strength (S ij ) for multipole transitions in terms of a reduced matrix element, are defined by the formula given below: where Q pλq k is the corresponding transition operator in the length or velocity gauge and ψ α pPJMq and ψ α `P1 J 1 M 1 ˘are initial and final atomic state functions.In Table 2, we list the values of the transition wavelength with line strengths, oscillator strengths and radiative rates in the Babushkin and Coulomb gauge for E1 transitions from the ground state for core-valence correlation of Cu XVII, Zn XVIII, Ge XX, Rb XXVI and Sr XXVII.From Table 2 and Figure 1a, we can view that with the increase in nuclear charge, the transition wavelength of E1 EUV transitions from the ground state decreases.Similarly, from Table 2 and Figure 1b, it is revealed that transition probability increases with nuclear charge for all E1 EUV transitions from the ground state, except transition No. 4. In Table 2, we have also tabulated the values of δT, which is an accuracy indicator for electric transitions, and it is the deviation of the ratio of the length and velocity form of line strengths from unity.One can see that the maximum value of δT is not more than 0.133, and hence, this confirms that our presented result is reliable.We have also made a comparison of our computed transition wavelengths with wavelengths from experimental measurement or other theoretical calculations in Table 2, and there is no major discrepancy between our results and other results available in the literature.
The transition parameters, such as radiative rates (Aji), oscillator strength (fij) and line strength (Sij) for multipole transitions in terms of a reduced matrix element, are defined by the formula given below: where Q k (λ) is the corresponding transition operator in the length or velocity gauge and ψ α (PJM) and ψ α (P ′ J ′ M ′ ) are initial and final atomic state functions.In Table 2, we list the values of the transition wavelength with line strengths, oscillator strengths and radiative rates in the Babushkin and Coulomb gauge for E1 transitions from the ground state for core-valence correlation of Cu XVII, Zn XVIII, Ge XX, Rb XXVI and Sr XXVII.From Table 2 and Figure 1a, we can view that with the increase in nuclear charge, the transition wavelength of E1 EUV transitions from the ground state decreases.Similarly, from Table 2 and Figure 1b, it is revealed that transition probability increases with nuclear charge for all E1 EUV transitions from the ground state, except transition No. 4. In Table 2, we have also tabulated the values of δT, which is an accuracy indicator for electric transitions, and it is the deviation of the ratio of the length and velocity form of line strengths from unity.One can see that the maximum value of δT is not more than 0.133, and hence, this confirms that our presented result is reliable.We have also made a comparison of our computed transition wavelengths with wavelengths from experimental measurement or other theoretical calculations in Table 2, and there is no major discrepancy between our results and other results available in the literature.2.

Line Intensity Ratio and Plasma Parameters
The variation in spectroscopic parameters leads to changes in plasma temperature, and the diagnosis of the elemental composition from various experimental spectroscopic techniques becomes simple and straightforward, if the plasma is optically thin and in Local Thermodynamic Equilibrium (LTE).The reason behind this is that the optically thick line produces a deformed or asymmetrical peak in the spectrum because of saturation and self-absorption in the line profile.This causes wrong and inaccurate measurement of electron density and plasma temperature [64][65][66][67][68].At higher values of temperature, kinematic excitement or movement of electrons increases, thereby increasing the number of collisions between electrons and, hence, LTE can be achieved very easily at higher values of temperature.Therefore, we have studied the characteristics of plasma at high values of temperature and the effect of plasma temperature and nuclear charge on the line intensity ratio for optically thin plasma.The line intensity ratio of two spectral lines of the same atom or ion is defined by the expression:  2.

Line Intensity Ratio and Plasma Parameters
The variation in spectroscopic parameters leads to changes in plasma temperature, and the diagnosis of the elemental composition from various experimental spectroscopic techniques becomes simple and straightforward, if the plasma is optically thin and in Local Thermodynamic Equilibrium (LTE).The reason behind this is that the optically thick line produces a deformed or asymmetrical peak in the spectrum because of saturation and self-absorption in the line profile.This causes wrong and inaccurate measurement of electron density and plasma temperature [64][65][66][67][68].At higher values of temperature, kinematic excitement or movement of electrons increases, thereby increasing the number of collisions between electrons and, hence, LTE can be achieved very easily at higher values of temperature.Therefore, we have studied the characteristics of plasma at high values of temperature and the effect of plasma temperature and nuclear charge on the line intensity ratio for optically thin plasma.The line intensity ratio of two spectral lines of the same atom or ion is defined by the expression: R " where I, g, λ and A are intensity, statistical weight, wavelength and transition probability, respectively.E, k and T are the energy of the upper level in eV, the Boltzmann constant and excitation temperature in K, respectively.The subscripts 1 and 2 denote the spectral lines of the same atom or ion.According to Equation ( 5), the line intensity ratio of two spectral lines is completely ruled by the exponential term.
In Table 3, we have shown the line intensity ratio (R) at high temperatures for Al-like ions.Currently, several heating methods are in progress at the International Thermonuclear Experimental Reactor (ITER) to boost the plasma temperature to the required 1.5 ˆ10 8 ˝C or 4 ˆ10 10 K [69].Therefore, we have considered the maximum plasma temperature of 1010 K in this paper.Since for a lower value of ∆E " E 1 ´E2 , exponential terms become very small, hence the line intensity ratio is very small for two spectral lines having a difference between upper energy levels being very small.Therefore, we have computed the line intensity for the ratio for two spectral lines, 1 and 7.The indices used for denoting the spectral lines are indicated in the first column "Transition No." in Table 2.In Figure 2a,b, we can see the behavior of the line intensity ratio with plasma temperature and nuclear charge.From Figure 2a and Table 3, we observe that the line intensity ratio increases with plasma temperature, and for a plasma temperature more than or equal to 10 9 K, the increase in the value of R is less than 0.087, which is very small, showing that the effect of higher temperatures on a line intensity ratio is almost negligible or the line intensity ratio becomes almost constant.From Figure 2b, we predict that the line intensity ratio decreases very rapidly with the nuclear charge for a plasma temperature of 10 10 K.By comparing our calculated line intensity ratio with the measured ratio at different delay times, experimentalists can achieve the time window where the plasma is in LTE and optically thin.In Figure 2a,b, we can see the behavior of the line intensity ratio with plasma temperature and nuclear charge.From Figure 2a and Table 3, we observe that the line intensity ratio increases with plasma temperature, and for a plasma temperature more than or equal to 10 9 K, the increase in the value of R is less than 0.087, which is very small, showing that the effect of higher temperatures on a line intensity ratio is almost negligible or the line intensity ratio becomes almost constant.From Figure 2b, we predict that the line intensity ratio decreases very rapidly with the nuclear charge for a plasma temperature of 10 10 K.By comparing our calculated line intensity ratio with the measured ratio at different delay times, experimentalists can achieve the time window where the plasma is in LTE and optically thin.2.
The total number of collisions between electrons must be large to attain LTE.For this objective, the McWhirter criterion [70] for a minimum or limiting value of electron density is defined by the inequality: In Equation ( 6), ne is the electron density, T is the plasma temperature in K and ΔE = E1 − E2 in eV.In Table 3, we have also tabulated the limiting values of electron density at different values of plasma temperature for Spectral Lines 1 and 7.The value of electron density is 10 21 cm −3 , and matter with this solid density is very popular due to its presence in all types of stars, such as white dwarf stars [71].The plasmas having such a solid density are inertial confinement fusion plasmas [72,73].
From Figure 3a, one can see that the value of electron density increases with plasma temperature for Al-like ions, which shows that the number of collisions in the plasma increases with the increase in plasma temperature.From Figure 3b and Table 3, we have also deduced that the electron density increases very expeditiously with nuclear charge.In the present paper, we have also presented the fitting values of electron density in Table 3, and our fitted values of electron density are in good agreement with the calculated electron density and, hence, confirm the authenticity and credibility of our fitted parameters given in Table 4.By employing fitting parameters K and C in the straight line  2.
The total number of collisions between electrons must be large to attain LTE.For this objective, the McWhirter criterion [70] for a minimum or limiting value of electron density is defined by the inequality: n e ě 1.6 ˆ10 12 T 1{2 p∆Eq 3 In Equation (6), n e is the electron density, T is the plasma temperature in K and ∆E = E 1 ´E2 in eV.In Table 3, we have also tabulated the limiting values of electron density at different values of plasma temperature for Spectral Lines 1 and 7.The value of electron density is 10 21 cm ´3, and matter with this solid density is very popular due to its presence in all types of stars, such as white dwarf stars [71].The plasmas having such a solid density are inertial confinement fusion plasmas [72,73].
From Figure 3a, one can see that the value of electron density increases with plasma temperature for Al-like ions, which shows that the number of collisions in the plasma increases with the increase in plasma temperature.From Figure 3b and Table 3, we have also deduced that the electron density increases very expeditiously with nuclear charge.In the present paper, we have also presented the fitting values of electron density in Table 3, and our fitted values of electron density are in good agreement with the calculated electron density and, hence, confirm the authenticity and credibility of our fitted parameters given in Table 4.By employing fitting parameters K and C in the straight line equation log 10 n e = Klog 10 T + C, one easily determines the limiting value of electron density for a particular value of plasma temperature for Al-like ions.Thus, this information may be advantageous for experimentalists for producing optically thin plasma in LTE at higher temperatures for Al-like ions.equation log10ne = Klog10T + C, one easily determines the limiting value of electron density for a particular value of plasma temperature for Al-like ions.Thus, this information may be advantageous for experimentalists for producing optically thin plasma in LTE at higher temperatures for Al-like ions.2. The relation between electron density and the electron plasma frequency with which electrons oscillate or the frequency of plasma oscillation is given by the expression: ω e = 8.979 × 10 4 n e 1/2 Hz (7) According to the above equation, plasma frequency is directly proportional to the square root of electron density and, hence, depends on the one-fourth power of the plasma temperature by Equation (6).In Table 3 and Figure 4a, we can see that with increasing electron density and plasma temperature, electrons oscillate with higher frequency due to thermal agitation.As can be seen from Table 3 and Figure 4b, the electron density increases with an increase in nuclear charge, and therefore, plasma frequency also increases with nuclear charge.In this work, the highest plasma frequency is 16.58 PHz at a plasma temperature of 10 10 K for Sr XXVII, which lies in the ultraviolet region.
The depth to which electromagnetic radiations can penetrate plasma, known as skin depth, is inversely proportional to the square root of electron density and is given by the following relation: δ = 5.31 × 10 5 n e −1/2 cm (8) From Equation ( 8), the plasma skin depth is inversely proportional to electron density and, hence, inversely proportional to the one-fourth power of the plasma temperature by Equation (6).Therefore, from Table 3 and Figure 5a, it is revealed that as plasma temperature and electron density increase, the skin depth of the plasma decreases rapidly for Al-like ions.From Table 3 and Figure 5b, we conclude that with the increment in the values of the nuclear charge of Al-like ions, i.e., with the increase in the number of protons w.r.t electrons, plasma skin depth decreases.In this paper, the minimum value of the plasma skin depth is 28.751 nm at T = 10 10 K for Sr XXVII or we can say that at 16.58 PHz, the skin depth is 28.751 nm.This implies that Al-like ions' plasma is highly conductive, and hence, we deduce that electron density and plasma temperature play a significant role in  2. The relation between electron density and the electron plasma frequency with which electrons oscillate or the frequency of plasma oscillation is given by the expression: ω e " 8.979 ˆ10 4 n 1{2 e Hz (7) According to the above equation, plasma frequency is directly proportional to the square root of electron density and, hence, depends on the one-fourth power of the plasma temperature by Equation (6).In Table 3 and Figure 4a, we can see that with increasing electron density and plasma temperature, electrons oscillate with higher frequency due to thermal agitation.As can be seen from Table 3 and Figure 4b, the electron density increases with an increase in nuclear charge, and therefore, plasma frequency also increases with nuclear charge.In this work, the highest plasma frequency is 16.58 PHz at a plasma temperature of 10 10 K for Sr XXVII, which lies in the ultraviolet region.
The depth to which electromagnetic radiations can penetrate plasma, known as skin depth, is inversely proportional to the square root of electron density and is given by the following relation: From Equation ( 8), the plasma skin depth is inversely proportional to electron density and, hence, inversely proportional to the one-fourth power of the plasma temperature by Equation (6).Therefore, from Table 3 and Figure 5a, it is revealed that as plasma temperature and electron density increase, the skin depth of the plasma decreases rapidly for Al-like ions.From Table 3 and Figure 5b, we conclude that with the increment in the values of the nuclear charge of Al-like ions, i.e., with the increase in the number of protons w.r.t electrons, plasma skin depth decreases.In this paper, the minimum value of the plasma skin depth is 28.751 nm at T = 10 10 K for Sr XXVII or we can say that at 16.58 PHz, the skin depth is 28.751 nm.This implies that Al-like ions' plasma is highly conductive, and hence, we deduce that electron density and plasma temperature play a significant role in reducing the skin depth of the plasma and controlling the conductivity of plasma.Since skin depth is in nm, therefore the size of the plasmon will be in nm; hence Al-like ions' plasma may be beneficial in several applications, such as solar cells, spectroscopy and cancer treatment.reducing the skin depth of the plasma and controlling the conductivity of plasma.Since skin depth is in nm, therefore the size of the plasmon will be in nm; hence Al-like ions' plasma may be beneficial in several applications, such as solar cells, spectroscopy and cancer treatment.2.
The plasma parameter (Λ) may be defined in terms of electron density and plasma temperature by the following expression: The Coulomb coupling parameter (Γ) and plasma parameter (Λ) for plasma are related to each other as follows The above two parameters play a vital role in differentiating between strongly-coupled plasma and weakly-coupled plasma and in the determination of plasma characteristics.From Table 3, we determine that for all values of plasma temperature and corresponding electron density, Λ < 1 and Γ > 1.Since for strongly-coupled plasma, the coupling parameter should be greater than unity, the optically thin plasma in LTE is a cold, dense and strongly-coupled plasma for Al-like ions, which also correspond to the conditions prevailing in white dwarf and neutron stars [74].Further, the ratio of potential energy due to Coulomb interaction between neighboring particles and the kinetic energy of particles is governed by the coupling parameter.In the present work, Γ > 1; this shows that the electrons are conducted by their interactions in the plasma.2.
Atoms 2016, 4, 22 14 of 18 reducing the skin depth of the plasma and controlling the conductivity of plasma.Since skin depth is in nm, therefore the size of the plasmon will be in nm; hence Al-like ions' plasma may be beneficial in several applications, such as solar cells, spectroscopy and cancer treatment.2.
The plasma parameter (Λ) may be defined in terms of electron density and plasma temperature by the following expression: The Coulomb coupling parameter (Γ) and plasma parameter (Λ) for plasma are related to each other as follows The above two parameters play a vital role in differentiating between strongly-coupled plasma and weakly-coupled plasma and in the determination of plasma characteristics.From Table 3, we determine that for all values of plasma temperature and corresponding electron density, Λ < 1 and Γ > 1.Since for strongly-coupled plasma, the coupling parameter should be greater than unity, the optically thin plasma in LTE is a cold, dense and strongly-coupled plasma for Al-like ions, which also correspond to the conditions prevailing in white dwarf and neutron stars [74].Further, the ratio of potential energy due to Coulomb interaction between neighboring particles and the kinetic energy of particles is governed by the coupling parameter.In the present work, Γ > 1; this shows that the electrons are conducted by their interactions in the plasma.2.
The plasma parameter (Λ) may be defined in terms of electron density and plasma temperature by the following expression: Λ " 4π ˆ7.43 ˆ10 2 ˆn´1{2 e T 1{2 The Coulomb coupling parameter (Γ) and plasma parameter (Λ) for plasma are related to each other as follows The above two parameters play a vital role in differentiating between strongly-coupled plasma and weakly-coupled plasma and in the determination of plasma characteristics.From Table 3, we determine that for all values of plasma temperature and corresponding electron density, Λ ă 1 and Γ ą 1.Since for strongly-coupled plasma, the coupling parameter should be greater than unity, the optically thin plasma in LTE is a cold, dense and strongly-coupled plasma for Al-like ions, which also correspond to the conditions prevailing in white dwarf and neutron stars [74].Further, the ratio of potential energy due to Coulomb interaction between neighboring particles and the kinetic energy of particles is governed by the coupling parameter.In the present work, Γ ą 1; this shows that the electrons are conducted by their interactions in the plasma.

Conclusions
In the present work, we have discussed the spectroscopic properties and plasma characteristics of Al-like ions in the plasma atmosphere by employing the GRASP2K package based on the MCDHF method, taking into account the contribution of QED and Breit corrections.We have also reported the energy levels and radiative data for E1 transitions from the ground state belonging to the configurations 3s 2 3p, 3s3p 2 and 3s 2 3d for Al-like ions.The energy levels and transition wavelengths presented in this paper were compared to the data available in the literature, and good agreement was found.We have graphically studied the behavior of transition wavelengths and radiative rates for EUV transitions with nuclear charge.We have also investigated and diagnosed graphically the behavior of the line intensity ratio, electron density, plasma frequency and skin depth of highly ionized Al-like ions with plasma temperature and nuclear charge for optically thin plasma in LTE.
As a conclusion, we conclude that the work presented in this paper is a reliable examination of the spectroscopic properties, and plasma characteristics will be useful in the modeling and characterization of plasma.Further, the atomic data may be useful in the designation and analysis of EUV spectral lines with the exploration of new data from fusion and astrophysical plasma sources.

Figure 1 .
Figure 1.(Color online) Variation of (a) transition wavelength and (b) the transition probability as a function of nuclear charge (Z) for Extreme Ultraviolet (EUV) Transitions 1-7 using Table2.

Figure 1 .
Figure 1.(Color online) Variation of (a) transition wavelength and (b) the transition probability as a function of nuclear charge (Z) for Extreme Ultraviolet (EUV) Transitions 1-7 using Table2.

Figure 2 .
Figure 2. (Color online) Variation of the line intensity ratio with (a) plasma temperature and (b) nuclear charge at T = 10 10 K for Al-like ions for Spectral Lines 1 and 7 using Table2.

Figure 2 .
Figure 2. (Color online) Variation of the line intensity ratio with (a) plasma temperature and (b) nuclear charge at T = 10 10 K for Al-like ions for Spectral Lines 1 and 7 using Table2.

Figure 3 .
Figure 3. (Color online) Variation of electron density with (a) plasma temperature and (b) nuclear charge at T = 10 10 K for Al-like ions for Spectral Lines 1 and 7 using Table2.

Figure 3 .
Figure 3. (Color online) Variation of electron density with (a) plasma temperature and (b) nuclear charge at T = 10 10 K for Al-like ions for Spectral Lines 1 and 7 using Table2.

Figure 4 .
Figure 4. (Color online) Variation of plasma frequency with (a) plasma temperature and (b) nuclear charge at T = 10 10 K for Al-like ions for Spectral Lines 1 and 7 using Table2.

Figure 5 .
Figure 5. (Color online) Variation of skin depth with (a) plasma temperature and (b) nuclear charge at T = 10 10 K for Al-like ions for Spectral Lines 1 and 7 using Table2.

Figure 4 .
Figure 4. (Color online) Variation of plasma frequency with (a) plasma temperature and (b) nuclear charge at T = 10 10 K for Al-like ions for Spectral Lines 1 and 7 using Table2.

Figure 4 .
Figure 4. (Color online) Variation of plasma frequency with (a) plasma temperature and (b) nuclear charge at T = 10 10 K for Al-like ions for Spectral Lines 1 and 7 using Table2.

Figure 5 .
Figure 5. (Color online) Variation of skin depth with (a) plasma temperature and (b) nuclear charge at T = 10 10 K for Al-like ions for Spectral Lines 1 and 7 using Table2.

Figure 5 .
Figure 5. (Color online) Variation of skin depth with (a) plasma temperature and (b) nuclear charge at T = 10 10 K for Al-like ions for Spectral Lines 1 and 7 using Table2.

Table 2 .
Radiative data for Al-like ions.

Table 3 .
Plasma temperature (T in K), line intensity ratio (R in 10 3 ), electron density (n e in 10 21 cm ´3), plasma frequency (ω e ) in PHz and skin depth (δ) in nm for optically thin plasma for Spectral Lines 1 and 7 of Al-like ions.

Table 4 .
Plasma fitting parameters K and C for Al-like ions.

Table 4 .
Plasma fitting parameters K and C for Al-like ions.