Carbon dioxide is one of the more important atmospheric infrared-absorbing gases due to its relatively high, and increasing, concentration. The spectral parameters of its bands are required for understanding radiative heat transfer in the atmosphere. The line intensities, positions, line half-widths, rotational constants, and band centers of three overlapping bands of CO2 near 2.2 microns are presented. Non-linear least squares (NLLS) regression procedures were employed to determine these parameters

Abubaker, M. S.

Shaw, J. H.

English

NASA Technical Reports Server

ANALYSIS OF CARBON DIOXIDE BANDS NEAR 2.2 pro
H. S. Abubakar and J. H. Shaw
Department of Physics, The Ohio State University,
Columbus, OH 43210
https://ntrs.nasa.gov/search.jsp?R=19840017711 2020-03-20T21:49:45+00:00Z
Abstract
The absolute intensities, positions, rotational constants and half-widths
-1
of the lines of three overlapping bands of CO near 4640 cm have been
2
determined by nonlinear, least-squares, fitting procedures. The intensities
0 0 1 1 16 12 18
of (00 2) «- 00 0 and (01 2) «- (01 0) of 0 C 0 are In good agreement
I I I
with the estimates in the AFGL listing. An improved value has been obtained
1 0 16 12 16
for the intensity of (31 0) «- 00 0 of 0 C 0.
Ill
1. INTRODUCTIOM
Carbon dioxide is one of the more important, atmospheric, infrared-
absorbing gases due to its relatively high and increasing concentration. The
spectral parameters of its bands are required for understanding radiative
transfer in the atmosphere and for many other applications. In this paper,
the line intensities and self-broadened widths as well as the band centers of
three overlapping bands of CO near 2.2 ym are reported. These were obtained
2
by nonlinear, least-squares (NLLS) regressions based on the method of Lin et
al. (1) as modified by Hoke and Shaw (2,3).
2. DATA
The experimental conditions of the CO samples are summarized in Table
2
-1
I. The bands occur between about 4545 and 4662 cm , Analyses of similar
-1
bands near 2600 cm in these same spectra were reported recently (2,3) and
these references give descriptions of the spectra and the calibration
procedures. Because of the higher detectivity of the InSb detector at longer
wavelengths the SNR of the spectra previously analyzed was in excess of 500.
Fig. 1 shows a portion of Spectrum I. The main band (centered around
-1
4640 cm ) has no Q-branch. Because there is a weak unresolved Q-branch near
-1 -1
4591 cm , the region between 4590 and 4592 cm was assigned a weight of zero
in all of the analyses. The data consisted of signal values obtained from
unapodized interferograms.
3. THEORY
A spectrum similar to the observed spectrum is calculated from model
functions with spectral quantities as adjustable parameters. In this analysis
the models consisted of the upper and lower state energy levels, the line
strengths and the dependence of the collision-broadened half-widths on
rotational quantum number. The influences of the source and the observing
instrument on the spectrum were also taken into consideration by modeling the
effects of the retardation A of the interferometer mirror, the uncorrected
phase error 4> and the angular spread of the rays passing through the optical
system. The frequency dependence of the baseline was described by a single
parameter but, for the background, a five-degree polynomial was needed.
The calculated signal values S'(v) are convolved with an instrument
spectral response function ISRF(v) as expressed by the relation:
S(v ) = /S'(v) ISRF(v ,v) dv. [1]
The spectral response function used in this analysis is of the form (4),
2,ISRF(A,4>,v,d) Acos4> sinc(2uA) + sin4» sin (uA)/uA [2]
where u = «(v - v ), and the sine and cosine functions model the uncorrected
o
phase error 4>. ¥(d) is a rectangular function whose half-width d depends on
the angular spread of the rays passing through the interferometer. This has
been described as the 'extended source effect' (4). Unlike the retardation,
the phase error and extended source parameters are frequency dependent.
The line positions are given by the difference of term values
[3]
where the primes and double primes Indicate the upper and lower levels
respectively. The term values are
F(v,J,l) = Gir 0 + Bv 0[J(J+1) - I2] - D 0[J(J+1) - I2]2 + ... , [4]v.Jfc v, x. v,*
where the G's are the vibrational terms and the B's and D's are the rotational
parameters.
Individual line intensities are expressed (3, 6), by;
S(m) = [8*3CjR(0)|2/3hcQ_] v (m)F(m)C (m) exp[-E"(m)/kT], [5]N n p o
where m = -J for the P-branch and m = J + 1 for the R=branch, and
F(m) =
2 2C (m) = (m - l)/|m|
o
for the * - « and
bands.
F(m) = (1
for the » - J band.
CQ(m) = |m
C Includes the vibrational partition sum and the natural abundance of the
N
isotope. The other symbols have their usual meaning. This relationship
yields Intensities which can be compared directly with those in the AF6L
Listing (7). A natural CO sample was used and the isotopic abundances are
2
assumed known.
As expected, the Lorentz and Voigt profiles yielded essentially identical
results for the line widths and intensities. The |m| dependence of the widths
was modeled by using the empirical relationship of Winters et al.(8).
a(m) = <»0(P.T) |Wo + W1 exp(W2|m|) + W3|m| exp[-Bw|m| (|m| - D/kT]] , [6]
where aQ(P,T) = P/PQ(1Q/T)%, PQ = 760 torr, TQ = 296 K, and the V^ are
regression parameters. This expression was found to be appropriate in the
previous analyses (2).
The regression program compares the calculated signal values with the
experimental values. The set of parameters which gives the minimum value for
the sum of squares of the differences between these signal values is
considered to give the best estimates for the spectral quantities. The total
number of data points included in the analysis is 3883 and the number of
parameters usually retrieved is between 30 and 40.
4. RESULTS
It is convenient to define a "signal-to-difference" ratio (SDR) by
analogy with the signal-to-noise ratio (SNR), where the "difference" is
defined as the standard deviation of the differences between the calculated
and the experimental signal values. The best values of the SDR were 110 for
Spectrum I and 220 for Spectrum II. The corresponding values of the SNR are
130 and 260. The differences between the SDR and SNR are primarily due to
very weak lines of unmodeled hot bands.
The experimental and the calculated signal values of a portion of band I
from Spectrum II are compared in Fig. 2. A plot of the residual is also shown
below on an expanded scale. A /noise spectrum' obtained from a region with no
significant absorption is also given together with its corresponding residual
spectrum which, in this case, is the difference from a straight line through
the noise. The appearances of these two residual spectra are very similar.
a. Vibration-Rotation Constants
The vibration-rotation parameters in Table II were only determined from
Spectrum I since the broader lines of Spectrum II are unsuitable for precise
line-position-parameter determination. The numbers of lines modeled in each
band are given in Table III. The quoted uncertainty in each parameter is 3 0,
where a is the standard deviation estimated by the regression program. Two
sets of rotational constants were obtained for band I and band II; one from a
retrieval with no constraints and the other with the lower state rotational
constants fixed to some recently reported values (7, 11, 12). In neither case
was it possible to obtain significant estimates of the H's and these
parameters have -been assigned the value zero.
The band centers in Table II include the contribution due to the
vibrational angular momentum quantum number, so that
v .. = AG , „ * (B-l"2 + DMl"4 - B'l'2 - D'l'4) , [7]
center v*,v"
where AG is the difference between the upper and lower vibrational
v* ,v"
levels. Other estimates of band parameters are also included in Table II.
Different estimates for the relative line positions for band I are
compared in Fig. 3. The differences are given by v - v where the v are the
a r r
line positions calculated with the constants obtained from the best
unconstrained retrieval and v are the positions obtained by assuming a common
a
band center and either the parameters reported by other investigators or
obtained from other retrievals assuming a common band center. The three
unlabeled curves near the zero line represent parameters from three different
unconstrained retrievals with essentially the same SDR values. Their
agreement is a measure of the internal consistency of the results. Line
position differences for the other bands are less consistent since the bands
are weaker and hence their parameter estimates are less precise than those of
band I.
The rotational parameters from Spectrum I were used to calculate the line
positions of Spectrum II but new values for the band centers of Spectrum II
-1
were retrieved. These showed a shift of about 0.01 cm from the low pressure
positions. Shifts of this magnitude are within the range routinely observed
on spectra measured on different days with our equipment. They are attributed
to small changes in the optical alignment of the spectrometer (2) and they
make it necessary to calibrate each spectrum individually (13).
b. Intensities
The band intensity is the sum of the line intensities S (v):
CO
s° = Z s (v) ,
i=l
The dipole moment matrix elements and band intensities obtained from both
spectra are given in Table IV. A strong Coriolis interaction in band II
causes the intensity asymmetry of the P- and R-branches shown in Fig. 4. Line
intensities for this band vanish near m = -6. The existence of similar
asymmetries in other w - £ bands of CO has also been reported (14, 15, 16,
2
17). The subbands of band III were treated as separate bands with a common
band center and dipole moment. The reported band intensity is the sum of the
intensities of both sets of lines. It was found that vibration-rotation
8interaction parameters could only be retrieved for band II.
c. Half-widths
The Lorentz widths as functions of |m| shown in Fig. 5 were obtained from
the parameter values of Table V. There is good agreement for |m| > 10. The
same width parameters were used for all bands in each spectrum. In Fig. 6 the
widths from the high pressure spectrum are compared with values obtained from
-1 -1
the 2600 cm region of the same spectrum and from near 2760 cm in another
spectrum recorded at 741 torr and 292 K (2, 18). There is some evidence of a
weak dependence of the widths on the vibrational transition although the
differences between the previously estimated widths and those reported in this
work are within 3 o, where a is the standard deviation in a(m) estimated from
the standard deviations of the parameters V of Eq. (6).
d. On background, baseline and instrument effects
A five-degree polynomial was required to model the background. The
baseline was more difficult to determine because of the small absorptances of
the lines (average peak absorptance value ~0.25). For Spectrum II a value of
-0.100(2) was found. This is about 10% of the observed signal for zero
absorptance. In Spectrum I the retrieved value of the baseline shift was
smaller than its uncertainty and hence it was set to zero. The retrieved
values for the retardation shown in Table VI agree with those estimated
previously (4).
5. DISCUSSION
Despite the decreased SNR the overall precision obtained in the
determination of intensities and rotational constants is comparable with that
-1
reported for the 2600 - 2800 cm region in the previous analyses. As
/
expected, the most precise constants and intensities were obtained from the
strongest band (band I). The intensity of band II appears to be significantly
greater than the estimate given in the AFGL listing (7). We are not aware of
any other reported experimental values for these band intensities.
The dipole moment matrix element values in Table IV agree within 1% for
band I and 2% for band II and band III. This is within the estimated
experimental errors.
It appears, from Table V, that, although there is some uncertainty in the
width parameters, the CO widths from different spectral regions and different
2
spectra are essentially consistent within the experimental uncertainty.
10
References
1. C. L. Lin, J. H. Shaw and J. 6. Calvert, "Band Analysis by Spectral Curve
Fitting," J. Quant. Spectrosc. Radiat. Transfer, 23. 387 (1980).
-1
2. H. L. Hoke and J. H. Shaw, "Analysis of CO bands near 2600 cm ,M Appl.
2
Opt. 21, 929 (1982).
3. H. L. Hoke and J. H. Shaw, "Rotational Analysis of CO Bands near 4 ym,"
2
Appl. Opt. 21, 935 (1982).
-1
4. H. L. Hoke, "Analysis of CO Bands near 2600 cm ."Ph.D. Dissertation,
2
Ohio State University (1982).
5. S. S. Penner, "Quantitative Molecular Spectroscopy and Gas Emissivities,"
Addison-Vesley Reading, Mass (1959).
13 16
6. C. P. Rinsland, A. Baldacci and K. N. Rao, "Strengths of C 0 Lines at
2
4.3 ym," J. Molec. Spectrosc. 81, 256 (1980).
7. L. S. Rothman and L. D. 6. Young, "Infrared Energy Levels and Intensities
of Carbon Dioxide," J. Quant. Spectrosc. Radiat. Transfer, 25. 505 (1981).
8. B. H. Winters, S. Silvennan and V. S. Benedict," Line Shape in the Wing
beyond the Band Head of the 4.3 ym Band of CO ," J. Quant. Spectrosc.
2
Radiat. Transfer, 4, 527 (1964).
9. J. T. Mandin, "Interpretation of the CO Absorption Bands Observed in the
2
Venus Spectrum between 1 and 2.5 ym," J. Molec. Spectrosc. 67. 304 (1977).
10. A. Chedin, "The Carbon Dioxide Molecule: Potential, Spectroscopic, and
Molecular Constants from its Infrared Spectrum," J. Molec. Spectrosc. 76.
430 (1979).
11
11. 6. Guelachvili, "High-Resolution Fourier Spectra of Carbon Dioxide and
Three of its Isotopic Species near 4.3 pro," J. Holec. Spectrosc. 79. 72
(1980).
12 16 -1 -1
12. H. Sakai, "Molecular Constants of C 0 Bands in 1900 cm ~2150 cm ,"
2
Paper MET, 38th Symposium on Molecular Spectroscopy, Ohio State
University, Columbus. OH 43210 (1983).
13. R. L. Hawkins, M. L. Hoke, and J. H. Shaw, "Vavenumber Calibration of
Fourier Transform Spectra," Appl. Spectrosc. 37. 134 (1983).
14. Ph. Areas, E. Arie, Ph. Cardinet, A. Valentin, and A. Henry, "Intensity
1 1 0
of CO Bands in the 4.8- to 5.3-pm Region. The (11 0, 03 0) *- 00 0
2 II
Band," J. Molec. Spectrosc. 81, 262 (1980).
19. C. P. Rinsland, D. C. Benner, D. J. Richardson, and R. A. Toth, "Absolute
1 12 16
Intensity Measurements of the (11 0) «- 00°0 band of C 0 at 5.2 pm,"
II 2
Appl. Opt. 22, 3805 (1983).
16. H. 0. Downing, B. J. Krohn, and R. H. Hunt, "Coriolis Intensity
Perturbations in * - £ Bands of CO ," J. Molec. Spectrosc. 55, 66 (1975).
2
17. Ph. Areas, B. Arie , A. Valentin, and A. Henry, "Intensity of CO Bands
2
1 1
in the 4.8 - 5.3 |im Region. The (11 0, 03 0) «• 00°0 Band," J. Molec.
Spectrosc. 96. 288 (1982).
18. M. L. Hoke and J. H. Shaw, "Parameters of CO Bands near 3.6 pm," Appl.
2
Opt. 22, 328 (1983).
12
Table I
Experimental conditions of the CO samples analyzed.
2
Condition
Pressure (torr)
Temperature (K)
Pathlength (m)
SNR
Spectrum I
177
294.2
476
-130
Spectrum II
658.3
294.2
131
-260
13
Table II
Retrieved rotational constants compared with other reported values. The
letters at the end of rows Identify the constants according to the following
assignments; N: Handin (Ref. 9). C: Chedin (Ref. 10), R: AFGL listing
(Ref. 7), F: Lower state rotational constants fixed to the values reported by
Guelachvlli (Ref. 11) for band I, by Sakai (Ref. 12) for band II and by AFGL
(Ref. 7) for band III. The upper state constants are retrieved, U: Constants
retrieved with no constraints.
The figures in parentheses are the uncertainty in the last digits.
14 TABLE II
Band
I
II
III
c
ZIId
Isotope
: 16012C180
16012C160
1G012C180
16012C180
Transition
(00°2) -*- 00°0
(3110)III •*- 00°0
(0112)I «- (0110)I
(0112)I ^ -(0110)I
Band Center
4639.5081(8)
4639.5082(3)
4639.502
4639.512
4639.5031
4591.1192(28)
4591.1195(28)
4591.118
4591.027
4591.1188
4614.7792(60)
4614.780
4614.787
4614.7794
4614.7792(60)
4614.780
4614.787
4614.7808
B'
0.3623911(27)
0.3623894(14)
0.362372
0.362384
0.36239118
0.38996(22)
0.389939(28)
0.389925
0.389927
0.38992250
0.362857(54)
0.362848
0.362849
0.3628716
0.363414(36)
0.363356
0.363376
0.36339483
B"
0.3681879(17)
0.36818571
0.3681771
0.368186
0.3681834
0.39025(26)
0.39021894
0.39021894
0.390221
0.3902215
0.3685824
0.3685824
0.368604
0.3685996
0.3691366
0.3691366
0.369156
0.3691715
D' x 107
1.146 (36)
1.1693 (93)
1.11
1.18
1.1967
1.52 (14)
1.585 (99)
1.53
1.54
1.5284
1.27 (60)
1.06
1.19
1.2561
1.41 (48)
1.06
1.20
1.2033
D" x 107
1.170 (29)
1.19030
1.17
1.19
1.1800
1.30 (72)
1.33281
1.33373
1.33
1.3373
1.14
1.14
1.20
1.1302-
1.14
1.14
1.21
1.2559
U
F
R
C
M
U
F
R
C
M
F
R
C
M
F
R
C
M
15
Table III
Number of P and R lines which can be distinguished
above the noise level and analyzed for each band.
Band
I
II
III c
III d
Number of P lines
57
26
34
34
Number of R lines
49
30
34
34
16
Table IV (a)
Rotationless Dipole Moment Matrix Elements
Band
I
II
III
|R(0)| x 10* Debye
Spectrum I
136 (1)
1.21 (9)
147 (4)
Spectrum II
137 (1)
1.19 (3)
149 (4)
Table IV (b)
Band intensities compared with the AFGL listing (Ref. 7)
Band
16o12c18o
00°2 «• 00°0
16o12c16o
(3110)III «- 00°0
16o12c18o
(0112)I «- (0110)I
S° x 1022 (cm 1/mol-cm~2)
Spectrum I
0.122 (1)
0.019 (6)
£ = 0.164 (4)
0.0108 (5)
Spectrum II
0.124 (2)
0.019 (2)
5 = 0,168 (2)
0.0111 (6)
AFGL
0.1302
0.00205
5 = 0.18
0.01042
17
Table V
Self-broadened Lorentz width parameters
for one atmosphere, 296 K.
Parameter
Wi
2 -1
W x 10 cm
o
2 -1Wj x 10 cm
W2
W x 103 cm"1j
Spectrum Hoke and Shaw (Ref . 18)
I
8.2 (2)
4.8 (1.0)
-0.15 (4)
3.3 (3)
II
8.34 (9)
6.8 (2)
-0.171 (9)
3.1 (1)
2614 cm"1
8.34 (2)
5.83 (6)
-0.1539 (4)
1.82 (2)
2757 cm"1
8.0732 (8)
6.15 (4)
-0.179 (9)
1.89 (9)
18
Table VI
Values of maximum optical path difference (retardation)
obtained by various analyses.
REF
Hoke (Ref. 4)
This work
Retardation (cm)
Spectrum
I
16.51 (10)
16.48 (15)
II
16.64 (10)
16.47 (9)
19
Figure Captions
Fig. 1
-1
The portion of Spectrum I analyzed. Lines of the bands centered at 4639.5 cm
-1
and 4591.1 cm are easily identified but lines of the much weaker band,
-1
centered at 4614.8 cm , can hardly be distinguished above the noise.
Fig 2.
Corresponding portions of the residual spectrum and the noise spectrum are
shown. The upper spectrum is the part of the experimental spectrum shifted by
0.2 units with respect to the calculated spectrum.
Fig. 3
Comparison of line position differences. Differences are plotted against
rotational index m. The letters labeling the plots are the same as those used
in Table II.
Fig. 4
1 0 16 12 16
Calculated intensities of the lines of band (31 0) «- 00 0 of 0 C 0
III
plotted against rotational index at.
Fig. 5
The |m| dependence of the line widths for Spectrum I and Spectrum II. All
widths have been reduced to 296 K and 760 torr.
Fig 6
Comparison of the |m| dependence of the widths of lines of Spectrum II with
other results from (Ref. 18). All widths reduced to 296 K and 760 torr.
1.0
JQ
<
"6
•c
S?
CO
I I I I
4550 4600
Wave Numbers (cm*1)
4650
1.00*=
0.80
0.60
at
'E 0.40
S
is 0.20
o>
(75
0.05
-0.05
\^f\^
4540 4550 4625 4630 4635
Wove Numbers (cm'1)
0.03
o
<D
0.02
0.01
0
-0.01
-0.02
-0.03
-50 -30 -10 10 30 50
M
• J -
0.12
^ 0.10
— 0.08
CJI
O
"o
H 0.06
•
E
O
.• 0.04
v>
Q)
0.02
-40 -20 0
M
20 40 60
O.I6
0.14
i
& 0.12
JC|5
£ 0.10
cO)
O
0.08
0.06
C
* ' 1 660 Torr
X
* 1 2 177 Torr
X '
^ *xx
O XM«
^^^tf
^^^M
*JK
****
*fc«
***Ia
1 1 1 I I
) 10 20 30 40 50
M
0.15
_ 0.12
e
I
o 0.09
.C
•5
M
•*-
0)
0.06
0.03
0 10 20
1 2614 cm'1
2 2757 cm'1
3 4640cm-'
30 40 50
M


Analysis of carbon dioxide bands near 2.2 micrometers

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