**1. Introduction**

The problem of the microfield dynamics effect on a spectral line shape was recognized many years ago [1–3]. There are different methods which were developed to solve this problem [4–6]. However, the best agreemen<sup>t</sup> with the results of Molecular Dynamics (MD) simulations provides the Frequency Fluctuation Model (FFM) [7]. It is based on dividing a spectral line contour in a static field into separate regions, between which there is an exchange of intensities due to thermal motion. This model is widely used for spectral line shape calculations in plasmas (see, e.g., [8–13]). It was shown that the FFM is equivalent to the method of the quantum kinetic Equation [14]. This approach makes it possible to reformulate the FFM in terms of analytical expressions. Namely, the FFM spectral line shape can be considered as the functional of the static profile. This circumstance allows one to use simple analytical expressions and perform fast calculations of a spectral line shape for arbitrary values of temperatures and densities.

The FFM was formulated for spectral line shape calculations in plasmas. It was never used for the Stark broadening by neutral particles—to the best of our knowledge. Most of the material on the van der Waals broadening accumulated by the end of the 1960s is presented in the famous review of S. Chen and M. Takeo [15]. Moreover, in this paper, the authors present the line shape calculation method (based on the work of P.W. Anderson and J.D. Talman [16]) for arbitrary values of temperatures and densities. The Chen and Takeo results (CT) are very old. Nevertheless, they are in good agreemen<sup>t</sup> with experimental data and can be used for the examination of the FFM theory. Moreover, the analytical expressions for the line shape presented in [15] are very cumbersome. The direct use of the CT theory in numerical calculations requires significant computational resources. Therefore, the FFM may provide simple analytical expressions for the description of the van der Waals broadening.

The FFM has a drawback. It does not reproduce the impact width correctly. This circumstance was noted in the original paper [7]. Furthermore, it was directly shown

**Citation:** Letunov, A.; Lisitsa, V.; Astapenko, V. The Frequency Fluctuation Model for the van der Waals Broadening. *Foundations* **2021**, *2*, 200–207. https://doi.org/ 10.3390/foundations1020015

Academic Editor: Eugene Oks

Received: 6 October 2021 Accepted: 25 October 2021 Published: 29 October 2021

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in [12,17]. However, this problem was solved for the linear Stark effect. The resolution consists of the dependence of the jumping frequency, which characterizes the rate of change of the microfield on the energy shift. This dependence is obtained by comparing the analytical calculations with MD in the paper [12]. The authors of the work [17] overcame this problem in the alternative way. They used the asymptotic expression of the jumping frequency obtained by S. Chandrasekhar and J. von Neumann [18], who derived this for the description of the stellar dynamics.

The detailed analysis of the FFM results for different plasma parameters is presented in the paper [17]. It is demonstrated that the accounting for the dependence of the jumping frequency on the energy shift leads to the correct behaviour of the spectral line width in the impact limit for the linear Stark effect. The difference between the modified and the original versions of the FFM for moderate values of temperatures and densities is insignificant. Therefore, accounting for the jumping frequency dependence on the field strength yields the simple analytical theory which provides the correct results for a wide range of plasma parameters. The situation is more complicated for the quadratic Stark effect. In this case, the problem of reproducing the impact width is not solved in this way. However, for realistic plasma parameters, the FFM with the constant jumping frequency provides quite satisfactory results.

Usually, the van der Waals broadening can be described in terms of the impact approximation [19]. Furthermore, there are modern works related to the spectral line broadening by neutral atoms and molecules (see, e.g., [20,21]). Therefore, the main goal of the present paper is the examination of the FFM for the van der Waals interaction. The second aim of this work is to provide the simple analytical algorithm for the line shape calculations for low temperatures (less that 300 K) or high densities.

### **2. Description of the Method**

In the present paper, we focused on the spectral broadening by neutral atoms and molecules. The detailed description of the analytical FFM formulation and the discussion of the dynamics effect on the spectral line formation are given in [17]. For convenience, an abridged version was presented below.

The effect of a multiparticle electric field on molecular and atomic spectra is characterized by the ratio of the jumping frequency and the Stark shift:

> *ν*

$$\mathbf{I} = \frac{\boldsymbol{\Omega}\_{\text{I}}}{\boldsymbol{\Omega}\_{\text{S}}^{\text{n}}} \, \tag{1}$$

where

$$
\Omega\_I = N^{1/3} \upsilon\_{T\_{I'}} \tag{2}
$$

$$
\Omega^{\rm n}\_{\rm S} = \mathbb{C}^{\rm S}\_{n} N^{\rm \frac{\mu}{\cdot}}.\tag{3}
$$

Here, *N* and *vT* are, respectively, the density and the thermal velocity of interacting particles; *C<sup>S</sup> n* is the constant of the Stark effect; <sup>Ω</sup>*J* is called the jumping frequency. The potential of binary interacting particles has the following form:

$$V(r) = \frac{\mathbb{C}\_n^S}{\mathbb{R}^n}.\tag{4}$$

In the present paper we considered the case of *n* = 6, which corresponds to a wide class of van der Waals interactions. The interaction between atoms and molecules often defers from (4). However, we focused on the examination of the FFM by the comparison with CT results, which were obtained for the potential (4).

The transition from binary to non-binary type of interactions is often characterized by the number of particles is the Weisskopf sphere *h*. For *n* = 6 in Expression (4), it equals to [19]

$$h = \left(\frac{3\pi}{8} \frac{C\_6^S}{\upsilon\_T}\right)^{3/5} N. \tag{5}$$

Parameter (5) also determines the transition from the static theory to the impact limit. The number of particles in the Weisskopf sphere is connected with *ν* by the simple relation

$$
\hbar \sim \nu^{-3/5}.\tag{6}
$$

Using the results from [19], we could estimate the parameter *ν*. For *T* ≈ 300 − 5000 K, *ν* approximately equals to:

$$
\nu \approx (10^{35} - 10^{38}) N^{-5/3} \,\mathrm{J} \tag{7}
$$

where *N* is expressed in cm<sup>−</sup>3.

All over the paper, atomic units were used (*e* = *m* = *h*¯ = 1, where: *e*—the elementary charge; *m*—the mass of an electron; *h*¯ is the Planck constant). Furthermore, for the sake of simplicity, we used the reduced detuning: *z* = <sup>Δ</sup>*ω*/Ω*nS*.

The FFM procedure is equivalent to the method of the kinetic equation with strongcollision integral, which describes an intensity exchange between different regions of a static profile [14]. The frequency of the exchange equals to *ν*. The solution of this equation led to the following expression for the resulting profile:

$$I(z) = \frac{1}{\pi} \text{Re} \frac{\int \frac{W(z')dz'}{\nu + i(z - z')}}{1 - \nu \int \frac{W(z')dz'}{\nu + i(z - z')}},\tag{8}$$

where *<sup>W</sup>*(*z*) is the normalized static profile. Note, that when *ν* → 0, *<sup>I</sup>*(*z*) turns into *<sup>W</sup>*(*z*). The Formula (8) can be rewritten in more convenient form:

$$I(z) = \frac{\nu}{\pi} \frac{J\_0(z)J\_2(z) - J\_1^2(z)}{J\_2^2(z) + \nu^2 J\_1^2(z)},\tag{9}$$

where

$$J\_k(z) = \int\_{-\infty}^{+\infty} \frac{\mathcal{W}(z') (z - z')^k dz'}{\nu^2 + (z - z')^2},\tag{10}$$

There is a useful relation between the functions *J*0(*z*) and *J*2(*z*):

$$J\_2(z) = 1 - \nu^2 J\_0(z). \tag{11}$$

S. Chen and M. Takeo presented the analytical expression for the intensity profile for an arbitrary value of *ν* for the van der Waals broadening [15]:

$$I^{(\mathcal{C}T)}(z) = \mathcal{C} \cdot \operatorname{Re} \left\{ \int\_{-\infty}^{\infty} \exp \left[ iz\tau - \frac{1}{\nu^{3/5}} \psi \left( \nu^{6/5} \tau \right) \right] d\tau \right\},\tag{12}$$

where

$$\mathcal{C} = \left[ \int\_{-\infty}^{+\infty} \text{Re} \left\{ \int\_{-\infty}^{\infty} \exp \left[ i z \tau - \frac{1}{\nu^{3/5}} \Psi \left( \nu^{6/5} \tau \right) \right] d\tau \right\} dz \right]^{-1}.$$

The function *ψ*(*x*) can be approximated by the simple functions:

$$\begin{cases} \psi(\mathbf{x} \ll 1) = \frac{(2\pi)^{3/2}}{3} \sqrt{\mathbf{x}} (1 - i),\\ \psi(\mathbf{x} \gg 1) = \frac{2\pi}{5} \left(\frac{3\pi}{8}\right)^{-3/5} \Gamma\left(\frac{2}{5}\right) \left(\cos\frac{\pi}{5} - i\sin\frac{\pi}{5}\right) \mathbf{x} + 1.91 + 2.63i, \end{cases} \tag{13}$$

where <sup>Γ</sup>(*z*) is the gamma function.

Expression (12) turned into the static profile *<sup>W</sup>*(*z*) when *ν* approached zero. In order to derive *<sup>W</sup>*(*z*), it was necessary to use the approximation for *ψ*(*x*) for small values of *x*.

Using the first relation of (13) and the stationary phase approximation, one obtains the following formula for the static profile:

$$\mathcal{W}(z) = \begin{cases} 0, z \ge 0\\ \frac{2\pi}{3|z|^{3/2}} \exp\left[\frac{4\pi^3}{9z}\right], z < 0 \end{cases} \tag{14}$$

Result (14) is in agreemen<sup>t</sup> with the well-known formula derived by H. Margenau [22]. The examination of the FFM consisted of the comparison of the CT Formula (12) with Expression (9). In the integrand of (10), we substituted the static profile (14). In the impact limit (*ν* ! 1), we could perform a simple estimation of the impact width. Namely, for large values of *ν*, the spectral line shape turned into the Lorentzian:

$$L(z) = \frac{1}{\pi} \frac{\gamma}{\gamma^2 + (z - z\_0)^2},\tag{15}$$

where *γ* is the width of the profile and *z*0 is the coordinate of the centre of the Lorentz profile.

Note, that the CT theory reproduced the impact limit. Indeed, the usage of the second relation from (13) led to Expression (15).

In order to estimate the impact width for the FFM, we used the following property of a wide class of normalized profiles:

$$F(0) \sim \frac{1}{\gamma}.\tag{16}$$

Relation (16) was valid here, because in the impact limit, *γ* and *z*0 always have the same dependence on *ν* (see, e.g., [19]). Therefore, it was obvious that Property (16) worked for the Lorentz profile (15). Using the asymptotic behaviour of Formula (14), we could estimate *Jk*(0):

$$f\_0(0) \sim \frac{1}{\upsilon^2},$$

$$f\_1(0) \sim \int\_\upsilon^{+\infty} dz' \frac{1}{z'^{1/2}(z'^2 + \upsilon^2)} \sim \frac{1}{\upsilon^{3/2}},$$

$$f\_2(0) \sim \int\_\upsilon^{+\infty} dz' \frac{z'^{1/2}}{z'^2 + \upsilon^2} \sim \frac{1}{\upsilon^{1/2}},$$

$$I(0) \sim \frac{1}{\upsilon^{1/2}}.\tag{17}$$

From Relations (16) and (17), it was easy to see that *γ* = *γFFM* ∼ *ν*1/2. Thus, the FFM did not reproduce the behaviour of the profile width in the impact limit. Indeed, it was the well-know result of the impact theory [19], which was *γ* = 8.16*ν*3/5. The difference in the dependence of the impact width on the power of velocity was very small *<sup>γ</sup>impact*/*γFFM* ∼ *ν*1/10.

### **3. The Results of Numerical Calculations**

In order to test the FFM procedure, we compared it with the CT results. The numerical calculations showed that, for *ν* ∼ 1, the FFM was in good agreemen<sup>t</sup> with CT. The example of it is demonstrated in Figure 1, where one can see the comparison of two theories for *ν* = 1.5. However, the FFM did not reproduce the impact width correctly (Figure 2) as it was shown in the previous section by the analytical estimations. The comparison of the CT theory (in the impact limit) with the FFM profile is presented in Figure 2. The full width at half maximum of the FFM profile was approximately 1.5 as large as the impact width for *ν* = 100. With an increase in the parameter *ν*, the FFM profile slowly became wider than the impact line shape. This circumstance was connected with the slow growth of the ratio: *<sup>γ</sup>impact*/*γFFM* ∼ *ν*1/10. Moreover, we only knew how *γFFM* depended on *ν*. Uncertainty in the value of the prefactor played an important role in the impact width determination. However, the derivation of the certain value of the FFM impact width prefactor by analytical calculations is a complex problem. It might be determined numerically as in the work of [17], but it is not reasonable to do so until the dependence of the jumping frequency on the detuning is determined. The discrepancy between two graphs became insignificant for *ν* ∼ 1000 (Figure 3). According to Estimation (7), this value of *ν* for *T* = 300–5000 K corresponded to *N* = 1019–1021 cm<sup>−</sup>3.

The discrepancy in the impact limit might have most likely been eliminated by accounting for the dependence of the jumping frequency on the detuning. It was shown that, for the linear Stark effect, it led to the correct result for large values of *ν* [17]. However, in order to determine this dependence, it was necessary to describe complicated dynamics of the van der Waals forces.

**Figure 1.** The normalized intensity profile of one spectral component as the function of the reduced energy shift. Comparison of the FFM profile with the Chen and Takeo theory; *ν* = 1.5.

**Figure 2.** The normalized intensity profile of one spectral component as the function of the reduced energy shift. Comparison of the FFM profile with the impact theory; *ν* = 100.

**Figure 3.** The normalized intensity profile of one spectral component as the function of the reduced energy shift. Comparison of the FFM profile with the impact theory; *ν* = 1200.
