# Stark Broadening from Impact Theory to Simulations

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## Abstract

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## 1. Introduction

## 2. Impact Broadening

_{β}line in an arc plasma with a density N = 2.2 × 10

^{22}m

^{−3}and a temperature T = 10,400 K. The remaining discrepancies concerned the central part of the line and the far line wings, two regions that required an improvement of the model.

## 3. Simulations of Impact Theory and Ion Dynamics

_{α}) line [19] showed later that the experimental profile was a factor 2.5 broader than the theoretical line using static ions in arc plasma conditions. This was a strong motivation for developing a technique able to retain ion dynamics in a context where the electric field is created by numerous ions in motion. Since perturbative approaches were unable to account for multiple strong collisions, a computer simulation has been proposed for describing the motion of the ions. The effect on the emitter of the time dependent ion electric field is obtained by a numerical integration of the Schrödinger equation. Early calculations showed the effect of ion dynamics in the central part of the line, and were able to strongly reduce the difference between experimental and simulation profiles [20,21,22].

_{B}the Boltzmann constant, and ${\mathsf{\epsilon}}_{0}$ the permittivity of free space. If we simulate only the ion perturbers, we assume that each particle creates a Debye shielded electric field, in an attempt to retain ion-electron correlations. Random number generators are used to obtain the uniform positions and Maxwell-Boltzmann distributed velocities of the charged particles. If an ion leaves the cubic box, it is replaced by a new one created near the cube boundaries. For the weak coupling conditions assumed, a large number of particles (several thousand commonly) is retained in a cube with a size larger than the Debye length. Such a model provides a good approximation for the time-dependent electric field in a weakly coupled ion plasma at equilibrium, although it suffers from inaccuracies, especially if the size of the box is not large enough [23]. We show in Figure 1 the time dependence of one component of the ionic electric field calculated at the center of the box for an electron density N

_{e}= 10

^{19}m

^{−3}, and a temperature T = 40,000 K. The electric field is expressed in units of ${E}_{0}=1/\left(4{\mathsf{\pi}\mathsf{\epsilon}}_{0}{\mathrm{r}}_{0}^{2}\right)$, where ${r}_{0}$ is the average distance between particles defined by ${r}_{0}^{3}=3/\left(4\mathsf{\pi}{N}_{e}\right)$. The time interval of 5 ns used in Figure 1a is the L

_{α}time of interest for such plasma conditions. The validity condition of the binary collision approximation requests that the Weisskopf radius ${\mathsf{\rho}}_{w}=\hslash {n}^{2}/{m}_{e}{v}_{i}$, with n the principal quantum number of the Lα upper states (n = 2), and ${v}_{i}=\sqrt{2{k}_{B}T/{m}_{p}}$ the thermal ion velocity (${m}_{e}$ and ${m}_{p}$ are resp. the electron and proton mass), is much smaller than the average distance between particles. This ratio is for L

_{α}and protons of the order of 0.04, enabling the use of an impact approximation. The electric field in Figure 1a clearly exhibits several large fields that are well separated in time during the 5 ns of the L

_{α}time of interest. During this time interval, only a few fields (3 in Figure 1a) have a magnitude larger than 50 E

_{0}, but about 20 have a magnitude of 10 E

_{0}or more. A piece of the same field history is shown in Figure 1b during a time interval equal to the time of interest for the Balmer-β (H

_{β}) line. For this time interval of 0.3 ns, the electric field shows much fewer fluctuations, the atom is no longer submitted to a sequence of sharp collisions, and we can no longer use the impact approximation. This is confirmed by a value of 0.16 for the ${\mathsf{\rho}}_{w}/{r}_{0}$ ratio, making the use of an impact approximation for this line problematic. Looking now at Figure 1a,b, we can see a background of electric field fluctuations with a small magnitude of about ${E}_{0}$, and a typical time scale longer than the collision time ${r}_{0}/{v}_{i}$. Such fluctuations correspond to the sum of electric fields of distant particles with a magnitude on the order of ${E}_{0}$. For hydrogen lines affected by the linear Stark effect, it is well known that this effect of weak collisions is dominant in near impact regimes [10], and results from the long range of the Coulomb electric field.

_{α}for the same plasma conditions as in Figure 1. The ab-initio DAF (solid line) is obtained by a simulation of the ions retaining also the effect of electrons with an impact approximation. We observe that this simulation is close to an impact calculation for both ions and electrons (dashed line). For the same condition and the H

_{β}line, the decay of the ab-initio DAF is significantly smaller than for the impact calculation, indicating again a deviation from ion impact broadening for this line.

## 4. Effect of Plasma Waves

_{L}is the electric field magnitude of the wave. For values of W much smaller than 1, we expect a linear behavior of the waves. In a linear regime, electronic Langmuir waves oscillate at a frequency close to the plasma frequency ${\mathsf{\omega}}_{p}=\sqrt{{N}_{e}{e}^{2}/{m}_{e}{\mathsf{\epsilon}}_{0}}$, and can be excited even by thermal fluctuations. We assume that the numerous emitters on the line of sight are submitted to different Langmuir waves, each with the same frequency ω

_{p}, but a different direction and phase chosen at random, and a magnitude sampled using a half-normal probability density function (PDF). In the following, we have used this half-normal PDF for the reduced electric field magnitude F = E/E

_{0}:

_{L}of E by writing ${E}_{L}=\mathsf{\sigma}{E}_{0}\sqrt{2/\mathsf{\pi}}$. Each Langmuir wave has a different electric field history, and we obtain the DAF by an average over about a thousand such field histories. For a plasma with a density N

_{e}= 10

^{19}m

^{−3}, and a temperature T = 10

^{5}K, we first calculated the L

_{α}DAF for Langmuir waves with a mean electric field magnitude corresponding to W = 0.01 (E

_{L}= 15E

_{0}). The response of the DAF is a periodic oscillation with a period equal to 2π/ω

_{p}, but with an amplitude much smaller than 1 for this average field magnitude of 15E

_{0}. After a product with an impact DAF for retaining the effect of the background electron and ion plasma, there remains no visible effect of the waves on the convolution DAF for the value W = 0.01. This ratio can take much larger values, however; especially if an external energy source such as a beam of charged particles is present. As W increases, nonlinear phenomena start showing up, enabling, for instance, wave-wave couplings. Although only recently investigated in plasmas, the occurrence of rogue waves has been raised in various plasma conditions [29,30,31]. Rogue waves have been studied in many dynamical systems, and are known to the general public by the observation and study of rogue or freak waves that suddenly appear in the ocean as large isolated waves. In oceanography, rogue waves are defined as waves whose height is more than twice the mean of the largest third of the waves in a wave record. Rogue waves appear to be a unifying concept for studying localized excitations that exceed the strength of their background structures. They are studied in nonlinear optics [32], Bose-Einstein condensates [33], and many other fields outside of physics. For our line shape problem in plasmas, we postulate that nonlinear processes create rogue waves from a random background of smaller Langmuir waves. The physical mechanism at play is the coupling of the Langmuir wave with ion sound and electromagnetic waves; density fluctuations of the sound waves affect the high frequency waves through ω

_{p}. The first Zakharov equation [34] shows how density fluctuations affect Langmuir waves, and a second equation how a Langmuir wave packet can produce a density depression via the ponderomotive force [35]. We will not discuss these equations here, which are particularly useful for a study of wave collapse. Most present rogue wave studies rely on the nonlinear Schrödinger equation (NLSE), which is obtained in the adiabatic limit (slowly changing density perturbations) of the Zakharov equations [35]. A one-dimensional solution of the NLSE is commonly used to approximate the response of nonlinear media. Stable envelope solitons are possible solutions of the 1D NLSE. We will assume that there is a contribution of a stable envelope soliton for each history of the microfield, similarly to what we did for the background Langmuir wave. Using a ratio W = 0.1, the average peak magnitude of such solitons will be 3 times the amplitude of background Langmuir waves, fitting them in the category of rogue waves. A possible shape for the envelope is a Lorentzian, with a time dependence that bears some similarity with the celebrated Peregrine soliton [36]. We observe in Figure 3 that the DAF of L

_{α}obtained with a product of the impact DAF and the Langmuir rogue wave DAF for W = 0.1 is affected by oscillations at the plasma frequency.

## 5. Conclusions

_{α}in an impact regime.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Electric field component in units of ${E}_{0}$ in a plasma with a density N

_{e}= 10

^{19}m

^{−3}, and a temperature T = 40,000 K, during (

**a**) a time interval of the order of the time of interest for the L

_{α}line, and (

**b**) a time interval of the order of the time of interest for the H

_{β}line.

**Figure 2.**Dipole autocorrelation functions with an ab-initio simulation (solid line) and in the impact limit (dotted line) in a plasma with a density N

_{e}= 10

^{19}m

^{−3}, and a temperature T = 40,000 K, for (

**a**) the L

_{α}Lyman transition, and (

**b**) the H

_{β}Balmer transition.

**Figure 3.**L

_{α}dipole autocorrelation function in a plasma with a density N

_{e}= 10

^{19}m

^{−3}, and a temperature T = 10

^{5}K, calculated with a product of the impact DAF and the Langmuir rogue wave DAF for W = 0.1.

**Figure 4.**L

_{α}in a plasma with a density N

_{e}= 10

^{19}m

^{−3}, and a temperature T = 10

^{5}K, calculated with an impact approximation (dashed line), and with a Fourier transform of the DAF in Figure 3 (solid line).

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**MDPI and ACS Style**

Stamm, R.; Hannachi, I.; Meireni, M.; Godbert-Mouret, L.; Koubiti, M.; Marandet, Y.; Rosato, J.; Dimitrijević, M.S.; Simić, Z. Stark Broadening from Impact Theory to Simulations. *Atoms* **2017**, *5*, 32.
https://doi.org/10.3390/atoms5030032

**AMA Style**

Stamm R, Hannachi I, Meireni M, Godbert-Mouret L, Koubiti M, Marandet Y, Rosato J, Dimitrijević MS, Simić Z. Stark Broadening from Impact Theory to Simulations. *Atoms*. 2017; 5(3):32.
https://doi.org/10.3390/atoms5030032

**Chicago/Turabian Style**

Stamm, Roland, Ibtissem Hannachi, Mutia Meireni, Laurence Godbert-Mouret, Mohammed Koubiti, Yannick Marandet, Joël Rosato, Milan S. Dimitrijević, and Zoran Simić. 2017. "Stark Broadening from Impact Theory to Simulations" *Atoms* 5, no. 3: 32.
https://doi.org/10.3390/atoms5030032