Line mixing effects in the 15 μm Q-branches of CO2 in helium: Theoretical analysis

J. Quatzt. Spectrosc. Radiat. Transfer Vol. 56. No. 6, pp. 835-853. 1996

Pergamon 800224073(96)OOO!W3

Copyright 8 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022~4073/96 $15.00 + 0.00

LINE MIXING EFFECTS IN THE 15 pm Q-BRANCHES IN HELIUM: THEORETICAL ANALYSIS

OF CO2

J. BOISSOLES,” F. THIBAULT,“1_ and C. BOULET’ “Dipartement de Physique Atomique et Moliculaire, URA 1203 du CNRS, Universite de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex and bLaboratoire de Physique Moleculaire et Applications, (Laboratoire associe aux Universites Paris&d et P. et M. Curie), UPR 136 du CNRS. Universite de Paris-Sud, Centre d’orsay, Bat. 350, 91405 Orsay Cedex, France (Received

7 February

1996)

Abstract-Starting from the formalism proposed by Green [J. Chem. Phys 90, 3603 (1989)] and based on the infinite order sudden approximation, we describe here an improved model, based on the energy corrected sudden approximation, which allows the calculation of all types of coupling cross-sections i.e. for stretching as well as bending bands of COz starting from a single set of basic rates. Predictions of that model with various experiments made on stretching bands in samples of CO2 in He had been shown to be rather good. The present paper shows that it also predicts the evolution of bending bands in these samples reasonably well, particularly that of Q branches, over a wide range of perturber density. Copyright 0 1996 Elsevier Science Ltd

1. INTRODUCTION

There has been much recent interest, both experimental and theoretical, in the effects of line mixing

on the shapes of the vibrational bands of COz. Interference effects have been shown to be very important in the wings of stretching bands and also in the shape of Q branches in bending bands, where the linespacing is small. The influence of line mixing on the shape of stretching bands of COZ in He has received considerable attention in our group. In the case of the 3v, band an extensive experimental study of both the wing and central region has been made for densities up to about 100 Amagat’ and all the departures from the lorentzian shape successfully explained by an impact line mixing approach in which the relaxation operator was modeled with the energy corrected sudden (ECS) approximation.‘,3 Then the analysis was extended to the v3 band and to densities up to about 1000 Amagat4 There are also a number of investigations in the literature that deal with the shape of CO? Q branches.’ Very recently Tonkov et al6 have reported measurements made in the 15 pm region, which are of particular interest. For instance it is well known that line mixing has to be included in the calculation of atmospheric transmittance in the 15 pm v2 Q branch.’ However, from a theoretical point of view, the main interest of their measurements with He was that they cover a wide variety of Q branches belonging to l&X and l&A bands, namely the Ol’&OO”Oband at 667.38 cm-‘, the (lO”O)r-Ol’Oand (lO”O)n-Ol’Obands at 720.804 and 618.028 cm-’ and (1 1’0)r(0220)1 and (1 1’0),r(0220)1 bands at 741.724 and 597.538 cm-‘. It is known since the work of Green’ that the coupling cross-sections strongly depend on the values of the vibrational angular momentum involved in the transitions. However the calculations of Green while properly accounting for the angular momentum coupling between radiation rotation and vibration were limited by the infinite order sudden (10s) approximation.’ The purpose of this article which is an extension of both our previous ECS analysis24 of stretching bands and the work of Green8 on

tTo whom all correspondence

should be addressed. 835

836

J. Boissoles et al

bending bands is to provide an ECS formalism valid for every type of vibrational band allowing the modelling of all the available experimental data starting from a single set of basic rates determined once for all at the beginning of the work from line broadening data. The paper is organized as follows. In Sec. 2 we present the basic approximations which allow an approximate ECS expression to be proposed for the calculation of the coupling cross-sections valid for all types of vibrational bands. Some results are then discussed. Section 3 is first devoted to a brief summary of the lineshape calculation followed by a comparison with the experimental data of Ref. 6 and a discussion of the theoretical results. 2. LINE

MIXING

CROSS-SECTIONS:

ECS FORMALISM

2.1. Basic approximations As stated in the introduction, we will follow the formalism presented earlier by Green’ and based on the infinite order sudden approximation. In fact, as the vibrational degrees of freedom are still treated as coupled channels, the resulting formalism should be called “vibrational close coupling-rotational I.O.S.” model. To simplify notation in the following a given transition k will be noted as: k E (UijiE,Urjfcf)where U,stands for the complete set of vibrational quantum numbers UliuzllziUji,j, is the rotational angular momentum quantum number, and ti is the parity index (when /li # 0) (a similar expression stands for the final level of the transition). In Green’s formalism, all the cross-sections coupling two lines k E UijiLiUfjfcfand I = UJ”E:UF~;E; can be expressed in terms of generalized I.O.S. basic rates: Q(L, Ui,SMa; urvrMb) where L is the order in the expansion of the intermolecular potential in spherical harmonics and the “allowed” values for the M are lzi + Lr and lh - 1111.In the calculations of Green the stretching vibrational modes were frozen, leading therefore to basic rates which only depend on the bending mode. However, starting from a simple but reasonable potential, Green has shown that: (i) The Q(L. v,, UiMaUfUrMb) with M, or Mb greater than zero are several orders of magnitude smaller than those with M, = Mb = 0 since the bending vibration only weakly breaks the cylindrical symmetry of CO*. (ii) The Q(L, Di,v,OvrufO)values are relatively insensitive to vibrational quantum numbers (at the 10% level), at least for not too high values of the u and 12. Therefore a very reasonable approximation, 81 is:

already mentioned by Green [cf. Eq. (24) of Ref.

Q(L, u,o,M,uru~M~)z

Q(L)G,,o&,o.

(1)

2.2. Cross-sections Equation (1) will be the key-approximation of the present work since all the coupling cross-sections, at least for transitions involving the lowest vibrational levels of CO,,, can be expressed-whatever the type of vibrational band-in terms of the set Q(L) rates we already used in our previous studies on stretching bands [v~ and 3~~1.~~ At that level of approximation, the cross-section coupling too allowed infrared rotational components labelled in the following by 1,j,ifjf E k and Iij:lrjf’ = I is given by [note that only even L are required, due to Eq. (l), for this system and 1, means in fact /li].

“’ C [L]F .!!,!, L

/iJfJi

Q’(L) Jf

(2)

where the spectroscopic coefficients are defined by F

,llilfL .

= I

.I

JIJfJn Jr

(_l)‘+‘,+‘f

Kid[jXjf1~ji7Y” x

and the Q’(L) have been defined in terms of the Q(L) in Eq. (5) of Ref. 3.

(3)

Line mixing

effects of CO? in He

837

In Eqs. (2) and (3), [Xl = 2X + 1, the large parentheses denote the usual 3-j angular-momentumcoupling symbols, and the large curly brackets represent a 6-j symbol. Equation (2) is valid for every type of coupling cross-section (intra-branch as well as interbranch coupling within a given vibrational band) and may be easily obtained from Eqs. (1) and (33) of Ref. 10. At this point it must be emphasized that Eq. (2) was also the starting point of an alternative model developed by the group in Besancon.“-‘3 The tetradic coupling cross-sections are first expressed, within the 10s approximation, in terms of dyadic generalized cross-sections” partly depending on various diagonal contributions which have to be deduced within the ECS formalism from a realistic description of the relaxation of the J rotational angular momentum and of higher order tensors [J2. . . ] associated to J.” It has been shown in Ref. 3 that the two ECS formalisms, namely that of Boissoles et al’ originated from the work of Green and De Pristo et alI4 and the recent one of Bonamy et al’* deduced from the work of Temkin et al” are rather equivalent at least for successfully predicting the evolution of the stretching bands of CO2 over an extended range of He densities. Therefore in the continuity of our previous studies, we have chosen to directly apply ECS corrections to Eq. (2). Indeed, as is now well known, IOS cross-sections do not satisfy the detailed balance principle which should not be ignored. De Pristo et alI4 have suggested an ECS scheme for enforcing detailed balance, which moreover includes corrections (of minor importance for C02-He) to 10s approximation by taking into account rotational inelasticity. In the ECS model Eq. (2) becomes: ~ . !l,lf

J,Jrj:$

=-

“*

(4)

In Eq. (4) RL is an adiabaticity factor defined in terms of a scaling length I, and the difference in angular frequency mL.L z between levels L and L - 2:

*Lc(l+*)I

(5)

where d is the mean relative velocity for colliding molecules. Note that Eq. (4) is used for “downward” cross section [j, > j,‘] only, with upward ones obtained from detailed balance: p&k-&)

= p,a’(l+k)

(6)

where pk is the population of the initial level of line k. Following our previous analysis of line mixing effects in CO? stretching bandsZA the Q’(L) basic rates have been expressed through a simple analytical law:

Q'(L) =

,L(LA+ l)l’.

The ECS parameters A, a, L had been determined from fits of both line broadening data in stretching bands and absorption in the near frequency wing of the 3v3 band. They were taken from column 1 of Table 2 of Ref. 3. It must be emphasized that they have been used in the present analysis devoted to bending bands without any modification. Finally, let us note that the coupling cross-sections will strongly depend on the vibrational angular momenta I, and lf [see Eqs. (4) and (3)] but of course not on the other vibrational quantum numbers. Therefore they are not affected by the large Fermi resonance existing in CO? since this interaction only couples states belonging to a given polyad, i.e. corresponding to a given value of the vibrational angular momentum &. 2.3. Discussion As already mentioned line coupling cross-sections (intrabranch coupling, i.e. P-P, R-R, Q-Q, as well as interbranch coupling P-R, P-Q, R-Q) have been calculated from Eq. (4) for various vibrational bands of CO*. As an example we will only present and discuss in this section the

838

J. Boissoles et al

cross-sections coupling two Q lines belonging to a Ii+Zf vibrational band. According to Eq. (4) these cross-section are given by (for the case j’ < j): o’[Q(j)+Q(j’)]

= (- l)h+‘f(2j’ + 1)“‘(2j + l)“*

Figure 1 gives the cross-sections coupling Q(16) to the other Q(j’) components as a function ofj’ for two types of bending bands: a X-II band (h = O+l,r = 1) and a A-II band (Izi= 2+12r = 1). Let us recall that due to 1 doubling, Q branches in A-II transition of CO2 have two subbranches. One of them consists of the Q(q lines with even J and the other corresponding to odd J values [cf. Fig. 113. For these bands, the oscillating form of the cross-section distribution is rather similar to that previously observed by Clary15 for the inelastic rotational cross sections within the vibrationally excited u2 = h = 1 state a’(l’, 16+17’) and may be related to the behavior of 3-j coefficients like

for largej.15 It seems therefore, from ECS calculation, that an odd Q(j) component (which means odd j) is only coupled to all the other odd Q lines (and of course a similar result for the even components). However the propensity toward the decoupling of the two Q subbranches is only approximate. Figure 2 displays similar cross-sections couplings now Q(2) to all the other Q(j’) components. As may be seen the propensity toward conservation of the parity ofj’ is not at all satisfied. This result will be of some importance when we will consider the evolution of the lineshape of the A-II Q branch with helium density (see Sec. 3.3). As expected from various previous studies,8,9.‘3cross-sections strongly depend on the type of vibrational band through the dependence of the cross-sections on the values of the vibrational angular momentum. As will be demonstrated in Sec. 3, the ability of the formalism derived from the Green’s model to correctly describe mixing effects in vibrational bands of different symmetries is mostly due to the fact that it takes into account that strong dependence of the cross-sections, of particular importance at low J values. The i.r. cross-section coupling Q(2) to Q(j’) are also compared in Fig. 2 with similar cross-sections for isotropic Raman diffusion, demonstrating also a strong dependence with the type of spectroscopy (rank of the tensor coupling matter to radiation).‘.‘*

0

10

20

Ji’

30

Fig. 1. Cross sections (in AZ) in coupling Q(16) to the other Q(j’) components.

839

Line mixing effects of CO? in He

12 -

10 -

0

--a--

\ \

\ \ \ \

5

isotropic Raman diision

-

z+n

---

A+lJ

10

15

J.’

I

20

Fig. 2. Comparison between i.r. and isotropic Raman cross-sections (in A*) in coupling Q(2) to the other Q( j’) components.

2.4. ECS calculation of the diagonal cross-sections

In the present ECS formalism, Eq. (4) is only valid for the calculation of coupling cross-sections (i.e. non-diagonal in the linespace). Diagonal elements, i.e. pressure broadening cross-sections, have to be deduced from the sum rule”,16 (exact for a rigid rotor): a’(k+k)

= -

c

2 a’(k-4)

l#k k

(9)

where dk is the dipole reduced matrix element of line k:

Since vibrational effects are negligible for COTHe (insignificant lineshifts), Eq. (9) is believed to be rather accurate.4 As an example for k E Q(j) Eq. (9) gives:

4oc,dK?W-+Q(i)1 = - 1

00"+A

-

&wo'[Q(j)-+Q(i')l

c drRci,la’[Q(i)~R(j’)l NJ”)

(11) Figure 3 presents a comparison between calculated values of diagonal pressure elements for P, Q and R lines and some available experimental data for different vibrational bands (C-Z, Z-lI, fI--X and II-A). As may be seen, Eq. (9) leads to diagonal elements, which only slightly depend on the vibration, although off diagonal elements strongly depend on the vibrational angular momentum, leading to a reasonable agreement with experiment.

840

J. Boissoles et al 3. LINESHAPE

CALCULATION

3.1. Summary of the theory

Within the framework of the impact theory, and neglecting any coupling between different vibrational bands (which are much smaller than intraband coupling) absorption at wavenumber cr, is given by”,‘*

a(a)=zn.(l-exp(-hccr/k,T))

c

lRV1’xImCd,dkpk

vibrational band

(12)

k.l

where n, is the CO2 density, and o. the diagonal matrix of line wavenumbers for a given vibrational band. The squares of the vibrational transition dipole moments were taken from Table XXI of Ref. 19, and all other individual line spectroscopic information from the Hitran molecular data base.” In the linespace, matrix elements of the relaxation operator are related to the cross-sections by:

(13)

0

10

20

30

40

m

50

Fig. 3. Comparison between ECS linewidths [cf. Eq. (9)] and experimental data for various vibrational bands (in IO-‘cm-’ atm-’ at 296 K). Experimental values are represented by symbols while calculated ones are connected by lines. (a) (0) Z-E transition {Ref. 3); (m) P lines of the Il-I: transition centered at 618 cm-‘; (A) R lines of the fJ+E transition centered at 720 cm-’ (&) Ref. 22. (-) Z-C transition; (--) R branch of the II-E transition centered at 720cm-I; (---) P branch for the II-E transition centered at 618 cm-‘. (b) (0) Q lines of the vt band centered at 667 cm-’ (Ref. 6); (-) Q lines of the v~band @*II); (---) Q lines of the A-+fJ transition centered at 597 cm-‘. (The vertical bars indicate an absolute error of 5%).

Line mixing effects of CO2 in He

841

where n,, is the perturber density. The elements of the (a - o. - iw)-’ matrices were calculated from the eigenvalues ,4, and eigenvectors X of (Q,,+ iW) as outlined in Ref. 18, leading to (for a given band): a(a) = gn,(l

- exp(-hca/kbT))]R,lZ

with

Gklr=

c (X-‘)~,~d,d,&

(15)

where .X’,,is a matrix whose columns are the normalized eigenvectors X of (a0 + iw). As may be seen from Eq. (14), whatever the amount of line overlap, the absorption profile can be expressed as a sum of “effective lines”, each of them having a lorentzian and a dispersive component. However, as will be seen in the following the “intensity” of effective line k: Re Gklr- $” will strongly depend (not necessarily linearly) on the perturber density as well as the halfwidth Im Ak, line center Re A,, and interference or more exactly asymmetry parameter Kff = Im Glk/Re Glrk. 3.2. Results and discussion for Z-ll

bands

In this section, predictions of ECS formalism are compared with various experiments made on the v: band of CO1 on a rather wide range of He density.6 We first consider spectra of the Q-branch recorded under moderate He densities, which means high enough to suppress the rotational structure, but low enough to let the Q-branch be relatively well isolated from the P and R branches. From a theoretical point of view the criterion of “isolated Q-branch” may be defined as follows: no significant difference in the calculation of the parameter of the effective lines between the “exact” calculation-i.e. including interbranch coupling-and a calculation without them. However, this criterion is not absolute: it varies not only with the perturber density but also with the spectral region investigated. Therefore in the following we will mostly consider “exact” calculations. Experimental and computed spectra are plotted in Fig. 4 for two characteristic values of moderate He pressure. For these densities only the rotational structure of the P and R branches is well resolved and for these regions the absorption can be well described by a sum of isolated lorentzian lines broadened by collisions. On the contrary, the rotational structure is totally washed out in the Q-branch where a sum of isolated lorentzian lines is particularly inaccurate. In both cases ECS predictions account quite satisfactorily for the experiments. In order to follow the evolution of the Q lineshapes with density it may be of some interest to look at the calculation of the parameters of the effective lines. Figure 5 gives the intensities 9”k , halfwidths Im Ak and products S;” . Y;” versus line centers Re A, as deduced from the calculation of the eigenvalues and eigenvectors of go + iW. They are also compared with the Hitran data base values (for intensities and wavenumbers) valid for isolated lorentzian lines. At 2 atm [cf. Fig. 5(a)] positions of lines are quite similar, but Q(J = 2) and Q(J = 4) have disappeared and their intensity transferred to the other lines, particularly to the first intense equivalent line which is located at about the frequency of Q(6). Note that this component which has a positive intensity has also a large positive asymmetry parameter which, according to Eq. (14) transfers intensity from the low wavenumber side of the Q-branch to the high one, explaining the important sublorentzian behavior observed in that wing of the v? Q-branch-around P(2)-as shown in Fig. 6. At IO atm [cf. Fig. 5(b)] equivalent lines are no longer identifiable to Q(j) components. All Q(j) lines with j = 2-12 have merged into a single component which contains the major part of the

842

J. Boissoles

et al

-

Exp.

-

-

Lor. E.C.S. talc.

P[CO,]=2.5

Torr

P[He]=2 Atm L=4.15 cm

cm-l

670

-

Exp. -

Lor.

O

E.C.S. celc

(W P[CO2]=1 Torr P[He]=9.85 Atm L=3.85 cm

0.0 650 Fig. 4. Absorption experiment, (- --)

660 coefficient in the additive Lorentzian

670

675

680

spectral region of 00’%-01’0 Q-branch lines, (000) present ECS calculation. Pk z 10 atm.

Cm-1

690

(T = 296 K): (---) (a) PH* = 2 atm; (b)

.

Line mixing effects of CO, in He

843

2 0.15 5 2 0.00 0.5 -

9

L

v, 0.0

I”“““’

o.511 ’

’

’

’

’

’

666

I

’

’

667

- . . I

’

I

I

I

*

I

669

008

.

*

4

I

cm-l

670

I

I

2 w” i= 5 z P

1

0 Y 5 0.5 9 0.0 2i-’ 2

l0

” uqu

u,u

668

”

u,”

:

“,

-

,989

,

Cm-l Fig. 5. Line parameters for the effective lines in the 009-01’0 Q-branch as function of line centers (Re ,4,) [cf. Eq. (14)]. The bar spectrum corresponds to the values deduced from diagonalization of uo + i IV; the dots correspond to the Hitran data base values (i.e. valid for isolated lorentzian lines). (a) PHI = 2 atm; (b) PW x 10 atm.

670

intensity of the Q branch. This component has a width smaller than that of an isolated line and an asymmetry parameter which now decreases with increasing density (approx. as l/I%). This phenomenon is strengthened at 50 atm (which is no longer a moderate density as defined above) as illustrated in Fig. 7. All the Q branch is nearly reduced by line mixing to a single lorentzian symmetric Iine [Ykff+O]. (More precisely two equivalent lines with weak negative intensities compensate for the fact that the equivalent intensity of the single line equates to 120% the whole Q-branch intensity.) As shown in Fig. 8 the ECS approach gives satisfactory agreement

844

J. Boissoles et al

0.15 h Ti 0 t) 0.10

0.05 0

664

E.C.S. talc.

cm-l

Fig. 6. Absorption coefficient in the negative wing of the OOWOl’O Q-branch.

experiment at this pressure. By using higher path length it has been possible to investigate a perturber density domain and a spectral region where interbranch coupling becomes of some importance. From Fig. 9 it appears that the ECS model is in reasonable agreement with experiment, particularly in the spectral regions between the P-Q and R-Q branches which are known as particularly sensitive to interbranch coupling. However the absorption is underestimated in the with

-1

’

I

666

I

I

66s

cm-l

Fig. 7. Same as Fig. 5 for PH*z 50 atm. For clarity only the lines parameters corresponding to the most intense effective lines are drawn.

670

845

Line mixing effects of CO? in He 0.3

I

I

I

I

h F

-

k s a

-

Exp. o

Lor. E.C.S. talc.

0.2

P[CO2]4.2 0.1

Torr

P[He]=49.6 Atm L=3.65 cm

J

640

660

680

cm-l

700

Fig. 8. Same as Fig. 4 for he = 50 atm.

around the maximum of the P and R branches, which may be due to some inaccuracy in the calculation of interbranch mixing cross-sections since it has been shown4 that at high perturber density spectra are particularly sensitive to even small uncertainties on these cross-sections. Note also in this spectrum the good agreement obtained for two other weaker Z-II transitions characterized by their “intense” Q branches located at 618 cm-’ [(lO”O)n-O1’O]and 720 cm-’ [( 10°O)r-O1IO], and which appears more clearly in Fig. 10. region

3.3. Results and discussion for II-A bands It was therefore interesting to undertake a similar analysis of line coupling mechanism collisionally induced by rotational energy transfers in the case of the Q-branches of the 597.3 and 741.7 cm-‘. The peculiarities in the rotational structure of odd and even sub-branches are more pronounced in the case of the 597.3 cm-’ band (cf. Fig. 11). Indeed the even sub-branch exhibits a band head at (J = 24) and the most intense lines of the J even sub-branch are located in a very narrow spectral range of about 0.01 cm-’ promoting a total collapse and very important mixing effects even at low perturber density. 3.3.1. Line mixing evolution versus pressure for the 597.3 cm-’ Q-branch. Experimental6 and computed spectra are plotted in Fig. 12 for two typical values of He pressure (2 and 10 atm). Once more, agreement between theory and experimental is reasonable. Note on Fig. 12(a) the presence in the spectral region of the Q-branch, of three P lines belonging to another vibrational band and which are practically not affected by any mixing effect since they are relatively well isolated at P b x 2 atm. On the contrary we observe for the Q-branch, in a regime of strong overlapping of its components, the characteristic enhancement of the profile due to mixing effects. As in the previous section the evolution of the lineshape with He density will be easily understood from the analysis of the parameters of the effective lines. At low pressure (i.e. 2 atm) the two sub-branches overlap weakly, leading to a very weak efficiency of the inter sub-branch coupling. From Fig. 13(a) we observe a total mixing of the even J components which merge into a single component, while at the same time, the odd components are slightly affected (with the exception of the low J values).

J. Boissoleset al

-

Exp. o

Lor. E.C.S. talc.

P[C02]=120 TOIT P[He]=39.3 Atm L=3.85 cm

640

700

720

cm-l

740

Fig. 9. Absorption coefficient in a wide spectral range including the Q-branch of the (ltYO)~r(Ol’O)~band at 618cm-‘, the (01’0~0000) band at 667 cm-‘, the (looO)~~Ol’O)band at 721 cm-’ and the (01’0)-(00’%) band of the W602 centered at 648 cm-‘. PW z 40 atm.

The inefficiency of the inter sub-branch mixing at low pressure can be corroborated as follows: in the calculation of the eigenvalues and eigenvectors of (~0 + iw), we first neglect the blocks coupling the Q branch to the P and R ones (interbranch couplings are negligible for that pressure). Then we suppress the inter sub-branches cross-sections, keeping only two diagonal blocks corresponding to intra-Q-sub-branch mixing. As it appears from Fig. 13(a) the parameters of the resulting effective components are not very different from those previously obtained in the “exact” calculation. At high pressures [lo atm: Fig. 13(b); 50 atm: Fig. 13(c)] calculation shows that this uncoupled sub-branches model becomes wrong. It leads to two principal effective lines (one for each sub-branch) having almost the same intensity whereas the “exact” calculation (including the sub-branch coupling terms) leads to a very different intensity distribution which proves that inter sub-branch couplings are now efficient. As outlined in Sec. 2.3 in the case of the A-II band, decoupling between odd and even sub-bands is obvious for high J, but, it has been emphasized that this is no longer the case at low J. Although sub-band coupling only comes from these low J values, it appears to be of major importance as perturber density increases. At 50 atm [cf. Fig. 13(c)] the two sub-branches even collapse into a single quasi-symmetrical lorentzian effective line and not into two lines as expected by neglecting the off diagonal blocks coupling the two sub-branches. Since this vibrational band is very weak as compared to the v2band the good agreement obtained at 10 atm for the spectral region reproduced in Fig. 12(b) comes not only from an accurate calculation of line mixing effects within the A-II band but also from a reasonable ECS description of the negative wings of the more intense bands located at higher wavenumbers (of course, particularly the wing of the v2 band and the band located at 618 cm-‘). The importance of inter sub-branch coupling may also be analysed from the study of the total width at half maximum SV,,~of a given Q branch (or sub-branch) in a regime of complete mixing (single lorentzian symmetric line containing the whole intensity). At low pressure the even

847

Line mixing effects of CO1 in He

I

I

I

1

I

v

-

Exp.

-

- Lor. o

>

E.C.S. celc.

(a)

0.2

-

f

P[C02]=120 Torr P[He]=4.8 Atm L=3.85 cm

0.a 600

610

620

cm-l

630

640

I

0.4

-

h T

Exp. a

Lor. E.C.S MC.

k z% 8

0.2 P[COz]=120 Torr P[He]=4.8 Atm L=3.85 cm

0.0 t 700

730 720 740 710 cm-l Fig. 10. Two other examples of lI +Z Q-branches absorption coefficient: (a) for the band located at 618 cm-’ and for PHc= 4.8 atm; (b) for the band located at 720 cm-’ and for PHI = 4.8 atm.

J. Boissoles et al

848

(4

dc

-I cm-l

even J (b)

\

,dc J

I -

ti 596

.

-

-r

-

-i cm-l

Fig. Il. Bar spectra showing the relative intensities (as given by the Hitran database) of Q lines in the two following II-8 transitions: (a) (1110)~-(0220)~ band; (b) (11’0)1~-(02~0)~ band.

Line mixing effects of CO: in He I

I-

I

849 I

’

P (a)

-

Exp.

P[C02]=80 Torr -

-

Lor.

>

PfHe]=1.87 Atm

0

E.C.S. cak.

L=26 cm

cm-l

599

ON P[CO,]=360

Torr

P[He]=9.4 Atm L=22.8 cm

-

Exp.

0

600

605

E.C.S. cak.

610

cm-l

615

Fig. 12. Absorption coefficient in the region of the (I 1’0)~~-(02~0), Q-branch: (a) f& z 2 atm [the three P lines belong to the (lO%)~~-(ol’O)band and are practically not affected by line mixing at that perturber density]; (b) PHI= 9.4 atm.

850

J. Boissoles et al

J sub-branch is relatively well isolated and the slope of its width 6vliz versus Pb may be related to ?

k.l

E

c

I

Pkdkd,fi 1 pk4 Z

even Q

1.6 ,

23.10-’ cm-’ atm-’

(16)

k E CL,,

I

I

I

I

(4

"f 1.2 z v) F 0.8 (0 5 I- 0.4 z 0.0

f

4

-

1

^

596.9

0.0 -

597.0

I

I

Ij

1

I

___t-

~

j

_ j..r_,,&Jl

0

I

I

I

597.1

597.2

597.3

I_-____ ..--4L_L_

Fig. 13(a, b)

851

Line mixing effects of COZ in He

I

0 I

,

3-

I

I

j 0 .

L__.-___A-

I

I

597.0

596.5

Fig. 13. Line intensities versus line centers as deduced from diagonalization of 00 + iW [the dots correspond to Hitran database values (isolated Lorentzian lines)]. Upper curve: calculation including inter sub-branch mixing, lower curve: without sub-branch mixing. (a) PH== 2 atm; (b) PH<= 10 atm: (c) PHc= 50 atm.

At a higher pressure the sub-branches cannot be separated and the slope of the width h~,,~of the single resulting line arising from the interferences between the components of the two sub-branches is approximately equal to: (17) So the variation of ;5vli2versus the perturber pressure is characterized by two different slopes, in agreement with the experimental results of Ref. 6 (cf. Tables II and III). 3.3.2. Line mixing evolution versus pressure for the 741.7 cm-’ branch. Comparison between the ECS results and experimental measurements of the absorption and the lorentzian profile is shown in Fig. 14 for this Q branch. Agreement between theory and experiment is once again reasonable despite the fact that the rotational structure of this branch is very different from the structure of the one located at 597.3 cm-‘. Here too since this band is very weak the good results obtained with the ECS formalism are also related to its ability to correctly describe the positive wings of the most intense bands located at lower wavenumbers. 4.

CONCLUSION

The present study has confirmed the accuracy of the ECS model derived from the IOS formalism of Greem8 Thanks to the fact that low lying bending modes are still nearly linear, it has been possible to propose an ECS formalism valid for both stretching and bending bands in which all the cross-sections are described by a single set of basic rates Q’(L). It would be worthwhile comparing now the predictions of the present ECS formalism for bending bands with those of the ECS model of Bonamy et al,” and simultaneously investigated the numerical influence of the contributions neglected in Eq. (2) [due to Eq. (l)] which is the starting point of both formalisms. However for COrHe we feel that the next and most interesting challenge should be the comparison between the large amount of available experimental data and quantum mechanical scattering*’ calculations based on an accurate intermolecular potential energy surface, which is not available yet for C02-He.

852

J. Boissoles et al

I

1

(a) P[C02]=l 50 Torr

-

Exp. 0

PfHe]=3.75 Atm L=22.8 cm

Lor. E.C.S. talc.

I

745 6 r

I

I

ib)

‘E 0

”

I

I

I

cm-l

P[C02]=360

0 c 6

Ton

P[He]=9.4 Atm L=22.8 cm

4

-

0

Lor. E.C.S. celc.

I

I

I

I

725

730

735

740

I

b

cm-l

745

Fig. 14. Absorption coefficient in the spectral region around the (1 110)1_(0220)1 Q-branch (which is located into the R branch of the lI-+Z transition centered at 720 cm-‘). (a) PH. = 3.75 atm; (b) PHI = 9.4 atm.

Line mixing effects of CO1 in He

853

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15.

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16. M. 0. Bulanin, A. B. Dokuchaev, M. V. Tonkov, and N. N. Filippov, JQSRT 31, 521 (1984). 17. A. Ben Reuven, Phy.s. Rev. 145, 7 (1966); R. G. Gordon, J. Chem. Phys. 45, 1649 (1966). 18. R. G. Gordon and R. P. MC Ginnis, J. Chem. Phys. 49, 2455 (1968); E. W. Smith, J. Chem. Phys. 74,

6658 (1981). 19. J. W. C. Johns and J. Vander Auwera, J. Molec. Spectrosc. 140, 71 (1990). 20. 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, JQSRT 48, 469 (1992). 21. S. Green, in Status and Future Development in Transport Properties, W. A. Wakeman, ed., Kluwer, New York (1992); S. Green, J. Boissoles, and C. Boulet, JQSRT 39, 33 (1988). 22. F. Thibault, unpublished results.