Journal of the OPTICAL of VOLUME 46, NUMBER 4 SOCIETY AMERICA APRIL, 1956 Infrared Transmission of Synthetic Atmospheres.* II. Absorption by Carbon Dioxide J. N. HOWARDI D. E. BURCH, AND DUDLEY WILLIAMS Department of Physics and Astronomy, The Ohio State University, Columbus, Ohio (Received October 11, 1955) The total absorption fAdv has been determined for the CO2 bands at 15, 5.2, 4.8, 4.3, 2.7, 2.0, 1.6, and 1.4A under stimulated atmospheric conditions. The absorber concentrations w ranged from 1 to 1000 atmos cm of CO 2 for the strong bands and from 100 to 8600 atmos cm for the weak bands. Nitrogen was added to give total pressures ranging up to atmospheric; the pressure effects of oxygen were found to be similar to those of nitrogen. The observed data can be satisfactorily represented by two types of empirical relations. (1) For small values of total absorption, f Ayd=cwI(P+p)k; (2) For large values of total absorption, f A,,d= C+D Logw+K Log(P+p), where w is the C02 absorber concentration, p is the C02 partial pressure, and P is the total pressure. Values of the constants c, k, C, D, and K are given for each region of characteristic absorption. The present results are compared with those obtained in earlier studies. The use of the empirical relations in calculating atmos- pheric absorption is discussed. A PREVIOUS article' has described a technique for the experimental measurement of the total ab- sorption fA dv by entire bands of CO 2 under simulated atmospheric conditions. The present paper gives the results obtained for each spectral region in the near infrared in which CO2 has characteristic absorption. Data were collected for the absorption bands of CO 2 at 15, 5.2, 4.8, 4.3, 2.7, 2.0, 1.6, and 1.4A under various conditions of CO 2 partial pressure p, total pressure P, and geometrical path length 1. The range of variation of these parameters is indicated in Table I, which also gives the number of independent "runs" taken for each "band." Instead of listing the total range of geometrical path lengths, the table lists a quantity w called "ab- sorber concentration," which for a given temperature * The research reported in this document has been made possible through support and sponsorship extended to the Ohio State University Research Foundation by the Geophysics Research Directorate of the Air Force Cambridge Research Center. It is published for technical information only and does not represent recommendations or conclusions of the sponsoring agency. t Present address: Air Force Cambridge Research Center. I Howard, Burch, and Williams, J. Opt. Soc. Am. (to be pub- ished). is proportional to the product of the partial pressure p and geometrical path length 1. This quantity w, which is a measure of the total number of CO 2 absorbers per unit area traversed by the beam of radiation, can be expressed as an equivalent thickness of undiluted ab- sorbing gas under stated conditions of pressure and temperature. A convenient unit for measuring CO 2 absorber concentration is the atmos cm. An atmos cm is a measure of the amount of gas in a column of arbitrary length, pressure, and temperature and is defined as the path length in cm which would contain the same number of molecules if the gas were at NTP. The quantity w listed in Table I is the number of atmos cm actually present in the absorption cell multiplied by the number of times the radiation traverses the cell.$ In order to compare the range of absorber concentra- tions listed in Table I with CO2 absorber concentra- tions in the atmosphere, it is useful to note that a 1 km t It might be noted that w, which is sometimes called "optical depth," has analogies in nuclear physics, where absorber thick- nesses are sometimes measured in "mg/cm 2 " or in terms of an equivalent thickness of aluminum. 237
Transcript
Journal of the
OPTICALof
VOLUME 46, NUMBER 4
SOCIETYAMERICA
APRIL, 1956
Infrared Transmission of Synthetic Atmospheres.* II. Absorption by Carbon Dioxide
J. N. HOWARDI D. E. BURCH, AND DUDLEY WILLIAMSDepartment of Physics and Astronomy, The Ohio State University, Columbus, Ohio
(Received October 11, 1955)
The total absorption fAdv has been determined for the CO2 bands at 15, 5.2, 4.8, 4.3, 2.7, 2.0, 1.6, and1.4A under stimulated atmospheric conditions. The absorber concentrations w ranged from 1 to 1000 atmoscm of CO2 for the strong bands and from 100 to 8600 atmos cm for the weak bands. Nitrogen was added togive total pressures ranging up to atmospheric; the pressure effects of oxygen were found to be similar tothose of nitrogen. The observed data can be satisfactorily represented by two types of empirical relations.
(1) For small values of total absorption,
f Ayd=cwI(P+p)k;
(2) For large values of total absorption,
f A,,d= C+D Logw+K Log(P+p),
where w is the C02 absorber concentration, p is the C02 partial pressure, and P is the total pressure. Valuesof the constants c, k, C, D, and K are given for each region of characteristic absorption. The present resultsare compared with those obtained in earlier studies. The use of the empirical relations in calculating atmos-pheric absorption is discussed.
A PREVIOUS article' has described a technique forthe experimental measurement of the total ab-
sorption fA dv by entire bands of CO 2 under simulatedatmospheric conditions. The present paper gives theresults obtained for each spectral region in the nearinfrared in which CO2 has characteristic absorption.
Data were collected for the absorption bands of CO 2at 15, 5.2, 4.8, 4.3, 2.7, 2.0, 1.6, and 1.4A under variousconditions of CO2 partial pressure p, total pressure P,and geometrical path length 1. The range of variationof these parameters is indicated in Table I, which alsogives the number of independent "runs" taken for each"band." Instead of listing the total range of geometricalpath lengths, the table lists a quantity w called "ab-sorber concentration," which for a given temperature
* The research reported in this document has been made possiblethrough support and sponsorship extended to the Ohio StateUniversity Research Foundation by the Geophysics ResearchDirectorate of the Air Force Cambridge Research Center. It ispublished for technical information only and does not representrecommendations or conclusions of the sponsoring agency.
t Present address: Air Force Cambridge Research Center.I Howard, Burch, and Williams, J. Opt. Soc. Am. (to be pub-
ished).
is proportional to the product of the partial pressure pand geometrical path length 1. This quantity w, whichis a measure of the total number of CO 2 absorbers perunit area traversed by the beam of radiation, can beexpressed as an equivalent thickness of undiluted ab-sorbing gas under stated conditions of pressure andtemperature. A convenient unit for measuring CO 2absorber concentration is the atmos cm. An atmos cmis a measure of the amount of gas in a column ofarbitrary length, pressure, and temperature and isdefined as the path length in cm which would containthe same number of molecules if the gas were at NTP.The quantity w listed in Table I is the number of atmoscm actually present in the absorption cell multiplied bythe number of times the radiation traverses the cell.$In order to compare the range of absorber concentra-tions listed in Table I with CO2 absorber concentra-tions in the atmosphere, it is useful to note that a 1 km
t It might be noted that w, which is sometimes called "opticaldepth," has analogies in nuclear physics, where absorber thick-nesses are sometimes measured in "mg/cm2" or in terms of anequivalent thickness of aluminum.
237
HOWARD, BURCH, AND WILLIAMS
TABLE I. Range of parameters studied.
C02 Absorber C02 Partial TotalBand Concentration, Pressure, p Pressure, P Number
air path at ground level has a CO2 absorber concentra-tion of 30 atmos cm.
FORM OF THE EMPIRICAL RELATIONS
The purpose of the present study is to develop rela-tions between the total absorption of CO2 in each regionof characteristic absorption and the absorber concentra-tion w, the CO2 partial pressure p, and the total pres-sure P. Although the final relations are admittedlyempirical, previous theoretical work can serve as a guideto the form of the relations.
The infrared absorption bands of H20 and CO2, whenexamined under high resolution, show a complex finestructure of absorption lines caused by simultaneouschanges in molecular vibrational and rotational energy.Therefore, in formulating the laws of absorption, onemust begin with a consideration of individual absorp-tion lines. The absorption A, at any frequency in thevicinity of an absorption line can be expressed as
IIA,>= 1--= -- k~wt
1o(1)
where I, is the intensity of the radiation of frequency transmitted by a sample having absorber concentrationw, o is the intensity of the incident radiation of fre-quency v and is assumed in the subsequent discussionto be constant for all frequencies in the vicinity of theabsorption line, and k, is the absorption coefficient atfrequency v. The natural widths and Doppler widthsof spectral lines in the infrared are extremely small ascompared with the widths due to the collision processesdiscussed below.
For H2 0 and CO2, under the temperature and pres-sure conditions of the lower atmosphere, the finitewidths of the spectral lines are due chiefly to molecularcollisions. The spectral line as described by k has theso-called Lorentz shape given by
S a
(j,- vo)2+a2' (2)
where S is the line strength, a is the half-width at half-maximum k, and o is the frequency of the line center.Because of finite spectrometer slit widths, it is prac-tically impossible to determine A, or k by direct meas-
urements. However, Ladenberg and Reiche2 havetreated the absorption of a single line of Lorentz shapeand have obtained a general expression giving the rela-tionship between an experimentally measurable quan-tity,1 total absorption Adi of a single line, and S,a, and w. In two limiting cases, the general expressionreduces to simple expressions that can be subjected todirect experimental check:
I. For weak absorption lines (Sw<'(a)
fAdv=Sw. (3)
II. For strong absorption lines (Sw>>a)
fA dv= 2 (Saw)?. (4)
The line strength S has been found to be nearly inde-pendent of pressure, but the half-width a is a function ofpressure P and absolute temperature T and dependsupon the collision cross sections and masses of the ab-sorbing and nonabsorbing molecules. One can expressa at any temperature and pressure in terms of ao atstandard conditions (To, P) as
P /To\ ia=ato-t-) * (5)
If it is possible to neglect temperature effects and self-broadening effects, the half-width can thus be con-sidered proportional to total pressure:
a=BP. (6)
In this latter case, the total absorption of a single strongline of Lorentz shape is given by
f Adv= 2 (Saw) = 2 (S#) (wP)I.
If a "weak" band can be considered to consist ofmany "strong" lines, which are far enough apart tomake the effects of overlapping negligible, the totalabsorption for the band can be written
fAdv= (wP)122(j3Si)1.f ~~~~~i (7)
Many investigators have shown that total absorptionvaries as w, but experiments show that total absorptionvaries more slowly than Pi as the total pressure is in-creased. Falkenberg,3 Schnaidt, 4 and Strong andWatanabe5 all found that total absorption is propor-
2 R. Ladenberg and F. Reiche, Ann. Physik 42, 181 (1913).3 G. Falkenberg, Met. Z. 55, 174 (1938); 56, 72 (1939).4 F. Schnaidt, Gerlands Beit. Geophys. 54, 203 (1939).5 J. Strong and K. Watanabe, 57, 203 (1939).
238 Vol. 46
April1956 INFRARED TRANSMISSION OF SYNTHETIC ATMOSPHERES. II. 239
tional to w and approximately proportional to Pi.Rauscher6 ascribes the failure of the Pi relation to theoverlapping of lines.
In view of the partial success of (7), an equation ofthe form
fAdv= cw1P"
was first tried in fitting the experimental results. Thisequation was satisfactory except in cases where the CO 2
partial pressure was comparable with the total pressureand self-broadening effects became important. It wasfound that better agreement with experiment was ob-tained if a double weight was given to the part of thetotal pressure due to CO 2. Therefore, the empiricalequation for weak bands was given by
f Adv=cw(P+p)" [Weak Band] (8)
where values of k between - and I are to be expected.It is obvious that Eqs. (7) and (8), which picture a
band as set of well separated lines, cannot continue toapply indefinitely as the absorption path is increased,since the effects of overlapping become more and morepronounced. In other words, for sufficiently largeabsorber concentrations portions of absorption bandsbecome sensibly opaque, and further increase in ab-sorber concentration produces no observable effect.
For such a case, the following approach has beensuggested by Elder and Strong7 in their recent reviewarticle. All observers of total absorption under lowresolution agree that, at constant total pressure P, asmall increase Aw in absorber concentration results in adecrease Al in transmitted radiation that is proportionalto the incident radiation intensity Io, associated withany relatively wide frequency interval. Furthermore, fora given value of lo the incremental decrease Al pro-duced by a small absorber concentration increment Awbecomes smaller as w becomes larger. These experi-mental results for large values of w can be expressed inthe approximate form
dI Io
dw wand
Integration gives
dl dwdA = -- =D"-.
Io w
A =D' Logw+K'. (9)
The above equations have been set up on the assump-tion that Io is the intensity of the radiation in a fre-quency interval sufficiently wide to include severalabsorption lines. If this frequency interval is sufficiently
I E. Rauscher, Z. Physik 7,418 (1949).7T. Elder and J. Strong, J. Franklin Inst. 255, 189 (1953).
wide to include the entire absorption band, the valueof A in (9) becomes the mean fractional absorption Afor the entire band, which is related to the measurabletotal absorption,
fI A',dv= (v2- v')A.Pi
In the present study, it was found that, for a givenvalue of absorber concentration w, variations in totalpressure P produced changes in absorption that couldbe expressed by a relation similar to (9) with (P+p)replacing w. Therefore, it seemed reasonable to use thefollowing type of empirical expressions for the totalabsorption of strong bands
fA dv=C+D Logw
+K Log(P+p). Strong Band]. (10)
It is difficult, and for present purposes unnecessary, togive clear physical interpretations of the empiricalconstants C, D, and K.
Qualitative arguments have been presented to justifythe form of Eqs (8) and (10). Although relations of thisform satisfactorily express total absorption in terms ofw, P, and p, the form of the relations is in no senseunique. It may be that other types of relations will alsofit the observed data and prove more satisfactory froma theoretical point of view. For example, by giving alarger weight to the effects of self-broadening, Benedict8
has been able to substitute for (8) an equation in whichw and the differently weighted pressure have the sameexponent.
DETERMINATION OF THE EMPIRICAL RELATIONS
After the experimental data had been collected for agiven band, appropriate band limits were establishedand the total absorption was determined for each "run"in the manner described in the previous paper.' InTable II, band limits are given for each region of charac-teristic absorption.
For each band an attempt was first made to fit theobserved data to (8). For some bands such as the 5.2wband, this "weak-band" relation was adequate for allvalues of total absorption. Most of the bands followedthe "weak band" relation for small values of totalabsorption and the "strong band" relation (10) forlarge values; for these bands it is possible to choose a"transition value" of total absorption, below which (8)applies and above which (10) applies. The selection ofthe transition value is somewhat arbitrary; in its im-mediate vicinity neither (8) nor (10) applies and aninterpolation function must be used. The strong-bandrelation (10) applies even to the smallest values of totalabsorption observed for the 4.3,g band. The values of
aw in atmos cm; P p, in mm Hg; logarithms are to base 10. Band limitsapply also to "Strong" Fit. "Weak" Fit applies for fA,dv less than"Transition A,dv", "Strong" Fit for higher values.
b These band limits were arbitrarily chosen to "Separate" overlappingbands.
the constants c, k, C, D, and K, and the transition valuesfor each of the major "bands" of CO2 are tabulated inTable II.
An example of the method used in determining theconstants for a strong band is shown in Figs. 1 and 2,which present plots for the strong 4.3k band. In Fig. 1,total absorption is plotted as a function of the logarithmof absorber concentration. In the plot, straight lineshave been drawn through the points obtained for con-stant total pressure. The best set of parallel lines fittingthe data gives a value of 34 for the slope. Thus,
f Avdv= 34 Logw+f(P),
where f(P) is a function of pressure which remains to beevaluated. Figure 2 shows a plot of "reduced total ab-sorption," fA,4d-34 Logw, as a function of thelogarithm of the weighted pressure (P+p). The straightline best fitting the points in this plot has a slope of31.5. Consideration of this plot shows the final form of
30
z5?Fa.0Inco
-JII50
20
10
l0 100 ,000
CONCENTRATION, w, (atmo.- cm)
FIG. 1. The observed total absorption data for the 4.31 CO2band plotted against absorber concentration.
the (10) for the 4.3p band to be
f Adv=27.5+34 Logw+31.5 Log(P+p),
where total absorption is in cm-l, w in atmos cm, pres-sures P and p in mm Hg, and logarithms are to thebase 10.
Figure 3 shows a comparison of the observed valuesof total absorption for various values of w, P, and pwith the total absorption as calculated from the aboveexpression. Similar curves have been plotted for theother bands. On the average, the empirical relations(8) and (10) with the constants given in Table IIpredict the observed total absorption to -t3%.
For only one absorption region, the 2 .7 , bands ofCO2 , did (8) and (10) fail to fit all the observed data.For this region, Eq. (8) with the constants given in TableII was satisfactory for total absorption less than 50
0
C7"
10 100 3000
WEIGHTED PRESSURE, (P+p), (mm.Hg)
FIG. 2. The "reduced total absorption" plotted against "weightedpressure" for the 4 .3,M CO 2 band.
cm'l; Eq. (10) applied above this transition value forabsorber concentrations up to 100 atmos cm. For higherabsorber concentrations, the following equation applied
IfAdv= -173+D(P) Logw+92 LogP,
where D(P) is a function of total pressure given by
D(P) = 107-16 LogP.
In atmospheric applications this complication is largelyavoided, since the 2.7,u CO2 absorption is overlapped bymuch stronger H20 vapor absorption. For ordinaryconditions of temperature and humidity, absorptionby water vapor is nearly complete before an absorptionpath is sufficiently long to include 100 atmos cm of CO2.
COMPARISON WITH OTHER STUDIES
Howard and Chapman9 in an earlier study investi-gated the total absorption of the 2.7 g and 4 .3,t bands for
9 J. N. Howard and R. M. Chapman, J. Opt. Soc. Am. 42, 856(1952).
M U I I I I I III I I 1 1 1I I I I I 11 1
140 4.3p
120 _ COP
100 I
60 C_
40 SLOPE = 31.5 _
201 l l I I I I I I I I . -1 I Io I I 1
I I il1 I I llliJ I I 1o0 TOTAL
PRESSURE 43(mm.Hg) Go 2
o740v 400
o _ 200* 100
SLOPE 34
o 1 1 1 il I I I I I I!II, , I I 1 I
240 Vol. 46
April1956 INFRARED TRANSMISSION OF
absorber concentrations in the range 4 to 50 atmos cm;their data are satisfactorily represented by the empiricalrelations given in this paper. The same is true for thedata obtained by Cloud' 0 in his studies of the 15p band.Gebbie et al."1 have measured the transmission of nearinfrared radiation through a long path in the open air.From their published figure, the total absorption of the4.3S CO2 band is estimated to be about 200 cm'l,while the empirical relations in the present reportpredict 179 cm7-. It is possible that the 4.5,u funda-mental band of N2 0, which is present in small amountsin the atmosphere, may have contributed slightly tothe rather broad CO2 band appearing in the Gebbiecurve.
USE OF RESULTS IN CALCULATINGATMOSPHERIC TRANSMISSION
It should be pointed out that the empirical relationsobtained in the present study are directly applicableto calculations of the absorption of infrared radiationby CO2 in horizontal paths through the lower atmos-phere, since the pressure broadening effects of N2 and 02have been found to be similar. Since the atmospherecontains only 0.03% of CO2 by volume, the CO2
partial pressure p is negligible as compared with totalatmospheric pressure P; thus the term involving (P+p)in (10) can be replaced by one involving P alone.
For approximate calculations; one can assume thatCO2 absorbs only radiation of frequencies included inthe band limits given in Table II. Thus, an averagefractional absorption A can be defined for each region
10 W. H. Cloud, Johns Hopkins University Report, January1952, Office of Naval Research Contract.
tl Gebbie, Harding, Hilsum, Pryce, and Roberts, Proc. Roy.Soc. (London) A206, 87 (1951).
250
T.E
.0
.0
0
0I-
Mu
iJn_
n
200.
150.
100.
50
50 100 150 200
TOTAL ABSORPTION. JApdv, Calculated, (cm.-')
FIG. 3. The validity of the empirical relationsfor the 4 .3,u CO2 band.
of characteristic absorption:
I=Ad
V2- V1
250
(11)
Thus, the fraction A of all radiation within the fre-quency range V2- VI will be absorbed by CO2. Hence, ifthe spectral distribution of the radiation from a givensource is known, the fraction of the total radiationabsorbed by CO2 in a given atmospheric path can becomputed.
Absorption by H20 vapor has also been investigated.Results on the subject will be presented in a subsequentpaper.
