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Li Xiang-Fu, Jiang Gang, Wang Hong-Bin, Sun Qian. Atomic structure and transition properties of H-like Al in hot and dense plasmas. Chinese Physics B, 2017, 26(1): 013101  
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Atomic structure and transition properties of H-like Al in hot and dense plasmas
Li Xiang-Fu1, 2, Jiang Gang1, 3, †, Wang Hong-Bin1, Sun Qian2
Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
College of Electrical Engineering, Longdong University, Qingyang 745000, China
The Key Laboratory of High Energy Density Physics and Technology of Ministry of Education, Chengdu 610065, China

 

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

Abstract

The atomic structure and transition properties of H-like Al embedded in hot and dense plasmas are investigated using modified GRASP2K code. The plasma screening effect on the nucleus is described using the self-consistent field ion sphere model. The effective nuclear potential decreases much more quickly with increasing average free electron density, but increases slightly with increasing electron temperature. The variations of the transition energies, transition probabilities, and oscillator strengths with the free electron density and electron temperature are the same as that of the effective nuclear potential. The results reported in this work agree well with other available theoretical results and are useful for plasma diagnostics.

PACS: 31.15.ac;31.15.ag;31.15.xr;52.27.Gr
Keyword:H-like Al;strongly-coupled plasmas;atomic structure;transition properties
1. Introduction

The study of the structural properties of ions within various external environments, especially in hot and dense plasmas, became an important area of research many decades ago because of its potential applications in astrophysical systems, laser-produced plasmas, inertial confinement fusion, x-ray lasers, plasma spectroscopy, etc.[1–11] In addition, compared to free systems, many interesting phenomena have been observed in plasmas, particularly in strongly coupled plasmas (SCPs), such as energy level shifts, changes in line shape, broadening of spectral lines, ionization potential depression (IPD), changes in transition properties, and line merging.[12–16]

The intensity of the influence of the plasma on the tested ion can be measured using the coupling strength , which is defined as the ratio of the average Coulomb energy between pairs of particles and their kinetic energy. denotes weakly coupled plasmas (WCPs), which can be described by the standard Debye–Hückel model.[17] In early works, only the plasma screening effect on the nucleus–electron interaction was taken into account, for example, see Refs. [18]–[20] Recently, the plasma screening effect on the electron–electron interaction has also been included, as in the works of Li et al.[5] and Xie et al.[21] refers to SCPs, which can be described by the ion sphere model (ISM).[22] The ISM assumes that free electrons in the ion sphere are distributed uniformly, but in actual plasmas, the distribution of free electrons is not homogeneous. Consequently, a better model, the self-consistent-field ion sphere model (SCFISM),[23] was proposed. It considers a self-consistent distribution between free electrons and bound electrons within the ion sphere, and the free electron distribution is described by the Boltzmann or Fermi–Dirac distribution function. In this work, the SCFISM is applied to describe the influence of strongly coupled plasmas on ions.

The experimental observations[12,24–27] of Al explicitly demonstrated the effects of an SCP on the spectral properties. Saemann et al.[12] used a laser-produced plasma, which caused the laboratory plasma conditions to change rapidly, so local thermodynamic equilibrium was not maintained. Consequently, the experimental measurements became extremely complicated, leading to a loss of accuracy. Several remarkable improvements[24–27] have been made because Linac coherent light sources (LCLS) can create relatively long-lived high-density plasmas with homogeneous temperatures. Ciricosta et al.[26] found that the measured IPDs of highly charged Al were not consistent with the predictions of the most widely used theoretical model of Stewart and Pyatt (SP),[28] but in good agreement with the earlier model of Ecker and Kröll (EK).[29] However, this measurement was questioned in a subsequent theoretical study by Preston et al.,[29] who used both the SP and EK models for the spectral lines of H-like Al. ISM potentials were used in both the SP model and the EK model. Hoarty et al.[31,32] further observed the K-shell spectra of Al plasmas using the Orion laser device; the values of the IPDs were more consistent with the SP model than the EK model. This situation clearly requires extensive and accurate theoretical study of atomic structures for Al. The energy eigenvalues of ns (n = 1, 2) and np (n = 2, 3) and the transition energies of the Lyman lines for within the framework of the ISM were investigated by Bhattacharyya et al.[33,34] Das[35] investigated the variation of the transition energy with the free electron density of the 1s–2p transition for H-like Al using the ISM. Salzmann et al.[36] presented the variations of the atomic properties and the transition probabilities with the free electron density for the ion.

GRASP2K[37,38] is a general-purpose relativistic atomic structure package. Many atomic structure and transition properties of neutral atoms or ions in a vacuum can be accurately calculated using this program, but these calculations cannot be performed for atoms or ions in external environments, such as plasma environments. To estimate the influence of hot and dense plasmas on the atomic structure and transition properties of H-like Al, we use both the ISM potential and the SCFISM potential instead of the original potential in the GRASP2K code to describe the plasma screening effect on the nucleus. Using the modified program, we estimate the effective nuclear potential and transition properties of the – (n = 2–4) transitions for H-like Al within SCP environments. The plasma parameters chosen in this paper are from 100 to 1000 eV for the electron temperatures and from to for the average free electron densities.

2. Theoretical method

The SCFISM potential is added to the GRASP2K code to describe the screening effect of plasmas on the nucleus. The theory behind the SCFISM has already been discussed in the literature,[23,39,40] so it will only be introduced briefly here.

The SCFISM is based on the following principles: the atom is represented by a point-like nucleus with charge Z embedded at the center of a spherical cavity containing enough electrons to ensure global neutrality. The spherical cavity is called an ion sphere or Winger–Seitz sphere. The ion sphere radius R 0 is determined by the formula , where is the average free electron density and is the number of bound electrons. The plasma is assumed to produce an electrically neutral background beyond the ion sphere radius R 0. Thus, the problem of atomic structure within dense plasmas depends on a self-consistent solution of the Dirac and Poisson equations related to bound electrons, free electrons, and charged particles in the ion sphere.

For an N-electron atom, the Dirac–Coulomb Hamiltonian containing all of the dominant interactions can be written as

(1)
where the first term is the contribution from one body, and the second term in the sum is the interaction between bound electrons. H i is defined as
(2)
where the first and the second terms in Eq. (2) are the relativistic kinetic energy of a bound electron, and the last term is the effective nuclear potential experienced by this bound electron. We will discuss how to get U i in the following paragraphs.

We begin with the Poisson equation for the potential experienced by a tested ion with nuclear charge Z. The total potential , which is produced by nuclear charge Z, free electron density , bound electron density , and other plasma ions with density , satisfies the Poisson equation[39]

(3)
Other ions here are characterized by an average charge . In the SCFISM, only one nucleus is located at the center of the ion sphere, and the perturbations from other ions are ignored in Eq. (3). The bound electron density can be obtained in terms of the radial wavefunctions and , which are the large and small component radial wavefunctions, respectively. For a given bound state, can be written as
(4)
where q i stands for the general occupation number of bound electrons in subshell i and M is the number of subshells. Equation (4) represents the spherically averaged bound electron density. We assume that free electrons in the ion sphere follow a Fermi–Dirac distribution in the plane wave momentum k space[40–44] and that the free electron density can be defined as
(5)
where . The chemical potential μ is determined by
(6)
with regard to the electrically neutral conditions supposed by the SCFISM. The boundary condition for the total potential is set as follows:
(7)
For , the solution of the Poisson equation takes the form[39]
(8)
which includes the contribution from the nucleus, free electrons, and bound electrons. The potential of the free electrons can be calculated by
(9)
and the potential of the bound electrons is similar to that of the free electrons
(10)
The effective nuclear potential is defined as
(11)
where is equal to U i in Eq. (2). To solve the bound electron wavefunctions in the framework of the SCFISM, the radial wavefunctions and are assumed to satisfy the boundary condition and normalization condition
(12)
(13)
If the electron temperature is so high that the kinetic energy completely overcomes the potential energy, equation (5) shows that the free electron density is spatially independent. This means that free electrons are uniformly distributed in the ion sphere. According to Eqs. (9) and (11) the effective nuclear potential can be reduced to the following uniform electron-gas model (UEGM):
(14)
This UEGM potential is the same as the ISM potential. However, the electron temperature in the actual plasma is finite, so the spatial distribution of free electrons is not uniform but similar to a Fermi–Dirac distribution. Once the contributions of the 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.

In fact, our calculation has some deficiencies. First, the free electron exchange and correlation effects are neglected because they have only minor influences on the results.[45] Second, the neighboring-ion correlations are simply included by assuming an electrically neutral background outside of the sphere rather than using a pair correlation function. The method will be inadequate for very strongly coupled plasmas,[46] for example, a lattice-type structure builds up in the plasma. In this work, the ion–ion coupling parameter , and a homogeneous ion distribution outside the ion sphere is still a plausible approximation.

3. Results and discussion
3.1. Spatial distribution of free electrons

The spatial distributions of free electrons surrounding the H-like Al ion at one electron temperature and three free electron densities are shown in Fig. 1. As seen from the figure, there is a sharp increase in the spatial density of free electrons near the nucleus, while the distribution is almost homogeneous far away from the nucleus. This result indicates that the spatial variation of the free electron distribution in the ion sphere is dominated by the strong nuclear charge attraction, but the repulsions among electrons play a less important role. The figure also shows that the free electron density tends to extremely large values around the nucleus when the free electron density is very high. This is not a physical problem, but in the actual calculation, this singularity does not cause notable effects on the final results.

Fig. 1. (color online) Spatial distribution of free electrons surrounding the H-like Al ion at one electron temperature and three average free electron densities , , and .

Figure 2 shows the spatial distributions of free electrons surrounding the H-like Al ion at an average free electron density and three electron temperatures , 500, 1000 eV. It can be seen from Fig. 2(a) that the spatial distribution of free electrons decreases significantly around the nucleus with increasing electron temperature, and it becomes more homogeneous and approaches that of the ISM. Because the kinetic energies of free electrons increase with increasing electron temperature, the probability that free electrons are far from the nucleus is increased. That is, free electrons move more freely and tend to distribute uniformly.

Fig. 2. (color online) (a) Spatial distribution of free electrons surrounding the H-like Al ion at one average free electron density and three electron temperatures , 500, and 1000 eV, as well as that of ISM ( a.u.); (b) the same as panel (a) ( a.u.).

It is shown in Fig. 2(b) that the free electron density close to the surface of the ion sphere is lower than that of the ISM. This results from the electrical neutrality of the whole ion sphere. In other words, the number of electrons inside the ion sphere is equal to the nuclear charge Z.

As mentioned above, due to the strong nuclear charge attraction of highly charged ions, the spatial distribution of free electrons is not uniform except when the electron temperature is extremely high. Therefore, the SCFISM is more reasonable than the ISM for describing the plasma screening effect on the tested ion, although the free electron density close to the nucleus is too large to have any meaning at very high average free electron densities or very low electron temperatures.

3.2. Effective nuclear potential

The effective nuclear potentials of the H-like Al ion at four average free electron densities and one electron temperature are displayed in Fig. 3. The figure shows that the effective nuclear potential decreases with increasing r, and its minimum value is equal to the number of bound electrons. For the H-like Al ion, this value is 1.

Fig. 3. (color online) Effective nuclear potential of the H-like Al ion at one electron temperature and four average free electron densities , , and cm−3.

It can also be seen from the figure that the effective nuclear potential decreases much more quickly as the average free electron density increases. Because the total electrical neutrality of the ion sphere is preserved, the radius of the ion sphere decreases with the increase in average free electron density. That is, more and more free electrons move around the nucleus as the average free electron density increases. The effective nuclear potential versus the average free electron density obtained in this work is similar to that of Be-like ions.[40]

The effective nuclear potentials of the H-like Al ion at two electron temperatures and 1000 eV with an average free electron density are shown in Fig. 4; the effective nuclear potential of the ISM is also plotted. The effective nuclear potential of the SCFISM approaches that of the ISM as the electron temperature increases because the spatial distribution of the free electrons becomes more homogeneous as the electron temperature increases. The other properties, such as the transition energies, transition probabilities, and oscillator strengths obtained by the SCFISM, also approach that of the ISM as the electron temperature increases. Therefore, in hot and dense plasmas, the screening effect of free electrons on the nucleus due to temperature is so important that it cannot be neglected.

Fig. 4. (color online) Effective nuclear potential of the H-like Al ion at one average free electron density and two electron temperatures and 1000 eV, as well as that of ISM.
3.3. Transition properties of free H-like Al

Our calculations and NIST[47] both show that the energy level intervals between the and and , and and atomic states are 1.301, 0.385, and 0.163 eV, respectively. The intervals of these energy levels are very small. The transition probabilities of – (n = 2–4) transitions are slightly larger than those of the corresponding – (n = 2–4) transitions. Therefore, only the transition properties of the – (n = 2–4) transitions are considered in this work. The electric dipole line strength S, transition probability A and oscillator strength gf in the length gauge is defined by[48]

(15)
(16)
(17)
where is the transition matrix element of transiting between atomic state i and atomic state j; E ij is the transition energy (in a.u.) between the initial and final states with statistical weights g i and g j . The value of A is proportional to , the value of gf is proportional to E ij , and they are all proportional to line strength S.

The transition energies, transition probabilities, and oscillator strengths of – (n = 2–4) transitions for the free H-like Al ion are displayed in Table 1, and the data collected by NIST[47] and the theoretical results of Jitrik et al.[49] are also listed. These data indicate that our results are in good agreement with NIST and Jitrik et al.

Table 1.

The transition energies E (in eV), transition probabilities A (in ), and oscillator strengths gf of – (n = 2–4) transitions for the free H-like Al ion. The values indicate .

.
3.4. Transition energies in plasmas

The transition energies of – (n = 2–4) transitions for the H-like Al ion in plasmas are displayed in Tables 2, 3, and 4, respectively. In addition, the results of the ISM are also listed. The change trends of the transition energies for – (n = 2, 4) transitions versus the average free electron density and electron temperature are the same as those of – transition. Therefore, the – transition is taken as an example to illustrate the influences of the free electron density and the electron temperature on the transition energies.

Table 2.

Transition energies (in eV) of – transition for the H-like Al ion in plasmas. The figures from 100 to 1000 are the electron temperatures (in eV). ISM is the result based on the ISM.

.
Table 3.

Transition energies (in eV) of – transition for the H-like Al ion in plasmas. The figures from 100 to 1000 are the electron temperatures (in eV). ISM is the result based on the ISM.

.
Table 4.

Transition energies (in eV) of – transition for the H-like Al ion in plasmas. The figures from 100 to 1000 are the electron temperatures (in eV). ISM is the result based on the ISM.

.

Figure 5 shows the transition energies of – transition for the H-like Al ion at different average free electron densities and one electron temperature . It can be seen from the figure that the transition energies almost linearly decrease with increasing average free electron density because the screening effect of free electrons on the nucleus becomes stronger with increasing free electron density. For example, the transition energies obtained from the ISM are 2047.994, 2044.382, and 2039.688 eV at the average free electron densities of , , and , respectively. The results at these densities obtained by Bhattacharyya et al.[34] in the relativistic framework are 2047.985, 2044.336, and 2039.619 eV, respectively, which indicates that our results agree well with those of Bhattacharyya et al.

Fig. 5. Transition energies of – transition for the H-like Al ion at different average free electron densities and one electron temperature .

The transition energies of – transition for the H-like Al ion at an average free electron density and different electron temperatures are displayed in Fig. 6, and the result from the ISM is also included. The figure shows that the transition energies of the SCFISM increase and approach those of the ISM as the electron temperature increases because the higher the electron temperature, the more uniform the free electron distribution. Therefore, the screening effect of the free electrons on the nucleus becomes weaker, which leads to an increase in the transition energies.

Fig. 6. Transition energies of – transition for the H-like Al ion at an average free electron density and different electron temperatures, as well as that of ISM.
3.5. Ionization potential depression

The ionization potentials (IPs) of [n = 2–4] atomic states for the H-like Al ion at one electron temperature and different free electron densities are displayed in Fig. 7. As shown in the figure, for all atomic states, as the free electron density increases, the ionization potentials decrease and approach zero. If the IP is equal to zero at a given free electron density and electron temperature, the spectral line corresponding to that transition disappears because the electron in the higher orbital has been ionized. For example, the spectral line corresponding to – transition disappears at the free electron density and electron temperature because the IP of atomic state decreases to zero. Table 5 shows the highest free electron densities and the lowest electron temperatures for which different spectral lines of the H-like Al ion can be observed, and the results of Bhattacharyya et al.[33] are also listed. The data in the table indicate that our relativistic results based on the ISM agree well with the non-relativistic results based on the ISM of Bhattacharyya et al., and the highest free electron densities obtained from the SCFISM are smaller than those from the ISM. The results of the SCFISM may be more accurate than those from the ISM because the electron temperature is finite in actual plasmas, but the electron temperature is assumed to be in the ISM.

Fig. 7. Ionization potentials of (a) 4p , (b) 3p , (c) 2p atomic states for the H-like Al ion at one electron temperature eV and different free electron densities.
Table 5.

The highest free electron densities (in ) and the lowest electron temperatures (in eV) for which different spectral lines of the H-like Al ion can be observed. The notation indicates .

.
3.6. Transition probabilities and oscillator strengths in plasmas

The transition probabilities and oscillator strengths of – (n = 2–4) transitions in the length gauge and velocity gauge are almost equal in our calculation, but the velocity results tend to be more sensitive to the accuracy of the wavefunctions than the length results. Therefore, only the results in the length gauge are presented in the appendix. The change trends of the transition probabilities and oscillator strengths versus the average free electron densities and electron temperatures are the same as that of the transition energies.

4. Conclusion

In this work, the atomic structure and transition properties of H-like Al embedded in hot and dense plasma environments are investigated. The SCFISM is better for describing the plasma screening effect on the nucleus than the ISM because the spatial distribution of free electrons is not uniform in actual plasmas except when the electron temperature is extremely high. There is a sharp increase in the spatial density of free electrons near the nucleus, while the distribution is almost homogeneous far from the nucleus. The spatial distribution of free electrons decreases significantly around the nucleus and becomes more homogeneous with increasing electron temperature. The effective nuclear potential decreases much more quickly as the average free electron density increases but only increases slightly as the electron temperature increases. The transition energies, transition probabilities, and oscillator strengths versus the free electron density and electron temperature are the same as the effective nuclear potential. The critical free electron densities above which different spectral lines disappear that are obtained using the SCFISM may be more accurate than those obtained using the ISM because the electron temperature is finite in actual plasmas, but the electron temperature is assumed to be in the ISM. The present SCFISM based on the GRASP2K code can be used to calculate relatively accurate atomic structure and transition properties for highly charged ions in SCP environments. The results reported in this work are useful for plasma diagnostics.

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