Research Article  Open Access
Retrieval of Total Scattering Cross Sections of Molecules from Inhomogeneously Broadened Absorption Lines
Abstract
A new method of retrieving quantitative information on the hard, soft, and diffraction collision’s frequencies from inhomogeneously pressurebroadened line profiles was proposed and tested. The essence of the method lies in the processing of recorded profiles using various profiles’ models containing these frequencies treated as adjustable parameters in different manners. Three selfbroadened H_{2}O absorption lines free of an instrumental function were processed with the use of four model line profiles. Retrieved set of pressure line broadening and narrowing constants allows finding out the sought frequencies. The estimation of the total scattering cross section, ∼630 Å^{2}, in pure water vapor was made.
1. Introduction
Various models for spectral line profiles allow retrieving information on collision processes that affect the line shape. The simplest Lorentzian and Voigt profiles provide retrieving only the homogeneous collision line halfwidth γ and the line shift Δ. With the aid of more complicated profiles [1–4], where the Dicke line narrowing [5, 6] is taken into account, it is possible to find out an additional parameter—the frequency of elastic velocitychanging collisions. In the conventional approach [1–6], this frequency is expressed via the diffusion coefficient D aswhere is the most probable thermal speed of absorbing molecules. Thus, exploitation of the collision line narrowing makes it possible to obtain the data on from processing recorded line profiles, which is still nonconventional information for spectroscopy. Similarly, involvement of other physical mechanisms of forming the line shape increases the number of the profile’s adjustable parameters. Certain parameters may have a direct physical meaning, and thus they can serve as a source of valuable additional information extracted from a line shape. Such are the frequencies of soft and hard velocitychanging collisions.
Let us detail the latter assertion. The matter is that the structure of the collision integral implies division of its kernel onto three parts related to hard (hard collisions are mainly largeangle scattering collisions [9] that lead to establishing the equilibrium distribution of molecules over velocities after each collision, soft collisions are caused by classical scattering on small angles less 0.3 rad [10], and the diffractionscattering collisions occur when scattering angles are ∼0.01 rad [8–10].), soft, and diffractionscattering collisions [7–11]. These three types of collisions function to a great extent additively due to considerable difference in widths of the corresponding differential scattering cross sections. Consequently, the total output frequency of the collision integral ν_{t} can be represented as a sum [7–11].where , , and are the input collision integral frequencies of the elastic hard, soft, and diffraction scattering collisions, respectively, and the Lorentzian collision halfwidth at halfheight contains the contributions of elastic and inelastic collisions. If the above frequencies enter into a model collision integral explicitly, then the line profile derived from the respective muster equation for a density matrix will contain these frequencies as adjustable parameters [9, 10]. This opens a possibility to retrieve the newly introduced parameters from processing experimental line profiles.
It was shown earlier [12–14] that the output frequency can be measured directly as a collisionbroadening constant of nonlinear resonances at sufficiently low gas pressures. The theory [14, 15] allows expressing the frequency through the total collision cross section. Thus, qualitatively new fundamental information can be obtained from nonlinear spectroscopic measurements. On the contrary, in this paper, we intend to show that the value of also may be obtained by means of linear spectroscopy. Namely, correct account of the combined effect of hard and soft collisions [8–11] performed on the basis of the quantummechanical collision integral kernel [7] and the inversepower intermolecular interaction potentials provides the presence of all the frequencies specified in (2) in the model line profiles [8–11]. Therefore, in principle, it is possible to retrieve these frequencies from processing recorded spectra with the profiles [8–11]. Though, such retrieval is not guaranteed to be a priori successful because of a strong correlation between some of the profile’s parameters [10]. At the same time, certain parameters of the collision integral can be calculated theoretically for different intermolecular interaction potentials [9, 10]. In particular, such is the ratio of soft to hard collision frequencies, which also can be used in the fitting. Thus, we can assume that the ambiguity caused by marked above correlation of the parameters can be eliminated, and the retrieving different collision integral frequencies will be possible. Ascertainment of a principal scheme of the estimation of the discussed frequencies and testing its operability form the goals of this paper.
2. Experimental Setup
The highresolution spectra of H_{2}O were recorded with the use of the photoacoustic (PhA) spectrometer based on a singlemode Tisapphire laser pumped by an Arion laser. The Tisapphire laser had the output power of 1 W and the 50 kHz linewidth. It was able to tune in the 11300–12800 cm^{−1} spectral range with the 20 MHz frequency step. This provided the spectral resolution better than 0.001 cm^{−1} and the negligible contribution of the laser bandwidth to recorded line profiles (the Doppler HWHM was ∼0.018 cm^{−1} or 542 MHz). The high power of the laser provided a high threshold absorption sensitivity of the spectrometer, ∼1⋅10^{−8} cm^{−1}. The body of the PhA cell was made of stainless steel. The length of the cell was 10 cm. The acoustic signal in the PhA cell was detected by highsensitive laboratorymade capacitor microphone with a preamplifier. A signal of the photoacoustic detector is directly proportional to the absorption coefficient of the gas as opposed to the spectrophotometric method where a signal is proportional to the transmittance of the medium. This circumstance allows recording a weak absorption with a higher accuracy. Wavelength calibration was carried out with the use of the Fourier spectrometer “Bruker DA 003”. The measurement technique is described in more detail in [16, 17].
Three H_{2}O lines centered at 12415.204, 12413.977, and 12411.404 cm^{−1} (≈805 nm) were used to perform the estimations outlined in Introduction. Parameters of these lines taken from [18] are presented in Table 1. The selfbroadening of these lines was registered within the pressure range 1 Torr ≤ ≤ 20 Torr.
 
Single and double strokes denote lower and upper molecular states, respectively; S is the line intensity. 
3. Data Processing
The following line profiles have been used in data processing:(i)K_{HC}(Ω)—the conventional profile in the hard collision model [3, 4] (calculated with equations (3) and (8) in [19]; Ω is the frequency detuning from a line center).(ii)K_{RS3}(Ω)—the profile based on the quantummechanical Rautian−Shalagin collision integral kernel [7] including the speed dependence [20, 21] and calculated for the dipoledipole intermolecular interaction potential α r^{−3}, where r is the distance between colliding molecules (equations (A1) and (A2) in [9]).(iii)K_{SH3}(Ω)—the profile accounting for the speeddependence, soft, and hard velocitychanging collisions and based on a collision integral partly represented by a differential form with the use of some fixed parameters calculated for dipoledipole intermolecular interaction (equation (9) and Table 1 in [10]; equations (5) and (6) in Ref. [22]).(iv)K_{SH3r} (Ω)—the same profile as the previous one where the ratio of soft to hard collision frequencies r is set as an adjustable parameter (equation (9) and Table 1 in [10]; equations (5) and (6) in [22]).
This group of profiles may provide retrieving all the collision frequencies presented in (2). Really, the halfwidth γ is obtained with each of the profiles listed above. If γ is defined as a quantity averaged over velocities of absorbing molecules, then, as follows from the practice [23], its values obtained with the use of all the profiles will be equal with a satisfactory accuracy. The dimensionless collision linenarrowing parameter is as follows:which is the fitting parameter of the profiles (i), (iii), and (iv), while the profile (ii) contains the substantially different narrowing parameter
As is seen from the comparison of (3) and (4), the inequality α_{tot} > α always holds under the natural supposition of ν_{h}, ν_{s}, and ν_{d} > 0. The relative difference α_{tot}/α − 1 may be significant, and it can amount hundreds of percent [11]. Note that α describes an actual line narrowing caused by collisions with a molecular scattering on predominantly large angles [9]. Due to this reason, the values of α obtained with the aid of the profiles (i), (iii), and (iv) must be approximately equal. In contrast to this, the parameter α_{tot} was introduced in [8] as the ratio of the sum frequency of elastic velocitychanging collisions, ν_{h} + ν_{s} + ν_{d}, to a homogeneous line halfwidth γ. Accepted in [8, 9], parametrization allows interpretation of α_{tot} as a fitting parameter and further retrieving the values of this narrowing parameter from spectra processing with profiles [8, 9].
The ratiocan be calculated theoretically [9, 10] and then used as a fixed parameter in profile (iii). Otherwise, r is a fitting parameter of profile (iv).
We note that the contour (i) does not contain parameters that are complementary to the parameters of the remaining three contours, and its application in processing is justified by controlling the values of γ and α obtained with the help of the profiles (ii)−(iv) (see Table 2).

The knowledge of values of γ, α, α_{tot}, and r retrieved with the aid of fitting the profiles (i)−(iv) to experimental ones makes it possible to solve the system of Equations (2)−(5) with respect to the sought frequencies ν_{t}, ν_{h}, ν_{s}, and ν_{d}. This solution gives a fundamental scheme of quantitative estimation of these frequencies. The respective expressions for the frequencies are
An additional testing these frequencies can be done by means of comparison of the ratiosfound from the experiment and relations (6) with the calculated values of R, presented in Figure 2 of [9].
It should be noted that the values of the collision line halfwidth γ retrieved from an experiment with the use of profiles (i)−(iv) distinguish due to different manners of accounting the speed dependence in them. But in cases when the speed dependence is small pronounced (heavy absorbing and/or light perturbing molecules), discrepancies between γ are minor [23], Table 2. Therefore, the main source of errors in the suggested method of retrieving frequencies ν_{t}, ν_{h}, ν_{s}, and ν_{d} lies in an inaccuracy of the narrowing parameters α and α_{tot}, as well as the ratio of the frequencies of soft and hard collisions r, which are more sensitive characteristics of a line shape.
An example of a recorded line with the bestfitted profile K_{HC}(Ω) is shown in Figure 1. The quality of fitting all the rest of listed above profiles is practically the same because of the account of the line narrowing in each of them, which makes these profiles to be flexible enough. The collision line broadening found from the processing is presented in Figure 2. Straight lines in this figure are plotted as a result of leastsquares fitting and their, slopes represent the pressurebroadening coefficients γ_{0} ≡ γ/p. A complete set of the parameters retrieved from the processing is presented (hard collisions are mainly largeangle scattering collisions [9] that lead to establishing the equilibrium distribution of molecules over velocities after each collision; soft collisions are caused by classical scattering on small angles less 0.3 rad [10], and the diffraction scattering collisions occur when scattering angles are ∼0.01 rad [8–10].) (Except for line intensities S disclosing wide range of values, which is typical for PhA measurements.) in Table 2. From this table and Equations (6) and (7), we have obtained the estimated values of the collision integral frequencies (Table 3).

4. Discussion and Conclusions
It is seen from Table 2 that the pressurebroadening coefficients γ_{0} for individual lines obtained with different line profiles are in a good agreement. Also it should be noted that the narrowing parameters α (2) are small compared with the values found with the aid of the expression for ν_{h} (1) via the selfdiffusion coefficient D. In particular, the use of the literature data D = 0.196 cm^{2}/s [24] and γ_{0} = 18 MHz/Torr taken from Table 2 leads to α = 0.51 that is two and a half times greater than the values of α∼0.2 retrieved from the experiment. This fact can be understood on the basis of simple physical arguments. Indeed, for longrange intermolecular interactions (i.e., dipoledipole and dipolequadrupole ones), the contribution of the molecular scattering on small angles in collision integral frequencies increases relatively to the scattering on large angles (see Figure 2 in Ref. [9]). In other words, the largeangle scattering is suppressed. Correspondingly, as is seen from Table 2, the values of the parameter α_{tot} (4) including frequencies of scattering on small angles are large compared with the values of α (3). Due to the decrease in the hard velocitychanging collisions, the Dicke line narrowing controlled by the parameter α also reduces. Thus, it is evident that relation (1) cannot be used for a reliable quantitative retrieving D from experimental values of narrowing parameters in the case of longrange intermolecular interactions.
Note that the parameters r (5) in Table 2 retrieved with the line profile K_{SH3r} have the same order of magnitude as the relating calculated values of r [10] used in fitting with the profile K_{SH3}.
Discussing Table 3, we mark that the main contribution to ν_{t} is made by ν_{d}. In accordance with the above discussion, this fact can be imputed to the longrange action of the dipoledipole intermolecular interaction potential of the water molecules having a large constant dipole moment.
There is an interesting opportunity to estimate the total scattering cross section σ from data on ν_{t} presented in Table 3, namely, equations (15)−(18) in [15], where the collision integral frequencies are expressed through the parameters of the intermolecular potential U_{jn} (r) = − C_{jn}/r^{n} (here j denotes a molecular state), allow expressing the real part of ν_{t} via σ averaged over velocities. Let us assume that constants C_{jn} are approximately equal for the upper and lower states of the considered vibrationalrotational transition [24]. Then, the index j can be omitted, and we have the following expression from equations (15) and (16) of [15] and the optical theorem of scattering [25]:where is the amplitude of elastic scattering on zero angles, is the Weisskopf radius, u is the relative velocity of molecules in the centerofmass coordinate system, and is the gamma function.
Averaging σ (8) over u with the Maxwellian distribution of relative velocities giveswhere and are masses of absorbing and perturbing molecules, respectively.
Analogous averaging real part of the output collision integral frequency given by equation (18) in [15] over velocities of absorbing molecules leads towhere is the density of perturbing molecules.
After combining (9) and (10), we have the expression of the average total scattering cross section through the output collision integral frequency:
In particular, in the considered case of selfbroadening H_{2}O lines (dipoledipole interaction with and β = 1), it follows from (11):
Calculations performed on the basis of (12) and the data of Tables 2 and 3 give the following estimations for the total scattering cross section for H_{2}OH_{2}O collisions: σ_{tot} ∼ 630 ± 150 Å^{2}. For comparison, the typical values of gaskinetic cross sections of many molecules are of the order 50–70 Å^{2} [26]. Relatively large errors (95% confidence intervals) in σ_{tot} may be caused by insufficiently accurate experiment as well as by a rough net used in numerical integration over velocities in the vicinity of a character point determining diffraction scattering [9]. The above estimation of σ_{tot} for H_{2}O molecule is close to total cross sections of three other molecules obtained by direct measurements of ν_{t} with the aid of nonlinear spectroscopy methods [12, 13, 27–30], namely, analogous calculations made in [27] ascertain that in pure gas of CH_{4}, σ_{tot} ≈ 380 [28], 350 [12], and 370 Å^{2} [29]; in CO_{2}σ_{tot} ≈ 500 Å^{2} [13], and in SF_{6}σ_{tot} ≈ 600 Å^{2} [30].
Coming to the conclusions, it can be stated that the proposed new method of retrieving input and output collision integral frequencies is mainly functional. It was shown that with the aid of this method, it is possible to get estimations of a fundamental quantity—total scattering cross section. At the same time, the results presented in this paper are only first steps in the outlined direction, and retrieving more accurate quantitative data on the frequencies and cross sections requires further improvement of experiment and theory, which can constitute a significant cycle of work in the spectroscopy of the line contour.
From the data processing, it was found that relation (1) between the frequency of hard collisions and the diffusion coefficient fails in the case of longranged intermolecular interactions.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
One of the authors (L. Sinitsa) expresses his gratitude to coworkers from the University of Science and Technology (Hefei, China) for the possibility to use Tisapphire laser. Dr. S. N. Mikhailenko is acknowledged for the help in compiling Table 1.
This work was partly supported by the RFBR (Grant No. 1643700492).
References
 L. Galatry, “Simultaneous effect of Doppler and foreign gas broadening on spectral lines,” Physical Review, vol. 122, no. 4, pp. 1218–1223, 1961. View at: Publisher Site  Google Scholar
 M. I. Podgoretsky and A. V. Stepanov, “On the Doppler emission and absorption line width,” Jurnal eksperimental'noi i teoreticheskoi fiziki, vol. 40, no. 2, pp. 561–566, 1961. View at: Google Scholar
 M. Nelkin and A. Ghatak, “Simple binary collision model for Van Hove's Gs(r,t),” Physical Review, vol. 135, no. 1A, pp. A4–A9, 1964. View at: Publisher Site  Google Scholar
 S. G. Rautian and I. I. Sobel'man, “The effect of collisions on the Doppler broadening of spectral lines,” Soviet Physics Uspekhi, vol. 9, no. 5, pp. 701–716, 1967. View at: Publisher Site  Google Scholar
 R. H. Dicke, “The effect of collisions upon the Doppler width of spectral lines,” Physical Review, vol. 89, no. 2, pp. 472473, 1953. View at: Publisher Site  Google Scholar
 J. P. Wittke and R. H. Dicke, “Redetermination of the hyperfine splitting in the ground state of atomic hydrogen,” Physical Review, vol. 103, no. 3, pp. 620–631, 1956. View at: Publisher Site  Google Scholar
 S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, Nonlinear Resonances in Spectra of Atoms and Molecules, Nauka, Novosibirsk, SB, Russia, 1979.
 V. P. Kochanov, “Manifestations of smallangle molecular scattering in spectral line profiles,” Journal of Experimental and Theoretical Physics, vol. 118, no. 3, pp. 335–350, 2014. View at: Publisher Site  Google Scholar
 V. P. Kochanov, “Combined effect of small and largeangle scattering collisions on a spectral line shape,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 159, pp. 32–38, 2015. View at: Publisher Site  Google Scholar
 V. P. Kochanov, “Speeddependent spectral line profile including line narrowing and mixing,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 177, pp. 261–268, 2016. View at: Publisher Site  Google Scholar
 V. P. Kochanov, “Effect of diffraction of molecules on collisional line narrowing,” Optics and Spectroscopy, vol. 89, no. 5, pp. 684–689, 2000. View at: Publisher Site  Google Scholar
 S. N. Bagayev, E. V. Baklanov, and V. P. Chebotayev, “Measurement of elastic scattering crosssections in a gas by laser spectroscopy methods,” ZhETF Pis. Red., vol. 16, no. 1, pp. 15–18, 1972. View at: Google Scholar
 V. P. Chebotayev, V. P. Kochanov, and L. S. Vasilenko, “Nonlinear dependence of optical resonance widths at CO_{2} transitions on pressure,” Optics Communications, vol. 20, no. 3, pp. 409–411, 1977. View at: Google Scholar
 V. P. Kochanov, S. G. Rautian, and A. M. Shalagin, “Broadening of nonlinear resonances by velocitychanging collisions,” Journal of Experimental and Theoretical Physics, vol. 45, no. 4, pp. 714–722, 1977. View at: Google Scholar
 V. P. Kochanov, “Line profiles for the description of line mixing, narrowing, and dependence of relaxation constants on speed,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 112, no. 12, pp. 1931–1941, 2011. View at: Publisher Site  Google Scholar
 V. Lazarev, Y. Ponomarev, L. Sinitsa, J.X. Han, L.Y. Hao, and Q.S. Zhu, “The Photoacoustic spectrometer with Ti: sapphire laser,” SPIE Proceedings, vol. 3090, pp. 245–248, 1997. View at: Google Scholar
 V. I. Serdyukov, L. N. Sinitsa, A. S. Dudaryonok, N. N. Lavrentieva, and A. P. Shcherbakov, “Investigation of the H_{2}O line broadening using an optoacoustic laser spectrometer in the range of 12411–12421 cm^{−1},” Russian Physical Journal, vol. 58, pp. 1–5, 2016. View at: Publisher Site  Google Scholar
 L. S. Rothman, I. E. Gordon, Y. Babikov et al., “The HITRAN2012 molecular spectroscopic database,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 130, pp. 4–50, 2013. View at: Google Scholar
 V. P. Kochanov, “Algebraic approximation of the spectral line profile with allowance for hard and soft speedrelated collisions,” Atmospheric and Oceanic Optics, vol. 28, no. 5, pp. 394–399, 2015. View at: Publisher Site  Google Scholar
 P. R. Berman, “Speeddependent collisional width and shift parameters in spectral line profiles,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 12, no. 9, pp. 1331–4132, 1972. View at: Publisher Site  Google Scholar
 J. Ward, J. Cooper, and E. W. Smith, “Correlation effects in the theory of combined Doppler and pressure broadening.−I. Classical theory,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 14, no. 7, pp. 555–590, 1974. View at: Publisher Site  Google Scholar
 V. P. Kochanov, “On parameterization of spectral line profiles including the speeddependence in the case of gas mixtures,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 189, pp. 8918–8923, 2017. View at: Publisher Site  Google Scholar
 V. P. Kochanov and I. Morino, “Methane line shapes and spectral line parameters in the 5647−6164 cm^{−1} region,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 206, pp. 313–322, 2018. View at: Publisher Site  Google Scholar
 V. P. Kochanov and V. I. Starikov, “Calculation and algebraic approximation of the diffusion and viscosity coefficients for polar gases as functions of the temperature and the parameters of the Stockmayer potential,” Optika Atmosfery i Okeana, vol. 14, no. 9, pp. 837–840, 2001. View at: Google Scholar
 J. R. Taylor, Scattering Theory. The Quantum Theory of Nonrelativistic Collisions, John Wiley & Sons, NY, London, Sydney, and Toronto, 1972.
 J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, NY, USA, 1954.
 V. P. Kochanov and V. P. Lopasov, “Investigation of an absorption line profile in molecular gases with methods of laser spectroscopy,” in Spectral Manifestations of Intermolecular Interactions in Gases, Y. S. Makushkin, Ed., pp. 142–172, Nauka, Novosibirsk, SB, Russia, 1982, in Russian. View at: Google Scholar
 R. L. Barger and J. L. Hall, “Pressure shift and broadening of methane line at 3.39μ, studied by lasersaturated molecular absorption,” Physical Review Letters, vol. 22, no. 1, pp. 4–8, 1969. View at: Publisher Site  Google Scholar
 W. Radloff and V. Stert, “Contrast of an inverted Lamb dip of a laser with an internal absorption cell,” Soviet Journal of Quantum Electronics, vol. 6, no. 7, pp. 817–821, 1976. View at: Publisher Site  Google Scholar
 N. G. Basov, O. N. Kompanets, V. S. Letokhov, and V. V. Nikitin, “Investigation of narrow resonances within the Doppler line of the rotationalvibrational transitions of the SF_{6} molecule during absorption saturation,” Soviet Physics Journal of Experimental and Theoretical Physics, vol. 32, pp. 214–219, 1971. View at: Google Scholar
Copyright
Copyright © 2018 V. P. Kochanov and L. N. Sinitsa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.