JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 46, NUMBER 5

Infrared Transmission of Synthetic Atmospheres.* IV. Applicationof Theoretical Band Models

J. N. HowARIf D. E. BURCH, AND DDLEY WILLIAMSDepartment of Physics and Astronomy, Te O/io State University, Columbuts, Ohio

(Received December 19, 1955)

Two types of theoretical models for an absorption band are described and applied to infrared absorptionbands of CO2 and H2 0. For CO2, which has a regular fine structure, the so-called Elsasser model, which as-sumes equally-spaced, equally-intense lines, can be applied. For H20, which has a highly irregular finestructure, the so-called statistical or Goody model, which assumes random spacings and intensity distri-bution, is applicable. The individual lines are assumed to have the Lorentz shape in both models. Themethod of application of these models to the experimentally observed bands is discussed in detail.

THE first paper' of this series described the ex-perimental techniques for measuring the total

absorption fA dv for entire infrared absorption bandsof atmospheric gases. The results obtained experi-mentally for carbon dioxide were discussed in the secondpaper2 ; in the third paper' results for water vapor werepresented. Because all the experimental work was doneat room temperature it was necessary, in order to avoidcondensation effects in the cell, not to exceed thesaturation vapor pressure of water vapor at roomtemperature. The intensity of the radiation sourceemployed, the sensitivity of the detector, and thereflection losses in the cell combined with this limitationto impose a maximum of about 2 cm of precipitablewater (hereafter abbreviated to pr. cm) which could bestudied in the absorption cell. For computations ofinfrared transmission in the lower atmosphere, it wouldbe desirable to extrapolate the results to air pathscontaining about 50 pr. cm H20. This prompted a studyof the application to the water vapor data of theoreticalmodels which have been proposed in the literature.

Carbon dioxide exists in such small quantities in theatmosphere (about 0.03% by volume) that extra-polation of the experimental data to greater absorberconcentrations than those studied experimentally wasnot really necessary; nevertheless, for completeness,the application of a theoretical band model to theabsorption bands of this molecule is also consideredhere.

ABSORPTION DUE TO ISOLATED LINES

As was pointed out in an earlier paper, for H2 0 andC02 under the temperature and pressure conditions ofthe lower atmosphere, the finite widths of the spectrallines are due chiefly to molecular collisions. The ab-

* The research reported in this paper 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. 46, 186 (1956).2 Howard, Burch, and Williams, J. Opt. Soc. Am. 46, 237 (1956).3 Howard, Burch, and Williams, J. Opt. Soc. Am. 46, 242 (1956).

sorption coefficient k of each spectral line has the so-called Lorentz shape given by

S ak,=- -

7r (V- V)2 +a 2(1)

where S is the line strength, a the half-width at half-maximum k, and o is the frequency of the line center.Because of the finite spectrometer slit widths, it isdifficult to determine k, or the related fractionalabsorption at the same frequency, A , by direct meas-urements, but one can determine experimentally thequantities S and a from measurements of the totalabsorption, fA ,dv, as the absorber concentration w isvaried.

The theory of the total absorption of a single linewas treated originally by Ladenberg and Reiche4 andis summaried in Elsasser's monograph.'

For the case of a band consisting of many lines ofLorentz shape which are sufficiently far apart to makethe effects of overlapping negligible, the total absorptionof the band can be written

f A ,dv= (wP)i E 2( 3Sj)1,i

(2)

where P is the total pressure and is a quantity relatedto the half-width.2 However, this case of non-over-lapping lines does not include the infrared bands ofatmospheric C0 2 and H2 0.

ELSASSER MODEL OF AN ABSORPTION BAND

Elsasser5 has considered the case of an idealizedabsorption band consisting of an infinite array ofequally intense, equidistant lines, each of the Lorentzshape, and each with the same half-width, a. Becausethe pattern is periodic in frequency, the fractionalabsorption in any period is the same as the fractionalabsorption for the entire band. If the line centers ofthe individual lines are located at v= 0, d, -i2d,

4R. Ladenberg and F. Reiche, Ann. Physik 42, 181 (1913).6 W. M. Elsasser, Harvard Meteorological Studies No. 6,Harvard University (1942).

334

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MAY, 1956

335May1956 INFRARED TRANSMISSION OF SYNTHETIC ATMOSPHERES

and each line has the Lorentz shape, then the absorptioncoefficient will be given by

X S ak,,= E -_ _

n=- 7 (V-nd)2

+aX2

(3)

This relation can be expressed in a more convenientform as

S sinh#k d=k(s)=- -coss

d cosh- coss(4)

where s= 27rv/d and j3= 2ar/d. If one assumes that the

incident intensity is constant throughout the spectrum,the fractional absorption for the entire band becomes

1 T

A=-J (1-exp-k(s)w)ds.27r _

(5)

For the case in which the lines are far apart comparedto their half-width (d>>a, or ( small), Eq. (4) becomes

S3k,=k(s)= so (6)

2d sin2(s/2)

and the expression (5) can be integrated to yield

(irSaw)lA = erf , (7)

d

where erf (x) designates the error function of x (some-times called the probability integral) given by

2 rZerf (x)=- f exp(-t2 )dt. (8)

0

One can represent the fractional absorption of theband by

A = erf (lw!2) , (9)

if a generalized absorption coefficient of the band isdefined by

27raS(10)

One can now relax somewhat the restrictions of theidealized band and consider the fractional absorptionA in Eq. (9) to be the absorption at the center of aninterval v, to V2 containing several lines that are ap-proximately equally spaced in this interval and ofapproximately equal intensity. If one assumes that

there is no contribution to the absorption at the centerof this interval from lines outside the interval, theexpression (9) can be written

A(Vl,v2)=erf ()' (11)

820 780 740 700 660 620 580WAVENUMBER In cm'

FIG. 1. The generalized absorption coefficient for the 15ju bandof CO2- Solid curve, Elsasser and King; broken curve, Kaplan.

where l(vI,V2), the generalized absorption coefficientdefined by Eq. (10), depends on the average linespacing, d(v1,v2), and the average line strength, S(vl,v2),in the interval. The applicable quantity I can be deter-mined empirically over the band, and is considered tobe a function of v, varying slowly over the band, therapid variation of the primary absorption coefficient k,from line to line having been "obliterated."

The CO2 absorption bands are sufficiently regular tobe treated by an Elsasser model. The average spacingbetween absorption lines is about 1.5 cm-l, and ao theaverage half-width of the individual absorption lineunder standard conditions in the atmosphere is about0.06 cm-l. It is seen that the special case of small ald,is satisfied for CO 2. The quantity I in Eq. (11) can bemodified for other temperatures and pressures by useof the relation (10) and

P To a1 = Yo- - )

Po T(12)

where ao, Po, and To refer to standard conditions.Figure 1 presents a plot of such a generalized ab-

sorption coefficient for the 15.4 band of CO 2. The solidcurve gives log (1/2) as determined by Elsasser andKing7 from a report by Cloud.8

The usefulness of such a generalized absorptioncoefficient for calculations of atmospheric transmissioncan best be illustrated by an example. For an atmos-pheric path at ground level (P= 760 mm Hg), andcontaining 100 atmos cm C0 2, one determines from

6 W. S. Benedict (private communication).7W. M. Elsasser and J. I. F. King, Rept. No. 9, Contract

AF 19(122)-392, University of Utah, (September 1, 1953).8 W. H. Cloud, "The 15 micron band of C0 2," Contract N-Onr-

248-01, The Johns Hopkins University (January 1, 1952).

,Z0

336 HO A n n TT C'iT A T A

Fig. 1 the value of /2 for various values of . Since /2represents the absorption coefficient per atmos cm,Iv/2 is then determined for w= 100, and A = erf (w/2)1is easily determined from mathematical tables. If nowA, is plotted against v and planimetered, the totalabsorption fAdv of the 15 CO 2 band is found to be,for this example, 173 cm-'. The empirical equation forthis band, given in the second paper of this series,2yields 177 cm-' for the same conditions of concentrationand pressure.

It should be remarked that in the Elsasser modelthe absorption A at a given frequency is a function ofthe product wP of the absorber concentration and thetotal pressure, as can be seen from Eqs. (10), (11), and(12). The empirical relations of the present studyhowever indicate2 that the total absorption is not aproduct of w and P, but that it increases somewhatmore rapidly with w than with P. Thus the Elsassermodel described here must be used with caution. If thegeneralized absorption coefficient is determined for atotal pressure of one atmosphere and the Elsasser modelused only to extrapolate the absorption to greater con-centration values, all at the same total pressure, nodifficulties will be encounted.

The dotted curve in Fig. 1 shows a more detailedabsorption coefficient for the 15,u CO2 band calculatedfor a temperature of 218'K by Kaplan.9 This workcomprises a rather complete theoretical reconstructionof the entire band, line by line. The agreement ofKaplan's theoretical results with Cloud's experimentalcurve is evidence that the assumption is correct, thatthe total transmission of a band can actually be ob-tained satisfactorily by superposing the transmissionsof all the individual lines, each at its proper place.

-

81. ~~~~~~/X

,2~~~~~~~I

z

LO WV N IN/ -0

001 2700 3100 3500 3900 4300

WAVENUMBER IN cm-,FIG. 2. The quantity w (the 20 concentration for a trans-mission of ) for the 3.2A and 2.7 u bands of H20. Solid curve, totalpressure 125 mm Hg; broken curve, total pressure 740 mm Hg.

IL. D. Kaplan, J. Chem. Phys. 18, 186 (1950); see also, JMeteorol. 9, 1 (1952).

I, A INI V L I 1 L LI A M Vol. 46

Kaplan's curve for the absorption coefficient clearlyshows a great deal more detail than does the generalizedabsorption coefficient which refers to spectroscopic datataken with very wide slits. For practical purposes, suchas constructing a CO2 radiation chart for the atmos-phere, the smaller details might be smoothed out. Thesimilarity of these two curves in Fig. 1 indicates thatthe effect of temperature on the course of the generalizedabsorption coefficient is rather small.

It is seen from the foregoing discussion that theElsasser model of an absorption band can be appliedsuccessfully to the infrared absorption bands of CO 2.Goody and Wormell 0 have also applied this modelsuccessfully to the 7 .8 u and 8.6 u bands of N20, whichbands also have a very regular structure. Attempts toapply this Elsasser model to molecules whose bandshave an irregular structure, however, have not provedsuccessful.

GOODY MODEL OF AN ABSORPTION BAND

In 1950, Cowling" took data on the line positionsand relative intensities for the pure-rotational absorp-tion lines of water vapor from Randall, Dennison,Ginsberg, and Weber, 2 assumed the Lorentz line shapefor the individual lines, and computed the absorptionas a function of absorber concentration for six wave-length regions. These absorption curves were so similar,however, that Cowling was led to the conclusion that"in atmospheric work complication is avoided, andremarkably little error is involved, if a single absorptioncurve is used at all wavelengths." This led Goody toconclude," from inspection of the diagrams of Randallel al., that the only common feature of these ranges is anearly random distribution of line intensities and posi-tions. Goody then postulated the so-called statisticalor random model for a disordered band, and showedthat this model fits quite well the earlier computationsof Cowling." Goody considered the fractional trans-mission at the center of an interval of frequencies n6wide, where n is the number of lines in the interval,and the mean line spacing. The interval is consideredsufficiently wide that there is no absorption at its centerdue to lines outside the interval. It is postulated thatall arrangements of line positions are equally probable,that is, the line spacings are random, and it is furtherpostulated that there is no correlation between linepositions and line strengths. The line strengths areassumed to be given by the following function:

1P (S) -exp (- SI,), (13)0,

0 R. M. Goody and T. W. Wormell, Proc. Roy. Soc. (London)A209, 178 (1951).

"T. G. Cowling, Phil. Mag. 41, 109 (1950).32 Randall, Dennison, Ginsberg, and Weber, Phys. Rev. 52,

160 (1937).'3 R. M. Goody, Quart. J. Roy. Meteorol. Soc. 78, 165 (1952).

.. 1 v *s *\ I L) 1

337May1956 INFRARED TRANSMISSION OF SYNTHETIC ATMOSPHERES

z0U)

U)

z4OrH

W

z0Ld

(Lw

1000

W/wo

FIG. 3. Fit of the H20 absorption data to the Goody relation. Left-hand curve, total pressure 740 mm Hg; 0, 6 .3,U band; A, 2 .7 , and3.2 A bands; *, 1.87/A, 1.38y, and I. 1 bands. Right-hand curve, total pressure 125 mm Hg; X, 6 .3 u band; A, 2. 7 A and 3.2M bands; o,

1.871, 1.3 8 g, and 1.11 bands.

where P(S)dS is the probability that a given line has a

strength in the range S to S+dS, and where a- is the

mean line strength in the interval. The individual linesof the band are assumed to have the Lorentz shape.

With these assumptions about the nature of the dis-ordered band, one can show that the transmission at

the center of the interval considered when the radiationtraverses an optical thickness w of absorbing gas isgiven by

r - wa-aT(a,wa-) = exp (14)

i a2+)

Thus, the fractional transmission in such an intervalof an absorption band depends on the mean line spacing,the mean line strength, and the half-width, as well ason the optical thickness of the absorber.

This random model can be applied to the near-infrared absorption bands of H20 in much the sameway that the Elsasser model was fitted to those of CO2.The half-width a was assumed to be the same for all of

the H2 0 individual absorption lines, and a the mean linespacing, was also assumed to have the same value forall the bands. Then, for a fixed value of w, the absorberconcentration, the transmission at a given frequencyis considered to depend on the mean line strength a- inthe neighborhood of this frequency. The mean line

strength is thought to be a slow function of frequencyover the band, and can be specified in terms of wo, the

absorber concentration necessary to yield a transmission

of at a given frequency. Thus, in the Goody model as

in the Elsasser model, the rapid variation over each

individual line of the primary absorption coefficient k,

has been "obliterated." The combination of the two

functions T(w,-) and a-(v) then describes the trans-

mission process for this particular band.The relation chosen in this study to represent the

variation of a over the band is a-= k/wo. Near the center

of the band, the mean line intensity is large, or only a

small value of absorber is required to obtain a trans-

mission of -. At the far wings of the band, a large wo

is required, and the mean line intensity is small. Then

Eq. (14) can be written as

T= exp-w/wo * ka

w/wo ka) 4o 2+

Or

I -w/wo-fl I= exp

i (1+)1

(15)

where 3= k/6b and -y= k/7ra. Thus, the transmission atany frequency is determined by the value of wo for thatfrequency, and the parameters 3 and y can be chosento yield the best fit of the data.

Figure 2 presents plots of wo as a function of fre-quency for the 3.2 yu and 2.74A bands of H20, as deter-mined from the experimental data of the present studyfor two values of total pressure, P= 740 and P= 125mm Hg. Figure 3 is a. composite plot for all of the H2 0data for the fractional transmission plotted against thelogarithm of w/wo for the same two values of totalpressure. From these two curves the parameters / and-y are determined to be /3=1.97 and y=6.57. No par-

HOWARD, BURCH, AND WILLIAMS Vol. 46

0

Cn

w0

n6 00 3200 3600 4000 4400

WAVEN UMBER I N cm-'FIG. 4. The 2 .7p and 3.2 bands of H2 0, showing predicted

curves for 50, 10, and 3 pr. cm. Total pressure 740 mm Hg.Reading down from top, concentration in pr. cm is: 50, 10, 3,1.68, 1.29, 1.12, 1.03, 0.60, 0.34, 0.17, 0.09, 0.04. The three topcurves are predicted; all lower curves are observed experimentally.Although the shape of the curves is determined by slit width, thetotal absorption fsAdv is of general significance.

ticular value need be assumed for the half-width a, butin order to correlate the data taken at the two differentvalues of total pressure it was assumed that a obeys therelation (12).

This final empirical fit,

- 1.97w/wo 17>= exp [ J . (16)(1 +6.5 7w/wo) 1

where wo is a function of frequency and total pressure,has been found to fit all of the H20 absorption bandsstudied. The foregoing empirical fit has been used tocompute the transmission (and hence the absorption)over the H20 absorption bands at 6.3, 3.2, 2.7, 1.87,1.38, and 1.1u for absorber concentrations of 3, 10, and50 pr. cm H20 at total pressures of 740 and 125 mm Hg.Figure 4 shows the predicted curves for the 3.2,u and2 .7g H20 absorption bands for a total pressure of 740mm Hg. Figures showing the application of the Goodymodel to the other bands of H20, as well as all of thedata, in figure or in tabular form, from which theempirical fit was derived, will appear in a forthcomingreport."

14 Howard, Burch, and Williams, Geophysical Research PaperNo. 40, Geophysics Research Directorate, Air Force CambridgeResearch Center, Bedford, Massachusetts (November, 1955).

338