Combined effect of small- and large-angle scattering collisions on a spectral line shape

Journal of Quantitative Spectroscopy & Radiative Transfer 159 (2015) 32–38

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Combined effect of small- and large-angle scattering collisions on a spectral line shape V.P. Kochanov a,b,n a V.E. Zuev Institute of Atmospheric Optics, Siberian Branch, Russian Academy of Sciences, 1 Academician Zuev Square, 634021 Tomsk, Russia b Tomsk State University, Tomsk 634050 Russia

a r t i c l e in f o

abstract

Article history: Received 3 January 2015 Received in revised form 17 February 2015 Accepted 19 February 2015 Available online 10 March 2015

Algebraic approximations for line profiles calculated on the basis of quantum-mechanical collision integral kernels for dipole–dipole, dipole–quadrupole, and quadrupole–quadrupole intermolecular interaction potentials were obtained. In derivation of the profiles velocity-changing collisions of molecules with scattering on small and large angles also with the speed-dependence of collision relaxation constants have been taken into account. It was shown that the relative contribution of small-angle collisions into the frequency of elastic velocity-changing collisions is much more pronounced for long-range potentials. A sensitive criterion for analysis of a line narrowing was proposed and tested. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Line profiles Soft collisions Line narrowing Speed-dependence

1. Introduction As is known, elastic velocity-changing collisions lead to Dicke's line narrowing [1,2] realizing at gas pressures low enough to provide noticeable Doppler line broadening. The physical mechanism of this phenomenon consists in compensation to some extent of the Doppler broadening due to random changes of the direction of a velocity of the absorbing molecule performing Brownian motion within the area whose size (the length of diffusion during light absorption) is restricted by a wavelength. In other words, standing molecule absorbs radiation in conditions when chaotic Doppler frequency shifts are minimized in average in the course of absorption. A spatial localization is needed for elimination of the phase modulation caused by

n Correspondence to: V.E. Zuev Institute of Atmospheric Optics, Siberian Branch, Russian Academy of Sciences, 1 Academician Zuev Square, 634021 Tomsk, Russia. Tel.: þ 7 3822491111, þ 7 3822491110; fax: þ 7 3822492086. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.jqsrt.2015.02.017 0022-4073/& 2015 Elsevier Ltd. All rights reserved.

translational motion of the absorbing molecule that gives rise to additional line broadening. If the interaction of molecule with light is long enough (i.e. the pressure broadening constant is small), then the line narrowing is pronounced. From here it is evident that the main contribution to this effect is caused by large-angle scattering collisions (LASC) with changes in velocity of the order of the most probable speed v. Conventional hard collision model [3,4] describes this situation whereas hard collisions leading by definition to the establishing of equilibrium Maxwell distribution over velocities after each collision give considerable changes in velocity in average. Collisions with a small-angle scattering (SASC) give much smaller changes in a velocity and much greater diffusion length that does not match the conditions for line narrowing. Therefore, in the case where such collisions prevail and the velocity-changing collisions with a largeangle scattering practically are absent, the narrowing must be negligible and hence the respective line profile will be flatter and closer to the Voigt one; scattering on zero angles gives the Voigt profile exactly.

V.P. Kochanov / Journal of Quantitative Spectroscopy & Radiative Transfer 159 (2015) 32–38

In this connection, it is significant to note that the widespread soft collision model [4–6] describes the line narrowing as well as the hard collision model does (e.g., see Ref. [7]), in spite of the given above qualitative argumentation of the absence of the collision line narrowing in the pure case of small-angle scattering. The reason for this lies in the fact that soft collision model [4–6] is based on the representation of the initial integral form of the collision integral in master equations for the density matrix as a right-hand part of the differential Fokker– Planck (diffusion) equation in the velocity space [4,6]. Though the supposition of light perturbing particles and hence small average changes in a velocity is exploited in this derivation (and the term “soft collisions” originates from here [4]), the obtained diffusion equation actually describes all sorts of collisions, SASC and LASC.1 Moreover, namely the large-angle scattering collisions play the major role in forming the diffusion coefficient [8] and the line narrowing. Consequently, the term “soft” as regards the model [4–6] is not quite adequate since it implies LASC as well as SASC, and due to this circumstance, this model is merely an alternative variant of the hard collision model. Indeed, the line profiles from both the models are hardly distinguished [7] and attempts to combine them [9,10] in order to improve the physical meaning of the line shape models gave no any results on a relationship between SASC and LASC in forming a line shape, taking into account that true soft velocity-changing collisions can be attributed only to the small-angle scattering. At the same time, the simultaneous account of soft and hard collisions (or SASC and LASC) in a spectral line profile is quite necessary because SASC make the line profile noticeably flatter as compared with the line profiles derived in the hard collision models. This action of SASC immediately follows from the above-considered mechanism of the collision narrowing and it was confirmed in Ref. [11] where a line profile was derived on the basis of the explicit introducing of SASC via a separate part of the collision integral additional to the common term representing hard collisions. In principal, this model allows determining the ratio of SASC and LASC frequencies by comparison of the line profile [11] with recorded line profiles. For all types of intermolecular interaction, except for dipole–dipole one, a line profile is affected by the wind effect caused by translational motion of molecules, which creates the anisotropy of perturbation and implies the dependence of collision relaxation constants on the molecular speed [12–16]. As distinct from Dicke's line narrowing, the wind effect is not associated with the Doppler line broadening, but it also produces the line narrowing [17]. Both these mechanisms of narrowing superpose in most frequently used range of gas pressures where inhomogeneous line broadening takes place. The speed-dependent Nelkin Ghatak line profile [15,18] describes this situation 1 It seems astonishing that the supposition of small changes in a velocity gives the equation that well describes also the opposite case of large velocity's changes. But due to this fortunate accidental operation of mathematics, the effect of LASC on the line profile [4–6] remains hidden up to the present.

33

in the framework of the hard collision model. But, since Dicke's narrowing is rather reduced due to the action of SASC, the line profile [15,18] overestimates the actual line narrowing. In order to overcome this imperfection of the theory revealing in comparison of the latter with an experiment, often the parameters characterizing the speed-dependence2 are treated as adjusting parameters in fitting of a model line profile to experimental ones (e.g., see Ref. [19]). Another approach, called “partially-correlated speed-dependent hard-collision model” (see Ref. [20] and references therein), consists in diminishing the speed dependence of the collision line broadening constant (and hence diminishing the line narrowing caused by the wind effect) by definition of the collision relaxation constant as a sum of speed-dependent and speed-independent parts, the ratio of which appears as an additional fitting parameter. The third manner to avoid disagreement between theory and experiment caused by disregard of SASC is ignoring Dicke's line narrowing and usage the speeddependent Voigt line profile [12]. This may be justified in the case when SASC dominates LASC and gas pressures are sufficiently high. As is shown in Section 2, such situation is plausible for long-range intermolecular potentials. Finally, the model line profile [11] that explicitly takes into account the SASC is valid only for dipole–dipole intermolecular interaction when the wind effect is absent. In the recent paper [17] numerical calculations of line profiles were performed that are based on the Rautian  Shalagin quantum-mechanical kernel of the collision integral [21,22] with the use of the differential scattering cross section calculated for the isotropic dispersion intermolecular interaction potential. Thus, the joint action of Dicke's line narrowing including SASC and LASC and the wind effect have been simultaneously taken into account in a form excluding artificial assumptions. The parameterized line profile was developed on the base of the calculated profiles that contains a minimal number of adjusting parameters. The goal of this paper is to carry out calculations of line profiles analogous to those of Ref. [17] for intermolecular interaction potentials proportional to r  n with n¼3, 4, and 5, where r is the distance between colliding molecules, and to develop respective analytical approximations for these line profiles suitable for experimental data processing. Also the questions on how the type of intermolecular potential affects the data processing and which are the indicators of line narrowing intrinsic to Dicke's and the wind effects in the presence of SASC are under consideration.

2. Line profiles for dipole–dipole, dipole–quadrupole, and quadrupole–quadrupole intermolecular interaction potentials Numerical calculations of the line profiles for intermolecular interaction potentials U(r) ¼ Cn/rn with n ¼3, 4, 2 Supposed to be fixed; there are the mass ratio of perturbing and absorbing molecules and the parameters characterizing the intermolecular interaction potential.

34

V.P. Kochanov / Journal of Quantitative Spectroscopy & Radiative Transfer 159 (2015) 32–38

and 5, where Cn are related constants, have been carried out following the detailed scheme of calculations presented in Ref. [17]. Computations were performed in the one-dimensional velocity approximation having a systematic erroro 1% [18]. Algebraic approximations for calculated profiles analogous to that obtained for the Van der Waals interaction with n ¼6 [17] are given in the appendix. Systematic errors of the approximations are of the same order of magnitude, r 0.5%, as for the case of n ¼6 (see Fig. 12 in Ref. [17]). Note, that the parameter α in Eqs. (A1)–(A5), in accordance with its definition (Eq. (15) in Ref. [17]), represents the ratio of the total elastic velocitychanging collision frequency including SASC and LASC to the effective collision line half-width, while the parameter αeff characterizes real line narrowing caused by both Dicke's and the wind effects. Thus, the difference α  αeff 40 indicates the contribution of SASC to forming a line profile. An example of the calculated kernels of the collision integral (defined by Eq. (13) in Ref. [17]) for the quadrupole–quadrupole interaction (n¼5) is presented in Fig. 1 and it illustrates the relation between cross-sections of SASC and LASC. The ratios of parts of the input collision integral frequencies formed by SASC with the velocity changes within jδvj r0:1v to the total input frequencies are plotted in Fig. 2 for interaction potentials of different types (n¼3–6). As it could be expected, these ratios enlarge for long-range potentials (n¼3 and 4) and light perturbing molecules. In particular, SASC evidently dominate in the case of the dipole– dipole intermolecular interaction potential that can be the reason of relatively small deviations of the speed-dependent Voigt line profile [12] from the line profile Eqs. (A1) and (A2) derived with simultaneous account of SASC and LASC. In order to clarify how does the type of intermolecular potential influence on the parameters retrieved from the data processing we have fitted the line profiles Eqs. (A1)– (A4) to the line profile Eqs. (A1) and (A5) calculated at specified values of the parameters. The base line was fixed

1

1.0 0.9 0.8 0.7

R

1

0.6 0.5

2

0.4 0.3

3

0.2 0.1

3

5

Fig. 2. The ratio of the small-angle part of the collision integral's input frequency restricted by changes of the velocity within 0.1v to the total input frequency as a function of the index n in the dependence 1/rn of the intermolecular interaction potential on the distance r between colliding molecules at the mass ratios β ¼ 0.2 (curve 1), 1 (2), and 4 (3).

Table 1 Parameters Γn and αn retrieved from fitting the line profiles Eq. (A1)–(A4) to the line profile Eqs. (A1) and (A5) calculated at various values of its parameters α, β, and Γ. The parameter αeff retrieved from fitting coincides with that calculated with Eq. (A5) at specified values of α, β, and Γ. β 0.25

α 2.5

10

1

2.5

Γ=kv

αeff

n

Γn/Γ

αn/α

0.25

0.263

1

0.311

0.25

0.838

1

0.976

0.25

0.923

1

0.991

0.25

3.098

1

3.206

0.25

1.726

1

2.039

0.25

5.742

1

6.344

3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5

1.045 1.007 1.009 1.065 1.026 1.028 0.997 0.941 0.943 1.057 1.018 1.021 1.205 1.073 1.076 1.222 1.089 1.091 1.148 1.023 1.026 1.218 1.085 1.088 1.612 1.233 1.235 1.606 1.228 1.231 1.638 1.253 1.255 1.615 1.268 1.270

27.54 4.162 2.441 27.68 4.300 2.437 21.96 3.496 1.822 21.72 3.656 1.956 26.92 5.063 2.945 27.31 5.239 2.959 22.58 4.558 2.577 22.08 4.721 2.744 24.44 5.935 3.659 30.14 7.453 4.604 20.37 5.492 3.408 23.47 19.48 12.56

1

-1

10

2

-2

3

-3 4

2.5

-4 -4

-3

-2

-1

0

v1 /v

1

2

3

4

Fig. 1. The kernels of the collision integral calculated for the dipole– quadrupole intermolecular potential (n¼5) at the mass ratio of perturbing and absorbing molecules β ¼1 as a functions of the velocity v1 of the absorbing molecule before the collision. The velocities after the collision are v¼0 (curve 1), v(2), and 2v (3). The vertical dashed lines demarcate the range of small-angle scattering.

6

n

0

log10 A, a.u.

4

10

V.P. Kochanov / Journal of Quantitative Spectroscopy & Radiative Transfer 159 (2015) 32–38

at zero. The results of fitting are presented in Table 1. As is seen from the table, for relatively light perturbing particles (β r1, where β is the mass ratio of perturbing and absorbing molecules) the difference between collision half-widths Γn found from fitting of the line profiles Eqs. (A1)–(A4) with n ¼3, 4, and 5 and the value of Γ used in calculations of the reference line profile Eqs. (A1) and (A5) is not too great. But for heavy perturbing molecules (β ¼4) it is large and reaches tens of percent. The parameters αn are much more sensitive to the type of an interaction potential, and the dipole–dipole interaction (n ¼3) is the most distinctive one in this regard.

3. The dependence of the narrowing parameter on a gas pressure Testing various model line profiles accounting the line narrowing can be performed by their comparison with an experiment made in the range of pressures where the inhomogeneous line broadening takes place. A standard manner of testing is checking residuals between experimental and best-fitted profiles. At the same time, the minimal residual is necessary but insufficient criterion since some line profiles contain redundant parameters with unclear physical meaning, which can provide minimal residuals without trends. Thus, additional criteria are desirable that can ascertain fine features of recorded line shapes caused by line narrowing arising due to Dicke's and the wind effects. To our opinion, the behavior of the narrowing parameter retrieved from fitting the line profile under examination to an experimental or to another model line profile as a function of a gas pressure can serve as such criterion. Indeed, in the case of full compliance between two taken for comparison profiles the narrowing parameter will depend on pressure linearly; otherwise, this dependence will be nonlinear. Hence, the marked nonlinearity indicates the intrinsic divergence between line profiles that may be hidden in view of satisfaction of the requirement for a minimal residual. Earlier this criterion was used in Ref. [18] where the effective dimensionless narrowing parameter obtained by fitting speedindependent line profile [3,4] to the calculated values of the speed-dependent Nelkin–Ghatak line profile [15,18] reveals the characteristic nonlinear dependence on a gas pressure (see Fig. 3b in Ref.[18]). The fitting both hard collision model profiles [3,4] and [15,18] to experimental profiles recorded for methane 6107.17 cm  1 line displays the nonlinear dependence of the narrowing parameter on pressure of a different kind as compared with the case mentioned above (see Fig. 11 in Ref. [23]) that may be associated with the disregard of SASC in the used profiles (not excluding other physical reasons). Since the observed deviations from linearity are well pronounced, the discussed criterion is sensitive enough. In this section, we examine the qualitative features of the dependencies of the effective narrowing parameter on a gas pressure caused by two mechanisms of line narrowing inherent in Dicke's and the wind effects. Taking the most simple line profile [3,4] as an indicator of narrowing, we will fit it with the aid of least squares technique to the

35

calculated line profiles that contain or not the effect of SASC and the dependence of relaxation constants on speed. First consider the pure action of SASC without the wind effect. The respective line profile is derived in Ref. [11] and reads   1 pffiffiffiffi 1 S K sh ðΩÞ ¼ pffiffiffiffi Re wsh  π νh =ðkvÞ ; π kv   Z 1   2 ν  iΩ νs wsh ¼ pffiffiffiffi exp t 2  2 tþ tan  1 2ϑt dt; kv ϑkv π 0

ϑ ¼ Δ=v; ð1Þ

where Ω is the frequency detuning from the line center, ν is the output collision integral frequency, νs and νh are the input collision integral frequencies for SASC and hard collisions respectively, Δ is the half-width at the 1/e height of the SASC integral kernel (Eq. (3) in Ref. [11]), k is the wave number, v is the most probable thermal velocity, and S is the line intensity. In the limit of νs or ϑ-0 the line profile in Eq. (1) transforms to the hard collision model speed-independent line profile [3,4]

 1 S Ω þ iν pffiffiffiffi K NG ðΩÞ ¼ pffiffiffiffi Re w  1  π νh =ðkvÞ ; kv π kv wðzÞ ¼ e  z erfcð  izÞ; 2

ð2Þ

where erfc(z) is the complex probability function [24]. Let us define the dimensionless narrowing parameters and the ratio of frequencies νs/ν as

νh ; Γ sh ¼ ν  νs  νh ; Γ sh ν αh ¼ h ; Γ h ¼ ν  ν h : Γh αsh ¼

ν ξ ¼ s; ν ð3Þ

Here the first line represents the parameters that determine the profile Ksh Eq. (1) and the second line is associated with the profile KNG Eq. (2). Fitting the profile KNG Eq. (2) with zero base line to the profile Ksh Eq. (1) calculated at different values of Γsh, ξ, and fixed αsh ¼1 gives the values of Γh E Γsh and the values of αh quite different from αsh (Fig. 3). Namely, due to the fact that the profile Ksh is flatter than the profile KNG Eq. (2) (see Fig. 1 in Ref. [11]), the best-fitted narrowing

h

1.0

1

0.8

2

0.6

3

0.4

4

0.2 0.0 0.0

0.5

1.0

1.5

2.0

2.5

/kv sh Fig. 3. The dependence of the best-fitted parameter αh in Eq. (2) on the dimensionless gas pressure Γsh/(kv) at the parameters of Eq. (1) ξ ¼0.5 (curves 1 and 3), 0.8 (curves 2 and 4); ϑ ¼0.1 (curves 1 and 2), 0.25 (curves 3 and 4), and αsh ¼1 (Eq. (3)).

36

V.P. Kochanov / Journal of Quantitative Spectroscopy & Radiative Transfer 159 (2015) 32–38

parameter αh is lesser than the value αsh ¼1 used in calculations of Ksh, and this difference enlarges with the increase in the relative contribution of SASC given by the parameters ξ and ϑ. Monotonous decrease in αh with the increase in a gas pressure (Γsh) can be explained as follows. When the pressure is low, the effective width of the distribution of a molecular polarization (light-induced dipole moment) over velocities  Γsh/k is lesser than Δ, and the small-angle scattering collisions act like the hard ones that leads to the line narrowing additional to that caused by hard collisions. Thus, the effective line narrowing parameter is greater than that at high pressures when Γsh/k⪢Δ because the narrow part of the collision integral kernel plays the role of a delta-function. The latter circumstance leads to the renormalization ν-ν  νs [11] and eliminates SASC from the process of line narrowing. In order to compare the behavior of the effective narrowing parameter αh as a function of a gas pressure determined for the discussed above case with that intrinsic to the line narrowing caused by the wind effect in the absence of SASC, we have fitted the line profile KNG Eq. (2) to calculated values of the speed-dependent hard collision model profile KsdNG (Eq. (4) in Ref. [18]). The narrowing parameter αsd in the latter was defined as αh Eq. (3) with replacement of νh and Γh to the respective quantities νsd and Γsd at zero speed of an absorbing molecule (see Eq. (3) in Ref. [18]). The results of fitting are presented in Fig. 4 from where it appears that the effective narrowing parameter αh increases with the increase in a gas pressure. The behavior αh (Γsd) qualitatively differs from that presented in Fig. 3. Thus, the considered criterion allows separating the cases of line narrowing caused by Dicke's effect in the pronounced presence of SASC and the wind effect. Note, that the deviations of αh (Γsd) from the linear dependence are greater for heavy perturbing molecules and for Van der Waals intermolecular interaction (n ¼6) as compared with the dipole–quadrupole interaction (n ¼4).

16 14 12 10 h

2

8 6 4

4

2

1 3

0 0.0

0.5

1.0

1.5

2.0

2.5

/kv sd Fig. 4. The best-fitted parameter αh obtained from fitting the profile KNG Eq. (2) to the speed-dependent profile KsdNG (Eq. (4) in Ref. [18]) versus the dimensionless gas pressure Γsh/(kv). The profile KsdNG have been calculated for the case of the Van der Waals intermolecular interaction with n¼ 6 (curves 1 and 2) and for the dipole–quadrupole interaction, n ¼4 (curves 3 and 4); β¼ 1 (curves 1 and 3), 5 (curves 2 and 4); αsd ¼1.

Now let us fit the profile KNG Eq. (2) to a line profile in which both the considered mechanisms of a line narrowing are taken into account. In order to vary the parameters determining the action of SASC in wide limits, we will use the model line profile that can be calculated by the method employed previously in Ref. [17] on the basis of the following equations: Z 1 Z   νs ðvÞ 1 νðvÞ iðΩ  kvÞ RðvÞ  νh ðvÞW ðvÞ Rðv1 Þdv1  2Δ  1 1   jv v1 j exp  ð4Þ Rðv1 Þdv1 ¼ iVn0 W ðvÞ;

Δ

Z 8π Ndω 1 ImRðvÞdv; cE 1 rffiffiffiffiffiffiffiffiffiffiffi 2kB T ; v¼ m

K sdsl ðΩÞ ¼ V¼

dE ; 2ℏ



1 v2 W ðvÞ ¼ pffiffiffiffi exp  2 ; πv v

νðvÞ=ν0 ¼ νs ðvÞ=ν0s ¼ νh ðvÞ=ν0h Z

¼

1 0

 2

 1 1 3 v ; ;  β 2 þ τ dτ ; e τΦ  þ 2 n1 2 v

τ ¼ v2? =v2 : Here R(v) is the time-independent part of the nondiagonal element of the density matrix or, with an accuracy to a factor, light-induced polarization at the active transition; ν0, ν0s, and ν0h are the values of the respective collision frequencies at zero speed of absorbing molecule; Φ(a, b, z) is the confluent hypergeometric function [24]; W(v) is the Maxwell distribution of absorbing molecules over the components of velocities parallel to the wave vector; N is the density of active molecules; n0 is the difference of equilibrium energy level populations; E and ω are the amplitude of the electric field and the frequency of a light wave, respectively; V is the Rabi frequency; d is the matrix element of the light-induced dipole moment at the given transition; kB is the Boltzmann constant; m is the mass of absorbing molecules; T is the gas temperature, v1 and v are the projections of an absorbing molecule's velocity before and after collision onto the wave vector; v⊥ is the absolute value of the component of the velocity perpendicular to the wave vector; ℏ is the Planck constant, and c is the light speed. The Eq. (4) is an immediate generalization of Eqs. (2) and (3) in Ref. [11] (from which follows Eq. (1)) made by attributing the same speed-dependence [16] adapted to the used one-dimensional velocity approach [18] to all the collision relaxation constants. The results of the fitting are presented in Fig. 5. As is seen from Fig. 5, the retrieved values of the effective narrowing parameter αh due to the action of SASC are lesser than the initial value of the narrowing parameter that follows from Eq. (4), αsdsl  ν0h/(ν0  ν0s  ν0h), that was put equal to unity in calculations. As distinct to the above considered cases, where the dependences of the best-fitted narrowing parameter on a gas pressure are monotonous (Figs. 3 and 4), the combined action of SASC and the wind effect can lead to the nonmonotonic dependences at certain sets of parameters (Fig. 5). The choice of the parameters ξ, ϑ, and β of the line profile Ksdsl Eq. (4) greatly affects the kind of dependencies αh (Γsh) and the

V.P. Kochanov / Journal of Quantitative Spectroscopy & Radiative Transfer 159 (2015) 32–38

0.9

1

0.8 0.7 h

0.6

5

0.5

3

0.4

2 4

0.3 0.0

0.5

1.0

1.5

2.0

2.5

/kv sh Fig. 5. The parameter αh retrieved from fitting the profile KNG Eq. (2) to the speed-dependent profile containing both the SASC and the hard collisions that was calculated on the basis of Eq. (4) for the dispersion intermolecular interaction with n¼6 as a function of the dimensionless gas pressure Γsh/(kv). The parameters used in calculations are: ξ ¼0.3 (curve 1), 0.5 (curves 2 and 3), 0.7 ξ¼ 0.5 (curves 4 and 5); ϑ ¼0.15 (curves 1, 4 and 5), 0.25 (curves 2 and 3); β ¼1 (curves 2 and 4), 4 (curves 1, 3 and 5); αsdsl ¼ 1.

values of αh. Thereby, the performed testing confirms the sensitivity of the proposed criterion. 4. Conclusions In this paper, approximated algebraic line profiles were obtained that are based on calculations performed with the use of quantum-mechanical kernels of the collision integral [21,22] for the cases of dipole–dipole, dipole–quadrupole, and quadrupole–quadrupole intermolecular interaction. These profiles have been derived with simultaneous account of velocity-changing collisions with small- and large-angle scattering and also with the speed-dependence of the collision relaxation constants, which is necessary in conditions of inhomogeneous line broadening. Simulations show that data processing gives somewhat different values of collision broadening constant retrieved with the use of the profiles related to different types of intermolecular potential and display considerable variation in the retrieved values of the parameter α (Table 1). The latter circumstance is caused by significant difference in the contributions of small-angle scattering (soft collisions) into the frequency of elastic collisions that is responsible for the line narrowing for the considered types of potentials (Fig. 2). Emphasize, that the parameter α in the approximated line profiles presented in the appendix has an explicit physical meaning and it determines the ratio of a total input frequency of the collision integral including SASC and LASC to the collision line broadening constant Γ [17]. At the same time, the effective narrowing parameter αeff is responsible for the actual line narrowing caused mainly by LASC implying the conventional Dicke's mechanism and the wind effect. Thus, the ratio of the parameter α retrieved from the processing the experimental data with the line profiles Eqs. (A1)–(A5) to the calculated parameter αeff (α, Γ, β) allows estimating the relationship between cross-sections of SASC and LASC, which is a

37

qualitatively new information with respect to the information retrieved from the typical data processing with the use of hard collision model line profiles. A sensitive criterion was proposed that consists in fitting the speed-independent hard collision model line profile Eq. (2) to recorded line profiles and further analysis of the behavior of the retrieved effective narrowing parameter as a function of a gas pressure. Namely, when Dicke's mechanism of narrowing prevails, then the dimensionless narrowing parameter αh (the ratio of the frequency of hard velocity-changing collisions to the collision line half-width) decreases with the increase of a gas pressure (Fig. 3). Its opposite behavior, i.e. increasing with pressure, testifies that the narrowing caused by the speeddependence of collision relaxation constants dominates (Fig. 4). In the case where both mechanisms of narrowing are significant, the parameter αh can be nonmonotonic function of a gas pressure (Fig. 5). Using another line profile that describes a line narrowing instead of the line profile Eq. (2) in the same manner of action allows its testing, since if the description of a narrowing is correct, then the dependence of the retrieved effective narrowing parameter on a gas pressure must be linear.

Acknowledgments This work was financially supported by the Russian Foundation for Basic Research (Project no. 13-02-00122). Appendix Approximations for line profiles calculated for intermolecular interaction potentials proportional to r  n are   S K n ðΩÞ ¼ pffiffiffi Re wa 1 Ω=ðkvÞ þ iΓ ef f ðΓ; α; β; nÞ π kv



ðβ þ 1Þð1=2Þ  ð1=ðn  1ÞÞ ðαeff ðΓ; α; β; nÞ þ 1Þ=ðkvÞ :

i1 pffiffiffi  π Γ ef f ðΓ; α; β; nÞðβ þ1Þð1=2Þ  ð1=ðn  1ÞÞ αeff ðΓ; α; β; nÞ=ðkvÞ ;

ðA1Þ wa ðzÞ ¼

0:15269784972008846  0:01429059277879871i  1:638566939130925 þ 1:390393937133246i þ z 1:022455279362302 þ 0:5341298895517378i   0:7327122194675937 þ 1:4139283787114099i þ z þ

0:6543564876039407 þ 1:5745917089047585i 0:10206925308122759 þ 1:3554420881163243i þ z

þ

0:2156138105591975  0:46143197880686115i : 1:0446775963500718 þ 1:2115613850263882i þ z

Here S is the line intensity and the effective narrowing parameters have the following expressions for different potentials: a) Dipole–dipole interaction, n ¼3: αef f ðΓ; α; β; 3Þ ¼ 0:011925 

0:0131729 1 þ 0:169416ð  1:050418 þ βÞ2

þα

 0:001285 þ 0:0213377β þ 0:00001854β2 1þ 0:489007β

þ

 0:0046187 þ

!

Γ 1þ 0:167894ð  1:255771 þ βÞ2 kv 0:0045196

38

V.P. Kochanov / Journal of Quantitative Spectroscopy & Radiative Transfer 159 (2015) 32–38

þ α  0:001396 þ

!

0:002366 1þ 0:372122ð 0:641495 þ βÞ

Γ ; kv

2

ðA2Þ Γ ef f ðΓ; α; β; 3Þ ¼ Γ; δ1=2 ¼ γ ¼ Γ:

δ1=2 ¼ γð1 þ βÞ  0:013 ¼ ð1 þ βÞ0:287 Γ: Here Γ is the collision relaxation constant at zero speed of the absorbing molecule, γ is the collision relaxation constant averaged over the velocities, and δ1=2 is the halfmaximum half-width of the line profiles in the high pressure limit.

b) Dipole–quadrupole interaction, n ¼4  0:00608351 þ0:0858759β  0:00439426β2 1 þ0:07857β 0:00922856 þ0:15447β 0:00149194β2 þα 1 þ1:1645β

References

αef f ðΓ; α; β; 4Þ ¼

0:0126749 þ 0:1422β þ 0:00230032β2 Γ kv 1 þ1:03845β

þ

þα

0:00485442 0:00675249β þ 0:000295302β2 Γ ; 1 þ0:181756β kv

ðA3Þ Γ ef f ðΓ; α; β; 4Þ ¼ Γ; δ1=2 ¼ γ ¼ Γð1 þβÞ0:167 : c) Quadrupole–quadrupole interaction, n ¼5  0:0564926 þ 0:0758507β  0:00482128β2 1  0:032727β  0:0164613 þ 0:325868β  0:00634591β2 þα 1 þ 1:51404β

αef f ðΓ; α; β; 5Þ ¼

þ þα

 0:0164418 þ 0:290953β  0:0286031β2 Γ 1 þ 0:697055β kv

0:00878946 0:0239543β þ 0:00112974β2 Γ ; kv 1 þ 0:527933β

ðA4Þ 0:99657 þ 0:32165β 0:0010456β2 Γ; 1 þ 0:39904β  0:084 0:166 ¼ ð1 þ βÞ Γ: δ1=2 ¼ γð1 þ βÞ

Γ ef f ðΓ; α; β; 5Þ ¼

d) Van der Waals interaction, n ¼6 [17] αef f ðΓ; α; β; 6Þ ¼

 0:0230161 þ 0:453369β þ 0:0301179β2 1 þ1:55321β

 0:0453981 þ 0:517833β  0:0190059β2 þα 1 þ 0:573681β  þ 0:00846344 þ 0:0552397β þ 0:00983371β2  0:00152409β3 þα

Γ kv

0:0213937  0:0217767β þ 0:00798925β2 Γ ; kv 1 þ 0:052263β

ðA5Þ Γ ef f ðΓ; α; β; 6Þ ¼ αð  0:00270024 þ 0:000779978β þ 0:0000483498β2  0:000011358β3 þ



1:00933 þ7:06383β  0:0067468β2 Γ kv 1 þ7:11816β

þ αð0:00151172þ 0:0000725962β  0:000161312β2 þ0:0000167866β3

Γ ; kv

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