Measurement of the Absorption Coefficient ofAtmospheric Dust

James D. Lindberg and Larry S. Laude

A method developed by previous workers for measuring the absorption coefficient of strongly absorbing

powdered materials has been applied to samples of atmospheric dust in the 0.3-1.1-lim wavelength inter-val. This work, which is based on the Kubelka-Munk theory of diffuse reflectance, provides an estimate

of the optical absorption coefficient. The corresponding imaginary refractive index is calculated fromthis value. Results are given for several samples of dry atmospheric dust collected in the desert of south-

ern New Mexico. A typical value for the imaginary refractive index was found to be 0.007 at 0.6 um,

with little dependence on wavelength in the spectral range investigated. These results are found to be ingood agreement with those of other workers obtained by different methods.

IntroductionThe quantitative measurement of the absorption

coefficient of atmospheric dust is a problem madedifficult by the complicated nature of the material.Since dust samples consist of mixtures of finelypowdered, highly scattering materials, conventionaltransmission spectroscopy is not directly applicable.The basic problem is to determine what fraction ofthe incident light attenuated by a sample of dust islost due to scattering and how much is absorbed bythe particles themselves. The measurement of ab-sorbed energy must then be related to the basic opti-cal absorption coefficient of the sample. This latterproblem is not trivial since the Bouguer-Lambertlaw, which normally solves this problem for nonscat-tering media, is not directly applicable here.

In spite of the difficulties in making such measure-ments, several workers have devised methods forsolving the problem, with varying degrees of success.Fischer' has obtained results based on a diffuse trans-mittance technique and the assumption that theBouguer-Lambert law can be used. Volz2 ' 3 has ap-plied variations of the potassium bromide presseddisk method, commonly used for qualitative work inthe ir region, to gain some information about the ab-sorption coefficient of dust. Grams et al.4 have esti-mated the imaginary refractive index of airborne flyash by inferring the value needed to account for theirmeasured values of laser radar backscatter cross sec-

The authors are with the Atmospheric Sciences Laboratory,U.S. Army Electronics Command, White Sands Missile Range,New Mexico 88002.

Received 12 October 1973.

tion and particle size distribution. Previous work inthis laboratory5 has developed a somewhat compli-cated method for applying the Kubelka-Munk theoryof diffuse reflectance to the problem. This reportdescribes a straightforward method of obtaining theabsorption coefficient from a single diffuse reflec-tance measurement. The final section is a discus-sion of the results obtained here in comparison tothose obtained by other workers.

Theory

In this work we regard a sample of atmosphericdust as a powdered material of unknown shape, sizedistribution, refractive index, and absorption coeffi-cient. The object is to infer the absorption coeffi-cient, or equivalently, the imaginary part of thecomplex refractive index of the material, from mea-surement of some optical property of the sample.

The most straightforward measurement that con-tains information about the absorption coefficient ofa powder is a measurement of its diffuse reflectanceR,,. It is well known that the diffuse reflectance of apowder is highly dependent on the absorption coeffi-cient. RB0. is high when the absorption is low and, ofcourse, is low for strongly absorbing materials. Forthis reason the diffuse reflectance of a powderedsample of a solid material plotted as a function ofwavelength has the character of a transmission spec-trum of the material itself. The resemblance is onlyqualitative, however.

To obtain quantitative information from a diffusereflectance measurement, it is necessary to havesome analytical relationship between the diffuse re-flectance and the desired optical property of thepowder, in this case the absorption coefficient. Sev-eral theories of diffuse reflectance provide such a

August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1923

Table I. Definitions of Symbols

c Concentration, in moles per liter, of a pure powder sample.c* Concentration, in moles per liter, of sample in a mixture of

sample and diluting agent.d Thickness, in cm, of a sample layer.

k Kubelka-Munk absorption coefficient in cm-.k* Kubelka-Munk absorption coefficient in cm-' for a mix-

ture of sample and diluting agent.n' The imaginary part of the complex refractive index of

atmospheric dust.R Diffuse reflectance of a layer of powder of thickness d.

R Diffuse reflectance of the flat black paint used to coat thebottom of a shallow aluminum sample dish.

R, Diffuse reflectance of a layer of powder with a thicknesslarge enough that no light passes through.

R.* Diffuse reflectance R measured for a mixture of sampleand diluting agent.

s Kubelka-Munk scattering coefficient in cm-'.s* Kubelka-Munk scattering coefficient in cm-' for a mix-

ture of sample and diluting agent.W The weight of pure atmospheric dust sample required to

fill completely the volume of the sample dish.W* The actual weight of atmospheric dust in the sample dish

when R,,* is measured.

connection. Some are very satisfactory quantita-tively, but require considerable a priori knowledgeabout the sample, such as size or real refractiveindex. For the work reported here the Kubelka-Munk theory was used, since it requires no otherknowledge about the sample at all. The price of thisconvenience is the assumption that the powder is anisotropically scattering medium, a point we willcome back to later in this section, and a definition ofabsorption coefficient slightly different from thatcommonly used in the Bouguer-Lambert law.

The Kubelka-Munk theory as it is applied in thiswork is well known and will not be discussed in de-tail here. The reader is referred to two excellentbooks6' 7 on the subject for details on its origin andderivation. The equations used here and the nota-tion, as much as possible, are taken from the workby Kortum.7 The notation is summarized in TableI.

The Kubelka-Munk theory treats a powder sampleas a continuous, turbid medium. In a layer of infini-tesimal thickness, fractions s and k per unit path-length of the incident light are scattered and ab-sorbed, respectively. The quantities s and k, inunits of cm-', are called the Kubelka-Munk scatter-ing and absorption coefficients. It is known that thecoefficient k is very closely related to the absorptioncoefficient in the familiar Bouguer-Lambert law.Discussions of this point are available in the ref-erences.6-8 The object of the work reported here isto determine the value of for a sample of atmo-spheric dust and to infer the imaginary refractiveindex from it.

One of the most important accomplishments of the

Kubelka-Munk theory is an equation relating thediffuse reflectance of a powder to the scattering andabsorption coefficients:

[(1 - R.)2]/(2R,) = k/s. (1)

This expression is known as the Kubelka-Munkfunction. This provides a determination of theratio k/s from one measurement of reflectanceR,. In order to arrive at values of k and s separately,more information is required.

The reflectance RC, by definition, is the diffuse re-flectance of an infinitely thick layer of the powder;this means a layer so thick that no detectable light istransmitted through the sample. In practice thisturns out to be a thickness of a few millimeters, de-pending on the value of k and s for the material.Suppose that we measure r- on a suitably thicklayer of sample and then also measure R, the diffusereflectance of a much thinner layer of thickness d.This can be done by placing the powder in a shal-low dish of depth d. Suppose also that we havemade the bottom of the dish nearly black and ha-veve previously measured its diffuse reflectance Rg.Kortum7 9 has shown from the Kubelka-Munktheory that s, R,, Rg, and d are related by the ex-pression

In (R - 1/BR.)(Rg R-a) 1 2sd(1/R - R.), (2)LR - 1,)(R R- )and that s can be determined satisfactorily frommeasurement of these quantities. Having obtained sin this manner, one can then solve for k in Eq. (1).This provides a means of obtaining the absorptioncoefficient of a powder from three reflectance mea-surements and one thickness measurement.

The method described above has been used suc-cessfully to obtain k and s for very weakly absorbing(nearly white) powders. But it is not a satisfactorysolution to the problem of interest here for three rea-sons. One is that for a sample of atmospheric dust,the absorption coefficient is relatively high, andtherefore even a thin layer appears infinitely thick,making it difficult to determine R, R . , and d to asatisfactory degree of accuracy. Second, it is noteasy to collect a sufficiently massive sample of atmo-spheric dust to carry out these kinds of measure-ments. The third problem arises from the assump-tion in the Kubelka-Munk theory that the samplepowder scatters light isotropically. Experimentshows that weakly absorbing powders are isotropicscatterers to a very good degree of approximation.But this is not necessarily true for strong absorberslike atmospheric dust.

The difficulties mentioned above arise from thefact that atmospheric dust is a relatively strong ab-sorber of light. Therefore, they can be avoided bydiluting the dust sample with some other powderthat has a trivially low absorption coefficient. Onecan then apply Eqs. (1) and (2) to the diluted sam-ple and obtain an absorption coefficient for the mix-ture. The remaining problem is to infer the absorp-

1924 APPLIED OPTICS / Vol. 13, No. 8 / August 1974

tion coefficient of the pure dust sample from that ofthe mixture and knowledge of the dilution ratio.

First, we look at the problem of obtaining the ab-sorption coefficient of a mixture of atmospheric dustand some other powder with low absorption coeffi-cient used as a diluting agent. We can then writeEq. (1) in the form

[(1 - B,,*)2]/(R*) =k*/s* (3)

where the asterisk indicates that we are dealing withthe properties of the diluted sample. Kortum hasshown that if the sample is highly diluted, the scat-tering coefficient s of the mixture is not affected bythe presence of the sample, so for a value of s* wecan use the scattering coefficient for the pure dilu-tant. This quantity can easily be determined bymeasuring R . , Ro, Rg, and d, then applying Eqs. (1)and (2) as discussed above, since the diluting agentis not a strong absorber like the pure dust sample.We can measure R . * directly for the diluted dustsample and determine k* from Eq. (3).

Now we have a valid measurement of k*, the ab-sorption coefficient of the diluted dust sample.What we really want, however, is the absorptioncoefficient of the pure dust sample, undiluted by airspaces or other white diluting agents. Previouswork7' 9 has shown that for strongly diluted samples,the absorption coefficient k* is directly proportionalto the sample concentration, in moles per liter, overseveral orders of magnitude of concentration. Thisis analogous to Beer's law in conventional transmis-sion spectroscopy of solutions. For the case of pow-dered materials, derivations from linearity occurwhen the sample is not sufficiently diluted, and theassumption that the scattering coefficient of themixture is that of the pure diluting agent is notvalid. Nonlinearities also occur for extremely weakdilutions because the absorption coefficient of themixture is so small that the intrinsic absorptioncoefficient of the diluting agent can no longer beconsidered trivial. In practice for the kinds of mate-rials dealt with here, the linear region extends typi-cally over a dilution range of one part sample in 102parts diluting agent to one part sample in 105 partsdiluting agent.

We now assume that we have measured R0 * for adiluted sample of concentration that falls in the lin-ear region and have calculated the correspondingvalue of k*. Then we can obtain the absorptioncoefficient k for a pure sample, free from dilution byany agent, from the expression

k = k*(c / c*), (4)

where c* is the concentration in moles per liter ofsample in one dilution, and c is the concentration, inmoles per liter, the sample would have if it were freeof air spaces or other diluting agents. Since concen-tration in moles per liter is directly proportional tothe weight per unit volume of sample, we can replacethe ratio c/c* in Eq. (4) with the ratio W/ W*, where

W* is the weight of atmospheric dust in our samplevolume, and W is the weight of dust that would berequired to fill the sample volume completely withthe dust. Making this change in Eq. (4) and substi-tuting for k* from Eq. (3), we have

k = [s*(1 -R*)W]1(2R*W*). (5)

This expression gives the Kubelka-Munk absorp-tion coefficient for the pure atmospheric dust sampleas a function of one reflectance measurement, aweight measurement, and a predetermined scatter-ing coefficient for the diluting agent.

Measurements

All the diffuse reflectance measurements weremade with a Cary 14 spectrophotometer equippedwith the manufacturer's 25-cm integrating sphere, asdescribed by Hedelman and Mitchell.10 In such aninstrument the quantity measured is the relative re-flectance, that is, the reflectance of the sample di-vided by the reflectance of some material used as areference standard. Thus, all measurements mustbe corrected by multiplying the measured reflectanceby the absolute reflectance of the reference materialto determine the absolute reflectance of the sample.In the work done here, highly refined BaSO 4 (Ref.11) was used as a reference standard, and the abso-lute reflectance of this material was obtained fromthe work of Grum and Luckey.' 2

As a diluting agent, the same BaSO 4 used as a ref-erence standard was chosen. A shallow aluminumdish a fraction of a millimeter deep, painted with flatblack enamel, was used to measure R, the reflec-tance of a thin layer of the white powder. Rg, thereflectance of the black dish, was also measured.The scattering coefficient was determined fromEqs. (1) and (2) and used as the value of s, thescattering coefficient of the diluted samples.

The dust samples used here were collected on thesurface of a 0.45-,um pore size membrane filter at theinput of a laboratory vacuum pump. Each samplewas collected over a period of 10 days, from a loca-tion 3 above the ground in a desert basin in southernNew Mexico. The dust collected on the filter wasgently scraped off, and an appropriate quantity wasmixed with 34 g of the BaSO4 diluting agent.' 3

Mixing of the BaSO 4 and dust was done in a poly-styrene vial in a vibrator mixer for a period of 5 min.The mixture was then pressed into a 1 cm deep by 5cm in diam lucite dish using a clean glass plate.The reflectance R . * was measured directly, and theweight W* of dust in the lucite dish was recorded.The quantity W, the weight of dust that wouldcompletely fill the dish, was calculated from theknown volume of the dish and the assumption thatthe average specific gravity of atmospheric dust ma-terials is 2.4, as reported by other workers.14

Different quantities of dust were used to insurethat the concentration chosen was in the linear rangeas discussed above and to determine over what

August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1925

Table II. Absorption Coefficient and Imaginary RefractiveIndex for Three Atmospheric Dust Samples

Sample 1 Sample 2 Sample 3X, um k, cm-' n' k, cm-, n' k, cm-' n'

0.35 5805 0.016 6312 0.018 5803 0.0160.40 3298 0.011 4239 0.013 2837 0.0090.45 2275 0.008 3239 0.012 1814 0.0070.50 1843 0.007 2432 0.010 1478 0.0060.55 1485 0.007 2243 0.010 1300 0.006

0.60 1328 0.006 1892 0.009 1047 0.0050.65 1119 0.006 1659 0.009 837 0.0040.70 1109 0.006 1537 0.009 968 0.0050.75 1100 0.006 1396 0.008 725 0.0040.80 933 0.006 1240 0.008 668 0.0040.85 919 0.006 1202 0.008 541 0.004

0.90 949 0.007 1098 0.008 494 0.0040.9) 978 0.007 1057 0.008 476 0.0041.00 100; 0.008 1058 0.008 702 0.0061.05 888 0.007 1045 0.009 852 0.0071.10 793 0.007 907 0.008 780 0.007

wavelength range the measurement could be made.It was found that a suitable linear relationship ex-isted for wavelengths from 0.3 m to 1.1 Acm. Out-side of this range, the intrinsic absorption coefficientof the BaSO 4 was too high to be considered negligi-ble. It was found that quantities of from 3 mg to 30mg of dust in the sample dish produced satisfactoryresults. From measurements of R . * at differentwavelengths and knowledge of W W*, and s* as de-termined earlier, k was calculated as a function ofwavelength from Eq. (5). Typical results are listedin Table II and discussed in the next section.

A programmable desk calculator was used to per-form the calculations required in this work. By esti-mating the errors in each of the input measurementsinvolved and determining the effect of each on thecomputed value of , an estimate of the experimen-tal error was made. The largest source of error wasfound to be in the determination of s* because of thedifficulty of preparing a uniform layer of powder afew tenths of a millimeter thick. It was estimatedthat s* might be in error by as much as 20%. Pho-tometric errors, assumed to be about 2%, were foundto contribute an error of about 5% to the determina-tion of k. Since s* is not determined individually foreach computation of k, its uncertainty affects theabsolute accuracy of the measurement, but not therepeatability from one sample to the next. There-fore, the experimental error in this determination ofthe Kubelka-Munk absorption coefficient of atmo-spheric dust is estimated to be on the order of 25%on an absolute basis, with a precision considerablygreater. This error could be reduced substantiallyby performing many determinations of s* and aver-aging the results.

Discussion of Results

The measurements made here give the Kubelka-Munk absorption coefficient for atmospheric dust, in

units of cm-'. In practice this number is found tobe nearly equivalent to the absorption coefficient asdefined by the Bouguer-Lambert law, usually differ-ing from it by a factor of about 2.6s In order tocompare the results of the work presented here withthat of previous workers, we will assume the two ab-sorption coefficients to be equivalent, so that a cor-responding imaginary refractive index n' can becomputed from the usual expression: n' = k/47r,where is the absorption coefficient in cm-' and A

is the wavelength in cm. In Table II the Kubelka-Munk absorption coefficient and correspondingimaginary refractive index are listed for differentwavelengths for several dust samples.

Fischer' has obtained a value of 0.01 for the imagi-nary refractive index in the visible region for atmo-spheric dust collected at a location in West Germa-ny. He also reported a sample-to-sample variationof about a factor of 3 and noted that there was littledependence on wavelength in the visible spectrum.Volz2,3"15 has reported 0.007 for the imaginary re-fractive index of atmospheric haze particles in thevisible spectrum and similar values, although with astrong wavelength dependence, for water solublerainwater residue.

It is known that the Kubelka-Munk function [Eq.(1)] is dependent on the particle size of the powderedmaterial being investigated. This means that k, asdetermined by the method described here, shoulddepend to some degree on the particle size of thesample. For this reason the atmospheric dust wasnot ground in any fashion (as is sometimes done inthe KBr pellet technique to improve the apparentintensity of ir spectra) and was thoroughly mixedwith the BaSO4 in an effort to keep the dust asmuch in its natural state as possible. Tests in thislaboratory on samples of ferric oxide with differentparticle size ranges show that k increases with de-creasing particle size, as expected.7 For Fe2O3, thiseffect is much more severe in the near uv, where k 105 cm-1, than it is in the NIR, where k _ 103cm-'. Based on these tests, it is expected that inthe case of atmospheric dust, where k 103 cm-,the value of k arrived at by this method should besomewhat dependent on the particle size distributionof the sample. Further investigations of this pointare in progress.

The method reported here gives results that areconsistent with those of other workers. An exactcomparison is of course difficult because identicalsamples were not used in all cases, and the measure-ment errors involved are not easy to assess. How-ever, it can be concluded from this work that diffusereflectance spectroscopy can give a useful measure-ment of the imaginary refractive index of atmospher-ic dust samples. This method clearly has the ad-vantage of simplicity and convenience, since it con-sists of a routine total diffuse reflectance measure-ment. Most laboratory spectrophotometers are ca-pable of making such measurements using accesso-ries designed for this purpose.

1926 APPLIED OPTICS / Vol. 13, No. 8 / August 1974

References1. K. Fischer, Beitr. Phys. Atmos. 43, 244 (1970).2. F. E. Volz, Appl. Opt. 11, 755 (1972).3. F. E. Volz, J. Geophys. Res. 77, 1017 (1972).4. G. W. Grams et al., J. Atmos. Sci. 29,900 (1972).5. J. D. Lindberg and D. G. Snyder, Appl. Opt. 12, 573 (1973).6. W. W. Wendlandt and H. G. Hecht, Reflectance Spectrosco-

py (Interscience, New York, 1966).7. G. Kortum, Reflectance Spectroscopy (Springer-Verlag, New

York, 1969).8. B. J. Brinkworth, Appl. Opt. 11, 1434 (1972).9. G. Kortum and D. Oelkrug, Z. Naturforsch. 19a, 28 (1964).10. S. Hedelman and W. N. Mitchell, in Modern Aspects of Re-

flectance Spectroscopy, W. W. Wendlandt, Ed. (Plenum,New York, 1968), p. 158.

11. This refined BaSO 4 is called Eastman White ReflectanceSiandard, available from Eastman Kodak Co., Rochester,New York.

12. F. Grum and G. W. Luckey, Appl. Opt. 7,2289 (1968).13. The choice of 34 g is somewhat arbitrary. All that is required

is that the powder be deep enough that no light passesthrough the powder, that is, deep enough that the diffuse re-flectance of an infinitely thick layer is measured. The sup-plier of the BaSO 4 specifies 34 g in the lucite dish in thepreparation of reflectance standards. For convenience here,these same dishes and the same quantity of BaSO 4 were usedfor the diluted dust samples.

14. D. C. Henley and G. B. Hoidale, J. Acoust. Soc. Am. 54, 437(1973).

15. F. E. Volz, Ann. Meteorol. 8, 34 (1957).

Alan J. Werner, a research associate in physics

in the Technical Staffs Division of Corning Glass Works,

retired June 1 after 37 years with the company. During

those years he was involved in programs concerned with

color and optical roperties of glasses, fields in which

he achieved a wide reputation. He earned the degrees

of B.S. and M.S. in physics at Franklin and Marshall

and Pennsylvania State Universities, respectively.

has been retained by Corning as a consultant in optical

engineering. Photo: (from left) Dr. Gail P. Smith,

director of Technical Staff Services; Alan J. Werner;

Charles J. Parker, manager of Physical Properties Research.

August 1974 / Vol. 13, No. 8 / APPLIED OPTICS 1927

He