- Email: [email protected]
Diode-laser spectroscopy: Temperature dependence of R (0) line in the ν4 band of CH4 perturbed by N2 and O2
Journal of Molecular Spectroscopy 233 (2005) 86–92 www.elsevier.com/locate/jms
Diode-laser spectroscopy: Temperature dependence of R (0) line in the m4 band of CH4 perturbed by N2 and O2 Muriel Lepe`re a,*,1, Alain Valentin b, Annie Henry b, Claude Camy-Peyret b Marc Lengele´ a, Jean-Claude Populaire a, Ghislain Blanquet a a
Laboratoire de Spectroscopie Mole´culaire, Faculte´s Universitaires Notre-Dame de la Paix, 61, rue de Bruxelles, B-5000 Namur, Belgium b Laboratoire de Physique Mole´culaire et Applications, CNRS, Universite´ Pierre et Marie Curie, Tour 13, Bte 76, 4 place Jussieu, F-75252 Paris cedex 05, France Received 30 March 2005; in revised form 1 June 2005 Available online 26 July 2005
Abstract In this paper, we present a line profile study of the R (0) line in the m4 band of methane diluted in nitrogen and oxygen, from room temperature to 153 K. The measurements were performed over a total pressure range from 14 to 128 mbar. The collisional broadening and narrowing (Dicke effect) coefficients are derived from a fit of the experimental spectra by using the soft and hard collision models, taking into account the Dicke effect. For higher pressures, we have fitted the data with a model taking into account simultaneously the Dicke narrowing and the speed dependence effect. Finally, we have deduced the parameter n of the temperature dependence (inverse power law) of the broadening coefficients for the CH4–N2 and CH4–O2 gas mixtures. 2005 Elsevier Inc. All rights reserved. Keywords: Tunable diode-laser; Methane; Line profile; Broadening coefficients; Temperature dependence
1. Introduction Methane is an important minor constituent in different atmospheres: Jupiter, Saturn, Titan, Uranus, and Neptune in the solar system as well as hot Jupiters in other planetary systems. On Earth, methane is the most abundant organic trace gas in our own atmosphere [1]. A good knowledge of collisional lineshapes in several bands of methane is thus important. Laboratory lineshape studies have been already extensively performed, mainly in the m3 band ([2] and references therein) and recently in the 2m3 band ([3] and references therein). The corresponding experiments allowed precise determinations of spectroscopic line parameters such as collisional broadening and narrowing. *
1
Corresponding author. Fax: +32 81 72 45 85. E-mail address: [email protected] (M. Lepe`re). Associate Researcher with FNRS, Belgium.
0022-2852/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2005.06.004
These measurements are very important in infrared remote sensing. For reliable atmospheric sounding of the Earth atmosphere, it is necessary to determine precisely methane collisional broadening coefficients by nitrogen and oxygen as well as their temperature dependence. Much of the previous efforts in determining broadening parameters for the m4 band have concentrated on precise measurements at room temperature [4–8] with only a few papers devoted to high resolution measurements at low temperature [9–12].
2. Experimental details The spectra were recorded at high spectral resolution using a diode-laser spectrometer described previously [13,14]. The diode-laser emission was thus actively controlled using a Michelson interferometer and the residual wavenumber fluctuations in the emitted laser
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
radiation were smaller than 4 · 105 cm1. An absorption line is scanned step-by-step with a spectral sampling step equal to 0.4845 · 103 cm1 at 1311.4315 cm1. The TDL beam used for absorption experiments is divided into two infrared beams; one beam is used as a reference signal, I0, and the other goes through the sample cell to provide a transmitted signal, It. At each step the beam intensities I0 and It are measured simultaneously, and the ratio It/I0 is determined with a precision of about 5 · 104. For low temperature measurements, we have used an absorption cell described in details elsewhere [15]. This cell can be cooled uniformly by a liquid–N2 flow at any temperature between 77 K and room temperature, with a temperature stability better than 0.5 K. In the experiments reported here, this absorption cell with an optical path length of 40.43 cm was cooled down to 223.2, 188.2, and 153.2 K. For each gas mixture (CH4–N2 and CH4–O2) and each temperature (296.0, 223.2, 188.2, and 153.2 K), we have recorded the spectra around the R (0) line at different (at least 4) gas mixture pressures between 14 and 128 mbar. For each pressure, three spectra were recorded. The total pressure of the gas mixture and the methane pressure were measured using two MKS Baratron gauges with 120 and 1.2 mbar full scales, respectively. The methane (N30 quality) with a stated purity of 99.9% was supplied by the Air Liquide company.
where kB is the Boltzmann constant, T is the temperature (296.0, 223.2, 188.2 or 153.2 K); c is the speed of light, m1 the molecular weight of the absorber or active molecule, and D12 the mass-diffusion coefficient for the CH4–N2 or CH4–O2 pair. This coefficient estimated from a Lennard-Jones potential is given at different temperatures for the CH4–N2 and CH4–O2 mixtures on Tables 1 and 2, respectively. Concurrently, the main effect of collisions is to broaden the lines of the active molecule. Considering the relative speed of the partner molecules as independent of the absorber speed, the collision broadening is described by a Lorentz function with a half-width at half maximum given by cL ¼ c0 ðP tot P CH4 Þ þ cself P CH4 ;
The measured absorption coefficient a (m) (in cm1 atm1) at wavenumber m of a homogeneous gas sample is defined by the Beer–Lambert law as I t ðmÞ P absorber LaðmÞ ¼ ln ; ð1Þ I 0 ðmÞ where I0 (m) and It (m) are the transmitted intensities obtained, respectively, with the cell of path length L under vacuum and filled with the absorbing gas at partial pressure Pabsorber (possibly with a buffer gas at total sample pressure Ptot). A synthetic spectrum is compared in a least squares fit procedure to the observed one to determine different spectroscopic parameters (e.g., S0, c0, b0, . . .). At very low pressure, the line profile is described by a Gauss function (Doppler effect). When the pressure increases, the Doppler line is progressively narrowed by the confinement of the active molecules in the buffer gas (Dicke effect [16]). This effect is characterized by the narrowing parameter b0 which can be compared to the dynamic friction coefficient bDiff deduced from [17]: bDiff ¼
kBT ; 2pcm1 D12
ð2Þ
ð3Þ
where c0 is the coefficient for collisional broadening by foreign molecules, Ptot is the total pressure of the gas mixture (CH4–N2 or CH4–O2), cself is the coefficient for self-broadening, and P CH4 is the partial pressure of methane. The narrowing effect of collisions on pure Doppler broadening and on Lorentzian collision broadening are taken into account by the uncorrelated hard colliTable 1 Collisional broadening c0 and narrowing b0 coefficients of R (0) line in the m4 band of CH4 perturbed in N2 T (K)
3. Data reduction
87
296.0 223.2 188.2 153.2
Parameters (103 cm1 atm1) c0a
b0a
bDiff
56.00 ± 0.68 68.37 ± 0.58 76.15 ± 0.47 87.52 ± 0.48
41.3 ± 3.5 56.7 ± 3.3 59.1 ± 5.6 68.1 ± 5.7
38.1 48.0 55.6 67.1
Comparison with parameter bDiff calculated by the diffusion theory. a c0 and b0 are obtained with the Galatry model for low pressures and with a model taking into account simultaneously Dicke narrowing and the speed dependence effect for higher pressures. Errors given represent one standard deviation obtained by averaging different measurements.
Table 2 Collisional broadening c0 and narrowing b0 coefficients of R (0) line in the m4 band of CH4 perturbed in O2 T (K)
296.0 223.2 188.2 153.2
Parameters (103 cm1 atm1) c0a
b0a
bDiff
53.05 ± 0.20 65.50 ± 0.57 74.72 ± 0.85 86.60 ± 0.38
31.0 ± 4.0 41.7 ± 4.0 46.6 ± 5.3 59.5 ± 6.7
37.6 47.8 55.5 67.3
Comparison with the parameter bDiff calculated by the diffusion theory. a c0 and b0 are obtained with the Rautian model for low pressures and with a model taking into account simultaneously Dicke narrowing and the speed dependence effect for higher pressures. Errors given here represent one standard deviation obtained by averaging different measurements.
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
88
sion lineshape model developed by Rautian and Sobelman [18] and the uncorrelated soft collision model proposed by Galatry [19]. The line profile is then equivalent to the convolution product of a Lorentz profile with a confinement narrowed profile instead of a Gaussian profile (Doppler effect) commonly used in the Voigt profile [20,21]. The Voigt profile is defined by [20,21] Z y þ1 expðt2 Þ aV ðx; yÞ ¼ A dt p 1 y 2 þ ðx tÞ2 ¼ A Re½W ðx; yÞ
ð4Þ
with W ðx; yÞ ¼
i p
where pffiffiffiffiffiffiffiffi S ln 2 pffiffiffi ; A¼ cD p
Z
þ1 1
y¼
expðt2 Þ dt; x þ iy t
ð5Þ
pffiffiffiffiffiffiffiffi m m0 pffiffiffiffiffiffiffiffi cL . ln 2 ; x ¼ ln 2 cD cD
ð6Þ
Here S is the intensity of the line under study (here in cm1/(atm cm) or usually in cm1/(molecule cm2) in atmospheric data bases), cL (cm1) is the collisional half-width, and m0 (cm1) is the line center wavenumber (pressure shifts are not considered in the present study). The Rautian and Sobelman model [18] is best suited when the perturber mass is larger than mass of the active molecule. This profile may be described as W ðx; y þ zÞ pffiffiffi aR ðx; y; zÞ ¼ ARe ; ð7Þ 1 pzW ðx; y þ zÞ where the parameters are the same as above and pffiffiffiffiffiffiffiffi b z ¼ ln 2 c cD
ð8Þ
4. Results As shown in a previous paper [24], it is necessary to select the most appropriate theoretical profile to reach the highest level of accuracy on the determination of spectroscopic line parameters. The most appropriate profile depends on experimental conditions, especially the pressure. The temperature has also a large impact on the physical processes which determine the molecular lineshape. To improve the reliability of our measurements, we have recorded spectra at different (4–8) pressures for each gas mixture at a given temperature. In addition, we have recorded three spectra at each pressure. In this way, the final value of each line broadening parameter is obtained from an average of the values derived from the different pressures. Fig. 1 illustrates the results obtained for the CH4–O2 mixture at 223.2 K. This figure is showing the coefficients of collisional broadening versus total pressure. These measurements were performed at eight different gas mixture pressures between 19.8 and 115.2 mbar, and we have recorded three spectra at each pressure. The solid line represents the average value (over 24 spectra) for the CH4–O2 collisional broadening coefficients. This average value includes those determined with the Rautian model at pressure below 70 mbar and those obtained with the model taking into account the speed dependence effect for higher pressures. Fitting different line profiles to the experimental ones, we have thus determined the coefficients of collisional broadening c0 and narrowing b0 at 296.0, 223.2, 188.2, and 153.2 K. At room temperature, the Rautian and Galatry models taking into account the Dicke effect are well adapted to the experimental profile while the usual Voigt profile
with bc (cm1) representing the average effect of collisions on Doppler broadening. The model proposed by Galatry [19] is best suited when the active molecule mass is much greater than the perturber one. It can be defined by Z þ1 A zt 1 þ ezt aG ðx; y; zÞ ¼ pffiffiffi Re exp ixt yt dt 2z2 p 1 ð9Þ with the same definition of the parameters. When the pressure increases (above about 120 mbar at room temperature), collisional broadening becomes progressively the dominant effect. It depends on the relative speed of the collision partners. Then it is necessary to take into account the different classes of speed from the Maxwell–Boltzmann distribution for the absorbing molecule [22]. The line profile is then described by a convolution between a Doppler profile narrowed by collisions and a weighted sum of Lorentzian profiles [23] (Rautian*WSL or Galatry*WSL).
Fig. 1. Collisional broadening coefficient c0 deduced by fitting different models to the observed spectra of the R (0) line in the m4 band of CH4 at 223.2 K perturbed by O2 at different pressures. The models are represented by: (·) Rautian and (s) Rautian*WSL.
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
89
presents a typical signature in the observed-minus-calculated (O C) residuals. This difference between the observed lineshape and the fit using a Voigt, Rautian or Galatry model is illustrated in Fig. 2. This figure gives an example of a fit performed to determine the collisional broadening coefficient of the R (0) line in the m4 band of methane. The spectrum was acquired for a mixture of 0.340 mbar of CH4 diluted in 60.44 mbar of oxygen at room temperature. The models including Dicke narrowing lead to a better agreement with the experimental profile than the Voigt profile. We observe differences in the derived values of the collisional broadening coefficients. The coefficient obtained with the Voigt profile is about 5% smaller than the others. The difference between the Galatry and Rautian models is due to their opposite limiting cases for the mass ratio. However, the standard deviations obtained with the Rautian profile, also called hard collision model, and the Galatry profile, also called soft collision model, are not quite the same. The mass ratio for the mixture (CH4–O2) under discussion here is more appropriate to the Rautian
hypothesis. Comparing the collisional narrowing parameters with the calculated value of the dynamic friction coefficient, we find that the narrowing parameter given by the Rautian model (hard collisions) is closer than the value, for the narrowing parameter, given by the Galatry model. For similar conditions (CH4–O2 mixture, Ptot 60 mbar and room temperature), we have thus used the Rautian model to determine with accuracy the collisional broadening coefficient. When the pressure increases, it is necessary to take into account the speed dependence effect. At room temperature, this effect is important for pressures exceeding 120 mbar, while at low temperature it appears for lower pressures. Fig. 3 presents an example of a fit performed to determine the collisional broadening coefficient of the R (0) line in in the m4 band of methane diluted in 36.20 mbar of nitrogen at 153.2 K. As previously, the (O C) residuals for the Voigt profile present a typical spectral signature; the Lorentz profile is in better agreement with the experimental lineshape. The models considering the Dicke effect (Rautian and Galatry) and
Fig. 2. Observed spectrum at room temperature of the R (0) line of m4 band of CH4 ðP CH4 ¼ 0.340 mbarÞ perturbed by N2 (Ptot = 60.44 mbar) and least squares fitted line described by a Voigt profile. The (O C) residuals are plotted for different least squares fits: Voigt profile, Rautian and Galatry models.
Fig. 3. Observed spectrum at 153.2 K of the R (0) line of m4 band of CH4 ðP CH4 ¼ 0.045 mbarÞ perturbed by O2 (Ptot = 36.20 mbar) and least squares fitted line described by a Voigt profile. The (O C) residuals are plotted for different least squares fits: Voigt and Lorentz profiles, Rautian, Galatry, Rautian*WSL, and Galatry*WSL models.
90
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
those taking into account both the Dicke and speed dependence effects are matching very well the experimental profile. Comparing the collisional broadening coefficients, large differences can be observed. The value obtained with the Voigt profile is about 5% smaller than what is obtained with models considering the Dicke effect. The value given by the Lorentz profile is about 7% larger than those obtained with the Rautian or Galatry models and those including speed dependence effect. The collisional broadening coefficient as well as the quality of the fits are quite the same for all the models including at least the Dicke effect. However, the narrowing coefficients are very different. The models considering the speed dependence effect produce a narrowing coefficient b0 very close to the calculated value of the dynamic friction coefficient, while the use of the Rautian or Galatry model results in a coefficient about 40% larger than the calculated value bDiff and thus with no physical significance. The mass ratio for the mixture (CH4–N2) under discussion here is more appropriate to the Rautian hypothesis. However, the narrowing coefficient obtained with the Rautian*WSL model is in slightly better agreement with the calculated value than the one obtained with the Galatry*WSL model. This difference has been observed previously for the same gas mixture in the case of the 2m3 band [3]. For similar conditions (CH4–N2 mixture, Ptot 36 mbar and T = 153 K, as shown in Fig. 3), we have thus used the Galatry*WSL model to describe the experimental lineshape and to determine precisely the line parameters. We have analyzed the collisional broadening coefficients of the R (0) line obtained with different line profile models as a function of the total pressure of the mixture. Fig. 4 illustrates these results for the CH4–N2 mixture at room temperature. The squares (n) represent the results obtained using the Lorentz profile; this profile overestimates the average value especially at low pressure and these values exhibit a large pressure dependence. The results obtained by using the Voigt profile, represented by triangles (m), depend also on the pressure: they increase with the pressure up to about 100 mbar and decrease for pressures above 120 mbar. These values are systematically smaller than the other determinations. The collisional broadening coefficients derived with the Rautian model (·) as well as with the Galatry model (+) are quite stable. Except for very low pressure, they decrease or increase only slightly when the pressure is increasing. As seen above, a soft model is preferable for the CH4–N2 mixture and, at room temperature, the Galatry model is thus used up to 120 mbar. For higher pressures, the collisional broadening coefficients derived using the Galatry as well as the Rautian model begin to decrease. For such pressures, if one takes into account simultaneously narrowing and speed dependence effects (Galatry*WSL model), the collisional broadening coefficient (s) is very close to what is given by the Galatry model
Fig. 4. Collisional broadening coefficient c0 deduced by fitting different models to the observed spectra of the R (0) line in the m4 band of CH4 at 296.0 K perturbed by N2 at different pressures. The models are represented by: (m) Voigt; (n) Lorentz; (+) Galatry; (·) Rautian; and (s) Galatry*WSL.
at lower pressure. The solid line represents the average value (56.00 ± 0.68 · 103 cm1 atm1) of the collisional broadening coefficient. This average includes the values obtained with the Galatry model for pressures below 120 mbar and with a model taking into account the Galatry profile and the speed dependence effect at higher pressures. We have made a similar analysis for the N2-collisional narrowing coefficients. Fig. 5 is an illustration of the derived values of b0 using the most suitable theoretical profiles with respect to the total pressure. The crosses (+) represent the results obtained when using the Galatry model and the open circle (s) is the result for the model taking into account the speed dependence effect. For pressures above 120 mbar, the narrowing coefficients obtained with the soft collision model increase rapidly and thus loose their physical significance (they are just fitting parameters). If we consider the speed dependence effect, the coefficients still keep a valid value. The average value (41.3 ± 3.5 · 103 cm1 atm1), represented by the solid line, includes the values based on the Galatry model at pressure below 120 mbar and the values of the model taking into account the speed dependence effect when the pressure is above 120 mbar; the dotted line represents the calculated value bDiff = 38.1 cm1 atm1. The agreement between our measurements and the diffusion theory is very satisfactory. Finally, Tables 1 and 2 give the collisional broadening and narrowing coefficients for the R (0) line in the m4 band of methane diluted in nitrogen and oxygen, respectively. In these tables, we present results obtained at room and low (223.2, 188.2, and 153.2 K) temperatures and we also compare the narrowing parameters with those calculated by the diffusion theory bDiff. It is more difficult to determine precisely the Dicke narrow-
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
91
Fig. 5. Collisional narrowing coefficient b0 deduced by fitting different models to the observed spectra of the R (0) line in the m4 band of CH4 at 296.0 K perturbed by N2 at different pressures. The models are represented by: (+) Galatry and (s) Galatry*WSL.
ing coefficient b0 than the broadening coefficient because the determination of b0 depends largely on the position of the baseline. The agreement between experimental and calculated narrowing coefficients is thus reasonable. The errors given in Tables 1 and 2 represent one standard deviation as obtained by averaging different measurements. For the CH4–N2 mixture, the broadening and narrowing coefficients given in Table 1 are the average values obtained with the Galatry model for low pressures and with a model taking into account simultaneously Dicke narrowing and the speed dependence effect (Galatry*WSL) for higher pressures. For the CH4–O2 mixture, the parameters given in Table 2 are the average values obtained with the Rautian model for low pressures and the Rautian*WSL model for higher pressures. The determination of collisional broadening coefficients at different temperatures is also needed for atmospheric applications as well as for testing theoretical models. The following empirical law is commonly used, for a given gas mixture, to describe the variation of the collisional broadening coefficients as a function of the temperature: n c0 ðT Þ T ¼ . ð10Þ c0 ðT 0 Þ T0 The logarithmic graphs of Fig. 6 present the c0 coefficients versus temperature for the two mixtures and illustrate the n values obtained for CH4–N2 (n = 0.676 ± 0.010) and CH4–O2 (n = 0.746 ± 0.007) mixtures. The errors given here represent 3 standard deviations. For the CH4–N2 mixture, it is possible to compare our results with previously measured values. Our determination (c0 = 56.00 ± 0.68 · 103 cm1 atm1) for the N2-collisional broadening coefficient of the R (0) line in
Fig. 6. O2- and N2-broadening coefficients of the R (0) line of the m4 band of CH4 versus temperature (logarithmic scales). The slope of the lines is derived from a linear regression for determining the exponent n of the inverse power law dependence with respect to temperature. Errors given represent three standard deviations.
the m4 band at room temperature and the measurement (c0 = 54.17 · 103 cm1 atm1) of Smith et al. [9] made by FT spectroscopy for the same line are in good agreement. Smith et al. have also performed low temperature measurements for this line of methane and they have deduced a temperature dependence exponent (n = 0.709) which is close to our determination (n = 0.676 ± 0.010).
5. Conclusion Using a diode-laser spectrometer, we have recorded spectra of the R (0) line in the m4 band of methane diluted in nitrogen and in oxygen. For each gas mixture, we have recorded about 20 spectra with different total pressures. These measurements were performed at four temperatures between 153.2 K and room temperature. The collisional broadening and narrowing coefficients are the average values of determinations made by fitting the most appropriate theoretical profile. This profile is chosen according to the experimental conditions: gas mixture, pressure, and temperature. From these results, we have finally determined the temperature dependence
92
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
of the broadening coefficient of the R (0) line in the m4 band of methane diluted in nitrogen and in oxygen.
Acknowledgments M. Lepe`re is acknowledging support from FNRS. This study was funded in part by the ‘‘Centre National dEtudes Spatiales’’ (CNES) and was performed within the ‘‘Programme National de Chimie Atmosphe´rique’’ (PNCA). The work at FUNDP was accomplished in the framework of the European associated laboratory HiRes.
References [1] D.J. Wuebbles, K. Hayhoe, Earth-Science Rev. 57 (2002) 177–210. [2] A.S. Pine, J. Quant. Spectrosc. Radiat. Transfer 57 (1997) 157–176. [3] G. Dufour, D. Hurtmans, A. Henry, A. Valentin, M. Lepe`re, J. Mol. Spectrosc. 221 (2003) 80–92. [4] V. Malathy Devi, C.P. Rinsland, M.A.H. Smith, D.C. Benner, Appl. Opt. 24 (1985) 2788–2791. [5] V. Malathy Devi, C.P. Rinsland, M.A.H. Smith, D.C. Benner, Appl. Opt. 24 (1985) 3321–3322. [6] C.P. Rinsland, V. Malathy Devi, M.A.H. Smith, D.C. Benner, Appl. Opt. 27 (1988) 631–651.
[7] V. Malathy Devi, C.P. Rinsland, M.A.H. Smith, D.C. Benner, Appl. Opt. 27 (1988) 2296–2308. [8] C.P. Rinsland, V. Malathy Devi, M.A.H. Smith, D.C. Benner, Appl. Opt. 28 (1989) 2111–2118. [9] M.A.H. Smith, C.P. Rinsland, V. Malathy Devi, D.C. Benner, Spectrochim. Acta A 48 (1992) 1257–1272. [10] J.S. Margolis, J. Quant. Spectrosc. Radiat. Transfer 50 (1993) 431–441. [11] J.S. Margolis, J. Quant. Spectrosc. Radiat. Transfer 55 (1996) 823–836. [12] M. Lepe`re, A. Valentin, A. Henry, C. Camy-Peyret, G. Blanquet, J.C. Populaire, A.W. Mantz, Spectrochim. Acta A 58 (2002) 2413–2419. [13] A. Valentin, Spectrochim. Acta A 52 (1996) 823–833. [14] A. Henry, A. Valentin, M. Margottin-Maclou, F. Rachet, J. Mol. Spectrosc. 166 (1994) 41–55. [15] Ch. Lerot, J. Walrand, G. Blanquet, J.P. Bouanich, M. Lepe`re, J. Mol. Spectrosc. 219 (2003) 329–334. [16] R.H. Dicke, Phys. Rev. 89 (1953) 472–473. [17] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1967. [18] S.G. Rautian, I.I. Sobelman, Sov. Phys. Usp. Engl. Transl. 9 (1967) 701–716. [19] L. Galatry, Phys. Rev. 122 (1960) 1218–1224. [20] B.H. Armstrong, J. Quant. Spectrosc. Radiat. Transfer 7 (1967) 61–88. [21] P.L. Varghese, R.K. Hanson, Appl. Opt. 23 (1984) 2376. [22] B. Lance, Ph. D. thesis, Faculte´s Universitaires N.D. de la Paix, Namur, Belgium, 1998. [23] H.M. Pickett, J. Chem. Phys. 73 (1980) 6090–6094. [24] M. Lepe`re, Spectrochim. Acta A 60 (2004) 3249–3258.
Diode-laser spectroscopy: Temperature dependence of R (0) line in the m4 band of CH4 perturbed by N2 and O2 Muriel Lepe`re a,*,1, Alain Valentin b, Annie Henry b, Claude Camy-Peyret b Marc Lengele´ a, Jean-Claude Populaire a, Ghislain Blanquet a a
Laboratoire de Spectroscopie Mole´culaire, Faculte´s Universitaires Notre-Dame de la Paix, 61, rue de Bruxelles, B-5000 Namur, Belgium b Laboratoire de Physique Mole´culaire et Applications, CNRS, Universite´ Pierre et Marie Curie, Tour 13, Bte 76, 4 place Jussieu, F-75252 Paris cedex 05, France Received 30 March 2005; in revised form 1 June 2005 Available online 26 July 2005
Abstract In this paper, we present a line profile study of the R (0) line in the m4 band of methane diluted in nitrogen and oxygen, from room temperature to 153 K. The measurements were performed over a total pressure range from 14 to 128 mbar. The collisional broadening and narrowing (Dicke effect) coefficients are derived from a fit of the experimental spectra by using the soft and hard collision models, taking into account the Dicke effect. For higher pressures, we have fitted the data with a model taking into account simultaneously the Dicke narrowing and the speed dependence effect. Finally, we have deduced the parameter n of the temperature dependence (inverse power law) of the broadening coefficients for the CH4–N2 and CH4–O2 gas mixtures. 2005 Elsevier Inc. All rights reserved. Keywords: Tunable diode-laser; Methane; Line profile; Broadening coefficients; Temperature dependence
1. Introduction Methane is an important minor constituent in different atmospheres: Jupiter, Saturn, Titan, Uranus, and Neptune in the solar system as well as hot Jupiters in other planetary systems. On Earth, methane is the most abundant organic trace gas in our own atmosphere [1]. A good knowledge of collisional lineshapes in several bands of methane is thus important. Laboratory lineshape studies have been already extensively performed, mainly in the m3 band ([2] and references therein) and recently in the 2m3 band ([3] and references therein). The corresponding experiments allowed precise determinations of spectroscopic line parameters such as collisional broadening and narrowing. *
1
Corresponding author. Fax: +32 81 72 45 85. E-mail address: [email protected] (M. Lepe`re). Associate Researcher with FNRS, Belgium.
0022-2852/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2005.06.004
These measurements are very important in infrared remote sensing. For reliable atmospheric sounding of the Earth atmosphere, it is necessary to determine precisely methane collisional broadening coefficients by nitrogen and oxygen as well as their temperature dependence. Much of the previous efforts in determining broadening parameters for the m4 band have concentrated on precise measurements at room temperature [4–8] with only a few papers devoted to high resolution measurements at low temperature [9–12].
2. Experimental details The spectra were recorded at high spectral resolution using a diode-laser spectrometer described previously [13,14]. The diode-laser emission was thus actively controlled using a Michelson interferometer and the residual wavenumber fluctuations in the emitted laser
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
radiation were smaller than 4 · 105 cm1. An absorption line is scanned step-by-step with a spectral sampling step equal to 0.4845 · 103 cm1 at 1311.4315 cm1. The TDL beam used for absorption experiments is divided into two infrared beams; one beam is used as a reference signal, I0, and the other goes through the sample cell to provide a transmitted signal, It. At each step the beam intensities I0 and It are measured simultaneously, and the ratio It/I0 is determined with a precision of about 5 · 104. For low temperature measurements, we have used an absorption cell described in details elsewhere [15]. This cell can be cooled uniformly by a liquid–N2 flow at any temperature between 77 K and room temperature, with a temperature stability better than 0.5 K. In the experiments reported here, this absorption cell with an optical path length of 40.43 cm was cooled down to 223.2, 188.2, and 153.2 K. For each gas mixture (CH4–N2 and CH4–O2) and each temperature (296.0, 223.2, 188.2, and 153.2 K), we have recorded the spectra around the R (0) line at different (at least 4) gas mixture pressures between 14 and 128 mbar. For each pressure, three spectra were recorded. The total pressure of the gas mixture and the methane pressure were measured using two MKS Baratron gauges with 120 and 1.2 mbar full scales, respectively. The methane (N30 quality) with a stated purity of 99.9% was supplied by the Air Liquide company.
where kB is the Boltzmann constant, T is the temperature (296.0, 223.2, 188.2 or 153.2 K); c is the speed of light, m1 the molecular weight of the absorber or active molecule, and D12 the mass-diffusion coefficient for the CH4–N2 or CH4–O2 pair. This coefficient estimated from a Lennard-Jones potential is given at different temperatures for the CH4–N2 and CH4–O2 mixtures on Tables 1 and 2, respectively. Concurrently, the main effect of collisions is to broaden the lines of the active molecule. Considering the relative speed of the partner molecules as independent of the absorber speed, the collision broadening is described by a Lorentz function with a half-width at half maximum given by cL ¼ c0 ðP tot P CH4 Þ þ cself P CH4 ;
The measured absorption coefficient a (m) (in cm1 atm1) at wavenumber m of a homogeneous gas sample is defined by the Beer–Lambert law as I t ðmÞ P absorber LaðmÞ ¼ ln ; ð1Þ I 0 ðmÞ where I0 (m) and It (m) are the transmitted intensities obtained, respectively, with the cell of path length L under vacuum and filled with the absorbing gas at partial pressure Pabsorber (possibly with a buffer gas at total sample pressure Ptot). A synthetic spectrum is compared in a least squares fit procedure to the observed one to determine different spectroscopic parameters (e.g., S0, c0, b0, . . .). At very low pressure, the line profile is described by a Gauss function (Doppler effect). When the pressure increases, the Doppler line is progressively narrowed by the confinement of the active molecules in the buffer gas (Dicke effect [16]). This effect is characterized by the narrowing parameter b0 which can be compared to the dynamic friction coefficient bDiff deduced from [17]: bDiff ¼
kBT ; 2pcm1 D12
ð2Þ
ð3Þ
where c0 is the coefficient for collisional broadening by foreign molecules, Ptot is the total pressure of the gas mixture (CH4–N2 or CH4–O2), cself is the coefficient for self-broadening, and P CH4 is the partial pressure of methane. The narrowing effect of collisions on pure Doppler broadening and on Lorentzian collision broadening are taken into account by the uncorrelated hard colliTable 1 Collisional broadening c0 and narrowing b0 coefficients of R (0) line in the m4 band of CH4 perturbed in N2 T (K)
3. Data reduction
87
296.0 223.2 188.2 153.2
Parameters (103 cm1 atm1) c0a
b0a
bDiff
56.00 ± 0.68 68.37 ± 0.58 76.15 ± 0.47 87.52 ± 0.48
41.3 ± 3.5 56.7 ± 3.3 59.1 ± 5.6 68.1 ± 5.7
38.1 48.0 55.6 67.1
Comparison with parameter bDiff calculated by the diffusion theory. a c0 and b0 are obtained with the Galatry model for low pressures and with a model taking into account simultaneously Dicke narrowing and the speed dependence effect for higher pressures. Errors given represent one standard deviation obtained by averaging different measurements.
Table 2 Collisional broadening c0 and narrowing b0 coefficients of R (0) line in the m4 band of CH4 perturbed in O2 T (K)
296.0 223.2 188.2 153.2
Parameters (103 cm1 atm1) c0a
b0a
bDiff
53.05 ± 0.20 65.50 ± 0.57 74.72 ± 0.85 86.60 ± 0.38
31.0 ± 4.0 41.7 ± 4.0 46.6 ± 5.3 59.5 ± 6.7
37.6 47.8 55.5 67.3
Comparison with the parameter bDiff calculated by the diffusion theory. a c0 and b0 are obtained with the Rautian model for low pressures and with a model taking into account simultaneously Dicke narrowing and the speed dependence effect for higher pressures. Errors given here represent one standard deviation obtained by averaging different measurements.
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
88
sion lineshape model developed by Rautian and Sobelman [18] and the uncorrelated soft collision model proposed by Galatry [19]. The line profile is then equivalent to the convolution product of a Lorentz profile with a confinement narrowed profile instead of a Gaussian profile (Doppler effect) commonly used in the Voigt profile [20,21]. The Voigt profile is defined by [20,21] Z y þ1 expðt2 Þ aV ðx; yÞ ¼ A dt p 1 y 2 þ ðx tÞ2 ¼ A Re½W ðx; yÞ
ð4Þ
with W ðx; yÞ ¼
i p
where pffiffiffiffiffiffiffiffi S ln 2 pffiffiffi ; A¼ cD p
Z
þ1 1
y¼
expðt2 Þ dt; x þ iy t
ð5Þ
pffiffiffiffiffiffiffiffi m m0 pffiffiffiffiffiffiffiffi cL . ln 2 ; x ¼ ln 2 cD cD
ð6Þ
Here S is the intensity of the line under study (here in cm1/(atm cm) or usually in cm1/(molecule cm2) in atmospheric data bases), cL (cm1) is the collisional half-width, and m0 (cm1) is the line center wavenumber (pressure shifts are not considered in the present study). The Rautian and Sobelman model [18] is best suited when the perturber mass is larger than mass of the active molecule. This profile may be described as W ðx; y þ zÞ pffiffiffi aR ðx; y; zÞ ¼ ARe ; ð7Þ 1 pzW ðx; y þ zÞ where the parameters are the same as above and pffiffiffiffiffiffiffiffi b z ¼ ln 2 c cD
ð8Þ
4. Results As shown in a previous paper [24], it is necessary to select the most appropriate theoretical profile to reach the highest level of accuracy on the determination of spectroscopic line parameters. The most appropriate profile depends on experimental conditions, especially the pressure. The temperature has also a large impact on the physical processes which determine the molecular lineshape. To improve the reliability of our measurements, we have recorded spectra at different (4–8) pressures for each gas mixture at a given temperature. In addition, we have recorded three spectra at each pressure. In this way, the final value of each line broadening parameter is obtained from an average of the values derived from the different pressures. Fig. 1 illustrates the results obtained for the CH4–O2 mixture at 223.2 K. This figure is showing the coefficients of collisional broadening versus total pressure. These measurements were performed at eight different gas mixture pressures between 19.8 and 115.2 mbar, and we have recorded three spectra at each pressure. The solid line represents the average value (over 24 spectra) for the CH4–O2 collisional broadening coefficients. This average value includes those determined with the Rautian model at pressure below 70 mbar and those obtained with the model taking into account the speed dependence effect for higher pressures. Fitting different line profiles to the experimental ones, we have thus determined the coefficients of collisional broadening c0 and narrowing b0 at 296.0, 223.2, 188.2, and 153.2 K. At room temperature, the Rautian and Galatry models taking into account the Dicke effect are well adapted to the experimental profile while the usual Voigt profile
with bc (cm1) representing the average effect of collisions on Doppler broadening. The model proposed by Galatry [19] is best suited when the active molecule mass is much greater than the perturber one. It can be defined by Z þ1 A zt 1 þ ezt aG ðx; y; zÞ ¼ pffiffiffi Re exp ixt yt dt 2z2 p 1 ð9Þ with the same definition of the parameters. When the pressure increases (above about 120 mbar at room temperature), collisional broadening becomes progressively the dominant effect. It depends on the relative speed of the collision partners. Then it is necessary to take into account the different classes of speed from the Maxwell–Boltzmann distribution for the absorbing molecule [22]. The line profile is then described by a convolution between a Doppler profile narrowed by collisions and a weighted sum of Lorentzian profiles [23] (Rautian*WSL or Galatry*WSL).
Fig. 1. Collisional broadening coefficient c0 deduced by fitting different models to the observed spectra of the R (0) line in the m4 band of CH4 at 223.2 K perturbed by O2 at different pressures. The models are represented by: (·) Rautian and (s) Rautian*WSL.
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
89
presents a typical signature in the observed-minus-calculated (O C) residuals. This difference between the observed lineshape and the fit using a Voigt, Rautian or Galatry model is illustrated in Fig. 2. This figure gives an example of a fit performed to determine the collisional broadening coefficient of the R (0) line in the m4 band of methane. The spectrum was acquired for a mixture of 0.340 mbar of CH4 diluted in 60.44 mbar of oxygen at room temperature. The models including Dicke narrowing lead to a better agreement with the experimental profile than the Voigt profile. We observe differences in the derived values of the collisional broadening coefficients. The coefficient obtained with the Voigt profile is about 5% smaller than the others. The difference between the Galatry and Rautian models is due to their opposite limiting cases for the mass ratio. However, the standard deviations obtained with the Rautian profile, also called hard collision model, and the Galatry profile, also called soft collision model, are not quite the same. The mass ratio for the mixture (CH4–O2) under discussion here is more appropriate to the Rautian
hypothesis. Comparing the collisional narrowing parameters with the calculated value of the dynamic friction coefficient, we find that the narrowing parameter given by the Rautian model (hard collisions) is closer than the value, for the narrowing parameter, given by the Galatry model. For similar conditions (CH4–O2 mixture, Ptot 60 mbar and room temperature), we have thus used the Rautian model to determine with accuracy the collisional broadening coefficient. When the pressure increases, it is necessary to take into account the speed dependence effect. At room temperature, this effect is important for pressures exceeding 120 mbar, while at low temperature it appears for lower pressures. Fig. 3 presents an example of a fit performed to determine the collisional broadening coefficient of the R (0) line in in the m4 band of methane diluted in 36.20 mbar of nitrogen at 153.2 K. As previously, the (O C) residuals for the Voigt profile present a typical spectral signature; the Lorentz profile is in better agreement with the experimental lineshape. The models considering the Dicke effect (Rautian and Galatry) and
Fig. 2. Observed spectrum at room temperature of the R (0) line of m4 band of CH4 ðP CH4 ¼ 0.340 mbarÞ perturbed by N2 (Ptot = 60.44 mbar) and least squares fitted line described by a Voigt profile. The (O C) residuals are plotted for different least squares fits: Voigt profile, Rautian and Galatry models.
Fig. 3. Observed spectrum at 153.2 K of the R (0) line of m4 band of CH4 ðP CH4 ¼ 0.045 mbarÞ perturbed by O2 (Ptot = 36.20 mbar) and least squares fitted line described by a Voigt profile. The (O C) residuals are plotted for different least squares fits: Voigt and Lorentz profiles, Rautian, Galatry, Rautian*WSL, and Galatry*WSL models.
90
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
those taking into account both the Dicke and speed dependence effects are matching very well the experimental profile. Comparing the collisional broadening coefficients, large differences can be observed. The value obtained with the Voigt profile is about 5% smaller than what is obtained with models considering the Dicke effect. The value given by the Lorentz profile is about 7% larger than those obtained with the Rautian or Galatry models and those including speed dependence effect. The collisional broadening coefficient as well as the quality of the fits are quite the same for all the models including at least the Dicke effect. However, the narrowing coefficients are very different. The models considering the speed dependence effect produce a narrowing coefficient b0 very close to the calculated value of the dynamic friction coefficient, while the use of the Rautian or Galatry model results in a coefficient about 40% larger than the calculated value bDiff and thus with no physical significance. The mass ratio for the mixture (CH4–N2) under discussion here is more appropriate to the Rautian hypothesis. However, the narrowing coefficient obtained with the Rautian*WSL model is in slightly better agreement with the calculated value than the one obtained with the Galatry*WSL model. This difference has been observed previously for the same gas mixture in the case of the 2m3 band [3]. For similar conditions (CH4–N2 mixture, Ptot 36 mbar and T = 153 K, as shown in Fig. 3), we have thus used the Galatry*WSL model to describe the experimental lineshape and to determine precisely the line parameters. We have analyzed the collisional broadening coefficients of the R (0) line obtained with different line profile models as a function of the total pressure of the mixture. Fig. 4 illustrates these results for the CH4–N2 mixture at room temperature. The squares (n) represent the results obtained using the Lorentz profile; this profile overestimates the average value especially at low pressure and these values exhibit a large pressure dependence. The results obtained by using the Voigt profile, represented by triangles (m), depend also on the pressure: they increase with the pressure up to about 100 mbar and decrease for pressures above 120 mbar. These values are systematically smaller than the other determinations. The collisional broadening coefficients derived with the Rautian model (·) as well as with the Galatry model (+) are quite stable. Except for very low pressure, they decrease or increase only slightly when the pressure is increasing. As seen above, a soft model is preferable for the CH4–N2 mixture and, at room temperature, the Galatry model is thus used up to 120 mbar. For higher pressures, the collisional broadening coefficients derived using the Galatry as well as the Rautian model begin to decrease. For such pressures, if one takes into account simultaneously narrowing and speed dependence effects (Galatry*WSL model), the collisional broadening coefficient (s) is very close to what is given by the Galatry model
Fig. 4. Collisional broadening coefficient c0 deduced by fitting different models to the observed spectra of the R (0) line in the m4 band of CH4 at 296.0 K perturbed by N2 at different pressures. The models are represented by: (m) Voigt; (n) Lorentz; (+) Galatry; (·) Rautian; and (s) Galatry*WSL.
at lower pressure. The solid line represents the average value (56.00 ± 0.68 · 103 cm1 atm1) of the collisional broadening coefficient. This average includes the values obtained with the Galatry model for pressures below 120 mbar and with a model taking into account the Galatry profile and the speed dependence effect at higher pressures. We have made a similar analysis for the N2-collisional narrowing coefficients. Fig. 5 is an illustration of the derived values of b0 using the most suitable theoretical profiles with respect to the total pressure. The crosses (+) represent the results obtained when using the Galatry model and the open circle (s) is the result for the model taking into account the speed dependence effect. For pressures above 120 mbar, the narrowing coefficients obtained with the soft collision model increase rapidly and thus loose their physical significance (they are just fitting parameters). If we consider the speed dependence effect, the coefficients still keep a valid value. The average value (41.3 ± 3.5 · 103 cm1 atm1), represented by the solid line, includes the values based on the Galatry model at pressure below 120 mbar and the values of the model taking into account the speed dependence effect when the pressure is above 120 mbar; the dotted line represents the calculated value bDiff = 38.1 cm1 atm1. The agreement between our measurements and the diffusion theory is very satisfactory. Finally, Tables 1 and 2 give the collisional broadening and narrowing coefficients for the R (0) line in the m4 band of methane diluted in nitrogen and oxygen, respectively. In these tables, we present results obtained at room and low (223.2, 188.2, and 153.2 K) temperatures and we also compare the narrowing parameters with those calculated by the diffusion theory bDiff. It is more difficult to determine precisely the Dicke narrow-
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
91
Fig. 5. Collisional narrowing coefficient b0 deduced by fitting different models to the observed spectra of the R (0) line in the m4 band of CH4 at 296.0 K perturbed by N2 at different pressures. The models are represented by: (+) Galatry and (s) Galatry*WSL.
ing coefficient b0 than the broadening coefficient because the determination of b0 depends largely on the position of the baseline. The agreement between experimental and calculated narrowing coefficients is thus reasonable. The errors given in Tables 1 and 2 represent one standard deviation as obtained by averaging different measurements. For the CH4–N2 mixture, the broadening and narrowing coefficients given in Table 1 are the average values obtained with the Galatry model for low pressures and with a model taking into account simultaneously Dicke narrowing and the speed dependence effect (Galatry*WSL) for higher pressures. For the CH4–O2 mixture, the parameters given in Table 2 are the average values obtained with the Rautian model for low pressures and the Rautian*WSL model for higher pressures. The determination of collisional broadening coefficients at different temperatures is also needed for atmospheric applications as well as for testing theoretical models. The following empirical law is commonly used, for a given gas mixture, to describe the variation of the collisional broadening coefficients as a function of the temperature: n c0 ðT Þ T ¼ . ð10Þ c0 ðT 0 Þ T0 The logarithmic graphs of Fig. 6 present the c0 coefficients versus temperature for the two mixtures and illustrate the n values obtained for CH4–N2 (n = 0.676 ± 0.010) and CH4–O2 (n = 0.746 ± 0.007) mixtures. The errors given here represent 3 standard deviations. For the CH4–N2 mixture, it is possible to compare our results with previously measured values. Our determination (c0 = 56.00 ± 0.68 · 103 cm1 atm1) for the N2-collisional broadening coefficient of the R (0) line in
Fig. 6. O2- and N2-broadening coefficients of the R (0) line of the m4 band of CH4 versus temperature (logarithmic scales). The slope of the lines is derived from a linear regression for determining the exponent n of the inverse power law dependence with respect to temperature. Errors given represent three standard deviations.
the m4 band at room temperature and the measurement (c0 = 54.17 · 103 cm1 atm1) of Smith et al. [9] made by FT spectroscopy for the same line are in good agreement. Smith et al. have also performed low temperature measurements for this line of methane and they have deduced a temperature dependence exponent (n = 0.709) which is close to our determination (n = 0.676 ± 0.010).
5. Conclusion Using a diode-laser spectrometer, we have recorded spectra of the R (0) line in the m4 band of methane diluted in nitrogen and in oxygen. For each gas mixture, we have recorded about 20 spectra with different total pressures. These measurements were performed at four temperatures between 153.2 K and room temperature. The collisional broadening and narrowing coefficients are the average values of determinations made by fitting the most appropriate theoretical profile. This profile is chosen according to the experimental conditions: gas mixture, pressure, and temperature. From these results, we have finally determined the temperature dependence
92
M. Lepe`re et al. / Journal of Molecular Spectroscopy 233 (2005) 86–92
of the broadening coefficient of the R (0) line in the m4 band of methane diluted in nitrogen and in oxygen.
Acknowledgments M. Lepe`re is acknowledging support from FNRS. This study was funded in part by the ‘‘Centre National dEtudes Spatiales’’ (CNES) and was performed within the ‘‘Programme National de Chimie Atmosphe´rique’’ (PNCA). The work at FUNDP was accomplished in the framework of the European associated laboratory HiRes.
References [1] D.J. Wuebbles, K. Hayhoe, Earth-Science Rev. 57 (2002) 177–210. [2] A.S. Pine, J. Quant. Spectrosc. Radiat. Transfer 57 (1997) 157–176. [3] G. Dufour, D. Hurtmans, A. Henry, A. Valentin, M. Lepe`re, J. Mol. Spectrosc. 221 (2003) 80–92. [4] V. Malathy Devi, C.P. Rinsland, M.A.H. Smith, D.C. Benner, Appl. Opt. 24 (1985) 2788–2791. [5] V. Malathy Devi, C.P. Rinsland, M.A.H. Smith, D.C. Benner, Appl. Opt. 24 (1985) 3321–3322. [6] C.P. Rinsland, V. Malathy Devi, M.A.H. Smith, D.C. Benner, Appl. Opt. 27 (1988) 631–651.
[7] V. Malathy Devi, C.P. Rinsland, M.A.H. Smith, D.C. Benner, Appl. Opt. 27 (1988) 2296–2308. [8] C.P. Rinsland, V. Malathy Devi, M.A.H. Smith, D.C. Benner, Appl. Opt. 28 (1989) 2111–2118. [9] M.A.H. Smith, C.P. Rinsland, V. Malathy Devi, D.C. Benner, Spectrochim. Acta A 48 (1992) 1257–1272. [10] J.S. Margolis, J. Quant. Spectrosc. Radiat. Transfer 50 (1993) 431–441. [11] J.S. Margolis, J. Quant. Spectrosc. Radiat. Transfer 55 (1996) 823–836. [12] M. Lepe`re, A. Valentin, A. Henry, C. Camy-Peyret, G. Blanquet, J.C. Populaire, A.W. Mantz, Spectrochim. Acta A 58 (2002) 2413–2419. [13] A. Valentin, Spectrochim. Acta A 52 (1996) 823–833. [14] A. Henry, A. Valentin, M. Margottin-Maclou, F. Rachet, J. Mol. Spectrosc. 166 (1994) 41–55. [15] Ch. Lerot, J. Walrand, G. Blanquet, J.P. Bouanich, M. Lepe`re, J. Mol. Spectrosc. 219 (2003) 329–334. [16] R.H. Dicke, Phys. Rev. 89 (1953) 472–473. [17] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1967. [18] S.G. Rautian, I.I. Sobelman, Sov. Phys. Usp. Engl. Transl. 9 (1967) 701–716. [19] L. Galatry, Phys. Rev. 122 (1960) 1218–1224. [20] B.H. Armstrong, J. Quant. Spectrosc. Radiat. Transfer 7 (1967) 61–88. [21] P.L. Varghese, R.K. Hanson, Appl. Opt. 23 (1984) 2376. [22] B. Lance, Ph. D. thesis, Faculte´s Universitaires N.D. de la Paix, Namur, Belgium, 1998. [23] H.M. Pickett, J. Chem. Phys. 73 (1980) 6090–6094. [24] M. Lepe`re, Spectrochim. Acta A 60 (2004) 3249–3258.