Kubelka-Munk Optical Coefficients for a Barium Sulfate White Reflectance Standard James B. Gillespie, James D. Lindberg, and Larry S. Laude

Atmospheric Sciences Laboratory, U.S. Army Electron­ics Command, White Sands Missile Range, New Mexico 88002. Received 30 September 1974. The dilution method of the Kubelka-Munk theory has

been shown to be valuable for determining the absorption coefficient of strongly absorbing powdered materials.1 A recent example of an application of this method was the determination of the absorption coefficient of atmospheric dust by Lindberg and Laude.2 This method requires as a diluent a very weakly absorbing material, i.e., a very white substance. Lindberg and Laude used a highly purified powder of barium sulfate white reflectance standard.3 A description of this material and its absolute reflectance data have been presented by Grum and Luckey.4

For use as a reference standard, the absolute reflectance of the barium sulfate white reflectance standard is all that is needed. However, to use the dilution method to deter­mine quantitatively the absorption coefficient of a pow­dered substance, it is necessary to have knowledge of the Kubelka-Munk optical coefficients of the diluting material. The scattering coefficient, s, is used directly in the calcula­tion of the absorption coefficient of the sample material. The Kubelka-Munk absorption coefficient, k, does not fig­ure into the calculation of, the absorption of the strongly absorbing powder; but it does provide knowledge of the ap­plicability of the material as a diluting agent. It is the pur­pose of this Letter to provide the Kubelka-Munk optical coefficients of the barium sulfate reflectance standard be­cause of their usefulness in determining the absorption coefficient of strongly absorbing powders. The range in­vestigated was 0.3-2.1 μm, covering the range for which barium sulfate is normally useful in reflectance work.

In the Kubelka-Munk theory the scattering and absorp­tion coefficients, s and k, are defined as the fraction of light scattered or absorbed per unit path length in an infinitesi-mally thin layer of material. Later in the derivation of the theory, a constant factor of 2 arises.5 This factor of 2 is

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Fig. 1. The wavelength dependence of the scattering coefficient of the barium sulfate white reflectance standard.

sometimes eliminated from being carried through the equa­tions by redefining the scattering and absorption coeffi­cients as S ≡ 2S and K ≡ 2k. In this Letter we have cho­sen to use the more fundamentally defined coefficients s and k.

The Kubelka-Munk function

relates the absolute reflectance R∞ of a powder to the Ku­belka-Munk scattering and absorption coefficients, s and k, in units of cm-1.

The R∞ of a powder is the reflectance from an infinitely thick layer of sample, or a layer so thick that no light is transmitted through the sample. If this value is deter­mined and a value R, the reflectance of a thin layer of thickness d of the sample, is measured on a background of reflectance Rg, then the Kubelka-Munk coefficients can be experimentally determined. Kortum5 has shown that s is related to R∞, R, d, and Rg by

where R0 is the value for R on a background for which Rg = 0.

To determine s experimentally, a layer of barium sulfate must be supported by a background whose reflectance is diffuse and nearly zero. In this work the barium sulfate was uniformly distributed onto a piece of opaque black glass plate whose surface had been ground to 400 grit to ap­proximate these criteria. By pressing the barium sulfate between two of these coarse plates, a thin uniform layer was obtained. The reflectance R0 of this thin layer was measured using a Cary 14 spectrophotometer equipped with a 25-cm diam integrating sphere as described by He-delman and Mitchell.6 The reflectance data were recorded on-line over the 0.3-2.1-μm spectral interval using a Cary Spectrosystem 100 minicomputer to collect the data and correct them for the baseline and white reflectance stan­dard.

The reflectance of the black background, Rg, was mea­sured and found to be between 3% and 4% throughout the spectral interval, a value close enough to zero to allow the use of Eq. (3), while introducing an error of only 0.1-0.2%.

Twelve different layers of the barium sulfate were used as samples. The thickness of the twelve layers ranged from 50 μm. to 200 μm. These thicknesses were measured with a micrometer microscope. The volumes and weights of each layer were determined and used to calculate the density. All sample layers had a density close to 2 g/cm3, as recommended by Grum and Luckey.4

The values of s were calculated at wavelength intervals of 0.01 μm for each of the twelve samples; the average is plotted in Fig. 1. From the average values of s, Eq. (1) was used to calculate the Kubelka-Munk absorption coeffi­cient, k, of the barium sulfate. The average values deter­mined for s and k are presented in Table I, along with Grum and Luckey's4 published values of R∞, which are in­cluded for the convenience of the reader.

There are several possible sources of error in the deter­minations of s and k. As can be seen from Eq. (3), s is de­pendent on the accuracy of R∞, R0, and d. The error in R∞ of the barium sulfate powder was reported to be ±0.001, a variation that introduces an error in s of only ±1% or less.

For a black background (i.e., Rg 0), this reduces to Table I. Kubelka-Munk Optical Coefficients of

Barium Sulfate

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The error due to the uncertainty in Ro was controlled somewhat by making the sample layers of the barium sul­fate very thin, thereby making R∞ - R0 fairly large com­pared with the variation in R0. For our work, the photo­metric variation in R0 resulted in about a ± 1 % uncertainty in s. The error resulting from the thickness, d, of the sam­ple was caused by the levelness of the layer, the definition of the layer surface, and the depth of field of the micro­scope. In an attempt to offset these possible sources of error in the measurement of d, at least 10 measurements of each layer thickness were made and averaged. However, an error of 5-10% was still possible for the individual values of d. The experimental standard deviations from the aver­age value of s were calculated for each wavelength interval and were on the order of 6-8% throughout the entire spec­tral region. This deviation was caused primarily by photo­metric variation in R0, the measurement uncertainty in d, and sample handling.

Figure 1 shows approximately an s = s0λ-a dependence as expected from the theory; however, there is a peak at about 0.42 μm; and the s value drops rapidly as wavelength decreases. This apparently occurs because of the increase of the absorption coefficient of the barium sulfate as the wavelengths approach the uv. There are also slight local minima in the s coefficient at 1.4 and 1.9 μm, which are probably due to the presence of water in the barium sul­fate. Values of a changed from 0.5 in the visible, to 1.0 at 1.0 μm, to 1.95 at 2.0 μm. This is consistent with the 0.05-3.0-μm particle size range of the barium sulfate.

The authors wish to acknowledge Orlando Perez for his contribution in the sample preparation and data reduction.

References 1. G. Kortum and D. Oelkrug, Z. Naturforch. 19a, 28 (1964). 2. J. D. Lindberg and L. S. Laude, Appl. Opt. 13, 1923 (1974). 3. This highly refined barium sulfate is called Eastman White Re­

flectance Standard 6091 available from Eastman Kodak Com­pany, Rochester, N.Y.

4. F. Grum and G. W. Luckey, Appl. Opt. 7, 2289 (1968). 5. G. Kortum, Reflectance Spectroscopy (Springer, New York,

1969). 6. S. Hedelman and W. N. Mitchell, Modern Aspects of Reflec­

tance Spectroscopy (Plenum, New York, 1968).

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