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Line mixing and line broadening in CO2 bands perturbed by helium at 193 K
CHEMICAL PHYSICS ttwi-:nS
27 December 1996
ELSEVIER
Chemical Physics Letters 263 (1996) 811-816
i
Line mixing and line broadening in CO 2 bands perturbed by helium at 193 K B. Khalil
a
O. Cisse b G. Moreau
a
F. Thibault
a
R. Le Doucen
a
j. Boissoles
a
a Unit[" Mixte de Recherche P.A.L.M.S. (Physique des Atomes, Lasers, Molecules et Surfaces), Universit~ de Rennes 1. campus de Beaulieu. 35042 Rennes cedex, France b Universit~ de Ouagadougou, Ouagadougou, Burkimt Faso
Received 11 July 1996; in final form 22 October 1996
Abstract
The 1'2 and 3 v 3 bands of CO 2 in helium baths at 193 K have been studied with a Fourier transform interfe~ometer. The behavior of the band shapes has been explored at moderate densities. The energy corrected sudden (ECS) approximation is used to model the relaxation matrix in order to account for line mixing effects. The basis cross-sections were ca!culated with the simple power law (P). Computed spectra are in good agreement with the observed ones. Measuredl broadening coefficients are also comparable with the ones derived from the ECS-P model.
1. I n t r o d u c t i o n
Early experiments and calculations on spectroscopic lines shapes were mainly concerned with Lorentzian parameters, and particularly with the broadening coefficients because of their practical and theoretical importance. Infrared experimental data on C O 2 - H e linewidths come from studies undertaken at room temperature (see [1] which provides a smooth fit of various results) or at elevated temperatures [2-5]. Smith, Giraud and Cooper [6], have been rather successful in applying their semi-classical theory for line broadening to this system. A first quantum mechanical calculation was performed within the infinite order sudden (IOS) approximation by Pack [7], who investigated the pressure broadening of the dipole and Raman lines of CO 2 by He (and Ar) in a wide range of temperature. From these
studies it appears that the C O 2 - H e linewidths (at constant pressure) decrease more rapidly, with the temperature, than T-1/2 ( _ 1 / 2 is the iimit of the hard sphere collisions model or the classical value expected), this temperature dependence Varying between T -°'58 [5] to T -°88 [2]. More recently He pressure broadening cross-sections in bending bands of CO 2 at a n equivalent temperature of 300 K were calculated within the IOS approximation by Green [8]. In addition, he improved this model to properly take into account line mixing effects regardless of the band type for symmetric linear triatomics such as CO 2. Concurrently with the determination of broadening coefficients, line couplings have r~ceived increasing interest as the theory developed and as numerical calculations became more feasible. It is well known [9] that there are spectral regions, with
0009-2614/96/$12.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0009-2614(96)01276-6
812
B. Khalil et al. / Chemical Physics Letters 263 (1996) 811-816
very dense rotational line structure or with lines spaced in such a way that collisional broadening gives rise to significant overlap, where band shape cannot be obtained as a superposition of Lorentzian profiles. Indeed, line coupling effects due to rotational energy transfer may occur. Due to its role in atmospheric transmission CO 2 has been extensively studied [9], but only a few studies have been devoted to the CO2-He system which is of theoretical importance. The contribution to line broadening and line mixing in CO 2 bands in He of the St Petersburg group is notable [1,2,10-13]. In the past few years our group had investigated [14-20] various CO 2 bands in He, both theoretically and experimentally. While the theoretical methods used by the St Petersburg group are based on the strong collision model [1,10-12] or on the classical impact model [13], we use [17-20] the Energy Corrected Sudden (ECS) approximation to take line coupling effects into account. This work extends our previous study [16-18] of line interferences between the rotational components of the 00°3-00°0 band of CO 2 pressurized by helium. For this stretching band ( E - E transition) we have already shown that the theoretical framework of the Energy Corrected Sudden (ECS) approximation is rather successful in predicting the evolution of the spectral profile over an extended range of perturber pressure. This is particularly true at high pressures where the rotational structure has disappeared, line mixing effects leading to an important enhancement of the R branch due to its narrowing. One can also refer, to reference [21] which is an extension of this study at higher helium densities for the u 3 and 3 u 3 bands of CO 2. In recent studies [19,20], the same E.C.S. formalism was successfully applied to various CO 2 Qbranch shapes of bending modes near 15 and 4.7 txm. These previous studies were undertaken at room temperature. In the present Letter we investigate the 3v 3 and u 2 bands of CO 2 in helium at a temperature of 193 K. The same ECS-P model is tested. In addition, measured linewidths are compared with the ECS-P predicted ones. Section 2, 3 are devoted to a brief presentation of experimental details and to a rdsum~ of the theoretical framework. The data used and the results are presented in Section 4.
2. Experimental procedure The experimental apparatus around the FTS and the data reduction procedures have been described in refs. [14-16]. For the 3u 3 band the White-type coolable cell with a baseline of 2 m was used. Its temperature is monitored within +__0.2 K through a circulation of methylcyclohexane cooled by liquid nitrogen. The resolution was 40 × 10 -3 cm - I and interferograms were averaged over 64 scans. A single path cell with 7.68 mm length has been built to study the u 2 band. This cell is made of a double-walled stainless-steel tube which allows the circulation of the cryogenic fluid and is surrounded by several dozens of superinsulation layers. The input windows are made of ZnSe and the output ones of KBr disks: the external ones are at room temperature and the internal ones are at the sample temperature. The temperature is controlled by platinum resistive thermal probes. The resolution varied from 10 × 10 -3 to 60)< 10 -3 cm - l as the helium pressure increased from 0.4 to 10 atm.
3. Theory In the impact formulation of spectral line shape theory, the effect of collisions is described by a relaxation matrix W, the rows and columns of which are labelled by spectral lines [9]:
-~" Wlk = 2'1TC O'q( V i f V f j f ; v i J i v f j f ) - nbWi ° ,
(l)
where n b is the density of perturbers, v is the mean relative velocity for colliding molecules, q is the tensor order of the radiative interaction (e.g. q = 1 for IR spectra), k refers to the spectral line of jr v i j ~, and primes indicate post-collision values. By invoking binary completed collisions, the cross section matrix o- can be expressed in terms of the S-matrices of scattering theory. Green has recently
B. Khalil et al./ Chemical Physics Letters 263 (1996) 811-816
applied the Infinite Order Sudden (lOS) approximation to bending bands of CO 2 [8]. This approximation is expected to be fairly accurate for CO z [22], and it has the advantage of separating the dependence of cross sections on spectral level, tensor order and angular momentum coupling, from the dependence on dynamical factors. Within this framework the IOS formalism applied to symmetric rotors leads to a convenient factorisation [23] of the cross-sections into dynamical terms or basic rates Q'(L) (cf. Eq. (5) of ref. [! 8]) and spectroscopic coefficients: O'"!i!"JiJ,%J,~-
__ (\I-('--~ [ Jt Jli]. I)
`/2
11 rt/r t (L), E[L]Fjd;j,~,Q
(2)
where the spectroscopic coefficients are defined by: lrir,~,L = ( - - l ) Fjjd~j,
×
li
I + r'+"([j~l[Jil[Jfl[ff])'/2
x{Ji Ji
0
-- I i
Jr
1}
Jl
L
- - If
0
If
dr -5-Wtk.
Wkk = -- •
(6)
uk
where d~ (respectively d r) are the dipole reduced matrix elements of transition k -= f ~ i (respectively l - f' *-- i'):
d~=(-1)lr+r'÷l~
1i
If-I i
-lf
"
(7) "
(3)
In Eq. (2) and Eq. (3), [j] = 2 j + 1, the large parentheses denote the usual 3-j angular-momentum-coupling symbols, and the large curly brackets represent a 6-j symbol. For the sake of simplicity following Green, only M i = Mf = 0 has been retained in Eq. (2), since low-lying bending modes are still nearly linear (cf. Eq. (24) of ref. [8]). However within the lOS approximation the detailed balance principle is not respected. So, for the present work, as in the case of stretching [17] and bending [19,20] bands of CO 2, we used: i) the power fitting law to characterize the energy gap dependence for the fundamental rates Q~, which introduces two parameters A(T) and a:
A(T) Qt(L)=-Q'(L~O,T)
ticity, are expressed in terms of an adjustable impact parameter/c, of the mean relative velocity and of the energy gap between rotational level j and the next lower level. Note that in this scheme, the ECS method, Eq. (2) is used only for downward rates with upward rates being obtained from detailed balance. Uising these two modifications in Eq. (2), we have calculated the off-diagonal elements Wrk. If neglecting vibrational effects, the diagonal elements have to be deduced from the sum rule:
l:# k
L
813
[ t ( L + 1)],~
(4)
with the temperature dependence written as [24]: N
Within the framework of the impact theory, at the wavenumber ~, the absorption coefficient a(o-) is given by [26-29]: 8~3oa ( o - ) = na---~-c (1 - exp(
-hco-/kbT))lR~l 2
1
×--Im~d/(ll "IT
( ~ r - tr0 - i W ) - l l k )
k,l
×d* Pk,
(8)
where n a is the CO 2 density, [Rv[ 2 is the square of the vibrational transition dipole moment, Pk the population of the initial level of transition k - f ~ i, and o-0 line wavenumbers. The elements of the ( o - - o-o - iW )-~ matrices were calculated from the eigenvalues A 1 and eigenvectors X of (or0 + iW) as outlined in refs. [28,29], so: 8v3oOt ( O" ) = t / a - ~ c
(1 - -
exp( -
hctr/kbT ))
1
ii) the adiabaticity factors as suggested by De Pristo et al. [25] which introduce a third parameter 1c. These factors, which account for rotational inelas-
1
× - - }--"Re Gk* (tr 'rr k - Re Ak) 2 + (Im Ak) 2
×[Imak+Yk(o'-ReAk)],
(9)
814
B. Khalil et a l . / Chemical Physics Letters 263 (1996) 811-816
with
G,, = Y'. ( X - ' ) ,tptRtRyXi * , jl
(10)
where Xjk is a matrix w h o s e c o l u m n s are the norm a l i z e d e i g e n v e c t o r s X o f (o- 0 + i W ) and R i is the r o o t - s q u a r e w i t h its s i g n o f the t r a n s i t i o n probability-squared as g i v e n by the Hitran data base [30]. Eq. (9) takes account o f line m i x i n g effects. W h a t e v e r the a m o u n t o f line overlap, the absorption profile can be expressed as a sum o f e f f e c t i v e lines, labelled k, characterized by intensities R e G~k, w a v e n u m b e r s Re A k, and widths Im A k. E a c h lorentzian contribution:
i
i
6986
6988
2
1
//
1.5
Im A k
:
l/
(11) (tr-
cm "1
(b)
Re Ak) 2 + ( I m A t ) 2 1.0
enters in partnership with a dispersive contribution that leads to an a s y m m e t r i c line shape profile and redistributes its strength inside the band without m o d i f y i n g its integrated intensity. T h e interference
,
105
f ~
~
95
i
i
r
O,O 6968
J
6;70
6972
iiii iiL
6974j /
6990
6992 cm-1 6994
Fig. 2. Comparison between experimental ( - - ) , Lorentzian (. - •) and ECS (O) absorption coefficients in the 31) 3 band region. (a) P[CO~]= 46 Ton., P[He] = 2.5 atm., L = 56 m; (b) P[CO2] = 159 Ton-, P[He] = 5 atm., L = 96 m.
,
90 [ \ l l ,
•
og.
p a r a m e t e r Yk associated with an e f f e c t i v e line k is defined by
.......
1v v" •• "
v v• " v v" " v v•
Im Gkk Y, = - Re Gkk
7:':- ° I --
(12)
75
70
I
I
f
I
10
20
30
40
Iml
50
Fig. 1. Measured and calculated half width at half maximum (in 10 -3 era-i atm-i at 193 K) versus Jml: m = J +1 in the R branch and m= - J in the P branch - see text. • = v2, measured values in the R and P branches; • = 3v3, measured values in the P branch; - 3v3, ECS values for the P branch; . . . . v2, ECS values for the P branch. The vertical bars indicate an absolute error of 5%.
T o bring to the fore line m i x i n g effects we should also c o m p a r e the m e a s u r e d spectra with those calculated as a sum o f Lorentzian contours: = na V ' ~r ( 1 -
exp(-hco'/kT))
O/Lot( O" ) nbT O
×S°(o.
o. )2 +(nbTO)2 ,
(13)
B. Khalil et at. / Chemical Physics Letters 263 (1996) 8 1 1 - 8 1 6 I
i
i
o
t
667
i
i
Ca)
668
669
cm.1
670
(b)
640
650
660
670
680
cm -1
690
Fig. 3. C o m p a r i s o n between experimental ( - - ) , Lorentzian (. - • ) and ECS (C)) absorption coefficients in the v 2 band region. (a) P [ C O 2 ] = 3 Tort, P[He] = 0.5 atm., L = 7.7 mm; (b) P[CO2] = 7.5 Ton-, P[He] = 9.4 aim., L = 7.7 ram.
where the normalized (per atmosphere) half widths at half maximum (HWHM) were computed from the empirical relation [31 ]: n
and other line parameters come from the Hitran database [30].
815
v 2 band they result from the slope of measured widths at 9 helium pressures while for the 3 v 3 band they result from only one spectrum with a CO 2 pressure of 47 Tort and a He one of 2.5 arm (at 193 K). One can also compare these broadening coefficients with the derived ones from the given values in Fig. 4 of ref. [18] by using Eq. (14) with an average value of n = 0.7. Due to inaccuracies no attempt has been made to explore the J dependence of n. These are the data which were used in the next siep. Since the basis Q~ rates are not presently available, such an analysis requires their model!ing by an analytical form. Our approach uses the simple polynomial or power law (P) resulting [17,19] in A ( T o = 0 2 296 K) = 23.4 A , a = 1.083 and Ic = 0.66 ,~ where following Ref. [32] the two last parameters are assumed to be temperature independent. The temperature dependent parameter A ( T = 193 K) has been deduced from a mean least squares fit of the broadening coefficients given by Eq. (6), the sum rule. The resulting value leads to N = 0.J40 +_ 0.02. The ECS-P broadening coefficients are in rather good agreement with the experimental ohes taking into account a mean absolute error of 5% for the experimental HWHM (see Fig. 1). Applied to the computation procedures (ECS-P and Lorentzian) these parameters give results similar to those obtain at ambient temperature [17-21]. Fig. 2a and b, in case of the 3v 3 band, sho~ that the absorption is superlorentzian just before the band head and sublorentzian in the high frequeficy wing as well as in the central part of the band. "]~e ECS-P model success fully predicts the band profile contrary to the Lorentzian method. Fig. 3a and b prove again, in case of the v 2 band for two different helium pressures, the ability of the ECS-P formalism to predict the collapsed Q-branch despite the uncertainties in the input parameters. It should be noticed that the width of the Q-branch broadens when the helium pressure rises [15] as in the case of anisotropic Raman spectra contrary to the isotropic case [for a recent discussion of these differences see Ref. [33]].
4. Data used and results
In a first step we measured the linewidths at the chosen temperature ( T = 193 K). The normalized broadening coefficients are given in Fig. 1. For the
5. Conclusion
Using the same modelling of the absorption as in the study of the stretching band and berlding bands
816
B. Khalil et al. / Chemical Physics Letters 263 (1996) 811-816
of C O z in He at room temperature, the present work has demonstrated the validity of the ECS formalism in accounting for the temperature dependence profile, at least in the restricted range of temperature investigated. Additional measurements would be of interest in order to extend this domain to investigate high temperatures. References [1] M.O. Bulanin, A.B. Dokuchaev, M.V. Tonkov and N.N. Filippov, J. Quant. Spectrosc. Radiat. Transfer 31 (1984) 521. [2] M.O. Bulanin, V.P. Bulychev and E.B. Khodos, Opt. Spectrosc. 48 (1980) 403. [3] A.M. Robinson and J.S. Weiss, Can. J. Phys. 60 (1982) 1656. [4] A.M. Robinson, Appl. Opt. 22 (1983) 718. [5] R.K. Brimacombe and J. Reid, IEEE J. Quantum Electron. QE-19 (1983) 1668. [6] E.W Smith, M. Giraud and J. Cooper, J. Chem. Phys. 65 (1976) 1256. [7] R.T. Pack, J. Chem. Phys. 70 (1979) 3424, and references therein for previous results. [8] S. Green, J. Chem. Phys. 90 (1989) 3603. [9] For a review see for instance: A. L~vy, N. Lacome and C.C. Chackerian Jr., Collisional Line Mixing, in: Spectroscopy of the Earth's atmosphere and Interstellar medium, ed. K.N. Rao and A. Weber (Academic Press, 1992); S. Green, Calculation of pressure broadened spectral line shapes including collisional transfer of intensity, in: Status and future developments in Transport Properties, ed. W.A. Wakeham (Kluwer Academic Publishers, 1992). [10] A.B. Dokuchaev, A. Yu. Pavlov and M.V. Tonkov, Opt. Spectrosc. 58 (1985) 769. [11] I.M. Grigorev, V.M. Tarabukhin and M.V. Tonkov, Opt. Spectrosc. 58 (1985) 147. [12] V.M. Tarabukhin and M.V. Tonkov, Opt. Spectrosc. 62 (1987) 199. [13] N.N. Filippov and M.V. Tonkov, J. Quant. Spectrosc. Radiat. Transfer 50 (1993) 111.
[14] F. Thibault, J. Boissoles, R. Le Doucen, J.-P. Bouanich, Ph. Areas and C. Boulet, J. Chem. Phys. 96 (1992) 4945. [15] M.V. Tonkov, J. Boissoles, R. Le Doucen, B. Khalil and F. Thibault, J. Quant. Spectrosc. Radiat. Transfer 55 (1996) 321. [16] F. Thibault, J. Boissoles, R. Le Doucen, V. Menoux and C. Boulet, J. Chem. Phys. 100 (1994) 210. [17] J. Boissoles, F. Thibault, R. Le Doucen, V. Menoux and C. Boulet, J. Chem. Phys. 100 (1994) 215. [18] J. Boissoles, F. Thibault, R. Le Doucen, V. Menoux and C. Boulet, J. Chem. Phys. 101 (1994) 6552. [19] J. Boissoles, F. Thibault and C. Boulet, J. Quant. Spectrosc. Radiat. Transfer, in press. [20] J. Boissoles, F. Thibault, F. Rachet, A. Valentin and C. Boulet, J. Quant. Spectrosc. Radiat. Transfer, in press. [21] L. Ozanne, Nguyen-Van-Thanh, C. Brodbeck, J.P. Bouanich, J.M. Hartmann and C. Boulet, J. Chem. Phys. 102 (1995) 7306. [22] C. Boulet and J. Boissoles, in: Spectral line shapes, Vol. 8, 12th ICLS, eds. A.D. May, J.R. Drummond and E. Oks (AlP Press, New York, 1995) p. 265, and references therein. [23] S. Green, J. Chem. Phys. 70 (1979) 816; S. Green, J. Chem. Phys. 70 (1979) 4686. [24] L. Bonamy, J. Bonamy, D. Robert, B. Lavorel, R. Saint-Loup, R. Chaux, J. Santos and H. Berger, J. Chem. Phys. 89 (1988) 5568. [25] A.E. De Pristo, S.D. Augustin, R. Ramaswamy and H. Rabitz, J. Chem. Phys. 71 (1979) 850. [26] A. Ben Reuven, Phys. Rev. 145 (1966) 7. [27] R.G. Gordon, J. Chem. Phys. 45 (1966) 1649. [28] R.G. Gordon and R.P. Mc Ginnis, J. Chem. Phys. 49 (1968) 2455. [29] E.W. Smith, J. Chem. Phys. 74 (1981) 6658. [30] L.S. Rothman, R.R. Gamache, R.H. Tipping, C.P. Rinsland, M.A.H. Smith, D.C. Benner, V. Malathy Devi, J.M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S.T. Massie, L.R. Brown and R.A. Toth, J. Quant. Spectrosc. Radiat. Transfer 48 (1992) 469. [31] P. Varanasi and S. Sarangi, J. Quant. Spectrosc. Radiat. Transfer 15 (1975) 473. [32] G. Millot, J. Chem. Phys. 93 (1990) 8001. [33] L. Bonamy, J. Bonamy, D. Robert, S.I. Temkin, G. Millot and B. Lavorel, J. Chem. Phys. 101 (1994) 7350.
27 December 1996
ELSEVIER
Chemical Physics Letters 263 (1996) 811-816
i
Line mixing and line broadening in CO 2 bands perturbed by helium at 193 K B. Khalil
a
O. Cisse b G. Moreau
a
F. Thibault
a
R. Le Doucen
a
j. Boissoles
a
a Unit[" Mixte de Recherche P.A.L.M.S. (Physique des Atomes, Lasers, Molecules et Surfaces), Universit~ de Rennes 1. campus de Beaulieu. 35042 Rennes cedex, France b Universit~ de Ouagadougou, Ouagadougou, Burkimt Faso
Received 11 July 1996; in final form 22 October 1996
Abstract
The 1'2 and 3 v 3 bands of CO 2 in helium baths at 193 K have been studied with a Fourier transform interfe~ometer. The behavior of the band shapes has been explored at moderate densities. The energy corrected sudden (ECS) approximation is used to model the relaxation matrix in order to account for line mixing effects. The basis cross-sections were ca!culated with the simple power law (P). Computed spectra are in good agreement with the observed ones. Measuredl broadening coefficients are also comparable with the ones derived from the ECS-P model.
1. I n t r o d u c t i o n
Early experiments and calculations on spectroscopic lines shapes were mainly concerned with Lorentzian parameters, and particularly with the broadening coefficients because of their practical and theoretical importance. Infrared experimental data on C O 2 - H e linewidths come from studies undertaken at room temperature (see [1] which provides a smooth fit of various results) or at elevated temperatures [2-5]. Smith, Giraud and Cooper [6], have been rather successful in applying their semi-classical theory for line broadening to this system. A first quantum mechanical calculation was performed within the infinite order sudden (IOS) approximation by Pack [7], who investigated the pressure broadening of the dipole and Raman lines of CO 2 by He (and Ar) in a wide range of temperature. From these
studies it appears that the C O 2 - H e linewidths (at constant pressure) decrease more rapidly, with the temperature, than T-1/2 ( _ 1 / 2 is the iimit of the hard sphere collisions model or the classical value expected), this temperature dependence Varying between T -°'58 [5] to T -°88 [2]. More recently He pressure broadening cross-sections in bending bands of CO 2 at a n equivalent temperature of 300 K were calculated within the IOS approximation by Green [8]. In addition, he improved this model to properly take into account line mixing effects regardless of the band type for symmetric linear triatomics such as CO 2. Concurrently with the determination of broadening coefficients, line couplings have r~ceived increasing interest as the theory developed and as numerical calculations became more feasible. It is well known [9] that there are spectral regions, with
0009-2614/96/$12.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PII S0009-2614(96)01276-6
812
B. Khalil et al. / Chemical Physics Letters 263 (1996) 811-816
very dense rotational line structure or with lines spaced in such a way that collisional broadening gives rise to significant overlap, where band shape cannot be obtained as a superposition of Lorentzian profiles. Indeed, line coupling effects due to rotational energy transfer may occur. Due to its role in atmospheric transmission CO 2 has been extensively studied [9], but only a few studies have been devoted to the CO2-He system which is of theoretical importance. The contribution to line broadening and line mixing in CO 2 bands in He of the St Petersburg group is notable [1,2,10-13]. In the past few years our group had investigated [14-20] various CO 2 bands in He, both theoretically and experimentally. While the theoretical methods used by the St Petersburg group are based on the strong collision model [1,10-12] or on the classical impact model [13], we use [17-20] the Energy Corrected Sudden (ECS) approximation to take line coupling effects into account. This work extends our previous study [16-18] of line interferences between the rotational components of the 00°3-00°0 band of CO 2 pressurized by helium. For this stretching band ( E - E transition) we have already shown that the theoretical framework of the Energy Corrected Sudden (ECS) approximation is rather successful in predicting the evolution of the spectral profile over an extended range of perturber pressure. This is particularly true at high pressures where the rotational structure has disappeared, line mixing effects leading to an important enhancement of the R branch due to its narrowing. One can also refer, to reference [21] which is an extension of this study at higher helium densities for the u 3 and 3 u 3 bands of CO 2. In recent studies [19,20], the same E.C.S. formalism was successfully applied to various CO 2 Qbranch shapes of bending modes near 15 and 4.7 txm. These previous studies were undertaken at room temperature. In the present Letter we investigate the 3v 3 and u 2 bands of CO 2 in helium at a temperature of 193 K. The same ECS-P model is tested. In addition, measured linewidths are compared with the ECS-P predicted ones. Section 2, 3 are devoted to a brief presentation of experimental details and to a rdsum~ of the theoretical framework. The data used and the results are presented in Section 4.
2. Experimental procedure The experimental apparatus around the FTS and the data reduction procedures have been described in refs. [14-16]. For the 3u 3 band the White-type coolable cell with a baseline of 2 m was used. Its temperature is monitored within +__0.2 K through a circulation of methylcyclohexane cooled by liquid nitrogen. The resolution was 40 × 10 -3 cm - I and interferograms were averaged over 64 scans. A single path cell with 7.68 mm length has been built to study the u 2 band. This cell is made of a double-walled stainless-steel tube which allows the circulation of the cryogenic fluid and is surrounded by several dozens of superinsulation layers. The input windows are made of ZnSe and the output ones of KBr disks: the external ones are at room temperature and the internal ones are at the sample temperature. The temperature is controlled by platinum resistive thermal probes. The resolution varied from 10 × 10 -3 to 60)< 10 -3 cm - l as the helium pressure increased from 0.4 to 10 atm.
3. Theory In the impact formulation of spectral line shape theory, the effect of collisions is described by a relaxation matrix W, the rows and columns of which are labelled by spectral lines [9]:
-~" Wlk = 2'1TC O'q( V i f V f j f ; v i J i v f j f ) - nbWi ° ,
(l)
where n b is the density of perturbers, v is the mean relative velocity for colliding molecules, q is the tensor order of the radiative interaction (e.g. q = 1 for IR spectra), k refers to the spectral line of jr v i j ~, and primes indicate post-collision values. By invoking binary completed collisions, the cross section matrix o- can be expressed in terms of the S-matrices of scattering theory. Green has recently
B. Khalil et al./ Chemical Physics Letters 263 (1996) 811-816
applied the Infinite Order Sudden (lOS) approximation to bending bands of CO 2 [8]. This approximation is expected to be fairly accurate for CO z [22], and it has the advantage of separating the dependence of cross sections on spectral level, tensor order and angular momentum coupling, from the dependence on dynamical factors. Within this framework the IOS formalism applied to symmetric rotors leads to a convenient factorisation [23] of the cross-sections into dynamical terms or basic rates Q'(L) (cf. Eq. (5) of ref. [! 8]) and spectroscopic coefficients: O'"!i!"JiJ,%J,~-
__ (\I-('--~ [ Jt Jli]. I)
`/2
11 rt/r t (L), E[L]Fjd;j,~,Q
(2)
where the spectroscopic coefficients are defined by: lrir,~,L = ( - - l ) Fjjd~j,
×
li
I + r'+"([j~l[Jil[Jfl[ff])'/2
x{Ji Ji
0
-- I i
Jr
1}
Jl
L
- - If
0
If
dr -5-Wtk.
Wkk = -- •
(6)
uk
where d~ (respectively d r) are the dipole reduced matrix elements of transition k -= f ~ i (respectively l - f' *-- i'):
d~=(-1)lr+r'÷l~
1i
If-I i
-lf
"
(7) "
(3)
In Eq. (2) and Eq. (3), [j] = 2 j + 1, the large parentheses denote the usual 3-j angular-momentum-coupling symbols, and the large curly brackets represent a 6-j symbol. For the sake of simplicity following Green, only M i = Mf = 0 has been retained in Eq. (2), since low-lying bending modes are still nearly linear (cf. Eq. (24) of ref. [8]). However within the lOS approximation the detailed balance principle is not respected. So, for the present work, as in the case of stretching [17] and bending [19,20] bands of CO 2, we used: i) the power fitting law to characterize the energy gap dependence for the fundamental rates Q~, which introduces two parameters A(T) and a:
A(T) Qt(L)=-Q'(L~O,T)
ticity, are expressed in terms of an adjustable impact parameter/c, of the mean relative velocity and of the energy gap between rotational level j and the next lower level. Note that in this scheme, the ECS method, Eq. (2) is used only for downward rates with upward rates being obtained from detailed balance. Uising these two modifications in Eq. (2), we have calculated the off-diagonal elements Wrk. If neglecting vibrational effects, the diagonal elements have to be deduced from the sum rule:
l:# k
L
813
[ t ( L + 1)],~
(4)
with the temperature dependence written as [24]: N
Within the framework of the impact theory, at the wavenumber ~, the absorption coefficient a(o-) is given by [26-29]: 8~3oa ( o - ) = na---~-c (1 - exp(
-hco-/kbT))lR~l 2
1
×--Im~d/(ll "IT
( ~ r - tr0 - i W ) - l l k )
k,l
×d* Pk,
(8)
where n a is the CO 2 density, [Rv[ 2 is the square of the vibrational transition dipole moment, Pk the population of the initial level of transition k - f ~ i, and o-0 line wavenumbers. The elements of the ( o - - o-o - iW )-~ matrices were calculated from the eigenvalues A 1 and eigenvectors X of (or0 + iW) as outlined in refs. [28,29], so: 8v3oOt ( O" ) = t / a - ~ c
(1 - -
exp( -
hctr/kbT ))
1
ii) the adiabaticity factors as suggested by De Pristo et al. [25] which introduce a third parameter 1c. These factors, which account for rotational inelas-
1
× - - }--"Re Gk* (tr 'rr k - Re Ak) 2 + (Im Ak) 2
×[Imak+Yk(o'-ReAk)],
(9)
814
B. Khalil et a l . / Chemical Physics Letters 263 (1996) 811-816
with
G,, = Y'. ( X - ' ) ,tptRtRyXi * , jl
(10)
where Xjk is a matrix w h o s e c o l u m n s are the norm a l i z e d e i g e n v e c t o r s X o f (o- 0 + i W ) and R i is the r o o t - s q u a r e w i t h its s i g n o f the t r a n s i t i o n probability-squared as g i v e n by the Hitran data base [30]. Eq. (9) takes account o f line m i x i n g effects. W h a t e v e r the a m o u n t o f line overlap, the absorption profile can be expressed as a sum o f e f f e c t i v e lines, labelled k, characterized by intensities R e G~k, w a v e n u m b e r s Re A k, and widths Im A k. E a c h lorentzian contribution:
i
i
6986
6988
2
1
//
1.5
Im A k
:
l/
(11) (tr-
cm "1
(b)
Re Ak) 2 + ( I m A t ) 2 1.0
enters in partnership with a dispersive contribution that leads to an a s y m m e t r i c line shape profile and redistributes its strength inside the band without m o d i f y i n g its integrated intensity. T h e interference
,
105
f ~
~
95
i
i
r
O,O 6968
J
6;70
6972
iiii iiL
6974j /
6990
6992 cm-1 6994
Fig. 2. Comparison between experimental ( - - ) , Lorentzian (. - •) and ECS (O) absorption coefficients in the 31) 3 band region. (a) P[CO~]= 46 Ton., P[He] = 2.5 atm., L = 56 m; (b) P[CO2] = 159 Ton-, P[He] = 5 atm., L = 96 m.
,
90 [ \ l l ,
•
og.
p a r a m e t e r Yk associated with an e f f e c t i v e line k is defined by
.......
1v v" •• "
v v• " v v" " v v•
Im Gkk Y, = - Re Gkk
7:':- ° I --
(12)
75
70
I
I
f
I
10
20
30
40
Iml
50
Fig. 1. Measured and calculated half width at half maximum (in 10 -3 era-i atm-i at 193 K) versus Jml: m = J +1 in the R branch and m= - J in the P branch - see text. • = v2, measured values in the R and P branches; • = 3v3, measured values in the P branch; - 3v3, ECS values for the P branch; . . . . v2, ECS values for the P branch. The vertical bars indicate an absolute error of 5%.
T o bring to the fore line m i x i n g effects we should also c o m p a r e the m e a s u r e d spectra with those calculated as a sum o f Lorentzian contours: = na V ' ~r ( 1 -
exp(-hco'/kT))
O/Lot( O" ) nbT O
×S°(o.
o. )2 +(nbTO)2 ,
(13)
B. Khalil et at. / Chemical Physics Letters 263 (1996) 8 1 1 - 8 1 6 I
i
i
o
t
667
i
i
Ca)
668
669
cm.1
670
(b)
640
650
660
670
680
cm -1
690
Fig. 3. C o m p a r i s o n between experimental ( - - ) , Lorentzian (. - • ) and ECS (C)) absorption coefficients in the v 2 band region. (a) P [ C O 2 ] = 3 Tort, P[He] = 0.5 atm., L = 7.7 mm; (b) P[CO2] = 7.5 Ton-, P[He] = 9.4 aim., L = 7.7 ram.
where the normalized (per atmosphere) half widths at half maximum (HWHM) were computed from the empirical relation [31 ]: n
and other line parameters come from the Hitran database [30].
815
v 2 band they result from the slope of measured widths at 9 helium pressures while for the 3 v 3 band they result from only one spectrum with a CO 2 pressure of 47 Tort and a He one of 2.5 arm (at 193 K). One can also compare these broadening coefficients with the derived ones from the given values in Fig. 4 of ref. [18] by using Eq. (14) with an average value of n = 0.7. Due to inaccuracies no attempt has been made to explore the J dependence of n. These are the data which were used in the next siep. Since the basis Q~ rates are not presently available, such an analysis requires their model!ing by an analytical form. Our approach uses the simple polynomial or power law (P) resulting [17,19] in A ( T o = 0 2 296 K) = 23.4 A , a = 1.083 and Ic = 0.66 ,~ where following Ref. [32] the two last parameters are assumed to be temperature independent. The temperature dependent parameter A ( T = 193 K) has been deduced from a mean least squares fit of the broadening coefficients given by Eq. (6), the sum rule. The resulting value leads to N = 0.J40 +_ 0.02. The ECS-P broadening coefficients are in rather good agreement with the experimental ohes taking into account a mean absolute error of 5% for the experimental HWHM (see Fig. 1). Applied to the computation procedures (ECS-P and Lorentzian) these parameters give results similar to those obtain at ambient temperature [17-21]. Fig. 2a and b, in case of the 3v 3 band, sho~ that the absorption is superlorentzian just before the band head and sublorentzian in the high frequeficy wing as well as in the central part of the band. "]~e ECS-P model success fully predicts the band profile contrary to the Lorentzian method. Fig. 3a and b prove again, in case of the v 2 band for two different helium pressures, the ability of the ECS-P formalism to predict the collapsed Q-branch despite the uncertainties in the input parameters. It should be noticed that the width of the Q-branch broadens when the helium pressure rises [15] as in the case of anisotropic Raman spectra contrary to the isotropic case [for a recent discussion of these differences see Ref. [33]].
4. Data used and results
In a first step we measured the linewidths at the chosen temperature ( T = 193 K). The normalized broadening coefficients are given in Fig. 1. For the
5. Conclusion
Using the same modelling of the absorption as in the study of the stretching band and berlding bands
816
B. Khalil et al. / Chemical Physics Letters 263 (1996) 811-816
of C O z in He at room temperature, the present work has demonstrated the validity of the ECS formalism in accounting for the temperature dependence profile, at least in the restricted range of temperature investigated. Additional measurements would be of interest in order to extend this domain to investigate high temperatures. References [1] M.O. Bulanin, A.B. Dokuchaev, M.V. Tonkov and N.N. Filippov, J. Quant. Spectrosc. Radiat. Transfer 31 (1984) 521. [2] M.O. Bulanin, V.P. Bulychev and E.B. Khodos, Opt. Spectrosc. 48 (1980) 403. [3] A.M. Robinson and J.S. Weiss, Can. J. Phys. 60 (1982) 1656. [4] A.M. Robinson, Appl. Opt. 22 (1983) 718. [5] R.K. Brimacombe and J. Reid, IEEE J. Quantum Electron. QE-19 (1983) 1668. [6] E.W Smith, M. Giraud and J. Cooper, J. Chem. Phys. 65 (1976) 1256. [7] R.T. Pack, J. Chem. Phys. 70 (1979) 3424, and references therein for previous results. [8] S. Green, J. Chem. Phys. 90 (1989) 3603. [9] For a review see for instance: A. L~vy, N. Lacome and C.C. Chackerian Jr., Collisional Line Mixing, in: Spectroscopy of the Earth's atmosphere and Interstellar medium, ed. K.N. Rao and A. Weber (Academic Press, 1992); S. Green, Calculation of pressure broadened spectral line shapes including collisional transfer of intensity, in: Status and future developments in Transport Properties, ed. W.A. Wakeham (Kluwer Academic Publishers, 1992). [10] A.B. Dokuchaev, A. Yu. Pavlov and M.V. Tonkov, Opt. Spectrosc. 58 (1985) 769. [11] I.M. Grigorev, V.M. Tarabukhin and M.V. Tonkov, Opt. Spectrosc. 58 (1985) 147. [12] V.M. Tarabukhin and M.V. Tonkov, Opt. Spectrosc. 62 (1987) 199. [13] N.N. Filippov and M.V. Tonkov, J. Quant. Spectrosc. Radiat. Transfer 50 (1993) 111.
[14] F. Thibault, J. Boissoles, R. Le Doucen, J.-P. Bouanich, Ph. Areas and C. Boulet, J. Chem. Phys. 96 (1992) 4945. [15] M.V. Tonkov, J. Boissoles, R. Le Doucen, B. Khalil and F. Thibault, J. Quant. Spectrosc. Radiat. Transfer 55 (1996) 321. [16] F. Thibault, J. Boissoles, R. Le Doucen, V. Menoux and C. Boulet, J. Chem. Phys. 100 (1994) 210. [17] J. Boissoles, F. Thibault, R. Le Doucen, V. Menoux and C. Boulet, J. Chem. Phys. 100 (1994) 215. [18] J. Boissoles, F. Thibault, R. Le Doucen, V. Menoux and C. Boulet, J. Chem. Phys. 101 (1994) 6552. [19] J. Boissoles, F. Thibault and C. Boulet, J. Quant. Spectrosc. Radiat. Transfer, in press. [20] J. Boissoles, F. Thibault, F. Rachet, A. Valentin and C. Boulet, J. Quant. Spectrosc. Radiat. Transfer, in press. [21] L. Ozanne, Nguyen-Van-Thanh, C. Brodbeck, J.P. Bouanich, J.M. Hartmann and C. Boulet, J. Chem. Phys. 102 (1995) 7306. [22] C. Boulet and J. Boissoles, in: Spectral line shapes, Vol. 8, 12th ICLS, eds. A.D. May, J.R. Drummond and E. Oks (AlP Press, New York, 1995) p. 265, and references therein. [23] S. Green, J. Chem. Phys. 70 (1979) 816; S. Green, J. Chem. Phys. 70 (1979) 4686. [24] L. Bonamy, J. Bonamy, D. Robert, B. Lavorel, R. Saint-Loup, R. Chaux, J. Santos and H. Berger, J. Chem. Phys. 89 (1988) 5568. [25] A.E. De Pristo, S.D. Augustin, R. Ramaswamy and H. Rabitz, J. Chem. Phys. 71 (1979) 850. [26] A. Ben Reuven, Phys. Rev. 145 (1966) 7. [27] R.G. Gordon, J. Chem. Phys. 45 (1966) 1649. [28] R.G. Gordon and R.P. Mc Ginnis, J. Chem. Phys. 49 (1968) 2455. [29] E.W. Smith, J. Chem. Phys. 74 (1981) 6658. [30] L.S. Rothman, R.R. Gamache, R.H. Tipping, C.P. Rinsland, M.A.H. Smith, D.C. Benner, V. Malathy Devi, J.M. Flaud, C. Camy-Peyret, A. Perrin, A. Goldman, S.T. Massie, L.R. Brown and R.A. Toth, J. Quant. Spectrosc. Radiat. Transfer 48 (1992) 469. [31] P. Varanasi and S. Sarangi, J. Quant. Spectrosc. Radiat. Transfer 15 (1975) 473. [32] G. Millot, J. Chem. Phys. 93 (1990) 8001. [33] L. Bonamy, J. Bonamy, D. Robert, S.I. Temkin, G. Millot and B. Lavorel, J. Chem. Phys. 101 (1994) 7350.