Products of Strength Times Self-Broadened Half-Widthsof Absorption Lines in the V2 Band of Water Vapor

D. K. Prinz

Products of the strengths times the self-broadened half-widths of eight lines in the 2 band of watervapor were obtained by the curve of growth method. The experimental conditions were room tempera-ture (300'K), low pressure (0.5-2.2 torr), long path length (1.98 cm X 104 cm), and an observed resolu-tion of 0.5 cm-'. Corrections for the absorptance from overlapping of near lines, for the neglectedwing absorptance of the main line, and for the absorptance due to wings of distant lines were made usingthe Lorentz profile and the results of previous theoretical calculations. The results are generally higherthan those previously reported.

1. IntroductionRecent interest in the strengths and half-widths of

lines in the vibration-rotation bands of molecules whichare constituents of planetary atmospheres has beenstimulated by the advent of space research and highspeed computers. The applications to space researchhave included determining the amount of water vaporon Venus' and the amount of water vapor betweenthe upper atmosphere of the earth and the sun2 ; thehigh speed computers have made theoretical calcula-tions in order to determine such quantities as the wavefunctions for such asymmetric top molecules moretractable, thus necessitating more accurate experi-mental values of the strengths and half-widths for com-parison.

Three experimental investigations have been made ofthe strengths and half-widths of either a single line orgroups of lines in the 2 fundamental of water vapor.Goldman and Oppenheim3 have published a value forthe strength of the 11-1-1 line centered at 1616.7 cm-'for water vapor at a temperature of 523 0K with anaccuracy of 10% to 15%. Ben-Aryeh 4 has obtainedthe strengths and self-broadened half-widths (or theproduct of these) for twelve lines in the wing of the Pbranch and eight lines in the wing of the R branch ofthis band at a temperature of 4000K with an optimisticuncertainty of 15%. Sakai' has studied the strengthsand air-broadened half-widths of eighteen lines in theregion 1570 cm-' to 1405 cm-' at room temperaturewith no quoted uncertainties because of the difficulties

The author was with The Johns Hopkins University, Balti-more, Maryland when this work was done; she is now with TheNaval Research Laboratory, Washington, D.C. 20390.

Received. 18 September 1967.

in determining the humidity accurately and the as-sumption of the functional form of the slit function.In addition, Benedict' has used the low resolutionspectra of Burch et al.7 coupled with theoretical calcula-tions to deduce the strengths of 3935 lines in the wave-number region 700 cm-' to 2544 cm-' with uncer-tainties of about 20% for the strong lines. My re-search was an attempt partially to fill in gaps left bythe first three investigations given above by coveringthe region 1430 cm-' to 1763 cm-' with eight lines,having at least one line coinciding with that previouslyinvestigated. The conditions under which it was car-ried out-pure water vapor at room temperature andlow pressure (0.5 torr to 2.2 torr)-contrasted in varyingways with the conditions under which the previous ex--periments were done.

I1. TheoryThe total equivalent width W of a single absorption,

line is defined by:

W= f A()dv. (1)"

A (v) is the absorptance at a wavenumber , i.e., A (v) =1 - ek()L, where L is the geometric optical pathlength and k(v) is the absorption coefficient, which for ascollision-broadened (Lorentz) line of strength S andhalf-width y is given by:

k(V) = (z- vo) + .2 (2)

If SL/27ry > 10, the total equivalent width of a linewith a Lorentz profile simplifies to the form

W = 2(SyL)i. (3)

April 1968 / Vol. 7, No. 4 / APPLIED OPTICS 689u

For sufficiently low densities of the absorbing gas andwith no foreign gas broadening, the strength and half-width are proportional to the number of absorbingmolecules. In accordance with the literature we definethe quantities So and yo through the relations:

S = Sop, (4)

y = yoP/760. (5)

p is the density of absorbing gas in units of g cm- 3 ;P is the pressure of absorbing gas in units of torr. Sothus has the units cm g'l; yo has the units cm-' atm-'.In terms of Soyo, Eq. (3) takes the form:

W = 2P[So-y,3ML/(760 RT)1. (6)

M is the gram molecular weight; R is the universal gasconstant; T is the absolute temperature. A plot of Vvs P should then be a straight line with a slope propor-tional to (Soyo)1.

Because accurate measurements of the absorptancecannot be made over the limits of 0 to - for v, it is acommon practice to integrate the absorptance over thefinite range v = o- g to v = o + g and to make a cor-rection for the neglected wing absorptance. If islarge enough so that 1 - e-k(,)L k l(v)L beyondPo g, and if the line has a Lorentz profile, the wingcorrection is 2S-yL/(7rg). Thus, for a line in the square.root region of the curve of growth,

W'= W - (W2/27rg), (7)

where

' = f|+ A(v)dv.0-

Suppose that instead of a single line, there are ll lineswith their centers within the interval 2g and N - /lines with their centers exterior to the interval. Welabel the line of interest with the subscript 1. Neglect-ing the scanning function, the overlapping of the lineslabeled 2, 3, . . ., ll with each other and the overlap-ping of the lines in 2g with those exterior to the interval(a1 = voi - ol)

W'- W W( + Eri- (1 Er2)i=2 2 7rg i =2 (

-22-2I"k ri(1 + ri)- 22 - (2)11 WI Z

i=2 (1 + ri)2 + (4aj1/ W)'

W' is that part of the total integrated absorptance lyingin 2g; w is the integrated absorptance in 2g due to theN - M11 lines exterior to the interval. ri = Wi/IlV.The last summation is derived from Sakai's expressionby assumming that all ll lines lie in the square root re-gion of the curve of growth and that all of the over-lapping considered takes place within 2g. W' and gcan be obtained from the spectra. Unless the completeband is scanned, the ri's, a,'s, and w must be calculatedwith the aid of information from other sources, e.g.,Benedict's table' for the Soi's and voi's and Benedict andKaplan's calculations9 for the yoi's. Equation (8) canthen be solved graphically to obtain W, for each pres-

sure. According to Eq. (6), a plot of W, vs P shouldthen be a straight line; Soiyo, can then be obtainedfrom the slope.

The effect of the scanning function is to decrease theintegrated absorptance in 2g due to the M lines withinthe interval, and to increase the integrated absorptancedue to lines exterior to the interval. Whether the netresult is an increase or decrease in W' above or belowthat of a single line before convolution with the scan-ning function, depends upon the relative positions of theline centers, the line shape, and the functional form ofthe scanning function. The exact form of the scanningfunction is not easy to determined unless the ratio ofthe geometric slit width to the diffraction width isgreater than about three,' when the triangular func-tion begins to be approached.

III. Apparatus

The large (30.5-m long) absorption cell, located atThe Johns Hopkins University when this research wasdone, is of the Pfund type and has an optical pathlength of 1.98 cm X 104 cm. After evacuating the cellto approximately 10-' torr, triply distilled water whichhad previously been pumped on was admitted to thecell by gently warming its container. The pressure inthe cell was monitored at the center of the cell by a pres-sure transducer which had been calibrated against aMcLeod gauge with dry air in the cell. The output ofthe transducer circuitry was in volts dc. The linearcalibration plot had a slope of 2.0 V/torr. The tem-perature over the length and height of the cell was keptconstant to within 10C by means of fans that circu-lated the air both around the circumference and alongthe length of the cell. The vapor within the cell wascirculated with a Roots blower.

The spectrometer was a Perkin-Elmer model 210Bwith special reducing gears attached that permittedscanning speeds as slow as four hours per revolution tothe drive shaft. The grating was a Bausch & Lombreplica ruled with 101.1 lines/mm. The spectrometerwas calibrated in two ways. The mercury line at5461 A was used in nine orders as reference lines. Inaddition, twenty-five of the water vapor lines them-selves, in the region 1770 cm-' to 1400 cm-', were used;the wavenumber for each center was taken from Bene-dict,6 corrected to air. Each method was done fivetimes. A least-squared fit of the total points impliedthe result 0.2118 cm-'/min for the slowest scanningspeed.

Radiation from the globar source was chopped at 13Hz, amplified and rectified by standard Perkin-Elmercomponents, and recorded both visually on a Honey-well Brown recorder and digitally on punched papertape via Perkin-Elmer digital equipment. A smallphotocell placed beside one of the source mirrors wasused to monitor the output of the globar. The signalfrom the photocell was electronically processed suchthat the change in the resultant for a small change inthe globar voltage was approximately 7.5 times thechange in the recorder signal from the spectrometer at a

690 APPLIED OPTICS / Vol. 7, No. 4 / April 1968

Table I. Curve of Growth Data for Eight Water Linesa

Po

(vacuum) g (SO7O)L SONa C X 103 Voi(cm-') Side (cm-,) (g-' atm-) (g-' atm-') (cm-' torr-2) (cm-') ri

'1761.8 L 1.57 960 1500 3.4 1762.8 0.01691762.8 0.03261762.7 0.0136

1761.8 S 1.90 780 1120 4.7 1760.9 0.03241761.1 0.06141761.1 0.02661761.2 0.0154

1761.8 B 1.57, 1.90 950 1380 4.051756.8 S 1.9 610 960 6.2 1756.1 0.0186

1756.4 0.01931739.9 L 1.74 1340 1830 8.6 1740.2 0.0145

1740.8 0.02921739.9 S 1.45 1160 1440 8.8 1738.5 0.0123

1738.7 0.01641739.3 0.1113

1739.9 B 1.74, 1.45 1230 1610 8.71616.7 B 2.33 3330 4490 4.1 1616.9 0.0263

1617.4 0.00831609.5 L 1.08 150 210 2.9 1610.0 0.1811

1609.5 S 1.32 225 310 3.91609.5 B 212 250 3.41496.3 L 1.27 2120 2550 12.4 1496.4 0.0204

1496.3 S 2.10 2500 3330 10.3 1494.5 0.01431494.7 0.02621496.1 0.01201496.2 0.0092

1496.3 B 1.27, 2.10 2260 2890 11.351464.9 B 1.28 1290 1780 2.7 1464.5 0.0185

1465.2 0.06011465.7 0.0108

1430.0 B 1.03 380 650 2.1 1429.8 0.0377

a Calculations are based upon symmetrized lines using long (L), short (S) wavenumber side, or both (B). Where two g values appearin column 3, the unsymmetrized line was used; Voi is from Ref. 6; C and ri are calculated from data obtained from Ref. 6 and Ref. 9;So-yo is calculated from data of the present investigation using the observed T = 1 level, while (SOya)L is calculated using the bottom ofnear troughs as the T = 1 level; g is one-half of the integration interval when the line was symmetrized. For the unsymmetrized line,the first value of g is the long v integration interval; the second value is the short v integration interval.

given spectrometer setting with the slit width usedduring the experiment. The spectrometer and sourceoptics were contained within a box that was contin-uously purged with 1.4 liters/sec of dry air at a dew-point of -730 C.

IV. Experimental ProcedureThe 100% transmittance data were obtained over

approximately 20-cm-' intervals centered on each ofthe lines of interest with the cell evacuated and thesource optics and spectrometer housing purged. Thestrongest line of interest, centered at 1616.7 cm-', had aresidual equivalent width of about 0.01 cm-'. Watervapor was then admitted to the cell and permitted toundergo adsorption to the walls for about 1 h. It wasthen pumped on for a few minutes, after which thepump was closed off from the cell. Thus, equilibrium,defined as a constant pressure transducer output, wasachieved more rapidly. Two scans over each of thelines of interest were taken. The cell was then evacu-ated and a rapid scan was taken to check the 100%

transmittance data to be sure the globar monitor wasworking properly. The zero transmittance level waschecked several times during the experiment.

The data were converted from punched tape to IBMcards and were processed with one of several Fortran IIprograms on an IBM 7094. The programs permittedthe use of either both sides of a line or only one side andsymmetrizing it.

V. Reduction of DataW' was calculated for each interval 2g centered about

each of the eight lines by the computer; g was chosen sothat the integration interval terminated in the vicinityof the nearest trough. The input data consisted of theencoder position corresponding to zero transmittance(T = 0), averaged over the noise, the encoder positionfor the 100% transmittance (T = 1) scan punched outat approximately 0.002-cm-l intervals, the encoderposition for the scan taken with water vapor in the cell(also at 0.002-cm-1 intervals), the location of the cen-ter of the line, the width of the integration interval,

April 1968 / Vol. 7, No. 4 / APPLIED OPTICS 691

w

1.5 -

1.0 -

0.5 -

_ (cm )

L

P

0.5 1.0 1.5 2.0 (torr)

Fig. 1. The equivalent width (V) vs pressure (P) for the 1616.7-cm'- line. Line L is lowest limit of values obtained from Eq.(13) using bottom of troughs to establish T = 1 level. Results of

calculationsu using Eq. (8).

Table II. Values of Sy All Referred to 300K0"

so

(cm-') A B C D E

1761.8 1330 14691756.8 960 7911739.9 1630 15871616.7 4490 3855 33781609.5 260 2351496.3 2920 23991464.9 1780 16741430.0 650 506 379 339

a A-Averages values from Table I; B-So from Ref. 6 and-yo from Ref. 9; C-So from Ref. 3 and yo from Ref. 9; D-Sofrom Ref. 5 and yo from Ref. 9; E-From Ref. 4. All values arein units of g-' atm-'.

and a character to signify that either both sides or onlyone side of a line was to be integrated.

References 6 and 9 were used to calculate the ri'sand w. Within 2 g, only those neighboring lines in thesquare root region of the curve of growth were used.The values of the r's are given in Table I. w was calcu-lated using the Lorentz line shape by including all thelines given in Ref. 6 in about a 50-cm-' interval oneither side of vol -4 g in addition to all the lines withstrength over 50 cm g-' for the remainder of the listingin that reference. The total number of lines used wasabout 450. This was done for each of the integrationintervals centered about a line of interest at a pressureof 2 torr. Because of the cost of computer time, theextrapolation to other pressures for each line was per-formed by assuming that w = CP2. The values thusobtained are given in the sixth column of Table I.

Equation (8) was used to obtain W for each line ateach pressure. The square of the best straight linevisually fitted through the points on the plot of W vs P

was then set equal to 1.0035 X 10-4 Soao. The valuesof So-yo thus obtained are in column five of Table I.Figure 1 is an example of the W vs P plots.

VI. ConclusionsThe uncertainty in Soyo due to the uncertainty in the

location of the center of a line was different for eachline. Extremum values for the centers of the 1616.7-cm-' and 1609.5-cm-' lines (the strongest and theweakest, respectively) showed that the effect of thisuncertainty on Soya was less than i 8%. The uncer-tainty in the location of the T = 1 level (=1%) affectsTV most at the lowest pressure, decreasing W as P in-creases. The effect is also more pronounced for weaklines than for strong lines. Thus, the uncertainty dueto this cause alone is greatest for the 1609.5-cm-' linein Soy. It is + 20%. The I = 0 line (averaged) wasvery stable; it could not introduce more than 2% tothe uncertainty in So-yo. The over-all uncertainty inSo-y is least for the 1616.7-cm-' line, it being 4 6%.

The effect of the scanning function is difficult to esti-mate quantitatively because the ratio of the geometricslit width to the diffraction width was about 2 and not 3,as would be required for the triangular approximationsto be valid. Some order of magnitude of the effect onw at P = 2 torr was attempted by convoluting a tri-angular function with a half-base of the order of the ob-served resolution in each region with the true ab-sorptance. The increase to w was of the order of 10%to 20% for most of the regions. Certainly, the scanningfunction plays a major role in causing the difference be-tween the values of Soyo for the two sides of a line whenthey are calculated separately. For this reason,the averages of the L, S, and B results were taken forcomparison with the results of other investigations.These averages appear in column A of Table II. Theresults are generally higher than those previously re-ported. In fact, the results in columns C, D, and E arelower than (SO'yo)L given in Table I. Whether thesediscrepancies are due to larger half-widths than pre-viously supposed, to a wing falloff more gradual thanthe Lorentz profile, or to the distortion of the scanningfunction cannot be ascertained since all three couldcause greater Soya products.

I would like to thank John Strong for his encourage-ment, William S. Benedict for the use of his unpub-lished calculations, and William Mankin for his gener-ous assistance in writing the computer programs.

This research was supported by Cwb and ONR con-tracts and by a grant from ARPA.

References1. M. Bottema, W. Plummer, and J. Strong, Astrophys. J.

139, 1021 (1964).

2. R. Zander, J. Geophys. Res. 71, 3775 (1966).

3. A. Goldman and U. P. Oppenheim, Appl. Opt. 5, 1073(1966).

4. Y. Ben-Aryeh, J. Quantum Spectrosc. Radiat. Transfer 7,211 (1967).

692 APPLIED OPTICS / Vol. 7, No. 4 / April 1968

5. H. Sakai, "Line Widths and Strengths in the 2 WaterBand," Final Rept. CWB, The Johns Hopkins University(1964).

6. W. S. Benedict and R. F. Calfee, "Line Parameters for the1.9 and 6.3 Micron Water Vapor Bands," ESSA ProfessionalPaper 2 (1967).

7. D. E. Burch, E. B. Singleton, W. L. France, and D. Wil-liams, "Infrared Absorption by Minor Atmospheric Con-stituents," Final Rept. AFCRL-412 (1960).

8. H. Sakai, J. Opt. Soc. Amer. 56, 127 (1966).

9. W. S. Benedict and L. D. Kaplan, J. Quantum Spectrosc.Radiat. Transfer 4, 453 (1964).

10. L. Sica, Jr., Dissertation, The Johns Hopkins University(1966).

11. H. Sakai, Dissertation, The Johns Hopkins University(1962).

Meetings Calendar continued from page 686

27 SPSE Mtg., So. California Chap. K. T. Lassiter,Eastman Kodak Co., Dept. 942, Rochester, N.Y.14650

27-29 Am. Ophthalmological Soc., Hot Springs S. D.McPherson, 1110 W. Main St., Durham, N.C.27701

27-31 Course in ir Interpretation, Part II, PhiladelphiaLynn Bass, Sadtler Res. Labs., Inc., 3316 SpringGarden St., Philadelphia, Pa. 19104

27-31 Course in Mass Spectrometry, Philadelphia LynnBass, Sadtler Res. Labs., Inc., 316 Spring GardenSt., Philadelphia, Pa. 19104

29 SPSE Mtg., Twin Cities Chap. K. T. Lassiter,Eastman Kodak Co., Dept. 942, Rochester, N.Y.14650

June? Summer Clinic in X-ray Spectroscopy, Albany

Henry Chessin, SUN Y, Albany, N.Y. 122033 SPSE Mtg., Chicago Chap. K. T. Lassiter, Eastman

Kodak Co., Dept. 942, Rochester, N.Y. 146503-7 Course in Gas Chromatography, Philadelphia Lynn

Bass, Sadtler Res. Labs., Inc., 3316 Spring GardenSt., Philadelphia, Pa. 19104

3-7 Course in Techniques of ir Spectroscopy, Phila-delphia Lynn Bass, Sadtler Res. Labs., Inc., 3316Spring Garden St., Philadelphia, Pa. 19104

7 SPSE Mtg., Connecticut Chap. K. T. Lassiter,Eastman Kodak Co., Dept. 942, Rochester, N.Y.14650

10 SPSE Mtg., Washington Chap. K. T. Lassiter,Eastman Kodak Co., Dept. 942, Rochester, N.Y.14650

10-14 Course in ir Interpretation, Part I, PhiladelphiaLynn Bass, Sadtler Res. Labs., Inc., 3316 SpringGarden St., Philadelphia, Pa. 19104

10-14 Course in uv Spectroscopy, Philadelphia LynnBass, Sadtler Res. Labs., Inc., 3316 Spring GardenSt., Philadelphia, Pa. 19104

10-14 SPSE Ann. Conf., Boston SPSE, Main P.O. Box1609, Wash., D.C.

10-14 Micro-68 W. C. McCrone, McCrone Res. Inst., 451E. 31st St., Chicago, Ill. 60616

10-21 Course in Gaussian Optics and Aberrations, LondonRegistrar, Imperial College, London SW7, England

10-21 Course in Image Forming Systems, London Regis-trar, Imperial College, London SW7, England

10-21 Course in Interferometry and Testing of OpticalInstruments, London Registrar, Imperial College,London SW7, England

10-21 Course in Physical Theory of Image Formation, WithApplications, London Registrar, Imperial College,London SW7, England

10-21 Course in Polarization, London Registrar, ImperialCollege, London SW7, England

10-21 Course in Sources and Detectors, London Registrar,Imperial College, London SW7, England

10-21 Course in Thin Films, London Registrar, ImperialCollege, London SW7, England

10-21 Course in Visual Optics and Photometry, LondonRegistrar, Imperial College, London SW7, England

12 SPSE Mtg., Dayton Chap. K. T. Lassiter, EastmanKodak Co., Dept. 942, Rochester, N.Y. 14650

14 SPSE Mtg., No. California Chap. K. T. Lassiter,Eastman Kodak Co., Dept. 942, Rochester, N.Y.14650

14 SPSE Mtg., New York Chap. K. T. Lassiter, East-man Kodak Co., Dept. 942, Rochester, N.Y. 14650

17-19 APS Mtg., Los Alamos W. Whaling, Caltech, Pasa-dena, Calif. 91109

17-19 5th AFCRL Scientific Balloon Symp., PortsmouthL. A. Grass, Hanscom Field, Bedford, Mass. 01730

17-21 4th Internatl. Materials Symp. on the Structure andChemistry of Solid Surfaces, Berkeley C. V.Peterson, Lawrence Radiation Lab., Bldg. 62, Rm.205, U. of Calif., Berkeley, Calif. 94720

17-21 Course in ir Interpretation, Part II, PhiladelphiaLynn Bass, Sadtler Res. Labs., Inc., 3316 SpringGarden St., Philadelphia, Pa. 19104

17-21 Course in Techniques of ir and Raman Spectroscopy,Los Angeles Program Promotion, UCLA Ext.,P.O. Box 24902, Los Angeles, Calif. 90024

17-21 Thermoanalysis Workshop, Philadelphia Lynn Bass,Sadtler Res. Labs., Inc., 3316 Spring Garden St.,Philadelphia, Pa. 19104

17-28 Quantum Electronics Tutorial Symp., FlagstaffS. F. Jacobs, U. of Ariz., Tucson, Ariz. 85271

18 OSA Greater New York Sec., Oddities and Problemsin Color Measurements H. Judin, P.O. Box 772,Melville, N.Y. 11 746

23-28 ASTM 71st Ann. Mtg. and 18th Materials TestingExhibit, San Francisco 1916 Race St., Philadelphia,Pa. 19103

23-29 8th Internatl. Congr. on High Speed Photography,Stockholm T. Ramqvist, c/o Research Inst. ofNatl. Defence, FAQ 2, Stockholm, Sweden

24-26 AIAA Conf. on Fluid and Plasma Dynamics, LosAngeles 1290 6th Ave., New York, N. Y. 10019

24-27 4th Internatl. Symp. on Bioastronautics and theExploration of Space, San Antonio Jack Harmon,P.O. Box 2296, San Antonio, Tex. 78206

continued on page 69

Our file of miscellany is beginning to bulge, and perhaps we canhide some of it here. We have always had a personal preferencefor an editorial that has a theme and that can be written more orless as an essay or a sermon; for one thing, sermon writing is goodpractice in case that time should come when we are washed up asan editor and go off to become a missionary among the aborigines.Our collection of essays on the simplicity of the metric system oron the advantages of the universal use of the English languagewill then no doubt come in handy.

But in the meantime we should dispose of some of these odds

and ends that do not conveniently file in any of our major cate-gories of editorial topics. Last summer a reader wrote: "Iwould like to see photos and short biographical sketches of yourauthors (as is done by Proc. IEEE) rather than the present prac-tice of filler pictures of the 'in' group of the optics community.An authorship in your journal would seem to be a less arbitraryand more consistent method of selecting those whose photos are toappear. An obvious reason for your present practice is that itundoubtedly takes less effort to print the pictures that happen tobe available, rather than making a formal effort to obtain author's

April 1968 / Vol. 7, No. 4 / APPLIED OPTICS 693