# Spectral-Kinetic Coupling and Effect of Microfield Rotation on Stark Broadening in Plasmas

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Spectral-Kinetic Coupling

_{0}is the detuning of cyclic frequency from the line center, the outer angle brackets correspond as usual to an averaging over the microfield values F (F is the absolute value of microfield), is the operator of density matrix, is the operator of dipole moment, is the operator of resolvent, is the operator of the linear Stark shift between sublevels of the upper n and lower level n’ in the line space (the direct product of subspaces of the upper and lower energy levels with the principal quantum numbers n and n’), is the electron broadening operator, indexes αβ and α’β’ designate quantum states of the upper and lower levels in the bra and ket vectors of the line space, respectively.

_{1}, n

_{2}, m) form the natural zero basis of the problem due to the cylindrical symmetry [17,18]. For sufficiently large values of the microfield, the impacts of electrons could not equate populations of sublevels [17,18], and thus the assumption of population equipartition [17,18,21], made as an initial condition, becomes violated. Evidently, since the density matrix could not have the diagonal equipartition form in the two different bases, the initial assumption of the theory [17,18,21] becomes invalid (see [42,43,44,45]). This simple example thus shows that a more consistent approach should simultaneously consider atomic kinetics and formation of spectral line profiles, that just signifies the spectral-kinetic coupling [42,43,44,45,50,53]. Those drawbacks in constructing Stark profiles under assumption of diagonal form of density matrix are weakened partially, if one introduces the dependence of electron impact broadening operator on the Stark levels splitting in the ion microfield [17,18]. Then the non-diagonal matrix elements will be the next order corrections in comparison with the impact widths for large values of the ion microfield, and terms responsible for the line mixing will drop down more rapidly in the line wings [17,18]. On the other hand, for small values of the ion microfield, for which the Stark splitting is of the order of the non-diagonal matrix elements of the electron-impact-broadening operator, the Stark components collapse to the center, thus becoming effectively degenerate [16,17,18], as it should be due to the straightforward physical reasoning (it is worth reminding that it was academician V.M. Galitsky who pointed out this effect to the authors of [16]). This collapse phenomenon is characterized by the appearance of the dependence of the decay constants and intensities of the “redefined” inside the collapse region “Stark components” on the microfield, while the energy splitting of these “Stark components” disappears. These “redefined” Stark components appear in the process of solution of the secular equation for the resolvent operator in the line space [13,14,15]. At F = 0, the intensity of one of the “redefined” components becomes equal to zero, while the other one gives contribution to the center of the profile identical to the contribution of the two symmetrical lateral Stark components without their redefinition during the solution of secular equation for the diagonalization of resolvent [14,15,16]. Therefore the collapse phenomenon of Stark components signifies the necessity to change the wave functions basis from the parabolic to spherical wave functions, or vice versa, depending on whether ion microfield value decreases or increases. Simultaneously, of course, this means the region of singularity for the ST assumption of the density matrix diagonality [21,51]. From this consideration it is obvious that the collapse phenomenon has the kinetic character and in fact is one of the examples of spectral-kinetic coupling. Thus the existence of the collapse phenomenon of Stark components at the same time means the necessity of more complete consideration of the Stark broadening within the formalism of kinetic equations for the density matrix [42,43,44,45,50,55], or in other terms, the necessity of application of the kinetic theory of Stark broadening.

## 3. Ion Dynamics in Statistical and Spectral Characteristics of Stark Profiles

_{0}(F

_{0}is the Holtsmark normal field value [1,5,6,7,8,9,12,21,23,24,25,48,67] and F is the current microfield value), assuming for simplicity (but without loss of generality) that ions produce the Coulomb electric field. In the case of β ≪ 1, the ion microfield is formed by many distant ions and due to isotropy the following relation takes place [5,6,24,25]:

_{0}definition). The definitions of the mean values for composition of various ion species “s” with the charges Z

_{s}and thermal velocity v

_{i,s}are given by the following relations:

_{i}≤ 1 [34,61,75]. This greatly facilitates the study of ion dynamics in simulations, since the consistent consideration of radiator motion effects in MD is quite time-consuming. The expressions for the fluctuation rates, Equation (4) and Equation (6), also show that for plasmas with complex ion composition there could be some deviations from the RM model, caused by peculiar distributions of ion charges. As the main precision experiments have, up to now, been conducted for simple charge distributions, the expressions in Equation (5) and Equation (6) could be greatly simplified and the terms, corresponding to the ion friction, could be omitted. Hence the charge distribution is neglected below.

_{1−α}(x) function describes the central component contribution to the Stark profile due to amplitude modulation, while the f

_{2−α}(x) function describes the contribution of lateral components to the Stark profile, related to the combined action of the amplitude modulation and non-adiabatic effects. Below, the explicit expressions for f

_{1−α}(x), f

_{2−α}(x) are given (Γ(z) is the gamma function):

_{⊥}(0) = 1, while the corresponding constant in the limit of β ≪ 1 was included in the definition of the f

_{2−α}(x) function. The behavior of f

_{1−α}(x) and f

_{2−α}(x) is presented in the Figure 2 and Figure 3, respectively. The parameter γ in Equations (9) and (10) is the impact electron width of the central component in the parabolic basis. As it follows from the results of [24,25], the corrections due to the ion dynamics effects are negative in the center of a line with the central components, corresponding to decreasing of the intensity in the line center due to the ion dynamics effects and its increasing in the shoulders (the transient region between approximately the half width and the nearest line wings). As the thermal corrections have perturbative character, functions f

_{1−α}(x) and f

_{2−α}(x) have zero integrals. So, due to ion dynamics effects, the intensity is redistributed from the line center, increasing the total width of the lines with central Stark components [24,25]. These general features are confirmed below in the next Section 4 using MD simulations (see also [75]), that are believed to be not limited by applicability conditions of the perturbation approach [24,25]. It should be noted that within the approach of [24,25], an exact analytical expression due to the ion-dynamics corrections for Ly-alpha, accounting for the collapse of the lateral Stark components [13,14,15], was also derived [24]. It has a rather complex structure and is not presented here, but the comparison of its functional behavior with the approximation of isolated individual Stark components f

_{2−α}(x) is shown in Figure 3.

**Figure 3.**Function f

_{2−α}(x)—dashed line. The solid line is the behavior of f

_{2−α}(x;ε = 1), obtained in numerical calculations accounting for the collapse effect [24].

_{2−α}(x;ε), which takes into account the collapse effect of lateral Stark components (where ε is the ratio of the non-diagonal matrix element of the electron impact broadening operator to the electron impact width of the central component of Ly-alpha). Neglecting dependence of the non-diagonal matrix element of the electron impact broadening operator on the value of the ion microfield F according to [17,18], which mainly is important for large F for transition from the overlapping to isolated broadening regime of the Stark components (since ε(β) → 0 for β → ∞), corresponds to ε = 1. The ratio of the lateral component electron impact width to the central component electron impact width is equal to 2 for Ly-alpha in parabolic basis [17,18]. This is reflected in the argument of , whose value is taken in the pole of resolvent , corresponding to the one lateral component. So, remembering that the central component is more intense than the lateral ones, its strong influence on the Ly-alpha Stark shape becomes obvious. Comparing f

_{2−α}(x;ε = 0) with f

_{2−α}(x;ε = 1) at x = 0, their ratio comes out to be about 1.26 [24]. At first glance, putting ε = 0 in f

_{2−α}(x;ε) allows obtaining the limit of isolated Stark components, but it turns out that f

_{2−α}(x;ε = 0) ≠ f

_{2−α}(x). This means that there is no commutativity in the sequence of performed mathematical operations, since the f

_{2−α}(x;ε) function is obtained in the solution of secular problem and inverting the resolvent. It is seen in Figure 3, that the difference between the approximation of isolated components (f

_{2−α}(x)) and the exact solution accounting for the collapse effect (f

_{2−α}(x;ε = 1)) at x = 0 is noticeably smaller than that between f

_{2−α}(0;ε = 1) and f

_{2−α}(0;ε = 0), as their ratio f

_{2−α}(0;ε = 1)/f

_{2−α}(0) is only about ~1.14. Thus this comparison demonstrates an acceptable accuracy of the approximation of isolated individual Stark components [18,19] for calculations of the thermal corrections to the Ly-alpha Stark profile.

_{2−β}(x) describes the lateral components contribution to the Stark profile, comprising to the action of only non-adiabatic effects in the case of lines without the central component [24,25],

_{2−β}(x) is presented in the Figure 4. The case of Lyman-beta illustrates that the ion dynamics effect increases the intensity in the center of the line without the central components and slightly decreases its width due to the lowering intensity in the nearest wings that is clearly seen in Figure 4.

_{⊥}(z) in Equations (9) and (11) was substituted for small values of argument near zero, corresponding to the line center. Moreover, as M

_{⊥}(z) varies very slowly on the characteristic frequency scale in the line center, in [24,25] its variation in the numerical results of Equations (9) and (11) was neglected. It allowed to neglect the difference of values of standing beside the terms, obtained due to residue in the various poles, and equate them in fact to the common constant due to the smallness of argument. Then summation of terms due to perturbation expansion leads to more simple formulas, which are expressed as the functions f, and finalized by introduction of some general scales for the Stark constants C and impact widths w. Moreover, the significant simplification of the result also is due to the constant output of the M

_{⊥}(z) function in the region near the line center, where its argument is small. Then all the derivatives of M

_{⊥}(z) turn out to be zero (or could be considered as higher order terms of expansion), thus one is left only with the derivatives of the dispersion functions in perturbation series (see [24,25]). However, from the principal point of view, it is important that functional behavior of M

_{⊥}(z) is proportional to the fluctuation of microfield component perpendicular to the microfield direction in Equations (4) and (10). It could be kept in the final result which would then look more cumbersome in this case than expressions (11) and (12). Indeed, the result would contain the sum of contributions of each Stark component, determined by its values of the electron impact widths and Stark constants, being multiplied by M

_{⊥}(z) from the different arguments, as explained just above (see explicit formulas for perturbation expansion, presented in [24,25]).

_{R}(Δλ), corresponding to the two different values of the reduced mass, were also considered. Within the notion of thermal corrections this difference is

_{D}and ρ

_{W}designate the Debye and Weisskopf radii, respectively (see [21,45,58]). It is seen that condition (16) (and other criteria from [24,25]) is difficult to fulfill, which somewhat limits the practical applicability of the theory. For the line wings, the results of [24,25] reproduce the results of the earlier work [12] under the criterion for the separate Stark components

^{(th)}(Δω/CF

_{0}) is the thermal correction profile that represents, within the assumptions of [18,19], a sum of contributions from the amplitude modulation, non-adiabatic effects, and phase modulation, and H (Δω/CF

_{0}) is a microfield distribution function.

## 4. Ion Dynamics Modeling and Statistical-Dynamical Coupling

_{0}is again the Holtsmark normal field for singly charged ions [1].

^{2}-times larger reduced mass. We note that by reusing the field histories generated only once, any possible inaccuracy due to a finite statistical quality of the simulations, such as a deviation from the Holtsmark distribution of the field magnitudes [1], should be present in all calculations and thus, cancel out when the difference profiles are evaluated. The parameters of the base run (s = 1) were selected to correspond to protons (i.e., μ

_{0}= 0.5) with the particle density N = 10

^{17}cm

^{3}and temperature T = 1 eV, while the additional runs with s = 2, 4, and 8 corresponded to μ = 2, 8, and 32, respectively.

_{0}/s

^{2}. The observed dependence is, thus, qualitatively similar to the Ly-α T-dependence inferred in an ion-dynamics study [76]. We note that the shape of the central component is practically Lorentzian.

_{i}plays the role of the electron impact one in Equation (9).

**Figure 5.**CS Ly-α profiles, broadened by an OCP, assuming N = 10

^{17}cm

^{−3}and T = 1 eV. The ion radiator reduced mass μ = s

^{2}μ

_{0}, where μ

_{0}= 0.5. (

**a**) Line shapes influenced by the rotational field component (18); (

**b**) Line shapes influenced by the vibrational field component (19).

**Figure 6.**(

**a**) CS full Ly-α profiles, broadened by an OCP, assuming N = 10

^{17}cm

^{−3}and T = 1 eV. The ion radiator reduced mass μ = s

^{2}μ

_{0}, where μ

_{0}= 0.5; (

**b**) Profiles differences between line shapes, calculated with s and sʹ = s + δs = 5/4s (i.e., δμ/μ = 9/16). The profile differences are scaled to the lineshape HWHM w

_{i}.

## 5. Discussion

_{0}. These profiles are characterized by well distinguished properties that are consequences of atom dynamics in the rotating microfield [10,13,75]. When the profile patterns are averaged over all microfield directional histories, the fixed value of the microfield leads to a statistical-dynamical coupling through specifics of solutions of the Schrödinger equation, i.e., coupling between the microfield statistics and specific dynamics of the atomic system [10,13]. This is illustrated by the instructive detailed patterns presented in [75] for the rotational contribution of ion dynamics effects. Let us now consider the proposed separation of phase modulation or vibrational effects. Here only the microfield orientation is assumed to be constant, while its magnitude as a function of time is preserved. The solution of the Schrödinger equation in this case reflects the specifics of a fixed orientation, and after averaging over microfield histories this imposes very characteristic features of the Stark profiles [7,8,75]. These profiles are also subject to statistical-dynamical coupling caused only by the fixed microfield orientation.

_{0}~ n

^{2}N

_{i}

^{2/3}, and in the impact regime it is proportional to ~n

^{4}(T

_{i/}/µ)

^{−1/2}N

_{i}, from the dimension consideration it follows that in the regime of Stark broadening controlled by the ion dynamics the characteristic scale w

_{i}could be just proportional to (T

_{i/}/µ)

^{1/2}N

_{i}

^{1/3}as there could be no dependence on the dipole atomic moment or microfield. On the right hand side in Figure 6b the difference profiles for the same set of artificial values of reduced mass are plotted. Their qualitative behavior is similar to the ones discussed in the Section 2 functions f

_{1−α}(x), f

_{2−α}(x). From the performed analysis in the previous section it could be deduced that the characteristic scale of HWHM is proportional to (2T

_{i}/µ)

^{1/2}, and at the same time the analysis, given in [67], has shown that in this range of parameters (N

_{e}~ 10

^{17}cm

^{−3}, T~ 1 eV) HWHM is proportional to N

_{i}

^{1/3}. Combining these two dependences lead to the conclusion that, for the chosen plasma parameters, the HWHM needs to be proportional to the typical ion microfield frequency: w

_{i}~ (2T

_{i}/µ)

^{1/2}N

_{i}

^{1/3}.

_{i }∝ v

_{i}N

_{i}

^{1/3}.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Demura, A.V.; Stambulchik, E. Spectral-Kinetic Coupling and Effect of Microfield Rotation on Stark Broadening in Plasmas. *Atoms* **2014**, *2*, 334-356.
https://doi.org/10.3390/atoms2030334

**AMA Style**

Demura AV, Stambulchik E. Spectral-Kinetic Coupling and Effect of Microfield Rotation on Stark Broadening in Plasmas. *Atoms*. 2014; 2(3):334-356.
https://doi.org/10.3390/atoms2030334

**Chicago/Turabian Style**

Demura, Alexander V., and Evgeny Stambulchik. 2014. "Spectral-Kinetic Coupling and Effect of Microfield Rotation on Stark Broadening in Plasmas" *Atoms* 2, no. 3: 334-356.
https://doi.org/10.3390/atoms2030334