Issue 
A&A
Volume 634, February 2020



Article Number  A117  
Number of page(s)  6  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/201937235  
Published online  19 February 2020 
Screening potential and continuum lowering in a dense plasma under solarinterior conditions^{⋆}
^{1}
College of Science, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, PR China
email: jiaolongzeng@hotmail.com
^{2}
Graduate School of China Academy of Engineering Physics, Beijing 100193, PR China
email: jmyuan@nudt.edu.cn
^{3}
Department of Physics, College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, Hunan 410073, PR China
Received:
3
December
2019
Accepted:
9
January
2020
An accurate description of the screening potential induced by a hot, dense plasma is a fundamental problem in atomic physics and plasma physics, and it plays a pivotal role in the investigation of microscopic atomic processes and the determination of macroscopic physical properties, such as opacities and equations of state as well as nuclear fusion cross sections. Recent experimental studies show that currently available analytical models of plasma screening have difficulty in accurately describing the ionizationpotential depression, which is directly determined by the screening potential. Here, we propose a consistent approach to determine the screening potential in dense plasmas under solarinterior conditions from the freeelectron microspace distribution. It is assumed that the screening potential for an ion embedded in a dense plasma is predominately determined by the free electrons in the plasma. The freeelectron density is obtained by solving the ionizationequilibrium equation for an averageatom model to obtain the average degree of ionization of the plasma. The proposed model was validated by comparing the theoretically predicted ionizationpotential depression of a soliddensity Si plasma with recent experiments. Our approach was applied to investigate the screening potential and ionizationpotential depression of Si plasmas under solarinterior conditions over a temperature range of 150–500 eV and an electrondensity range of 5.88 × 10^{22}–3.25 × 10^{24} cm^{−3}. It can be easily incorporated into atomicstructure codes and used to investigate basic atomic processes, such as photoionization, electronion collisional excitation and ionization, and Auger decay, in a dense plasma.
Key words: atomic data / atomic processes / Sun: interior / stars: interiors / dense matter
Tables of the numerical data used for the figures are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/cat/J/A+A/634/A117
© ESO 2020
1. Introduction
The Sun is a prototypical star that can be used for investigating stellar processes and for testing theories of stellar structure and evolution. In the solar interior, the plasma is dense, and stellar processes are thus significantly influenced by plasma screening. Although the solar interior is not directly accessible to observations, plasma screening shows its effects in astrophysical phenomena, such as solar oscillations (helioseismology) (Antia 2009) because it affects the opacity and equation of state. The screening potential induced by the plasma environment plays pivotal roles in both microscopic atomic processes and macroscopic physical properties (Basu & Antia 2008; Bahcall et al. 2004). Microscopically, it affects the atomic structure and all kinds of radiative and collisional atomic processes, such as photoexcitation and photoionization, as well as electronimpact excitation and the ionization of ions (Vinko et al. 2015; van den Berg et al. 2018). Macroscopically, the ionization balance, and hence, the physical properties of equations of state, opacities, and thermal and electronic conductivities can be significantly modified by the dense plasma environment (Bailey et al. 2015; Rogers & Iglesias 1994; Iglesias & Rogers 1991; Rose 1992; Hansen et al. 2005; Vinko et al. 2012). Moreover, in the dense plasma, the nuclear fusion cross section and rate can be significantly enhanced over the binary Gamow (1928) results because of the plasma screening effect (Yakovlev & Shalybkov 1989; Ichimaru 1993); therefore, they can affect the stellar evolution.
One key physical quantity is continuum lowering, which is also referred to as ionizationpotential depression (IPD) and is directly determined by the plasma screening potential. Recent technological advances have made it possible to measure the IPD of dense plasmas directly (Ciricosta et al. 2012, 2016; Hoarty et al. 2013; Fletcher et al. 2014). However, the predictions of widely used analytical models (Stewart & Pyatt 1966; Ecker & Kröll 1963; Rozsnyai 1972; Debye & Hückel 1923) disagree with the experimental measurements. The first experiment (Ciricosta et al. 2012), which was performed at the Linac Coherent Light Source (LCLS), shows that the EckerKröll model (Ecker & Kröll 1963) is in much better agreement with the observations than the StewartPyatt model (Stewart & Pyatt 1966). However, laserdriven shock experiments at the ORION laser system are in better agreement with the StewartPyatt model (Stewart & Pyatt 1966) than the EckerKröll model (Ecker & Kröll 1963). Moreover, a later experiment at LCLS shows that the predictions of the EckerKröll model for the IPD of Mg and Al compounds disagree with the experimental results (Ciricosta et al. 2016). Comparisons of theoretical predictions with recent experiments have further revealed that the analytical models (Stewart & Pyatt 1966; Ecker & Kröll 1963; Rozsnyai 1972; Debye & Hückel 1923) have difficulty in capturing the essential features of hot, dense plasmas at solid and abovesolid densities. This lack of a consistent theoretical formulation for the IPD of hot dense plasmas has stimulated a variety of theoretical investigations (Preston et al. 2013; Iglesias 2014; Hansen et al. 2014; Crowley 2014; Son et al. 2014; Calisti et al. 2015; Vinko et al. 2014; Stransky 2016; Lin et al. 2017; Rosmej 2018; Kasim et al. 2018; Ali et al. 2018; Hu 2017; Röpke et al. 2019; Kraus et al. 2018).
In the solar interior, the plasma is dense, and has an electron density above 1.0 × 10^{22} cm^{−3}, and thus the plasma screening potential strongly influences the ionization balance and atomic processes. Physical properties, such as the opacities and equations of state, play important roles in the determination of the internal solar structure. However, the current standard solar model (Bahcall & Ulrich 1988; Guenther et al. 1992) disagrees with helioseismic studies (Bahcall et al. 2004; Basu & Antia 2008) that employ the most updated Rosseland mean opacity for a solar mixture. This open problem of the “missing opacity” in the solar interior can be resolved if we increase the opacity. Accurate calculations of the radiative opacity require a consistent description of the plasma screening potential. However, atomic data that take the density effect into account are extremely lacking for dense plasmas under solarinterior conditions. Very recently, Deprince et al. have investigated plasmaenvironment effects on the K lines of oxygen (Deprince et al. 2019a) and iron (Deprince et al. 2019b) ions in accretion disks around black holes. Magnetohydrodynamic simulations show that the electron temperatures and densities span the ranges of 10^{5}–10^{7} K and 10^{18}–10^{22} cm^{−3}, respectively (Schnittman et al. 2013). The electron density in the accretion disk of a black hole is much lower than that in the solar interior, and Deprince et al. used the DebyeHückel model to describe the plasmascreening effect. Such an analytical model is not valid for solar and stellarinterior plasmas, as mentioned above. There is thus an urgent need to improve the treatment of density effects in higher density plasmas.
In this work, we aim to develop a consistent theoretical formulation of the plasma screening potential for the dense plasmas under solarinterior conditions. The screening potential can be further incorporated into an atomicstructure code to investigate microscopic atomic processes, including the excitation and ionization caused by photons and electrons. As an application of our theory, we investigate the plasma screening potential and IPD of Si plasmas under solarinterior conditions. It is thought that the atomic processes that occur in the dense environment of the solar interior may show interesting and novel phenomena, which may help to resolve the current discrepancy between the internal solar structure predicted by the standard solar model (Bahcall & Ulrich 1988; Guenther et al. 1992) and the constraints from helioseismic observations (Bahcall et al. 2004; Basu & Antia 2008).
2. Theoretical method
For an isolated atom and ion, the wave functions and atomic structure are determined by solving quantum mechanical equations [see for example (Jönsson et al. 2007; Gu 2008)] based on the Hamiltonian for which atomic units are used unless otherwise specified
where H_{D}(i) is the singleelectron Dirac Hamiltonian for the potential due to the nuclear charge and N denotes the number of electrons. The basis states of the atomic system are constructed from products of N oneelectron Dirac spinors,
where P_{nκ}(r) and Q_{nκ}(r) are the large and small components of the radial wave functions, respectively, and χ_{κm}(θ, ψ, σ) is a twocomponent spherical spinor. The indices n, κ, and m denote the principal, relativistic angular, and magnetic quantum numbers of the orbital, respectively.
In the centralpotential approximation, the large and small components of the radial wave function satisfy coupled Dirac equations in the standard DiracFockSlater method:
where α denotes the finestructure constant and ε_{nκ} is the energy eigenvalue of the electron orbital. The local central potential V(r) includes contributions from the nuclear charge and from electronelectron interactions (Jönsson et al. 2007; Gu 2008), which can be determined by selfconsistent iteration.
In a dense plasma characterized by an electron temperature T and density n_{e}, an electron that is bound to an ion feels a screening potential induced by the plasma environment (Li et al. 2008),
where Z is the nuclear charge and the radius of the ion sphere is determined by the ion number density: . The last term in Eq. (4) represents the Slater exchange potential. For r > R_{0}, the potential is taken to have the constant value as it reaches r = R_{0} (Son et al. 2014). The freeelectron charge density ρ(r) obeys the FermiDirac distribution
where c denotes the speed of light in vacuum, k is the momentum of the electron, , and μ denotes the chemical potential of the plasma.
The chemical potential is determined by the condition of electrical neutrality:
where Z_{f} denotes the average degree of ionization, which is determined by the ionization balance of the plasma. This is usually obtained by solving the Saha ionizationequilibrium equation (Gao et al. 2013) for plasmas in local thermal equilibrium. In this code, the effect of IPD is considered when solving the Saha equation. When the ionization potential depression is properly considered, as in this work, this code correctly predicts the hydrogen ionization fraction in the solar interior (ChristensenDalsgaard & Döppen 1992). Here, we used the averageatom model instead to determine the average degree of ionization and the chemical potential by predicting the average orbital properties and occupation numbers at a finite temperature (Yuan 2002). We used the finitetemperature exchangecorrelation functionals of DharmaWardana and Taylor (Dharmawardana & Taylor 1981) to construct the selfconsistent potential of the averageatom.
Once the radial wave functions are obtained, the energylevel structure of an isolated and screened ion can be determined, and then the atomic processes initiated by photons and electrons can be investigated. In this work, we focus mainly on the IPD because this physical quantity has been satisfactorily measured for soliddensity plasmas. In order to calculate the IPD, it is necessary to solve the Dirac equation both for the isolated and screened ions. New functions that incorporate the screening potential into the atomicstructure calculation have also been implemented based on the flexible atomic code (Gu 2008).
3. Results and discussion
Plasma screening effects can be reflected in atomic processes, such as electronion collisional excitation and ionization, photoionization, and Auger decay. Both the reaction thresholds and the crosssections can be affected. However, it is difficult to separate a particular pathway from the myriad transitions that occur in a dense plasma, and thus experimental investigations of these processes are challenging. However, it is possible to measure physical properties, such as the radiative opacity for a dense plasma as a whole (Bailey et al. 2015). Advances in Xray laser technology have now made it possible to measure the IPDs of dense plasmas quantitatively both for pure elements and for mixtures (Ciricosta et al. 2012, 2016; Fletcher et al. 2014). Ultraintense Xray radiation is used to heat soliddensity matter to the hotdense regime. By increasing the photon energy of the laser across the Kshell ionization threshold, the Kedge, and thus the continuum lowering, can be measured by observing the onset of fluorescence emitted from the ions. We thus utilized the IPD to check the validity of our proposed model for the plasma screening potential.
In Fig. 1, we compare our theoretical IPDs for a soliddensity Si plasma with the experimental measurements of Ciricosta et al. (2016). In their experiment, the laser photon energy was varied over the range 1290–1985 eV to open the Kedges of the ions. By observing the onset of the K_{α} fluorescence, they were able to measure the IPD for increasing charge states by increasing the photon energy. They determined the electron temperature and density of the plasma by solving the timedependent rate equations that take into account the processes involved in the interaction between the Xrays and the Si plasma; detailed information can be found in their work (Ciricosta et al. 2016). They also give the theoretical results obtained from the analytical models of Ecker & Kröll (Ecker & Kröll 1963), the ion sphere (Rozsnyai 1972), and Stewart & Pyatt (Stewart & Pyatt 1966) in order to evaluate their predictive capabilities in dense plasmas. It is well known that the DebyeHückel model (Debye & Hückel 1923) is only valid for plasmas of lower densities, and thus they are not given here.
Fig. 1. Comparison of IPDs for soliddensity Si plasmas predicted by our model with the experimental results (Expt) (Ciricosta et al. 2016). Theoretical results from the analytical models of Ecker & Kröll (EK) (Ecker & Kröll 1963), the ionsphere model (IS) (Rozsnyai 1972), and Stewart & Pyatt (SP) (Stewart & Pyatt 1966) are given as well in order to evaluate their validity. The electron temperatures and densities are taken from a simulation (Ciricosta et al. 2016). 

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By inspecting Fig. 1, we find that our predictive IPDs are in good agreement with the experimental values for the lower ionization stages of Si^{4+}–Si^{6+}, although our theoretical result is somewhat smaller than the experimental value for Si^{7+}. There are several possible reasons for the discrepancy between theory and experiment for the higher ionization stages. First, the higher charge states are produced at a later time during the interaction with the Xray laser, and the energy of the photons is only slightly above the ionization threshold of Si^{7+}. Thus the energies of the free electrons around Si^{7+} that are produced by the Xray laser are very low. The highly ionized ion of Si^{7+} is thus surrounded by relatively “colder” free electrons, which changes the screening from the plasma. Second, the free electrons are not uniformly distributed in space; more free electrons follow the more highly ionized ions in order to maintain electrical neutrality. Finally, the plasma produced by the Xray laser deviates considerably from thermal equilibrium, which also influences the screening.
Among the analytical models, the one of EckerKröll (Ecker & Kröll 1963) seems to be in much better agreement with the measurements than the ionsphere (Rozsnyai 1972) and StewartPyatt (Stewart & Pyatt 1966) models. But it is known that the EckerKröll model overestimates the IPD for the higher ionization stages (Hoarty et al. 2013), as in the present case. For dense plasmas, the analytical ionsphere (Rozsnyai 1972) and StewartPyatt (Stewart & Pyatt 1966) models have been widely used in various fields to compute the basic atomic processes and physical properties of dense plasmas. They have been extensively applied to investigate atomic structure (Janev et al. 2016), photoionization (Qi et al. 2009), electronimpact excitation and ionization (Zhang et al. 2010; Li et al. 2017), and momentumtransfer crosssections (Khrapak et al. 2003). Experimental elastic Xray scattering suggests that plasma screening is important for the determination of the denseplasma static structure factor (Saiz et al. 2008). The analytical models have also been used to calculate opacities and equations of state (Bailey et al. 2015; Rogers & Iglesias 1994; Iglesias & Rogers 1991; Rose 1992; Hansen et al. 2005; Vinko et al. 2012; Porcherot et al. 2011; Blancard et al. 2011; Zeng & Yuan 2006, 2007; Gao et al. 2011). Moreover, the IPDs are used to diagnose plasma conditions, such as the density or temperature. Therefore an accurate description of the IPD is very important for dense plasmas. From the comparison shown in Fig. 1, we are confident that our model of plasma screening is valid for hot, dense plasmas.
Having validated our model by utilizing the IPD, we then used it to investigate the plasma screening potential for dense plasmas under solarinterior conditions. By using Si as an example, we show the screening potential produced by the plasma as a function of the distance from the center of the involved ion in Fig. 2. The electron temperature and density are taken from the standard solar model (Bahcall & Ulrich 1988; Guenther et al. 1992). The highest temperature and density are 496.42 eV and 3.25 × 10^{24} cm^{−3}, respectively, and they correspond to the plasma conditions at 0.3529 R_{⊙} (where R_{⊙} is the solar radius), whereas the lowest values, 156.90 eV and 5.88 × 10^{22} cm^{−3}, are the conditions at 0.7523 R_{⊙}. We note that the electron temperature and density at the solar radiation and convection zone boundary, which is located at 0.7133 R_{⊙} (Bahcall et al. 1982), are determined to be 189.35 eV and 8.08 × 10^{22} cm^{−3}.
Fig. 2. Screening potential V_{scr}(r) in Si plasmas under solarinterior conditions. The temperature and electrondensity (T/n_{e}) values for the Si plasmas are taken from the standard solar model (Bahcall & Ulrich 1988; Guenther et al. 1992): 156.90 eV/5.88 × 10^{22} cm^{−3} (black longdasheddotted line), 189.00 eV/8.0 × 10^{22} cm^{−3} (black shortdasheddotted line), 205.12 eV/9.89 × 10^{22} cm^{−3} (black longdashed line), 227.53 eV/1.36 × 10^{23} cm^{−3} (black shortdashed line), 244.94 eV/1.7 × 10^{23} cm^{−3} (black dotted line), 275.80 eV/2.76 × 10^{23} cm^{−3} (violet dasheddasheddotted line), 304.23 eV/4.06 × 10^{23} cm^{−3} (turquoise dasheddotteddotted line), 334.40 eV/6.12 × 10^{23} cm^{−3} (brown longdasheddotted line), 373.18 eV/9.79 × 10^{23} cm^{−3} (yellow shortdasheddotted line), 407.65 eV/1.43 × 10^{24} cm^{−3} (blue longdashed line), 438.70 eV/1.96 × 10^{24} cm^{−3} (green shortdashed line), 468.84 eV/2.57 × 10^{24} cm^{−3} (red dotted line), and 496.42 eV/3.25 × 10^{24} cm^{−3} (black solid line). 

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Inspection of Fig. 2 reveals features of the additional potential exerted by the plasma environment on the involved Si ion. First, at any given temperature and density, the potential induced by the environment increases rapidly from a negative value at a distance very near the nucleus of an ion to a positive maximum and then gradually decreases with increasing distance from the ionic center. The negative potential that is very close to the nucleus means that the contributions from the exchange interaction exceed the Coulombic interaction, which is completely an effect of quantum mechanics. Thus plasma screening results in an increase in the energies of bound states and a depression of the ionization threshold. Secondly, at lower electron densities, the induced potential decreases very slowly with r. The induced potential becomes larger and the decreasing trend becomes stronger with increasing electron density, which means stronger screening for a denser plasma. Finally, the induced potential does not increase linearly with electron density; the temperature of the plasma also has an effect on it. At the lowest density that we consider, 5.88 × 10^{22} cm^{−3} (T_{e} = 156.90 eV), the peak value of the induced potential is 2.66 a.u. This peak value increases by factors of 1.11 and 3.20 at the densities of 4.06 × 10^{23} cm^{−3} (304.23 eV) and 3.25 × 10^{24} cm^{−3} (496.42 eV), respectively.
Figure 3 shows the calculated IPDs for Si plasmas under the same conditions as in Fig. 2. According to the calculations of ionization equilibrium determined by the DLAYZ code (Gao et al. 2013), the dominant ionization stages are predicted to be Si^{8+}–Si^{13+} under the given plasma conditions. The DLAYZ code was developed to investigate the population kinetics and radiative properties of plasmas by solving the Saha ionization equilibrium equation or coupled rate equations. For dense plasmas, the plasma screening effect is included in the calculations of the basic atomic data. The first ionization potentials are 351.28 eV, 401.38 eV, 476.27 eV, 523.42 eV, 2437.66 eV, and 2673.18 eV for isolated ions of Si^{8+}–Si^{13+}, respectively (Kramida et al. 2019). However, if these ions are immersed in a denseplasma environment, the ionization potential is significantly decreased. For example, at the electron temperature and density of 496.42 eV and 3.25 × 10^{24} cm^{−3}, the ionization potentials of Si^{8+}–Si^{13+} are lowered to 250.22 eV, 284.91 eV, 316.27 eV, 348.32 eV, 379.62 eV, and 413.43 eV by the plasma environment. These are significant decreases in the ionization potentials, which will evidently modify the wave functions of the continuum states and thus affect photoionization and electronion collisional ionization processes. For a plasma at a given temperature and density, the IPDs increase nearly linearly with an increasing ionization stage, particularly at lower densities. Only when arriving at some critical plasma condition is such a linear dependence on the ionization stage slightly modified. For example, at the density of 4.06 × 10^{23} cm^{−3} (304.23 eV), the IPD of Si^{9+} deviates slightly from a linear relation. This is because the bound 4s orbital predicted at an electron density lower than 4.06 × 10^{23} cm^{−3} begins to dissolve into continuum states at higher densities. The lines representing the IPDs become steeper with increasing density. At a density of 5.88 × 10^{22} cm^{−3} (156.90 eV), the IPDs for Si^{6+} and Si^{13+} are 52.0 eV and 113.6 eV, respectively. The corresponding values are 187.6 eV and 413.4 eV at 3.25 × 10^{24} cm^{−3} (496.42 eV), which increase 135.6 eV and 299.8 eV, respectively. The slopes of the IPD lines are thus 8.8 and 32.3 at the lowest and highest densities given, respectively.
Fig. 3. Ionizationpotential depressions of Si plasmas at the electron densities of 5.88 × 10^{22} cm^{−3} (black longdasheddotted line), 8.0 × 10^{22} cm^{−3} (black shortdasheddotted line), 9.89 × 10^{22} cm^{−3} (black longdashed line), 1.36 × 10^{23} cm^{−3} (black shortdashed line), 1.7 × 10^{23} cm^{−3} (black dotted line), 2.76 × 10^{23} cm^{−3} (violet dasheddasheddotted line), 4.06 × 10^{23} cm^{−3} (turquoise dasheddotteddotted line), 6.12 × 10^{23} cm^{−3} (brown longdasheddotted line), 9.79 × 10^{23} cm^{−3} (yello shortdasheddotted line), 1.43 × 10^{24} cm^{−3} (blue longdashed line), 1.96 × 10^{24} cm^{−3} (green shortdashed line), 2.57 × 10^{24} cm^{−3} (red dotted line), and 3.25 × 10^{24} cm^{−3} (black solid line). They are taken from the standard solar model (Bahcall & Ulrich 1988; Guenther et al. 1992). 

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In Fig. 4 we compare our predicted IPDs with those from the analytical models of Ecker & Kröll (Ecker & Kröll 1963), the ion sphere (Rozsnyai 1972), Stewart & Pyatt (Stewart & Pyatt 1966), and Debye & Hückel (Debye & Hückel 1923) at a few selected electron temperatures and densities: 156.90 eV and 5.88 × 10^{22} cm^{−3}, 227.53 eV and 1.36 × 10^{23} cm^{−3}, 407.65 eV and 1.43 × 10^{24} cm^{−3}, and 496.42 eV and 3.25 × 10^{24} cm^{−3}. We find that the EckerKröll model (Ecker & Kröll 1963) always predicts stronger screening and thus higher IPDs than our nonanalytical model, while the DebyeHückel model (Debye & Hückel 1923) always predicts weaker screening and lower IPDs than our model. The IPDs predicted by the ion sphere (Rozsnyai 1972) and StewartPyatt (Stewart & Pyatt 1966) models are very similar to our results for the lower ionization stages, but they deviate from our results for the increasing ionization stages; the higher the ionization stage, the larger the deviation. In addition, the results obtained from the ionsphere (Rozsnyai 1972) and StewartPyatt (Stewart & Pyatt 1966) models agree reasonably well with each other for the lower ionization stages (less than nine), but the differences become larger for more highly charged states.
Fig. 4. Comparison of our theoretical ionizationpotential depression with the analytical models of DebyeHückel (DH) (Debye & Hückel 1923), StewartPyatt (SP) (Stewart & Pyatt 1966), ion sphere (IS) (Rozsnyai 1972), and EckerKröll (EK) (Ecker & Kröll 1963) for a few selected plasma conditions of (a) 156.90 eV/5.88 × 10^{22} cm^{−3}, (b) 227.53 eV/1.36 × 10^{23} cm^{−3}, (c) 407.65 eV/1.43 × 10^{24} cm^{−3}, and (d) 496.42 eV/3.25 × 10^{24} cm^{−3}. 

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An accurate description of the plasma screening potential is of crucial importance for investigations of the atomic processes in dense plasmas (Liu et al. 2018). Recent experimental investigations on electronimpact ionization show that the crosssections and rates are enhanced by denseplasma effects and cannot be explained by currently available atomic collision theory (Vinko et al. 2015; van den Berg et al. 2018). Photoionization crosssections are also increased, and thus the radiative opacity is increased as well for hot, dense plasmas (Bailey et al. 2015). The IPD is partially responsible for the enhancement of these continuum atomic processes.
4. Summary and conclusions
In this work, we propose a theoretical formalism for investigating the plasma screening potential of dense plasmas under solarinterior conditions. This screening is determined by the freeelectron microspace distribution obtained from ionization balance, which yields the average ionization in the plasma for an averageatom model. The contribution to the screening from ions in the plasma is assumed to be trivial and hence is neglected. The ionizationpotential depression is directly determined by the screening potential, and hence, we use it to check the validity of our model of the screening potential. We obtain the ionizationpotential depression by computing the energy shifts of the involved electron orbitals with and without including plasma screening effects. A comparison with an experimental measurement of a soliddensity Si plasma shows that our predicted ionizationpotential depression agrees with the measured values, while the widely applied analytical ionsphere (Rozsnyai 1972) and StewartPyatt (Stewart & Pyatt 1966) models usually underestimate the screening for highly charged states. We applied the proposed model to investigate the screening potential and ionizationpotential depression of Si plasmas under solarinterior conditions from 0.7523 R_{⊙} to 0.3529 R_{⊙}. The plasma screening potential provides the basis for further studies of atomic processes in dense plasma, including radiative transition, photoionization, and electronimpact excitation and ionization. We expect for it to find wideranging applications in astrophysics, plasma physics, atomic and molecular physics, highenergydensity physics, fusion science, and nonequilibrium plasmas produced by Xray freeelectron lasers.
Acknowledgments
This work was supported by Science Challenge Project No. TZ2018005, by the National Key R&D Program of China under the grant No. 2017YFA0403202, and by the National Natural Science Foundation of China under Grant Nos. 11674394.
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All Figures
Fig. 1. Comparison of IPDs for soliddensity Si plasmas predicted by our model with the experimental results (Expt) (Ciricosta et al. 2016). Theoretical results from the analytical models of Ecker & Kröll (EK) (Ecker & Kröll 1963), the ionsphere model (IS) (Rozsnyai 1972), and Stewart & Pyatt (SP) (Stewart & Pyatt 1966) are given as well in order to evaluate their validity. The electron temperatures and densities are taken from a simulation (Ciricosta et al. 2016). 

Open with DEXTER  
In the text 
Fig. 2. Screening potential V_{scr}(r) in Si plasmas under solarinterior conditions. The temperature and electrondensity (T/n_{e}) values for the Si plasmas are taken from the standard solar model (Bahcall & Ulrich 1988; Guenther et al. 1992): 156.90 eV/5.88 × 10^{22} cm^{−3} (black longdasheddotted line), 189.00 eV/8.0 × 10^{22} cm^{−3} (black shortdasheddotted line), 205.12 eV/9.89 × 10^{22} cm^{−3} (black longdashed line), 227.53 eV/1.36 × 10^{23} cm^{−3} (black shortdashed line), 244.94 eV/1.7 × 10^{23} cm^{−3} (black dotted line), 275.80 eV/2.76 × 10^{23} cm^{−3} (violet dasheddasheddotted line), 304.23 eV/4.06 × 10^{23} cm^{−3} (turquoise dasheddotteddotted line), 334.40 eV/6.12 × 10^{23} cm^{−3} (brown longdasheddotted line), 373.18 eV/9.79 × 10^{23} cm^{−3} (yellow shortdasheddotted line), 407.65 eV/1.43 × 10^{24} cm^{−3} (blue longdashed line), 438.70 eV/1.96 × 10^{24} cm^{−3} (green shortdashed line), 468.84 eV/2.57 × 10^{24} cm^{−3} (red dotted line), and 496.42 eV/3.25 × 10^{24} cm^{−3} (black solid line). 

Open with DEXTER  
In the text 
Fig. 3. Ionizationpotential depressions of Si plasmas at the electron densities of 5.88 × 10^{22} cm^{−3} (black longdasheddotted line), 8.0 × 10^{22} cm^{−3} (black shortdasheddotted line), 9.89 × 10^{22} cm^{−3} (black longdashed line), 1.36 × 10^{23} cm^{−3} (black shortdashed line), 1.7 × 10^{23} cm^{−3} (black dotted line), 2.76 × 10^{23} cm^{−3} (violet dasheddasheddotted line), 4.06 × 10^{23} cm^{−3} (turquoise dasheddotteddotted line), 6.12 × 10^{23} cm^{−3} (brown longdasheddotted line), 9.79 × 10^{23} cm^{−3} (yello shortdasheddotted line), 1.43 × 10^{24} cm^{−3} (blue longdashed line), 1.96 × 10^{24} cm^{−3} (green shortdashed line), 2.57 × 10^{24} cm^{−3} (red dotted line), and 3.25 × 10^{24} cm^{−3} (black solid line). They are taken from the standard solar model (Bahcall & Ulrich 1988; Guenther et al. 1992). 

Open with DEXTER  
In the text 
Fig. 4. Comparison of our theoretical ionizationpotential depression with the analytical models of DebyeHückel (DH) (Debye & Hückel 1923), StewartPyatt (SP) (Stewart & Pyatt 1966), ion sphere (IS) (Rozsnyai 1972), and EckerKröll (EK) (Ecker & Kröll 1963) for a few selected plasma conditions of (a) 156.90 eV/5.88 × 10^{22} cm^{−3}, (b) 227.53 eV/1.36 × 10^{23} cm^{−3}, (c) 407.65 eV/1.43 × 10^{24} cm^{−3}, and (d) 496.42 eV/3.25 × 10^{24} cm^{−3}. 

Open with DEXTER  
In the text 
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