Review of recent advances in the analytical theory of Stark broadening of spectral lines in plasmas: applications to laboratory discharges and astrophysical plasmas

We present an overview of latest advances in the analytical theory of Stark broadening of spectral lines and their applications to various types of laboratory and astrophysical plasmas. They include: 1) in-depth study of intra-Stark spectroscopy in the x-ray range in relativistic laser-plasma interactions; 2) effect of diamagnetism on the number of observable hydrogen lines in plasmas; 3) influence of magnetic-field-caused modifications of trajectories of plasma electrons on the width of hydrogen/deuterium spectral lines: applications to white dwarfs; 4) Stark broadening of hydrogen/deuterium spectral lines by a relativistic electron beam: analytical results and applications to magnetic fusion; 5) counterintuitive dependence of the dynamical Stark width of hydrogenic spectral lines on the electron density.


In-depth study of intra-Stark spectroscopy in the x-ray range in relativistic laser-plasma interactions
Over the last 40 years there were numerous experimental studies of structures/modulations in spectral line profiles caused by Langmuir waves and called L-dips. The practical significance of studies of the L-dips is threefold. First, they provide the most accurate passive spectroscopic method for measuring the electron density Ne in plasmas, e.g. more accurate than line broadening measurements. Second, they provide the only one non-perturbative method for measuring the amplitude of Langmuir waves in plasmas. Third, in laser-produced plasmas they help to reveal the rich physics of the laser-plasma interactions.
In one of the most recent studies [1] performed at Vulcan Petawatt facility in the UK, thin Si foils were irradiated by relativistic laser pulses (intensities ~10 21 W/cm 2 ). In addition to the Langmuir waves, there were discovered spectroscopically, for the first time, ion acoustic waves in laser-produced, highdensity plasmas. The simultaneous production of both kinds of these waves was attributed to the development of the parametric decay instability at the surface of the relativistic critical density.
Here we present the latest study of the relativistic laser-plasma interactions from paper [2] (never reported at ICSLS). In this experiment at the Vulcan facility a plasma mirror was used to significantly improve the laser contrast. As a result, there was a much smaller laser pre-plasma (compared to the previous experiment), allowing the main laser pulse to interact with a higher density plasma. This 2 improvement enabled us to perform an in-depth spectroscopic study of the simultaneous production of the Langmuir waves and of the ion acoustic turbulence at the surface of the relativistic critical density. This was achieved by the spectroscopic analysis of the same spectral line (Si XIV Ly-beta) in three different shots of about the same laser intensity 2×10 20 W/cm 2 . It demonstrated reliable reproducibility of the L-dips at the same locations in the experimental profiles, as well as of the deduced parameters (fields) of the Langmuir waves and ion acoustic turbulence in all three shots presented in Fig. 1. In the profiles from shots A, B and C, there are clearly seen 'bump-dip-bump' structures typical for the L-dips phenomenon. The best fit: A) Ne = 2.2×10 22   In shot D the laser intensity and therefore the electron density were significantly lower than in shots A, B and C. So, the damping of the Langmuir waves was significantly lower, which could allow the Langmuir waves to reach a significantly higher amplitude. It caused the ratio E0/Fres to be > 0.5, i.e., beyond the range where the L-dips can form according to the theory. Thus, we showed that different interplay of the Langmuir wave field with the field of the ion acoustic turbulence lead to distinctly different manifestations in the spectral line profiles, including the disappearance of the L-dips.

Effect of the diamagnetism on the number of observable hydrogen lines in plasmas
Under a strong magnetic field B, the most important center-of-mass effect is that the diamagnetic potential term in the Hamiltonian for the relative motion can become responsible for the formation of an additional potential well (hereafter, B-well) -far away from the hydrogen nucleus (proton). In paper [3] we analyzed the effect of the B-well on the number of observable hydrogen lines in strongly magnetized plasmas.
For hydrogen energy levels below the top of the potential barrier (E < Etop), the atomic electron is confined in a relatively narrow potential well (see Fig. 2). However, for the energy levels at or above the top of the potential barrier, the width of the potential well increases by several orders of magnitude -for sufficiently large values of the pseudomomentum K, which is the constant arising from the separation of the center-of-mass and internal motions (see Fig. 2). According to the uncertainty relation, 3 this means that in the latter case, the spacing of the energy levels decreases by several orders of magnitude. Since these energy levels have a finite width (e.g., due to the Stark broadening in plasmas and the natural broadening), the dramatic decrease of the spacing between these energy levels creates a quasi-continuum out of them and thus makes the atomic electron to be practically free from the proton. This is equivalent to the ionization of the atom. Therefore, this mechanism limits the number of the discrete energy levels. In all of the above examples, the principal quantum number of the last observable hydrogen line is controlled by the diamagnetism-caused B-well, rather than by any other effect. Thus, the primary effect of the diamagnetic term in the Hamiltonian is the creation of the B-well causing the decrease of the number of observable hydrogen lines. Compared to this primary effect, other effects of the diamagnetic term -those discussed in Rosato paper [4] -are just minor, secondary outcomes.
Our results open up an avenue for the experimental determination of the pseudomomentum K: K(a.u.) ≈ (nmax,B/72) 4 /B(Tesla). Thus, from the experimental values of the magnetic field B and the number nmax,B of the last observable hydrogen line, it is possible to deduce the value of the pseudomomentum. This is the first proposed method for the experimental determination of the psudomomentum -to the best of our knowledge.

Analytical study of the influence of magnetic-field-caused modifications of trajectories of plasma electrons on the width of hydrogen/deuterium spectral lines: applications to white dwarfs
In paper [5] we studied analytically the influence of magnetic-field-caused modifications of trajectories of plasma electrons on shifts and relative intensities of Zeeman components of hydrogen/deuterium spectral lines (reported also at the ICSLS-23). In our later papers [6,7] the focus was on how the same cause affects widths of these spectral lines. We concentrated on the case of a strong magnetic field B, such that the so-called non-adiabatic Stark width practically vanishes and only the so-called adiabatic Stark width remains B > Bthreshold = 4c(meNe) 1/2 , Bthreshold(Tesla) = 3.62x10 -7 [Ne(cm -3 )] 1/2 , Such strong magnetic fields encountered, e.g., in white dwarfs.
We showed analytically that for the range of plasma parameters typical for DA white dwarfs, the neglect for the actual, helical trajectories of perturbing electrons can lead to the overestimation of the Stark width by up to one order of magnitude for the alpha-and beta-lines of the Lyman and Balmer series, or to the underestimation of the Stark width by several times for the delta-and higher-lines of the Balmer series (see Fig. 3). Therefore, our results should motivate astrophysicists for a very significant revision of all existing calculations of the broadening of hydrogen lines in DA white dwarfs. Fig. 3. The ratio of "helical" width to the "rectilinear" width versus the electron densities for the Lymanalpha line (the lower curve), for the two intense π-components of the Balmer-beta line (the middle curve), and for the two most intense π-components of the Balmer-delta line (the upper curve).
After we published the 1 st (out of 2 papers on this) in 2017, Rosato et al in 2018 [8] published just one simulation of the effect of helical trajectories of perturbing electrons on the adiabatic width of the π-component of the Lyman-alpha line only at one set of plasma parameters: B = 2000 Tesla, Ne = 10 17 cm -3 , Te = 1 eV. There are 2 conceptual errors in Rosato et al paper [8].
For their chosen parameters, the allowance for the helical trajectories of perturbing electrons decreases the adiabatic width compared to the corresponding result for rectilinear trajectories, as it was shown analytically in our paper [14] in 2017 (a year before Rosato et al paper [8]). When Rosato et al [8] in 2018 tried to give a physical explanation of this kind of the result of their simulation (while claiming erroneously that they were the first to discover this effect), they referred to the upper cutoff (for the integration over impact parameters) ρmax = min(v/ωpe , v/ΔωB) = v/ΔωB in case of ΔωB > ωpe. However, in reality this cutoff is effective only for the non-adiabatic contribution to the width, but has no effect on the adiabatic contribution to the width, the latter being the subject of their paper [8] (and of our paper [7]).
Another conceptual error by Rosato et al [8] was to imply that in strongly magnetized plasmas, the allowance for helical trajectories of perturbing electrons always decreases the adiabatic width regardless of Ne and Te. However, in reality for relatively large principal quantum number n (delta and higher lines) and/or relatively large Ne and/or relatively small Te, the width is greater than if it were calculated for  [8] stems from the general inferiority of simulations compared to the analytical theory: simulations are unable to provide the functional dependence of the effect under consideration on various input parameters (in distinction to the analytical theory) and thus often miss the big picture.
Alexiou in 2019 [9] performed more accurate simulations of the effect of helical trajectories of electrons on the width of Ly-α Zeeman components. His primary result was basically that at about the same parameters as used by Rosato et al [8], the width decreases. So, he also missed the fact (shown analytically in our paper [6]) that for relatively large principal quantum number n (delta and higher lines) and/or relatively large Ne and/or relatively small Te, the width is greater than if it were calculated for rectilinear trajectories of perturbing electrons.

Stark broadening of hydrogen/deuterium spectral lines by a relativistic electron beam: analytical results and applications to magnetic fusion
Developing diagnostics of a Relativistic Electron Beam (REB) is important for tokamaks, where runaway electrons can become a REB, what should be timely detected to allow the mitigation of the problem. In papers [10,11] we used an advanced analytical formalism to develop a theory of the Stark broadening of hydrogen/deuterium spectral lines by a REB. In particular, we obtained an analytical result for the width function of ANY hydrogen/deuterium spectral line. Figure 4 shows our calculated ratio Γσ/Γπ of the REB-caused widths of the σ-and π-components of the Ly-alpha line versus the relativistic factor γ = 1/(1 -v 2 /c 2 ) 1/2 at Ne = 10 15 cm -3 and Te = 2 eV. Separate measurements of the widths of the σ-and π-components (and thus of the ratio Γσ/Γπ) can be performed for the observation perpendicular to the REB velocity by placing a polarizer into the optical system. By monitoring the dynamics of the ratio Γσ/Γπ, it would be possible, at least in principle, to detect the development of a REB in tokamaks and to engage the mitigation of the problem.
After we published the 1 st paper on this in 2016 [10], Rosato et al in 2017 [12] attempted studying the Stark broadening of the hydrogen Ly-α line by a REB in magnetic fusion edge plasmas. However, they used the quasistatic approximation, which is totally inappropriate for the broadening by fast electrons of a REB (it is inappropriate even for the broadening by thermal electrons in such plasmas).
Another issue is a possible broadening of hydrogen/deuterium (H/D) spectral lines by Langmuir waves. Let's clarify this. When Langmuir waves are quasimonochromatic, their primary manifestation is L-dips. L-dips are a multifrequency resonance phenomenon caused by the coupling of a "single frequency" (quasimonochromatic) electric field with a quasistatic electric field in plasmas. However, when the spectrum of Langmuir waves broadens due to various nonlinear phenomena, then they could cause an additional broadening of H/D spectral lines. This effect was first estimated by Sholin in 1970 [13]; then calculated more accurately in subsequent works of his group (e.g., [14]). This effect was first in a beam-plasma experiment as early as in 1971 [15]. Despite both the analytical theory of the broadening of ANY H/D line by non-quasimonochromatic Langmuir waves and the observation of this effect were achieved already in 1970s, in several recent publications their authors tried to "re-invent the wheel" [16][17][18][19][20]. Of course, these simulations of selected H/D line did not discover anything new compared to the analytical theory from 1970s, The authors of those simulations seem to be totally unaware of both the corresponding analytical theory and the observation of this effect from 1970s by Sholin's group: they did not make any reference -just as in paper [21], the authors did not refer to about half-a-hundred theoretical and experimental studies of lineshape-based diagnostics of plasma turbulence in laboratory and astrophysical plasmas published between 1970 and (including) 2017, despite this was the subject of paper [21].

On the counter-intuitive dependence of the dynamical Stark width of hydrogenic spectral lines on the perturbers density and temperature
There is a very good paper by Stambulchik and Demura of 2016 [22]. They showed by simulations that for the Ly-α line the Stark width is ~ Np at relatively small density of perturbers Np (the impact limit), but at large Np becomes ~ Np 1/3 (practically no quasistatic result, which would be ~ Np 2/3 ). They correctly explained this result by the fact that for the central Stark component of Ly-α (which contributes overwhelmingly to the total intensity of Ly-α), the dominating broadening mechanism for high Np is the amplitude modulation. So, the Ly-α width at high Np is proportional to the rotation frequency of the perturbers ~ vpNp 1/3 . Similarly, Stambulchik & Demura in the same paper showed that for the Ly-α line the Stark width is ~ 1/T 1/2 at relatively large temperature T (impact result), but at small T becomes ~ T 1/2 (i.e., proportional to the rotation frequency of the perturbers ~ vpNp 1/3 ).
But what about the lines without the central Stark component, e.g., Ly-β, Ly-δ, H-β, H-δ and so on? The corresponding results were obtained analytically for any such line and presented in papers [23,24]. For example, for the ion dynamical broadening, the Stark width is ~Ni at relatively small ion density Ni (the impact limit), but at large Ni becomes ~ Ni 1/2 (rather than ~Ni 1/3 as for the Ly-α). The dynamic width becoming ~ Ni 1/2 is a non-binary result. The temperature dependence: for such lines, e.g., for the ion dynamical broadening, the Stark width is ~ 1/T 1/2 at relatively high T (the impact limit), but at small T becomes ~T 1/4 , (rather than ~ T 1/2 as for the Ly-α). Figure 5 shows a typical profile of ANY Stark component for the ion broadening: bold solid line is our analytical result. Our analytically calculated profile demonstrates the transition from the Lorentzian at the center to the Gaussian in the near wings. The quasistatic part is only at the far wings (not shown here). Thus, the combination of these analytical results [23,24] with Stambulchik-Demura's simulation results [22] constitute the complete set for any hydrogenic line for any density of perturbers and for any temperature.
6. Conclusions 1. The latest study of the relativistic laser-plasma interactions from paper [2] at the Vulcan facility demonstrated the reliable reproducibility of the L-dips at the same locations in the experimental profiles of the Si XIV Ly-beta line, as well as of the deduced parameters (fields) of the Langmuir waves and ion acoustic turbulence, in different shots. 2. The number nmax,B of the last observable hydrogen lines is controlled by the diamagnetism by sufficiently large pseudomomentum. From the experimental values of the magnetic field B and of the number nmax,B, it is possible to deduce the value of the pseudomomentum. This is the first proposed method for the experimental determination of the psudomomentum -to the best of our knowledge. 3. The neglect for the actual, helical trajectories of perturbing electrons can lead to the overestimation of the Stark width by up to one order of magnitude for the alpha-and beta-lines of the Lyman and Balmer series, or to the underestimation of the Stark width by several times for the delta-and higherlines of the Balmer series. Therefore, our results should motivate astrophysicists for a very significant revision of all existing calculations of the broadening of hydrogen lines in DA white dwarfs. 4. Our analytical results on the broadening of any hydrogen or deuterium spectral line by a Relativistic Electron Beam (REB) allow detecting the development of a REB in tokamaks and engaging the mitigation of the problem. 5. For the dependence of the dynamical Stark width of hydrogenic spectral lines on the perturbers density and temperature, the combination of our analytical results [23,24] with Stambulchik-Demura's simulations [22] constitute the complete set for any hydrogenic line for any density of perturbers and for any temperature.