Ion-sphere Model Research Articles

Nonideal effects on ionization potential depression and ionization balance in dense Al and Au plasmas.

For low-density plasmas, the ionization balance can be properly described by the normal Saha equationin the chemical picture. For dense plasmas, however, nonideal effects due to the interactions between the electrons and ions and among the electrons themselves affect the ionization potential depression and the ionization balance. With the increasing of plasma density, the pressure ionization starts to play a more obvious role and competes with the thermal ionization. Based on a local-density temperature-dependent ion-sphere model, we develop a unified and self-consistent theoretical formalism to simultaneously investigate the ionization potential depression, the ionization balance, and the charge states distributions of the dense plasmas. In this work, we choose Al and Au plasmas as examples as Al is a prototype light element and Au is an important heavy element in many research fields such as in the inertial confinement fusion. The nonideal effect of the free electrons in the plasmas is considered by the single-electron effective potential contributed by both the bound electrons of different charge states and the free electrons in the plasmas. For the Al plasmas, we can reconcile the results of two experiments on measuring the ionization potential depression, in which one experiment can be better explained by the Stewart-Pyatt model while the other fits better with the Ecker-Kröll model. For dense Au plasmas, the results show that the double peak structure of the charge state distribution appears to be a common phenomenon. In particular, the calculated ionization balance shows that the two- and three-peak structures can appear simultaneously for denser Au plasmas above ∼30g/cm^{3}.

Nonideal effect of free electrons on ionization equilibrium and radiative property in dense plasmas.

The thermodynamic as well as optical properties of strongly coupled plasmas depend crucially on the average degree of ionization and the ionic state composition, which, however, cannot be determined by using the normal Saha equationusually used for the ideal plasmas. Hence, an adequate treatment of the ionization balance and the charge state distribution of strongly coupled plasmas is still a challenge for theory due to the interactions between the electrons and ions and among the electrons themselves. Based on a local density temperature-dependent ion-sphere model, the Saha equationapproach is extended to the regime of strongly coupled plasmas by taking into account the free-electron-ion interaction, the free-free-electron interaction, the nonuniform free-electron space distribution, and the free-electron quantum partial degeneracy. All the quantities, including the bound orbitals with ionization potential depression, free-electron distribution, and bound and free-electron partition function contributions, are calculated self-consistently in the theoretical formalism. This study shows that the ionization equilibrium is evidently modified by considering the above nonideal characteristics of the free electrons. Our theoretical formalism is validated by the explanation of a recent experimental measurement of the opacity of dense hydrocarbon.

A superconfiguration calculation of opacity with consistent bound and continuum electron treatments using green’s functions

One of the challenges in calculating the opacity of dense plasmas is the difficulty in consistently modeling electrons bound to nuclei and those that exist within the continuum of free states in electronic structure models. We address this issue by adapting the green’s function approach, originally developed for use in average atom calculations, to the determination of superconfiguration (SC) electronic structure. The spectra created using these SCs indicate that a consistent treatment of continuum electronic structure is important for phenomena involving electrons near ionization thresholds, such as the pressure ionization of bound states and the opacity due to transitions near bound-free edges. Though important for dense plasmas, the detailed incorporation of continuum electrons into structure calculations does not have significant impact on the recent discrepancies between the predicted and measured opacity of hot, dense iron (Bailey et al 2015 Nature 517 56). We find that the inclusion of plasma effects through an ion-sphere model along with our treatment of continuum electronic states gives a description of pressure ionization in hot, dense aluminum that is in better agreement with experiment than methods that rely on perturbative descriptions of the plasma environment (Hoarty et al 2013 Phys. Rev. Lett. 110 265003).

Analysis of electronic properties of dense plasma-embedded highly charged ions using temperature-dependent ion sphere model

Analysis of electronic properties of dense plasma-embedded highly charged ions using temperature-dependent ion sphere model

Combined effect of plasma screening and electric field on the He atom

The combined effect of plasma screening and electric field on the structural properties of atoms/ions is investigated with the help of the relativistic computational method. Two concepts, the ion-sphere model potential and the electric field perturbation operator, which are crucial to perform the current investigation, are introduced to solve the Dirac–Fock equations based on the self-consistent field theory. The developed algorithm is applied to calculation of the plasma-embedded He atom in the case of the field strength 0 ≤F≤ 0.1 a.u. The influences of plasma screening and applied electric field strength, separately and combined, on the energies and radiative transition properties are analyzed in detail. Our results reveal that adding shielding effect destabilizes the atomic system (reduces the energy eigenvalues), and the applied electric field counteracts this effect, thereby increasing the stability of the atomic system. On the whole, the effect of electric field on the electronic structures and spectral properties of the guest atom is a deflection shielding effect, implying that the latter effect is stronger. The present results are in good agreement with the values from the previous literature.

Ionization potential depression in dense iron plasmas near solid density

In a dense hot plasma, the energy levels of an embedded ion are lowered by the induced screening potential, which is termed as the ionization potential depressions (IPDs) for the continuum levels. Recent experiments on the precision measurements of the IPDs in solid-density plasmas made it feasible to validate the widely used models. Yet, to date, most of the renewed experimental and theoretical verifications were focused on the low Z dense plasmas. In this work, the IPDs of heavier Fe plasmas were studied near solid density using a recently developed self-consistent temperature dependent ion-sphere model. The present results show reasonable agreement with the b4-potential method at some densities and temperatures, yet there exist large discrepancies between our predicted IPDs and those by the widely used simple closed analytical models. Namely, for Fe plasmas in a density range of 1–16 g cm−3 and a temperature range of 100–1000 eV, a simplified modification of Ecker–Kröll model predicts a much higher IPD than our model, whereas the Stewart–Pyatt, the ion-sphere, and the Debye–Hückel models predict much lower IPDs, which indicates the need of further study on the plasma screening potential. The results obtained in this work should be helpful for the astrophysical and inertial-fusion applications where the IPDs are vital parameters in determining the ionization balance.

Comment on “Stark shift and width of x-ray lines from highly charged ions in dense plasmas”

We point out in this Comment a misleading remark made by Gu and Beiersdorfer [Phys. Rev. A 101, 032501 (2020)] that the Debye-H\"uckel (DH) model ``performs rather poorly'' compared with the ion sphere model is due to their questionable use of the Debye length by assuming the same mobility for the plasma electrons and ions. Actually, our recent works based on a judicious application of the DH approximation, which meets the spatial and temporal criteria, have led to good agreements with all the available line shift experimental measurements on $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ emission lines of He-like ions.

Satellite lines of helium-like ions in strongly coupled plasma environment

We report the transition energies and probabilities of satellite lines due to 2p2(3P0,1,2)→1s2p(3P0,1,2) transitions of He-like C,Ne,Al and Ar ions embedded in strongly coupled plasma (SCP) environment. The SCP environment is represented by the ion-sphere (IS) model and the IS radius is varied in such a way that plasma electron densities (∼1020−1026 per cm3) cover a wide range of experimental interest. Both relativistic and non-relativistic energies of 2p2(3P) and 1s2p(3P) states are reported. The relativistic MCDF results are obtained by modified GRASP2K code and the non-relativistic variational results are done by using explicitly correlated Hylleraas basis. The relativistic results are in good agreement with experimental data available in literature. The positions of intercombination lines in presence of dense plasma can be calculated from present results which are comparable with experimental measurements. The partial pressures experienced by the plasma-embedded ions are analyzed in the purview of available experimental observations. An empirical relation is reported to estimate partial pressure on ions in dense plasma environment.

Effects of plasma screening on radiative transition and photoionization of Si[formula omitted]–Si[formula omitted] in a dense plasma environment

The screening induced by a dense plasma environment is crucially important for determining radiative atomic data including bound–bound radiative decay rates and photoionization cross sections, which are vital for obtaining accurate opacity. Previous experimental investigations of ionization potential depression have shown that the widely used analytical models of screening struggle to capture the essential physics in the solid-density plasma. In the present work, we investigate how plasma screening affects the radiative atomic data of Si10+–Si13+ embedded in a dense plasma environment using a self-consistent field finite temperature ion sphere model. The single-electron effective potential of an embedded atom has a short-range feature that determines the available number of bound states. Plasma screening is effective in modifying the matrix elements of the radiative dipole transitions, thereby generally increasing the photoionization cross section near the ionization threshold and decreasing the bound–bound radiative decay rates. The enhancement of the photoionization processes stems from lowering the ionization threshold and the shape and virtual-state resonances, where the upper bound levels become delocalized after a certain critical screening strength. With increasing plasma density, the Rydberg series for the bound–bound radiative transitions disappear successively.

Strongly coupled plasma effect on excitation energies of O-like ions and photoionization of F-like ions

The main goal of the present study is to analyze and estimate the plasma screening effect on excitation energies O-like ions and photoionization process of F-like ions under the influence of strongly coupled plasma. We have employed an ion sphere model (ISM) in flexible atomic code (FAC) to study and analyze plasma effect in atomic structure of O-like ions and photoionization of F-like ions. We have presented shift in excitation energies of lowest 10 levels of Fe XIX, Co XX, Ni XXI, Cu XXII and Zn XXIII at electron densities ranges 1022–1024 cm−3 under plasma environment. We have also compared our transition wavelengths for Fe XIX with wavelengths observed in ENEA laser facility and calculated from Hebrew University Lawrence Livermore Atomic Code (HULLAC) code at electron density 1021 cm−3. The lowering in ionization potentials and effect on photoionization cross sections for ground and first excited states of Fe XVIII, Co XIX, Ni XX, Cu XXI and Zn XXII also studied at electron densities ranges 1022–1024 cm−3 at five photo-electron energies ranges 100–500 eV.

Shannon, Rényi entropies, and Fisher information calculations of the Li1+ and Be2+ ions screened by the ion-sphere plasma model

We show results for the Shannon entropy, Rényi entropy, Fisher information, and their relative relations in the position space for the Li1+ and Be2+ ions embedded in a strong plasma. This medium is simulated by the ion-sphere screening model as a function of the plasma electron density, N e . We solve the Hartree–Fock equations using a numerical grid method in a self-consisting manner and the simple Hylleraas-type correlated wave function is applied. Our findings show a reduction of the total energy for both ions as the electron density is increased reaching the ionization limit. We find interesting features in the description of the Shannon entropy and Fisher information where the correlation effects of the embedded ions are relevant. The Shannon-Fisher product results indicate that the electron-electron correlation decreases as N e increases and similar behavior is found for the Rényi-Fisher product. Finally, in the absence of the plasma medium, our results are in excellent agreement with those available in the literature.

Transition parameters of Li-like ions (Z = 7 – 11) in dense plasmas* *Project supported by the National Natural Science Foundation of China (Grant No. 11847163), the Natural Science Foundation of Gansu Province (Grant No. 20JR10RA131), the Doctoral Science Foundation of Longdong University (Grant No. XYBY202005), the Natural Science Foundation of Shaanxi Province (Grant No. 2021JQ-698), and the Special Project

The energy levels, transition energies, transition probabilities, weighted oscillator strengths, and line strengths of Li-like ions (Z = 7–11) in dense plasmas are investigated in this work. The relativistic effects and electron correlation effects are described by the MCDHF method. The ion sphere model is applied to include the dense plasma screening effect. The ground configuration 1s22s and the excited 1s22p, 1s23l (l = 0–2) are considered. The configuration sets are enlarged until n = 7 where the calculated energy levels have converged. The critical free electron densities of 1s23d states are estimated. Except for 1s23s–1s23p transitions, the transition energies for Δ n = 0 increase, and for Δ n ≠ 0 decrease with increasing free electron densities. For 1s23s–1s23p transitions, the spectra show blue-shift at lower free electron densities and red-shift at higher free electron densities, and the energy level crossing phenomens are observed at higher free electron densities.

Atomic structure and radiative properties of He-like Ni26+ ion in dense plasma

The influence of plasma screening on the excitation energies and transition properties of He-like Ni26+ ion under strongly coupled plasma background has been analyzed. To perform the analysis, the multi-configuration Dirac–Fock method has been adopted by incorporating the ion sphere model potential as a modified interaction potential between the electron and the nucleus. For comparison purposes, parallel calculations have been carried out using the modified relativistic configuration interaction method. It is found that the transition energies corresponding to principal quantum number conserving transitions (Δn = 0) are blue shifted, whereas they are redshifted for the transitions where the principal quantum number is not conserved (Δn ≠ 0). The variation of transition probabilities and weighted oscillator strengths with free electron densities has also been studied. The present results should be advantageous in the modeling and diagnostics of astrophysical and laboratory plasmas.

Ionization potential depression and ionization balance in dense carbon plasma under solar and stellar interior conditions

Recent quantitative experiments on the ionization potential depression (IPD) in dense plasma show that the observational results are difficult to explain with the widely used analytical models for plasma screening. Here, we investigate the effect of plasma screening on the IPD and ionization balance of dense carbon plasma under solar and stellar interior conditions using our developed consistent nonanalytical model. The screening potential can be primarily attributed to the free electrons in the plasma and is determined by the microspace distribution of these free electrons. The ionization balance is determined by solving the Saha equation, including the effect of IPD. The predicted IPD and average ionization degree are larger than those obtained using the Stewart–Pyatt model for mass densities that are greater than 3.0 g cm−3. Under solar interior conditions, our results are in better agreement with the Ecker–Kröll model at electron temperatures and densities lower than 250 eV and 2.1 × 1023 cm−3 and in the best agreement with the ion-sphere model at 303 eV and 4.3 × 1023 cm−3. Finally, our results are compared with those obtained via a recent experiment on a CH-mixture plasma that has been compressed six times. The predicted average ionization degree of C in a CH mixture agrees better with the experiment than the Stewart–Pyatt and Thomas–Fermi models when the screening from free electrons contributed by hydrogen atoms is included. Our results provide useful information concerning the ionization balance and can be applied to investigate the opacity and equations of state for dense plasma under the solar and stellar interior conditions.

Neutron star crust in Voigt approximation: general symmetry of the stress–strain tensor and an universal estimate for the effective shear modulus

ABSTRACT I discuss elastic properties of neutron star crust in the framework of static Coulomb solid model when atomic nuclei are treated as non-vibrating point charges; electron screening is neglected. The results are also applicable for solidified white dwarf cores and other materials, which can be modelled as Coulomb solids (dusty plasma, trapped ions, etc.). I demonstrate that the Coulomb part of the stress–strain tensor has additional symmetry: contraction Bijil = 0. It does not depend on the structure (crystalline or amorphous) and composition. I show as a result of this symmetry the effective (Voigt averaged) shear modulus of the polycrystalline or amorphous matter to be equal to −2/15 of the Coulomb (Madelung) energy density at undeformed state. This result is general and exact within the model applied. Since the linear mixing rule and the ion sphere model are used, I can suggest a simple universal estimate for the effective shear modulus: $\sum _Z 0.12\, n_Z Z^{5/3}e^2 /a_\mathrm{e}$. Here summation is taken over ion species, nZ is number density of ions with charge Ze. Finally, ae = (4πne/3)−1/3 is electron sphere radius. Quasi-neutrality condition ne = ∑ZZnZ is assumed.

Analysis of circular polarization of the emitted photon in plasmas

In this paper, a new relativistic distorted wave approximation for cross section describing longitudinally-polarized electron impact excitation of ions in the presence of a strongly coupled plasma is proposed and implemented based on the flexible atomic code. The uniform electron gas model is used to incorporate the plasma effects, which includes the effect of static plasma screening on nuclear charge and the effect of confinement due to the neighbouring ions. For the bound and continuum wavefunctions, the Dirac equation with the ion-sphere potential is solved to account for relativistic effects. In order to check the consistency of the modifications, a self consistent theoretical attempt using the multiconfiguration Dirac-Fock method is also carried out to get the energy levels accounting for the plasma environment within the ion-sphere model. For the collision dynamics, as examples of applications, we investigate the total cross sections, the magnetic sublevels cross sections, and the degrees of circular polarizations of fluorescence radiation after 1s2s01→1s2ss03 and 1s2s01→1s2pp13,1 excitations of Helium-like Fe XXV ion by longitudinally-polarized electron impact with the ion-sphere potential. It is shown that the behavior of the cross sections for a pure Coulomb potential changes rapidly due to the plasma screening effects described by the ion-sphere potential in the process of calculating scattering matrix elements. A detailed, quantitative, comparison between the present results and other available theoretical values is presented for the plasma-free cases, showing good agreement. Our results contribute to a better understanding of the fundamentals of fluorescence polarization in strongly coupled plasmas. Results obtained in this work are beneficial for the diagnostic determination of laboratory plasmas.

Analytical approach to level delocalization and line shifts in finite temperature dense plasmas

X-ray line shifts and delocalization of levels of He-like ions are studied within the framework of the self-consistent multi-configuration relativistic finite temperature ion sphere model. A set of analytical equations for an N-electron bound atomic system to determine X-ray line shifts and level delocalization is proposed that provides very good agreement with numerical calculations, newly high precision line shift measurements of the 1s3pP11−1s2S01Heβ transition in Cl15+ and the 1s3pP11-level delocalization in Al11+. We re-analyzed the recently claimed discrepancy between the predictions from the ion sphere model, analytical approaches and the data. The analytical b-potential approach that imitates lattice effects via a perturbation of the free electron distribution in the in sphere is developed and contrasted with currently unexplained line shift measurements of the 1s2pP11−1s2S01Heα transition in Al11+.

Effect of Dense Plasma Environment on the Spectroscopic Properties of He-like Ca18+ Ion

Plasma shielding effects on the energy levels and radiative properties of He-like Ca\(^{18+}\) ion under dense plasma conditions have been studied. For this purpose, the multiconfiguration Dirac-Fock method has been implemented by incorporating the ion sphere model potential as a modified interaction potential between the electron and the nucleus. To check the authenticity of our results, independent calculations have been performed using the modified relativistic configuration interaction method. It is observed that the transition energies associated with \(\Delta n = 0\) transitions are blue shifted, whereas red shifted for \(\Delta n \neq 0\) transitions. The variation in transition probabilities and weighted oscillator strengths with plasma densities has also been analyzed. The present results should be beneficial in the plasma modeling and fusion plasma research applications.

Calculation of the energies and oscillator strengths of Cl[formula omitted] in hot dense plasmas

The high-resolution measurements of Beiersdorfer et al. [Phys. Rev. A 100, 012511 (2019)] for the shifts of the 1s3p1P1 → 1s2 1S0 He-β transition of Cl15+ are in significant disagreement with a generalized analytical approach calculations of Li and Rosmej [Europhys. Lett. 99, 33001 (2012)]. The multi-configuration Dirac–Fock method incorporating the average-atom ion-sphere model is applied to the problem. Our results are in good agreement with the experimental data across the full range of electron density measured. Oscillator strengths are also given. Systematic trends are observed for the properties under study regarding to increased temperature and electron density.

Influence of plasma screening on the energies and transition properties of the Al9+ ion using Debye and ion-sphere models

Investigations on the energies and radiative properties of a highly charged ion embedded in plasma in the weak- and strong-coupling regimes are made, taking the Al9+ ion as an example. The Debye and ion-sphere (IS) potentials are adopted to describe the plasma screening effects, respectively. The multiconfiguration Dirac-Fock (MCDF) method is employed to describe the relativistic and electronic correlation effects in the above atomic system. The screening effects on the ionization potential (IP), transition energies, and radiative rates are estimated. Results obtained are compared with the available results in the literature. Comparisons between the Debye and IS models results are also carried out considering similar plasma conditions. The present data are useful for diagnostics of astrophysical and laboratory plasmas.