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

Abstract

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.

1. 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 [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37], 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].

2. Theoretical Method

2.1. 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:

$${\text{H}}_{\text{DC}}={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}\left({\text{c}\mathsf{\alpha }}_{\mathrm{i}}.{\mathrm{p}}_{\mathrm{i}}+\left({\mathsf{\beta }}_{\mathrm{i}}-1\right){\mathrm{c}}^2+{\mathrm{V}}_{\mathrm{i}}^{\mathrm{N}}\right)+{\displaystyle \sum }_{\mathrm{i}>j}^{\mathrm{N}}\frac{1}{{\mathrm{r}}_{\text{ij}}}$$

(1)

where α and β are 4 × 4 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).

$$|{\mathsf{\psi }}_{\mathsf{\alpha }}\left(\text{PJM}\right)\rangle ={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{i}}\left(\mathsf{\delta }\right)|\mathsf{\varphi }({\mathsf{\gamma }}_{\mathrm{i}}\left(\text{PJM}\right))\rangle$$

(2)

In Equation (2), $\mathsf{\varphi }$ is CSF, which represents a particular state with a given parity and angular momentum ($\mathrm{J}$, M). $\mathsf{\delta }$ represents the orbital occupation numbers, population, coupling, etc. The configuration state functions ${\mathsf{\gamma }}_i$ 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:

$${\text{H}}_{\text{Breit}}=-{\displaystyle \sum }_{\mathrm{i}>j}^{\mathrm{N}}\left[{\mathsf{\alpha }}_{\mathrm{i}}.{\mathsf{\alpha }}_{\mathrm{j}}\frac{\cos ({\mathsf{\omega }}_{\text{ij}}{\mathrm{r}}_{\text{ij}}/\mathrm{c})}{{\mathrm{r}}_{\text{ij}}}+\left({\mathsf{\alpha }}_{\mathrm{i}}.{\nabla }_{\mathrm{i}}\right)\left({\mathsf{\alpha }}_{\mathrm{j}}.{\nabla }_{\mathrm{j}}\right)\frac{\cos ({\mathsf{\omega }}_{\text{ij}}{\mathrm{r}}_{\text{ij}}/\mathrm{c})-1}{{\mathsf{\omega }}_{\text{ij}}^2{\mathrm{r}}_{\text{ij}}/{\mathrm{c}}^2}\right]$$

(3)

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.

2.2. 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 3s²3p ground state, we perform Single and Double (SD) excitations from the 3s²3p set to the active space, and for terms related to 3s3p² and 3s²3d, we have done SD excitations from the {3s3p², 3s²3d} 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 = {n = 3, l = 0–2}

AS2 = AS1 + {n = 4, l = 0–3}

AS3 = AS2 + {n = 5, l = 0–4}

AS4 = AS3 + {n = 6, l = 0–4}

AS4 = AS3 + {n = 7, l = 0–4}

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 {3s²3p, 3s3p3d, 3s3p4s} and {3s3p², 3s²3d} 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.

3. Results and Discussion

3.1. Energy Levels and Radiative Data

The accomplishment of our computations relies on judicious selection of the multi-reference set. In

Table 1. Aluminum-like isoelectronic sequence: excitation energies (in cm⁻¹) from Multi-Configuration Dirac–Hartree–Fock (MCDHF) calculations, experiments and other theoretical calculations, Diff. (in cm⁻¹) with NIST, Diff.^ref (in cm⁻¹) with other results. VV, Valence-Valence; CV, Core-Valence.

Cu
Configurations	VV				CV			NIST	Ref. [41]	Diff.	Diff.ref
	n = 4	n = 5	n = 6	n = 7	n = 4	n = 5	n = 6
3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	0	0	0	0	0	0	0	0	0	0	0
3s23p 2${\mathrm{P}}_{3/2}^{\mathrm{o}}$	33,012	33,103	33,117	33,121	33,215	33,021	33,233	33,239	33,231	6	2
3s3p2 4P1/2	275,539	276,003	276,098	276,121	276,393	276,507	277,136	277,231	277,018	95	118
3s3p2 4P3/2	290,024	290,481	290,577	290,602	290,854	291,014	291,649	291,810	291,749	161	100
3s3p2 4P5/2	305,982	306,418	306,512	306,535	306,764	306,961	307,584	307,708	307,482	124	102
3s3p2 2D3/2	372,211	372,306	372,353	372,358	373,159	372,845	373,184	372,236	371,135	948	2049
3s3p2 2D5/2	377,646	377,742	377,790	377,795	378,581	378,293	378,648	377,783	376,849	865	1799
3s3p2 2S1/2	448,218	447,924	447,922	447,904	448,298	447,345	447,383	444,759	444,858	2624	2525
3s3p2 2P1/2	484,224	483,980	483,986	483,973	483,784	482,863	482,937	480,016	480,269	2921	2668
3s3p2 2P3/2	495,323	495,091	495,102	495,094	494,167	493,288	493,397	490,467	490,612	2930	2785
3s23d 2D3/2	578,586	577,747	577,748	577,742	577,777	576,221	576,056	574,180	575,511	1876	545
3s23d 2D5/2	582,476	581,641	581,648	581,642	581,821	580,260	580,102	578,243	579,601	1859	501
Zn
Configurations	VV				CV			NIST	Ref. [12,38]	Diff.	Diff.ref
	n = 4	n = 5	n = 6	n = 7	n = 4	n = 5	n = 6
3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	0	0	0	0	0	0	0	0	0	0	0
3s23p 2${\mathrm{P}}_{3/2}^{\mathrm{o}}$	39,252	39,332	39,352	39,356	39,467	39,247	39,346	39,483	39,482 a 39,441 b	137	136 a 95 b
3s3p2 4P1/2	293,690	294,143	294,240	294,262	294,565	294,650	295,273	295200	295,565 b	73	292 b
3s3p2 4P3/2	311,359	311,812	311,911	311,935	312,214	312,358	312,990	312,993	313,401 b	3	411 b
3s3p2 4P5/2	329,845	330,275	330,370	330,394	330,647	330,828	331,445	331,359	331,743 b	86	298 b
3s3p2 2D3/2	398,396	398,483	398,534	398,539	399,352	399,004	399,347	398,390	401,063 a 398,202 b	957	1716 a 1145 b
3s3p2 2D5/2	405,589	405,684	405,737	405,743	406,541	406,229	406,592	405,760	407,005 a 405,567 b	832	413 a 1025 b
3s3p2 2S1/2	476,279	475,983	475,990	475,974	476,211	475,232	475,293	472,601	472,583 a 472,400 b	2692	2710 a 2893 b
3s3p2 2P1/2	517,541	517,277	517,288	517,274	517,162	516,197	516,278	513373	513,350 a 513,114 b	2905	2928 a 3164 b
3s3p2 2P3/2	529,352	529,111	529,129	529,121	528,134	527,222	527,339	524,382	524,354 a 524,114 b	2957	2985 a 3225 b
3s23d 2D3/2	613,765	612,920	612,925	612,918	612,900	611,308	611,167	609,252	609,219 a 608,425 b	1915	1948 a 2742 b
3s23d 2D5/2	618,568	617,725	617,735	617,729	617,874	616,275	616,141	614,272	614,214 a 613,395 b	1869	1927 a 2746 b
Ge
Configurations	VV				CV			NIST		Diff.
	n = 4	n = 5	n = 6	n = 7	n = 4	n = 5	n = 6
3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	0	0	0	0	0	0	0	0		0
3s23p 2${\mathrm{P}}_{3/2}^{\mathrm{o}}$	54,325	54,374	54,417	54,422	54,568	542,98	54,572	54,564		8
3s3p2 4P1/2	331,139	331,559	331,668	331,692	332,052	332,079	332,689
3s3p2 4P3/2	356,988	357,425	357,538	357,564	357,895	358,004	358,630
3s3p2 4P5/2	381,052	381,453	381,562	381,586	381,895	382,033	382,635
3s3p2 2D3/2	453,950	454,013	454,080	454,087	454,922	454,513	454,858	453,869		989
3s3p2 2D5/2	466,163	466,252	466,324	466,332	467,154	466,808	467,185	466,407		778
3s3p2 2P1/2	534,573	534,269	534,300	534,287	534,241	533,224	533,314	530,521		2793
3s3p2 2S1/2	589,387	589,081	589,111	589,098	589,143	588,103	588,190	585,257		2933
3s3p2 2P3/2	602,113	601,848	601,888	601,881	600,802	599,833	599,958	597,020		2938
3s23d 2D3/2	686,908	686,054	686,070	686,062	685,949	684,312	684,193	682,246		1947
3s23d 2D5/2	693,914	693,062	693,082	693,076	693,164	691,518	691,408	689,533		1875
Rb
Configurations	VV				CV			NIST		Diff.
	n = 4	n = 5	n = 6	n = 7	n = 4	n = 5	n = 6
3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	0	0	0	0	0	0	0	0		0
3s23p 2${\mathrm{P}}_{3/2}^{\mathrm{o}}$	110,871	110,922	110,947	110,962	111,202	110,804	110,952	111,140		188
3s3p2 4P1/2	431,229	431,578	431,694	431,728	432,222	432,068	432,657
3s3p2 4P3/2	492,049	492,464	492,589	492,626	493,091	493,100	493,730
3s3p2 4P5/2	532,051	532,378	532,494	532,527	532,436	532,976	533,550
3s3p2 2D3/2	615,166	615,202	615,278	615,295	616,172	615,618	615,968	614,900		1068
3s3p2 2D5/2	653,652	653,781	653,868	653,890	654,237	654,394	654,824	654,160		664
3s3p2 2P1/2	697,265	696,970	697029	697,032	696,475	695,384	695,519	692,660		2859
3s3p2 2S1/2	807,338	806,983	807,035	807,034	807,429	806,248	806,353	803,460		2893
3s3p2 2P3/2	817,637	817,345	817,405	817,410	816,252	815,150	815,286	812,510		2776
3s23d 2D3/2	892,285	891,474	891,470	891,465	891,150	889,467	889,357	887,290		2067
3s23d 2D5/2	906,272	905,468	905,460	905,457	905,395	903,767	903,662	901,780		1882
Sr
Configurations	VV				CV			NIST		Diff.
	n = 4	n = 5	n = 6	n = 7	n = 4	n = 5	n = 6
3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	0	0	0	0	0	0	0	0		0
3s23p 2${\mathrm{P}}_{3/2}^{\mathrm{o}}$	126,134	126,200	126,215	126,231	126,484	126,054	126,211	126,410		199
3s3p2 4P1/2	452,427	452,799	452,880	452,914	453,304	453,345	453,759	453,140		619
3s3p2 4P3/2	523,935	524,036	524,127	524,165	524,529	524,751	525,212	525,050		162
3s3p2 4P5/2	566,888	567,248	567,319	567,352	567,184	567,905	568,303	568,030		273
3s3p2 2D3/2	652,579	652,194	652,229	652,248	653,000	652,654	652,833	651,640		1193
3s3p2 2D5/2	699,145	699,350	699,379	699,402	699,591	700,025	700,297	699,540		757
3s3p2 2P1/2	733,800	733,548	733,575	733,580	732,821	731,952	731,924	728,900		3024
3s3p2 2S1/2	859,193	858,873	858,892	858,892	859,219	858,251	858,189	855,010		3179
3s3p2 2P3/2	867,966	867,223	867,243	867,249	865,973	865,077	865,040	862,070		2970
3s23d 2D3/2	939,073	938,203	938,169	938,162	937,675	936,228	935,948	933,830		2118
3s23d 2D5/2	954,081	953,288	953,279	953,273	953,083	951,677	951,397	949,440		1957

$\text{Diff}=\left|\text{CV}\left(\mathrm{n}=6\right)-\text{NIST}\right|;\text{Diff}=\left|\text{CV}\left(\mathrm{n}=6\right)-\text{Ref}.\left[12,38\right]\right|$; ^a [12]; ^b [38].

, we present our fine structure energies of 12 levels belonging to the configurations 3s²3p, 3s3p² and 3s²3d 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 3s²3d ²D_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 3s²3p ²${\mathrm{P}}_{3/2}^{\mathrm{o}}$, 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:

$$\langle {\mathsf{\psi }}_{\mathsf{\alpha }}\left(\text{PJM}\right)||{\mathrm{Q}}_{\mathrm{k}}^{\left(\mathsf{\lambda }\right)}||{\mathsf{\psi }}_{{\mathsf{\alpha }}^{\prime }}\left({\text{P}}^{\prime }{\text{J}}^{\prime }{\text{M}}^{\prime }\right)\rangle$$

(4)

where ${\mathrm{Q}}_{\mathrm{k}}^{\left(\mathsf{\lambda }\right)}$ is the corresponding transition operator in the length or velocity gauge and ${\mathsf{\psi }}_{\mathsf{\alpha }}\left(\text{PJM}\right)$ and ${\mathsf{\psi }}_{\mathsf{\alpha }}\left({\text{P}}^{\prime }{\text{J}}^{\prime }{\text{M}}^{\prime }\right)$ are initial and final atomic state functions. In

Table 2. Radiative data for Al-like ions.

Transition No.	Cu
	Transition		λc (in Å)	λ (in Å)	A		gf		S		dT
	i	j			B	C	B	C	B	C
1	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 4P1/2	361.00 d 361.00 g 361.16 h	360.83	8.08E +07	9.33E +07	3.15E − 03	3.64E − 03	3.75E − 03	4.33E − 03	0.134
2	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 4P3/2	342.80 d 342.80 g 342.70 h	342.88	1.73E +06	1.72E +06	1.22E − 04	1.21E − 04	1.38E − 04	1.37E − 04	0.006
3	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2D3/2	269.40 g 268.65 h	267.96	3.84E +09	3.93E +09	1.65E −01	1.69E −01	1.46E −01	1.49E −01	0.022
4	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2S1/2	224.80 g 224.84 h	223.52	2.99E +10	3.13E +10	4.48E −01	4.70E −01	3.29E −01	3.46E −01	0.047
5	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2P1/2	208.20 g 208.33 h	207.07	1.16E +10	1.16E +10	1.49E −01	1.49E −01	1.02E −01	1.02E −01	0.001
6	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2P3/2	203.80 g 203.88 h	202.68	1.09E +10	1.11E +10	2.69E −01	2.75E −01	1.79E −01	1.83E −01	0.023
7	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s23d 2D3/2	173.54 f 173.80 g 174.17 h	173.59	4.47E +10	4.48E +10	8.07E −01	8.09E −01	4.61E −01	4.62E −01	0.002
Transition No.	Zn
	Transition		λc (inÅ)	λ (in Å)	A		gf		S		dT
	i	j			B	C	B	C	B	C
1	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 4P1/2	338.50 e	338.67	1.15E +08	1.32E +08	3.95E −03	4.54E −03	4.40E −03	5.07E −03	0.131
2	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 4P3/2	319.20 e	319.93	2.41E +06	2.42E +06	1.47E −04	1.48E −04	1.55E −04	1.55E −04	0.006
3	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2D3/2		250.75	4.55E +09	4.43E +09	1.71E −01	1.66E −01	1.41E −01	1.37E −01	0.023
4	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2S1/2	211.60 a	210.40	3.44E +10	3.61E +10	4.57E −01	4.80E −01	3.17E −01	3.32E −01	0.047
5	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2P1/2	194.79 a	193.69	1.07E +10	1.07E +10	1.20E −01	1.20E −01	7.67E −02	7.68E −02	0.002
6	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2P3/2	190.71 a	189.89	1.26E +10	1.23E +10	2.71E −01	2.65E −01	1.69E −01	1.65E −01	0.023
7	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s23d 2D3/2	164.14 a 163.60 f	163.84	4.79E +10	4.78E +10	7.69E −01	7.67E −01	4.14E −01	4.13E −01	0.002
Transition No.	Ge
	Transition		λc (inÅ)	λ (in Å)	A		gf		S		dT
	i	j			B	C	B	C	B	C
1	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 4P1/2		300.58	2.21E +08	2.53E +08	5.98E −03	6.86E −03	5.92E −03	6.78E −03	0.127
2	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 4P3/2		278.84	4.52E +06	4.54E +06	2.11E −04	2.12E −04	1.93E −04	1.94E −04	0.005
3	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2D3/2		219.85	5.85E +09	6.04E +09	1.70E −01	1.75E −01	1.23E −01	1.27E −01	0.031
4	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2S1/2		170.01	9.06E +09	9.08E +09	7.85E −02	7.87E −02	4.39E −02	4.40E −02	0.002
5	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2P1/2		187.51	4.40E +10	4.62E +10	4.64E −01	4.87E −01	2.86E −01	3.01E −01	0.047
6	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2P3/2		166.68	1.57E +10	1.62E +10	2.62E −01	2.69E −01	1.44E −01	1.48E −01	0.027
7	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s23d 2D3/2		146.16	5.42E +10	5.43E +10	6.94E −01	6.96E −01	3.34E −01	3.35E −01	0.003
Transition No.	Rb
	Transition		λc (inÅ)	λ (in Å)	A		gf		S		dT
	i	j			B	C	B	C	B	C
1	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 4P1/2		231.13	8.76E +08	9.95E +08	1.40E −02	1.59E −02	1.07E −02	1.21E −02	0.120
2	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 4P3/2		202.54	1.77E +07	1.80E +07	4.35E −04	4.44E −04	2.90E −04	2.96E −04	0.021
3	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2D3/2	162.63 h	162.35	1.14E +10	1.19E +10	1.80E −01	1.88E −01	0.96E −01	1.00E −01	0.042
4	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2S1/2	124.46 h	124.02	6.43E +09	6.46E +09	2.97E −02	2.98E −02	1.21E −02	1.22E −02	0.004
5	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2P1/2	144.37 h	143.78	7.19E +10	7.56E +10	4.45E −01	4.69E −01	2.11E −01	2.22E −01	0.049
6	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2P3/2	123.07 h	122.66	3.56E +10	3.66E +10	3.21E −01	3.30E −01	1.30E −01	1.33E −01	0.026
7	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s23d 2D3/2	112.71 h	112.44	6.64E +10	6.66E +10	5.03E −01	5.05E −01	1.86E −01	1.87E −01	0.003
Transition No.	Sr
	Transition		λc (inÅ)	λ (in Å)	A		gf		S		dT
	i	j			B	C	B	C	B	C
1	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 4P1/2		220.38	1.10E +09	1.26E +09	1.61E −02	1.83E −02	1.17E −02	1.33E −02	0.118
2	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 4P3/2		190.40	2.26E +07	2.32E +07	4.91E −04	5.04E −04	3.08E −04	3.16E −04	0.024
3	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2D3/2	153.47 h	153.18	1.29E +10	1.35E +10	1.82E −02	1.90E −01	9.18E −02	9.60E −02	0.043
4	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2S1/2	116.92 h	116.52	6.10E +09	6.12E +09	2.48E −02	2.49E −02	9.53E −03	9.56E −03	0.003
5	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2P1/2	137.19 h	136.63	7.85E +10	8.26E +10	4.39E −01	4.62E −01	1.98E −01	2.08E −01	0.049
6	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s3p2 2P3/2	115.98 h	115.60	4.42E +10	4.53E +10	3.54E −01	3.63E −01	1.35E −01	1.38E −01	0.024
7	3s23p 2${\mathrm{P}}_{1/2}^{\mathrm{o}}$	3s23d 2D3/2	107.10 h	106.84	6.60E +10	6.61E +10	4.52E −01	4.52E −01	1.59E −01	1.59E −01	0.001

^a [12]; ^d [14]; ^e [15]; ^f [40]; ^g [41]; ^h NIST.

, 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.

3.2. 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:

$$\mathrm{R}=\frac{{\mathrm{I}}_1}{{\mathrm{I}}_2}=\frac{{\mathsf{\lambda }}_2}{{\mathsf{\lambda }}_1}\frac{{\mathrm{A}}_1}{{\mathrm{A}}_2}\frac{{\mathrm{g}}_1}{{\mathrm{g}}_2}\exp \left(-\frac{{\mathrm{E}}_1-{\mathrm{E}}_2}{\text{kT}}\right)$$

(5)

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. Plasma temperature (T in K), line intensity ratio (R in 10³), electron density (n_e in 10²¹ cm⁻³), plasma frequency (${\mathsf{\omega }}_e$) in PHz and skin depth (δ) in nm for optically thin plasma for Spectral Lines 1 and 7 of Al-like ions.

Cu
T (in K)	R	ne	nfit	${\mathsf{\omega }}_{\mathit{e}}$	δ
2 × 106	0.927	0.115	0.115	0.96289	495.1601
4 × 106	1.033	0.162	0.162	1.14284	417.193
6 × 106	1.070	0.199	0.199	1.266644	376.4159
8 × 106	1.090	0.230	0.230	1.361733	350.131
1 × 107	1.102	0.257	0.257	1.439443	331.2287
1 × 108	1.145	0.813	0.813	2.560196	186.2298
1 × 109	1.149	2.570	2.570	4.551919	104.7437
1 × 1010	1.150	8.128	8.129	8.095055	58.89829
Zn
T (in K)	R	ne	nfit	${\mathsf{\omega }}_{\mathit{e}}$	δ
2 × 106	0.686	0.135	0.135	1.043266	457.012
4 × 106	0.768	0.192	0.192	1.244167	383.2162
6 × 106	0.798	0.235	0.235	1.376455	346.3862
8 × 106	0.813	0.271	0.271	1.47813	322.5595
1 × 107	0.823	0.303	0.303	1.562965	305.0515
1 × 108	0.857	0.959	0.959	2.780592	171.4688
1 × 109	0.860	3.033	3.034	4.944976	96.41804
1 × 1010	0.861	9.593	9.598	8.794379	54.21473
Ge
T (in K)	R	ne	nfit	${\mathsf{\omega }}_{\mathit{e}}$	δ
2 × 106	0.392	0.187	0.187	1.22786	388.3057
4 × 106	0.444	0.264	0.264	1.458915	326.8079
6 × 106	0.464	0.324	0.324	1.61622	295.0000
8 × 106	0.473	0.374	0.374	1.736456	274.5736
1 × 107	0.480	0.418	0.418	1.835761	259.7206
1 × 108	0.501	1.321	1.322	3.263468	146.0976
1 × 109	0.504	4.179	4.178	5.804491	82.14069
1 × 1010	0.504	13.210	13.211	10.31999	46.20012
Rb
T (in K)	R	ne	nfit	${\mathsf{\omega }}_{\mathit{e}}$	δ
2 × 106	0.112	0.410	0.410	1.801505	262.2422
4 × 106	0.132	0.580	0.580	2.142681	220.4857
6 × 106	0.140	0.710	0.710	2.370679	199.2807
8 × 106	0.143	0.820	0.820	2.547713	185.4332
1 × 107	0.146	0.916	0.917	2.692721	175.4473
1 × 108	0.155	2.899	2.899	4.790355	98.62123
1 × 109	0.156	9.167	9.166	8.518384	55.46013
1 × 1010	0.156	28.990	28.986	15.14843	31.18677
Sr
T (in K)	R	ne	nfit	${\mathsf{\omega }}_{\mathit{e}}$	δ
2 × 106	0.087	0.482	0.482	1.971294	241.8639
4 × 106	0.104	0.682	0.682	2.344877	203.3304
6 × 106	0.110	0.835	0.835	2.594605	183.7601
8 × 106	0.113	0.965	0.964	2.789277	170.9349
1 × 107	0.115	1.079	1.078	2.949434	161.653
1 × 108	0.123	3.411	3.411	5.244074	90.91879
1 × 109	0.124	10.789	10.786	9.326497	51.12154
1 × 1010	0.124	34.110	34.115	16.58322	28.75105

, 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 × 10⁸ °C or 4 × 10¹⁰ K [69]. Therefore, we have considered the maximum plasma temperature of 1010 K in this paper. Since for a lower value of $\Delta \mathrm{E}={\mathrm{E}}_1-{\mathrm{E}}_2$, 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 10⁹ 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 10¹⁰ 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.

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:

$${\text{n}}_{\mathrm{e}}\ge 1.6\times {10}^{12}{\mathrm{T}}^{1/2}{\left(\Delta \mathrm{E}\right)}^3$$

(6)

In Equation (6), n_e 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 10²¹ 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. Plasma fitting parameters K and C for Al-like ions.

Z	K	C
29	0.500	16.909
30	0.500	16.979
32	0.499	17.121
37	0.500	17.462
38	0.500	17.532

. By employing fitting parameters K and C in the straight line equation log₁₀n_e = 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.

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:

$${\mathsf{\omega }}_{\mathrm{e}}=8.979\times {10}^4{\mathrm{n}}_{\mathrm{e}}^{1/2}\mathrm{Hz}$$

(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 10¹⁰ 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:

$$\mathsf{\delta }=5.31\times {10}^5{\mathrm{n}}_{\mathrm{e}}^{-1/2}\mathrm{cm}$$

(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 = 10¹⁰ 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.

The plasma parameter (Λ) may be defined in terms of electron density and plasma temperature by the following expression:

$$\Lambda =4\mathsf{\pi }\times 7.43\times {10}^2\times {\mathrm{n}}_{\mathrm{e}}^{-1/2}{\mathrm{T}}^{1/2}$$

(9)

The Coulomb coupling parameter (Γ) and plasma parameter (Λ) for plasma are related to each other as follows

$$\Gamma \approx {\Lambda }^{-2/3}$$

(10)

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, $\Lambda <1$ and $\Gamma >1$. 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, $\Gamma >1;$ this shows that the electrons are conducted by their interactions in the plasma.

4. 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 3s²3p, 3s3p² and 3s²3d 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.