- Email: [email protected]
Line-mixing, finite duration of collision, vibrational shift, and non-linear density effects in the ν3 and 3ν3 bands of CO2 perturbed by Ar up to 1000 bar
Vol. 58. No. 2. pp. 261-277. 1997 0 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain (X22-4073/97 $17.00 + 0.00
J. Quont. Specrrosc. Radial. Transfer
Pergamon PII: S0022-4073@7)00007-1
LINE-MIXING, FINITE DURATION OF COLLISION, VIBRATIONAL SHIFT, AND NON-LINEAR DENSITY EFFECTS IN THE v3 AND 3v, BANDS OF CO, PERTURBED BY AR UP TO 1000 BAR L. OZANNE,” J. P. BOUANICH,”
Q. MA,b NGUYEN-VAN-THANH,” C. BRODBECK,” J. M. HARTMANN,“? C. BOULET” and R. H. TIPPING’
‘Laboratoire de Physique Moleculaire et Applications, Unit& propre du CNRS (UPR 136) associee aux Universitts P. et M. Curie et Paris-Sud (bitiment 350), 91405, Orsay, Cedex, France, hDepartment of Applied Physics, Columbia University and Institute for Space Studies, Goddard Space Flight Center, New York, NY 10025, U.S.A. and ‘Department of Physics and Astronomy, University of Alabama, Tuscaloosa. AL 35487, U.S.A. (Received
8 July
1996; receioed for publication
7 January
1997)
Abstract-We present high-density experimental and theoretical results on C02-Ar gas-phase absorption in the v1 and 315 infrared bands. Measurements have been made at room temperature for pressures up to 1000 bar in both the central and wing regions of the bands. A non-linear perturber density dependence of the absorption, clearly shown in the far wing, is attributed to the finite volume of the molecules. Furthermore, experiments show vibrational dephasing and narrowing effects. We have performed line-mixing computations based on the Energy Corrected Sudden approximation (ECS impact model). Significant discrepancies between experimental and calculated spectra appear when pressure increases. We then tested the influence of the finite duration of collision by using interpolations between ECS and quasi-static calculations, and we have evaluated the sensitivity of the band profiles to the interbranch mixing effects. Finally, an effective width is used in order to take other effects into account. c 1997 Elsevier Science Ltd
1. INTRODUCTION
The calculation of accurate band-center and band-wing profiles for molecular rovibrational transitions over wide ranges of thermophysical parameters (e.g., number density, temperature, etc.) and wavenumber remains a difficult problem. The validity of impact models based on fitting and scaling laws has been widely demonstrated in central regions of CO, infrared bands under conditions where the contribution of far line wings is negligible. Departures from the Lorentzian shape owing to line-mixing effects in CO, have been successfully analyzed in Q-branches (see, for instance,lm3) as well as in P- and R-branches”‘. Other approaches, based on the quasi-static approximation, enable computation of absorption in the far wings where influence of the collision duration is significant; their accuracy has been demonstrated in the particular case of CO, in Refs 8-10. The main problem remaining is the prediction of absorption in the intermediate wavenumber detuning region (mid-wing) where both the impact and quasi-static approximations break down. The present paper is an experimental and theoretical study of absorption by C02-Ar gas mixtures. It is basically similar to previous works for C02-He6 and C02-Ar at lower densities.’ Measurements have been in the regions of the v3 and 3v, bands at room temperature for total pressures in the range 100-1000 bar. Measured absorptions in the far wing of the v3 band, as for C02-He,h display a non-linear density effect which is attributed to the volume occupied by the molecules. Theoretical analysis of experimental spectra are presented in both the wing and central regions of the two bands. The model used assumes binary collisions and is based on the impact and quasi-static approximations in the near and far wings, respectively, whereas a simple interpolation approach is proposed for the intermediate region. It is used for the test of the influences of a number of processes on high-density absorption which are: tTo whom all correspondence
should be addressed. 261
L. qzanne et al
262
1. The finite duration of collisions, which clearly affects the far wings of the bands but has negligible effects on the central region at elevated density. 2. Significant spectral shifts of the bands whose values are in good agreement with purely vibrational shifts computed with a semi-classical model. 3. Interbranch (P-R) mixing, which has large effects on the band shape in the entire density range. 4. Excluded volume effects, which are evidenced from absorption in the far wings and accounted for through use of an effective density. 5. Finally let us note that the binary approximation, which is assumed in the model used, probably explains part of the discrepancies observed at high density: the present results indicate that it is roughly valid up to about 400 Am. Above this point, an empirical broadening parameter of the band is proposed in order to obtain agreement with measured spectra which accounts for all neglected processes. The remainder of the paper is divided into three sections. The theoretical model and data used are described in Sec. 2. Experimental details and results are presented in Sec. 3, where non-linear density and vibrational shift effects are analyzed. Comparisons between measured and computed absorptions are the subject of Sec. 4. 2. THEORETICAL
MODEL
AND
DATA
USED
2.1. A theoretical model 2.1.1. Absorption coeficient. We consider the case of a mixture of an absorbing gas “a” with a transparent perturber “b” with densities n, and nb at temperature T. Within the binary collision approximation, the absorption coefficient c( at wavenumber Q in the infrared is given by:“?
.(Ql=%g
$)]xp*(nQd,xsh($$)’
sh(&)x{$l-exp(-
x
Im{<<~~[Z-Lo-iW(o.n,,nb,T)l'lk))}). (1)
where Im{. . .} denotes the imaginary part and the subscripts k and I refer to lines Ik)) = IUiji,U,j,)) and IZ)) = IuiJ’,u,j,‘))in th e L’touville space; dk and d, are their dipole-reduced matrix elements and pk is the population of the initial level of line k. C and L, represent diagonal operators associated with the scanning wavenumber cr and the spectral positions (ok) of the transitions, respectively. The relaxation operator W is a bath-averaged complex matrix which contains all the influence of collisions on the spectral shape. Within the binary collision approximation it is related to those associated with the “a’‘-“a” and “a’‘--“b” collisions for unit density by: W(o,na,b,,,T) = n, x “Wa-a(a,T) + nb x ‘Wa_b(C,T).
(2)
Prediction of a full relaxation operator including its wavenumber dependence in a large spectral range is a difficult problem. On the other hand, efficient models are available within the impact and quasi-static approximations, which enable calculations near line centers and in the far wings, respectively. 2.1.2. Near wing-impact approach. The impact approximation is valid at moderate density and when wavenumbers relatively close to the positions of transitions are considered. A criterion can be deduced from a typical value r, of the collision duration; indeed, the impact approximation is accurate for wavenumber detunings Aa which satisfy: IAal I AalMP<< (2ncr,)-’ .
In this case a number of models are available for the computation
(3) of the real part of the relaxation
tWhen the relaxation operator Wdepends on frequency, the distribution of levels changes. ‘I This effect is taken into account by renonnalizing the band surface instead of calculating the new distribution of levels.
High-density COrAr
263
gas-phase absorption
operator, which is then independent of wavenumber [simultaneously, both hyperbolic sines must be omitted in Eq. (I)]. That presented in Refs 6, 7 is used in the following; it is based on the Energy Corrected Sudden approximation and enables prediction of Re( WMp)only by assuming that CO? is a rigid rotor. Possible vibrational effects and the contribution of the imaginary elements of W are discussed at the end of this section. 2.1.3. Far wings-quasi-static approach. The quasi-static approximation is valid for wavenumbers far away from the positions of the optical transitions, which satisfy: ]Ao] 2 AaQSC>>(2nc~,)-’
(4)
The theoretical approach used in the present work is described in detail in Ref. 10. The matrix elements of Re( We”‘) are then computed directly from knowledge of the interaction potential and depend strongly on wavenumber as shown in Refs 8,9. Again, vibrational effects are neglected and only the real part of W is predicted. 2.1.4. Mid wings-interpolation model. Both the impact and quasi-static approximations break down in the intermediate region which corresponds to the mid wing of the lines. In order to predict the elements of the real part of the relaxation operator we have thus used a simple interpolation approach. Since Ref. 12 shows that the element ((k]lRe{ WQSC(a)}lll)) mainly depends on the detuning, I [a - (a, + cr,)/2] 1, we have used the following interpolation procedure in order to compute Re{ W} over the entire wavenumber region:
AalMPI (0 - 91
=
Kk IIW o-
I A~Qsc*((kIIRe(W(o))IIl))
w”‘(cr)}llI)) + a.#-
(Tk+ 0,
-- 2
~l-AdMp~+&,[~~-
?I
_A,lMPl’,
~A~QSC~((kIIRe{W(~))lll)) = ((kllRe(WoSC(a)}lJI)),
where Re( WiMp) and Re(WQSC) are operators predicted within the impact and quasi-static approximations, respectively. The a,, and b,, parameters of the intermediate region are determined in order to respect the continuity of the matrix elements of Re( W) and their first derivative versus r~for the detuning Ao Qsc The impact and quasi-static approaches are thus assumed for detunings lower than AcrIMPand greater than AcQsc, respectively, whereas interpolation is used in the intermediate range. 2.1.5. Imaginary part of W and vibrational eflects. The models presented above neglect vibrational effects and only predict the real part of W. Available measurements of the individual line shifts” [diagonal elements of Im(WMP)] indicate values of the order of -0.01 cm-’ Am-‘, which cannot be neglected. Nevertheless, prediction of the off-diagonal elements of Im( W), which are very sensitive to vibrational effects, is intractable. Fortunately, as shown in appendix A, the shift of spectra at high density is mainly dependent on the sum of the elements of Im( W). As is commonly done in the treatment of Raman Q-branch spectra,14 we have thus assumed that the imaginary part of W is diagonal and governed by an effective overall shift, AeH,i.e., Im(((kl WV))) = - Aefx 4, .
(6)
Note that appendix A shows that the effective shift in both the band wing and the central region of the band at elevated density can be identified as the purely vibrational dephasing. Vibrational effects on the real elements of Ware also very difficult to predict. Nevertheless, again, an empirical correction can be made, as for the shift and in Raman spectra,14 by adding a diagonal contribution
264
L. Ozanne et al
to Re( IV), governed by an effective broadening value, ycR.This point is discussed in the last part of the paper. 2.1.6. Binary absorption coefficients in the far wings. Within the present approach, and assuming the binary collision approximation, a linear development of Eq. (1) in the far wing leads to: -40) = n, x B,_,
+
nb
X
Bamb[a
-
&&Lnb,T),
r]
,
n,
where the density normalized absorption coefficient, B,_x, is given by:
B&a
- AenT) = $
sh($$x
{z “‘lrJ[l
-exp(
- s)]
sh Z$
x (a -
AeR- bk)(o - A,K- 6,) ’
Note that when the influence of the volume occupied B,_Ja - A,(n,,q,T),TJ should be independent of density.
(8)
by the molecules6 is negligible,
2.2. Data used The line positions and dipole-reduced elements (deduced from line intensities) have been taken from the 1992 edition of the HITRAN database.‘S*‘6 The real elements of the impact relaxation operator have been computed by using the equations and parameters given in Ref. 6 and in Table 2 of Ref. 7. The real elements of the quasi-static relaxation operator have been computed by using the equations and data of Ref. 10. The duration of the efficient collision for CO,-Ar at room temperature is about 0.2 psec, so that (27rcr,)- is of the order of 25 cm-‘. On the other hand, it has been shownI that the lower limit of the purely quasi-static approximation is of the order of 50 cm-‘. We have thus retained the values: AtY
= 10 cm-‘,
AcQsc = 70 cm-’ ,
(9)
Fig. 1 presents typical results obtained for some of the elements of Re( IV) connecting the RI6 line to others. The increase of these elements with increasing detuning is characteristic of the influence of the collision duration and has been observed previously.’ 3. EXPERIMENTAL
3.1. A set-up and treatment
The spectra were recorded on the same apparatus and following the same procedure as done in a previous study of CO,-He mixtures,6 where all experimental details can be found. Measurements have been made at room temperature (293-297 K) for total pressures in the range 100-1000 bar. High-purity CO, and Ar gases (99.998% for COZ and 99.9999% for Ar) were supplied by Air Liquide and used without further purification. Measurements have been made for adapted CO, mole fractions and about six total pressures in the range from 100 to 1000 bar (note that the absorption in the v3 band is so intense that “pure” Argon gas contained enough CO, for the study of the central part of the band; in this case, the unknown CO* density was determined from the integrated band intensity [Eq. (10) below] as done in Ref. 6). For given thermodynamic conditions (p,, pb, r) the Ar and CO2 densities n, and nb have been computed by using the equations of state given in Refs 17, 18, respectively. As is described in our previous work on C0,-He,6 a very careful error analysis was made in order to quantify experimental uncertainties. Absorption in the central regions of the v3 and 3v3 ‘V’O, bands shows contributions of some other weak bands, among which are transitions of other isotopes as well as hot transitions (such as the mv, + v2 - v2, m = integer). In order to simplify the analysis these contributions were removed from experimental spectra in the central regions as follows. Let us consider a weak band
High-density COrAr O.lO.
265
gas-phase absorption
(a)
,c----________________-_________. .
0.04 -
:
*.
0.02 -
*. ..
:
..
:
:
:
.’
I 0.00 -
I
I
I
I
,_.* *... . . _._.-.- ._._.___.; ~--~~__________---______ -.-.~__~_.________~-.-.---
-0.02 -0.03 _ -o&t-
(b) .
1 -50.
0.
1 50. Au
I 100.
1 150.
I 200.
(cm“)
Fig. 1. Wavenumber dependences of the interpolated Re(” W) elements connecting the R,, line to some others. (a) IV,,,,,,: - - -, ECS model; ., quasi-static model; -, interpolation model. (b) IV,,, R,4: - - _, ECS model; ., quasi-static model; -, interpolation model. W,,, R14:- -, interpolation model.
of central wavenumber cW and integrated intensity SWlying near the vj band, for instance; its absorption was determined by shifting experimental spectra of the quantity cW- (z,.? and multiplying it by the ratio SW/S,,. The spectrum obtained was then subtracted from the measured absorption. This procedure, which assumes that all bands have the same shape, leads to small errors since the removed bands have intensities that are,15,16at the most, about 6% of that of the main (v, or 3vJ transition. 3.2. Results A test of the experimental procedure can be made in the central regions of the bands by integrating the absorption coefficient over the spectral range of significant absorption. Indeed, one has:
(10) where alnf and asUPare wavenumbers on both sides of the band where absorption is negligible and S, are individual line intensities. The results obtained in the 3v, band, which are given in Table 1, are a first validation of our measurements. Furthermore, the agreement with the HITRAN database value and the lack of dependence on perturber density indicate the absence of any induction effect within the present experimental uncertainties. Table I. Integrated absorptions in the 3v, central region [see Eq. (IO)]. n. (Amagat) 3.25 3.27 3.24 3.24 3.21 3.26 Average HITRAN” 16 Ref. 6
nb
(Amagat) 144.6 283. I 319.2 449.9 503.3 545.5
lOO/+,,
~$$a(b,nro2,n,,,)
du (cm _’ Amagat - ‘) 4.57 4.55 4.53 4.57 4.50 4.51 4.54 (a = 0.27) 4.64 4.71 (a = 0.11)
266
L. Ozanne
8. J-* 9.
et al
cm-’
a=2520
7.-
5
m" 6.-
4. TI
-r
T
itii
0.
100.
300.
200.
500.
400.
Ar density nb (Am) Fig. 2. Experimental
Y) band wing parameter
Eco,_a,(u,n.,nb) versus argon density for three wavenumbcrs.
3.2.1. Excluded volume effects in the wing of the v3 band. The values of B,,,_,,(o) in the v3 band wing were deduced from the present measurements by using Eq. (7) removing the small C02-C02 contribution by using the data of Ref. 19; in doing this, the effective shift A,&ra,nb,T) was first determined for each density, as explained in the next paragraph. The values of BCo2_,,(a)obtained, plotted in Fig. 2, show a dependence on density that is qualitatively similar to that obtained previously for CO,-He6 and which is attributed to excluded volume effects.20.2’In order to quantify this, the values of B have been fitted, following Ref. 6, by the expression:
&-~r@ - Ac,d = B&da) x y
= B&&Y)
x 1+
kO,-A,@)
x
.
nb
(11)
1
[
The values of the “zero density” binary absorption coefficients, B&+ are given in Table 2, where Table 2. Binary
absorption
coefficients
B&,.,,, and density
10680C0,.,,(u) (cm -
(r (cm-‘) 2415 2480 2485 2490 2495 2500 2505 2510 2515 2520 2525 2530 2535 2540 2545 2550 2555 2560 2565 2570
effect parameter
’ Am - ‘)
Ref. 9
This work
32.3 k 1.6 27.5 f 1.4 23.8 & 1.2 20.0 + 1.0 17.3 f 0.90 12.5 k 0.60 11.4 + 0.60 11.1 * 0.55 9.90 f 0.50 8.45 k 0.42 6.56 f 0.33 5.43 f 0.27 4.44 f 0.22 3.84 f 0.19 3.38 f 0.17 2.97 It 0.15 2.72 k 0.14 2.32 f 0.12 1.98 f 0.99 1.80 f 0.90
34.8 f 2.1 28.1 f 2.0 22.8 f 1.4 18.5 + 1.1 15.1 * 0.83 12.4 f 0.68 10.2 + 0.58 8.46 + 0.53 6.96 f 0.48 5.73 f 0.45 4.14 f 0.43 3.95 + 0.42 3.22 f 0.40 2.65 k 0.38 2.21 + 0.36 1.81 + 0.34 1.53 f 0.33 1.27 + 0.31 1.05 + 0.29 0.86 f 0.27
cc++
[see Eq.
(I I)] in the v, band wing.
1~cco,-&) (Am - ‘1 This work 9.6 8.8 8.4 8.2 8.2 1.9 7.7 7.4 7.4 7.4 7.2 6.9 7.3 7.6 1.6 7.9 6.7 6.9
& 2.1 f 1.9 f 1.8 + 1.8 f 1.8 f 1.7 f 1.7 f 1.7 f 1.8 f 1.8 + 1.8 f 1.9 f 2.7 f 3.4 f 3.9 f 4.8 k 5.0 & 5.8 .. ...
High-density COrAr
gas-phase absorption
267
they can be compared with previous determinations. The good agreement obtained for the smallest wavenumbers where previous experiments are believed to be accurate is a further validation of our experimental procedure. The values of cCO1+,given in the same table are, as expected, practically independent of the wavenumber and equal to cCO,+ = 7.4 x 10m4f 2.0 x 10e4 Am-‘. It is worth noting that the theoretical value of cCo2+ estimated*’ from the CO,-CO, and CO,-Ar collision diameters is 12.4 x 10m4Am-‘, in satisfactory agreement with the experimental determination. In all subsequent parts of this paper, spectra computations have been made by replacing n,, by the effective density &,(n,,)= nb x [l + 7.4 x 10e4 x n,], where nb is expressed in Amagat units. 3.2.2. Vibrational shifts. The plots in Fig. 3 compare measured spectra in the 3v, band with results of computations from the impact model with AeR= 0 [Im( IV) = 01. It is clear, as expected from individual line shift valuesI that the imaginary part of W cannot be neglected. The effective shift AeKwas then determined, for each spectrum, from the wavenumber difference between the position of the measured maximum and the location of the maximum of a theoretical calculation with AeR= 0; this procedure is valid since the location of the theoretical maximum is practically independent of the model (impact or including some frequency dependence). The values obtained in the vj and 3v, bands are plotted versus corrected density fi, in Fig. 4. The linearity is satisfactory and the values of the slopes obtained from a linear fit [A&&) = (fi,, x S,,] are: &(v,) = -5.4 + 1.0 x lo--’ cm-’ Am-‘, &,(3v,) = -17.0 + 1.0 x lo-’ cm-’ Am-‘. It is worth noting that these values show the factor of three expected from their physical origin which is the purely vibrational dephasing (see appendix B). Furthermore, theoretical predictions from a semi-classical model and knowledge of the interaction potential (see appendix B) lead to results (-4.6 and - 14.0 x 10e3 cm-’ Am-’ in the v3 and 3~2,bands, respectively) which agree well with the measurements. 3.2.3. Narrowing effects. Finally, the narrowing of the vibrational bands with increasing Ar density can be revealed, as done in Ref. 22, by defining the half-width r’(fi,,) of the band through:
(12) where c,, is the band origin and ~(6~) is the perturber-density-dependent position” defined by:
“average rotational
(13)
The values of ?(E,) deduced from measured spectra are plotted corrected for excluded volume effects. They clearly show the line-mixing processes, They also indicate that computations with predict too much narrowing. The reasons for this discrepancy 4. THEORETICAL
in Fig. 5 versus perturber density narrowing effect due to efficient the present ECS impact approach are discussed in the next section.
ANALYSIS
4.1. Band wings 4.1.1. High-frequency wing of the 3v, band. A test of the theoretical models in the 3~1,wing is plotted in Fig. 6. Since the spectral region is relatively close (from 10 to 40 cm-‘) to the most intense R-lines the impact approximation is reasonable and leads to satisfactory results as assumed in Ref. 7. For the highest wavenumbers, the uncertainty of measurements prevents a definitive conclusion. 4.1.2. High-frequency wing of the v, band. Comparisons between measured and computed absorptions in the high-frequency wing of the v3 band are plotted in Fig. 7; experimental values include our results and those of Ref. 9 in the far wing beyond the band head (a > 2400 cm-‘) as well as previous measurements23 in the troughs between R-lines. Accounting for line-mixing effects within the impact approximation leads to satisfactory results in the near wing (a < 2410 cm-‘), as
268
L. Ozanne et al 3.5 3.0 2.5 2.0 1.5
-i
E
1.0
u :
.41 ‘a g
0.5 0.0
6920.0
6940.0
6980.0
6960.0
7000.0
u(cm-I) Fig. 3. Absorption coefficients in the central region of the 3vl band: 0, measured; -, calculated with the ECS model. (a) ncO, = 3.26 Am and nAr= 283.1 Am (& = 342.4 Am); (b) nco, = 3.26 Am and nAr= 545.5 Am (n-Ar= 765.6 Am).
0. -
-2. -4.-6. -8. -10. -12.1 -14. -
(b) 200.
6tiO.
corrected A%mity
8dO.
(Am)
Fig. 4. Effective shift LeRdetermined (see text) from high-pressure spectra: 0, from spectra; -, law [A&,,) = fib x a,,]. (a) v1 band; (b) 3vl band.
linear
High-density COrAr
269
gas-phase absorption
14.
8.8
10.
1000.
Fig. 5. Overall band half-widths k(&$ of the bands versus density [Eqs (12) and (13)J. Measured values values computed with the ECS impact model. in the v1 band (0) and 3vl band (0); p,
expected, but fails in the mid and far wing regions. In this region the interpolation scheme proposed in Sec. 2.1 is too crude and overestimates the absorption by a factor of about 2, while the impact scheme underestimates it by roughly the same factor. In the far wing, according to the interpolation procedure [Eqs (5),(9)], the profile is purely quasi-static and a very good agreement between computed and measured values is obtained. This was expected since the potential was adjusted” in order to obtain good agreement with the measured data of Ref. 9. Note, however, that the agreement seems to be much better with the present experimental results. In summary, even if the interpolation scheme seems too crude to give a precise description in the mid wing, it gives us a “tool” to test the influence of the finite duration of collision on the central region of the vibrational bands at very high perturber densities.
6990.
6995.
7000.
7005. u
7010.
7015.
7020.
(cm-‘)
Fig. 6. 3v, band wing parameters B&, A,.Experimental results:’ l , from St. Petersburg; 0, from Rennes. the ECS impact model; ., the impact/quasi-static interpolation model: - - -, Calculations with: -, the Lorentzian model.
210
L. Ozanne et al
t . -.
1
--__
I
2390.0 .,.,,I., 2400.0 2410.0 2420.0 2430.0. 2440.0 . 2450.0 I 2460.0
--s_
..
--S_
*.
I”‘.
o(cm-‘) ---_
--__
l;-.
\
2350.0
--__
;_
I 2400.0
”
”
1
““I”’
----__
l
2500.0
2450.0
---___
1
---
’
”
2550.0
a(cm-l) Fig. 7. v1 band wing parameters $o,.~,. Experimental results from: 0, Refs 9, 23; 0, this work. the ECS impact model; ., the impact/quasi-static interpolation model; - - -, Calculations with: p, the Lorentzian model.
4.2. Band centers
In the following, different factors which affect the absorption in the central regions of the bands are discussed. The spectra in the vj and 3v, regions, which lead to similar conclusions, are used alternately. 4.2.1. Finite duration of collision eflects. The influence of the collision duration on the central region of the 3v, band is illustrated by the plots in Fig. 8 (similar results are obtained when the vj band is considered). As expected, absorption at moderate density is practically insensitive to the frequency dependence of the relaxation matrix since absorption in the line wings is small. Even at elevated density the finite duration of collision does not seem to be of great importance. This result may be understood from the strong narrowing of the band already mentioned (see Fig. 5). Indeed, at 750 Am, the full width of the band is about 20 cm-’ which is much smaller than AgQsc. Note that these plots clearly show the great inaccuracy of the Lorentzian approach already noticed for CO,-He mixtures.6 4.2.2. Znterbranch mixing effects. The influence of the amount of interbranch (R-P) mixing on the spectral shape at elevated density has been discussed in details in Refs 67. Such effects are responsible for the filling of the trough between the R- and P-branches,22 as is demonstrated by the results of Fig. 9 in the case of the v1 band (considering the 3v, region would lead to similar conclusions). Measured spectra are compared with those predicted when multiplying the R-P couplings by 1 - &. and modifying the intrabranch (R-R and P-P) couplings in order to respect the sum rule as explained in Ref. 6. Even small values of BRPlead to significant modifications of the theoretical lineshapes. It is clear that the increase of Rt+P transfers (negative &.) fills the trough between branches and narrows the band. 4.2.3. Other e&m. A number of effects were not accounted for in the present computations. Among these are the influence of vibration on the real part of Was well as the breakdown of the binary collision approximation. Such phenomena are very difficult to model correctly but can be
High-density COT-Ar gas-phase absorption
271
empirically accounted for through the introduction of an effective broadening, ycR.Starting from the previous calculations (with &. = 0.1) satisfactory results for both the v, and 3v, bands can be obtained by using yeR= 3.6 x 10e3 cm-’ Am-’ as may be seen from Fig. 10. The physical meaning of yea remains unclear. Indeed, a first (tempting) possible attribution is vibrational dephasing (yJ since this phenomenon quantitatively explains the effective shift, AeR (Sec. 3). However, even if the calculations presented in appendix B are not very accurate owing to the uncertainties in the vibrational dependence of the isotropic potential, the fact that yVvaries with u: does not depend on these uncertainties is in strong disagreement with the independency on u3. Furthermore, a question remains about neglecting the dependence of the anisotropic of 3’eR part of the potential on the vibrational quantum number; nevertheless, since no potential including that dependence is presently available for CO,-Ar, it is not possible to test its influence on the
4. -
h ‘;
3.-
E 22
.
______--
u
(cm-‘)
u
(cm-‘)
6. ,
5.
I.
Fig. 8. Absorption coefficients in the central region of the 3v, band; 0, experimental. Calculated with: -, the ECS impact model; . ., the impact/quasi-static interpolation model; - - -, the Lorentzian model. (a) nco, = 3.25 Am and nAr= 144.6 Am (&, = 160.1 Am); (b) nco, = 3.26 Am and nA, = 545.5 Am (&, = 765.6 Am). JQSRT 5812-E
212
L. Ozanne et al
2300.
2310.
2320.
2330.
2340.
2350.
2370. 2380. 2390. 2400.
(cm-‘)
0
1.44
2360.
b)
2300. 2310. 2320. 2330. 23
)O.
u
(cm-‘)
Fig. 9. Absorption coefficients in the central region of the vj band; 0, experimental, Theoretical results with the ECS impact model and various values of the parameter PRPmodifying the R++P coupling: - -, ~RP=0.25;---,~Rp=0.1;-, b.p = 0; . ., /&P= - 0.2. (a) ttAr= 305.9 Am (AAr= 375.1 Am); (b) nAr= 527.0 Am (AAI= 732.5 Am).
elements of the relaxation operator. A number of other mechanisms can be responsible for yen which, unfortunately, cannot be quantitatively checked; among these is clearly the breakdown of the binary collision approximation. Indeed, above 350 Am, the time interval between successive collisions is smaller that of the collision duration. yea must thus be considered as an empirical parameter taking into account the insufficiencies of the theory, including, of course, the approximate approach of the impact relaxation matrix W since yeRdepends, for instance, on BRP.
Table 3. Parameters used for calculations of broadening and shifting coefficients. B, (cm - I)“. I6 CO* Ar
0.39021889
Br (cm - ‘)15.16
0.38099334
E (cm-‘)
o (A)
177.F0
4.631”” 3.505”
117.7”’
213
High-density COrAr gas-phase absorption
r (a)
0.00
6900.
6920.
6940.
6960.
6980.
7000
2300.
2320.
2340.
2360
2380
2400.
0
(cm-‘)
Fig. 10. Normalized coefficient of absorption in the central region. ., Measured values; ~ computed results with the ECS model with /I& = 0.1, including an effective width 0; 3.6 x IO-’ cm-’ Am-‘. (a) V, band and the densities tiA,= 109.3 Am, 375.1 Am, and 732.5 Am; (b) 3v, band and the densities CA,= 112.9 Am, 342.4 Am, and 765.6 Am.
REFERENCES B. and Strow, L. L., J. Chem. Phys., 1987, 86, 5722. Bonamy, L., Bonamy, J., Temkin, S., Robert, D. and Hartmann, J. M., J. Chem. Phys., 1993, 98, 3747. Rachet, F., Margottin-Maclou, M., Henry, A. and Valentin, A., J. Molec. Spectrosc., 1996, 175, 3 15. Filippov, N. N. and Tonkov, M. V., JQSRT, 1993, 50, 111. Boissoles, J., Thibault, F., Le Doucen, R., Menoux, V. and Boulet, C., J. Chem. Phys., 1994, 101,6552. Ozanne, L., Nguyen-Van-Thanh, Brodbeck, C., Bouanich, J. P., Hartmann, J. M. and Boulet, C., J. Chem. Phys., 1995, 102, 7306.
1. Gentry,
2. 3. 4. 5. 6.
L. Ozanne et al
274
7. Filippov, N. N., Bouanich, J. P., Hartmann, J. M., Ozanne, L., Boulet, C., Tonkov, M. V., Thibault, F. and Le Doucen, R., JQSRT, 1996, 55, 307. 8. Boulet, C., Boissoles, J. and Robert, D., J. Chem. Phys., 1988, 89, 625. 9. Boissoles, J., Menoux, V., Le Doucen, R., Boulet, C. and Robert, D., J. Chem. Phys., 1989, 91, 2163. 10. Ma, Q. and Tipping, R. H., J. Chem. Phys., 1994, 100, 8720. 11. Davies, R. W., Tipping, R. H. and Clough, S. A., Phys. Rev. A, 1982, 26, 3378. 12. Ma, Q., Tipping, R. H., Hartmann, J. M. and Boulet, C., J. Chem. Phys., 1995, 102, 3009. 13. Thibault, F., Boissoles, J., Le Doucen, R., Bouanich, J. P., Arcas, Ph. and Boulet, C., J. Chem. Phys., 1992, 96, 4945. 14. Lavorel, B., Millot, G., Saint-Loup, R., Berger, H., Bonamy, L., Bonamy, J. and Robert, D., J. Chem. Phys., 1990, 93, 2185. 15. Rothman, L. S., Gamache, R. R., Tipping, R. H., Rinsland, C. P., Smith, M. A. H., Benner, D. C., Malathy Devi, V., Flaud, J. M., Camy-Peyret, C., Perrin, A., Goldman, A., Massie, S. T., Brown, L. R. and Toth, R. A., JQSRT, 1992, 48, 469. 16. Rothman, L. S., Gamache, R. R., Tipping, R. H., Rinsland, C. P., Smith, M. A. H., Benner, D. C., Malathy Devi, V., Flaud, J. M., Camy-Peyret, C., Perrin, A., Goldman, A., Massie, S. T., Brown, L. R. and Toth, R. A., JQSRT, 1992, 48, 537. 17. Angus, S., Armstrong, B. and de Reuck, K. M., International Thermodynamic Tables of the Fluid State: Argon, Pergamon, Oxford, 1976. 18. Angus, S., Armstrong, B. and de Reuck, K. M., International Thermodynamic Tables of the Fluid State: Carbon Dioxide, Pergamon, Oxford, 197 1. 19. Le Doucen, R., Cousin, C., Boulet, C. and Henri, A., Appl. Opt., 1985, 24, 897. 20. Delalande, C. and Gale, G. M., J. Chem. Phys., 1980, 73, 19 18. 21. Hirschfelder, J. O., Curtiss, C. F., Byron Bird, R., Mofecular Theory of Gasesand Liquids, John Wiley & Sons Inc., New York, 1964. 22. Hartmann, J. M. and L’Haridon, F., J. Chem. Phys., 1995, 103, 6467. 23. Bulanin, M. O., Dokuchaev, A. B., Tonkov, M. V. and Filippov, N. N., JQSRT, 1984, 31, 521. 24. Bulgakov, Yu. I. and Strekalov, M. L., Opt. Spectrosc., 1994, 76, 516. 25. Monchick, L., J. Chem. Phys., 1991, 95, 5047. 26. Monchick, L., J. Chem. Phys., 1995, 102, 3009. 27. Fano, U., Phys. Rev., 1963, 131, 259. 28. Robert, D. and Bonamy, J., J. Phys. (Paris), 1974, 40, 923. 29. Garand, A., These de 3eme Cycle, Paris, 1972. 30. Bouanich, J. P., JQSRT, 1992, 47, 243. 31. Sherwood, A. E. and Prausmitz, J. M., J. Chem. Phys., 1964, 41, 429. 32. Tejeda, G., Mate, B. and Montero, S., J. Chem. Phys., 1995, 103, 568. 33. Weast, R. C., Astle, M. J. and Beyer, W. H., Handbook of Chemistry and Physics, CRC Press, Inc., Boca Raton, FL, 1985-l 986.
APPENDIX
A
Influence of the Imaginnry Part of the Relaxation Operator
Starting from Eq. (I), we consider asymptotic density situations in order to study the influence of the imaginary part of the relaxation operator on the absorption coefficient. At very high densities,when the real elements of the relaxation operator are much larger than the spreading of the optical transitions, it is known that the whole vibrational band becomes symmetric with a Lorentzian shape; its effective shift A*,, (with respect to the central wavenumber of the unperturbed band) is then exactly the vibrational shift, A,, since one has
AcK= -T 2 ((~~lm(W)~k)) = A,
(Al)
At moderate densities, we consider now a perturber density such that the P- and R-branches remains relatively distinct but with a complete overlapping of the R- (or P-) lines. The shift of the maximum of the R-branch, for instance, is then given by:
(I&R) - x$’ ((rlIm(W)lk)) = I$’ ((~IImtWllk)) + A,.
ER !.
EP A
W)
The last sum, which involves R++P couplings, is expected to be small according to Refs 25,26; furthermore, when applied to the real elements of W, it is negligible since the most populated levels of CO? correspond to relatively high J values for which interbranch mixing vanishes. Eq. (A2) therefore shows that the effective shift A.#,of the R- (or P-) branch maximum is also given, with a good approximation, by A.,.
275
High-density COI-Ar gas-phase absorption 0.10
0.08
Lh.__
h ‘; 2 ‘;E S 2
0.06
l l
.
0.04
0
-10
-20
-40
-30
m Fig. Al. Ar-broadening
coefficients y0 for CO, lines at 296 K in the 3v, band. 0, Measured values;” -. calculated results
-0.006
-0.008
h ‘;<
-0.010
7 E 2 u?
-0.012
-0.014
-0.016 1-
0
-20
m Fig. A2. Argon-shifting coefficients Sofor CO* lines at 296 K in the P-branch of the 3vr band. 0. Measured values;‘J -, calculated results.
Absorption
in the near wing at high density
Following the method proposed by Fano ,I7 the operator in Eq. (1) may be written up to the second order in a density expansion as: 1
~_Lo_iw
-- [ 1 - z_r,
I-iW=
1
+(-iW&(-iW&
+...
1.
(A3)
where only the imaginary part must be retained. The only contribution of Im( IV) then comes from the last term and is given by: (A4)
where
*dd = -T
$E
(
(A3
276
L. Ozanne et al
Since ((nlIm(?V)lk)) is expected to vanish for lines n and k far apart, the small frequency dependence in Eq. (AS) can be neglected. b(a) is then equal to the purely vibrational shift, A, [Eq. (Al)]. Eq. (A4) is then identical to the second-order term of the development of the operator: 1 Z-&-iW
z Z - (&I- A#)
+... '-iRe(mz _(Lo'_ A,Jd) 1
C
Comparison between Eq. (A6) and Eqs (7) and (8) shows that the effective shift AcRin the far wing can again be approximated by the purely vibrational value, A,. These three characteristic situations show that the present experiments are not sensitive to the details of the imaginary part of W and justify, in some way, the approximate treatment used in the present work.
APPENDIX
B
Widths and Shifrs in COrAr
Within the semi-classical model developed by Robert and Bonamy,‘* the broadening (uO)and shifting (15~) coefficients for an isolated Lorenttian line are given by: y0 _ id, = Z!! z 2nb S’(b) - iSs(b) 2nc o s [
1
db ,
@I)
where N is the molecule density of the perturbing gas for 1 Am, Vis the mean relative velocity, and b is the impact parameter. By neglecting weak rotational contributions to the shifts, the differential broadening and shifting cross-sections, P(b) and P(b), are given by S’(b) = 1 - exp[ - Rep(b)] P(b) = exp[ - ReSy(b)]
cos[Sy(b)]
,
WI
sin[$“(b)],
where Sp is the vibrational phase shift arising from the difference of the isotropic part of the potential in the initial and final states of the transition; the real part of SF is derived from the anisotropic part of the potential. When the vibrational dependence of the anisotropic potential is neglected, one can show that?
where the pure vibrational width and shift are given by
Since we are only interested here in the order of magnitude of yVand a,, the CO?-Ar potential has been simply represented by a two-term expansion of Legendre polynomials such that:
where r is the intermolecular distance between the perturbers, 6, is the angle between the axis of the absorber and the intermolecular axis, and P2 is a second-order Legendre polynomial; the Lennard-Jones (LJ) parameters E and Q for CO?-Ar have been determined from those for CO,CO, and Ar-Ar and the usual combination rules; Rz and Al have been considered as adjustable parameters in the fit of experimental CO?--Ar broadening coefficients. The parameters used in the computation are listed in Table 3. Figure Al shows that they lead to satisfactory agreement with widths measured by Thibault et al’) in the 3v3 band. The isotropic part of the potential in Eq. (BS) is a 612 LJ potential given by
with C, = 4s~~ and C,, = 4.s&. Assuming that its vibrational dependence arises from the polarizability matrix elements a, of CO* leads to: 1Ic, Aal c, =7.
(B6)
The value of Aa,/a, was first calculated by using the polar&ability derivatives for CO: given in Ref. 32: da,/Sq, = 0.1 I I7 A’, a’a,/aq: = 0.00404 A-l, ahjaq: = 0.00135 A’, a mean polarizability (a, = 2.911 A’)ji and Aq, = (odq,lvr) - (uilqllv,) = 0.5438 for the 3vr band;z9 this calculation leads to AC,/C, = 0.0218. By considering in addition’> y _ AC,1 AG
CI?
C,
= 1.69,
the computed lineshifts are about two times larger than the experimental data given in Ref. 13.
High-density COrAr
gas-phase absorption
217
Owing to the approximate nature of Eq. (B6), we have considered AG,/C, and _Vas adjustable parameters. The values ACJC, = 0.010 and y = 1.2provide lineshift coefficients for the 3v, band in satisfactory agreement with the experimental data, as shown in Fig. A2. The corresponding pure vibrational broadening is very weak, y,(3v,) = 0.79 x IO- 3cm - ’ Am - I, whereas the vibrational lineshift value is 6,(3v,) = - 14.0 x IO-’ cm-’ Am- ‘, in reasonable agreement with the effective shift &&3vJ = - 17.0 x IO-’ cm-’ Am- ’ derived from high-pressure measurements in the 3v, band (see text). Calculations in the 11) band were then made, considering (hC,/C,),,,/(AC,/C,),,, = (Aa,/a,),,/(A~,/ol,)i,,, leading to (ACJC,), , = 0.0032. This value associated with v = 1.20 leads to v,(v,) and ,.~ -I = 0.081 x IO-’ cm-’ Am-’ a,(~,) = 14.6 x 10-l cm-’ Am-‘, again in reasonable agreement effective shift with the experimental 6,,dvl) = - 5.4 x IO-’ cm- ’ Am- ‘. Finally, let us recall thaP Aq, is proportional to v,; since a second-order expansion of the exponential function is valid in Eq. (B4), it leads to &av, and ~,avv,.z The above numerical results and the experimental values of &Asupport these conclusions.
J. Quont. Specrrosc. Radial. Transfer
Pergamon PII: S0022-4073@7)00007-1
LINE-MIXING, FINITE DURATION OF COLLISION, VIBRATIONAL SHIFT, AND NON-LINEAR DENSITY EFFECTS IN THE v3 AND 3v, BANDS OF CO, PERTURBED BY AR UP TO 1000 BAR L. OZANNE,” J. P. BOUANICH,”
Q. MA,b NGUYEN-VAN-THANH,” C. BRODBECK,” J. M. HARTMANN,“? C. BOULET” and R. H. TIPPING’
‘Laboratoire de Physique Moleculaire et Applications, Unit& propre du CNRS (UPR 136) associee aux Universitts P. et M. Curie et Paris-Sud (bitiment 350), 91405, Orsay, Cedex, France, hDepartment of Applied Physics, Columbia University and Institute for Space Studies, Goddard Space Flight Center, New York, NY 10025, U.S.A. and ‘Department of Physics and Astronomy, University of Alabama, Tuscaloosa. AL 35487, U.S.A. (Received
8 July
1996; receioed for publication
7 January
1997)
Abstract-We present high-density experimental and theoretical results on C02-Ar gas-phase absorption in the v1 and 315 infrared bands. Measurements have been made at room temperature for pressures up to 1000 bar in both the central and wing regions of the bands. A non-linear perturber density dependence of the absorption, clearly shown in the far wing, is attributed to the finite volume of the molecules. Furthermore, experiments show vibrational dephasing and narrowing effects. We have performed line-mixing computations based on the Energy Corrected Sudden approximation (ECS impact model). Significant discrepancies between experimental and calculated spectra appear when pressure increases. We then tested the influence of the finite duration of collision by using interpolations between ECS and quasi-static calculations, and we have evaluated the sensitivity of the band profiles to the interbranch mixing effects. Finally, an effective width is used in order to take other effects into account. c 1997 Elsevier Science Ltd
1. INTRODUCTION
The calculation of accurate band-center and band-wing profiles for molecular rovibrational transitions over wide ranges of thermophysical parameters (e.g., number density, temperature, etc.) and wavenumber remains a difficult problem. The validity of impact models based on fitting and scaling laws has been widely demonstrated in central regions of CO, infrared bands under conditions where the contribution of far line wings is negligible. Departures from the Lorentzian shape owing to line-mixing effects in CO, have been successfully analyzed in Q-branches (see, for instance,lm3) as well as in P- and R-branches”‘. Other approaches, based on the quasi-static approximation, enable computation of absorption in the far wings where influence of the collision duration is significant; their accuracy has been demonstrated in the particular case of CO, in Refs 8-10. The main problem remaining is the prediction of absorption in the intermediate wavenumber detuning region (mid-wing) where both the impact and quasi-static approximations break down. The present paper is an experimental and theoretical study of absorption by C02-Ar gas mixtures. It is basically similar to previous works for C02-He6 and C02-Ar at lower densities.’ Measurements have been in the regions of the v3 and 3v, bands at room temperature for total pressures in the range 100-1000 bar. Measured absorptions in the far wing of the v3 band, as for C02-He,h display a non-linear density effect which is attributed to the volume occupied by the molecules. Theoretical analysis of experimental spectra are presented in both the wing and central regions of the two bands. The model used assumes binary collisions and is based on the impact and quasi-static approximations in the near and far wings, respectively, whereas a simple interpolation approach is proposed for the intermediate region. It is used for the test of the influences of a number of processes on high-density absorption which are: tTo whom all correspondence
should be addressed. 261
L. qzanne et al
262
1. The finite duration of collisions, which clearly affects the far wings of the bands but has negligible effects on the central region at elevated density. 2. Significant spectral shifts of the bands whose values are in good agreement with purely vibrational shifts computed with a semi-classical model. 3. Interbranch (P-R) mixing, which has large effects on the band shape in the entire density range. 4. Excluded volume effects, which are evidenced from absorption in the far wings and accounted for through use of an effective density. 5. Finally let us note that the binary approximation, which is assumed in the model used, probably explains part of the discrepancies observed at high density: the present results indicate that it is roughly valid up to about 400 Am. Above this point, an empirical broadening parameter of the band is proposed in order to obtain agreement with measured spectra which accounts for all neglected processes. The remainder of the paper is divided into three sections. The theoretical model and data used are described in Sec. 2. Experimental details and results are presented in Sec. 3, where non-linear density and vibrational shift effects are analyzed. Comparisons between measured and computed absorptions are the subject of Sec. 4. 2. THEORETICAL
MODEL
AND
DATA
USED
2.1. A theoretical model 2.1.1. Absorption coeficient. We consider the case of a mixture of an absorbing gas “a” with a transparent perturber “b” with densities n, and nb at temperature T. Within the binary collision approximation, the absorption coefficient c( at wavenumber Q in the infrared is given by:“?
.(Ql=%g
$)]xp*(nQd,xsh($$)’
sh(&)x{$l-exp(-
x
Im{<<~~[Z-Lo-iW(o.n,,nb,T)l'lk))}). (1)
where Im{. . .} denotes the imaginary part and the subscripts k and I refer to lines Ik)) = IUiji,U,j,)) and IZ)) = IuiJ’,u,j,‘))in th e L’touville space; dk and d, are their dipole-reduced matrix elements and pk is the population of the initial level of line k. C and L, represent diagonal operators associated with the scanning wavenumber cr and the spectral positions (ok) of the transitions, respectively. The relaxation operator W is a bath-averaged complex matrix which contains all the influence of collisions on the spectral shape. Within the binary collision approximation it is related to those associated with the “a’‘-“a” and “a’‘--“b” collisions for unit density by: W(o,na,b,,,T) = n, x “Wa-a(a,T) + nb x ‘Wa_b(C,T).
(2)
Prediction of a full relaxation operator including its wavenumber dependence in a large spectral range is a difficult problem. On the other hand, efficient models are available within the impact and quasi-static approximations, which enable calculations near line centers and in the far wings, respectively. 2.1.2. Near wing-impact approach. The impact approximation is valid at moderate density and when wavenumbers relatively close to the positions of transitions are considered. A criterion can be deduced from a typical value r, of the collision duration; indeed, the impact approximation is accurate for wavenumber detunings Aa which satisfy: IAal I AalMP<< (2ncr,)-’ .
In this case a number of models are available for the computation
(3) of the real part of the relaxation
tWhen the relaxation operator Wdepends on frequency, the distribution of levels changes. ‘I This effect is taken into account by renonnalizing the band surface instead of calculating the new distribution of levels.
High-density COrAr
263
gas-phase absorption
operator, which is then independent of wavenumber [simultaneously, both hyperbolic sines must be omitted in Eq. (I)]. That presented in Refs 6, 7 is used in the following; it is based on the Energy Corrected Sudden approximation and enables prediction of Re( WMp)only by assuming that CO? is a rigid rotor. Possible vibrational effects and the contribution of the imaginary elements of W are discussed at the end of this section. 2.1.3. Far wings-quasi-static approach. The quasi-static approximation is valid for wavenumbers far away from the positions of the optical transitions, which satisfy: ]Ao] 2 AaQSC>>(2nc~,)-’
(4)
The theoretical approach used in the present work is described in detail in Ref. 10. The matrix elements of Re( We”‘) are then computed directly from knowledge of the interaction potential and depend strongly on wavenumber as shown in Refs 8,9. Again, vibrational effects are neglected and only the real part of W is predicted. 2.1.4. Mid wings-interpolation model. Both the impact and quasi-static approximations break down in the intermediate region which corresponds to the mid wing of the lines. In order to predict the elements of the real part of the relaxation operator we have thus used a simple interpolation approach. Since Ref. 12 shows that the element ((k]lRe{ WQSC(a)}lll)) mainly depends on the detuning, I [a - (a, + cr,)/2] 1, we have used the following interpolation procedure in order to compute Re{ W} over the entire wavenumber region:
AalMPI (0 - 91
=
Kk IIW o-
I A~Qsc*((kIIRe(W(o))IIl))
w”‘(cr)}llI)) + a.#-
(Tk+ 0,
-- 2
~l-AdMp~+&,[~~-
?I
_A,lMPl’,
~A~QSC~((kIIRe{W(~))lll)) = ((kllRe(WoSC(a)}lJI)),
where Re( WiMp) and Re(WQSC) are operators predicted within the impact and quasi-static approximations, respectively. The a,, and b,, parameters of the intermediate region are determined in order to respect the continuity of the matrix elements of Re( W) and their first derivative versus r~for the detuning Ao Qsc The impact and quasi-static approaches are thus assumed for detunings lower than AcrIMPand greater than AcQsc, respectively, whereas interpolation is used in the intermediate range. 2.1.5. Imaginary part of W and vibrational eflects. The models presented above neglect vibrational effects and only predict the real part of W. Available measurements of the individual line shifts” [diagonal elements of Im(WMP)] indicate values of the order of -0.01 cm-’ Am-‘, which cannot be neglected. Nevertheless, prediction of the off-diagonal elements of Im( W), which are very sensitive to vibrational effects, is intractable. Fortunately, as shown in appendix A, the shift of spectra at high density is mainly dependent on the sum of the elements of Im( W). As is commonly done in the treatment of Raman Q-branch spectra,14 we have thus assumed that the imaginary part of W is diagonal and governed by an effective overall shift, AeH,i.e., Im(((kl WV))) = - Aefx 4, .
(6)
Note that appendix A shows that the effective shift in both the band wing and the central region of the band at elevated density can be identified as the purely vibrational dephasing. Vibrational effects on the real elements of Ware also very difficult to predict. Nevertheless, again, an empirical correction can be made, as for the shift and in Raman spectra,14 by adding a diagonal contribution
264
L. Ozanne et al
to Re( IV), governed by an effective broadening value, ycR.This point is discussed in the last part of the paper. 2.1.6. Binary absorption coefficients in the far wings. Within the present approach, and assuming the binary collision approximation, a linear development of Eq. (1) in the far wing leads to: -40) = n, x B,_,
+
nb
X
Bamb[a
-
&&Lnb,T),
r]
,
n,
where the density normalized absorption coefficient, B,_x, is given by:
B&a
- AenT) = $
sh($$x
{z “‘lrJ[l
-exp(
- s)]
sh Z$
x (a -
AeR- bk)(o - A,K- 6,) ’
Note that when the influence of the volume occupied B,_Ja - A,(n,,q,T),TJ should be independent of density.
(8)
by the molecules6 is negligible,
2.2. Data used The line positions and dipole-reduced elements (deduced from line intensities) have been taken from the 1992 edition of the HITRAN database.‘S*‘6 The real elements of the impact relaxation operator have been computed by using the equations and parameters given in Ref. 6 and in Table 2 of Ref. 7. The real elements of the quasi-static relaxation operator have been computed by using the equations and data of Ref. 10. The duration of the efficient collision for CO,-Ar at room temperature is about 0.2 psec, so that (27rcr,)- is of the order of 25 cm-‘. On the other hand, it has been shownI that the lower limit of the purely quasi-static approximation is of the order of 50 cm-‘. We have thus retained the values: AtY
= 10 cm-‘,
AcQsc = 70 cm-’ ,
(9)
Fig. 1 presents typical results obtained for some of the elements of Re( IV) connecting the RI6 line to others. The increase of these elements with increasing detuning is characteristic of the influence of the collision duration and has been observed previously.’ 3. EXPERIMENTAL
3.1. A set-up and treatment
The spectra were recorded on the same apparatus and following the same procedure as done in a previous study of CO,-He mixtures,6 where all experimental details can be found. Measurements have been made at room temperature (293-297 K) for total pressures in the range 100-1000 bar. High-purity CO, and Ar gases (99.998% for COZ and 99.9999% for Ar) were supplied by Air Liquide and used without further purification. Measurements have been made for adapted CO, mole fractions and about six total pressures in the range from 100 to 1000 bar (note that the absorption in the v3 band is so intense that “pure” Argon gas contained enough CO, for the study of the central part of the band; in this case, the unknown CO* density was determined from the integrated band intensity [Eq. (10) below] as done in Ref. 6). For given thermodynamic conditions (p,, pb, r) the Ar and CO2 densities n, and nb have been computed by using the equations of state given in Refs 17, 18, respectively. As is described in our previous work on C0,-He,6 a very careful error analysis was made in order to quantify experimental uncertainties. Absorption in the central regions of the v3 and 3v3 ‘V’O, bands shows contributions of some other weak bands, among which are transitions of other isotopes as well as hot transitions (such as the mv, + v2 - v2, m = integer). In order to simplify the analysis these contributions were removed from experimental spectra in the central regions as follows. Let us consider a weak band
High-density COrAr O.lO.
265
gas-phase absorption
(a)
,c----________________-_________. .
0.04 -
:
*.
0.02 -
*. ..
:
..
:
:
:
.’
I 0.00 -
I
I
I
I
,_.* *... . . _._.-.- ._._.___.; ~--~~__________---______ -.-.~__~_.________~-.-.---
-0.02 -0.03 _ -o&t-
(b) .
1 -50.
0.
1 50. Au
I 100.
1 150.
I 200.
(cm“)
Fig. 1. Wavenumber dependences of the interpolated Re(” W) elements connecting the R,, line to some others. (a) IV,,,,,,: - - -, ECS model; ., quasi-static model; -, interpolation model. (b) IV,,, R,4: - - _, ECS model; ., quasi-static model; -, interpolation model. W,,, R14:- -, interpolation model.
of central wavenumber cW and integrated intensity SWlying near the vj band, for instance; its absorption was determined by shifting experimental spectra of the quantity cW- (z,.? and multiplying it by the ratio SW/S,,. The spectrum obtained was then subtracted from the measured absorption. This procedure, which assumes that all bands have the same shape, leads to small errors since the removed bands have intensities that are,15,16at the most, about 6% of that of the main (v, or 3vJ transition. 3.2. Results A test of the experimental procedure can be made in the central regions of the bands by integrating the absorption coefficient over the spectral range of significant absorption. Indeed, one has:
(10) where alnf and asUPare wavenumbers on both sides of the band where absorption is negligible and S, are individual line intensities. The results obtained in the 3v, band, which are given in Table 1, are a first validation of our measurements. Furthermore, the agreement with the HITRAN database value and the lack of dependence on perturber density indicate the absence of any induction effect within the present experimental uncertainties. Table I. Integrated absorptions in the 3v, central region [see Eq. (IO)]. n. (Amagat) 3.25 3.27 3.24 3.24 3.21 3.26 Average HITRAN” 16 Ref. 6
nb
(Amagat) 144.6 283. I 319.2 449.9 503.3 545.5
lOO/+,,
~$$a(b,nro2,n,,,)
du (cm _’ Amagat - ‘) 4.57 4.55 4.53 4.57 4.50 4.51 4.54 (a = 0.27) 4.64 4.71 (a = 0.11)
266
L. Ozanne
8. J-* 9.
et al
cm-’
a=2520
7.-
5
m" 6.-
4. TI
-r
T
itii
0.
100.
300.
200.
500.
400.
Ar density nb (Am) Fig. 2. Experimental
Y) band wing parameter
Eco,_a,(u,n.,nb) versus argon density for three wavenumbcrs.
3.2.1. Excluded volume effects in the wing of the v3 band. The values of B,,,_,,(o) in the v3 band wing were deduced from the present measurements by using Eq. (7) removing the small C02-C02 contribution by using the data of Ref. 19; in doing this, the effective shift A,&ra,nb,T) was first determined for each density, as explained in the next paragraph. The values of BCo2_,,(a)obtained, plotted in Fig. 2, show a dependence on density that is qualitatively similar to that obtained previously for CO,-He6 and which is attributed to excluded volume effects.20.2’In order to quantify this, the values of B have been fitted, following Ref. 6, by the expression:
&-~r@ - Ac,d = B&da) x y
= B&&Y)
x 1+
kO,-A,@)
x
.
nb
(11)
1
[
The values of the “zero density” binary absorption coefficients, B&+ are given in Table 2, where Table 2. Binary
absorption
coefficients
B&,.,,, and density
10680C0,.,,(u) (cm -
(r (cm-‘) 2415 2480 2485 2490 2495 2500 2505 2510 2515 2520 2525 2530 2535 2540 2545 2550 2555 2560 2565 2570
effect parameter
’ Am - ‘)
Ref. 9
This work
32.3 k 1.6 27.5 f 1.4 23.8 & 1.2 20.0 + 1.0 17.3 f 0.90 12.5 k 0.60 11.4 + 0.60 11.1 * 0.55 9.90 f 0.50 8.45 k 0.42 6.56 f 0.33 5.43 f 0.27 4.44 f 0.22 3.84 f 0.19 3.38 f 0.17 2.97 It 0.15 2.72 k 0.14 2.32 f 0.12 1.98 f 0.99 1.80 f 0.90
34.8 f 2.1 28.1 f 2.0 22.8 f 1.4 18.5 + 1.1 15.1 * 0.83 12.4 f 0.68 10.2 + 0.58 8.46 + 0.53 6.96 f 0.48 5.73 f 0.45 4.14 f 0.43 3.95 + 0.42 3.22 f 0.40 2.65 k 0.38 2.21 + 0.36 1.81 + 0.34 1.53 f 0.33 1.27 + 0.31 1.05 + 0.29 0.86 f 0.27
cc++
[see Eq.
(I I)] in the v, band wing.
1~cco,-&) (Am - ‘1 This work 9.6 8.8 8.4 8.2 8.2 1.9 7.7 7.4 7.4 7.4 7.2 6.9 7.3 7.6 1.6 7.9 6.7 6.9
& 2.1 f 1.9 f 1.8 + 1.8 f 1.8 f 1.7 f 1.7 f 1.7 f 1.8 f 1.8 + 1.8 f 1.9 f 2.7 f 3.4 f 3.9 f 4.8 k 5.0 & 5.8 .. ...
High-density COrAr
gas-phase absorption
267
they can be compared with previous determinations. The good agreement obtained for the smallest wavenumbers where previous experiments are believed to be accurate is a further validation of our experimental procedure. The values of cCO1+,given in the same table are, as expected, practically independent of the wavenumber and equal to cCO,+ = 7.4 x 10m4f 2.0 x 10e4 Am-‘. It is worth noting that the theoretical value of cCo2+ estimated*’ from the CO,-CO, and CO,-Ar collision diameters is 12.4 x 10m4Am-‘, in satisfactory agreement with the experimental determination. In all subsequent parts of this paper, spectra computations have been made by replacing n,, by the effective density &,(n,,)= nb x [l + 7.4 x 10e4 x n,], where nb is expressed in Amagat units. 3.2.2. Vibrational shifts. The plots in Fig. 3 compare measured spectra in the 3v, band with results of computations from the impact model with AeR= 0 [Im( IV) = 01. It is clear, as expected from individual line shift valuesI that the imaginary part of W cannot be neglected. The effective shift AeKwas then determined, for each spectrum, from the wavenumber difference between the position of the measured maximum and the location of the maximum of a theoretical calculation with AeR= 0; this procedure is valid since the location of the theoretical maximum is practically independent of the model (impact or including some frequency dependence). The values obtained in the vj and 3v, bands are plotted versus corrected density fi, in Fig. 4. The linearity is satisfactory and the values of the slopes obtained from a linear fit [A&&) = (fi,, x S,,] are: &(v,) = -5.4 + 1.0 x lo--’ cm-’ Am-‘, &,(3v,) = -17.0 + 1.0 x lo-’ cm-’ Am-‘. It is worth noting that these values show the factor of three expected from their physical origin which is the purely vibrational dephasing (see appendix B). Furthermore, theoretical predictions from a semi-classical model and knowledge of the interaction potential (see appendix B) lead to results (-4.6 and - 14.0 x 10e3 cm-’ Am-’ in the v3 and 3~2,bands, respectively) which agree well with the measurements. 3.2.3. Narrowing effects. Finally, the narrowing of the vibrational bands with increasing Ar density can be revealed, as done in Ref. 22, by defining the half-width r’(fi,,) of the band through:
(12) where c,, is the band origin and ~(6~) is the perturber-density-dependent position” defined by:
“average rotational
(13)
The values of ?(E,) deduced from measured spectra are plotted corrected for excluded volume effects. They clearly show the line-mixing processes, They also indicate that computations with predict too much narrowing. The reasons for this discrepancy 4. THEORETICAL
in Fig. 5 versus perturber density narrowing effect due to efficient the present ECS impact approach are discussed in the next section.
ANALYSIS
4.1. Band wings 4.1.1. High-frequency wing of the 3v, band. A test of the theoretical models in the 3~1,wing is plotted in Fig. 6. Since the spectral region is relatively close (from 10 to 40 cm-‘) to the most intense R-lines the impact approximation is reasonable and leads to satisfactory results as assumed in Ref. 7. For the highest wavenumbers, the uncertainty of measurements prevents a definitive conclusion. 4.1.2. High-frequency wing of the v, band. Comparisons between measured and computed absorptions in the high-frequency wing of the v3 band are plotted in Fig. 7; experimental values include our results and those of Ref. 9 in the far wing beyond the band head (a > 2400 cm-‘) as well as previous measurements23 in the troughs between R-lines. Accounting for line-mixing effects within the impact approximation leads to satisfactory results in the near wing (a < 2410 cm-‘), as
268
L. Ozanne et al 3.5 3.0 2.5 2.0 1.5
-i
E
1.0
u :
.41 ‘a g
0.5 0.0
6920.0
6940.0
6980.0
6960.0
7000.0
u(cm-I) Fig. 3. Absorption coefficients in the central region of the 3vl band: 0, measured; -, calculated with the ECS model. (a) ncO, = 3.26 Am and nAr= 283.1 Am (& = 342.4 Am); (b) nco, = 3.26 Am and nAr= 545.5 Am (n-Ar= 765.6 Am).
0. -
-2. -4.-6. -8. -10. -12.1 -14. -
(b) 200.
6tiO.
corrected A%mity
8dO.
(Am)
Fig. 4. Effective shift LeRdetermined (see text) from high-pressure spectra: 0, from spectra; -, law [A&,,) = fib x a,,]. (a) v1 band; (b) 3vl band.
linear
High-density COrAr
269
gas-phase absorption
14.
8.8
10.
1000.
Fig. 5. Overall band half-widths k(&$ of the bands versus density [Eqs (12) and (13)J. Measured values values computed with the ECS impact model. in the v1 band (0) and 3vl band (0); p,
expected, but fails in the mid and far wing regions. In this region the interpolation scheme proposed in Sec. 2.1 is too crude and overestimates the absorption by a factor of about 2, while the impact scheme underestimates it by roughly the same factor. In the far wing, according to the interpolation procedure [Eqs (5),(9)], the profile is purely quasi-static and a very good agreement between computed and measured values is obtained. This was expected since the potential was adjusted” in order to obtain good agreement with the measured data of Ref. 9. Note, however, that the agreement seems to be much better with the present experimental results. In summary, even if the interpolation scheme seems too crude to give a precise description in the mid wing, it gives us a “tool” to test the influence of the finite duration of collision on the central region of the vibrational bands at very high perturber densities.
6990.
6995.
7000.
7005. u
7010.
7015.
7020.
(cm-‘)
Fig. 6. 3v, band wing parameters B&, A,.Experimental results:’ l , from St. Petersburg; 0, from Rennes. the ECS impact model; ., the impact/quasi-static interpolation model: - - -, Calculations with: -, the Lorentzian model.
210
L. Ozanne et al
t . -.
1
--__
I
2390.0 .,.,,I., 2400.0 2410.0 2420.0 2430.0. 2440.0 . 2450.0 I 2460.0
--s_
..
--S_
*.
I”‘.
o(cm-‘) ---_
--__
l;-.
\
2350.0
--__
;_
I 2400.0
”
”
1
““I”’
----__
l
2500.0
2450.0
---___
1
---
’
”
2550.0
a(cm-l) Fig. 7. v1 band wing parameters $o,.~,. Experimental results from: 0, Refs 9, 23; 0, this work. the ECS impact model; ., the impact/quasi-static interpolation model; - - -, Calculations with: p, the Lorentzian model.
4.2. Band centers
In the following, different factors which affect the absorption in the central regions of the bands are discussed. The spectra in the vj and 3v, regions, which lead to similar conclusions, are used alternately. 4.2.1. Finite duration of collision eflects. The influence of the collision duration on the central region of the 3v, band is illustrated by the plots in Fig. 8 (similar results are obtained when the vj band is considered). As expected, absorption at moderate density is practically insensitive to the frequency dependence of the relaxation matrix since absorption in the line wings is small. Even at elevated density the finite duration of collision does not seem to be of great importance. This result may be understood from the strong narrowing of the band already mentioned (see Fig. 5). Indeed, at 750 Am, the full width of the band is about 20 cm-’ which is much smaller than AgQsc. Note that these plots clearly show the great inaccuracy of the Lorentzian approach already noticed for CO,-He mixtures.6 4.2.2. Znterbranch mixing effects. The influence of the amount of interbranch (R-P) mixing on the spectral shape at elevated density has been discussed in details in Refs 67. Such effects are responsible for the filling of the trough between the R- and P-branches,22 as is demonstrated by the results of Fig. 9 in the case of the v1 band (considering the 3v, region would lead to similar conclusions). Measured spectra are compared with those predicted when multiplying the R-P couplings by 1 - &. and modifying the intrabranch (R-R and P-P) couplings in order to respect the sum rule as explained in Ref. 6. Even small values of BRPlead to significant modifications of the theoretical lineshapes. It is clear that the increase of Rt+P transfers (negative &.) fills the trough between branches and narrows the band. 4.2.3. Other e&m. A number of effects were not accounted for in the present computations. Among these are the influence of vibration on the real part of Was well as the breakdown of the binary collision approximation. Such phenomena are very difficult to model correctly but can be
High-density COT-Ar gas-phase absorption
271
empirically accounted for through the introduction of an effective broadening, ycR.Starting from the previous calculations (with &. = 0.1) satisfactory results for both the v, and 3v, bands can be obtained by using yeR= 3.6 x 10e3 cm-’ Am-’ as may be seen from Fig. 10. The physical meaning of yea remains unclear. Indeed, a first (tempting) possible attribution is vibrational dephasing (yJ since this phenomenon quantitatively explains the effective shift, AeR (Sec. 3). However, even if the calculations presented in appendix B are not very accurate owing to the uncertainties in the vibrational dependence of the isotropic potential, the fact that yVvaries with u: does not depend on these uncertainties is in strong disagreement with the independency on u3. Furthermore, a question remains about neglecting the dependence of the anisotropic of 3’eR part of the potential on the vibrational quantum number; nevertheless, since no potential including that dependence is presently available for CO,-Ar, it is not possible to test its influence on the
4. -
h ‘;
3.-
E 22
.
______--
u
(cm-‘)
u
(cm-‘)
6. ,
5.
I.
Fig. 8. Absorption coefficients in the central region of the 3v, band; 0, experimental. Calculated with: -, the ECS impact model; . ., the impact/quasi-static interpolation model; - - -, the Lorentzian model. (a) nco, = 3.25 Am and nAr= 144.6 Am (&, = 160.1 Am); (b) nco, = 3.26 Am and nA, = 545.5 Am (&, = 765.6 Am). JQSRT 5812-E
212
L. Ozanne et al
2300.
2310.
2320.
2330.
2340.
2350.
2370. 2380. 2390. 2400.
(cm-‘)
0
1.44
2360.
b)
2300. 2310. 2320. 2330. 23
)O.
u
(cm-‘)
Fig. 9. Absorption coefficients in the central region of the vj band; 0, experimental, Theoretical results with the ECS impact model and various values of the parameter PRPmodifying the R++P coupling: - -, ~RP=0.25;---,~Rp=0.1;-, b.p = 0; . ., /&P= - 0.2. (a) ttAr= 305.9 Am (AAr= 375.1 Am); (b) nAr= 527.0 Am (AAI= 732.5 Am).
elements of the relaxation operator. A number of other mechanisms can be responsible for yen which, unfortunately, cannot be quantitatively checked; among these is clearly the breakdown of the binary collision approximation. Indeed, above 350 Am, the time interval between successive collisions is smaller that of the collision duration. yea must thus be considered as an empirical parameter taking into account the insufficiencies of the theory, including, of course, the approximate approach of the impact relaxation matrix W since yeRdepends, for instance, on BRP.
Table 3. Parameters used for calculations of broadening and shifting coefficients. B, (cm - I)“. I6 CO* Ar
0.39021889
Br (cm - ‘)15.16
0.38099334
E (cm-‘)
o (A)
177.F0
4.631”” 3.505”
117.7”’
213
High-density COrAr gas-phase absorption
r (a)
0.00
6900.
6920.
6940.
6960.
6980.
7000
2300.
2320.
2340.
2360
2380
2400.
0
(cm-‘)
Fig. 10. Normalized coefficient of absorption in the central region. ., Measured values; ~ computed results with the ECS model with /I& = 0.1, including an effective width 0; 3.6 x IO-’ cm-’ Am-‘. (a) V, band and the densities tiA,= 109.3 Am, 375.1 Am, and 732.5 Am; (b) 3v, band and the densities CA,= 112.9 Am, 342.4 Am, and 765.6 Am.
REFERENCES B. and Strow, L. L., J. Chem. Phys., 1987, 86, 5722. Bonamy, L., Bonamy, J., Temkin, S., Robert, D. and Hartmann, J. M., J. Chem. Phys., 1993, 98, 3747. Rachet, F., Margottin-Maclou, M., Henry, A. and Valentin, A., J. Molec. Spectrosc., 1996, 175, 3 15. Filippov, N. N. and Tonkov, M. V., JQSRT, 1993, 50, 111. Boissoles, J., Thibault, F., Le Doucen, R., Menoux, V. and Boulet, C., J. Chem. Phys., 1994, 101,6552. Ozanne, L., Nguyen-Van-Thanh, Brodbeck, C., Bouanich, J. P., Hartmann, J. M. and Boulet, C., J. Chem. Phys., 1995, 102, 7306.
1. Gentry,
2. 3. 4. 5. 6.
L. Ozanne et al
274
7. Filippov, N. N., Bouanich, J. P., Hartmann, J. M., Ozanne, L., Boulet, C., Tonkov, M. V., Thibault, F. and Le Doucen, R., JQSRT, 1996, 55, 307. 8. Boulet, C., Boissoles, J. and Robert, D., J. Chem. Phys., 1988, 89, 625. 9. Boissoles, J., Menoux, V., Le Doucen, R., Boulet, C. and Robert, D., J. Chem. Phys., 1989, 91, 2163. 10. Ma, Q. and Tipping, R. H., J. Chem. Phys., 1994, 100, 8720. 11. Davies, R. W., Tipping, R. H. and Clough, S. A., Phys. Rev. A, 1982, 26, 3378. 12. Ma, Q., Tipping, R. H., Hartmann, J. M. and Boulet, C., J. Chem. Phys., 1995, 102, 3009. 13. Thibault, F., Boissoles, J., Le Doucen, R., Bouanich, J. P., Arcas, Ph. and Boulet, C., J. Chem. Phys., 1992, 96, 4945. 14. Lavorel, B., Millot, G., Saint-Loup, R., Berger, H., Bonamy, L., Bonamy, J. and Robert, D., J. Chem. Phys., 1990, 93, 2185. 15. Rothman, L. S., Gamache, R. R., Tipping, R. H., Rinsland, C. P., Smith, M. A. H., Benner, D. C., Malathy Devi, V., Flaud, J. M., Camy-Peyret, C., Perrin, A., Goldman, A., Massie, S. T., Brown, L. R. and Toth, R. A., JQSRT, 1992, 48, 469. 16. Rothman, L. S., Gamache, R. R., Tipping, R. H., Rinsland, C. P., Smith, M. A. H., Benner, D. C., Malathy Devi, V., Flaud, J. M., Camy-Peyret, C., Perrin, A., Goldman, A., Massie, S. T., Brown, L. R. and Toth, R. A., JQSRT, 1992, 48, 537. 17. Angus, S., Armstrong, B. and de Reuck, K. M., International Thermodynamic Tables of the Fluid State: Argon, Pergamon, Oxford, 1976. 18. Angus, S., Armstrong, B. and de Reuck, K. M., International Thermodynamic Tables of the Fluid State: Carbon Dioxide, Pergamon, Oxford, 197 1. 19. Le Doucen, R., Cousin, C., Boulet, C. and Henri, A., Appl. Opt., 1985, 24, 897. 20. Delalande, C. and Gale, G. M., J. Chem. Phys., 1980, 73, 19 18. 21. Hirschfelder, J. O., Curtiss, C. F., Byron Bird, R., Mofecular Theory of Gasesand Liquids, John Wiley & Sons Inc., New York, 1964. 22. Hartmann, J. M. and L’Haridon, F., J. Chem. Phys., 1995, 103, 6467. 23. Bulanin, M. O., Dokuchaev, A. B., Tonkov, M. V. and Filippov, N. N., JQSRT, 1984, 31, 521. 24. Bulgakov, Yu. I. and Strekalov, M. L., Opt. Spectrosc., 1994, 76, 516. 25. Monchick, L., J. Chem. Phys., 1991, 95, 5047. 26. Monchick, L., J. Chem. Phys., 1995, 102, 3009. 27. Fano, U., Phys. Rev., 1963, 131, 259. 28. Robert, D. and Bonamy, J., J. Phys. (Paris), 1974, 40, 923. 29. Garand, A., These de 3eme Cycle, Paris, 1972. 30. Bouanich, J. P., JQSRT, 1992, 47, 243. 31. Sherwood, A. E. and Prausmitz, J. M., J. Chem. Phys., 1964, 41, 429. 32. Tejeda, G., Mate, B. and Montero, S., J. Chem. Phys., 1995, 103, 568. 33. Weast, R. C., Astle, M. J. and Beyer, W. H., Handbook of Chemistry and Physics, CRC Press, Inc., Boca Raton, FL, 1985-l 986.
APPENDIX
A
Influence of the Imaginnry Part of the Relaxation Operator
Starting from Eq. (I), we consider asymptotic density situations in order to study the influence of the imaginary part of the relaxation operator on the absorption coefficient. At very high densities,when the real elements of the relaxation operator are much larger than the spreading of the optical transitions, it is known that the whole vibrational band becomes symmetric with a Lorentzian shape; its effective shift A*,, (with respect to the central wavenumber of the unperturbed band) is then exactly the vibrational shift, A,, since one has
AcK= -T 2 ((~~lm(W)~k)) = A,
(Al)
At moderate densities, we consider now a perturber density such that the P- and R-branches remains relatively distinct but with a complete overlapping of the R- (or P-) lines. The shift of the maximum of the R-branch, for instance, is then given by:
(I&R) - x$’ ((rlIm(W)lk)) = I$’ ((~IImtWllk)) + A,.
ER !.
EP A
W)
The last sum, which involves R++P couplings, is expected to be small according to Refs 25,26; furthermore, when applied to the real elements of W, it is negligible since the most populated levels of CO? correspond to relatively high J values for which interbranch mixing vanishes. Eq. (A2) therefore shows that the effective shift A.#,of the R- (or P-) branch maximum is also given, with a good approximation, by A.,.
275
High-density COI-Ar gas-phase absorption 0.10
0.08
Lh.__
h ‘; 2 ‘;E S 2
0.06
l l
.
0.04
0
-10
-20
-40
-30
m Fig. Al. Ar-broadening
coefficients y0 for CO, lines at 296 K in the 3v, band. 0, Measured values;” -. calculated results
-0.006
-0.008
h ‘;<
-0.010
7 E 2 u?
-0.012
-0.014
-0.016 1-
0
-20
m Fig. A2. Argon-shifting coefficients Sofor CO* lines at 296 K in the P-branch of the 3vr band. 0. Measured values;‘J -, calculated results.
Absorption
in the near wing at high density
Following the method proposed by Fano ,I7 the operator in Eq. (1) may be written up to the second order in a density expansion as: 1
~_Lo_iw
-- [ 1 - z_r,
I-iW=
1
+(-iW&(-iW&
+...
1.
(A3)
where only the imaginary part must be retained. The only contribution of Im( IV) then comes from the last term and is given by: (A4)
where
*dd = -T
$E
(
(A3
276
L. Ozanne et al
Since ((nlIm(?V)lk)) is expected to vanish for lines n and k far apart, the small frequency dependence in Eq. (AS) can be neglected. b(a) is then equal to the purely vibrational shift, A, [Eq. (Al)]. Eq. (A4) is then identical to the second-order term of the development of the operator: 1 Z-&-iW
z Z - (&I- A#)
+... '-iRe(mz _(Lo'_ A,Jd) 1
C
Comparison between Eq. (A6) and Eqs (7) and (8) shows that the effective shift AcRin the far wing can again be approximated by the purely vibrational value, A,. These three characteristic situations show that the present experiments are not sensitive to the details of the imaginary part of W and justify, in some way, the approximate treatment used in the present work.
APPENDIX
B
Widths and Shifrs in COrAr
Within the semi-classical model developed by Robert and Bonamy,‘* the broadening (uO)and shifting (15~) coefficients for an isolated Lorenttian line are given by: y0 _ id, = Z!! z 2nb S’(b) - iSs(b) 2nc o s [
1
db ,
@I)
where N is the molecule density of the perturbing gas for 1 Am, Vis the mean relative velocity, and b is the impact parameter. By neglecting weak rotational contributions to the shifts, the differential broadening and shifting cross-sections, P(b) and P(b), are given by S’(b) = 1 - exp[ - Rep(b)] P(b) = exp[ - ReSy(b)]
cos[Sy(b)]
,
WI
sin[$“(b)],
where Sp is the vibrational phase shift arising from the difference of the isotropic part of the potential in the initial and final states of the transition; the real part of SF is derived from the anisotropic part of the potential. When the vibrational dependence of the anisotropic potential is neglected, one can show that?
where the pure vibrational width and shift are given by
Since we are only interested here in the order of magnitude of yVand a,, the CO?-Ar potential has been simply represented by a two-term expansion of Legendre polynomials such that:
where r is the intermolecular distance between the perturbers, 6, is the angle between the axis of the absorber and the intermolecular axis, and P2 is a second-order Legendre polynomial; the Lennard-Jones (LJ) parameters E and Q for CO?-Ar have been determined from those for CO,CO, and Ar-Ar and the usual combination rules; Rz and Al have been considered as adjustable parameters in the fit of experimental CO?--Ar broadening coefficients. The parameters used in the computation are listed in Table 3. Figure Al shows that they lead to satisfactory agreement with widths measured by Thibault et al’) in the 3v3 band. The isotropic part of the potential in Eq. (BS) is a 612 LJ potential given by
with C, = 4s~~ and C,, = 4.s&. Assuming that its vibrational dependence arises from the polarizability matrix elements a, of CO* leads to: 1Ic, Aal c, =7.
(B6)
The value of Aa,/a, was first calculated by using the polar&ability derivatives for CO: given in Ref. 32: da,/Sq, = 0.1 I I7 A’, a’a,/aq: = 0.00404 A-l, ahjaq: = 0.00135 A’, a mean polarizability (a, = 2.911 A’)ji and Aq, = (odq,lvr) - (uilqllv,) = 0.5438 for the 3vr band;z9 this calculation leads to AC,/C, = 0.0218. By considering in addition’> y _ AC,1 AG
CI?
C,
= 1.69,
the computed lineshifts are about two times larger than the experimental data given in Ref. 13.
High-density COrAr
gas-phase absorption
217
Owing to the approximate nature of Eq. (B6), we have considered AG,/C, and _Vas adjustable parameters. The values ACJC, = 0.010 and y = 1.2provide lineshift coefficients for the 3v, band in satisfactory agreement with the experimental data, as shown in Fig. A2. The corresponding pure vibrational broadening is very weak, y,(3v,) = 0.79 x IO- 3cm - ’ Am - I, whereas the vibrational lineshift value is 6,(3v,) = - 14.0 x IO-’ cm-’ Am- ‘, in reasonable agreement with the effective shift &&3vJ = - 17.0 x IO-’ cm-’ Am- ’ derived from high-pressure measurements in the 3v, band (see text). Calculations in the 11) band were then made, considering (hC,/C,),,,/(AC,/C,),,, = (Aa,/a,),,/(A~,/ol,)i,,, leading to (ACJC,), , = 0.0032. This value associated with v = 1.20 leads to v,(v,) and ,.~ -I = 0.081 x IO-’ cm-’ Am-’ a,(~,) = 14.6 x 10-l cm-’ Am-‘, again in reasonable agreement effective shift with the experimental 6,,dvl) = - 5.4 x IO-’ cm- ’ Am- ‘. Finally, let us recall thaP Aq, is proportional to v,; since a second-order expansion of the exponential function is valid in Eq. (B4), it leads to &av, and ~,avv,.z The above numerical results and the experimental values of &Asupport these conclusions.