Absorption Line Broadening in the Infrared

Darrell E. Burch, Edgar B. Singleton, and Dudley Williams

The effects of various gases on the absorption bands of nitrous oxide, carbon monoxide, methane, carbondioxide, and water vapor have been investigated. Self-broadening effects for each of these gases havebeen compared with the effects of nitrogen in broadening the rotational lines within various vibration-rotation bands; the results can be expressed in terms of self-broadening coefficients. The effects pro-duced by various foreign gases have also been compared with those of nitrogen; the results are expressedin terms of relative foreign broadening coefficients and relative collision diameters. The foreign gasesstudied include the nonabsorbing gases helium, oxygen, argon, hydrogen, and nitrogen and also carbonmonoxide, carbon dioxide, and methane in spectral regions where there are no overlapping bands.

The fractional absorption A'(v) of monochromaticradiation of frequency v can be expressed in terms ofLambert's Law: A'(v) = 1 - exp [(-k(v)w)] wherek(v) is the absorption coefficient and w is the opticalthickness or absorber concentration. In the case ofgases k(v) is a rapidly varying function of frequency inthe infrared, and the measured fractional absorption A (v)is strongly influenced by the spectral slit widths of thespectrographs employed. However, it has been foundthat the measured total absorption f' A (v)dv is equal tothe value of Jf A 'G()dv, provided the limits of integrationinclude an entire absorption band.' The total absorp-tion of atmospheric absorption bands is a subject ofconsiderable importance in studies of the earth's heatbalance and in the development of infrared signalingsystems. The present paper deals with the effects ofvarious gases on the widths of the absorption lineswithin atmospheric bands and hence on the total ab-sorption of atmospheric bands.

The Lorentz line shape

S a!k(P) = - ~~~~~~~(1)r (v - o)' + a 2

The authors were in the Laboratory of Molecular Spectroscopyand Infrared Studies, Department of Physics and Astronomy,The Ohio State University, Columbus, Ohio. Darrell E. Burchis presently with Aeronutronic Division, Ford Motor Company,Ford Road, Newport Beach, California. Edgar B. Singletonis presently at Bowling Green State University, Bowling Green,Ohio.

Received September 1961.Supported by Geophysics Research Directorate, AFCRL,

Bedford, Mass.

where the line strength

S f k(V)dv

is constant, gives a fairly satisfactory approxima-tion of absorption line shapes for the wide range of pres-sures in which "collision broadening" is encountered. 2

With certain approximations, it can be shown that forsmall values of Sw/27ra ("weak lines") the total ab-sorption of a line is given by Sw, while for large valuesof Sw/2ro ("strong lines") the total absorption is givenby 2/Sw.3 Experimental results for limiting casesare in agreement with these predictions; total absorp-tion for entire bands can in certain cases be correlatedwith the total absorptions for individual componentlines. 4

The half-width a of a collision-broadened spectral lineis proportional to the molecular collision frequency.From kinetic theory5 it can be shown that

1 F /ii V 1~~~~1 1 2a - ENi(Dai)l 2lkT (1 +4r I ma mii (2)

where Ni is the number of molecules of the ith type perunit volume, Dai is the sum of the optical collisiondiameters of the absorbing molecule and a molecule ofthe ith type, ma is the mass of the absorbing molecule,and m the mass of molecule of the ith type.

The present paper gives the results obtained in anexperimental study of the relative abilities of variousmolecules to broaden the infrared absorption lines ofatmospheric gases such as N 20, CO, CH4, C02 , andH20 vapor. In the course of the experimental work,various binary mixtures of absorbing gases and "broad-ening gases" were employed. For binary mixtures,

May 1962 / Vol. 1, No. 3 / APPLIED OPTICS 359

Eq. (2) reduces to

a = (2rkT)112 Na(Da.a)2 1/+

Nb(Dab)2 (nm, + ifb)]1. (3)\ inimb

where the subscripts a and b refer to absorbing andbroadening gases, respectively. Substitution for Naand Nb in terms of the partial pressures Pa and pb

gives

1r k 2 /2 C ,P° =~ 4 (kT) (Cawapa + C.,bPb), (4)

where Caa and Cab are constants involving the massesand optical collision diameters of the absorbing andbroadening gases. Recalling that the total pressureP = Pa + Pb, one may obtain from (4) an expressionfor a in terms of P and pa:

a = (.) Ca,b [(Pa + Pb) + C ) pa]

J_ {7 \1/2= ( )1/2 Ca,b[P + (B-l)pal, (5)

where B = Caan/Cab is called the self-broadening co-efficient of the absorbing gas and represents the ratioof the "self-broadening ability" of the absorbing gas tothe "broadening ability" of the broadening gas, whichwas N2 in the present study. The term in brackets in(5) is called the equivalent pressure P, of the example

P. = P + (B -1)p, (6)

where P is the total pressure of the sample and p is thepartial pressure of the absorbing gas in binary mixturewith N2 .

It should be remarked that the theory involved in theabove discussion may be expected to apply to lineswhose width is determined primarily by collision broad-ening; it is not applicable to extremely low pressures,for which natural widths and Doppler broadening aredominant, or to extremely high pressures, at which ab-sorption during collision or "close approach" leads toStark effects and asymmetrical pressure shifts. Ex-perimental studies to be reported later have shown thatthe total absorption J A (v)dv for an entire band in theinfrared can be expressed in terms of the two parametersP. and w for the range of pressures in which collisionbroadening is dominant.

Self-Broadening Coefficients

The experimental determination of the self-broaden-ing coefficient B involves a study of the transmission ofradiation through cells of different length in which theabsorber concentration w is the same. The sample inthe short cell, called the "reference cell," consists of apure sample of the absorbing gas; in view of (6) the

equivalent pressure in the reference cell is Pe = BPref.

The sample in the longer cell, called the "sample cell,"initially consists of the pure absorber at the partial pres-sure required to produce an absorber concentrationequal to that in the reference cell. Various amounts ofdry nitrogen N 2 are then added to produce equivalentpressures PO = N2 + Bp, where PN2 is the partial pres-sure of the nitrogen. For the situation in which theabsorptions in the sample and reference cells are mostnearly equal, it is assumed that the equivalent pressuresin the two cells are the same. The value of the self-broadening coefficient B can then be determined fromthe relation

B = p2/(prcf - p) (7)

In some experiments the reference cell also contains N 2

at a partial pressure pN 2 ref; for this situation the ex-pression for B becomes:

B = (N2 - N2rc)/(Prf - p). (7')

The value of B was determined for several absorptionbands by making use of the double-beam feature of aPerkin-Elmer Model 21 spectrometer. In some cases, a1.55-cm cell was used in the reference beam with a6.35-cm cell in the sample beam; in other cases a 400-cm reference cell was used in conjunction with an 800-cm or a 1600-cm sample cell. Typical comparisons ob-tained with the double-beam spectrometer are shownin the upper panel of Fig. 1. The spectral record givesthe ratio of the transmission of the sample Tsam to thatof the reference Trer; the dashed line represents the re-corder pen position when the two beams were "bal-anced" with Tam Trf. It is seen that the ratioTsam/Tref decreases with increasing pressure in thesample cell, as would be expected, and that some curvesare above the balance line for some frequencies andbelow the balance line for other frequencies. The pres-sure corresponding to balance at any given frequencycan be determined by interpolation between curveslying above and below the balance line.

Since the values of N2 corresponding to balancevary with frequency, it is apparent that the value of Bvaries throughout the band. Points giving the ex-perimentally determined values of B are shown in thecenter panel of Fig. 1. The corresponding ratios ofcollision diameters are indicated on the right-handordinate scale; these were determined from the relation

B = ( 2mb )1/2

\n a + mb

(D., 2

kDa,b/(8)

where the subscript a refers to the absorbing gas (N20)and the subscript b refers to the broadening gas (N2).A typical absorption curve of an N20 sample is shownin the lower panel of Fig. 1 in order that the "structure"of the absorption curve can be correlated with 'the"structure" of the curve relating B to frequency.

360 APPLIED OPTICS / Vol. 1, No. 3 / May 1962

T Sam.

T ref.

B

HbLJ0Of0d

z0

0l0(I)cr3

1.4 , |- II.2- a , .

1.0 A0.8I01

200

40 _2300 2200

1.2 DN20, N20

al DN2O, N,

1.0

2100

WAVENUMBER in cm-l

Fig. 1. The self-broadening coefficient B for the 2224 cm-'N20 band. I-Spectra obtained with w = 0.30 atm cm of N20in both reference and sample cells; total pressure in referencecell = 164 mm Hg. Total pressure in sample cell for each spec-trum is indicated. II-Plot of B and collision cross sectionratio. The four different geometrical points correspond to theresults of the set of spectra shown in I as well as to three othersets of samples having different values of w. III-A typicalabsorption curve which is shown for comparison.

Studies similar to the one summarized in Fig. 1 weremade for various absorption bands of CH4, CO, C02 ,and H20 vapor as well as for other N20 bands. Variouscell lengths and absorber concentrations were used.A single-beam Perkin-Elmer Model 99 spectrometermounted in a vacuum tank was used for certain bands ofC02 and H20 vapor in order to eliminate possible errorsdue to the effects of atmospheric absorption. For thesebands the value of B was determined by comparingthe total absorption of samples having the same valueof absorber concentration, but in cells of differentlengths.

Much current interest in atmospheric transmission isinvolved with the total absorption associated withvarious bands rather than with absorption at isolatedfrequencies. It was therefore desirable to determinenominal values of B for various entire bands of differentgases. Nominal values of B are listed in Table I andrepresent weighted averages of B, with extra weightgiven to frequency ranges at which absorption is great-est. It is believed that the listed values of B are ac-curate to d 6% when applied to entire bands, exceptin the case of H20 vapor, for which the uncertainty is

± 25%. Possible adsorption effects along with the

limited range of partial pressures of H20 vapor con-tribute to the large uncertainty given to the B valuefor H20 vapor.

Included in Table I are also values of B obtained byearlier investigators. 5 Agreement of the values ob-tained by different methods is fair. Some of the resultslisted by other investigators are based on studies of thetotal absorption of single lines; other results listed arebased on measured transmission at one or more fre-quencies within absorption bands.

Foreign Gas Broadening

The relative line broadening abilities of two non-absorbing gases can be found by comparing thepartial pressures required to produce equal total ab-sorption f A(v)dv when the two gases are added toequal samples of an absorbing gas. The method used inthe present work is illustrated in Fig. 2. A sample ofCO having a partial pressure of 100 mm Hg was intro-duced into a 6.35-cm cell; the total absorption of the

Table I. Self-Broadening Coefficients

Self-broadeningBand Investigator coefficient B

N 202224 cm-', 1285 Present study 1.12 i 0.07

cm-', and 1167cm-l

1285 cm Goody and Wormell 1.27 4t 0. 041167 cm-, Goody and Wormell 1.35 ±4 0.072224 cm-' Cross and Daniels 1.29

CO2143 cm-' Present study 1.02 4 0.064260 cm- Present study 1.08 0. 06

CH 43020 cm-' Present study 1.30 4t 0.081306 cm-' Present study 1.38 4 0.08

CO23716 cm-', 3609 Present study 1.30 0.08

cm-', 2350 cm-',961 cm-' bands,and the 875-495cm-' region

Same CO2 bands Edwards 2 + 0. 5p(where p isCO2 partialpressure inatmospheres)

H205332 cm-', 3756 Present Study 5 -4 1.5

cm-', and 1595cm'1

20 micron region Palmer 11 - 6T, whereT is the frac-tional trans-mission

4025.4 cm-' H20 Vasilovsky and Ne- 6line porant

4 lines in 500-600 Izatt 3.6-5.5cm region

M1962 / V, 1,,No. .3 / APPLIED QPTICS 361

w = 0. 1 48 <

\ainos cm /20mm Hg

N2 0 -

ALONE -

I I I

Ii - - I I I I I liii I I

-2143 cm-' CO BAND , B

w= 0.75 ATMOS CM

- I G~~~0H4N2

-~~~ 8 ~~~. H ?

o He

tOO 200l I I I I I I I I

500 1000 2000

TOTAL PRESSURE in mm Hg

Fig. 2. Total absorption versus total pressure for 2143 cm-' CO band. Partial pressure p of CO 10 mm Hg.

2143-cm-' band was measured and is represented by thepoint at the lower end of the curves in Fig. 2. One of theforeign gases, such as He, was then added to the cell;the total absorptions for He pressures of 200, 400, 760,1500, and 3000 mm Hg were measured and plotted onthe lower curve in Fig. 2. The same procedure was re-peated for other foreign gases; great care was takento start with the same quantity of CO in the absorptioncell. From the curves in Fig. 2, it can be seen that witha sample containing CO and N2 the total absorption is50 cm-' when the total pressure is 960 mm Hg; thesame total absorption is obtained with a CO and Hesample at a total pressure of 1450 mm Hg. Since thepartial pressure of CO is 100 mm Hg in both cases, itcan be concluded that 1350 mm Hg of He are requiredto produce the same broadening as 860 mm Hg of N 2 .

A quantity called the relative foreign broadening co-efficient F of a nonabsorbing gas b is defined by

F = N2/Pb, (9)

where Pb is the partial pressure of gas b required to givethe same total absorption as that produced by a partialpressure PN2 of nitrogen when the absorber concentrationw is the same in the two samples. Thus, for the case ofHe cited in Fig. 2, F = 860/1350 = 0.64. Values of Ffor various foreign gases are listed in Table II forvarious absorbing gases; all values of F are referred toN2 as in (9). Values of F for different values of w anddifferent values of total absorption are nearly the same,and there is no significant variation of F for the rangesof total absorption and total pressure covered in thepresent work.

Also listed in Table II are ratios of the collisiondiameter ratios Da,b/Da,N which are related to F bythe expression

D.-b F,/2 [(28 +Ma b(10)D,,N2 L\Mb M a 28 (

where Ma and Mb are the molecular weights of the ab-sorbing and broadening gases, respectively.

Table II. Foreign-Broadening Coefficients and RelativeMolecular Cross Sections

Partialpres-

sure ofab- Total

sorber' pressure Broad- Dab

Absorber (mm range ener Fb DbN2

and band Hg) (mm Hg) b (mean) (mean)

N20 10 25-1000 He 0.73 0.582224 cm-' 02 0.83 0.93

A 0.78 0.93H2 1.23 0.64CH4 1.08 0.94

N20 100 180-1000 He 0.70 0.571285 cm-' H2 1.21 0.64

A 0.83 0.9602 0.72 0.87CO2 1.17 1.14CO 0.97 0.99

CO 100 200-3000 He 0.64 0.562143 cm-' A 0.78 0.92

H2 0.85 0.56CH4 1.12 0.98

CO2 50 80-700 He 0.59 0.522350 cm-' 02 0.81 0.92

H2 1.17 0.62A 0.78 0.93

C2 H63000 cm- 10 50-1000 He 0.52 0.51bands

CH 4 50 100-2500 He 0.56 0.563020 cm-' A 0.82 0.93

CO2 1.25 1.15

a Cell length = 6.35 cm.Tabulated values of F are believed accurate to less than

t6%.

362 APPLIED OPTICS / Vol. 1, No. 3 / May 1962

80-

60-.

.

C

-o__

40

20

I l l III

5000

I I , I I .,

k-- - - I .I I I I I I 1 1

Some of the present values of collision diameterratios can be compared with earlier results of others.Some years ago Cross and Daniels5 studied the effectsof He, 112, 02, and A on the absorption of N 20 at a givenfrequency in the region of its 2 2 24-cm-l band; theirvalues of collision diameter ratios are approximately7% higher than the present values, which are based ontotal absorption. Their values of collision diameterratios for He and H2 in CO are approximately 12%higher than the values obtained in the present work.There is fair agreement between the present results andthose reported by Benesch and Elder,6 who studied theeffects of He, A, N2, and CO2 on the J = 4 5 line ofthe 3020 cm-' band of CH4 . Although the results ofthe two studies are reported in different forms, it wouldappear that they agree to within 10% for CO2 and towithin 5% for the other gases.

Since both the self-broadening coefficients B and theforeign-broadening coefficients F listed in the tables arereferred to nitrogen, the equivalent pressure Pe fora sample consisting of several broadening gases can bewritten

P. = Bp + pN2 + E Fipbi, (11)

where p is the partial pressure of the absorbing gas andFi is the foreign-broadening coefficient of the ithbroadening gas whose partial pressure is Pbi. Thisequation is useful in studies of atmospheric gases. Ithas been demonstrated in work scheduled for earlypublication that at a given temperature the total ab-sorption of a given band depends only on P8 and ab-sorber concentration w.

References

1. J. N. Howard, D. Burch, and D. Williams, J. Opt. Soc. Am.46, 186, 237, 242, 334, 452 (1956). For a theoretical treat-ment see J. Rud Nielsen, V. Thornton, and E. Brock Dale,Revs. Modern Phys. 16, 307 (1944).

2. W. Benedict, R. Herman, G. Moore, and S. Silverman, Can.J. Phys. 34, 830, 850 (1956).

3. R. Ladenberg and F. Reiche, Ann. Physik 42, 181 (1913).4. G. N. Plass, J. Opt. Soc. Am. 48, 690 (1958).5. R. M. Goody and T. W. Wormell, Proc. Roy. Soc. A209, 178

(1951); P. C. Cross and F. Daniels, J. Chem. Phys. 2, 6(1934); D. K. Edwards, J. Opt. Soc. Am. 50, 617 (1960);C. H. Palmer, J. Opt. Soc. Am. 50, 1232 (1960); K. P.Vasilovsky and B. S. Neporant, Optics and Spectroscopy 7,353 (1959); J. Izatt, Ph.D. Dissertation, Johns HopkinsUniv., 1960.

6. W. Benesch and T. Elder, Phys. Rev. 91, 308 (1953).

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