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Li Xiang-Fu, Jiang Gang. Relativistic calculations of fine-structure energy levels of He-like Ar in dense plasmas . Chinese Physics B, 2018, 27(7): 073101  
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Relativistic calculations of fine-structure energy levels of He-like Ar in dense plasmas
Li Xiang-Fu1, 2, Jiang Gang1, 3, †
Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
College of Electrical Engineering, Longdong University, Qingyang 745000, China
Key Laboratory of High Energy Density Physics and Technology, Ministry of Education, Chengdu 610065, China

 

† Corresponding author. E-mail: gjiang@scu.edu.cn

Project supported by the National Natural Science Foundation of China (Grant No. 11474208) and the Doctoral Science Foundation of Longdong University, China (Grant No. XYBY1704).

Abstract

The fine-structure energy levels of 1s2s and 1s2p atomic states for the He-like Ar ion immersed in dense plasmas are calculated. The ion sphere model is used to describe the plasma screening effect on the tested ion. The influences of the hard sphere confinement and plasma screening on the fine-structure energy levels are investigated respectively. The calculated results show that the confined effect of the hard sphere on the fine-structure energy levels increases with decreasing hard sphere radius, and the plasma screening effect on the fine-structure energy levels increases with the increase of free electron density. In dense plasmas, the confined effect of the hard sphere on the fine-structure energy levels can be neglected generally, compared with the contribution from free electron screening. An interesting phenomenon about the energy level crossing is found among 1s2s (1S0) and 1s2p (3P0,1) atomic states. The results reported at the present work are useful for plasma diagnostics.

PACS: 31.15.ac;31.15.ag;31.15.xr;52.27.Gr
Keyword:He-like Ar;fine-structure energy levels;hard sphere confinement;free electron screening
1. Introduction

The dense plasma state is a common phase of matter in the universe and can be found in all types of stars and within giant planets.[1] Dense plasmas are commonly created in experiments involving high-power light sources, such as the National Ignition Facility,[2] recently developed x-ray free-electron laser Linac Coherent Light Source (LCLS) (USA)[3] and SACLA (Japan).[4] In dense plasmas, the potential in and near an ion is influenced by its bound electrons, free electrons, and neighboring ions. Consequently, the phenomena, such as spectral line shifts, ionization potential depression (IPD), line merging, fine-structure energy level changes, energy level crossing, and continuum states lowering, could be observed. Such properties can effectively be utilized for inertial confinement fusion (ICF) plasma diagnostics and the investigation of x-ray opacity of matter under conditions prevailing in stellar interiors.

Several dense plasma experiments[5–9] about IPD have been performed recently by intense short-pulse laser irradiation. However, the widely used theoretical expression for the IPD given by Ecker and Kröll (EK)[10] or Stewart and Pyatt (SP)[11] cannot give a satisfied description for all of these experiments. The experimental results[5] about the critical electron densities of the He-like Al ion, after which spectral lines disappear, seem more consistent with SP model, but Ciricosta’s experiments[6,7] about the direct measurements on the ionization energy of the K-shell in aluminum and the subsequent Kα lines tend to confirm EK model. It is inexplicable that the reported experimental results by Kraus et al.[9] could not be explained using either of the two theoretical models. One important reason for this circumstance is the lack of knowledge on the response of the fine-structure of the ions embedded in dense plasmas, e.g., the changes of the relative energy level order and the changes of transition probabilities. These effects play a crucial role for precise simulation of the final radiation from the plasma (spectral distribution) and for the precise diagnostic of plasma parameters.[12]

The fine-structure energy levels of 1s4l (l = s, p, d) atomic states for the He-like Al ions immersed in dense plasmas were calculated by Belkhiri et al.[13] using the ion sphere model (ISM).[14] The impact of dense plasma environments on the 1s3l (l = s, p, d) fine-structure levels of He-like ions (Z = 7–12) and thirty bound fine-structure levels of He-like Al ions were calculated by Li et al.[12,15] using the self-consistent-field ion sphere model (SCFISM).[16] De et al.[17] studied the effects of oscillatory quantum plasma screening on the fine-structure splitting between the components of Lyman-α and β line doublets of atomic hydrogen and hydrgen-like argon ion within dense quantum plasmas. To our knowledge, the fine-structure energy levels of the He-like Ar ion immersed in dense plasmas so far have not been reported. There are only several theoretical studies about the atomic structures of the He-like Ar ion within dense plasmas. The nonrelativistic energy values of 1sns (1Se) (n = 1–3) and 1snp(1Po) (n = 2–4) states of the He-like Ar ion within dense plasma environment were estimated by Bhattacharyya et al.[18] based on the ISM. Sil et al.[19] used the ISM to estimate the effects of dense plasma on the He-like Ar ion within the non-relativistic as well as the relativistic framework, and the results showed that the relativistic scheme represented a relatively good agreement with the experiment.[20] The ionization potentials and excitation energies of the Ar16+ ion embedded in plasma environment were calculated by Das et al.[21] for the first time using the state-of-the-art coupled cluster-based linear response theory with the four-component relativistic spinors, and their results showed that the transition energies obtained from the relativistic calculations were in promising agreement with the experiment.[20]

In this work, the ISM is used to describe the plasma screening effect on the He-like Ar ion. It is electrical neutrality outside of the ion sphere, so the wavefunctions of bound electrons are not allowed to stretch outside of the ion sphere. That is, the ion sphere actually becomes an impenetrable cavity or hard sphere. Therefore, when the tested ion immersed in dense plasmas, it will experience not only the screening effect from free electrons, but also the confined effect from the hard sphere. We firstly calculate the fine-structure energy levels of the He-like Ar ion located in the hard sphere (not plasma environment), in order to estimate the confined effect of the hard sphere. We then calculate the fine-structure energy levels of the He-like Ar ion within dense plasmas, and discuss the influence of plasma environment on the fine-structure energy levels. Lastly, we compare the relative importance of the effects of hard sphere confinement and plasma screening on the fine-structure energy levels of the He-like Ar ion immersed in dense plasmas. The free electron density range chosen in this paper is from 2.06 × 1023 to 7.52 × 1025 cm−3.

2. Theoretical method

The ISM is based on the principles that the ion is represented by a point-like nucleus with charge Z embedded at the centre of a spherical cavity containing enough electrons to ensure global neutrality. The spherical cavity is called as an ion sphere or Wigner–Seitz sphere. The ion sphere radius R0 is determined by the formula , where nf is the free electron density and Nb is the number of bound electrons. The plasma is assumed to produce an electrically neutral background beyond the ion sphere radius R0. For a N-electron atom, the Dirac–Coulomb Hamiltonian containing all of the dominant interactions can be written as where the first term is the contribution from one-body and the second term in the sum is the interaction between bound electrons. Hi is defined as where the first and the second terms are the relativistic kinetic energy of a bound electron. The last term VIS(ri) is the modified potential as “seen” by one bound electron within the ion sphere, and is given by The radial wavefunctions of bound electrons are derived by the Dirac equations[22] where Pnk(r) and Qnk(r) are the large and small components of the radial wavefunctions, respectively. Here εnk is the orbital energy eigenvalue and n is the principal quantum number. The spin-orbital quantum number k = −l−1 for j = l + 1/2, or k = l for j = l − 1/2. XP(r) and XQ(r) are the exchange potentials which force the orthogonality between orbits of the same symmetry. With regard to the electrically neutral conditions supposed by the ISM, the radial wavefunctions Pnk(r) and Qnk(r) are assumed to satisfy the boundary conditions and normalization condition respectively, namely,

Once the contributions of nuclear charge, bound electrons, and free electrons are included in the total Hamiltonian, the single-electron wavefunction can be obtained via the self-consistent-field method, which is carried out using the modified GRASP2K code.[23,24] By the way, the theory for the calculated atomic structures under the hard sphere confinement (not plasma environment) is very similar to that of the ISM mentioned above, except that the second term in Eq. (3) is equal to zero, namely, the free electron density is equal to zero. Thus the theoretical method for calculating atomic structures under the hard sphere confinement is not repeated here.

3. Results and discussion
3.1. Fine-structure energy levels of free He-like Ar

In this work, the multiconfiguration Dirac–Fock (MCDF) method is used to describe the correlation effect and relativistic effect. An atomic state function (ASF) is constructed approximately by a linear combination of configuration state functions (CSFs) with the same parity and angular momentum. The CSFs are constructed by double exciting bound electrons in the occupied orbits of the reference CSFs to the unoccupied orbits. The configurations 1s2 and 1s2p are double excited to orbits nl (n = 1, 2, l = s, p), in order to form the CSFs.

The energy eigenvalues and energy levels of 1s2, 1s2s, and 1s2p atomic states for the free He-like Ar ion are listed in Table 1. The table shows that our calculated energy levels of 1s2s and 1s2p atomic states agree well with those of NIST.[25] In addition, one point should be mentioned here that the correlation effect between bound electrons may not be considered very enough. The reasons are explained below. When the atomic structures of He-like Ar ions immersed in dense plasmas are calculated and the free electron densities are greater than 9.39 × 1024 cm−3 (R0 < 1.4 a.u.), the bound electrons in the occupied orbits of the reference CSFs could not be excited to the higher unoccupied orbits, such as 3l (l = s, p, and d) orbits, due to the confinement from the hard sphere. The CSFs for one ASF in both plasmas and vacuum should be the same as each other, or else the estimated intensity of plasma screening effect on the atomic structures is meaningless. Therefore, the only orbits nl (n = 1, 2, l = s, p) are used to construct CSFs in both plasmas and vacuum, in order to describe the correlation effect between bound electrons.

Table 1.

The energy eigenvalues Eb (in a.u.) and energy levels El (in cm−1) of 1s2, 1s2s, and 1s2p atomic states for the free He-like Ar ion.

.
3.2. Fine-structure energy levels under the hard sphere confinement (not plasma environment)

The energy eigenvalues of 1s2, 1s2s, and 1s2p atomic states for the He-like Ar ion under the hard sphere confinement are listed in Table 2. It can seen that all energy eigenvalues change very slightly with the hard sphere radius, and all of them shift to continuum state with decreasing hard sphere radius. The energy eigenvalues of all atomic states for the He-like Ar ion under the hard sphere confinement increase with the reduction of hard sphere radius, which is consistent with the reported results of the other confined two electron systems.[26–28] Figure 1 shows that the sequence of fine-structure energy levels of the He-like Ar ion under the hard sphere confinement is the same as that of the free case (see Table 1), but all energy levels move slightly. In order to more clearly display the energy level changes, the energy level shift is defined here, which is the difference between the energy level under hard sphere confinement and that in free case. The energy level shifts of 1s2s and 1s2p atomic states for the He-like Ar ion under the hard sphere confinement are displayed in Fig. 2. It can be seen that all energy levels shift upward. The smaller radius of the hard sphere is, the greater extent of the energy level shifts. The reason for this result showed from Table 2 and Fig. 3 is that the energy eigenvalue of 1s2 is nearly unchanged but those of 1s2s and 1s2p atomic states increase relatively greater with the decrease of hard sphere radius. In addition, the energy eigenvalues of 1s2 (1S0), 1s2s (1S0, 3S1), and 1s2p (1P1, 3P0,1,2) atomic states for the He-like Ar ion under hard sphere confinement are not reported for R0 ≤ 0.7 a.u., because the calculated data by our program become unstable for R0 ≤ 0.7 a.u., namely, the calculated data change with step size. The calculated data must be stable for different step size, otherwise the calculated data is unreliable.

Fig. 1. (color online)The energy levels of 1s2s and 1s2p atomic states for the He-like Ar ion under the hard sphere confinement.
Fig. 2. (color online) The energy level shifts of 1s2s and 1s2p atomic states for the He-like Ar ion under the hard sphere confinement.
Fig. 3. The energy eigenvalues of 1s2 (1S0) and 1s2s (3S1) atomic states for the He-like Ar ion under the hard sphere confinement. By the way, the variation trend of 1s2s (1S0) and 1s2p (3P0,1,2 and 1P1) atomic states with hard sphere radius is the same as that of 1s2s (3S1) atomic state.
Table 2.

The energy eigenvalues E (in a.u.) of 1s2, 1s2s, and 1s2p atomic states for the He-like Ar ion under the hard sphere confinement.

.

It is also shown from Fig. 1 that the separation between the adjacent energy levels is slightly changed with the hard sphere radius. To more clearly investigate the variation of energy level separation with the hard sphere radius, the energy level separation shift is used here. The energy level separation shift is the difference between energy level separation under the hard sphere confinement and that in free case. As shown in Fig. 4, when the hard sphere radius is gradually decreased, the separations between 1s2s (3S1) and its lower adjacent level 1s2 (1S0), 1s2s (1S0) and its lower adjacent level 1s2p (3P1) are remarkably increased, the energy level separations between 1s2p (3P1), 1s2p (1P1) and their adjacent lower atomic states are slightly increased respectively, but the energy level separations between 1s2p (3P0), 1s2p (3P2) and their adjacent lower atomic states are significant decreased respectively. In a word, the energy level separation shift under the hard sphere confinement is different for different energy levels.

Fig. 4. (color online) The energy level separation shifts of 1s2s and 1s2p atomic states for the He-like Ar ion under the hard sphere confinement. 1s2s (3S1), 1s2s (1S0), 1s2p (3P0), 1s2p (3P1), 1s2p (3P2), and 1s2p (1P1) represent the energy level separations between 1s2s (3S1) and 1s2 (1S0), 1s2s (1S0) and 1s2p (3P1), 1s2p (3P0) and 1s2s (3S1), 1s2p (3P1) and 1s2p (3P0), 1s2p (3P2) and 1s2s (1S0), and 1s2p (1P1) and 1s2p (3P2) atomic states, respectively.

It can be concluded from the above discussions that the confinement effect of the hard sphere on the fine-structure energy levels of the He-like Ar ion increases with decreasing the hard sphere radius.

3.3. Fine-structure energy levels in dense plasmas

The energy eigenvalues of 1s2, 1s2s, and 1s2p atomic states for the He-like Ar ion in dense plasmas are displayed in Table 3. By comparing the data in Tables 2 and 3, it can be concluded that the energy eigenvalues are upshifted by free electrons, compared with the hard sphere confinement, very significantly. The energy levels of 1s2s and 1s2p atomic states for the He-like Ar ion in dense plasmas are listed in Table 4 and displayed in Fig. 5. It is shown from Fig. 5 that all energy levels decrease with increasing free electron densities. The reasons for this result are expressed below. The energy eigenvalue shift is the difference between energy eigenvalue of the ion immersed in plasmas and that in free case. As shown from Fig. 6, although energy eigenvalue shifts of all atomic states increase with increasing free electron densities, the extents of energy eigenvalue shifts for 1s2 atomic state are always greater than those of the other atomic states. Thus the energy levels of all atomic states decrease with the rise of free electron density.

Fig. 5. (color online) The energy levels of 1s2s and 1s2p atomic states for the He-like Ar ion in dense plasmas.
Fig. 6. (color online) The energy eigenvalue shifts of 1s2s and 1s2p atomic states for the He-like Ar ion in dense plasmas.
Table 3.

The energy eigenvalues E (in a.u.) of 1s2, 1s2s, and 1s2p atomic states for the He-like Ar ion in dense plasmas.

.
Table 4.

The energy levels El (in cm−1) of 1s2s and 1s2p atomic states for the He-like Ar ion in dense plasmas.

.

An interesting phenomenon about “energy level crossing” is found from Fig. 5. The energy level of 1s2s (1S0) state lies above that of 1s2p (3P0,1) states for free electron density nf ≤ 3.22 × 1024 cm−3 (R0 ≥ 2.0 a.u.), including the free case for nf = 0 (R0 = ∞). However, the energy level of 1s2s (1S0) state lies below that of 1s2p (3P0,1) after free electron density nf ≥ 7.64 × 1024 cm−3 (R0 ≤ 1.5 a.u.). This interesting phenomenon can be interpreted using Fig. 7. It is shown in Fig. 7 that the amount of energy level shift (decrease) of 1s2s (1S0) state is greater than those of 1s2p (3P0,1) states after free electron density nf ≥ 7.64 × 1024 cm−3 (R0 ≤ 1.5 a.u.), thus leading to that 1s2s (1S0) level lies blow 1s2p (3P0,1) levels. The “energy level crossing” phenomenon shows that an incidental degeneracy[29] has taken place among the 1s2s (1S0) and 1s2p (3P0,1) atomic states of the Ar16+ ion at some free electron density 3.22 × 1024 cm−3 < nf < 7.64 × 1024 cm−3 (1.5 a.u. < R0 < 2.0 a.u.), and then a level crossing occurs among these three atomic states having different symmetry properties. By the way, such incidental degeneracy and subsequent level crossing phenomenon have been found in case of cage confined H− ion[30] and He atom,[31] He+ ion in quantum dot,[32] and He-like ions within strongly coupled plasma environment.[12,13,15,18]

Fig. 7. (color online) The energy level shifts of 1s2s and 1s2p atomic states for the He-like Ar ion in dense plasmas.

As shown in Fig. 5, the energy level gap between states having the same symmetry decreases, but that between states having different symmetry increases, with increasing free electron densities. That is, the energy level gaps between states 1s2s (3S1) and 1s2s (1S0), 1s2p (3P0,1,2) and 1s2p (1P1) decrease respectively, but that between states 1s2s(1S0) and 1s2p (3P0) increases, with increasing free electron densities.

It is also shown in Figs. 7 and 2 that the direction of energy level shift caused by free electron screening is different from that caused by hard sphere confinement, namely, free electrons decrease energy levels but hard sphere confinement lifts energy levels. Therefore, the impact of the hard sphere confinement on the energy levels in dense plasmas is discussed here. It can be seen from Fig. 8 that the energy level shifts caused by the hard sphere confinement as a percentage of total energy level shifts of 1s2s and 1s2p atomic states for the He-like Ar ion in dense plasmas increase with decreasing the ion sphere radius (increasing the free electron density), and the highest percentage is −1.10% for 1s2s (1S0) state at R0 = 0.7 a.u. (free electron density nf = 7.52 × 1025 cm−3). Thus we can conclude that the contribution of the hard sphere confinement to the fine-structure energy levels of the He-like Ar ion in dense plasmas, compared with that of free electron screening, is relative small and can be neglected generally.

Fig. 8. (color online) The energy level shifts caused by the hard sphere confinement as a percentage of total energy level shifts of 1s2s and 1s2p atomic states for the He-like Ar ion in dense plasmas. Esh is the energy level shift caused by the hard sphere confinement and Est is the total energy level shift caused by the hard sphere confinement and free electron screening.

However, there will not exist hard confinement but some kind of penetrable confinement, if the boundary condition that the radial wavefunctions gradually decay to zero outside of the ion sphere is used. This boundary condition has been used by Belkhiri et al.[13,33] and Li et al.[34] Of course, the most widely used boundary condition is that the radial wavefunctions are directly set as zero outside of the ion sphere (present case).[18,19,35–41] In addition, the two boundary conditions both have shortcomings. The boundary condition that the radial wavefunctions always equal to zero outside of the ion sphere is corresponding to that the electrostatic potential is equal to ∞ outside of the ion sphere, but the real potential is equal to zero outside of the ion sphere. The boundary condition that the radial wavefunctions gradually decay to zero outside of the ion sphere could not well maintain electrical neutrality of the whole ion sphere, because the bound electrons can move to outside of the ion sphere.

4. Conclusion

In this work, the fine-structure energy levels of 1s2s and 1s2p atomic states of the He-like Ar ion are calculated in the hard sphere confinement (no free electrons) and in dense plasmas (hard sphere confinement together with free electrons), respectively. The results show that under the only hard sphere confinement, all energy eigenvalues are shifted to continuum states, all energy levels are upward moved, the energy level separation shift is different between different energy levels, and the confinement effect of the hard sphere increases with decreasing the hard sphere radius. In dense plasmas, all energy levels decrease with increasing free electron densities; an “energy level crossing” phenomenon takes place among 1s2s (1S0) and 1s2p (3P0,1) atomic states while they are effectively bound states; the energy level gap between states having the same symmetry decreases, but that between states having different symmetry increases, with increasing free electron densities. The confined effect of the hard sphere on the fine-structure energy levels, compared with the contribution from free electron screening, can be neglected generally.

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