Journal of Quantitative Spectroscopy & Radiative Transfer 63 (1999) 31}48
Model, software, and database for computation of line-mixing e!ects in infrared Q branches of atmospheric CO : II Minor and asymmetric isotopomers K.W. Jucks , R. Rodrigues, R. Le Doucen, C. Claveau, W.A. Traub , J-M. Hartmann * Smithsonian Astrophysical Observatory, Cambridge, MA 02138, USA Laboratoire de Physique Mole& culaire et Applications, UPR 136 du CNRS associe& e aux Universite& s P. et M. Curie et Paris-Sud, Universite& Paris-Sud (baL t. 350), 91405 Orsay Cedex, France Laboratoire de Physique des Atomes, Lasers, Mole& cules, et Surfaces, Unite& Mixte de Recherche, Universite& de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France Received 23 April 1998
Abstract The in#uence of line-mixing on the shape of infrared Q branches of minor isotopomers of CO is studied for the "rst time. Laboratory spectra of isotopically enriched CO }N mixtures have been measured in the 15 lm region at the temperatures of 200 and 300 K for total pressures between 1 and 10 atm. Comparisons with measurements for the l and (l }l ) Q branches of the six isotopomers O-C}O ' demonstrate the quality of the theoretical approach presented in the preceding paper (Part I). The model is used to generate a set of numerical data (available by ftp) for the prediction of absorption by OCO, OCO, OCO, OCO, and OCO infrared Q branches under atmospheric conditions; this database completes that proposed in the preceding paper and now includes 271 bands considering the eight most abundant CO isotopomers. Its quality is tested by comparisons with atmospheric limb emission measured by a balloon-borne high resolution Fourier transform instrument. The l Q branches of OCO, OCO, OCO, and OCO recorded above Alaska have been used. This shows that line mixing has signi"cant e!ects, even for minor isotopomers, and is correctly accounted for by the model and data proposed. 1999 Elsevier Science Ltd. All rights reserved.
***** * Corresponding author. Tel.: 00 33 169157514; fax: 00 33 16915730; e-mail:
[email protected] 0022-4073/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 9 8 ) 0 0 1 3 3 - 2
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1. Introduction Line-mixing e!ects in infrared Q branches of CO have been the subject of many experimental and theoretical studies based on laboratory and atmospheric spectra (see Ref. [1] and those quoted therein). Nevertheless, except for our previous work [1] in which the l band of CO was investigated using atmospheric emission recordings, available results are limited to CO . Although the other isotopomers have small abundances on Earth, correct modeling of their absorption is required. Indeed, as shown in this paper, the l Q branches of CO , OCO, and OCO appear in atmospheric emission spectra and can be a!ected by line mixing. The present paper follows detailed laboratory experimental and theoretical studies [2}4] of line-mixing e!ects in CO infrared Q branches; it is an extension of the work of Ref. [1] (referred to as I in the following) which was limited to symmetric isotopomers of CO . Laboratory experiments have been made in the 14}15 lm region using isotopically enriched CO , and absorption of the l and (l !l ) Q branches of CO , CO , CO , OCO, ' OCO, and OCO has been studied. Measurements were made for CO }N mixtures in the 1-10 atm total pressure range for the temperatures 200 and 300 K. The results of computations using the model presented in I (and no adjusted parameter) are compared with experimental values; this proves the quality of the theoretical approach under conditions where line-mixing has large in#uence on the Q-branch shape. The model is then used to generate a comprehensive set of data for computation of line-mixing e!ects in atmospheric CO Q branches, that completes that proposed in I. The "nal database includes all (eight) CO isotopomers of signi"cant absorption in the atmosphere and most bands (271) for which the HITRAN-96 list [5] provides information on Q lines. The proposed parameters and model, which are available to potential users, are tested by using balloon-borne measurements of atmospheric emission in the l Q branches of CO , CO , OCO, and OCO. Comparison between measured and computed radiances show that line mixing signi"cantly a!ects limb emission spectra for all isotopomers, and is correctly accounted for by our model. 2. Experimental 2.1. Laboratory spectra CO spectra near 15 lm have been recorded in Rennes with a high-resolution Fourier transform spectrometer. Details on the experimental apparatus can be found in Refs. [2, 6]. Measurements have been made for CO }N mixtures near the l (645}670 cm\, for the temperatures 200 and 300 K) and (l }l ) (695}725 cm\, for the temperature of 300 K) bands for total pressures ranging ' from 1 to 10 atm. Small mixing ratios of CO have been used and the spectral resolution was high enough that the e!ect of the instrument shape is negligible. Isotopically enriched CO provided by Matheson was used in which the relative amounts of oxygen atoms, determined by mass spectroscopy after dissociation of the molecules, are 38.0% O, 47.9% O, and 14.1% O. The volume mixing ratios of the CO isotopomers can then be estimated assuming statistical combination of the oxygen atoms, i.e. x(OCO)"(2!d );[x(O)x(C);x(O)]
(1)
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Table 1 Fractions of the most abundant CO isotopomers Isotopomer
O}C}O O}C}O O}C}O O}C}O O}C}O O}C}O
From Eq. (1) & mass spectrosc. % in CO bottle
14.3 10.6 36.0 2.0 13.4 22.7
From "t of spectra in the l bands
l }l bands
% in CO bottle
% in N bottle
% in CO bottle
16.2$0.7 10.4$0.3 37.9$0.8 1.4$0.2 11.5$0.6 21.6$0.5
0.6$0.06 10\ Negligible Negligible Negligible Negligible Negligible
14.3$0.4 9.10$0.5 38.9$0.3 * 14.6$0.6 25.6$0.9
This leads, for species involving C, to the values given in the second column of Table 1 (those involving C are about one hundred times less abundant). Note that, in the l region, small amounts (about 0.6 ppmv) of CO in the N gas used made signi"cant contributions to measured absorption under the experimental conditions investigated. The preceding fractions thus had to be corrected in order to account for these impurities. Furthermore, since our interest is in line-mixing e!ects (i.e., in the absorption shape), and in order to get rid of possible uncertainties on the dipole transition moments used (see Section 3), we have "tted the mole fractions of the various isotopomers. This was done, for each spectrum, by a least-squares "t of measured absorption using the results of computations with the line-mixing model described in the next section. This procedure, when applied to all spectra (OP and T) in both bands studied, leads to the results displayed in the last two columns of Table 1. The agreement with predictions using Eq. (1) and the consistency of values retrieved from absorption in the l and l }l regions are quite satisfactory considering all uncertainties (i.e. on the measurement, the model, and the spectroscopic data). For clarity, as done previously [1, 2], all contributions of P and R lines have been numerically removed from measured spectra using the spectroscopic parameters from the HITRAN-96 database. Voigt line shapes were used for this procedure; indeed, for the pressures and wavenumbers investigated here, the isolated line and impact approximations are both valid for the P and R transitions. The absorption coe$cients presented thereafter are thus due to Q-lines only. Since all measurements have been made with small CO fractions, pressure normalized absorption in cm\ atm\) are presented for simplicity. coe$cients (i.e. a/p !- 2.2. Atmospheric spectra. The atmospheric data used in the following are limb emission spectra which were recorded with the balloon-borne FIRS-2 instrument of the Smithsonian Astrophysical Observatory. This ***** The uncertainties in Table 1 are the scatter of results obtained from specra at di!erent ¹, P conditions (10 and 5 in the l and l }l bands, respectively).
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instrument, which is described in Refs. [7, 8], is based on a high resolution (0.004 cm\) Fourier transform spectrometer carried by stratospheric balloons. The #ight whose data are used was launched from Fairbanks (Alaska, 683N-1533W) on the 30th of April 1997. Measurements were made at #oat (balloon altitude 37 km), for various limb viewing geometries (tangent heights ranging from 10 to 37 km). The associated stratospheric temperature range is 214}248 K and is slightly narrower than that of the mid-latitude measurements used in Ref. [1]. Details on the treatment used to deduce radiances from the atmospheric recordings can be found in Ref. [8]. 3. Theoretical model and data used Since the theoretical approach used here is similar to that detailed in Section 2 of I, only the main elements are recalled here. Values of the absorption coe$cient a(p, p , P,T ) have been computed with and without (purely !- Lorentzian) the inclusion of line-mixing using Eqs. (1) and (7) of Ref. [1], including Doppler e!ects when necessary. The required data (see Section 2.3 of I) are the spectroscopic parameters of absorption lines (positions, level populations, dipole transition moments, and pressure broadened widths) and the o!-diagonal elements of the relaxation operator. The "rst have been generated by using data stored in the 1996 version of the HITRAN database [5] and (when missing) recent results by Te!o et al. [9] obtained as in Ref. [10]. For example, line positions and energies for OCO, which is disregarded in HITRAN due to its small natural abundance, were generated using values from Ref. [9] and the dipole transition moments were assumed equal to those for OCO. Fortunately, errors in the spectroscopic constants have small in#uence on the shape of Q branches at the considered pressures and little a!ect the following analysis of line-mixing e!ects. The line-broadening coe$cients (diagonal elements of the relaxation operator) have been determined as in I [see Eq. (16)] and the pressure shifts have been neglected. The relaxation operator o!-diagonal elements have been computed from Eqs. (9)}(14) and the parameters of Table 1 of Ref. [1], thus assuming no in#uence of the isotopomer on the basis parameters of the Energy Corrected Sudden model used. Note that Eq. (11) of I is an approximation when applied to Q branches of asymmetric isotopomers. In this case, the contribution of odd values of L should be included in the sum in Eq. (11), as shown in the Appendix. However, since the considered molecules are only slightly asymmetric, the interaction potential is likely dominated by even rank contributions; the basis rates Q!-\6(odd L,T) are thus expected to be small and we have neglected them. Hence (see the Appendix), each Q branch is composed of two &&sub-branches&& within which coupling occurs but which are isolated from each other. This approximation is con"rmed by the quality of results presented in the next section but its further check in %}* bands would be of interest. 䉴 Fig. 1. Absorption in the region of the l Q branches for ¹"300 K and P"0.997 atm: 䢇 measured values, }}} com puted results accounting for line-mixing and corresponding Meas-Calc residuals. Fig. 2. Absorption in the region of the l Q branches: 䢇 measured values, - - - computed results neglecting line-mixing, computed results accounting for line-mixing and corresponding Meas-Calc residuals. (a) P"2.96 atm and ¹"300 K. (b) P"7.95 atm and ¹"300 K.
}}}
K.W. Jucks et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 63 (1999) 31}48
Fig. 1. See caption opposite.
Fig. 2. See caption opposite.
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4. Results 4.1. Laboratory spectra The computations have been made as explained in I, using the data described above and in Ref. [1]. Since detailed analysis of the e!ects of physical parameters (i.e., total pressure, temperature, band structure, 2) has been made previously [2}4], only a brief description of the results is made hereafter. The structure of the 660 cm\ region and results obtained at moderate pressure are shown in Fig. 1 where the l Q-branches of all six isotopomers OCO appear. As expected [10], the increase of molecular weight shifts the band towards low frequencies and those of OCO and OCO, which have the same molecular weight, are quite close to each other. Note that the contributions of the 2l }l transitions are discernible on the spectra (centered about 0.3 cm\
Fig. 3. Same legend and symbols as in Fig. 2. (a) P"5.01 atm and ¹"198 K. (b) P"9.87 atm and ¹"201 K.
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37
higher than the fundamental) and that the quality of calculations indicates that the precision of spectroscopic data is satisfactory. Results obtained at higher pressures are plotted in Figs. 2 and 3 for the temperatures of 300 and 200 K, respectively. The quality of our approach and the large discrepancies obtained when line-mixing is neglected are clear. Indeed, our model correctly predicts absorption regardless of the P/¹ conditions and isotopomer, whereas addition of Lorentzian contributions largely overestimates (resp. underestimates) the widths (resp. peak absorption) of the Q branches, as widely observed previously for CO . The structure of the 100 Q010 region and the results obtained at moderate pressure are ' ' plotted in Fig. 4. Again, the Q branches of OCO and OCO are quite close to each other and the quality of the calculation is satisfactory. Most discrepancies (particularly on the left side of the plot) result from inaccurate removing of the P and R line contributions (due to improper or lack of spectroscopic data). Results obtained at higher pressures, plotted in Fig. 5, con"rm the accuracy of our approach and the need to account for line mixing. In order to have a synthetic view of the model quality, we have determined the ''width&& of Q branches. This was done by "tting each spectrum (measured and calculated) by a sum of Lorentzian &&lines'' (one per Q branch); this is valid at elevated pressure since the Q branch then has a Lorentzian shape [2, 6]. The half-widths ! obtained were then "tted by a linear law versus total / pressure, giving the slope c and zero pressure intersect *p . The "rst parameter results from / / collisions whereas the second is due to the spectral distribution of the line intensities [2, 6]. The
Fig. 4. Absorption in the region of the (l }l ) Q branches for ¹"300 K and P"0.987 atm: 䢇 measured values, ' }}} computed results accounting for line-mixing and corresponding Meas-Calc residuals.
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Fig. 5. See caption opposite.
Fig. 6. See caption opposite.
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39
pressure dependent term (! !*p ) deduced from measured and computed room temperature / / spectra in the l branch of the various isotopomers for all studied total pressures (about 1, 3, 5, 8, and 10 atm) is plotted in Fig. 6. The quality of the line-mixing model and the large inaccuracy of computations using purely Lorentzian pro"les are con"rmed (the ratios of computed to measured values are 1.02$0.06 and 3.4$0.3, respectively). Note that the e!ect of pressure on the Q branch width is similar for all isotopomers, justifying the use of a single set of parameters of the Energy Corrected Sudden model for all species. 4.2. Atmospheric spectra The equations and procedures used here to compute atmospheric emission are similar to those described in I (i.e., Eqs. (1), (7), (22) of Ref. [1]). The data used to calculate CO Q-branch absorption are those described in the next section. Results obtained in the vicinity of the l Q branch of the OCO, OCO, OCO, and OCO isotopomers are illustrated in Figs. 7, 8, 9, and 10, respectively. As observed previously using mid latitude spectra [1], line mixing signi"cantly a!ects emission by CO and CO and e!ects are correctly accounted for by our model (Figs. 7 and 8). The relatively poor agreement observed in Fig. 8 is unexpected since more satisfactory results were obtained for CO in Ref. [1]. Among possible explanations are experimental uncertainty, improper geo physical parameters, and errors in the modeling of the contribution of the nearby P24 and P26 lines of the CO l band. Figs. 9 and 10 present more original results since they demonstrate, for the "rst time, that line mixing also a!ects absorption by OCO, and OCO. Furthermore, they con"rm the quality of our model. As analyzed in details in I, the decrease of tangent height signi"cantly reduces the in#uence of line mixing on radiance. As a result, e!ects in weak transitions (e.g. the minor isotopomers) are only discernible for high altitude spectra. Indeed, low tangent heights involve &&high'' pressures and absorption in the (weak) Q-branch wing is masked by the contribution of nearby intense lines (l P lines of CO ). In the case of OCO, and OCO, the in#uence of line mixing becomes smaller than the noise on measured value for tangent heights below about 20 km. In order to have an overall view of the quality of computations, we have thus retained the CO Q-branch. Relative deviations between measured and computed emission have been 䉳 Fig. 5. Absorption in the region of the (l }l ) Q branches: 䢇 measured values, - - - computed results neglecting ' line-mixing, }}} computed results accounting for line- mixing and corresponding Meas-Calc residuals. (a) P"2.96 atm and ¹"299 K. (b) P"9.86 atm and ¹"299 K. Fig. 6. Pressure dependent contributions to the &&width'' of the l Q branches (see text). Values deduced from computed spectra versus those determined from measured absorption with the conventions: 䊐 OCO, * OCO,
OCO and OCO, * OCO; open and full symbols are results obtained when line- mixing is accounted for and neglected, respectively. ***** Values for the OCO isotopomer are not given since absorption is to weak for correct determination of the width. The same value was attributed to the OCO and OCO isotopomers whose Q branches are very close.
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Fig. 7. Atmospheric radiance (normalized to the blackbody value at 277 K) in the (010)Q(000) Q branch of CO for a balloon altitude z "37 km looking horizontally (tangent height H g"37 km). The upper part gives measured values and the deviations with respect to calculations accounting for and neglecting line- mixing. The lower part presents details with the convention: 䢇 measured values; }}} computation accounting for line-mixing - - - computation neglecting for line-mixing.
determined for each spectrum at some speci"c spectral points: the latter have been selected in troughs between lines in the wing of the Q-branch (666}667 cm\). The results are plotted versus tangent height and wavenumber in Fig. 11. They con"rm the quality of the computation and the analysis of the e!ect of the optical path thickness made in Ref. [1]. Finally, comparisons are made between the present high latitude spectra (683N) and those used in the previous paper, which were recorded at mid latitude (343N). This comparisons is interesting since Fig. 12 shows that these #ights involve quite di!erent temperature pro"les. The consequences in terms of emission in the CO Q-branch wing are plotted in Fig. 13 which shows that our model correctly accounts for the in#uence of geophysical parameters. In the saturated region of strong absorption (right side of the plots), emission is governed by the temperature ¹(z ) at the balloon altitude; radiances in the Alaska #ight are about 8% lower than in the New Mexico
K.W. Jucks et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 63 (1999) 31}48
41
Fig. 8. Same as Fig. 7 but for CO .
experiment since the temperature at #oat is about 5 K lower. In the wing, results depend on the tangent height: for high altitudes (b and b in Fig. 13), results are quite similar since the number densities of both #ights are practically identical. For low tangent heights (a and a in Fig. 13), the higher temperature in the tangent layer (Fig. 13a ) leads to a signi"cantly steeper decrease with detuning from the Q branch head and a quicker vanishing of line mixing e!ects. This is the result of the fact that the increase of temperature at constant pressure leads to less collisions and thus to a reduction of absorption in the Q branch wing [2].
5. Data for atmospheric applications The Energy Corrected Sudden model (see I) has been used to compute a set of numerical data which is su$cient for calculation of atmospheric absorption by all signi"cant CO Q branches.
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Fig. 9. Same as Fig. 7 but for OCO and the tangent height H "33 km.
This database is similar to that described in Section 5 of I but now includes both symmetric and asymmetric isotopomers of CO . Only the vibrational transitions for which HITRAN-96 provides data on Q-lines have been treated, thus disregarding the OCO, OCO, OCO, and OCO isotopomers whose abundance and absorption in the atmosphere are very small. For each band, three data "les have been generated, giving the spectroscopic parameters, the "rst order line-coupling coe$cients, and the relaxation operator elements, respectively [1]. The main characteristics of the "nal database, which represents about 16 Megabytes of ASCII data and includes 271 Q branches, are summarized in Table 2. These data and some software (see I), are available by ftp Anonymous in the directory /PUB/CO2/ of the computer batz.lpma.u-psud.fr (IPC 194.57.26.165).
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43
Fig. 10. Same as Fig. 9 but for OCO.
Potential users should get the ReadMe.txt "le, read it and proceed according to instructions (Ref. [1] also brings useful information).
6. Conclusion The present and previous works have presented the results of a wide experimental and theoretical investigation of line-mixing e!ects in CO infrared Q branches. A large number of new laboratory measurements have been made for many isotopomers and bands, covering wide ranges of pressure and temperature. Comparisons with these data have proved the quality of the model
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Fig. 11. Relative deviations between measured and computed radiances in the wing of the CO l Q branch versus wavenumber and tangent height. (a) and (b) refer to computation respectively accounting for and neglecting line mixing.
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Fig. 12. Temperature and pressure vertical pro"les of the atmospheric measurements. The arrows indicate the tangent heights of the spectra retained in Fig. 13. }}} and - - - are the pro"les in the high (Alaska, 683N}1533W) and mid latitude (New Mexico, 343N, 2543W) experiments, respectively. The arrows indicate the tangent heights of the spectra in Fig. 13.
proposed which has be used to generate data and software for the prediction of CO Q-branch contributions to atmospheric spectra. This database, which is available to users, has been validated by comparisons with many transmission and emission recordings collected by stratospheric balloon-borne instruments. Recall that limitations and opened questions remain. The main limitation, discussed in details in the conclusion of I, is that the present model should be used only in the vicinity ($5}10 cm\) of Q branches. An other question concerns the neglecting of the basis rates Q(odd L). This approximation seems valid in view of the results obtained in the &% transitions studied here but further tests in *% Q branches would be of interest. Nevertheless, we expect the approximation to be satisfactory and, furthermore, concerned bands have very small absorption in the atmosphere.
Acknowledgements The authors from LPMA are grateful to the Centre National d'Etudes Spatiales for supporting part of this work under contract No. 96/CNES/0248 in the frame of spectroscopic studies in support of the IASI mission.
Appendix. Required basis rates for CO2 Q-branches Consider the case of the v vJ v Qv vJ v vibratonal band. Within the IOS approxima tion, the o!-diagonal element of the relaxation operator coupling the Q and Q lines, for CO }X ( (Y
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Fig. 13. Atmospheric radiance (normalized to the blackbody value at 277 K) in the (010)Q(000) Q branch of CO for a balloon altitude z "37 km and the tangent heights (a) 20 km, (b) 37 km. The left and right plots are for the New Mexico and Alaska experiments, respectively. 䢇 measured values; ** measured values; - - - computation accounting for the line-mixing.
Table 2 Number of Q branches for each isotopomer in the proposed database. Those marked with a * have been treated in Ref. [1]. The "les include those of spectroscopic, "rst order, and relaxation operator parameters and the eventual separation into two &&sub branches' Isotopomer
No.
Nat. Abund (%)
Nb of Bands
Nb of Files
O}C}O O}C}O O}C}O O}C}O O}C}O O}C}O O}C}O O}C}O
1 2 3 4 5 6 7 8
98.4 1.1 3.9;10\ 7.3;10\ 4.4;10\ 8.2;10\ 4.0;10\ 1.5;10\
*147 *53 33 23 6 5 *3 1
3;147"441 3;53"159 3;(14#2;19)"156 3;(12#2;11)"102 3;(3#2;3)"27 3;(3#2;2)"21 3;3"9 3;1"3
All
271
918
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collisions, is then given by [11] 1Q "WJ J } "Q 2"(2J #1)(2J#1 (!1)J>J [N N N N ] [1#(!1)*>(>(Ye Ge G ] (Y !- 6 ( J J J J ( J (Y J ( J ¸ J J ¸ J J J 1 ;[1#(!1)*>(>(Ye e ] ( J (Y J !l 0 !l !l 0 l J J ¸ G G (A1) ;(2¸#1)Q (¸), !- \6 where (2) and +2, are 3J and 6J coe$cients [12] N and e depend on the parity of the J ( J considered rovibrational levels and are given by
level type & (l"0), N
"1, e "0 J ( J e O " 1 for levels of parity e, ( J level typeO& (lO0), N O "1/(2, (A2) J e O "!1 for levels of parity f. ( J Note that contributions Q(¸,MO0, M O0) [11] have been neglected in Eq. (A1) [13] and, within this approximation, [1, 11, 14], lead to identical results when coupling within the Q branch is considered. For symmetric isotopomers in a (lO0) vibrational state, odd J levels have parity e and even J levels have parity f, or vice versa. The product e ;e is then (!1)(>(Y so that the associated ( J (Y J quantity [1#(!1)*>(>(Ye e ] in Eq. (A.1) is [1#(!1)*]. Only even values of ¸ contribute to ( J (YJ Eq. (A.1) which reduces to
J ¸ J l 0 !l * O0 G G J J 1 (2¸#1)Q } (¸), !- 6 J J ¸
1Q "WJ J } "Q 2"(2J #1)(2J#1 (!1)J>J (Y !- 6 (
;
J ¸ J
(A3) l For bands of asymmetric isotopomers the situation is di!erent since, for lO0, two levels (e and f ) are associated with each value of J. In this case, both odd and even values of ¸ contribute to Eq. (A1). Indeed, let us consider, for instance, the case of the (010) Q(000) band studied in the ' present work. All levels in the lower vibrational state (fundamental) have parity e whereas all "nal levels of the Q transitions (ef selection rule) have parity f. The product [1#(!1)*>(>(Ye Ge G ] [1#(!1)*>(>(Ye e ] is then [1#(!1)*>(>(Y] and the is divided ( J (Y J ( J (Y J into two sub branches associated with the even and odd values of J, respectively. Coupling within each of them is determined by the even values of ¸ and given by: !l
0
for "J!J " even: 1Q "WJ J } "Q 2"(2J #1)(2J#1 (!1)J>J (Y !- 6 (
;
J ¸ l
J
* O0 J ¸ J J
0 !l !l G ;(2¸#1)Q } (¸), !- 6
0
l
J
1
J J ¸
(A4)
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whereas odd ¸ values lead to interferences between the two sub-branches, i.e. for "J!J " odd: 1Q "WJ J - "Q 2"(2J #1)(2J#1 (!1)J>J (Y !- 6 (
;
J ¸
J
* O0 J ¸ J J
J
1
l 0 !l !l 0 l J J ¸ G (A5) ;(2¸#1)Q } (¸), !- 6 The case of bands other than &!% (l ;l O0) is basically similar although they involve twice more lines. The Q branch is composed of two &&sub branches'': the "rst is associated with (even J of type e#odd J of type f in the lower state) and the second contains lines of (odd J of type e#even J of type f in the lower state). Rates Q } (¸) with even ¸ values lead to mixing within the !- 6 &&sub-branches'' whereas those associated with odd ¸ values generate mixing between the two &&sub-branches''. When Q } (odd ¸) is neglected, there is no coupling between the &&sub-branches'' !- 6 which are thus isolated.
References [1] Rodrigues R, Jucks KW, Lacome L, Blanquet Gh, Walrand J, Traub WA, Khalil B, Le Doucen R, Valentin A, Camy-Peyret C, Bonamy L, Hartmann JM. JQSRT (in press). [2] Rodrigues, R, Khalil B, Le Doucen R, Bonamy L, Hartmann JM. J Chem Phys 1997;107:4118. [3] Rodrigues R, Blanquet Gh., Walrand J, Khalil B, Le Doucen R, Thibault F, Hartmann JM. J Mol Spectrosc 1997;186:256. [4] Bouanich JP, Rodrigues R, Hartmann JM, Domenech JL, Bermejo D. J Mol Spectrosc 1997;186:256. [5] Rothman LS, (HAWKS 1996) The HITRAN atmospheric workstation, ASA meeting, Reims, France, 4}6 Sept. (1996). [6] Tonkov MV, Boissoles J, Le Doucen R, Khalil B, Thibault F. JQSRT 1996;55:32. [7] Traub WA, Chance KV, Johnson DG, Jucks KW. J Soc Photo Opt Instrm Engng 1991;1491:298. [8] Johnson DG, Jucks KW, Traub WA, Chance KV. J Geophys Res 1995;100D:3091. [9] Te!o JL, Spectroscopic constants for the l and l levels of minor isotopomers. Private communication. [10] Te!o JL, Claveau C, Valentin A. JQSRT 1998;59:151. [11] Green S. J Chem Phys 1979;70:816. [12] Edmonds AR. In: Hofstadter R, Editor. Angular momentum in quantum mechanics, Princeton: Princeton University Press, 1974. [13] Green S. J Chem Phys 1989;90:3603. [14] Bonamy L, Emond F, Phys Rev 1995;A51:1235.
