Test of Theoretical Absorption Band Model Approximations

Darrell E. Burch and Dudley Williams

Three approximations to theoretical absorption band models have recently been proposed by Plass.Experimental studies of total band absorptances for N20, CO2, C114, and CO show that only the "weakline approximation" is actually completely fulfilled; the "strong line approximation" is realized only forlarge values of total absorptance corresponding to near saturation. The validity of the "nonoverlappingapproximation" could not be determined easily from the data except under conditions for which thestrong line approximation would also be expected to apply. Considerable deviation from this approxima-tion was found and has been attributed to the presence of lines that remain weak even for large absorberconcentrations. Quantitative tests of this hypothesis were made on the 2224 cm-' N20 band and the COfundamental at 2143 cm-'. Absorptance near the R branch maximum of the CO fundamental repre-sented the only close approach to complete fulfillment of the strong line approximation. Since the Rbranch of CO is nearly free of secondary lines, it is concluded that the strong line approximation can berealized experimentally only for "clean" spectral regions.

In a series of recent papers1-4 the authors have giventhe results of extensive laboratory measurements of thetotal absorptance of the major near infrared absorptionbands of gases present in planetary atmospheres. Thetotal absorptance of each band was carefully measuredas a function of absorber concentration and total pres-sure. The results were given graphically by means oflogarithmic plots showing total absorptance for variousvalues of equivalent pressure as a function of absorberconcentration. On the basis of the plots, it is possible byinterpolation to give valid estimates of the total ab-sorptance to be expected for any gas sample havingany combination of absorber concentration andequivalent pressure within the extensive range of thesevariables covered in the actual laboratory measure-ments. Empirical equations relating total absorptanceto absorber concentration and equivalent pressure canalso be developed but have somewhat limited ranges ofvalidity. In the absence of exact theoretical expressionsto serve as guides, extrapolation of the experimentalresults summarized in logarithmic plots or empiricalequations to ranges of absorber concentration or

The authors were at the Laboratory of Molecular Spectroscopyand Infrared Studies, Department of Physics and Astronomy,The Ohio State University. D. E. Burch was a General MotorsPostdoctoral Research Fellow during a major portion of thisinvestigation; his present address is Aeronutronic, a Division ofFord Motor Company, Newport Beach, California. D. Williamsis now with the North Carolina State University, Raleigh, N. C.

Received 4 September 1962.Supported by Air Force Cambridge Research Laboratories.

equivalent pressure not covered in the actual laboratoryexperiments is hazardous.

However, in atmospheric applications extrapolationof laboratory results becomes necessary, since thetransmission of radiation along long paths may involvevalues of absorber concentration and equivalent pres-sure not readily attainable in controlled laboratoryexperiments. This is particularly true in the case ofslant paths over which the pressure varies. Severalidealized "band models" have been proposed for use insuch extrapolations. In his excellent recent article5

Plass has proposed a convenient method of employingexperimental data to select appropriate band modelsfor use in making valid extrapolations and has pro-posed three approximations that can be used to giverelatively simple analytic expressions for the absorp-tance predicted by the various band models. Thepurpose of the present paper is to discuss the possibilityof experimental realization of the conditions leading tothe Plass approximations. In cases where the approxi-mations can be realized it is possible to obtain simpleand direct connections between the experimentalparameters, absorber concentration and equivalentpressure, and certain other parameters appearing inthe theoretical expressions; in cases for which theapproximations cannot be realized experimentallyother methods of establishing relations between ex-perimental and theoretical parameters must be em-ployed before the idealized band models can be used inmaking extrapolations. Experimental realization ofthe Plass approximations would provide added confi-dence in the validity of theoretical band models.

January 1964 / Vol. 3, No. 1 / APPLIED OPTICS 55

Idealized Band Models

Present theoretical treatments of absorption of radia-tion by gases assumes the validity of Lambert's ex-

ponential absorption law, which states that the spectralabsorptance A'(v) of monochromatic radiation of fre-

quency v is given by the expression: A'(v) = 1 -

exp(-k(v)w] where w is the absorber concentration or

optical thickness of a sample, and the absorption co-

efficient k(v) is a rapidly varying function of frequencyfor gases at low pressure. The spectral absorptanceA(v) actually measured with a spectrograph is greatlyinfluenced by the effective slitwidth employed; how-

ever, it has been found that for an isolated line or for a

complete band the measured total absorptance fA (v)dv= fA'(v)dv provided the limits of integration includeall absorptance due to the isolated line or the entireband.

When the spectral slitwidth Av is sufficiently wide to

include several neighboring spectral lines in an absorp-

tion band, it is possible to use the measured value ofspectral absorptance A(v) as a valid measure of theaverage value of A'(v) within the spectral interval Av;

however, whenever such an averaging property of wideslits is employed one must be careful to employ the

same slitwidths for comparison of different absorbinggas samples.

The Lorentz line shape

S a

( ) r -Vo)2 + a2

where S = f> k(P)dv, has a convenient analytic form

and gives a fairly satisfactory approximation of actualline shape for the range of pressures in which collision

broadening is important. Ladenberg and Reiche6 suc-ceeded in obtaining a somewhat involved analytic ex-pression for the total absorptance JA (v)dv of a

Lorentzian line; their expression is in the form of Bessel

functions of imaginary argument involving the parame-

ter x = Sw/27ra. Elsasser7 obtained simple expressionsfor the total absorptance of lines in cases where theparameter x assumes limiting values. For "weaklines" x << 1, the linear approximation fA (v)dv = Sw

applies; for "strong lines" x >> 1, the square root ap-

proximation JA(v)dv = 2 w applies. In theseexpressions, w is a directly measurable quantity and theline intensity S is a constant in a gas sample at a given

temperature. The line half-width a of a collision-

broadened spectral line is proportional to the molecularcollision frequency and therefore, at a given tempera-

ture, to the equivalent pressure Pe, a pressure com-puted in such a way as to take account of differences

between self-broadening and foreign gas broadening. 8

It can be shown that, at absolute temperature T andequivalent pressure P6, a = a(Pe/Peo) T/T, where

ao is the half-width at standard conditions Peo and To.

Thus, for a given line in various gas samples at a con-stant temperature, the parameter x wIPe, and thetotal absorptance A (v)dv w in the linear ap-proximation and JA (v)dv NKP in the square rootapproximation. If the linear and the square root ap-proximations can be realized experimentally both S anda can be determined in terms of Pe and w, and hencethe parameter x can be expressed in terms of measuredquantities. Once this has been done, the total absorp-tance of an isolated line can be expressed in terms of wand P, for any combination of these variables by theuse of the Ladenberg-Reiche expressions or by the use ofvalid approximations to them.

In the near infrared isolated absorption lines arerare, and the major absorption is associated withvibration-rotation bands in which the rotational lineshave varying intensities and usually have varying spac-ings throughout the bands; overlapping of the spectralabsorptance due to neighboring lines may occur.Various band models have been proposed for use inpredicting absorptance. The statistical model,9 con-sisting of randomly spaced lines of equal strength orwith randomly varying strength, has been proposedas a representation for bands like those of water vaporin which there is little regularity in line spacing or linestrength. The Elsasser model,7 consisting of equallyspaced lines of equal strength, has been proposed as anapproximation for bands like those of carbon dioxidein which there are nearly equally spaced lines withstrengths that vary only slightly from line to line. Thetrue statistical model thus applies strictly to bandshaving complete "disorder"; the Elsasser model appliesstrictly to bands having a complete "order." A"random Elsasser model" for use in intermediate caseshas recently been proposed by Plass 0; this model,which consists of two or more overlapping Elsasserbands with different line spacings and line strengths,has been successfully applied to the 1040 cm- ozoneband.

The expression for the absorptance of a statisticalband involves integrals of Bessel functions of imaginaryargument together with other factors involving theparameter x defined above and, in addition, a factor, = 27ra/d, which is a measure of the ratio ofthe line half-width a to the mean line spacing d.The absorptance of an Elsasser or a random Elsasserband involves one or more integrals that cannot beevaluated in terms of elementary functions; the ex-pressions in these integrals involve the two parametersx and 3. As in the case of isolated lines, a K Pe in

studies of a given band in gas samples at a given tem-perature; hence x W/Pe and B cc Pe, as in the case oflines.

Plass' has recently discussed useful representations

for measurements of spectral band absorptance and

has proposed three approximations that permit predic-

56 APPLIED OPTICS / Vol. 3, No. 1 / January 1964

tion of band absorptance in terms of relatively simplemathematical expressions involving the parameters xand f. These simple expressions apply to certain limit-ing curves on logarithmic plots giving absorptance as afunction of x, /3, or various products of these twoparameters. The three Plass approximations are thefollowing:

1. Strong Line Approximation. In this approxima-tion, which is expected to apply when spectral absorp-tance is virtually complete near the centers of the linesin a band, the band absorptance is a function of thesingle variable fl'x = 27raSw/d' - Pew. The strongline approximation, which is valid even when the linesoverlap, is the top or limiting curve in a logarithmicplot of A, the average absorptance over the band or overa portion of it, for constant i3 vs. /3 x and correspondsto the smallest value of Pe. Plass has given simpleexpressions for A on the basis of the strong line ap-proximation for the three band models described above.The values of A given by the Elsasser model "saturate"for a smaller value of O3x than do the values of A givenby the statistical model. The shape of experimentallymeasured curves should therefore serve as a guide to theselection of an appropriate band model. The strongline approximation is useful in extrapolating laboratorydata to large values of w and small values of Pe, andshould therefore be of considerable importance in at-mosphere studies.

2. Weak Line Approximation. When the spectralabsorptance is small at all frequencies in a band includ-ing those at the centers of the strongest lines, the ab-sorptance is a function of the single variable /3x =Sw/d w. The top or limiting curve in a logarithmicplot of A for constant /3 P vs. x c w represents theweak line approximation; this curve applies for large /3.The absorptance for the weak line approximation isgiven by the simple expression A = - exp(-/3x) forthe Elsasser model and the statistical model withequally intense lines; the exponential must be slightlyaltered for a completely statistical model. The aboveexpression applies even when overlap of lines occurs.The weak line approximation is useful for extrapola-tions to small w and large Pe and is therefore useful inestimating band intensities.'

3. The Nonoverlapping Line Approximation. Whenthe value of /3e P is sufficiently small that there is nooverlapping of lines, the band absorptance is simply thesum of the absorptances of individual lines as given bythe Ladenberg-Reiche expressions. The nonover-lapping approximation is the top or limiting curve in alogarithmic plot of A3 for constant values vs x; thetop curve corresponds to small / c P. This curveshould have a linear region for weak nonoverlappinglines and a square root region for strong nonoverlappinglines; these correspond to the Elsasser approximationsfor single lines discussed earlier. The nonoverlapping

approximation is useful in extrapolations to small w andsmall Pe.

Plass has shown that one or more of the above ap-proximations should be valid over practically all rangesof x and /3 for the Elsasser model and over most rangesof x and / for statistical band models. He asserts thatover most ranges of validity it is possible to predictabsorptance to within -± 10% provided the "boundariesof validity" or the "validity charts" given in his paperare avoided.

In suggesting appropriate plans for laboratorystudies to serve as a basis for the use of the approxima-tions, Plass suggests that data be taken over ranges ofw and Pe that are sufficiently wide to establish thelimiting curves corresponding to the three approxima-tions listed above. In particular, it is necessary to havemeasurements made in the region where the slope of thestrong line approximation approaches zero, since theshape of the curve in this region determines the opti-mum band model for use in computing absorptance. Inaddition, the experimental measurements should clearlydefine the linear and square root regions of the non-overlapping line approximation, since the constants re-lating x and with w and Pe can be determined fromthese regions; the relative contributions of self-broad-ening can be determined-or checked-in this region.Finally, Plass suggests that the analysis can be simpli-fied if laboratory measurements are made at either aseries of constant pressures or a series of constant ab-sorber concentrations.

Comparison of Experimental Results with TheoryThe recent laboratory studies'- 4 of the major vibra-

tion-rotation bands of N20, CO, CH4, CO2, and H2Ovapor have covered fairly wide ranges of w and Pen al-though in the case of H2O vapor the range of w wassomewhat limited by the low saturated vapor pressureat laboratory temperatures. It is interesting to com-pare some of the experimental results with the resultsto be expected from idealized band models. In a gen-eral way, the plots of experimental results for total bandabsorptance are reasonably consistent with those to beexpected on the basis of band models. However, thereare certain serious disagreements that can best be il-lustrated by a comparison of the approach of experimen-tal results to the limiting curves representing the Plassapproximations.

As mentioned in the earlier papers, laboratory ab-sorptance measurements were made by varying theequivalent pressure Pe by addition of N for a series ofconstant absorber concentrations w in the same manneras that suggested by Plass, thereby simplifying analysis;in view of the extent of the data, based on a total ofsome 3000 records of spectral absorptance, it was alsopossible to present valid curves giving absorptance fordifferent constant equivalent pressures as a function of

January 1964 / Vol. 3, No. 1 / APPLIED OPTICS 57

Table I. Empirical Constants m and n for Ranges wheref A(v)dp cc WP,. a

Band m n

NO 2224 cm-' 0.53 0.37CO 2143 and 4260 cm-' 0.55 0.44CH4 3020 cm-' 0.55 0.22

1306 cm-' 0.45 0.27CO2 875-495 cm' _0. 5b 0.44

2350 cm-' 0.54 0.403716, 3609 cm-' 0.58 0.395000 cm-' ,. 0 5 b 0.39

H20 1595 cm-' 0 5b 0.303700 cm l 0 5b 0.325340 cm-' 0.5b 0.307240 cm - ..0 5 b 0.30

a For band consisting of nonoverlapping strong lines m = n =

0.5.b These values for CO2 and H20 are taken from results of

Howard, Burch, and Williams, J. Opt. Soc. Am. 46, 373, 242(1956). These authors gave equations in which in was assigned

the value 0.5. and a "best fit" was obtained for n for an effectivepressure different from the P, used in the present work; on thebasis of the analysis used in the present work, n would be some-

what larger than 0.5 and n would be somewhat smaller than thevalues listed above.

absorber concentration. This made it possible tostudy systematically the separate effects of Pe variationand w variation on fA (v)dv for the various bands.The values of w could be varied, in most cases, fromthose obtained at the minimum accurately measurablepartial pressures-1 or 2 mm Hg-with a cell 1.55 cmlong to those obtainable with a maximum partial pres-sure of 3000 mm Hg and an absorption cell length of 48m. The values of equivalent pressure could be variedfrom approximately 1 mm Hg to more than 3000 mmHg.

In the case of nearly every band studied it was possi-ble to employ pressures sufficiently large to obtainsatisfactory approaches to the weak line approxima-tion. The slope of this curve on a logarithmic plot offA (v)dv vs. w is linear and has a slope of unity forsmall values of w; this indicates that the linear regionhas been attained. The fact that the limiting curve isalso approached at low values of w by other curvescorresponding to somewhat lower values of Pe indicatesthat total absorptance becomes independent of Pe forlow values of w. No anomalies are observed in theplots, and their absence indicates that the expressionsselected for Pe 8 take adequate account of self-broaden-ing effects. In the case of one or two very intensebands, the linear region was not attained in the plots ofactual data but could be reached by a small extrapola-tion based on a limiting slope of unity and coincidenceat small w with curves for smaller Pe. Therefore, in

general, the experimental data provide the limitingcurve for the weak line approximation and cover thelinear region.

The approach to the square root region of the strongline approximation was less satisfactory. It will berecalled that for a strong line fA(v)dv = 2X/Saw;hence, for a band consisting of a set of strong non-overlapping lines, it is to be expected that fA (v)dv cc

/'i,, where the limits of integration include the entireband. In no case was this expectation realized experi-mentally for any appreciable range of w and Pe. Plotsof fA(v)dv for constant w vs. Pe showed that forconsiderable ranges fA(v)dv cc Pe"; correspondingplots of fA(v)dv for constant Pe vs. w showed thatfA (v)dv cc w over roughly the same ranges. Otherplots of fA(v)dv vs. Pe for samples in which Pe cc W

gave further evidence that JA(v)dv -c wP," overconsiderable ranges of these variables. In every caseit was found that m > n; i.e., fA (v)dv is morestrongly dependent on w than on Pe. Values of m and nobtained for various bands are summarized in Table I.

This type of dependence of fA (v)dv on w and Pe hasbeen noted in various less extensive previous investiga-tions. One explanation is that the conditions for thestrong line approximation have not been fulfilled. Sincetheory predicts and experiment verifies that fA(v)dvis directly proportional to w but independent of Pe forweak lines, the relation JA(v)dv -c W'Pe0 might bewritten for weak lines; for nonoverlapping strong lines

2224 cm- Snder . ,F ~ 10 - '- _- ' ' e 100~~~~~~~~~~~~~~~~~0

00

760id ,', to / . 3000

I , . ..

a e in-

mmHg

2010- ~~~~~~~~~~~~40

100250760

' v ~~~30001F ,0 I I0 lo0 lo,

P2 Inotmoscm, Pe mm Hq

Fig. 1. Strong line approximation test plots for the 2224cm' N20 band showing absorptance for various values ofequivalent pressure as a function of the product of absorberconcentration and equivalent pressure. The top panel showstotal band absorptance, the center panel shows measured spectralabsorptance at 2213 cm-' near the P branch maximum, and thebottom panel shows measured spectral absorptance at 2240

cm-' near the R branch maximum.

58 APPLIED OPTICS / Vol. 3, No. 1 / January 1964

Z

cc

'0 2 204 0

002<>'~~~~~~~~~~~0

703000

10? 1a I 10 100 0' 0o 6115nPe H 0 tmos cm. in mm Hg

Fig. 2. Strong line approximation test plots from the COfundamental band at 2143 cm-' showing absorptance for vari-ous values of equivalent presssure as a function of the productof absorber concentration and equivalent pressure. The toppanel shows total band absorptance, the central panel showsmeasured spectral absorptance at 2116 cm-' near the P branchmaximum, and the bottom panel shows measured spectral ab-

sorptance at 2168 cm-' near the R branch maximum.

theory predicts fA(P) cc w0-Pe0- 5 . Thus, in a transi-tion from weak lines to strong lines in the nonoverlap-ping approximation or when both weak and strong linesare present it would be expected that 1 > m >0.5 andO < n < 0.5. This interpretation would lead one toconclude that the strong line approximation is notrealized in the range of w and Pe for which the non-overlapping approximation applies. Thus, x and /3cannot be determined in terms of w and Pe in themanner suggested by Plass.

According to the Plass criterion, the strong line ap-proximation is given by the limiting top curve in afamily of curves giving A (v)dv for various constantvalues of Pe as a function of the product wPe. Thelimiting curve would correspond to the one approachedas Pe - 0; the approach to the limiting curve is ex-pected to be nearly complete for small but measurablevalues of Pe. The smallest values of Pe used extensivelyin the present studies were 10 mm Hg and 20 mm Hg;the shift between the curves giving A ()dv for thesetwo values of Pe as a function of WPe is appreciableexcept for large values of wPe, for which fA (v)dv ap-

proaches its maximum value; i.e., the values of WPecorresponding to the onset of saturation.

The lack of a firmly established limiting curve onplots of f A ()dv vs. wPe together with the absence ofany appreciable range of w and Pe for which fA(v)dvis a function of the product WPe, would indicate that novalid strong line approximation was established by ex-periment. Further evidence of the lack of approach tothis approximation was obtained in the case of the N20band at 2224 cm- by using certain data of Goody andWormell" to provide relations between x and /3 and and Pe and by using Plass' strong line approximationfor an Elsasser band to predict values of fA(P)dv forthe range of w and Pe where the strong line approxima-tion might be expected to apply; the difference' be-tween theoretical prediction and observed value was20% instead of the 5% that might have been antici-pated. Although the apparent regularity in line spacingand slow variation in line strength revealed by highresolution studies of the 2224 cm-' N2 0 band wouldseem to suggest strongly the appropriateness of theElsasser band model, it is possible that closer agreementmight have been obtained with a random Elsassermodel.

One possible reason for the failure of the strong lineapproximation for the bands investigated is the exis-tence of lines within absorption bands that remain"weak" even for extremely large values of w. Suchweak lines frequently exist in the extreme wings of thebands. Some evidence of this is revealed by a carefulstudy of plots of A (v)dv vs. wPe in the upper panelsof Figs. 1 and 2. No real limit is approached byJfA(P)dv even for enormous values of wPe. The lackof complete saturation and the slight continued growthof JA (v)dv may well be attributed to weak lines in theband wings.

Studies of Spectral Absorptance in P and RBranches

In order to check the suggested explanation of thefailure of the strong line approximation as due to weaklines, it was decided to study the variation of measuredspectral absorptance A(v) at the frequencies of maxi-mum absorptance in the P and R branches of the 2224cm- N20 band where the effect of the weak lines in thewings of the band is small. Certain weaknesses of thisprocedure are admitted, since measured values ofspectral absorptance are strongly influenced by effectiveslitwidth and valid comparisons can be made only forfixed values of effective slitwidth. Constant slit pro-grams were used for each band in the present studiesand the effective slitwidths were sufficiently large thatthe absorptance due to several neighboring rotationallines were included in the range of frequencies vpassed by the spectrograph. The spectral slitwidthswere approximately 25 cm-'.

January 1964 / Vol. 3, No. 1 / APPLIED OPTICS 59

mm Hg

20

40

0 100

250

760

6 30000.1

lug tvv

Table II. Emperical Constants for Ranges Where Absorp-tance cCxWmPen a

m n

NO 2224 cm-' bandfA(v)dv 0.53 0.37A(v) near P branch maximum 0.52 0.38A(v) near R branch maximum 0.51 0.41

CO 2143 cm-' bandfA(v)dv 0.55 0.44A (v) near P branch maximum 0.52 0.45A(v) near R branch maximum 0.51 0.48

a For nonoverlapping strong lines n = n = 0.5.

Strong line test plots showing measured values ofA(v) in the vicinity of the maximum spectral absorp-

tance in the P and R branches of the 2224 cm-' N20band as a function of WP, are shown in the lower panels

of Fig. 1. The general form of the curves is similar to

that of the corresponding plot of total band absorptancein the upper panel. However, the curves in the lowerpanels approach a definite limit as A(v) becomes com-plete, whereas as noted earlier the curves for fA (P)dvshow a continued slight rise even for the largest valuesof WPe. By additional plots of A (v) at constant Pe vs wand plots of A (v) at constant w vs. Pe, it was found thatthe values of A (v) in the two branches are proportionalto wmPen over a considerable range of these variables.The values of m and n for A (v) in the branches are listedin Table II along with the corresponding values basedon fA(v)dv; it will be noted that values of m and n

for the branches show only a slightly closer approachto a common value of 0.5, even though A (v) wasmeasured in the vicinity of the strongest lines in theband.

Since it is known that the major 2224 cm-' N20 fundamental is overlapped by much weaker bands,' it wasconcluded that continued failure of the strong line ap-proximation might be caused by weaker lines dis-

tributed throughout the major portion of the banditself; hence, it was decided to make a further test in theregions of maximum spectral absorptance in the COfundamental band at 2143 cm- '. The P branch of thisband is overlapped by weaker lines of the fundamentalband of C'3 0' 6 and other less abundant isotopicmolecules, but the R branch is relatively "clean" when

the gas sample is at room temperature. There are someweak lines of the C'3 0'6 isotopic band present at thefrequency of maximum absorptance in the R branch of

the main band; however, they are weaker than in theP branch as a result of the difference in location of theband centers of the two bands.

Strong line approximation test plots for the COfundamental are shown in the three panels of Fig. 2.The test plots for JA (v)dv in the upper panel approach

one another at large values of WPe but, as mentioned

earlier, do not approach complete saturation. Theplots of A (v) for the P and R maxima are shown in thelower panels; their general features are similar and bothapproach a limiting saturation value representing com-plete absorptance. However, it will be noted that theonset of complete saturation for the R branch occursat a much lower value of WPe than that for the P branch.There is also for the R branch a somewhat closer ap-proach to a true limiting curve of the type proposed byPlass. The variation of A(v) with independent varia-tions of w and Pe was investigated and it was foundagain that A(v) cc wmPen for considerable ranges ofthese variables; the values obtained for m and n arelisted in Table I. It will be noted that the values m =0.51 and n = 0.48 obtained for A(v) at the R branchmaximum, where the contribution of weak lines isknown from high-resolution spectra to be least, repre-sent the closest approach to a square root dependenceobserved in the course of the present investigations.

The deviation from the square root dependence isgreater for the P branch maximum of CO and is stillgreater in the case of the absorption maxima of N2 0 forwhich there are known to be several weak lines present.These results provide evidence that the presence ofweak lines in any portion of an absorption band can giverise to observed deviations from a strong line de-pendence. It is therefore essential that any theoreticalband model take proper account of weak lines which arepresent.

Summary of Results

On the basis of the present work it would appear thatthe laboratory results give satisfactory approaches tothe weak line approximation that has been proposed fortheoretical band models. However, the strong lineapproximation is approached only in cases where ab-sorption is nearly complete and, even in these cases, theobserved total band absorptance does not show thecomplete saturation to be expected. In no case studieddid the total absorptance of an entire band approachthe square root dependence expected for nonoverlappingstrong lines; only in the case of average spectral ab-sorptance given by A (v) near the region of maximumabsorptance in the R branch of the CO fundamental wasa close approach to the nonoverlapping approximationfor strong lines observed. It is suggested that thefailure of the band model approximations can be at-tributed to the existence of minor absorption lines thatcontinue to be weak under conditions of pressure andabsorber concentration that cause the major lines to bestrong. If this conclusion is correct, the strong lineapproximation will be complete only in the case ofbands completely free of secondary lines such as thoseproduced by isotopic molecules of low abundance,those associated with weak overlapping bands, or eventhose with high J values normally present in band

60 APPLIED OPTICS / Vol. 3, No. 1 / January 1964

wings. The approach of total band absorptanceJA (v)dv to complete saturation will never be exactlythat predicted by the strong line approximation.Therefore, it will usually be necessary to introduce re-sults of high-resolution studies in order to establish re-lations between the theoretical parameters x and ,3 andthe experimental parameters w and Pe and to selectappropriate band models.

The authors wish to express their appreciation toE. B. Singleton, Wilbur France, and David Gryvnak fortheir assistance in various parts of the experimentalwork and to the Institut d'Astrophysique of the Uni-versit6 de Liage for the use of library facilities during thepreparation of this paper.

References1. D. E. Burch and D. Williams, Appl. Opt. 1, 473 (1962).2. D. E. Burch and D. Williams, Appl. Opt. 1, 587 (1962).3. D. E. Burch, D. Gryvnak, and D. Williams, Appl. Opt. 1,

759 (1962).4. D. E. Burch, W. L. France, and D. Williams, Appl. Optics

2, 585 (1963).5. G. N. Plass, J. Opt. Soc. Am. 50, 868 (1960).6. R. Ladenberg and F. Reiche, Ann. Physik 42, 181 (1913).7. W. M. Elsasser, "Harvard Meteorological Studies No. 6,"

Harvard University, 1942.8. D. E. Burch, E. B. Singleton, and D. Williams, Appl.

Optics 1, 359 (1962).9. R. M. Goody, Quart. J. Roy. Meterol. Soc. 78, 165 (1952);

H. Mayer, Los Alamos Report, LA-647 (October 31, 1947).10. G. N. Plass, J. Opt. Soc. Am. 48, 690 (1958).11. R. M. Goody and T. W. Wormell, Proc. Roy. Soc. A 209,

178 (1951).

Meeting Reports continued from page 44

International Colloquium on PhotographicScience, Turin, Italy, 23-28 September 1963Reported by Henri Thiry, Institut de Physique, Liege

Some two hundred delegates from fifteen countries participatedin this colloquium and the sixty-five reports read there coveredmost aspects of fundamental photographic science. The paperswere grouped into four sections, and in this short report, wecan only cite briefly the numerous studies, but pay special atten-tion to new techniques and promising results.

Among the papers on properties of photographic emulsiongrains a few communications dealt with detailed investigationsof the double-jet precipitation of AgBr, nucleation, growth rate,crystal imperfection, size distribution, and morphology of AgBrcrystals. Recent and present studies indicate that this field isquickly becoming an important subject of interest for the emul-sion maker. The aim of this section is to describe the physicaland chemical properties of the photographic grains and also topoint out the different conditions of precipitation which lead todefinite size and morphology of AgBr microcrystals. The mainkinds of microcrystals discussed are the perfect cubes, tetra-decahedra, regular octahedra, tabular, and needle crystals.E. Moisar and E. Klein considered the influence of pAg and pBron the precipitation of AgBr crystals: nearly equal concentrationof Ag+ and Br- ions leads to regular cubes, an increasing Br-excess can produce tetradecahedra with different ratios of 100-111-face areas and finally regular octahedra. It is to be assumedthat the two-dimensional nucleation determines the growth rateand that the occurrence of tabular crystals or needle crystalscan be explained by either dendritic nuclei with screw dislocationsor stacking faults and twining. G. Danguy was able to producetabular crystals from globular crystals by the addition of smallcrystals to the first precipitate.

Some workers studied also the photographic properties of AgClor AgI microcrystals. A. Piron and F. Orban investigated thedevelopmental defect of AgI crystals. They found that the sur-face of AgI crystals can be developed easily but that the reductionprocess spreads in the mass with an incomparably lower speedthan in the case of AgBr or AgCl crystals. The influence of gela-tin or synthetic colloids on the crystal habitus particle size dis-tribution and lattice defects was also reported. We must pointout an electronic particle-sizing instrument designed by G. H.Klinger, F. C. Forsgard, and M. V. Cwikla and based on the in-crease in resistance which occurs when particles suspended in an

electrolyte pass through a small aperture with simultaneous flowof electric current. The discussions were polarized mainly bythe incidence of dislocations on crystal growth, the correlationbetween speed and crystallographic properties, and the adsorptionof cationic materials on silver halide emulsions.

The section on fundamental processes of latent image forma-tion contained twenty-nine communications and was the mostimportant from the point of view of quantity. The classicalphenomena, i.e., reciprocity failure, solarization, sensitization,latensification, continue to be investigated by many workers.No really new ideas concerning these phenomena were reportedbut they were perhaps better explained than previously. Newtechniques were reported, and very exciting results may be ex-pected in the near future. J. Eggers, E. Moisar, and S. Wagnerinvestigated the solarization effects which occur during a se-quence of flash exposures and found that both solarization regionscan be ascribed to a destruction of the superficial latent image.They also explained the repeated solarization on the basis ofelectron trap and positive hole migration. H. E. Spencer andR. E. Atwell explained the high-intensity reciprocity failure(HIRF) of sulfur-sensitized grains as a development phenomenondue to a wide distribution of the induction periods of the in-dividual grains. HIRF is also a function of the surface concen-tration of sulfide ions. Very interesting and promising were thecommunications dealing with the distribution of developmentcenters among the photographic grains. Some results indicatedthat the distribution of development centers is only random athigh-intensity exposure (H. E. Spencer, L. E. Brady, and J. F.Hamilton), and that the developed superficial silver centers pro-duced by low-intensity exposure are characterized by a high resist-ance to bleaching (W. E. Berg and H. Ueda).

For new techniques and new methods presented in this section,we can cite a computer investigation of a latent image model.This model contains many features common to the Gurney-Motttheory; B. E. Bayer and J. F. Hamilton were able to determinethe effects of changes in various physical parameters, i.e., numberand thermal depth of electron traps and ionic neutralization time,on photographic response. H. Frieser and D. Eschrich investi-gated the forming of the latent image by measurement of smalldifferences in the extinction during the exposure by meansof an apparatus previously described in the literature. The mostimportant improvement concerning the apparatus is mainly thesuppression of electron noise, which has been reached by a Hallmultiplicator. With these last reports, the colloquium reachedtwo very big points: the computer model involves film models

January 1964 / Vol. 3, No. 1 / APPLIED OPTICS 61

A Wide-Angle, Linear-Output Horizon Sensor Optical System

William E. Palser

The optical system described generates a linear output signal at small attitude errors when operatingov r a range of altitudes from 145 km to 640 km. This Honeywell-designed horizon sensor usesonly one bolometer and has no frictional moving parts.

The optical system described in this article was de-signed for a Honeywell horizon sensor. Over its operat-ing range it gives a linear output signal to approxi-mately 7 of attitude error and a usable polarity signalout to an attitude error greater than 1450.

System Description

The system is shown schematically in Fig. 1 andconsists of curved cone, lens, chopper and axisseparator, filter, and immersed bolometer. Thecurved cone reflects the infrared energy from the earthto the lens, which focuses the signal onto a singlebolometer flake. The radiation striking the bolometeris modulated and time-shared between the pitch androll channels by a combination chopper and axisselector.

The signal from the bolometer is amplified and thenseparated into respective pitch and roll channels.Signals are then demodulated to dc output to indicatethe polarity and amplitude of the attitude error.

The primary element of the system is the aluminumalloy, aluminum-coated reflecting cone. Its designfunction is discussed in detail in a later section of thisarticle. Situated behind the cone is a convexo-con-cave germanium f/1 lens with an aperture 2.54 cmin diameter. Both faces of the lens have an anti-reflection coating.

The incoming radiation is separated by axis, and theenergy is modulated by the chopper. The filter is abandpass device which limits the infrared signal tobetween 14 and 16 ,. This band, selected as the moststable area of the spectrum for an energy-measuringdevice, corresponds to one of the CO2 bands.

The active element of the bolometer was immersed

The author was with Minneapolis-Honeywell Regulator Com-pany, Los Angeles, California. He is presently with Nortron-ics, a division of Northrop Corporation, Palos Verdes Estates,California.

Received 29 September 1962.

to increase the signal-to-noise ratio. The immersionlens is germanium with an antireflective coating on thefront surface. (The immersion lens is omitted in Fig.2 in order to simplify the drawing.) Only those rayswhich enter the lens within one degree of being parallelto the optical axis will strike the bolometer. The re-flecting cone is larger in diameter than the apertureof the lens to prevent direct radiation from entering thelens and impinging on the bolometer flake. All use-ful information must be reflected by the cone.

The field of view is bounded by the angles V/min and,kmax (see Fig. 2), which are determined by theearth's subtended angle variation owing to altitudechanges and also by the required system's attitudelinearity. The system under discussion was designedfor a satellite altitude range of 145 km to 640 km. Atthese altitude extremes the earth subtends a half-angle of approximately 770 and 63°. The desiredlinearity of the system for attitude errors of 100 was10%. The field of view was made as small as possiblein order to obtain the largest error signal per degree.In this system, the field of view was limited to 250with extremes of 'min = 600 and max = 850. fbmax

was limited to 850 in order to give the sensor a clearfield of view.

The surface angles of the cone, varying betweenOmax and 0mint which are determined by the viewangle, are 42.50 at the apex and 300 at the base. Thecurvature of the surface must be such that the energystriking the bolometer is constant for each incrementof target subtence between the values of Qbmax and Qbmin-

In other words, since the signal output from the bolo-meter is the difference between the energy levels arriv-ing from opposing quadrants, a 1 difference oftarget subtence at the minimum altitude has the sameeffect as a 1 difference at the maximum altitude.

All radiation that strikes the bolometer must bereflected by the cone. Therefore, radiation from asegment of some target, 1 , within the field of view

January 1964 / Vol. 3, No. 1 / APPLIED OPTICS 63