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^ecUnical ilot& 332 CARBON DIOXIDE SPECTRAL LINE POSITIONS AND INTENSITIES CALCULATED FOR THE 2.05 AND 2.7 MICRON REGIONS ROBERT F. CALFEE AND WILLIAM S. BENEDICT U. S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS
Transcript of Carbon dioxide spectral line positions and intensities ... · ro in •JO so cO fM cr-fM \0o rO o...
^ecUnical ilot& 332
CARBON DIOXIDE SPECTRAL LINE POSITIONS AND INTENSITIES
CALCULATED FOR THE 2.05 AND 2.7 MICRON REGIONS
ROBERT F. CALFEE AND WILLIAM S. BENEDICT
U. S. DEPARTMENT OF COMMERCENATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDSTechnical Note 332
ISSUED MARCH 15, 1966
CARBON DIOXIDE SPECTRAL LINE POSITIONS AND INTENSITIES
CALCULATED FOR THE 2.05 AND 2.7 MICRON REGIONS
Robert F. Calfee
Institute for Telecommunication Sciences and Aeronomy *
Environmental Science Services Administration
Boulder, Colorado
and
William S. Benedict
Institute for Basic Standards
National Bureau of Standards
Washington, D.C.
This work sponsored by the Advanced Research Projects Agency,
under Contract Number 250-61; Code Number 7400-DO
* Formerly the Central Radio Propagation Laboratory of the
National Bureau of Standards; CRPL was transferred to ESSAin October 1965, but will temporarily use various NBS pub-
lication series pending inauguration of ESSA counterparts.
NBS Technical Notes are designed to supplement the Bu-reau's regular publications program. They provide a
means for making available scientific data that are of
transient or limited interest. Technical Notes may belisted or referred to in the open literature.
For sale by the Superintendent of Documents, U.S. Government Printing Office
Washington, D.C, 20402 Price 60*
CONTENTS
1. Introduction 1
2. Frequencies 1
3. Intensities 2
4. Discussion 12
5. References 110
in
CARBON DIOXIDE SPECTRAL LINE
POSITIONS AND INTENSITIES CALCULATED
FOR THE 2. 05 AND 2. 7 MICRON REGIONS
Robert F. Calfee and William S- Benedict
The spectral line positions and intensities for the carbon dioxide
bands occurring in the 2. 05 and 2. 7 micron regions of the infrared spectrumare listed. Lines with intensities down to 10" cm" /atm cm are listed for
a temperature of 296°K.
Keywords: 2. 05\i, 2. 7(jl, carbon dioxide, intensities and line positions.
1
.
Introduction
A precise calculation of the infrared absorption by carbon dioxide depends upon a
knowledge of the position and strength of the individual lines of the spectrum. The absorp-
tion spectrum of carbon dioxide has been obtained in considerable detail by Courtoy (1959).
He not only presents spectra of the gas as its isotopes occur in natural abundance, but also
includes results of enhancement by certain of the isotopic molecules. Lists of the identifi-
cation and position of the absorption lines are also tabulated.
However, in order to make quantitative calculations, it is necessary to have values
for the intensities of individual lines. Furthermore, in several instances Courtoy does not
give an exact value for the position of lines. For these reasons it was necessary to calculate
the intensities and positions of the CO? lines. This report concerns such calculations for
those bands which give rise to absorption in the 2. 05 and 2. 7u regions of carbon dioxide.
2. Frequencies
In this study centrifugal stretching effects are taken into account in calculating the
frequencies and intensities of the spectral lines. The energy values for the upper and lower
states are written in the following manner:
Upper level: v, ,
=u + J'(J' + 1)B'-J' (J'-Pl) D'
Lower level: v „=vl +J"(J"+1 )B"- J"2(J"+1 ) D",
V J V
where V and J denote the vibrational and rotational quantum numbers respectively of a
particular energy level. Here, v is the vibrational contribution; B and D are the usual
rotational constants. If we let a = B' + B", b = B'-B", c = -2(D'+D"), d = D"-D', then for the
P(A J = - 1 ) and R(A J = + 1 ) branches the frequency for a transition between energy levels is:
2 3 4u{V'J' - V"J") =15 +am + (b+d)m + cm + dm . (1)
3 3For the Q branch (A J =0), the terms am and cm are replaced by bm and dm ,
respectively. The possible branches depend upon the type of transition (E-E, II- IT, A -A, etc. ).
Values of m are determined by the branch: m = - J" for the P branch, m = J" in the Q branch,
and m = J" + l in the R branch. The possible values of J depend upon the parity of the lower
state; only even values of J are allowed for even parity and only odd values of J are permitted
for odd parity. In the case of molecules with different isotopes of oxygen, the listing of both
parities was included as an artificial means of forcing these bands through a computational
program relying on an even - odd scheme. In (1) V (band origin) is the vibrational energy
difference between the two states, (u - v ,,,,)• For a more thorough discussion of the be-
havior of linear polyatomic molecules, see Herzberg (1962).
3. Intensities
Line intensities are a combination of the vibrational and rotational contributions
determined by use of the following formulas:
S =(u/u )S S /Q ; (2)° v r r
where
S^gLexp (-E^'/kT); (3)
Qr=Sg(2J + l) exp(-E^/kT); (4)
E =B"J"(J" + 1) -D"J"2(J" + 1)
Z. (5)
Here values of S , the vibrational band intensities, have been chosen on a semi-v
theoretical basis attempting to fit the total observed intensities. The abundance factor for
the various isotopic species is taken into account in the values listed for S . Values of Sv v
for a temperature of 296°K used for each band are listed in the seventh column of tables
1 and 2.
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Determination of the rotational contribution, S , depends upon a weighting factor g,
based upon the parity; in the case of -even parity g = 1 for even values of J, and g = for odd
values of J. Just the opposite is true for odd parity. The rotational partition functions, Q ,
r
have nearly the same values in all the bands of a particular molecular species. The values
of Q for a temperature of 296°K are as follows: CCX,16
, Q =263; C016 17, Q =271 and
r * r r
C016 18, Q =279. For a linear molecule, the rotational partition function can be taken as
directly proportional to the temperature (Herzberg, 1962). The rotational line strength
factors, Li, depend upon the type of band as well as the values of m which in turn depend
upon J. The following table shows this dependence:
Table 3. Rotational Line Strength Factors
BandType
Branch
E- S
n- n
A - A
$ - $
n- s
s- n
P and R m
a- n
p
Q
R
P
Q
R
Im| -1/| m|
(Zm +1), m(m +1)
|m| - 4/|m|
4(2m + l)/m(m-j- 1)
(|mj - l)(|m[ -2)/|mj
(m + 2)(m- l)(2m+ l)/m(m + l)
(m + 2)(m +l)/m
10
It should be noted that in tables 1 and 2 each series of lines with a distinct frequency
formula has been listed as a "separate band" for convenience of computational programming.
Thus in a A- II transition of an asymmetric molecule there would be four sub-bands listed
separately: the P- and R- branches with odd J", the P- and R- branches with even J", and
the corresponding odd and even Q branches. Each "band" has been normalized so that the
sum of states is close to that for the most common case, that of the T.-T, band in the sym-
metric molecule, with only even J". The rotational line strength factors in table 3 reflect
this unconventional definition of a "band".
The values used for the various other parameters for COHdefined above and occur-
ring in ( 1) through (5) are listed for each band in tables 1 and 2. The first column gives a
band number used for identification purposes in this report, the second column shows the
band type, and the isotopic species of the molecule is shown in the third column. The fourth
column indicates the parity of J" (e for even, o for odd). The next three columns (fifth,
sixth, and seventh), give the band origin u (cm ), the vibrational energy of the lower state,
E (cm ), and the vibrational band strength determined for 296°K. The eighth through
thirteenth columns give appropriate values of the molecular constants B", D", a, b, c, and
d. Certain of the weaker bands (II- S, S- II) require the addition of a term, (1 + t, m) in
calculating intensities to take into consideration the Coriolis vibration-rotation interaction.
Values of the empirical constant, t, , depending on the Coriolis interaction and the vibra-
tional transition involved, which provide a satisfactory fit with the experimental data are
listed in the fourteenth column of tables 1 and 2.
Taking into account the values from tables 1 and 2, a program was developed for
computing the rotational energies, E", of the lower state, using values of J through 100,
and at the same time determining the rotational partition function, Q , according to (4).
The frequency and intensity are computed for all the lines in each of the branches again for
values of J through 100, according to (1) and (3).*
11
Table 4 shows a tabulation of the results of the computations made by an electronic
computer using the appropriate data for the 61 bands of table 1 for the 2. 05u region. The
listing is according to increasing frequency, in cm , in the first column. The second
column gives the computed intensity of the line in cm /(atm-cm) . Note that all numbers
-3listed are to be multiplied by 1 as indicated by the -3 at the head of this column. The
notation STP refers to the absorber concentration (atm-cm) and not the temperature for
which the values were computed. The energy (cm ), of the lower state including both
vibrational and rotational contributions, is found in the third column. Values of the
rotational line strength factor, L, are listed in the fourth column. The fifth column gives
the identification of the line in terms of the band number, the branch, and the value of J.
Table 5 gives similar information for the 2. 7u region based on the parameters listed in
table 2.
4. Discussion
Recently Burch, Gryvnak and Patty (19 64), have made extensive measurements of
the absorption spectrum in the 2. 05u region. Comparisons with these records have been
used to arrive at values of S , the vibrational band intensities, used to determine the linev
intensities computed for table 4. For the 2. 7|jl region, comparisons with the absorption
spectra published by Burch, Gryvnak and Williams (I960) have been used to assign the values
of S for these bands. With the exception of regions of low absorption, where bands weaker
than those included in the present computations may be expected to make a proportionately
large contribution, the transmission calculated from the line strength listed here agrees
with the measured values of Burch, Gryvnak and Patty (1964) to within a few percent. The
average Lorentz half-width parameter required to give the best agreement is approximately
0. 080 cm /atm for nitrogen broadening and 0. 104 cm /atm for self broadening.
In tables 4 and 5, the calculated line intensities for all lines having intensities of
-7 -110 cm /(atm-cm) and greater have been listed. Since the experimental spectra with
which comparisons were made were observed at a temperature of 296°K, the intensity values
were calculated and listed for this same temperature.
12
For extremely long optical paths, weaker lines due to higher J values and for other
bands not treated here will appear. However, for absorption paths through the earth's
atmosphere, all the significant carbon dioxide lines in the 2. 05 and 2 . 7(jl regions of the
spectrum will be found in the tables of this report.
13
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5. References
Burch, Darrell E. , David Gryvnak and Dudley-
Williams (I960), Infrared Absorption by Carbon
Dioxide, Scientific Report No. II(AFCRL-255).
Burch, Darrell E. , David Gryvnak and Richard
R. Patty (1964), Absorption by CO Between 4500
and 5400 cm" , Philco Scientific Report No. U2955.
Courtoy, Charles P. (1959), Spectre Infrarouge,
A Grand Dispersion et Constantes Moleculaires
Du CO , Annales de la Societe Scientifique de
Bruxelles, 73, 5-203.
Herzberg, G. (1962), Molecular Spectra and
Molecular Structure II Infrared and Raman Spectra
of Polyatomic Molecules (Van Nostrand, New York,
1962)
110
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