JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Pressure-Modulation Study of the Absorption of Non-Black Thermal Radiation*RICHARD R. PATTY AND DUDLEY WILLIAMS

Laboratory of Molecular Spectroscopy and Infrared Studies, Department of Physics and Astronomy,The Ohio State University(Received July 10, 1961)

The pressure-modulated near-infrared emission spectrum of a gas consists of a set of vibration-rotationbands. The intensity of the emitted radiation is a rapidly varying function of frequency. The transmissionof this radiation through various gas samples has been investigated for absorbing atomspheric gases suchas carbon dioxide, carbon monoxide, nitrous oxide, and methane; the effects of pressure-broadening of spec-tral lines by inert gases such as argon and nitrogen have been considered. The transmittance of radiantpower has been shown to depend on the composition of the emitting sample and the compression ratioemployed as well as on the composition of the absorbing sample. Possible application of pressure-modulationtechniques to atmospheric transmission studies is suggested.

THE transmission of non-black thermal radiation1by atmospheric gases is of considerable import-

ance in considerations of the earth's heat balance and inthe development of infrared signaling systems. Forexample, the radiation from the ozone layer traverseslayers of water vapor, carbon dioxide, and nitrous oxide

100M

80.N20 V = 1299 cm-'

60-(L \ -Calculated fromconventional transmissionZ \ 0 Observed

I- 40

20- \t 4zi

LII 2L0

0,2 0,4 0.6 1 12Sample In path (atmos cm)

W N 20 ) =1145 cm-'0 100- W

aCalculated from conventional transmissionI- Ad \ Observeda80 U)z

I-

z . 0. i

40

20

0 0.2 0,4 0.6 0,8 1.0 1.2

Sample in path (atmos cm)

FIG. 1. Plots of percent transmittance versus absorber concen-tration in path comparing the results for a conventional sourcewith a "black" and "non-black" gaseous source. Pure nitrous oxideused as source and absorbing sample. 2:1 compression ratio.12.6-cm absorption cell.

* Supported in part by the Geophysics Research Directorate,Research Division, Air Research and Development Command.

before reaching ground level, while the thermal radia-tion from warm carbon dioxide traverses colder layersof carbon dioxide and other gases in its upward passagethrough the atmosphere. Conventional laboratorystudies of non-black radiation through various gaseshave not been numerous but have usually employedsources consisting of heated gas cells or hot flames;undesired radiation from the walls and windows of theheated cells and large temperature gradients in flamescan present serious difficulties.

The present paper gives an account of an exploratorystudy in which the thermal radiation from a pressure-modulated gas serves as the "incident radiation"; thepercentage of this energy transmitted by various ab-sorbing gases is then measured. The experimental

z

U.1

0 100

80(I)

60

a:

- 40

1-J

w20

0100z

':

Q 80

U)

a-

PN IN PATH (cm Hg)

10 20 30 40 50 60 70 80

PN IN PATH (cm H)

FIG. 2. Plots of percent transmittance versus partial pressureof sample in path for three sources of radiation. 2:1 compressionratio. 12.6-cm absorption cell.

546

V=2168 cm 1

SOURCE:NERNST4cm CO+ARGON TO 15cm Hg

I 0.2 cm N0 +ARGON TO 15cm Hg

VOLUME 52, NUMBER 5 MAY, 1 962

NON-BLACK THERMAL RADIATION

SOURCE4 cm N20 +Argonto 15 cm Hgc = 5:1

SAMPLE IN PATH (cm Hg)P P

72.4 0i ~~~0.24\E ~~~1.0

4.1817.4

72.4FIG. 3. Spectral radiance of

nitrous oxide viewed throughmethane (upper curves), and spec-tral radiance of methane viewedthrough nitrous oxide (lowercurves) for the partial pressure pand the total pressure P shown.Compression ratio 5: 1.

SOURCE4 cm CH4+ Argon

to 15 cm Hg

SAMPLE IN PATH (cm Hg)P P

72.4 00.1180.491.0

2.058.54

35.572.4

1360 1290WAVENUMBER in cm-'

arrangements employed are the same as those describedin earlier reports",2 except for the insertion of an addi-tional absorption cell in the optical path between thepressure modulator and the entrance slit of a prismspectrograph. The gases investigated were nitrous oxide,carbon monoxide, methane, and carbon dioxide, alongwith nonabsorbing gases such as argon and nitrogenintroduced to broaden the emission and absorptionlines of the emitting and absorbing gases. In the pres-ent work, studies of self-absorption were made by usingthe same gas as emitter and absorber; other studies weremade in which the emitter and the absorber were dif-ferent gases with overlapping bands.

The near-infrared emission spectrum of a pressure-modulated gas is not continuous but consists of char-acteristic vibration-rotation bands, The spectral radi-ance is less than that emitted by a correspondinglytemperature-modulated blackbody except in spectralregions in which the modulated gas sample is "black"(i.e., a gas sample which would transmit no radiation)during all portions of the modulation cycle. Withinemission bands the spectral radiance is a rapidly varyingfunction of frequency except near the centers of ex-tremely intense bands.

The detection system used in the present work issuch that the recorder chart deflection Do(v) produced

' R. R. Patty and D. Williams, J. Opt. Soc. Am. 51,1351 (1961).2 R. R. Patty and D. Williams, J. Opt. Soc. Am. 52, 543 (1962).

of the modulated radiation reaching the detector fromthe modulated source. When a cell containing an ab-sorbing gas sample is placed between the source and thespectrometer, the recorded chart deflection becomesD (v). The measured transmittance D(v)/Do(v) is de-pendent on the nature of the source sample and on thenature of the absorbing sample; its value also dependson the spectrometer slitwidths. The spectral slitwidthsused in the present work were sufficiently large to includeseveral neighboring rotational lines, so that the meas-ured value of D(v)/Do(v) represents an average valueof the actual transmittance in the vicinity of fre-quency v.

It can be shown that the power emitted for an entireband is proportional to the recorded area JDdx, whereD is the recorder deflection and x is the distance alongthe chart paper. A significant quantity, which is inde-pendent of spectrometer slitwidths, is the absorptance,or fraction of the total radiant power absorbed by anabsorbing sample. For a given emission band, thisquantity is given by the expression

fJDodx-J fDdx

JfDodx

where the integration limits include the entire emissionband. As in the case of measured transmittance, thisfraction also depends on the natures of the emitting

1250

547Malt 1962

R. R. PATTY

CoSample in path (cm Hg

P P21.2 0

0.11R.$R

2.5.

210.

956592

2213 2143 2069WAVENUMBER in cm-'

FIG. 4. Spectral radiance of carbon monoxide viewed throughvarious absorber concentrations and total pressures of carbonmonoxide placed in the beam for the 2143-cm- 1 band. Source is4 cm CO+argon to 15 cm Hg. 2:1 compression ratio. 12.6-cmabsorption cell.

and absorbing samples. In spectral regions in which thesource sample is "black," the transmittance of the ab-sorbing sample would be expected to be the same as thetransmittance of the same sample when a Nernstglower or blackbody is used as a source; in spectralregions in which the source is not "black," the trans-mittance of the absorbing sample depends upon the way

in which the lines of the emitter and the absorber over-lap or "match."

TRANSMITTANCE

In Fig. 1 are shown some results obtained when purenitrous oxide is employed as a source and also as anabsorber. The upper part of the figure shows the meas-ured transmittance at 1299 cm-' in one branch of thestrong 1285-cm-' band; the transmittance decreasesas the absorber concentration (product of cell lengthand partial pressure of absorbing gas) in the absorptioncell is increased. The observed values of transmittanceare essentially the same as those obtained convention-ally with a Nernst glower used as a source; this resultindicates clearly that the emitting sample is essentiallyblack at 1299 cm-'. In the lower part of Fig. 1 are shownresults obtained at 1145 cm-' in a branch of the weakernitrous oxide band centered at 1167 cm-'. In this case,the observed values of transmittance are less than thosecalculated from conventional transmittance measure-ments; the source is obviously not black at this fre-quency and its emission lines "match" closely those ofthe absorber.

In Fig. 2 the results obtained with two differentpressure-modulated sources are compared with theresults obtained with a Nernst source. One source con-sisted of a carbon monoxide-argon mixture; the otherwas a nitrous oxide-argon mixture. The upper panelshows the measured transmittance at 2213 cm-' in aregion of strong absorption as the pressure of nitrousoxide in the absorption cell is increased; the lowerpanel gives corresponding results for 2168 cm1 in aregion of weaker absorption. In both regions thetransmittance curve, is lowest for the nitrous oxidesource, since with the nitrous oxide source, emissionand absorption lines most nearly "match." It is perhapssurprising that the transmittance plots obtained withthe carbon-monoxide source fall below those obtained

),02 0.05 0.1 0.5 1.0

PCIN PATH (cm Hg)

5.0 10

FIG. 5. Plots of the ab-sorptance versus partialpressure of carbon monox-ide placed in the path. Thesource is 4 cm Hg of COplus argon to 15 cm Hg. 2: 1compression ratio. 12.6-cmabsorption cell.

50

in path (cmHdC oP

10.0 00.16

0 -

'5

I0

Sample in path (cm Hg)JP p

co ~72.4

1926 9

1.00.80.6

0.4

Q2

0.10.080.06

0.04

0.02

0.01

w0Z

It0(I,co

- I *'. . '* I .I1 §

_ 2143 cm-' CO BAND e e- A P=72.4 cm Hg e/ . -

B P=21.1 cm Hg A /

C P= 1O.0 cm HgB- e//°0

,, I, ,,,11 , , I ,,,,11 , , I, ,,1, ,

I

l

548 ANAD D A\,I L L I A M S Vol. 52

SampleCO

M 1 NON-BLACK THERMAL RADIATION

with the Nernst glower; this result indicates consider-able overlap of the lines of carbon monoxide andnitrous oxide in the vicinity of the two frequenciesstudied. It should be recalled that curves of the typeshown in Figs. 1 and 2 are strongly dependent on ef-fective slitwidth; however, even with wide slits, thepresence of significant line overlap can be clearlydemonstrated.

In the case of nitrous-oxide and carbon-monoxidespectra near 2213 cm-l and 2168 cm-l, there are sets oflines which are regularly spaced. Quite a different situa-tion is evident from the spectra shown in Fig. 3. Theupper curves in this figure show the results obtainedwhen radiation from pressure-modulated nitrous oxideis transmitted by methane; the lower curves give theresults obtained with methane as the source and nitrousoxide as the absorber. The Q branch of the methaneis easily identified as an absorption maximum in theupper curves and an emission maximum in the lowercurves. Comparison of the curves in the lower and upperparts of Fig. 3 shows that in most of the spectralregions in which the bands of methane and nitrous oxideoverlap, methane does not absorb nearly so strongly asnitrous oxide; this is due to the larger line spacing inthe methane band. The behavior of the methane andnitrous oxide bands is an example of an extreme

.-0.8

0.6

,, 0.4

, 0.3

0.2

0.11.0 4 10

pco2 in path (cm Hg)

40 10

667 cm-' Region of CO2 Sample in path (cm Hg)C=2:1 P p= 0.0Slits 50 72.4 0.57

01-4 ~~~~~~~~~~~~1.53.9610.427.472.4

667 cm-,

FIG. 6. Spectral radiance of C0 2 viewed through various C02samples and plot showing the dependence of the absorptance forthe entire band on the compression ratio of the source which is4 cm Hg of C02 plus argon to 15 cm Hg. 12.6-cm absorption cell.

WAVENUMBER in cm-'

FiG. 7. Spectra of the 1285-cm-' and 1167-cm-' N 20 bandsshowing the effect of adding a nonabsorbing gas to various ab-sorber concentrations w, where w is the product of the partialpressure of absorber in atmospheres and the cell length. The sourceis 4 cm Hg of N 20 plus argon to 10 cm Hg. Compression ratio2:1. 6.3-cm absorption cell.

"mismatch" between the lines of emitter and absorber.Quite different results are encountered in cases of opti-mum "match" between emitter and absorber en-countered in "self-absorption."

SELF-ABSORPTION

The present investigation included a series of meas-urements of self-absorption by carbon monoxide in thevicinity of its fundamental; similar results were ob-tained for N 20. The source of radiation consisted of asample of carbon monoxide with argon added; themixture was selected to give a reasonably intense butnot a "black" emission spectrum when a 2: 1 compres-sion ratio was used. The emitted radiation was passedthrough various absorber concentrations of carbonmonoxide with nitrogen added to give three differenttotal pressures. The resulting spectra are shown in Fig.4; increasing the carbon-monoxide partial pressure pfor the absorber produces increased absorption for eachtotal pressure P. Curves of growth showing the fractionof absorbed power as a function of absorber concentra-tion or partial pressure are shown in Fig. 5. On the basisof an Elsasser treatment for strong, nonoverlappinglines, a square-root dependence of the fraction of ab-sorbed power on absorber concentration was antici-pated; the slopes of the curves have values of 0.42

r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -

* C=2:1*o P = 72.4 cm Hg

*C =5:1

I I A . I _ _

667 cm-' Region of 00, Sample in path (cm Hg)P po =000o=5:l 72.4 0,99

Slits 3 O"U 2.024.138.45

17.3ft 35.4

7__2.4

667 cm-'

549May 1962

R. R. PATTY AND D. WILLIAMS

0.6

0.4

0.2

0

1.0

0.8

0.6

0.4

1285 cm' N20 Bond

\ -w=~~~0.376(atmos cm)

w=0.75

w= 1.5w 3.0

.... . ...... ...,.....

10 100Total Pressure of Sample in path (cm Hg)

1167 cm-' N0 Bandw=0.376w=0.75w=1.5

w=3.0

.... ,.,,,1, . ,,.. . ,

10 100Total Pressure of Sample in Path (cm Hg)

FIG. 8. Plots showing the dependence of the transmittance onthe total pressure of the absorber for strong and weak bands.The source is 4 cm Hg of N2 0 plus argon to 10 cm Hg and the ab-sorber is N 20. 2:1 compression ratio. 6.3-cm absorption cell.

to 0.46, while a square-root dependence would be char-acterized by a slope of 0.5. The results given in Figs. 4and 5 are examples of the effects to be expected whenthe match between emission and absorption lines isclose; the fraction of the power absorbed by a givenabsorbing sample is much greater than that absorbedwhen a conventional source is employed.

When small compression ratios are employed, thereis a fairly good match between the emission lines of amodulated sample and the absorption lines of the samegas sample at room temperature. The match betweenemission spectrum and absorption spectrum becomespoorer when high compression ratios are used, becausemuch of the emitted radiation involves molecular energystates not appreciably populated at room temperature.The resulting mismatch results in a decrease in theabsorptance when the compression ratio is increased.These effects are illustrated in Fig. 6, which shows re-sults obtained near 667 cm-' in the spectrum of carbondioxide. In addition to the fundamental v2, there areseveral "hot bands" in this region; the sharp emissionpeaks shown in the observed spectra are Q branches ofthese hot bands. It will be noted that these are muchstronger in the spectrum obtained with a 5:1 compres-sion ratio than in the corresponding spectrum obtainedwith a 2: 1 compression ratio. It will also be noted thatabsorption at these peak positions and at the extremewings of the fundamental band is not nearly as strong

as at the center of the fundamental band. The curves ofgrowth shown in the top panel of Fig. 6 show that theabsorptance is greater for the radiation obtained at thelower compression ratio.

Another type of mismatch can be produced by pres-sure broadening. The addition of a nonabsorbing gassuch as argon or nitrogen to a sample of an absorbinggas produces a broadening of absorption lines as a resultof increased collision frequency. If the initial absorberconcentration in the sample is sufficiently low to produceweak absorption lines,3 addition of the nonabsorbing gasresults in a reduction of the absorption coefficient nearthe line centers. Therefore, if radiation from a gas at apressure sufficiently low to produce narrow lines ispassed through an absorbing sample having weak ab-sorption lines, an increased transmittance would beexpected to be produced by the addition of a nonabsorb-ing gas to the absorbing sample.

In order to investigate this effect, spectra of varioussamples were studied in the vicinity of the 1167-cm-lband of nitrous oxide, which is a fairly weak band forthe samples employed. The nearby 1285-cm-1 band,which is a strong band, was also studied in order toshow the contrast between the effects of line broadeningin the two types of bands. The source sample at maxi-mum volume contained nitrous oxide at a pressure of4 cm Hg with argon added to give a total pressure of10 cm Hg; a 2:1 compression ratio was employed. Aneven lower pressure was desirable but was not feasiblewith the present modulator. Absorbing samples withfour different absorber concentrations were studied;the total pressures of the absorbing samples were in-creased by the addition of dry nitrogen. Representativespectra for each of the four samples are shown in Fig. 7.For each absorber concentration, the addition of nitro-gen to the sample results in an increase in measuredtransmittance in the weak 1167-cm-l band and, ingeneral, a decrease in measured transmittance in thestrong 1285-cm-l band.

Plots of the transmittance for each of the absorberconcentrations are shown in Fig. 8 as a function of thetotal pressure in the absorbing samples. The trans-mittance increases with increasing total pressure forthe weak 1167-cm-' band as indicated by the lowerpanel of the figure; this is the reverse of the effects ob-served when radiation from a Nernst glower or othersource having a continuous spectrum is incident on anabsorbing gas sample. In the case of the strong 1285-cm'- band, the observed variation of transmittancewith total pressure variation more closely resembles theresults obtained in conventional studies involvingincident radiation from a blackbody or Nernst glower.The transmittance of blackbody radiation by a strongband decreases to some fairly well-defined limit as thetotal pressure of the sample is increased by the addition

I See W. M. Elasasser, Harvard Meteorological Studies No. 6,Harvard University, 1942, for discussion of weak and stronglines and G. N. Plass, J. Opt. Soc. Am. 50, 868 (1960) for discus-sion of weak and strong bands.

00

0I-

Vol. 52550

May 1962 NON-BLACK THERMAL RAD IATION 551

of a nonabsorbing gas; the center of the band is essenti-ally black and no important additional absorption isproduced by lines in the wings of the band. However, inthe present study the absorption by these weak linesin the band wings actually decreases with increasingtotal pressure and can result in a slight increase in thetransmitted power, as shown in the two lower curves ofthe upper panel of Fig. 8. The variation in the bandwings of the 1285-cm-l band can also be noted in thespectra shown in Fig. 7.

SUMMARY

The present work was undertaken with a view to ex-ploring the possibility and feasibility of employing

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

pressure-modulation techniques to problems involvingthe transmission of thermal radiation through atmos-pheric gases. The studies of self-absorption have led toresults that can be interpreted on a semiquantitativebasis in the special cases selected for investigation. Theother classes of absorption of non-black thermal radia-tion are understood on a qualitative basis. It is hopedthat pressure-modulation techniques can be developedfor use in quantitative studies of atmospheric trans-mission problems and also other more general problems.

ACKNOWLEDGMENT

The writers wish to express their appreciation to Dr.Darrell Burch for many helpful suggestions.

VOLUME 52, NUMBER 5 MAY, 1962

Light Scattering Functions for Concentric Spheres. Total ScatteringCoefficients, mi=2.1050, m2 =1.4821*

M. KERKER, J. P. KRATORVIL, AND E. MATIJEVI6Department of Chemistry, Clarkson College of Technology, Potsdam, New York

Total scattering coefficients for concentric spheres with inner sphere mi=2.1050 and concentric sphericalshell m2= 1.4821 have been computed for v=27rb/X over the interval 0.1 (.1) 23.0 (2) 53.0 and for a/v valuesof 0, 0.2,0.4,0.6,0.8,0.9,0.95,0.98,0.99, and 1.00, where a= 27ra/X, a and b are the radii of the inner and totalspheres, and X is the wavelength. The results are compared with those obtained by a small particle approxi-mation, the approximation suggested by Ryde, the Rayleigh-Gans method and an approximation based onusing the single sphere method with a volume averaged refractive index. The Rayleigh-Gans equations forconcentric spheres are derived. The small particle approximation permits accurate estimation of the totalscattering coefficient for any combination of a and P up to = 1.4.

ALTHOUGH the theory of scattering of electro-A magnetic radiation by a spherical particle encasedin a concentric spherical shell of a second material hasbeen available for some time,1 the complexity ofnumerical computation of the scattering functions hasprecluded extensive application. A very small numberof results obtained with a desk calculator were used toconsider problems in radar meteorology.2'3 Morerecently Herman and Battan4 obtained machinecomputations for this purpose in connection with theexperimental studies of Atlas, Harper, Ludlam, andMacklin,5 but these have been limited to back-scatteringintensity functions. In treating the light scattering ofglass spheres of colloidal dimensions coated with a water

* This work has been supported by the U. S. Atomic EnergyCommission, under contract.

I A. L. Aden and M. Kerker, J. Appl. Phys. 22, 1242 (1951).2 M. Kerker, P. Langleben, and K. L. S. Gunn, J. Meteorol. 8,

424 (1951).3 D. Atlas, M. Kerker, and W. Hitschfeld, J. Atm. Terrest. Phys.

3, 108 (1953).4 B. M. Herman and L. J. Battan, Institute of Atmospheric

Physics, University of Arizona, Tucson, Arizona, 1960, ScientificReport No. 15 (unpublished).

5 D. Atlas, W. G. Harper, F. H. Ludlam, and W. C. MacklinQuart. J. Roy. Meteorol. Soc. 86, 468 (1960).

film and immersed in a medium of the same refractiveindex of the glass, Debye6 used an approximate methodrather than the exact theory. The National Bureau ofStandards has programmed the theory for the IBM704 computer,7 possibly in connection with the scatter-ing of radio waves from the Echo satellite. Althoughthis program is general in that it generates both angularintensity functions and scattering functions for eitherreal or complex refractive indexes, its running time israther slow, making extensive computations quiteexpensive. It is rather interesting to note that in allthree areas, viz., the scattering of microwaves byhydrometeors, the scattering of light by colloids, and thescattering of radio waves by large objects, the ratios ofthe particle radius to wavelength normally encounteredare comparable, r/X is from 0.1 to 10.

We have been engaged in an experimental study oflight scattering by coated aerosols8 9 consisting of

6 P. Debye and L. K. H. van Beek, J. Chem. Phys. 31, 1595(1959).

7 H. Oser (private communication).8 E. Matijevi6, M. Kerker, and K. F. Schulz, Discussions

Faraday Soc. 30, 178, 223 (1960).9 E. Matijevi6, K. F. Schulz, and M. Kerker, J. Colloid Sci.

17, 26 (1962).