## 2. Spectral-Kinetic Coupling

Let us consider broadening of the hydrogen atom in the well-known setting of the so-called standard theory (ST), related to formulation given in [

17,

18,

21,

51]. Then the total Stark profile of spectral line

I(Δω) is represented as the convolution of the Stark subprofiles, corresponding to the transitions between upper and lower substates being broadened by electrons in the fixed value of ion microfield, and integrated over ion microfield values with the microfield probability distribution function as a weight:

In Equation (1) Δω = ω − ω

_{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.

Conventionally assuming that density matrix is diagonal in Equation (1), it could be factored out since in the region of small values of microfield all sublevels are degenerate and thus equally populated [

17,

18,

21]. For now, the fine structure splitting is neglected. In this region, the spherical quantum functions (labeled by the quantum numbers

n,

l,

m) form a natural zero basis of the problem due to the spherical symmetry [

17,

18,

21]. Suppose now that the value of the ion microfield is increased, thus the sublevels are split due to the linear Stark effect in the ion electric microfield, and the degeneracy is partially removed. Now the parabolic quantum functions (labeled by the parabolic quantum numbers

n_{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.

The spectral-kinetic coupling discussed above is a common thing in laser physics [

42]. Indeed, the lasing condition is directly connected with the difference of populations that, in turn, is proportional to the non-diagonal matrix elements of the density matrix (called coherences), describing the mixing of the upper and lower levels due to the interaction with the radiation field [

42]. In the density matrix formalism it is necessary to solve the kinetic equation that leads to the system of much larger rank in comparison with conventional amplitude approach that significantly complicates finding the solution [

42,

43,

44,

45,

50,

55]. Moreover, the construction of terms, describing sinks and sources, is not straightforward as during their derivation it is necessary to average over a subset of variables taking account of specific physical conditions [

37,

42,

43,

44,

45]. So, as a rule, these terms could be derivedin a more or less general form only in the impact limit, and their concrete expressions are rather arbitrary [

37,

42,

43,

44,

45,

50,

55]. Moreover, even the formula (1) should be changed to a more general and complex expression for the power that is absorbed or emitted by the medium [

42,

43,

44,

45,

50,

55].

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

Within the assumptions of ST the plasma ions are considered as static [

17,

18,

21], and hence the resultant Stark profiles are called static or quasistatic. We now consider the ion dynamics effects,

i.e., the deviations from the static Stark profiles due to the thermal motion (see, for example, [

7,

8,

9,

12,

17,

18,

20,

21,

23,

24,

25,

26,

27,

28,

29,

30,

31,

32,

33,

34,

35,

36,

74,

75,

76]). If these deviations are small enough (called in earlier works “thermal corrections” [

7,

8,

9,

12,

21,

23,

24,

25]), then it is possible to express them through the second moments of ion microfield time derivatives of the joint distribution functions

W(

,

,

) of the ion electric microfield strength vector

and its first

and second

time derivatives [

5,

6,

7,

8,

9,

12,

21,

23,

24,

25,

48,

67]. The basic idea of Markov construction of these joint distributions is that

,

and

are stochastic independent variables, formed by summation of electric fields or its derivatives over all individual ions of the medium [

1,

5,

6]. So, these probability distribution functions are the many body objects [

5,

6,

7,

8,

9,

12,

21,

23,

24,

25,

48,

67]. However, those joint distributions possess nonzero constraint moments over, for example, microfield time derivatives, the value and direction of electric ion microfield strength vector being fixed [

5,

6,

7,

8,

9,

12,

21,

23,

24,

25,

48,

67]. So, each value of the ion microfield corresponds, for example, in fact to a nonzero “mean” value of the square of its derivative. In other words, it means that by fixing one of the initially independent stochastic variables, the mean values of the other ones become nonzero and functionally dependent on the value of the fixed variable [

5,

6]. So, the direct correlations between fixed stochastic variables with the moments over the other ones under this condition are evident. Another kind of correlations appears if one considers large values of ion microfields, which are produced by the nearest particle (so called “nearest neighbor approximation”) [

5,

6,

7,

8,

9,

12,

21,

23,

24,

25,

48,

67]. In this case there is a direct proportionality between the value of the ion electric field and its time derivative, where the stochasticity is involved due to another stochastic variable—particle velocity. Indeed, the mean square value of particle velocities is a necessary factor in the second moments over the microfield time derivatives [

5,

6,

7,

8,

9,

12,

21,

23,

24,

25,

48,

67].

Consider now the components of the microfield time derivative that are perpendicular and parallel to the direction of the ion microfield strength vector [

5,

6,

7,

8,

9,

12,

21,

23,

24,

25,

48,

67]. By calculating the second moments of the perpendicular and parallel components of time derivative, it is possible to establish relations between them in the limits of small and large reduced microfields values β =

F/

F_{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]:

On the other hand, in the case of β ≫ 1, the ion microfield is formed by the nearest neighbor and the corresponding relation shows preferential direction of the microfield fluctuations along the microfield vector [

5,

6,

24,

25]:

The general expression for

in the case of a Coulomb electric field of point charges is

Here

e is the elementary charge and

designates the mean of the radiator velocity square. The complex ionization composition is accounted for in Equation (4) (note the generalization of the

F_{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:

Similarly to Equation (4), an expression for the parallel component

is

In Equations (4) and (6),

H(β) is the Holtsmark function (1). The other functions are related to it through integral or differential equations (see [

5,

6])

The properties of asymptotics for these functions are given by the relations [

3]

The results of [

5,

6] for joint distribution functions and their moments for the Coulomb potential were generalized within the Baranger–Mozer scheme, accounting for the electron Debye screening and the ion-ion correlations [

38,

48,

67]. The quantities

and

characterize the statistical properties of plasma microfield and play a key role within the idea of thermal corrections to Stark profiles [

7,

8,

9,

12,

24,

25]. The terms proportional to the mean of radiator velocity square in Equations (4) and (6), in square brackets, describe the

fluctuations of microfield, induced by the relative thermal motion of the radiator atoms. The other terms,

proportional to the mean of radiator velocity square in Equations (4) and (6) besides the factor p are due to the effects of ion-dynamics friction on the radiator motion. As one can see, the latter terms could not be made proportional to the reduced mass of the ion-radiator pair. However, the cases when the influence of the effects of ion-dynamics friction on Stark profiles is significant have not been revealed up to now, since the corresponding deviations of profiles turn out to be small. And indeed, the full scale Molecular Dynamics (MD) simulations confirmed that the effects of ion dynamics could be well described by a so-called “reduced mass” model (RM), where the motion of radiator is neglected for moderately coupled plasma with the ion couplingparameter

Γ_{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.

By analyzing the difference between profiles formed for two different reduced masses of the plasma ions, it is seen that within the idea of thermal corrections they are proportional to the second moments of parallel

and perpendicular

components of the ion microfield fluctuations. The general analysis, performed in different approximations [

7,

8,

9,

12,

24,

25], has shown that the ion dynamical perturbations of Stark profiles are controlled by three main mechanisms. The first mechanism is due to the amplitude modulation, induced by the rotation of the atomic dipole along with the rotating ion microfield [

12]. Due to the amplitude modulation, the projections of the atomic dipole on the coordinate axis, being at rest, are changing (or “modulated”), while the atomic dipole rotates together with the rotation of the electric microfield strength vector. The second mechanism exists due to the atomic dipole inertia with respect to the microfield rotation, and results in nonadiabatic transitions between states defined in the frame with the quantization axis along the rotating field direction [

12]. The third mechanism (historically considered the first) is the phase modulation related to changes in the microfield magnitude [

7,

8,

9,

12,

24,

25]. Only this mechanism was taken into account in the earlier works (in the 1950s) [

7,

8,

9], where the Stark broadening by ions was analyzed in the adiabatic approximation, i.e., only within the framework of phase modulation or frequency Stark shift [

21]. As it was demonstrated in the works [

24,

25] within the approach of thermal corrections, the amplitude modulation gives the largest contribution to the Stark contour deformation due to ion dynamics in the vicinity of the line center. The general ideas of amplitude modulation, non-adiabatic effects and usage of the electron broadening for extending the theory of thermal corrections to the line center were proposed by Gennadii V. Sholin.

**Figure 1.**
Function M_{⊥}(x).

**Figure 1.**
Function M_{⊥}(x).

Recall that the thermal corrections are defined as a difference between the total profile, calculated accounting for perturbations due to ion dynamics, and the Stark profile within the ST approach [

24,

25]. In the case of equal temperatures of plasma ions and radiators, as well as electrons, this gives for Ly-alpha (compare with [

24,

25]) in the approximation of isolated individual Stark components (

i.e., neglecting non-diagonal elements of the electron impact broadening operator)

The

f_{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):

In (9) and (10) the

dimensionless function is introduced (see

Figure 1), proportional to the second moment of the microfield time derivative component, perpendicular to the microfield direction according to Equation (4) and defining in fact the mean square of microfield rotation frequency. It is defined in such a way, that

M_{⊥}(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 2.**
Function f_{1−α}(x).

**Figure 2.**
Function f_{1−α}(x).

**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].

**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].

It is interesting to note several properties of the above-mentioned complex function

f_{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.

**Figure 4.**
The function f_{2−}_{β}(x).

**Figure 4.**
The function f_{2−}_{β}(x).

Similar to Equations (9) and (10), result for Ly-beta is [

24,

25].

where

f_{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],

and “

w” is the electron impact width of the Stark sublevel (002), designated by parabolic quantum numbers. The graph of

f_{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.

Expressions (9) and (12) are derived assuming the concrete numerical values of the Stark shifts and dipole matrix elements, calculated in the parabolic basis for corresponding Stark sublevels and components of considered transitions [

17,

18]. The above results are obtained analytically by the perturbation theory for non-Hermitian operators and with the analytical continuation of the microfield distribution function and the second moments of its time derivatives that were shown to possess analytical properties in the upper complex plane (see [

24,

25]). As it follows from the validity conditions of this approach, the main contribution to integrals, describing the amplitude modulation of lines with central components, give regions of detunings near the line center, where the argument of the universal functions, describing the second moments of derivatives, is small. Moreover, the principal term of the expansion, corresponding to the amplitude modulation of the central component, does not depend on the microfield, that allows for integrating over the microfield distribution analytically [

24,

25]. On the other hand, the principal terms of the expansion for the lateral components related to the amplitude modulation and non-adiabatic effects, are obtained analytically via integration in the upper half of the complex plane [

24,

25]. As it follows from the asymptotic properties of these functions in the region of small values of argument, the effective frequency of rotations is practically constant. Due to this, in [

24,

25] the value of the dimensionless function

M_{⊥}(

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]).

In the case of a line without the central component, the ion dynamics corrections are positive in the line center (see

Figure 4). Thus, the intensity in the center increases, while it is decreases in the nearest wings (see

Figure 4). The results of [

24,

25] qualitatively confirm the experimental patterns, observed in [

20,

22,

27,

31]. Also in [

24,

25] the difference profiles, corresponding to two different values of the reduced mass, were considered and compared with the results of experiments [

20,

22] for the Balmer-beta line. It was shown that relative behavior of the thermal corrections versus wavelength detuning from the line center

Δλ describes sufficiently accurately [

24,

25] the experimental results [

20,

22]. In [

24,

25] the difference profiles

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

In Equation (13) f(

x) is the relative behavior of the difference profile in the line center. In [

24,

25] it was shown, that the dependences

versus frequency detunings from the line center, spanned by the profile difference (10), coincide well with the corresponding experimental data for the Balmer-beta line, given in [

20,

22].

So, according to the results and ideas of [

24,

25] and the discussion above, the

difference profiles are proportional to the statistical characteristics of microfield and, more precisely, to the characteristics of the microfield fluctuations, related to the microfield rotations (compare with [

24,

25]).

That is why this property could in principal be used to study microfield statistics in experiments and simulations.

In short time the ideas of [

12,

24,

25] were accepted and the notion of dominating effect of microfield rotation into ion dynamics became widely spread [

30,

33].

Nowadays, the computer simulation technique has become a powerful tool for studying the physics of various non-stationary processes, and particularly plasma microfield ion dynamics effects. However, computer simulations are rather time-consuming and at present impractical for large-scale calculations. Thus, simultaneously with the computer simulations the development of model approaches, that accounted for the ion dynamical effects in an approximate manner, were carried on independently [

32,

33,

34,

35,

36,

37,

39,

41,

46,

49,

53,

54,

56,

59,

60,

61,

62,

63,

64,

68,

69,

70,

71,

72,

73,

74,

75,

76]. The first such model (actually, predating computer simulations) was MMM [

14,

15,

28,

29,

52], later followed by various applications of the BID [

37,

71] and Frequency Fluctuation Model (FFM) [

46,

56,

66] methods. Notably, neither of these models explicitly accounts for the effects of microfield rotation [

24,

25,

30,

33,

53].

It is necessary now to consider the formal conditions of validity of thermal corrections approach [

7,

8,

9,

12,

24,

25]. The results of reference [

12] are applicable only for the lateral Stark components and quasistatic ions, when

and in the spectral region of detunings from the line center Δω, corresponding to the line wings

On the other hand, the spectral region of applicability of the theoretical approach in [

24,

25] is expanded till the line center only due to the additional inclusion into consideration, besides the quasistatic ions -Equation (14), of the electron impact effect, that allowed to analyze the central Stark components, too. However, the applicability criteria of [

24,

25] are rather complicated and depend on the spectral region under consideration. For example, in the line center for the central Stark components the criterion of validity of [

24] has the form

where ρ

_{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

where

I^{(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.

The results of [

12,

24,

25] proved the numerical predominance of amplitude modulation and non-adiabatic effects contributions over the phase modulation one in the line wings, where the perturbation approach of [

12,

24,

25] is applicable practically for any plasma parameter. The magnitudes of amplitude modulation and non-adiabatic effects contributions are of the same order in this region of Stark profiles [

12]. Moreover, the amplitude modulation and non-adiabatic effects contributions have the same sign, which is opposite to the sign of the phase modulation correction [

12]. So, the cancellation of non-adiabatic and atom reorientation effects (amplitude modulation) does not take place, as was proposed earlier by Spitzer in his very instructive papers [

2,

3,

4], and they play the dominant role in the line wings.

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

The work during the preparation of SLSP workshops and along with their conduction revealed unexpected spread of results of various computational models, done for

ion perturbers only (see for example [

74,

75,

76]).

In this respect, the study of directionality correlations, presented at SLSP-1 [

65], inspired the authors to test whether the rotation effects really are responsible for a dominating contribution according to the predictions of [

24,

25].

To this end the Ly-α profile was calculated using a computer simulation (CS) method [

57]. A one-component plasma (OCP) was assumed, consisting only of one type of ions. Furthermore, to avoid effects of plasma non-ideality such as the Debye screening, the ions were assumed moving along straight path trajectories. Time histories of the electric field

(

t), formed by the ions, were stored, to be used as an input when numerically solving the time-dependent Schrödinger equation of the hydrogen atom.

It is instructive to separately analyze how changing the direction, and the magnitude of the microfield influence the line shape. Let us define “rotational” and “vibrational” microfields as

and

respectively (compare with [

75]), where

F_{0} is again the Holtsmark normal field for singly charged ions [

1].

The effect of varying reduced mass was modeled by enabling time dilation of the field histories. Evidently, the field that is changing slower by a factor of

s, corresponds to that, formed by particles moving

s times more slowly,

i.e., with an

s^{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.

The resulting Ly-α profiles are presented in

Figure 5. It is seen that the rotational microfield component has a significantly more pronounced effect on the total line shapes, while changing only the magnitude of the field while keeping its direction constant (the “vibrational” component) has only a minor effect on the shape of the lateral components. Evidently, with no change in the field direction, the central component remains the

δ-function (not shown on

Figure 5b). We note that the width of the central component due to the rotational microfield component increases when μ decreases. This is in a qualitative agreement with Equation (9).

The Ly-α profiles calculated with the full microfields (

Figure 6a) show a resembling behavior: the line HWHM, mostly determined by the central component, scales approximately inversely with s within the range of parameters assumed, while the shape of the lateral components remains mostly unchanged. We note that varying

s may alternatively be interpreted as scaling the temperature according to

T =

T_{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.

We now turn to analyzing the difference profiles defined in the spirit of Equation (13). However, the theory of thermal corrections [

24,

25] was derived perturbatively with the zero-order broadening due to the electron impact effect, while in the present CS calculations no electron broadening was included.

Therefore, it is the ion-dynamical broadening itself that fulfills this role, and one should expect to find a self-similar solution. We checked it by calculating difference between the line shapes calculated with the time-dilation factors s and s′ = s + δs, keeping the δs/s ratio constant, and normalizing the frequency axis of the resulting difference profiles to the line width. In other words, the ion-induced HWHM w_{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 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).

Although it is desirable to keep

δs/s as “infinitesimal” as possible, in practice too small a ratio results in rather noisy profiles due to a finite accuracy of the simulations; for this reason,

δs/s = 1/4, corresponding to

δμ/μ = 9/16, was used. The results are shown in

Figure 6b. Indeed, the normalized difference profiles remain practically the same over the 64-fold variation of

μ tested. Furthermore, the profiles in the central region are qualitatively similar to the prediction of the theory of thermal corrections [

24,

25] (

cf. Figure 2). It appears, however, that the functional form is rather close to

also shown in

Figure 6b. It is easy to see that such a functional form corresponds to a difference between two Lorentzians, confirming the shape of the Ly-α central component inferred from our CS calculations.

**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}.

**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}.