ABSTRACT
To well predict the emitted spectrum of ionized plasmas, the modern spectral synthesis codes need thermally averaged free–free Gaunt factors in a wide plasma temperature and density range. However, the plasma screening effects on the Gaunt factors have rarely been studied and taken into account in the emission calculations. In this work, we comprehensively investigate the free–free absorption Gaunt factors with the inclusion of plasma screening effects for weakly coupled (Debye) plasmas, met in the stellar atmospheres and their interiors and in the inertial confinement fusion research. The free–free Gaunt factors are presented over a wide range of screening lengths, covering the energy parameter space log10εi(Ry) = −8 to +8 and log10ωi(Ry) = −8 to +8. The temperature-averaged (over the Maxwellian electron energy distribution) Gaunt factors, as well as the frequency integrated Gaunt factors, are presented for a number of screening lengths in the range 10–500 a.u., and covering the parameter space log10γ2 = −4 to +6 and log10u = −4 to +12.
1 INTRODUCTION
The process of free–free transitions of an electron in the field of a positive ion, accompanied by continuous photon emission or absorption, plays an important role in a wide range of laboratory and astrophysical plasmas (plasma cooling, opacity, radiation transfer, etc.) (Johnston 1967; Lange & Schlüter 1985), and has been one of the oldest problems studied by the quantum mechanics (Kramers 1923; Wentzel 1924; Gaunt 1930). The quantum result for the process is usually presented by the free–free Gaunt factor, which is a dimensionless multiplicative factor to the classical result of Kramers (1923). In many astrophysical and plasma physics studies, the process of free–free continuum emission in ionized plasmas is described in terms of the non-relativistic electron motion in the pure Coulomb field of an ion. In this case, Sommerfeld (1953), Landau & Lifshitz (1951), and Biedenharn (1956), have provided analytic expressions for the non-relativistic Gaunt factors in the acceleration gauge, as reviewed in Karzas & Latter (1961), which were used in many numerical calculations (Karzas & Latter 1961; Carson 1988; Hummer 1988; Nicholson 1989; Janicki 1990; Sutherland 1998; van Hoof et al. 2014)
To well model the continuous spectra of astrophysical plasmas, the modern spectral synthesis codes, such as cloudy (Ferland et al. 2017), need accurate values for the Gaunt factors in a very wide range of plasma parameters space. However, in many laboratory and astrophysical plasmas, the many-body correlations of interacting charged particles introduce a collective screening effect on the Coulomb interaction and the motion of a continuum electron takes place not anymore in a pure Coulomb field. The plasma screening of Coulomb interaction has very important effects on the electron transition processes both in the discrete and continuum energy spectrum (Herman & Coulaud 1970; Roussel & O’connell 1974; Weisheit & Shore 1974; Shore 1975; Höhne & Zimmermann 1982; Lange & Schlüter 1985; Pavlov & Potekhin 1995; Qi, Wang & Janev 2008, 2009a; Ghoshal & Ho 2009; Qi, Wu & Wang 2009b; Lin & Ho 2010; Zammit, Fursa & Bray 2010; Zhang et al. 2010a; Zhang, Wang & Janev 2010b,c; Qi, Wang & Janev 2011; Zhang et al. 2011; Xie, Wang & Janev 2014; Jakimovski, Markovska & Janev 2016; Janev, Zhang & Wang 2016).
In this work, we shall study the free–free Gaunt factors in the screened Coulomb potential in weakly coupled plasmas present in the stellar atmospheres and their interiors, as well as in many laboratory plasmas (such as those in inertial confinement fusion, laser-solid interaction produced plasmas, etc.). The densities (ne) and temperatures (Te) in these plasmas span the ranges n ∼ 1015–1018 cm−3, T ∼ 0.5–5 eV (stellar atmospheres), n ∼ 1019– 1021 cm−3, T ∼ 50 – 300 eV (laser-produced plasmas), and n ∼ 1022– 1026 cm−3, T ∼ 0.5 – 10 keV (inertial confinement fusion plasmas).
Taking into account that plasma electrons have certain energy distribution, the Astrophysical Plasma Emission Data base (APED; Smith et al. 2001) requires more research on the temperature-averaged Gaunt factors and the total, frequency integrated Gaunt factors. The Maxwellian electron energy distribution is presently being broadly employed to calculate the thermally averaged results (Karzas & Latter 1961; Gayet 1970; Armstrong 1971; Feng et al. 1983; Carson 1988; Hummer 1988; Nicholson 1989; Janicki 1990; Sutherland 1998; van Hoof et al. 2014). This assumption will also be used in this work.
The structure of this paper is as follows. In the next section, the computational method and formulae will be briefly described. In Section 3, we present and discuss the results of our calculations. In Section 4, the conclusions of this work are given. Atomic units (a.u.) are used in the remaining part of this article. Note that in atomic units, the numerical values of electron mass me, elementary charge e, reduced Planck’s constant ℏ = h/(2π) and Coulomb’s constant 1/(4πε0) are all unity by definition, and 1 a.u. length and energy are 0.0529 nm and 27.211 eV, respectively (Shull & Hall 1959).
2 FREE–FREE GAUNT FACTOR
Note that under the transformations: ρ = Zr, δ = ZD, ε(δ) = (Z, D)/Z2, the radial Schrödinger equation with the potential (1) is scalable with respect to Z and reduces to the equation for the ion with Z = 1. For the sake of simplicity, the notations for the energy and the screening length will be those of the unscaled case (Z = 1).
3 RESULTS
As revealed in our previous study (Wu et al. 2019), the plasma screening plays an important role in the free–free Gaunt factors gff(εi, ω), especially in the low energy region where gff(εi, ω) can be significantly enhanced by resonances that appear when the screening length D is near the critical screening lengths Dnl for which the bound nl states in the potential (1) merge with the continuum. The resonance effect on gff(εi, ω) is more pronounced for the smaller screening lengths. This work intends to provide comprehensive data of gff(εi, ω) for a broad range of screening lengths D from 10 to 500 a.u. (documented in the supplemented data). In this paper, the gff(εi, ω) for the screening lengths D = 20 and 100 a.u. are taken as examples to illustrate the general features of the plasma screening effects on gff(εi, ω).
Fig. 1 shows the free–free Gaunt factors gff(εi, ω) in Debye plasmas of D = 20 and 100 a.u. in the scaled energy range of 10−8 ≤ εi(Ry) ≤ 108 and 10−8 ≤ ω(Ry) ≤ 108. As the figure shows, for |${\varepsilon _i} \gt {10^0}{\ \rm {Ry}}$|, gff(εi, ω) of the screened case gradually approach those of the pure Coulomb case. This can be understood from the fact that the electron with high energy passes through the interaction region very swiftly; its motion turns to be insensitive to the outer, exponentially decreasing part of interaction potential. For energies bellow |${\varepsilon _i} \sim {10^0}\ \rm {Ry}$|, the motion of the electron becomes fairly sensitive to the interaction potential, and can form either a virtual or quasi-bound state in the effective potential (broad s-type and narrow l > 0-shape resonances, respectively), as revealed in Wu et al. (2019). This results in the observed low-energy enhancement structures in gff(εi, ω) in the screened case (cf. Fig. 1). Note that the broad peaks in the case of D = 20 a.u. at |${\varepsilon _i} \simeq {10^{ - 6}}{\ \rm {Ry}}$| come from a broad, s-type resonance.
Non-relativistic free–free absorption Gaunt factors gff(εi, ω) for screening lengths of D = 20 and 100 a.u. in the ranges |${\varepsilon _i} = {10^{ - 8}} \!-\! {10^8}\rm {Ry}$| and |${\omega } = {10^{ - 8}} \!-\! {10^8}\rm {Ry}$|.
For a specific value of γ2, 〈gff(γ2, u)〉 generally decreases with the increasing of u. However, when log10γ2 ∼ >0 and log10u ∼ >0, corresponding to εi in the small energy region with enhanced values of gff(εi, ω), the integration of equation (3) over x or εi produces peak structures observed in Fig. 2. The peak structures at log10u ≈ 6.5 are very significant for D = 20 a.u., and are directly related to the broad s-type resonance shown in Fig. 1.
Temperature-averaged free–free Gaunt factor 〈gff(γ2, u)〉 over a Maxwellian electron distribution for screening lengths of D = 20 and 100 a.u. in the ranges u = 10−4–1012 and γ2 = 10−4–106.
The temperature averaged total free–free Gaunt factors 〈gff(γ2)〉 for a broad range of screening lengths are shown in Fig. 3. Fig. 3(a) shows very good agreement of our calculations of 〈gff(γ2)〉 for the pure Coulomb case with the previous result of van Hoof et al. (2014), verifying the accuracy of present calculations. As shown in Fig. 3(b), for log10γ2 < ∼ 0 the screened 〈gff(γ2)〉 are close to those of pure Coulomb case (this closeness increasing with increasing D), consistent with the fact that small γ2 correspond to large electron energies εi, when gff(εi, ω) becomes insensitive to the interaction screening. Note that for D = 500 a.u. the closeness of screened 〈gff(γ2)〉 to the pure Coulomb values extends to log10γ2 ∼ 1.5, reflecting the decrease of plasma screening with increasing D (cf. equation 1). Fig. 3(b) also shows that with increasing γ2 above the region where the screened and unscreened total free–free Gaunt factors are equal, the screened one decreases faster than the unscreened one until reaching a minimum at certain |$\gamma _0^2$|, after which it starts to increase again and exhibit peak structures (clearly observed for D = 20, 50, 100, and 500 a.u.). The values of 〈gff(γ2)〉 in the screened cases for most γ2 are smaller than that of the pure Coulomb case, indicating that the plasmas screening weakens the particle interactions and suppresses the free–free cooling process. In certain ranges of γ2, the values of total free–free Gaunt factors for the screening lengths D = 10, 20, 50, 200 a.u. are considerably larger than those in the pure Coulomb case, implying that free–free cooling by plasmas can also be enhanced. Obviously, the peak structures in the total free–free Gaunt factors at high-γ2 values are related to the low-energy resonances of free–free Gaunt factors (see Fig. 1).
Top panel: comparison between the total free–free Gaunt factor published by van Hoof et al. (2014, the red line). Bottom panel: Total free–free Gaunt factor 〈gff(γ2)〉 for coulomb potential and screening lengths of D = 10, 20, 50, 100, 200, 500 a.u. in the ranges γ2 = 10−4–106.
Note that with the increasing of |$\gamma ^{ 2 } = \frac{ Z^{ 2 } R y }{ k T _ { \mathrm{ e} } }$|, the effect of plasmas screening on 〈gff(γ2)〉 becomes more and more significant, especially for the region of log10γ2 > 0 and cold plasmas. The stellar atmospheres are located exactly in this region with log10γ2 ≈ 1; 〈gff(γ2)〉 of the screened case with D = 20 a.u. is about 85 per cent of the pure Coulomb case. The supplemented files to this paper provide a more complete presentation of the behaviour of total free–free Gaunt factors as a function of γ2 for a broad range of screening lengths.
4 CONCLUSIONS
In this work, we have studied the non-relativistic free–free Gaunt factors gff(εi, ω) in weakly coupled plasmas using the Debye–Hückel screened potential to represent the electron–ion interaction. The numerical calculations have been performed for a wide range of plasma screening lengths, covering the electron and photon energy space log10εi(Ry) = −8 to +8 and log10ω(Ry) = −8 to +8. For the same screening lengths, we have also calculated the temperature-averaged free–free Gaunt factors 〈gff(γ2, u)〉 (using the Maxwellian electron energy distribution) in the ranges log10γ2 = −4 to +6 and log10u = −4 to +12. Finally, the total (temperature-averaged and frequency integrated) free–free Gaunt factors 〈gff(γ2)〉 have also been calculated (by integrating 〈gff(γ2, u)〉 over the variables u) in the range log10γ2 = −4 to +6.
The most important result of this study is that in certain regions of γ2 the total temperature averaged and frequency integrated Gaunt factor is significantly smaller or larger than the one for the pure Coulomb case, indicating that the interaction screening can strongly suppress or enhance the plasma cooling.
On selected examples of calculated quantities, presented in Figs 1–3, we have discussed their dependence on the relevant variable, relating their specific features with the electron scattering resonances in the Debye–Hückel screened potential.
Supplemented files to this paper provide more complete information on the studied quantities.
SUPPORTING INFORMATION
Supplemented_files.zip
Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.
ACKNOWLEDGEMENTS
Grants from the National Basic Research Program of China (No. 2017YFA0403200), NSFC (No. 11604197), the Science Challenge Program of China (Nos. TZ2018005, TZ2016005), NSAF (No. U1530142), and the Organization Department of CCCPC are acknowledged.
