The Frequency Fluctuation Model for the van der Waals Broadening

: The effect of atomic and molecular microﬁed dynamics on spectral line shapes is under 1 consideration. This problem is treated in the framework of the Frequency Fluctuation Model (FFM). 2 For the ﬁrst time the FFM is tested for the broadening of a spectral line by neutral particles. The 3 usage of the FFM allows one to derive simple analytical expressions and perform fast calculations 4 of the intensity proﬁle. The obtained results was compared with Chen and Takeo theory (CT), 5 which is in a good agreement with experimental data. It was demonstrated that for moderate 6 values of temperature and density the FFM successfully describes the effect of the microﬁeld 7 dynamics on a spectral line shape. 8


Introduction
The problem of the microfield dynamics effect on a spectral line shape was recog- 11 nized many years ago [1,2]. The best agreement with the results of Molecular Dynamics 12 (MD) simulations provides the Frequency Fluctuation Model (FFM) [3]. It is based on 13 dividing of a spectral line contour in a static field into separate regions, between which 14 there is an exchange of intensities due to thermal motion. This model is widely used 15 for spectral line shape calculations in plasmas (see e.g. [4][5][6][7][8][9]). It was shown that the 16 FFM is equivalent to the method of the quantum kinetic equation [10]. This approach 17 makes it possible to reformulate the FFM in terms of analytical expressions. Namely, 18 the FFM spectral line shape can be considered as the functional of the static profile. 19 This circumstance allows one to use simple analytical expressions and perform fast 20 calculations of a spectral line shape for arbitrary values of temperatures and densities.

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The FFM was formulated for spectral line shape calculations in plasmas. It was 22 never used for the Stark broadening by neutral particles-to the best of our knowledge.

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Most of material on the van der Waals broadening accumulated by the end of 1960s is 24 presented in the famous review of S. Chen and M. Takeo [11]. Moreover, in this paper 25 the authors presented the line shape calculation method (based on the work of P.W. 26 Anderson and J.D. Talman [12]) for arbitrary values of temperatures and densities. The

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Chen and Takeo results (CT) are very old. Nevertheless, it is in a good agreement with 28 experimental data and can be used for the examination of the FFM theory. Moreover, the 29 analytical expressions for the line shape presented in [11] are very cumbersome. The  The FFM has a drawback. It doesn't reproduce the impact width correctly, as it was 34 directly shown in [8,13]. However, this problem was solved for the linear Stark effect.

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The resolution consists in the dependence of the jumping frequency, which characterizes 36 the rate of change of the microfield, on the energy shift. This dependence was obtained 37 by comparing of the analytical calculations with MD in the paper [8]. The authors of 38 the work [13] overcame this problem in the alternative way. They used the asymptotic 39 expression of the jumping frequency obtained by S. Chandrasekhar and J. von Neumann 40 [14], who derived this for the description of the stellar dynamics.

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The detailed analysis of the FFM results for different plasma parameters is presented 42 in the paper [13]. It was demonstrated that the account of the dependence of the jumping 43 frequency on the energy shift leads to the correct behaviour of the spectral line width  In this case the problem of reproducing the impact width wasn't solved in this way.

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However, for realistic plasma parameters the FFM with the constant jumping frequency 51 provides quite satisfactory results.

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Usually, the van der Waals broadening can be described in terms of the impact 53 approximation [15]. So, the main goal of the present paper is the examination of the FFM 54 for the van der Waals interaction. The second aim of this work is to provide the simple 55 analytical algorithm for the line shape calculations for low temperatures (less that 300K) 56 or high densities. In the present paper we are focusing on the spectral broadening by neutral atoms 59 and molecules. The detailed description of the analytical FFM formulation and the 60 discussion of the the dynamics effect on the spectral line formation are given in [13]. For 61 convenience, an abridged version is presented below.

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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 where Here, N and v T are, respectively, the density and the thermal velocity of interacting particles; C S n is the constant of the Stark effect; Ω J is called the jumping frequency. The potential of binary interacting particles has the following form In the present paper we consider the case of n = 6, which corresponds to a wide 63 class of van der Waals interactions.

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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 the expression (4) it equals to [15] The 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 Using the results from [15] we can estimate the parameter ν. For T ≈ 300 − 5000 K ν approximately equals to ν ≈ (10 35 − 10 38 )N −5/3 , where N is expressed in cm −3 .

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All over the paper atomic units are used 1 . Also, for the sake of simplicity, we use 66 the reduced detuning: z = ∆ω/Ω n S .

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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. [10]. The frequency of the exchange equals to ν. The solution of this equation leads to the following expression for the resulting profile where W(z) is the normalized static profile. Note, that when ν → 0 I(z) turns into W(z).

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The formula (8) can be rewritten in more convenient form where There is a useful relation between the functions J 0 (z) and J 2 (z) S. Chen and M. Takeo presented the analytical expression for the intensity profile 69 for an arbitrary value of ν for the van der Waals broadening [11] : The function ψ(x) can be approximated by the simple functions: where Γ(z) is the gamma-function.

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The expression (12) turns into the static profile W(z) when ν approaches zero. In 73 order to derive W(z) it is necessary to use the approximation for ψ(x) for small values 74 of x.

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Using the first relation of (13) and the stationary phase approximation one obtains the following formula for the static profile The result (14) is in agreement with the well-known formula derived by H.Margenau 76 [16].

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The examination of the FFM consists in the comparsion of the CT formula (12) with the expression (9). In the integrand of (10) we will substitute the static profile (14). In the impact limit (ν 1) we can do a simple estimation of the impact width. Namely, for large values of ν the spectral line shape turns into the Lorentzian where γ is the width of the profile and z 0 is the coordinate of the center of the Lorentz 78 profile.

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Note, that the CT theory reproduces the impact limit. Indeed, the usage of the 80 second relation from (13) leads to the expression (15).

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In order to estimate the impact width for the FFM we will use the following property of a wide class of normalized profiles: The relation (16) is valid here, because in the impact limit γ and z 0 always have the same dependence on ν (see e.g. [15]). So, it is obvious that the property (16) works for the Lorentz profile (15). Using the asymptotic behavior of the formula (14) we can estimate J k (0): From the relations (16) and (17) it is easy to see that γ = γ FFM ∼ ν 1/2 . Thus, the 82 FFM doesn't reproduce the behavior of the profile width in the impact limit. Indeed, it 83 is the well-know result of the impact theory [15] is γ = 8.   The comparsion of the CT theory (in the impact limit) with the FFM profile is presented