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Line mixing effects in the Q branch of the 1000 ← 0110 transition of CO2
IOURNAL
OF MOLECULAR
SPECTROSCOPY
138, 141-161
(1989)
Line Mixing Effects in the Q Branch of the 10’0 + 01’0 Transition of CO2 THIERRY
HUET,
NELLY
LACOME,
AND
ARMAND
LEVY
I
Laborutoire d ‘Infrarouge, associP au CNRS, Bat. 350. Universite’ de Paris-Sud. 91405 Orsay Cede.y. France Line mixing effects have been investigated in the Q branch of CO2 near 720 cm-‘, by means of a tunable diode laser. Spectra were obtained at room temperature, in the lOO- 1000 torr pressure range. In the high-frequency wing of the Q branch, below the Q(2) line, the observed absorption is nearly 50% less than that calculated on the assumption of purely additive Lorentz lines, whereas the Q branch itself is nearly insensitive to line coupling in the pressure range investigated. From the line broadening coefficients alone, a simple scaling law allows an estimate of the state-to-state rotationally inelastic transfer rates. Calculation of line mixing coefficients using these rates yields corrected absorption coefficients, in very good agreement with the observed ones, all over the analyzed spectral range. The results appear to be of importance for modeling outgoing radiances in temperature sounding channels near 14 @m. 0 1989 Academic Press, hc. I. INTRODUCTION
At moderate pressures, the vibration-rotation spectra of gaseous systems are described, in the vast majority of cases, as sums of isolated Lorentzian lineshapes. As pressure is increased, line overlapping creates interference effects resulting in an intensity redistribution among lines. This interference phenomenon, usually denoted as line mixing or line coupling, may thus produce significant deviations from directly additive individual line profiles. The principle of this coupling has long been identified in the theory of spectral shape of collision-broadened lines. Its theoretical description was developed, within the frame of the impact approximation, by Baranger ( 1)) Kolb and Griem (2) and later by Ben-Reuven (3) as an application of the general relaxation model introduced by Fano (4). However, very little experimental work was carried out on this subject until recent years, with the exception of a few low-resolution IR (5, 6) and Raman (7) studies. This can be easily understood, since line-mixing effects become significant only when two conditions are fulfilled: (i) the line spacing must be comparable to the collisional width y and (ii) the separation between rotational energy levels must be comparable to kT. In Q branches, with closely spaced lines, the first condition is most often met; and it is easily seen that, for systems like CO*, the rotationally inelastic collision frequency becomes of the same magnitude as the vibration-rotation interaction even for pressures near or below 1 atm. This would thus allow easier experimental observation of the influence of line interferences on the spectrum. Unfortunately, the com’ Present address: Laboratoire de spectrochimie VI, 4 place Jussieu 75230 Paris cedex 05. France.
moleculaire,
141
associe au CNRS.
0022-2852/89 Copyright 0
Bat.F, Universitt
Paris
$3.00
1989 by Academic Press. Inc.
All rights of reproduction in any form reserved.
142
HUET,
LACOME,
AND
LEVY
pactness of Q-spectral structures makes it difficult to derive accurate values of individual line parameters (line intensities, collision-broadened widths, and absorption coefficients). Only in the last decade have such measurements become more possible with the advent of high-resolution instruments: tunable diode lasers, Fourier transform spectrometers, difference-frequency laser systems, etc. In recent years, rotational and vibrational Raman spectra of N2 have received considerable attention (8-14) owing to their usefulness for diagnostic measurement of temperatures and pressures in harsh environmental conditions (flames, combustion engines, shock tubes, etc.). Several other simple systems such as CO ( 1 la), NO ( 12, IS), CO* and N20 (16)) and C2H4 (17)have also been the subject of recent investigations by Raman techniques. Continued effort has been focused on CO* infrared spectra because of their importance in atmospheric transmission. Braun (18) and Armstrong (19), starting from slightly different formalisms, have developed calculations incorporating line mixing in the u2 Q branch. They both predicted that significant effects should be observed in the 15-pm spectral region for pressures representative of atmospheric conditions. The far wings of lines also can be sensitive to the influence of line coupling. Bulanin et al. (20) investigated the contribution of line coupling to far wing absorption by studying the v3 band of CO2 (along with ~3in N20 and the fundamental in CO). The observed deviations of the band shape from Lorentzian behavior were accounted for in terms of intensity redistribution due to the interferences. Cousin et al. (21) emphasized on the various mechanisms involved in the observed absorption in the microwindows between lines and discussed the influence of the rotational distribution of line coupling. The first experimental investigation of line mixing for an IR (2 branch has been reported recently by Strow and Gentry (22). For the combination transition centered near 1932 cm-‘, a strongly sub-Lorentzian absorption in the low-frequency wing of the Q branch in self-broadened CO2 was clearly pointed out. Following the formalism previously used in Raman by Rosasco et al. ( 1 I ) , these authors modeled the collisional transfer rates by means of a simple energy gap law. A similar study for N2-broadened CO2 Q branches was also recently reported (2.3). An extension has also been made to the case of N,O, which exhibits the pecularity of I-type doubling in the upper vibrational level (24). The accurate calculation of CO2 absorption in spectral regions which involve the presence of Q branches is of prime importance for atmospheric studies. In particular, satellite infrared radiometers utilize the emission of CO2 in the 14- 15 /*rn region for the retrieval of atmospheric temperature profiles. It is now well established that several of the temperature sounding channels must be affected to some degree by line mixing (25). Account should therefore be taken of such effects when the corresponding radiances in atmospheric models are modeled. Otherwise, noticeable errors could result in the retrieved temperatures. The main goal of the present work is to investigate the effect of line mixing within the spectral range corresponding to one of these temperature sounding channels, centered near 720 cm-’ . This channel precisely covers the Q branch of the 10'0f 0 I'0 transition of “C r602. As clearly shown by Chedin and Scott (25), both the NESS (26) and 4A (27) models, when applied to this spectral region, yield computed radiances
LINE
MIXING
EFFECTS
IN CO1
143
that depart noticeably from the true values, the deviations being, for both, of similar magnitude and the same sign. Since the two models are based upon quite different mathematical approaches, the discrepancies may presumably be ascribed to the insufficient quality of the spectroscopic information used in the models rather than to the models themselves. In particular, neglect of fine effects such as interferences between lines could be responsible, at least partly, for the difficulty encountered in modeling correctly the outgoing atmospheric radiances. Another objective of our work is to compare our results with those derived for other CO2 Q branches. It is of importance to check to what extent the parameters involved in the fitting laws so far used for estimating transfer rates could be dependent on the transition considered, more especially on the symmetry of initial and final vibrational levels involved. After the experiments are described and the results obtained are discussed (Section II), the theory will be briefly recalled in Section III. The results of calculations are given in Section IV. We shall conclude with a discussion of some problems requiring further investigation. II. EXPERIMENT I.
Experimental Details and Data Collection
In a recent paper (28), we reported tunable diode laser measurements of individual intensities and collisional widths for 18 lines in the Q branch of the 10’0 + 01 ‘0 transition of COz. J values ranging from 2 to 48 were investigated in the O-20 Torr pressure range. The results obtained are utilized here for studying the evolution of absorption in the region centered near 720 cm-’ when pressure is increased and line coupling is manifest in the spectrum. All spectra were taken with a tunable diode laser spectrometer (Spectra Physics SP 5080) working in dual beam mounting. The device has been described elsewhere (28). We recall here only the main features of the operating mode. For the present experiments, the Ge etalon was replaced by an optical cell of 10 cm length containing a fraction of a Torr of COz . Two signals were recorded simultaneously: ( 1) the reference signal, consisting of a small number of well isolated lines which were used to frequency calibrate the spectrum. This signal served also to check the emission of the diode for stability; and (2) the sample signal, giving the absorption
to be studied as a function
of pressure.
The spectrum was scanned by successive narrow segments extending typically over 0.080-0.120 cm-‘, which is approximately the spectral separation of lines in this Q branch. Systematic care was taken to see that the successive portions of recorded spectra overlapped each other. Comparison of data for the spectral interval common to two adjacent recordings allowed us to check the reproducibility of the diode emission and to detect possible short-term drifts. The purity of laser modes was monitored by saturating the absorption to ensure that the observed saturation level was identical to the null transmission signal obtained when the sample beam was blocked. In a small number of cases, some weak parasitic modes remained uneliminated, which resulted thus in departures from the blocked
HUET, LACOME,
144
AND
L&Y
beam zero reaching sometimes a few percent. This was mainly the case in the range extending from Q( 8) to Q( 18). The corresponding settings of the diode were therefore discarded. Optical pathlengths of the samples were chosen to achieve absorptances of 50-60% in each portion of the spectrum. As was done previously for individual line measurements, all spectra were taken in the slow scan mode. For each spectral segment under study, 10 to 15 records were made for a series of increasing values of pressure in the range 1OO- 1000 Tort-. Absolute absorptions were determined by ratioing all scans to zero absorption background in order to remove the influence of the varying 100% transmission level. For this purpose, we applied the following procedure. Let S, ( a) and &( U) be respectively the reference signal and the sample signal, recorded when both cells are empty. r(a) denotes the ratio of &( a) to S, ( a). For a given setting of the diode, it is experimentally observed that the value taken by I( a) at a given wavenumber u remains constant even when the emission level of the diode varies. Thus for each setting of the diode, it can be considered that r(u) is negligibly affected by fluctuations of current. Accordingly, data were collected as follows. Prior to the run, S, ( u) and &( u) were recorded several times, then averaged, and the function r(u) determined within the spectral interval under study for the considered setting-up of the system. Then, the sample cell was filled to the desired pressure of COz and signals S’, and Sb were obtained. The transmitted intensity Zt( a) is then given by the sample signal S$( a), whilst the zero absorption is ZO(a) = S;(u) - r(u). Thus, the transmission is given by T(u) = S;(U)/[S;(~~)* r(u)]. All measurements were made with high purity (99.998%) CO2 and natural isotopic composition was assumed. Gas pressures were measured to better than 0.5% accuracy with a Datametrics capacitance gauge. Temperature of samples was continuously monitored with a platinum sonde. During the time interval required to make the lo12 different scans for each given segment of spectrum, the observed temperature variation never exceeded a few tenths of a degree. The spectra, obtained in the form of graphical records, were then digitized and the resulting data stored for subsequent processing.
2. Data Reduction The great number of raw data obtained at different pressures and temperatures were put in usable form by determining the values of the absorption coefficient at a given set of wavenumbers and pressures. First, for a given scan, recorded at working pressure pw, an interpolation procedure vs wavenumbers yielded adjusted values of the absorption coefficient for a number of previously fixed test frequencies Us. Then, for each one of these reference frequencies, all data corresponding to the various experimental pressures investigated were least-squares fitted vs. pw values. Smoothed values of the absolute absorption coefficient K( at) were thus derived for a series of test pressures pt, The process can be’schematized as follows:
mu,Pw,
T) * K(u*, Pw, T) * K(% Pt, 73.
LINE
MIXING
EFFECTS
145
IN CO2
3. Calculation of the Reference Profile The reference value of the absorption coefficient K(c) at wavenumber u is given by the simple summation of noninteracting purely Lorentzian lines. In the present case, lines from all the vibrational transitions lying in the spectral range 650-750 cm -’ were included in the calculation. These transitions are listed in Table I, while Table II summarizes all spectroscopic parameters needed for calculating line positions. For the Q branch of the 10’0 + 0 I ‘0 transition studied here, we adopted the values of individual line intensities and self-broadening parameters recently determined (28). For all other nearby transitions, line intensities were calculated from the values of vibrational intensities given in the AFGL compilation (29). Since all the vibrational transitions taken into account are perpendicular-type bands, we considered it inappropriate to take values of the broadening coefficients determined on parallel bands, as was discussed previously (28, 30). We therefore performed a series of new measurements of self-broadening coefficients for a number of perpendicular bands of COZ in the 4.8-5 pm region (31). Several P and R branches were investigated by very highresolution Fourier transform spectroscopy. The values so obtained appear systematically lower by 6-B% than the corresponding parameters for parallel bands. Therefore, we adopted these new determinations for all transitions of Table I, including the P and R branches of the 10’0 + 01 ‘0 transition. It may be asked whether it is really consistent to adopt for the collisional widths Q-branch values different from those taken for P and R lines. A similar situation has been recently reported by Lempert et al. ( 1.5). These authors carried out direct measurements of line broadening in the fundamental Q branch of NO and found values 10% lower than derived from Rbranch data (32). So they evoked, as one plausible explanation, the possibility of a small difference in Q-branch vs R-branch broadening. This question would thus deserve systematic investigation.
TABLE I Vibrational
Transitions
Considered
for Calculating
Band Center ( cm.-’ )
Transition
Ol’Ot
OO”0
02?0 toll0 11’0 t lo”0 t
IO”0 01’0
11’0 t0220 10°0t
01’0
the Absorption
near 720 cm-’ Dipole moment ( Debye )
667.380
0.1804
667.752
0.2560
-
688.671
0.2203
-
720.805
0.1191
-
741.724
0.1360
"PO,
721.584
0.1191
12c’60, -
L
Nore. For lo”0 + 01’0 transitions, dipole moment moment calculated from intensities of Ref. (29).
from Ref. (28).
For all other transitions,
dipole
146
HUET, LACOME, AND LI%Y TABLE II Spectroscopic Constants Required for Calculating Line Frequencies vibrational level
Isotope
‘ee60z
1
YYO,
I I
oo”O
01’0 lo”0
vibrational energy (cm-’ )
1
I 1
0
BC (cm’) 1 0.39021817
648.4784 1370.0626
I 1
B,
(cm’)
D,
DE (
10.’ cm-’
; ( 10“
cm-’
% --I-
HF
lo-”
cm-’
0.055
( 10.” cd
1
0.055
0.39021817
1.33204
0.39125388
1.35133
1.35900
0
0
0.39018823
1.14801
1.14801
1.846
1.846
0.39133321
1.2579
1.20968
0.960
0.442
0.39166614
1.3735
1.37919
-3.560
0.076
1.33204
0.39061023
0.39124371
1.34973
1.35754
0
0
0.38971761
0.38971761
1.20137
1.20137
2.083
2.083
Note. All values from Ref. (29).
Regarding 13C02, the broadening parameters were taken equal to those of ‘*CO2 and the transition moment of the 10’0 + 0 1’0 transition was assumed to be identical to the corresponding one in 12C02. To reduce computational time, we retained only those contributions to the absorption coefficient greater than an arbitrary cutoff value of IO-’ to 10e6 cm-‘, depending on the pressures. We estimate the uncertainties on calculated absorption coefficients to be approximately 6%, on account of uncertainties on the spectroscopic data introduced in the calculation. Before any conclusion was derived from the comparison of observed and “Lorentzian” spectra, a preliminary analysis of the relative magnitude of the various contributions to the absorption was attempted for some characteristic regions of the (2 branch. This is shown in Fig. 1: -in the range of higher J-values, for example near Q( 44)-Q(46), the Q branch itself contributes 70-80%. Due to its proximity, P( 3) provides lo-20% of the total absorption. This contribution however comes mainly from the core of the line (the line center falls between Q( 46) and Q( 48 )) and it is therefore correctly described by a Lorentz shape. All the remainder contributions amount to less than 8-10%. nearly
-near Q( 34)-Q( 95%. ’
32), the Q branch
dominates
-for lower J-values (J i 20), contributions itself become smaller than 2-3%.
as expected
and it contributes
other than that of the Q branch
Beyond the Q branch, in the wing of low J-value lines, the situation appears quite different. Three main contributions must be considered: (i) the absorption due to the wings of the Q-branch lines amounts to 70-80%; (ii) line R( 66) of the u2 band, centered at 72 1.4 cm-‘, also contributes significantly (near the line center, it represents 25 30% of the total absorption, but the corresponding contribution is known to be strictly Lorentzian; furthermore, at less than 0.3 cm -’ from the line center, it becomes almost negligible); (iii) the third contribution originates from the Q branch of the isotopic
)
LINE
MIXING
147
IN CO2
(A)
1
0.20
EFFECTS
K(o) (cm-‘)
0.15
total Q branch
0.10
P branch
other contributions
I
I
I
I
I
I
718.60
718.70
718.65
I
o(cm-‘)
718.75
t
t
Q(a)
Q(a)
(B)
K(o) (cm-‘)
0.08
0.06
0.04
branch 0.02
I
I
1
721.3
FIG. 1. Comparison cm-‘.
ofthe different contributions
I
721.4
to the absorption
I 721.5
a(cm-‘)
coefficient near 7 18.7 cm -’ and 72
1.21
transition. In particular, the spectral range 72 1.2-72 1.6 cm-’ investigated here spans the range from Q( 2) to Q( 14) so that line mixing should also be considered for these isotopic lines. However, as will be seen later, this range of strong Q lines is quite
HUET, LACOME, AND LfiVY
148
negligibly affected by line coupling effects. Even if these were explicitly taken into account for this branch, the corresponding correction would be of very small influence. Finally, it appears that the Q branch considered is particularly suitable for studying interference effects since all “foreign” contributions either remain of very small (if not negligible) magnitude or they result in strictly Lorentzian absorptions. The possible influence of sub-Lorentzian behavior of the far wings from remote lines can therefore be ignored. Thus, observed deviations from the reference profile will be safely attributed to line coupling. 4. Comparison of the Observed Absorption to the Lorentz Model Calculated are displayed
and observed values of the absorption in Fig. 2 for three typical cases.
coefficient as a function
of pressure
-At 718.79 cm-‘, between Q(44) and Q(46), the observed value of K(a) is systematically lower than the calculated one. However, the departure stands at the limit of uncertainties as shown by the error bars. This indicates that the region of higher J-values probably exhibits sub-Lorentzian behavior. But this requires further confirmation from investigations at higher pressures. -At 719.83 cm-‘, near Q(30), i.e., in the central region of the Q branch, no difference can be detected between the Lorentzian and the observed absorption. This region is insensitive to line coupling for pressures up to 1 atm. -At 72 1.4 cm-‘, beyond the Q branch, a strong departure creasing magnitude as pressure increases. The observed spectrum Lorentzian and the deviation exceeds 40% at 900 Torr.
is observed, of inis noticeably sub-
The present results are very similar to those reported by Strow and Gentry (22, 23) for the 11’0 + 00’0 transition near 5 pm. They corroborate the conclusion that the main effect of line coupling is located primarily in the low-Jwing beyond the Q branch when pressure values range from a few hundreds of torr to 1 atm. The overall narrowing of the whole Q branch would become apparent only at much higher pressures where the expected enhancement of the absorption at the center of the branch should be detectable to a certain extent. The remaining problem is now to attempt to account for all the above observed effects by means of an adequate line mixing model.
III. THEORY
1. Line Mixing Model Within the framework is expressed as (33-35)
of the impact approximation,
the absorption
coefficient
K(u)
149
LINE MIXING EFFECTSIN COz ( A ) :
0.16 0.14 0.12 0.10 0.08 200
0
400
600
800
Iwo
( B ) : (3 = 719.83 cm -’ ( near the Q(30) line ) 0.8
0.6 0.5 0.4 0.3 200
0
400
600
800
1000
( C ) : o = 721.38 cm -’ ( Beyond the Q(2) line )
0.16
.i
0.14
)
0.12 0.10 0.08 0.06 0.04 0.02
, 0
FIG. 2. Observed experiment 2; 4
i 200
and calculated absorption = Lorentz profile.
I 400
coefficients
pressure(=) I 800
I 600
versus pressure.
-<‘-
= experiment
, 1000
1: -o-
150
HUET,
LACOME,
AND LfiVY
where LTis the wavenumber (cm ), c the velocity of light, and 12the number density of absorber molecules. I j) and I k) are doubled state vectors denoting spectral transitions. In the present case, for example, I k) = I qZ)f, JJ) denotes the Q line ViJ + vfJ and I j) = 1ViVf, J’s) corresponds to ViJ’ + VfJ’. pk is given by a normalized Boltzmann distribution over the initial state population associated with the radiative transition I k) , dk and dj are the reduced matrix elements (as defined in Ref. (36)), (jlalk) = a6jk, and (jla,,lk) = kajk, where k is the vibration-rotation wavenumber of the radiative transition Ij) of the molecule. The diagonal elements W,, of the relaxation matrix are the broadening coefficients -rk while the off-diagonal terms (j I W 1k) = Wjk are the line-coupling parameters (as defined in Refs. 37 and 38). The effect of the off-diagonal elements Wjkcan be neglected for well separated lines and the absorption coefficient is given simply as the summation over all contributing Lorentz line profiles. In the case of a Q branch, these elements Wjk cannot be ignored, even for low pressures. Equation ( 1) takes into account line interferences but ignores the finite duration of collisions (impact approximation) which is known to affect both diagonal and offdiagonal elements of W (39). However, as long as the reciprocal of the collision time T, is larger than the separation between the centers of lines under consideration or larger than the distance between the line center and the wavenumber at which K(c) is calculated, this effect is negligible. For the collisional system C02-C02, the impact parameter b0 can be evaluated to 3 A, giving a collision time equal to about 0.6 ps. So the impact approximation should be valid over a distance of 9 cm-’ from line centers. This justifies the use of Eq. ( 1) over the frequency range spanned by the Qbranch lines. Accordingly, the only restriction introduced in our calculation was that only lines of the Q branch were allowed to mix and no mixing of Q- with P- and Rbranch lines was considered. An a priori calculation of the coefficients Wj, requires accurate knowledge of the intermolecular potential surface. For a collisional system like CO2-CO2 this is not available. Moreover, a calculation at each wavenumber would lead to a formidable computational task. To circumvent this difficulty, Strow and Gentry (22, 23) transposed to infrared absorption the method previously introduced in isotropic Raman scattering by Rosasco et al. (11, 1.5) for studying line mixing effects. In Raman spectroscopy, the off-diagonal elements W,, can be equated to - 1 times the rate at which collisions transfer amplitude from line k to linej (if elastic reorientation collisions do not contribute significantly to the widths). In other words, the coefficient W,, that ensures coupling of the Q(J) = line k to the Q( J’) = line j is assumed to be identical to the inelastic rotational transfer rate between levels J and J’, Wj, = -K(J+
J’).
(2)
This relation strictly holds for isotropic Raman spectra when the tensor responsible for the coupling of radiation and matter is of order zero. It becomes only approximate in infrared absorption (tensor of order one). The resulting approximation was discussed by Cousin et al. (21) who showed that such an assumption remains reasonably valid for the case of CO*.
LINE MIXING
Therefore,
the pressure-broadened -rk=
;[
half-width
K(J +
2 Jk,+
EFFECTS IN CO2
151
for a single line is written
J’) +
c
K(J-+
as (40, 41)
J’)],
(3a)
J&per+ J
J
where Yk is the pressure-broadening coefficient of line Q(J) and K(J + J’) is the collision rate of rotational transfer from J to J’, within a single vibrational state. In their recent study, Strow and Gentry (22, 23) suggested the introduction of a factor of 2 in the summation over the Z vibrational level, in order to reflect the fact that the density of rotational states is one half that of a II level. Applied to our transition, this leads to Yk=
I[
c
--*
K(J
Jk&
J’) + 2
c J&r+
J
K(J+
J’)].
(3b)
J
The introduction of such a factor is essentially empirical and clearly indicates the degree of approximation that is made in infrared absorption when one assumes the coefficients Wj, to be identical to inelastic rotational transfer rates. According to these authors, quantitive agreement with observed absorption cannot be obtained unless this factor is taken into account. In the present work, we performed both calculations, with and without this factor. The inelastic transfer rates K( J + J’) are also constrained to satisfy the detailed balance, through the usual relationship, pJ” K(J’ + J) = pJ- K(J --) J’), for an energetically downward transition (J’ < J). The approach adopted by most authors for estimating the off-diagonal elements, w;k, consists in making use of a scaling law for deriving the inelastic rotational rates for upward state to state transitions in each vibrational state. A detailed balance then gives the downward rates and an immediate identification yields the coefficients. We adopted in the present work the exponential power-gap law (E.P.G.L.) (42) wjk
K(J_,
J’)
=
a(!+~exp(-@k$dc)
(4)
where Elk = Ek - Ej = EJ - EJ!,
B is the rotational
constant
a, 6, c are adjustable
introduced
to get dimensionless
quantities,
and
parameters.
We also attempted to utilize either an exponential-gap law or a simple power-gap law. In the first case, only high J-values were correctly modeled, while the power-gap law yielded a correct fit only for the lower J-values, as was already noticed by Rosasco et
al. (11, 15). 2. Practical Calculation In the case of a completely overlapped COz Q branch, due to the great number of lines to be considered, inversion of the matrix given in Eq. ( 1) for each wavenumber under study would be excessively time-consuming. For instance, in our calculations, we considered elements in the relaxation matrix up to J = 50. In order to avoid such
HUET,
152
LACOME,
AND
L6VY
successive inversions at different wavenumbers, we follow the procedure discussed by Koszykowski et al. (43) and Gordon and McGinnis (44) and applied to the infrared by Strow and Gentry (22, 23). For this purpose, the expression of K( a) can be written as a function of the matrix product, K(a)
= MIm(d*.(a
- H)-‘-p-d),
(5)
where p is the diagonal matrix whose elements are the Pk previously introduced in relation ( I ) , d is the electric dipole moment vector (superscript t denotes its transpose), and Q is the product of the unit matrix and the wavenumber u. The H matrix is independent of (r and is defined as H = a0 + inW,
(6)
where u. is the diagonal matrix of the transition wavenumbers. For a given wavenumber, A4 is a constant which contains all other factors appearing in Eq. ( 1). Let A be the matrix which diagonalizes H. Then A-‘HA = L - A will also diagonalize u - H, which only differs from H by a constant added to the diagonal terms. The resulting expression for the absorption coefficient is given by K(u)
=
(d*A)i(A-‘*P*d)i
MC
’
(a-li)
i
where the I;‘s are the diagonal elements of matrix L. Therefore the practical calculation was done as follows: from the values of the linewidths, the three coefficients by performing a least square fit to relation (3a) or (3b); from these parameters,
all the elements
a, b, and c were calculated
wjk were obtained
the H matrix was then calculated and diagonalized absorption coefficient K(u) was computed for every
by using relation
2;
for each pressure and the required wavenumber via
Eq. (7). IV. RESULTS 1.
OF CALCULATIONS
Comparison of Experimental and Calculated Spectra
Figure 3 and Figure 4 give plots of the absorption spectral regions:
coefficient
K(u)
for three typical
(a) for the higher J-values, for example in the interval Q( 32)~Q( 34), the observed spectrum is correctly reproduced by the Lorentz model. It is important to notice that no difference results in the calculated spectrum when line mixing terms are included (Fig. 3a). Thus, in the pressure range investigated here, the absorption appears as strictly Lorentzian in the higher-J wing of the Q branch. (b) in the central region, near Q( 12)~Q( 14), the observed spectrum exhibits an absorption slightly enhanced as compared to Lorentz (Fig. 3b). However, the error bars show that the deviation remains at the limits of uncertainties, so that the present data suggest only a trend toward super-Lorentzian behavior in the center of the Q branch (as expected), but this should be corroborated by investigating higher pressures.
LINE
MIXING
EFFECTS
(A) 0.60
IN CO2
:Q(W-QQW
K(cti’)
0.55
0.50
0.45
0.40
0.35
(me’)
(T
0.30 119.54
719.59
719.64
(B) : Q (12) 2.50
-
719.69
Q (14)
K(c&
2.25
2.00
1.I5
1.50
1.25
0 1.00
I 720.56
720.58
I 720.60
I 720.62
I 720.64
(cm-‘) I 720.66
FIG. 3. Experimental and calculated absorption coefficients in two regions of self-broadened Q branch. The pressure is 700 Torr. t = experimental; X = Lorentz profile; A = line-mixing model, using relation (3a); o = line-mixing model, using relation (3b).
The line mixing model qualitatively accounts for a super-Lorentzian it overestimates the deviation (both calculations with and without similar results but the former lies closer to experimental data).
absorption, but the factor 2 yield
(c) the most spectacular result is obtained in the wing beyond the Q( 2) line where the magnitude of deviations between observed and Lorentz absorptions was quite large. Figure 4 gives a plot of the absorption in this region of the spectrum (which includes the R(66) line of the u2 band). It can easily be seen that a substantial improvement is obtained in the calculated spectrum by introducing line-mixing terms and that the remaining deviations are much smaller than the magnitude of experimental
HUET, LACOME, AND L&Y
154 _
K(o)
(cm-’1
0.12
0.10
0.08
0.06
0.04 0
0.02 721.:2
721.3
721.4
( observed - calculated ) / observed
( cti’ ) I
I
I
I
721.6
721.5
(%)
20 0 -20 -40 -60 -80 0 -100
! 721.2
I 721.3
t 721.4
I 721.5
(cm-‘) I 721.6
FIG. 4. Experimental and calculated absorption coefficients in the region beyond the Q( 2) line. The percent deviation of the Lorentz and line-mixing calculations from the observed absorption coefficient is shown below this plot. t = experimentak X = Lorentz profile; A = line-mixing model, using relation (3a); o = line-mixing model, using relation (3b).
uncertainties. Nevertheless, evidence for the necessity of introducing a factor of 2 in the relation between half-widths and inelastic rates cannot be clearly pointed out from our experiments. The two calculations, performed using relations (3a) or ( 3b), give results on both sides of the experimental curve. Thus, one calculation overestimates the correction while the second underestimates it. This, once again, shows the empirical character of the factor 2 and the need for further theoretical refinement of the line mixing model.
LINE
MIXING
EFFECTS
155
IN CO*
It is also of interest to plot the absorption coefficient K( cr) for a given wavenumber as a function of pressure. This is displayed in Fig. 5 for u = 72 1.2 1 cm-‘. This wavenumber is chosen well beyond the Q( 2) line and far enough from R(66) that the subLorentzian absorption should not be masked by the Lorentzian core of R ( 66). The line-mixing model very clearly accounts for the important deviations from Lorentz absorption in the whole range of investigated pressures. But it seems that the calculation
K(o) (cm-’ )
0.25
100
300
500
( observed - calculated ) /observed
700
900
1100
(%)
-20
r----=-~
-60
pressure ( torr ) -100 loo
t
I
I
300
500
I
700
I
I
900
1100
FIG. 5. Absorption coefficient versus pressure at wavenumber u = 721.21 cm-‘. = Lorentz profile; t = line-mixing model, using relation (2); 0 = line-mixing
(3).
X
= experimental; model, using relation
HUET,
156
LACOME.
AND
Lh’Y
that involves the factor of 2 more closely fits the observed spectrum than does the other calculation.
as pressure increases
2. Perturbation Model In order to get some insight on the basic mechanisms involved for each of the different spectral regions analyzed above, it is of interest to perform a perturbation calculation of the line-mixing contributions. Rozenkranz (37) and Smith (35) derived a first-order (in pressure) approximation for the inversion of the H matrix that has been used by a number of authors ( 19,21-23,45). Obviously, such an approximation is valid only at low pressures and it breaks down rapidly as pressure is increased and lines overlap more completely. However, in addition to being more convenient, it has the advantage of providing a simpler representation of line mixing and therefore illustrates more clearly the different effects observed. Rozenkranz (37) showed that K( a) can be approximated as a sum of Lorentz and mixing terms for each line significantly contributing to the absorption coefficient, K(a)
= P(a)
+ KM(a),
where
k
Yk (c - flk12 + -d
flk r
(8)
and
k
bk
In Eqs. (8) and (9) Sk is the line strength the Coefficients,
ak)2+ 7; ’
?r (a -
and the Yk are expressed
as a function
of
wjk
Yk=2cA
d. j+k dk
w,k (ak
-
(10) nj)
’
The influence of mixing is therefore essentially contained in the coefficients Yk. Thus, the analysis of the behavior of the dispersive components will enable us to understand the differences appearing throughout the Q branch. The mixing coefficients Yk as derived from Eq. ( 10) are plotted in Fig. 6. Due to the very small separation between Q(2) and Q(4), the influence of these two lines is markedly dominant as compared to the remainder of the Q-lines. It is clear that the behavior of Y2 and Y, will thus govern the magnitude and sign of the line mixing contributions. This is shown in Fig. 7, which gives a diagram of the Lorentz and dispersive components of all the Q-lines at different wavenumbers. At 7 19.6 cm-‘, near Q( 32), the dispersive component of Q( 2), which is positive, is balanced by the negative components of the other lines of the branch. Moreover, the magnitude of these mixing contributions is much smaller than the Lorentz terms: Lorentzian absorption dominates.
157
LINE MIXING EFFECTS IN CO2
0.5
( Mixing model : relation (2) )
Yk
0.0
-0.5
-1.0
-1.5
-2.0 -2.5 -3.0 -3.5
2
6
IO
14
18
22
26
30
34
38
42
46
50
54
58
FIG. 6. Parameters Ykfrom the perturbation model.
In the center of the branch (at u = 720.6 cm-') several positive dispersive components, in particular those of Q( 16), Q( 18),and Q( 20))contribute in addition to Q( 2) and Q(4). Therefore, the absorption should be super-Lorentzian but the magnitude of positive contributions is still small in comparison to the Lorentzian terms. At the pressures investigated here, the expected sub-Lorentzian behavior is, in some way, masked by the strong Lorentzian contributions. The situation is completely different beyond Q(2) at (r = 721.2 cm-‘. The very strong negative components of Q(2) and Q(4) are no longer compensated since all positive terms are much smaller. Furthermore, the magnitude of these negative contributions is also much larger than the Lorentzian absorption in this region. The resulting spectrum is clearly sub-Lorentzian and the line-mixing influence is much more pronounced than in any other part of the Q branch. V. CONCLUSIONS In agreement with previous observations of CO* Q branches made at 1932 cm-’ (22) and 2076 cm-’ (23), the present investigation has shown that line mixing modifies the absorption differently according to the region considered in the spectrum. For the Q branch of COZ analyzed here, a decrease in absorption as large as 50% at 1000 Torr
HUET, LACOME, AND LhY
158
0.20
1
2
0.50
0 = 7 19.60 cm+ (between Q(32) and Q(34))
cm-’
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44
cm-’
J
o = 720.60 cm-’
0.40
(between Q(12) and Q(14))
0.30 0.20 0.10 0.00 -0.10
I
.
. 2
0.02
,
. 4
. 6
. 8
. 10
. 12
. 14
. 16
_ 18 20
cm-’
Q(J)
. 22
24
26
28
30
32
34
Q = 721.21 cm-’
0.00
-0.02
-0.04
-0.06 -0.08 2
For each line :
4
EH
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
Lorentzian term ti (a)
Mixing term I
FIG. 7. Lorentzian and line-mixing contributions of Q lines to the absorption coefficient at different wavenumbers. Line-mixing terms are calculated with the first-order approximation.
LlNE MIXING EFFECTS IN CO>
159
is observed in the low-J wing, while other parts of the branch are less affected. An empirical model based on the identification of line-mixing coefficients to inelastic transfer rates correctly accounts for the observed variations. This is of particular interest in the remote sensing of atmospheric temperatures and more generally in all radiative transfer problems. We mentioned above the overestimate of the absorption in some of the sounding channels in the HIRS experiments. This is also observed in several other spectral regions. Up to now interference effects were ignored in the models and the observed sub-Lorentzian absorption was entirely attributed to the behavior of the far wings of lines. In order to remove the corresponding departures, one approach widely used consists in introducing an arbitrary cut-off at a smaller than usual distance to the center of lines. Such a drastic approach is rather dangerous as shown by Susskind and Sear1 (46) for the absorption of CO2 near 2400 cm-‘. Similarly Chedin and Scott (25) report an example of computed spectra in the spectral range 600-750 cm-i. Careful examination of Fig. 2 in Ref. (25) clearly shows that, for both the Q branch near 720 cm-’ and the u2 Q branch, the use of such a cutoff, while improving the agreement with observation on the high-frequency wing, results in a striking degradation of the fit on the low-frequency side, by calculating exceedingly large corrections. Accordingly, for the 10’0 f 01 ‘0 Q branch at 720 cm -‘, the present study (along with our preceeding paper (28)) demonstrates that the deviations in modeling radiances result in fact from the addition of two causes: (i) the band intensity of the 10’0 0 1 ‘0 used so far was obviously too large. Thus, by adopting the band intensity derived from our recent Q branch measurements (which is 10-l 5% smaller) the considered channel becomes more transparent over the entire spectral interval considered; (ii) the residual excess of absorption must then be attributed to the neglect of line mixing. When it is taken into account, an additional decrease in absorption should be obtained, but only on the high-frequency side of the branch (low J-values), whereas the rest of the spectrum should remain insensitive to the influence of mixing. Here, it should be noted that the smaller value for the band intensity is corroborated by very recent determinations made by Fourier transform spectroscopy for several P and R lines of this band (4 7). The derived vibrational transition moment agrees with ours to within the experimental uncertainty. This close agreement of two independent sets of data obtained by different methods (Q-branch diode laser and P-R-branches ITS) should provide confidence in the new value adopted for the band strength. A problem requiring additional investigation concerns the values of linewidths to be introduced in the mixing models. In order to test the sensitivity of the mixing calculation, we have introduced either the smoothed values for parallel bands (30) or the Q-branch values directly measured by Huet ez al. (28). In Table III, it clearly appears that parameter c is only affected by an amount of 20% (which is easily understood, since the exponential term in the scaling law accounts mainly for the higher J-values which do not differ markedly between the two sets of data). On the contrary, a and b are affected by a variation of 40%. Therefore. it is of importance to reinvestigate the problem of line width values in perpendicular transitions in order to clarify this point. All recent results (22-24) including the present ones raise finally the question of how to incorporate line mixing in atmospheric models. Future sounders are planned
160
HUET, LACOME, AND LEVY TABLE III Dependence of the EPGL Parameters a, 6, and c on Linewidth Values Scaling law coefficients
Mixing model Broadening coefficients
1 : relation (2) 2 : relation (3)
Y(*)
-f//c**)
y(***)
a (cm-’ at&
C
b )
(xl@)
I
1
0.05216
0.3952
2.131
2
0.04062
0.3987
2.130
1
0.03067
0.2382
2.592
2
0.02378
0.2422
2.581
2
0.02474
0.2410
2.587
( * ) Q-branch line widths (28).
(* *) smoothed values for parallel bands (30). (a * * ) values used by Strow and Gentry near 5 pm (22, 23).
operate at higher resolutions, which will require that explicit account be taken of line mixing effects. Even the empirical approach used up to now is still too complex and time consuming for practical atmospheric applications. Thus, one priority should be to set up a more tractable treatment easily usable in radiative transfer codes. A direct extension of the present work will be the study of collisional systems closer to atmospheric reality such as C02-N2 and, perhaps more important, an investigation of the temperature dependence of line mixing. to
ACKNOWLEDGMENTS One of the authors (T.H.) acknowledges a grant from the Direction des Recherches et etudes Techniques (DRET) Contract 86 / 1489. This work was performed within the frame of Contract ATP 90- 178 1 Physique de l’atmosphire. It has also benefited by financial support from EOARD (European Office for Aerospace Research and Developement). The authors thank Professor K. Narahari Rao and Doctor C. Chackerian, Jr., for reading and improving the manuscript. RECEIVED:
May 22, 1989 REFERENCES
1. 2. 3. 4. 5. 6.
M. BARANGER, Whys. Rev. 111,481-493 (1958); Phys. Rev. 111,494-504 (1958). A. C. KOLB AND M. C. GRIEM, Phys. Rev. 111,514-521 (1958). A. BEN-REUVEN, Phys. Rev. 145,7-22 ( 1966). U. FANO, Phys. Rev. 131,259-268 (1963). A. LIGHTMAN AND A. BEN-REUVEN, J. Chem. Phys. 50,351-353 (1969). R. G. GARDEN AND R. P. MCGINNIS, J. Chem. Phys. 55,4898-4906 ( 1971).
LINE
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EFFECTS
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7.(a)P. DION AND A. D. MAY, Cunad. J. Phys. 51,36-39 ( 1973); (b) A. D. MAY, J. C. STRYLAND,AND G.VARGHESE,Canad J Phys.48,2331-2335 (1970). 8. R.J.HALL,J.F.VERDIECK, ANDA.C. ECKBRETH,@. Cornmun.35,69-75 (1980). 9. R.J.HALL AND D.A. GREENHALGH, Opt. Commun. 40,417-420(1982). 10.D. A.GREENHALGH, F.M. PORTER, AND S.A. BARTON,J Quant. Spectrosc. Radiat. Transfer34,9599 (1985). 11. G. J. ROSASCO,W.LEMPERT,W.S.HURST,ANDA.FEIN,(~) Chem.Phys.Lett.97,435-440(1983); (b) ‘Spectral LineShapes" (K.Burnett, Ed.), Vol. 2, p. 635, DeGruyter, Berlin, 1983. 1-I. W.S. HURST,G.J.ROSASCO, AND W. LEMPERT,SPlE482,23-30(1984). 13. (a)J. P. SALA,J.BONAMY,D. ROBERT,B.LAVOREL,G.MILLOT,AND H. BERGER,J. Chem. Phyx 106,427_439(1986);(b)B. LAVOREL.G.MILLOT,J.BONAMY,ANDD.ROBERT, J Chem. PhJs. llS,69-78 (1987). 14. (a)M.L. KOSZYKOWSIU,L.A.RAHN,R.E. PALMER,AND M. E.COLTRAN.J. Phys. Chem. 91,4146(1987):(b)L.A. RAHN, R.E. PALMER,M.L.KOSZYKOWSKI,ANDD.A.GREENHALGH, Chem. Phys. Lett. 133, 513-516 (1987). 15. W. LEMPERT,G. J.ROSASCO,AND W. S. HURST,J. Chem. PhJs. 81,4241-4245 ( 1984). 16. R. J.HALL AND J.H. STUFFLEBEAM, Appl. Opt. 23,4319-4327 ( 1984). 17. J.T. HAYS AND R. L.ARMSTRONG, J. Chem. Phys. 78,5905-5912 (1983). 18. C. BRAUN. J. Mol. Spectrosc. 93, l-15 (1982). 19. R. L.ARMSTRONG, Appl. Opt. 21, 2141-2145 (1982). 20. M.O. BULANIN.A.B.DOKUCHAEV,M.V.TONKOV,ANDN.N.FILIPPOV. J.Quartz.Spectrosc. Radiat. Transfer 31, 521-543 (1984). 21. C. COUSIN,R. LE DOUCEN, C. BOULET,A. HENRY. AND D. ROBERT,J. Quant. Spectrosc. Radiat. Transfer 36, 521-538 (1986). 22. L. L. STROW AND B. GENTRY. J. Chem. Phys. 84, 1149-l 156 ( 1986). 23. B.GENTRY AND L.L.STROW, J. Chem. Phys. 86,5722-5730 ( 1987). 24. L. L. STROW AND A. S.PINE,J. Chem. Phys. 89, 1427-1434 (1988). 25. A. CHEDIN AND N. A. SCOTT, J. Quant. Spectrosc. Radiat. Transfer 32,453-46 1 ( 1984). 26. M. P. WEINREB,H.E.FLEMING,L.M. MCMILLIN, AND A.C. NEUENDORFER,unpublished. 27. N. A. SCOTT AND A. CHEDIN,J. Appl. Meteorol. 20, 802-8 12 ( 198 1). 28. T. HUET, N. LACOME,AND A. LEVY,J.Mol. Spectrosc. 128,206-215 (1988). 29. L. S. ROTHMAN, Appl. Opt. 26,4058-4097 (1986); Appt. Opt. 25, 1795-1816 (1986). 30. E. ARID, N. LACOME,P.ARCAS,AND A. LEVY,Appl. Opt. 25,2584-2591 ( 1986). 31. N. LACOME,E.AR& AND A. LEVY, to be published. 3-7. J.P. HOUDEAU, C. BOULET.J.BONAMY, A. KHAYAR, AND G. GUELACHVILI, J Chem. Phys. 79, 1634-1640(1983). 33. R. G. GORDON. Adv. Magn. Reson. 3, l-42 ( 1968). 34. K. S. LAM, J Quant. Spectrosc. Radial. Transfer 17, 351-383 (1977). 35. E. W. SMITH,J. Chem. Ph.vs. 74, 6658-6673 ( 1981). 36. D. M. BRINK ANDG. R. SATCHLER,“A~~~~~~ Momentum," Clarendon Press.Oxford, 1968. 37. P. W. ROZENKRANZ,I.E.E.E. Trans. Antennas Propag. 23,498-506 (1975). 38. R. SCHAFERAND R. G. GORDON, J. Chem. Phys. 58, 5422-5443 (1973). 39. J. P.HOUDEAU, C. BOULET, AND D. ROBERT,J. Chem. Phqs. 82, 1661-1673 (1985). 40. A. E. DE PRISTO AND H. RABITZ, J.Mol. Spectrosc. 70, 476-480 ( 1978). 41. A. E. DE PRISTOAND H. RABITZ,J. Quant. Spectrosc. Radial. Transfer 22, 65-79 ( 1979). 42. T. A. BRUNNER AND D. PRITCHARD, Adv. Chem. PhJls. 50, 589-641 (1982). 43. M. L. KOSZYKOWSKI,R. L.FARROW,AND R.E. PALMER,Opt. Lett. 1,478-480(1985). 44. R.G.GORD~NANDR.P.MCGINNIS, J Chem.Phys.49,2455-2457(1968). 45. B. LAVOREL, G. MILLOT.R. ST LOUP, C. WENGER. H. BERGER, J. P. SALA, J. BONAMY. AND D. ROBERT, J. Phvs. (Paris) 47. 417-425 ( 1986). 46. J. SUSSKIND AND J. E. SEARL,J. Quant. Spectrosc. Radiat. TransfP 18, 581-587 (1977). 47. V. DANA, private communication.
OF MOLECULAR
SPECTROSCOPY
138, 141-161
(1989)
Line Mixing Effects in the Q Branch of the 10’0 + 01’0 Transition of CO2 THIERRY
HUET,
NELLY
LACOME,
AND
ARMAND
LEVY
I
Laborutoire d ‘Infrarouge, associP au CNRS, Bat. 350. Universite’ de Paris-Sud. 91405 Orsay Cede.y. France Line mixing effects have been investigated in the Q branch of CO2 near 720 cm-‘, by means of a tunable diode laser. Spectra were obtained at room temperature, in the lOO- 1000 torr pressure range. In the high-frequency wing of the Q branch, below the Q(2) line, the observed absorption is nearly 50% less than that calculated on the assumption of purely additive Lorentz lines, whereas the Q branch itself is nearly insensitive to line coupling in the pressure range investigated. From the line broadening coefficients alone, a simple scaling law allows an estimate of the state-to-state rotationally inelastic transfer rates. Calculation of line mixing coefficients using these rates yields corrected absorption coefficients, in very good agreement with the observed ones, all over the analyzed spectral range. The results appear to be of importance for modeling outgoing radiances in temperature sounding channels near 14 @m. 0 1989 Academic Press, hc. I. INTRODUCTION
At moderate pressures, the vibration-rotation spectra of gaseous systems are described, in the vast majority of cases, as sums of isolated Lorentzian lineshapes. As pressure is increased, line overlapping creates interference effects resulting in an intensity redistribution among lines. This interference phenomenon, usually denoted as line mixing or line coupling, may thus produce significant deviations from directly additive individual line profiles. The principle of this coupling has long been identified in the theory of spectral shape of collision-broadened lines. Its theoretical description was developed, within the frame of the impact approximation, by Baranger ( 1)) Kolb and Griem (2) and later by Ben-Reuven (3) as an application of the general relaxation model introduced by Fano (4). However, very little experimental work was carried out on this subject until recent years, with the exception of a few low-resolution IR (5, 6) and Raman (7) studies. This can be easily understood, since line-mixing effects become significant only when two conditions are fulfilled: (i) the line spacing must be comparable to the collisional width y and (ii) the separation between rotational energy levels must be comparable to kT. In Q branches, with closely spaced lines, the first condition is most often met; and it is easily seen that, for systems like CO*, the rotationally inelastic collision frequency becomes of the same magnitude as the vibration-rotation interaction even for pressures near or below 1 atm. This would thus allow easier experimental observation of the influence of line interferences on the spectrum. Unfortunately, the com’ Present address: Laboratoire de spectrochimie VI, 4 place Jussieu 75230 Paris cedex 05. France.
moleculaire,
141
associe au CNRS.
0022-2852/89 Copyright 0
Bat.F, Universitt
Paris
$3.00
1989 by Academic Press. Inc.
All rights of reproduction in any form reserved.
142
HUET,
LACOME,
AND
LEVY
pactness of Q-spectral structures makes it difficult to derive accurate values of individual line parameters (line intensities, collision-broadened widths, and absorption coefficients). Only in the last decade have such measurements become more possible with the advent of high-resolution instruments: tunable diode lasers, Fourier transform spectrometers, difference-frequency laser systems, etc. In recent years, rotational and vibrational Raman spectra of N2 have received considerable attention (8-14) owing to their usefulness for diagnostic measurement of temperatures and pressures in harsh environmental conditions (flames, combustion engines, shock tubes, etc.). Several other simple systems such as CO ( 1 la), NO ( 12, IS), CO* and N20 (16)) and C2H4 (17)have also been the subject of recent investigations by Raman techniques. Continued effort has been focused on CO* infrared spectra because of their importance in atmospheric transmission. Braun (18) and Armstrong (19), starting from slightly different formalisms, have developed calculations incorporating line mixing in the u2 Q branch. They both predicted that significant effects should be observed in the 15-pm spectral region for pressures representative of atmospheric conditions. The far wings of lines also can be sensitive to the influence of line coupling. Bulanin et al. (20) investigated the contribution of line coupling to far wing absorption by studying the v3 band of CO2 (along with ~3in N20 and the fundamental in CO). The observed deviations of the band shape from Lorentzian behavior were accounted for in terms of intensity redistribution due to the interferences. Cousin et al. (21) emphasized on the various mechanisms involved in the observed absorption in the microwindows between lines and discussed the influence of the rotational distribution of line coupling. The first experimental investigation of line mixing for an IR (2 branch has been reported recently by Strow and Gentry (22). For the combination transition centered near 1932 cm-‘, a strongly sub-Lorentzian absorption in the low-frequency wing of the Q branch in self-broadened CO2 was clearly pointed out. Following the formalism previously used in Raman by Rosasco et al. ( 1 I ) , these authors modeled the collisional transfer rates by means of a simple energy gap law. A similar study for N2-broadened CO2 Q branches was also recently reported (2.3). An extension has also been made to the case of N,O, which exhibits the pecularity of I-type doubling in the upper vibrational level (24). The accurate calculation of CO2 absorption in spectral regions which involve the presence of Q branches is of prime importance for atmospheric studies. In particular, satellite infrared radiometers utilize the emission of CO2 in the 14- 15 /*rn region for the retrieval of atmospheric temperature profiles. It is now well established that several of the temperature sounding channels must be affected to some degree by line mixing (25). Account should therefore be taken of such effects when the corresponding radiances in atmospheric models are modeled. Otherwise, noticeable errors could result in the retrieved temperatures. The main goal of the present work is to investigate the effect of line mixing within the spectral range corresponding to one of these temperature sounding channels, centered near 720 cm-’ . This channel precisely covers the Q branch of the 10'0f 0 I'0 transition of “C r602. As clearly shown by Chedin and Scott (25), both the NESS (26) and 4A (27) models, when applied to this spectral region, yield computed radiances
LINE
MIXING
EFFECTS
IN CO1
143
that depart noticeably from the true values, the deviations being, for both, of similar magnitude and the same sign. Since the two models are based upon quite different mathematical approaches, the discrepancies may presumably be ascribed to the insufficient quality of the spectroscopic information used in the models rather than to the models themselves. In particular, neglect of fine effects such as interferences between lines could be responsible, at least partly, for the difficulty encountered in modeling correctly the outgoing atmospheric radiances. Another objective of our work is to compare our results with those derived for other CO2 Q branches. It is of importance to check to what extent the parameters involved in the fitting laws so far used for estimating transfer rates could be dependent on the transition considered, more especially on the symmetry of initial and final vibrational levels involved. After the experiments are described and the results obtained are discussed (Section II), the theory will be briefly recalled in Section III. The results of calculations are given in Section IV. We shall conclude with a discussion of some problems requiring further investigation. II. EXPERIMENT I.
Experimental Details and Data Collection
In a recent paper (28), we reported tunable diode laser measurements of individual intensities and collisional widths for 18 lines in the Q branch of the 10’0 + 01 ‘0 transition of COz. J values ranging from 2 to 48 were investigated in the O-20 Torr pressure range. The results obtained are utilized here for studying the evolution of absorption in the region centered near 720 cm-’ when pressure is increased and line coupling is manifest in the spectrum. All spectra were taken with a tunable diode laser spectrometer (Spectra Physics SP 5080) working in dual beam mounting. The device has been described elsewhere (28). We recall here only the main features of the operating mode. For the present experiments, the Ge etalon was replaced by an optical cell of 10 cm length containing a fraction of a Torr of COz . Two signals were recorded simultaneously: ( 1) the reference signal, consisting of a small number of well isolated lines which were used to frequency calibrate the spectrum. This signal served also to check the emission of the diode for stability; and (2) the sample signal, giving the absorption
to be studied as a function
of pressure.
The spectrum was scanned by successive narrow segments extending typically over 0.080-0.120 cm-‘, which is approximately the spectral separation of lines in this Q branch. Systematic care was taken to see that the successive portions of recorded spectra overlapped each other. Comparison of data for the spectral interval common to two adjacent recordings allowed us to check the reproducibility of the diode emission and to detect possible short-term drifts. The purity of laser modes was monitored by saturating the absorption to ensure that the observed saturation level was identical to the null transmission signal obtained when the sample beam was blocked. In a small number of cases, some weak parasitic modes remained uneliminated, which resulted thus in departures from the blocked
HUET, LACOME,
144
AND
L&Y
beam zero reaching sometimes a few percent. This was mainly the case in the range extending from Q( 8) to Q( 18). The corresponding settings of the diode were therefore discarded. Optical pathlengths of the samples were chosen to achieve absorptances of 50-60% in each portion of the spectrum. As was done previously for individual line measurements, all spectra were taken in the slow scan mode. For each spectral segment under study, 10 to 15 records were made for a series of increasing values of pressure in the range 1OO- 1000 Tort-. Absolute absorptions were determined by ratioing all scans to zero absorption background in order to remove the influence of the varying 100% transmission level. For this purpose, we applied the following procedure. Let S, ( a) and &( U) be respectively the reference signal and the sample signal, recorded when both cells are empty. r(a) denotes the ratio of &( a) to S, ( a). For a given setting of the diode, it is experimentally observed that the value taken by I( a) at a given wavenumber u remains constant even when the emission level of the diode varies. Thus for each setting of the diode, it can be considered that r(u) is negligibly affected by fluctuations of current. Accordingly, data were collected as follows. Prior to the run, S, ( u) and &( u) were recorded several times, then averaged, and the function r(u) determined within the spectral interval under study for the considered setting-up of the system. Then, the sample cell was filled to the desired pressure of COz and signals S’, and Sb were obtained. The transmitted intensity Zt( a) is then given by the sample signal S$( a), whilst the zero absorption is ZO(a) = S;(u) - r(u). Thus, the transmission is given by T(u) = S;(U)/[S;(~~)* r(u)]. All measurements were made with high purity (99.998%) CO2 and natural isotopic composition was assumed. Gas pressures were measured to better than 0.5% accuracy with a Datametrics capacitance gauge. Temperature of samples was continuously monitored with a platinum sonde. During the time interval required to make the lo12 different scans for each given segment of spectrum, the observed temperature variation never exceeded a few tenths of a degree. The spectra, obtained in the form of graphical records, were then digitized and the resulting data stored for subsequent processing.
2. Data Reduction The great number of raw data obtained at different pressures and temperatures were put in usable form by determining the values of the absorption coefficient at a given set of wavenumbers and pressures. First, for a given scan, recorded at working pressure pw, an interpolation procedure vs wavenumbers yielded adjusted values of the absorption coefficient for a number of previously fixed test frequencies Us. Then, for each one of these reference frequencies, all data corresponding to the various experimental pressures investigated were least-squares fitted vs. pw values. Smoothed values of the absolute absorption coefficient K( at) were thus derived for a series of test pressures pt, The process can be’schematized as follows:
mu,Pw,
T) * K(u*, Pw, T) * K(% Pt, 73.
LINE
MIXING
EFFECTS
145
IN CO2
3. Calculation of the Reference Profile The reference value of the absorption coefficient K(c) at wavenumber u is given by the simple summation of noninteracting purely Lorentzian lines. In the present case, lines from all the vibrational transitions lying in the spectral range 650-750 cm -’ were included in the calculation. These transitions are listed in Table I, while Table II summarizes all spectroscopic parameters needed for calculating line positions. For the Q branch of the 10’0 + 0 I ‘0 transition studied here, we adopted the values of individual line intensities and self-broadening parameters recently determined (28). For all other nearby transitions, line intensities were calculated from the values of vibrational intensities given in the AFGL compilation (29). Since all the vibrational transitions taken into account are perpendicular-type bands, we considered it inappropriate to take values of the broadening coefficients determined on parallel bands, as was discussed previously (28, 30). We therefore performed a series of new measurements of self-broadening coefficients for a number of perpendicular bands of COZ in the 4.8-5 pm region (31). Several P and R branches were investigated by very highresolution Fourier transform spectroscopy. The values so obtained appear systematically lower by 6-B% than the corresponding parameters for parallel bands. Therefore, we adopted these new determinations for all transitions of Table I, including the P and R branches of the 10’0 + 01 ‘0 transition. It may be asked whether it is really consistent to adopt for the collisional widths Q-branch values different from those taken for P and R lines. A similar situation has been recently reported by Lempert et al. ( 1.5). These authors carried out direct measurements of line broadening in the fundamental Q branch of NO and found values 10% lower than derived from Rbranch data (32). So they evoked, as one plausible explanation, the possibility of a small difference in Q-branch vs R-branch broadening. This question would thus deserve systematic investigation.
TABLE I Vibrational
Transitions
Considered
for Calculating
Band Center ( cm.-’ )
Transition
Ol’Ot
OO”0
02?0 toll0 11’0 t lo”0 t
IO”0 01’0
11’0 t0220 10°0t
01’0
the Absorption
near 720 cm-’ Dipole moment ( Debye )
667.380
0.1804
667.752
0.2560
-
688.671
0.2203
-
720.805
0.1191
-
741.724
0.1360
"PO,
721.584
0.1191
12c’60, -
L
Nore. For lo”0 + 01’0 transitions, dipole moment moment calculated from intensities of Ref. (29).
from Ref. (28).
For all other transitions,
dipole
146
HUET, LACOME, AND LI%Y TABLE II Spectroscopic Constants Required for Calculating Line Frequencies vibrational level
Isotope
‘ee60z
1
YYO,
I I
oo”O
01’0 lo”0
vibrational energy (cm-’ )
1
I 1
0
BC (cm’) 1 0.39021817
648.4784 1370.0626
I 1
B,
(cm’)
D,
DE (
10.’ cm-’
; ( 10“
cm-’
% --I-
HF
lo-”
cm-’
0.055
( 10.” cd
1
0.055
0.39021817
1.33204
0.39125388
1.35133
1.35900
0
0
0.39018823
1.14801
1.14801
1.846
1.846
0.39133321
1.2579
1.20968
0.960
0.442
0.39166614
1.3735
1.37919
-3.560
0.076
1.33204
0.39061023
0.39124371
1.34973
1.35754
0
0
0.38971761
0.38971761
1.20137
1.20137
2.083
2.083
Note. All values from Ref. (29).
Regarding 13C02, the broadening parameters were taken equal to those of ‘*CO2 and the transition moment of the 10’0 + 0 1’0 transition was assumed to be identical to the corresponding one in 12C02. To reduce computational time, we retained only those contributions to the absorption coefficient greater than an arbitrary cutoff value of IO-’ to 10e6 cm-‘, depending on the pressures. We estimate the uncertainties on calculated absorption coefficients to be approximately 6%, on account of uncertainties on the spectroscopic data introduced in the calculation. Before any conclusion was derived from the comparison of observed and “Lorentzian” spectra, a preliminary analysis of the relative magnitude of the various contributions to the absorption was attempted for some characteristic regions of the (2 branch. This is shown in Fig. 1: -in the range of higher J-values, for example near Q( 44)-Q(46), the Q branch itself contributes 70-80%. Due to its proximity, P( 3) provides lo-20% of the total absorption. This contribution however comes mainly from the core of the line (the line center falls between Q( 46) and Q( 48 )) and it is therefore correctly described by a Lorentz shape. All the remainder contributions amount to less than 8-10%. nearly
-near Q( 34)-Q( 95%. ’
32), the Q branch
dominates
-for lower J-values (J i 20), contributions itself become smaller than 2-3%.
as expected
and it contributes
other than that of the Q branch
Beyond the Q branch, in the wing of low J-value lines, the situation appears quite different. Three main contributions must be considered: (i) the absorption due to the wings of the Q-branch lines amounts to 70-80%; (ii) line R( 66) of the u2 band, centered at 72 1.4 cm-‘, also contributes significantly (near the line center, it represents 25 30% of the total absorption, but the corresponding contribution is known to be strictly Lorentzian; furthermore, at less than 0.3 cm -’ from the line center, it becomes almost negligible); (iii) the third contribution originates from the Q branch of the isotopic
)
LINE
MIXING
147
IN CO2
(A)
1
0.20
EFFECTS
K(o) (cm-‘)
0.15
total Q branch
0.10
P branch
other contributions
I
I
I
I
I
I
718.60
718.70
718.65
I
o(cm-‘)
718.75
t
t
Q(a)
Q(a)
(B)
K(o) (cm-‘)
0.08
0.06
0.04
branch 0.02
I
I
1
721.3
FIG. 1. Comparison cm-‘.
ofthe different contributions
I
721.4
to the absorption
I 721.5
a(cm-‘)
coefficient near 7 18.7 cm -’ and 72
1.21
transition. In particular, the spectral range 72 1.2-72 1.6 cm-’ investigated here spans the range from Q( 2) to Q( 14) so that line mixing should also be considered for these isotopic lines. However, as will be seen later, this range of strong Q lines is quite
HUET, LACOME, AND LfiVY
148
negligibly affected by line coupling effects. Even if these were explicitly taken into account for this branch, the corresponding correction would be of very small influence. Finally, it appears that the Q branch considered is particularly suitable for studying interference effects since all “foreign” contributions either remain of very small (if not negligible) magnitude or they result in strictly Lorentzian absorptions. The possible influence of sub-Lorentzian behavior of the far wings from remote lines can therefore be ignored. Thus, observed deviations from the reference profile will be safely attributed to line coupling. 4. Comparison of the Observed Absorption to the Lorentz Model Calculated are displayed
and observed values of the absorption in Fig. 2 for three typical cases.
coefficient as a function
of pressure
-At 718.79 cm-‘, between Q(44) and Q(46), the observed value of K(a) is systematically lower than the calculated one. However, the departure stands at the limit of uncertainties as shown by the error bars. This indicates that the region of higher J-values probably exhibits sub-Lorentzian behavior. But this requires further confirmation from investigations at higher pressures. -At 719.83 cm-‘, near Q(30), i.e., in the central region of the Q branch, no difference can be detected between the Lorentzian and the observed absorption. This region is insensitive to line coupling for pressures up to 1 atm. -At 72 1.4 cm-‘, beyond the Q branch, a strong departure creasing magnitude as pressure increases. The observed spectrum Lorentzian and the deviation exceeds 40% at 900 Torr.
is observed, of inis noticeably sub-
The present results are very similar to those reported by Strow and Gentry (22, 23) for the 11’0 + 00’0 transition near 5 pm. They corroborate the conclusion that the main effect of line coupling is located primarily in the low-Jwing beyond the Q branch when pressure values range from a few hundreds of torr to 1 atm. The overall narrowing of the whole Q branch would become apparent only at much higher pressures where the expected enhancement of the absorption at the center of the branch should be detectable to a certain extent. The remaining problem is now to attempt to account for all the above observed effects by means of an adequate line mixing model.
III. THEORY
1. Line Mixing Model Within the framework is expressed as (33-35)
of the impact approximation,
the absorption
coefficient
K(u)
149
LINE MIXING EFFECTSIN COz ( A ) :
0.16 0.14 0.12 0.10 0.08 200
0
400
600
800
Iwo
( B ) : (3 = 719.83 cm -’ ( near the Q(30) line ) 0.8
0.6 0.5 0.4 0.3 200
0
400
600
800
1000
( C ) : o = 721.38 cm -’ ( Beyond the Q(2) line )
0.16
.i
0.14
)
0.12 0.10 0.08 0.06 0.04 0.02
, 0
FIG. 2. Observed experiment 2; 4
i 200
and calculated absorption = Lorentz profile.
I 400
coefficients
pressure(=) I 800
I 600
versus pressure.
-<‘-
= experiment
, 1000
1: -o-
150
HUET,
LACOME,
AND LfiVY
where LTis the wavenumber (cm ), c the velocity of light, and 12the number density of absorber molecules. I j) and I k) are doubled state vectors denoting spectral transitions. In the present case, for example, I k) = I qZ)f, JJ) denotes the Q line ViJ + vfJ and I j) = 1ViVf, J’s) corresponds to ViJ’ + VfJ’. pk is given by a normalized Boltzmann distribution over the initial state population associated with the radiative transition I k) , dk and dj are the reduced matrix elements (as defined in Ref. (36)), (jlalk) = a6jk, and (jla,,lk) = kajk, where k is the vibration-rotation wavenumber of the radiative transition Ij) of the molecule. The diagonal elements W,, of the relaxation matrix are the broadening coefficients -rk while the off-diagonal terms (j I W 1k) = Wjk are the line-coupling parameters (as defined in Refs. 37 and 38). The effect of the off-diagonal elements Wjkcan be neglected for well separated lines and the absorption coefficient is given simply as the summation over all contributing Lorentz line profiles. In the case of a Q branch, these elements Wjk cannot be ignored, even for low pressures. Equation ( 1) takes into account line interferences but ignores the finite duration of collisions (impact approximation) which is known to affect both diagonal and offdiagonal elements of W (39). However, as long as the reciprocal of the collision time T, is larger than the separation between the centers of lines under consideration or larger than the distance between the line center and the wavenumber at which K(c) is calculated, this effect is negligible. For the collisional system C02-C02, the impact parameter b0 can be evaluated to 3 A, giving a collision time equal to about 0.6 ps. So the impact approximation should be valid over a distance of 9 cm-’ from line centers. This justifies the use of Eq. ( 1) over the frequency range spanned by the Qbranch lines. Accordingly, the only restriction introduced in our calculation was that only lines of the Q branch were allowed to mix and no mixing of Q- with P- and Rbranch lines was considered. An a priori calculation of the coefficients Wj, requires accurate knowledge of the intermolecular potential surface. For a collisional system like CO2-CO2 this is not available. Moreover, a calculation at each wavenumber would lead to a formidable computational task. To circumvent this difficulty, Strow and Gentry (22, 23) transposed to infrared absorption the method previously introduced in isotropic Raman scattering by Rosasco et al. (11, 1.5) for studying line mixing effects. In Raman spectroscopy, the off-diagonal elements W,, can be equated to - 1 times the rate at which collisions transfer amplitude from line k to linej (if elastic reorientation collisions do not contribute significantly to the widths). In other words, the coefficient W,, that ensures coupling of the Q(J) = line k to the Q( J’) = line j is assumed to be identical to the inelastic rotational transfer rate between levels J and J’, Wj, = -K(J+
J’).
(2)
This relation strictly holds for isotropic Raman spectra when the tensor responsible for the coupling of radiation and matter is of order zero. It becomes only approximate in infrared absorption (tensor of order one). The resulting approximation was discussed by Cousin et al. (21) who showed that such an assumption remains reasonably valid for the case of CO*.
LINE MIXING
Therefore,
the pressure-broadened -rk=
;[
half-width
K(J +
2 Jk,+
EFFECTS IN CO2
151
for a single line is written
J’) +
c
K(J-+
as (40, 41)
J’)],
(3a)
J&per+ J
J
where Yk is the pressure-broadening coefficient of line Q(J) and K(J + J’) is the collision rate of rotational transfer from J to J’, within a single vibrational state. In their recent study, Strow and Gentry (22, 23) suggested the introduction of a factor of 2 in the summation over the Z vibrational level, in order to reflect the fact that the density of rotational states is one half that of a II level. Applied to our transition, this leads to Yk=
I[
c
--*
K(J
Jk&
J’) + 2
c J&r+
J
K(J+
J’)].
(3b)
J
The introduction of such a factor is essentially empirical and clearly indicates the degree of approximation that is made in infrared absorption when one assumes the coefficients Wj, to be identical to inelastic rotational transfer rates. According to these authors, quantitive agreement with observed absorption cannot be obtained unless this factor is taken into account. In the present work, we performed both calculations, with and without this factor. The inelastic transfer rates K( J + J’) are also constrained to satisfy the detailed balance, through the usual relationship, pJ” K(J’ + J) = pJ- K(J --) J’), for an energetically downward transition (J’ < J). The approach adopted by most authors for estimating the off-diagonal elements, w;k, consists in making use of a scaling law for deriving the inelastic rotational rates for upward state to state transitions in each vibrational state. A detailed balance then gives the downward rates and an immediate identification yields the coefficients. We adopted in the present work the exponential power-gap law (E.P.G.L.) (42) wjk
K(J_,
J’)
=
a(!+~exp(-@k$dc)
(4)
where Elk = Ek - Ej = EJ - EJ!,
B is the rotational
constant
a, 6, c are adjustable
introduced
to get dimensionless
quantities,
and
parameters.
We also attempted to utilize either an exponential-gap law or a simple power-gap law. In the first case, only high J-values were correctly modeled, while the power-gap law yielded a correct fit only for the lower J-values, as was already noticed by Rosasco et
al. (11, 15). 2. Practical Calculation In the case of a completely overlapped COz Q branch, due to the great number of lines to be considered, inversion of the matrix given in Eq. ( 1) for each wavenumber under study would be excessively time-consuming. For instance, in our calculations, we considered elements in the relaxation matrix up to J = 50. In order to avoid such
HUET,
152
LACOME,
AND
L6VY
successive inversions at different wavenumbers, we follow the procedure discussed by Koszykowski et al. (43) and Gordon and McGinnis (44) and applied to the infrared by Strow and Gentry (22, 23). For this purpose, the expression of K( a) can be written as a function of the matrix product, K(a)
= MIm(d*.(a
- H)-‘-p-d),
(5)
where p is the diagonal matrix whose elements are the Pk previously introduced in relation ( I ) , d is the electric dipole moment vector (superscript t denotes its transpose), and Q is the product of the unit matrix and the wavenumber u. The H matrix is independent of (r and is defined as H = a0 + inW,
(6)
where u. is the diagonal matrix of the transition wavenumbers. For a given wavenumber, A4 is a constant which contains all other factors appearing in Eq. ( 1). Let A be the matrix which diagonalizes H. Then A-‘HA = L - A will also diagonalize u - H, which only differs from H by a constant added to the diagonal terms. The resulting expression for the absorption coefficient is given by K(u)
=
(d*A)i(A-‘*P*d)i
MC
’
(a-li)
i
where the I;‘s are the diagonal elements of matrix L. Therefore the practical calculation was done as follows: from the values of the linewidths, the three coefficients by performing a least square fit to relation (3a) or (3b); from these parameters,
all the elements
a, b, and c were calculated
wjk were obtained
the H matrix was then calculated and diagonalized absorption coefficient K(u) was computed for every
by using relation
2;
for each pressure and the required wavenumber via
Eq. (7). IV. RESULTS 1.
OF CALCULATIONS
Comparison of Experimental and Calculated Spectra
Figure 3 and Figure 4 give plots of the absorption spectral regions:
coefficient
K(u)
for three typical
(a) for the higher J-values, for example in the interval Q( 32)~Q( 34), the observed spectrum is correctly reproduced by the Lorentz model. It is important to notice that no difference results in the calculated spectrum when line mixing terms are included (Fig. 3a). Thus, in the pressure range investigated here, the absorption appears as strictly Lorentzian in the higher-J wing of the Q branch. (b) in the central region, near Q( 12)~Q( 14), the observed spectrum exhibits an absorption slightly enhanced as compared to Lorentz (Fig. 3b). However, the error bars show that the deviation remains at the limits of uncertainties, so that the present data suggest only a trend toward super-Lorentzian behavior in the center of the Q branch (as expected), but this should be corroborated by investigating higher pressures.
LINE
MIXING
EFFECTS
(A) 0.60
IN CO2
:Q(W-QQW
K(cti’)
0.55
0.50
0.45
0.40
0.35
(me’)
(T
0.30 119.54
719.59
719.64
(B) : Q (12) 2.50
-
719.69
Q (14)
K(c&
2.25
2.00
1.I5
1.50
1.25
0 1.00
I 720.56
720.58
I 720.60
I 720.62
I 720.64
(cm-‘) I 720.66
FIG. 3. Experimental and calculated absorption coefficients in two regions of self-broadened Q branch. The pressure is 700 Torr. t = experimental; X = Lorentz profile; A = line-mixing model, using relation (3a); o = line-mixing model, using relation (3b).
The line mixing model qualitatively accounts for a super-Lorentzian it overestimates the deviation (both calculations with and without similar results but the former lies closer to experimental data).
absorption, but the factor 2 yield
(c) the most spectacular result is obtained in the wing beyond the Q( 2) line where the magnitude of deviations between observed and Lorentz absorptions was quite large. Figure 4 gives a plot of the absorption in this region of the spectrum (which includes the R(66) line of the u2 band). It can easily be seen that a substantial improvement is obtained in the calculated spectrum by introducing line-mixing terms and that the remaining deviations are much smaller than the magnitude of experimental
HUET, LACOME, AND L&Y
154 _
K(o)
(cm-’1
0.12
0.10
0.08
0.06
0.04 0
0.02 721.:2
721.3
721.4
( observed - calculated ) / observed
( cti’ ) I
I
I
I
721.6
721.5
(%)
20 0 -20 -40 -60 -80 0 -100
! 721.2
I 721.3
t 721.4
I 721.5
(cm-‘) I 721.6
FIG. 4. Experimental and calculated absorption coefficients in the region beyond the Q( 2) line. The percent deviation of the Lorentz and line-mixing calculations from the observed absorption coefficient is shown below this plot. t = experimentak X = Lorentz profile; A = line-mixing model, using relation (3a); o = line-mixing model, using relation (3b).
uncertainties. Nevertheless, evidence for the necessity of introducing a factor of 2 in the relation between half-widths and inelastic rates cannot be clearly pointed out from our experiments. The two calculations, performed using relations (3a) or ( 3b), give results on both sides of the experimental curve. Thus, one calculation overestimates the correction while the second underestimates it. This, once again, shows the empirical character of the factor 2 and the need for further theoretical refinement of the line mixing model.
LINE
MIXING
EFFECTS
155
IN CO*
It is also of interest to plot the absorption coefficient K( cr) for a given wavenumber as a function of pressure. This is displayed in Fig. 5 for u = 72 1.2 1 cm-‘. This wavenumber is chosen well beyond the Q( 2) line and far enough from R(66) that the subLorentzian absorption should not be masked by the Lorentzian core of R ( 66). The line-mixing model very clearly accounts for the important deviations from Lorentz absorption in the whole range of investigated pressures. But it seems that the calculation
K(o) (cm-’ )
0.25
100
300
500
( observed - calculated ) /observed
700
900
1100
(%)
-20
r----=-~
-60
pressure ( torr ) -100 loo
t
I
I
300
500
I
700
I
I
900
1100
FIG. 5. Absorption coefficient versus pressure at wavenumber u = 721.21 cm-‘. = Lorentz profile; t = line-mixing model, using relation (2); 0 = line-mixing
(3).
X
= experimental; model, using relation
HUET,
156
LACOME.
AND
Lh’Y
that involves the factor of 2 more closely fits the observed spectrum than does the other calculation.
as pressure increases
2. Perturbation Model In order to get some insight on the basic mechanisms involved for each of the different spectral regions analyzed above, it is of interest to perform a perturbation calculation of the line-mixing contributions. Rozenkranz (37) and Smith (35) derived a first-order (in pressure) approximation for the inversion of the H matrix that has been used by a number of authors ( 19,21-23,45). Obviously, such an approximation is valid only at low pressures and it breaks down rapidly as pressure is increased and lines overlap more completely. However, in addition to being more convenient, it has the advantage of providing a simpler representation of line mixing and therefore illustrates more clearly the different effects observed. Rozenkranz (37) showed that K( a) can be approximated as a sum of Lorentz and mixing terms for each line significantly contributing to the absorption coefficient, K(a)
= P(a)
+ KM(a),
where
k
Yk (c - flk12 + -d
flk r
(8)
and
k
bk
In Eqs. (8) and (9) Sk is the line strength the Coefficients,
ak)2+ 7; ’
?r (a -
and the Yk are expressed
as a function
of
wjk
Yk=2cA
d. j+k dk
w,k (ak
-
(10) nj)
’
The influence of mixing is therefore essentially contained in the coefficients Yk. Thus, the analysis of the behavior of the dispersive components will enable us to understand the differences appearing throughout the Q branch. The mixing coefficients Yk as derived from Eq. ( 10) are plotted in Fig. 6. Due to the very small separation between Q(2) and Q(4), the influence of these two lines is markedly dominant as compared to the remainder of the Q-lines. It is clear that the behavior of Y2 and Y, will thus govern the magnitude and sign of the line mixing contributions. This is shown in Fig. 7, which gives a diagram of the Lorentz and dispersive components of all the Q-lines at different wavenumbers. At 7 19.6 cm-‘, near Q( 32), the dispersive component of Q( 2), which is positive, is balanced by the negative components of the other lines of the branch. Moreover, the magnitude of these mixing contributions is much smaller than the Lorentz terms: Lorentzian absorption dominates.
157
LINE MIXING EFFECTS IN CO2
0.5
( Mixing model : relation (2) )
Yk
0.0
-0.5
-1.0
-1.5
-2.0 -2.5 -3.0 -3.5
2
6
IO
14
18
22
26
30
34
38
42
46
50
54
58
FIG. 6. Parameters Ykfrom the perturbation model.
In the center of the branch (at u = 720.6 cm-') several positive dispersive components, in particular those of Q( 16), Q( 18),and Q( 20))contribute in addition to Q( 2) and Q(4). Therefore, the absorption should be super-Lorentzian but the magnitude of positive contributions is still small in comparison to the Lorentzian terms. At the pressures investigated here, the expected sub-Lorentzian behavior is, in some way, masked by the strong Lorentzian contributions. The situation is completely different beyond Q(2) at (r = 721.2 cm-‘. The very strong negative components of Q(2) and Q(4) are no longer compensated since all positive terms are much smaller. Furthermore, the magnitude of these negative contributions is also much larger than the Lorentzian absorption in this region. The resulting spectrum is clearly sub-Lorentzian and the line-mixing influence is much more pronounced than in any other part of the Q branch. V. CONCLUSIONS In agreement with previous observations of CO* Q branches made at 1932 cm-’ (22) and 2076 cm-’ (23), the present investigation has shown that line mixing modifies the absorption differently according to the region considered in the spectrum. For the Q branch of COZ analyzed here, a decrease in absorption as large as 50% at 1000 Torr
HUET, LACOME, AND LhY
158
0.20
1
2
0.50
0 = 7 19.60 cm+ (between Q(32) and Q(34))
cm-’
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44
cm-’
J
o = 720.60 cm-’
0.40
(between Q(12) and Q(14))
0.30 0.20 0.10 0.00 -0.10
I
.
. 2
0.02
,
. 4
. 6
. 8
. 10
. 12
. 14
. 16
_ 18 20
cm-’
Q(J)
. 22
24
26
28
30
32
34
Q = 721.21 cm-’
0.00
-0.02
-0.04
-0.06 -0.08 2
For each line :
4
EH
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
Lorentzian term ti (a)
Mixing term I
FIG. 7. Lorentzian and line-mixing contributions of Q lines to the absorption coefficient at different wavenumbers. Line-mixing terms are calculated with the first-order approximation.
LlNE MIXING EFFECTS IN CO>
159
is observed in the low-J wing, while other parts of the branch are less affected. An empirical model based on the identification of line-mixing coefficients to inelastic transfer rates correctly accounts for the observed variations. This is of particular interest in the remote sensing of atmospheric temperatures and more generally in all radiative transfer problems. We mentioned above the overestimate of the absorption in some of the sounding channels in the HIRS experiments. This is also observed in several other spectral regions. Up to now interference effects were ignored in the models and the observed sub-Lorentzian absorption was entirely attributed to the behavior of the far wings of lines. In order to remove the corresponding departures, one approach widely used consists in introducing an arbitrary cut-off at a smaller than usual distance to the center of lines. Such a drastic approach is rather dangerous as shown by Susskind and Sear1 (46) for the absorption of CO2 near 2400 cm-‘. Similarly Chedin and Scott (25) report an example of computed spectra in the spectral range 600-750 cm-i. Careful examination of Fig. 2 in Ref. (25) clearly shows that, for both the Q branch near 720 cm-’ and the u2 Q branch, the use of such a cutoff, while improving the agreement with observation on the high-frequency wing, results in a striking degradation of the fit on the low-frequency side, by calculating exceedingly large corrections. Accordingly, for the 10’0 f 01 ‘0 Q branch at 720 cm -‘, the present study (along with our preceeding paper (28)) demonstrates that the deviations in modeling radiances result in fact from the addition of two causes: (i) the band intensity of the 10’0 0 1 ‘0 used so far was obviously too large. Thus, by adopting the band intensity derived from our recent Q branch measurements (which is 10-l 5% smaller) the considered channel becomes more transparent over the entire spectral interval considered; (ii) the residual excess of absorption must then be attributed to the neglect of line mixing. When it is taken into account, an additional decrease in absorption should be obtained, but only on the high-frequency side of the branch (low J-values), whereas the rest of the spectrum should remain insensitive to the influence of mixing. Here, it should be noted that the smaller value for the band intensity is corroborated by very recent determinations made by Fourier transform spectroscopy for several P and R lines of this band (4 7). The derived vibrational transition moment agrees with ours to within the experimental uncertainty. This close agreement of two independent sets of data obtained by different methods (Q-branch diode laser and P-R-branches ITS) should provide confidence in the new value adopted for the band strength. A problem requiring additional investigation concerns the values of linewidths to be introduced in the mixing models. In order to test the sensitivity of the mixing calculation, we have introduced either the smoothed values for parallel bands (30) or the Q-branch values directly measured by Huet ez al. (28). In Table III, it clearly appears that parameter c is only affected by an amount of 20% (which is easily understood, since the exponential term in the scaling law accounts mainly for the higher J-values which do not differ markedly between the two sets of data). On the contrary, a and b are affected by a variation of 40%. Therefore. it is of importance to reinvestigate the problem of line width values in perpendicular transitions in order to clarify this point. All recent results (22-24) including the present ones raise finally the question of how to incorporate line mixing in atmospheric models. Future sounders are planned
160
HUET, LACOME, AND LEVY TABLE III Dependence of the EPGL Parameters a, 6, and c on Linewidth Values Scaling law coefficients
Mixing model Broadening coefficients
1 : relation (2) 2 : relation (3)
Y(*)
-f//c**)
y(***)
a (cm-’ at&
C
b )
(xl@)
I
1
0.05216
0.3952
2.131
2
0.04062
0.3987
2.130
1
0.03067
0.2382
2.592
2
0.02378
0.2422
2.581
2
0.02474
0.2410
2.587
( * ) Q-branch line widths (28).
(* *) smoothed values for parallel bands (30). (a * * ) values used by Strow and Gentry near 5 pm (22, 23).
operate at higher resolutions, which will require that explicit account be taken of line mixing effects. Even the empirical approach used up to now is still too complex and time consuming for practical atmospheric applications. Thus, one priority should be to set up a more tractable treatment easily usable in radiative transfer codes. A direct extension of the present work will be the study of collisional systems closer to atmospheric reality such as C02-N2 and, perhaps more important, an investigation of the temperature dependence of line mixing. to
ACKNOWLEDGMENTS One of the authors (T.H.) acknowledges a grant from the Direction des Recherches et etudes Techniques (DRET) Contract 86 / 1489. This work was performed within the frame of Contract ATP 90- 178 1 Physique de l’atmosphire. It has also benefited by financial support from EOARD (European Office for Aerospace Research and Developement). The authors thank Professor K. Narahari Rao and Doctor C. Chackerian, Jr., for reading and improving the manuscript. RECEIVED:
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