- Email: [email protected]
Spectrum of water vapor between 8050 and 9370 cm−1
JOURNAL
OF
MOLECULAR
75, 339-362 (1979)
SPECTROSCOPY
Spectrum of Water Vapor between 8050 and 9370 cm-l J.-M. Laboratoire
K.
FLAUD
AND C. CAMY-PEYRET
de Physique Mokulaire et d’optique Afmosphkrique, Campus d’orsay, 91405 Orsay Cedex. France
NARAHARI
Department
RAO,
of Physics,
DA-WUN
CHEN,
C.N.R.S.,
AND YAN-SHEK
The Ohio State University, Columbus,
BBt. 221,
HOH
Ohio 43210
AND
J.-P. Observatoire
de Meudon,
MAILLARD
I Place J. Janssen,
92190 Meudon,
France
Measurements of the line positions of H,‘“O in the 8050 to 9370 cm-’ region have been performed at a spectral resolution of 0.07 cm-‘. A grating spectrum of room-temperature water vapor and a Fourier transform spectrum of heated water vapor (T = WC) were used in the interpretation. A careful analysis of the bands V, + 3~~. 3~~ + vQ, 2~~ + Ye, v, + v2 + I+, and vz + 2~~ has led to a set of rotational levels belonging to the vibrational states (130), (031), (210), (ill), and (012). Many vibrorotational resonances were detected.
INTRODUCTION
The vibration-rotation spectrum of the water molecule begins to be understood in great detail in the 6.3~pm (Z-3), 2.7~pm (4-6), 1.9~pm (1,6,7), and 1.4~pm (8,9) regions. Toward the photographic infrared and the visible, the situation is not so favorable because the bands are weak and also because of the possibility that multiple resonances will make both the assignment and the calculation of spectra much more difficult. However, it is important to improve our knowledge of these spectral regions from both practical considerations (atmospheric studies) and theoretical reasons (determination of the potential function). A systematic effort to analyze the H,O spectra in these regions was performed earlier by Benedict (20). In the present paper, we report an experimental investigation of the spectrum of water vapor from 8050 to 9370 cm-‘; we have been able to assign about 1200 vibration-rotation lines belonging to the v1 + 3v,, 3~~ + vQ, 2v, + v2, v1 + v2 + v3, and V, + 2v, bands. From the observed line positions, an extensive set of experimental rotational levels has been obtained for the vibrational states (130), (031), (210), (ill), and (012).
339
0022-2852/79/060339-24$02.00/O Copyright
0
1979 by Academic
All rights of reproduction
Press.
Inc.
in any form reserved.
FLAUD ET AL.
340
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1. l-pm BANDS EXPERIMENTAL
OF I&.‘“0
34.5
DETAILS
For this study, we analyzed two different spectra. For the strongest lines, we used the line positions given by a room-temperature spectrum recorded from 8550 to 9024 cm-l. The positions of the weaker lines were deduced from a Fourier transform spectrum of heated water vapor extending from 8050 to 9370 cm-l. The cha~~te~stics of the two spectra are given below: Room-Tempera0.ue
Spectrum
A 10-m focal length Czerny-Turner vacuum spectrometer at the Ohio State University was used to record the spectrum. This spectrometer employed a 40 x 20-cm2 (16 x 8-in.2) echelle with 79 grooves per millimeter on its surface and blazed at an angle of about 63”. The source of continuous infrared radiation was a carbon rod furnace working at an input power of 3.4 kW (10 V at 340 A). A desc~pt~on of this spectrometer was given previously (t I). The infrared radiation was detected with two detectors -1nSb for the CO calibrating lines and a liquid-nitrogen-cooled photomultiplier for the water vapor spectrum. The signal was amplified by a Princeton Applied Research Model HR-8 phase-sensitive amplifier and then recorded on a strip chart recorder. The resolution was about 0.07 cm-’ and the precision of the positions of the lines is approximately eO.005 cm-l. Figure 1 shows a reproduction of this spectrum. The water vapor partial pressure used was 7 mm Hg, and the length of the abso~tion path was 100 m. Fourier
Transform
Specfrum
cell (T = 6O”C, by one of us (J.-P.M.) with a resolution 6a = 0.070 cm-‘. Under these conditions, the experimental linewidth at half height was approximately 0.11 cm-l, and the precision of the positions of the tines was t0.005 cm-‘. Figure 2 shows a reproduction of the observed spectrum. For unblended and unsaturated lines common to both spectra we have noticed that the line positions agree to within the stated experimental uncertainties, that is rtO.005 cm-‘. The water vapor was introduced
into a heated multiple-path
P = 90 Torr, total length L = 40 m) and the spectrum was recorded
DATA ANALYSIS
AND RESULTS
The identification of the lines of H,160 was facilitated by the previous work of Benedict (10) and with the help of the available rotational energy levels for the (000) and (111) vibrational states of H2160 (12,13). In the 1.1~pm region, the A-type band v, + v2 f v3 is much stronger (see the strong Q branch at 8807 cm-‘) than the bands y1 + 3v,, 3v, + vg, 2v, + v2, and v2 + 2~. Consequently, the number of levels determined for the (111) state is larger than the number determined for the (130), (031), (210), and (012) states. Also, since the I+ + vz + vQband center (8807.000 cm-l) is close to the 2u, + v2 band center (8761.579 cm-l), the lines of this weak band are often blended with lines of the strong band, which complicated the interpretation.
I f
8060
T = WC).
FIG. 2.Heated water vapor spectrum (Fourier transform spectrometer,
I I I I I1 11 I, I I I1 1 I I, 8110
8050
1,,111,~1”1~,1111,,,,,1,11,,,,,,,,,,,,,,,,,,~,
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resolution 0.07 cm-‘) between 8050 and 9370 cm-’ (L = 40 m, p 2 9@‘Torr,
I, i I 8120
8070
1. l-pm BANDS OF H2180
347
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--ts
t -8
--2
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-iiJ -2 -8
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--si 0
-4
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FLAUD ET AL.
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--E!
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-&
1. I-pm BANDS OF Hzx60
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351
1. l-pm BANDS OF H,160
-r: --fs
-s
-A?
F-
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-
F
-
9300
I
I--, i
I,,
H20
t
I
I,,
I, I
9310
I I
i,,
,
FIG.2--Continued.
3&T----
9190
1-r-r-r 9200
353
1. I-&m BANDS OF H,r60 TABLE I List of the Lines of Water Vapor between 8050 and 9370 cm-r a
il The columns mean: d,
”
Labels of the upper correspondence:
and lower
J‘K;K:JK,K, PI,
Observed wavenum~r
states
of the transition
1 = (OlO),
20 = (060),
15 = (130),
16 = (0311,
17 = (210).
18 = (ill),
19 = (012),
24 = (121).
0 = (OOO),
Sigma
vibrational
with the
of the transition (in cm-‘)
Rotational quantum numbers of the upper and lower levels p2
G I
Percentages of absorption at the center of the line for the heated and room-temperature spectra Statistical weight of the line Isotopic species with the notation 0 = H,160. 1 = H,‘rO, 2 = H,*80
354
FLAUD ET AL.
TABLE
I-Continued
“““L”““““”
. . . . ..o... ..**.*...o
NL”“_
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356
FLAUD ET AL.
1. l-pm BANDS OF H,160
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FLAUD ET AL.
358
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TABLE
I
1. I-pm BANDS OF H,160
100
3
1
1
100 100 100 100 100
0
0
TABLE I-Confinued
,a
0
19 10
0 0
360
FLAUD ET AL. TABLE II Energies (cm-‘) of the Rotational Levels for the Vibrational States (130), (031), (210), (11 l), (012), of H,‘*Oa LJO
03L
210
11,
012
I. I-/XII BANDS OF Hz’“0
361
a To the right of each energy level (in cm~‘) the statistical uncertainty (68% confidence interval in IO-:’ cm-l) is given. Additional level appearing through perturbation: (060) 161619400.6409 2.4.
Particular attention was paid to lines originating from resonant levels which, as already mentioned, are quite numerous.’ In general, perturbed lines were located by considering strong lines which cannot be attributed to normal lines of the strongest band, vl + v2 + z+. Among these perturbations are two interesting examples: -The perturbation between the levels (210)[221] and (111)[2 1l] leads to doublets of lines of nearly equal intensity, instead of the usual single lines belonging to the strongest A-type band v1 + v2 + vS (10). -An extreme example of resonance is the quadruple resonance between the levels (060)[616], (130)[652], (210)[634], and (111)[624]. In this case, the labeling of the levels will be definitive only when a theoretical calculation provides the mixing coefficients. It may be emphasized that the 6~~ band appears only through lines originating in the perturbed level (060)[616]; this behavior is similar to that of the 1These resonances can be classified for the hexad of the vibrational states {(OSO),(130). (031). (210). (11 l), (012)} under study as follows: Coriolis-type interactions between ( L’,upu3)and (u, - 1 L’~ o3 + 1): (CAL’& and (u,u* - 2u3 + 1); Fermi-type interaction between (v,v,v,) and (0, - luz + 2~~): and Darling-Dennison-type interaction between ( u, upus)and ( u, - 2 L’~cg + 2).
362
FLAUD ET AL.
4~~ band, in the 1.4-pm region, which appears only through lines originating in the perturbed level (040)[945], (14). Because of the large value of the P x L product for the heated spectrum, we have been able to detect lines belonging to the “hot” band v1 + 2~~ + v3 - r+ (relative intensity compared to v1 + v2 + v3 = 0.2%) and lines belonging to the y1 + v2 + v3 band of H2180 (relative isotopic abundance -0.2%). The assignment of this band of H2180 has been facilitated by knowledge of the ground state of H2180 (1.5,16). The complete list of the transitions with their assignments and the percent absorption at the center of the lines is given in Table I. From these lines and from the ground-state energy levels, we deduced the vibration-rotation energy levels of the upper states, which are given in Table II. Since the rotational levels of the (111) vibrational state were obtained previously from the study of a flame spectrum (13), we combined the data originating from the latter work and the present study to obtain what we consider to be the most reliable set of rotational energy levels for this state. In the present work, we did not observe lines involving high rotational quantum numbers, and, therefore, we limited the levels listed in Table II to J s 13. Reference (13) gives the information for./ > 13 of the (111) state. ACKNOWLEDGMENTS The authors wish to thank Professor W. S. Benedict for valuable comments about this paper. One of us (K.N.R.) expresses his gratitude to the Atmospheric Research Section of the National Science Foundation and the National Aeronautics and Space Administration for the support provided for this investigation. RECEIVED:
May 19, 1978 REFERENCES
I. W. S. BENEDICT AND R. F. CALFEE, ESSA Prof. Paper 2, 1%7. 2. C. CAMY-PEYRET AND J.-M. FLAUD, Mol. Phys. 32, 523-537, (1976). 3. J.-M. FLAUD, C. CAMY-PEYRET, J.-Y. MANDIN, AND G. GUELACHVILI, Mol. Phys. 34,413-426, (1977). 4. D. M. GATES, R. F. CALFEE, D. W. HANSEN AND W. S. BENEDICT, Nat. Bur. Stand. (U. S.) Monogr. 71, 1964. 5. J.-M. FLAUD AND C. CAMY-PEYRET, J. Mol. Spectrosc. 55, 278-310, (1975). 6. L. A. PUGH, Ph.D. dissertation, The Ohio State University, 1972. 7. C. CAMY-PEYRET, J.-M. FLAUD, AND R. A. TOTH, J. Mol. Spectrosc. 67, 117-131, (1977). 8. R. A. TOTH AND J. S. MARGOLIS, J. Mol. Spectrosc. 55, 229-251, (1975). 9. C. CAMY-PEYRET AND J.-M. FLAUD, These, CNRS, A0 11443, Universite Pierre et Marie Curie, Paris, 1975. IO. W. S. BENEDICT, Phys. Rev. 74, 1246-1247, (1948); J. W. SWENSSON, W. S. BENEDICT, L. DELBOUILLE, AND G. ROLAND, Mem. Sot. Roy. Sci. LiPge, Vol. Hors Ser., No. 5, 8, 1970. II. B. D. ALPERT, Ph.D. dissertation, The Ohio State University, 1970. 12. J.-M. FLAUD, C. CAMY-PEYRET,AND J.-P. MAILLARD,Mol. Phys. 32, 499-521, (1976). 13. C. CAMY-PEYRET,J.-M. FLAUD, J.-P. MAILLARD, AND G. GUELACHVILI,Mol. Phys. 33, 16411650, (1977). 14. W. S. BENEDICT,M. A. POLLACK,AND W. J. TOMLINSON,IEEEJ. Quantum Electron. 5,108-124, (1969). 15. R. A. TOTH, J.-M. FLAUD, AND C. CAMY-PEYRET,J. Mol. Spectrosc. 16. J.-M. FLAUD, C. CAMY-PEYRET,AND R. A. TOTH, J. Mol. Spectrosc.
67, 185-205,
(1977).
68, 280-287, (1977).
OF
MOLECULAR
75, 339-362 (1979)
SPECTROSCOPY
Spectrum of Water Vapor between 8050 and 9370 cm-l J.-M. Laboratoire
K.
FLAUD
AND C. CAMY-PEYRET
de Physique Mokulaire et d’optique Afmosphkrique, Campus d’orsay, 91405 Orsay Cedex. France
NARAHARI
Department
RAO,
of Physics,
DA-WUN
CHEN,
C.N.R.S.,
AND YAN-SHEK
The Ohio State University, Columbus,
BBt. 221,
HOH
Ohio 43210
AND
J.-P. Observatoire
de Meudon,
MAILLARD
I Place J. Janssen,
92190 Meudon,
France
Measurements of the line positions of H,‘“O in the 8050 to 9370 cm-’ region have been performed at a spectral resolution of 0.07 cm-‘. A grating spectrum of room-temperature water vapor and a Fourier transform spectrum of heated water vapor (T = WC) were used in the interpretation. A careful analysis of the bands V, + 3~~. 3~~ + vQ, 2~~ + Ye, v, + v2 + I+, and vz + 2~~ has led to a set of rotational levels belonging to the vibrational states (130), (031), (210), (ill), and (012). Many vibrorotational resonances were detected.
INTRODUCTION
The vibration-rotation spectrum of the water molecule begins to be understood in great detail in the 6.3~pm (Z-3), 2.7~pm (4-6), 1.9~pm (1,6,7), and 1.4~pm (8,9) regions. Toward the photographic infrared and the visible, the situation is not so favorable because the bands are weak and also because of the possibility that multiple resonances will make both the assignment and the calculation of spectra much more difficult. However, it is important to improve our knowledge of these spectral regions from both practical considerations (atmospheric studies) and theoretical reasons (determination of the potential function). A systematic effort to analyze the H,O spectra in these regions was performed earlier by Benedict (20). In the present paper, we report an experimental investigation of the spectrum of water vapor from 8050 to 9370 cm-‘; we have been able to assign about 1200 vibration-rotation lines belonging to the v1 + 3v,, 3~~ + vQ, 2v, + v2, v1 + v2 + v3, and V, + 2v, bands. From the observed line positions, an extensive set of experimental rotational levels has been obtained for the vibrational states (130), (031), (210), (ill), and (012).
339
0022-2852/79/060339-24$02.00/O Copyright
0
1979 by Academic
All rights of reproduction
Press.
Inc.
in any form reserved.
FLAUD ET AL.
340
-8 3 _ -
Y
5
_
--
b g
-
NOlldllOSBV
-3
341
iir
I-8
3
-
P
-
%
-
0
-
NOlld8OS8V
0
u
3
FLAUD ET AL.
342
8
-
MHldUOSUV
-I
0
-3
344
FLAW
-8
ET AL.
m
8
-5:
_E: 8
$
ii-
-s 8
I
5
*
NOlld10SW
I
ii
-
NOlldlOSiBV
1. l-pm BANDS EXPERIMENTAL
OF I&.‘“0
34.5
DETAILS
For this study, we analyzed two different spectra. For the strongest lines, we used the line positions given by a room-temperature spectrum recorded from 8550 to 9024 cm-l. The positions of the weaker lines were deduced from a Fourier transform spectrum of heated water vapor extending from 8050 to 9370 cm-l. The cha~~te~stics of the two spectra are given below: Room-Tempera0.ue
Spectrum
A 10-m focal length Czerny-Turner vacuum spectrometer at the Ohio State University was used to record the spectrum. This spectrometer employed a 40 x 20-cm2 (16 x 8-in.2) echelle with 79 grooves per millimeter on its surface and blazed at an angle of about 63”. The source of continuous infrared radiation was a carbon rod furnace working at an input power of 3.4 kW (10 V at 340 A). A desc~pt~on of this spectrometer was given previously (t I). The infrared radiation was detected with two detectors -1nSb for the CO calibrating lines and a liquid-nitrogen-cooled photomultiplier for the water vapor spectrum. The signal was amplified by a Princeton Applied Research Model HR-8 phase-sensitive amplifier and then recorded on a strip chart recorder. The resolution was about 0.07 cm-’ and the precision of the positions of the lines is approximately eO.005 cm-l. Figure 1 shows a reproduction of this spectrum. The water vapor partial pressure used was 7 mm Hg, and the length of the abso~tion path was 100 m. Fourier
Transform
Specfrum
cell (T = 6O”C, by one of us (J.-P.M.) with a resolution 6a = 0.070 cm-‘. Under these conditions, the experimental linewidth at half height was approximately 0.11 cm-l, and the precision of the positions of the tines was t0.005 cm-‘. Figure 2 shows a reproduction of the observed spectrum. For unblended and unsaturated lines common to both spectra we have noticed that the line positions agree to within the stated experimental uncertainties, that is rtO.005 cm-‘. The water vapor was introduced
into a heated multiple-path
P = 90 Torr, total length L = 40 m) and the spectrum was recorded
DATA ANALYSIS
AND RESULTS
The identification of the lines of H,160 was facilitated by the previous work of Benedict (10) and with the help of the available rotational energy levels for the (000) and (111) vibrational states of H2160 (12,13). In the 1.1~pm region, the A-type band v, + v2 f v3 is much stronger (see the strong Q branch at 8807 cm-‘) than the bands y1 + 3v,, 3v, + vg, 2v, + v2, and v2 + 2~. Consequently, the number of levels determined for the (111) state is larger than the number determined for the (130), (031), (210), and (012) states. Also, since the I+ + vz + vQband center (8807.000 cm-l) is close to the 2u, + v2 band center (8761.579 cm-l), the lines of this weak band are often blended with lines of the strong band, which complicated the interpretation.
I f
8060
T = WC).
FIG. 2.Heated water vapor spectrum (Fourier transform spectrometer,
I I I I I1 11 I, I I I1 1 I I, 8110
8050
1,,111,~1”1~,1111,,,,,1,11,,,,,,,,,,,,,,,,,,~,
1
J
1
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I,
I
I
I I,
I
/
I
--"I-*
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I I I1 : I I !
---I$7
8080
8140
resolution 0.07 cm-‘) between 8050 and 9370 cm-’ (L = 40 m, p 2 9@‘Torr,
I, i I 8120
8070
1. l-pm BANDS OF H2180
347
-Z
--ts
t -8
--2
-P -2 -Z --E
-iiJ -2 -8
--f2
-55 -4%
-3 -42
-S
--si 0
-4
348
FLAUD ET AL.
z2 2
-s
--E!
/
-=:
--2
-z
--3
-s
-42 -3 -43
-z
-4 -63 -23
-a
--E -B
-2
-s
-&
1. I-pm BANDS OF Hzx60
349
r
350
FLAUD ET AL.
G ..-%
23
--z
-3
--fs
-5:
-3
-53 -23
351
1. l-pm BANDS OF H,160
-r: --fs
-s
-A?
F-
F s=-
-
F
-
9300
I
I--, i
I,,
H20
t
I
I,,
I, I
9310
I I
i,,
,
FIG.2--Continued.
3&T----
9190
1-r-r-r 9200
353
1. I-&m BANDS OF H,r60 TABLE I List of the Lines of Water Vapor between 8050 and 9370 cm-r a
il The columns mean: d,
”
Labels of the upper correspondence:
and lower
J‘K;K:JK,K, PI,
Observed wavenum~r
states
of the transition
1 = (OlO),
20 = (060),
15 = (130),
16 = (0311,
17 = (210).
18 = (ill),
19 = (012),
24 = (121).
0 = (OOO),
Sigma
vibrational
with the
of the transition (in cm-‘)
Rotational quantum numbers of the upper and lower levels p2
G I
Percentages of absorption at the center of the line for the heated and room-temperature spectra Statistical weight of the line Isotopic species with the notation 0 = H,160. 1 = H,‘rO, 2 = H,*80
354
FLAUD ET AL.
TABLE
I-Continued
“““L”““““”
. . . . ..o... ..**.*...o
NL”“_
+..a.
-....
“““rrrrrr..rr
D.*P.....*.*m
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a
”
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a.
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00
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”
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LC””
+N~.=.W:O**W
UWY-Id-“D”*ON*
356
FLAUD ET AL.
1. l-pm BANDS OF H,160
1)es..e75. O655.I 055 0655.,055 66IIS.541.
:xFT*“,E: 66s?:‘6‘,
**6?.9,60 6l)66..6LO 6*66.?06‘ 6*66.*?30 6669.**,0 6M1.1219 6661.3430 666*.s33.¶ 6663*3..* i3t33.;w: 6666h6.9 ~~.~~=~ 6666:6339 6666.?3?? 66e.6.936, 6669.6?3, 66TO.3936 66?0.662S 66?,..766 II?,..?66 66?1.?6?2 6672.0396 ,6?*.,39, 66l3.6606 66?..3.36 66?..6?6. 66?..,?6. 66?6.266) 66??.666, 66??.6z6, 66?6+1661 66?&.,63 66?64,96 66?P.,L66 *66*.0*.* 6666.6W9 pg;*:o~$ 6662:ioss 666*.6?t* 666,.,76? 666‘*06.6 **-+‘*?73 6666*6,30 666?.?,?6
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2
423 :::
202 7P6 503 616 "j:: 821 62s
FLAUD ET AL.
358
I9
0
17
P
TABLE
I
1. I-pm BANDS OF H,160
100
3
1
1
100 100 100 100 100
0
0
TABLE I-Confinued
,a
0
19 10
0 0
360
FLAUD ET AL. TABLE II Energies (cm-‘) of the Rotational Levels for the Vibrational States (130), (031), (210), (11 l), (012), of H,‘*Oa LJO
03L
210
11,
012
I. I-/XII BANDS OF Hz’“0
361
a To the right of each energy level (in cm~‘) the statistical uncertainty (68% confidence interval in IO-:’ cm-l) is given. Additional level appearing through perturbation: (060) 161619400.6409 2.4.
Particular attention was paid to lines originating from resonant levels which, as already mentioned, are quite numerous.’ In general, perturbed lines were located by considering strong lines which cannot be attributed to normal lines of the strongest band, vl + v2 + z+. Among these perturbations are two interesting examples: -The perturbation between the levels (210)[221] and (111)[2 1l] leads to doublets of lines of nearly equal intensity, instead of the usual single lines belonging to the strongest A-type band v1 + v2 + vS (10). -An extreme example of resonance is the quadruple resonance between the levels (060)[616], (130)[652], (210)[634], and (111)[624]. In this case, the labeling of the levels will be definitive only when a theoretical calculation provides the mixing coefficients. It may be emphasized that the 6~~ band appears only through lines originating in the perturbed level (060)[616]; this behavior is similar to that of the 1These resonances can be classified for the hexad of the vibrational states {(OSO),(130). (031). (210). (11 l), (012)} under study as follows: Coriolis-type interactions between ( L’,upu3)and (u, - 1 L’~ o3 + 1): (CAL’& and (u,u* - 2u3 + 1); Fermi-type interaction between (v,v,v,) and (0, - luz + 2~~): and Darling-Dennison-type interaction between ( u, upus)and ( u, - 2 L’~cg + 2).
362
FLAUD ET AL.
4~~ band, in the 1.4-pm region, which appears only through lines originating in the perturbed level (040)[945], (14). Because of the large value of the P x L product for the heated spectrum, we have been able to detect lines belonging to the “hot” band v1 + 2~~ + v3 - r+ (relative intensity compared to v1 + v2 + v3 = 0.2%) and lines belonging to the y1 + v2 + v3 band of H2180 (relative isotopic abundance -0.2%). The assignment of this band of H2180 has been facilitated by knowledge of the ground state of H2180 (1.5,16). The complete list of the transitions with their assignments and the percent absorption at the center of the lines is given in Table I. From these lines and from the ground-state energy levels, we deduced the vibration-rotation energy levels of the upper states, which are given in Table II. Since the rotational levels of the (111) vibrational state were obtained previously from the study of a flame spectrum (13), we combined the data originating from the latter work and the present study to obtain what we consider to be the most reliable set of rotational energy levels for this state. In the present work, we did not observe lines involving high rotational quantum numbers, and, therefore, we limited the levels listed in Table II to J s 13. Reference (13) gives the information for./ > 13 of the (111) state. ACKNOWLEDGMENTS The authors wish to thank Professor W. S. Benedict for valuable comments about this paper. One of us (K.N.R.) expresses his gratitude to the Atmospheric Research Section of the National Science Foundation and the National Aeronautics and Space Administration for the support provided for this investigation. RECEIVED:
May 19, 1978 REFERENCES
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