Determination of the Optical Absorption Coefficient of Powdered Materials Whose Particle Size Distribution and Refractive Indices Are Unknown

James D. Lindberg; David G. Snyder

doi:10.1364/AO.12.000573

Applied Optics
Vol. 12,
Issue 3,
pp. 573-578
(1973)
•https://doi.org/10.1364/AO.12.000573

Determination of the Optical Absorption Coefficient of Powdered Materials Whose Particle Size Distribution and Refractive Indices Are Unknown

James D. Lindberg and David G. Snyder

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James D. Lindberg and David G. Snyder, "Determination of the Optical Absorption Coefficient of Powdered Materials Whose Particle Size Distribution and Refractive Indices Are Unknown," Appl. Opt. 12, 573-578 (1973)

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Abstract

A method has been devised, based on the Kubelka-Munk theory of diffuse reflectance, for obtaining an estimate of the optical absorption coefficient of a finely powdered material that has been collected on the surface of a membrane filter. The procedure does not require any knowledge of either the real refractive index or particle size distribution of the material. No mechanical or chemical processing is involved, and only one sample is required. Results of experimental tests on materials of known absorption coefficient are presented.

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a	An intermediate dimensionless parameter used to simplify the explicit form of the Kubelka-Munk equations. Defined as a = 1 + k/s, or a = 1/2(1/R_∞ + R_∞).
b	An intermediate dimensionless parameter used to simplify the explicit form of the Kubelka-Munk equations. Defined as b = (a² − 1)^1/2.
d	The thickness, in cm, of the layer of dust or powdered sample collected on a filter.
k	The Kubelka-Munk absorption coefficient, defined as the fraction of light absorbed per centimeter, in an infinitesimally thin layer of the powdered sample.
k′	The absorption coefficient of a material, as determined from the Beer-Lambert law, of the glass or minerals used to test the method described in this report.
M₁	Diffuse reflectance of a membrane filter lying on a black back-ground.
M₂	Diffuse reflectance of a membrane filter lying on a white back-ground.
M₃	Diffuse reflectance of a dust sample, collected on a membrane filter, lying on a black background.
M₄	Diffuse reflectance of a dust sample, collected on a membrane filter, lying on a white background.
R	Diffuse reflectance of a layer of powder lying on a background of reflectance R_g.
R_g	Diffuse reflectance of the background material that is supporting a layer of powdered sample.
R_o	Diffuse reflectance of a layer of powdered sample with a black background.
R_∞	Diffuse reflectance of a layer of powdered sample thick enough that no detectable light passes through it.
s	The Kubelka-Munk scattering coefficient, defined as the fraction of light scattered per centimeter in an infinitesimally thin layer of the powdered sample.
T	The diffuse transmittance of a layer of powder.

		Wavelength in Micrometers
		0.528	0.550	0.585	0.620
Sample 1	s*	343	333	339	323
	k	4.8	0.94	16.6	0.51
Sample 2	s	309	304	304	295
	k	4.3	0.65	15.2	0.36
Sample 3	s	328	332	330	316
	k	4.5	0.80	16.6	0.51
Sample 4	s	340	337	338	327
	k	4.7	0.88	16.9	0.62
Transmission measurement	k′	4.2	0.27	16.0	0.19

		Wavelength in micrometers
		0.88	1.29	1.66	1.94
Sample 1	s*	715	519	397	288
	k	7.1	24.6	16.9	15.1
Sample 2	s	693	487	357	225
	k	7.9	23.6	16.8	14.0
Transmission measurement	k′	4.3	17.2	11.6	9.3

a	An intermediate dimensionless parameter used to simplify the explicit form of the Kubelka-Munk equations. Defined as a = 1 + k/s, or a = 1/2(1/R_∞ + R_∞).
b	An intermediate dimensionless parameter used to simplify the explicit form of the Kubelka-Munk equations. Defined as b = (a² − 1)^1/2.
d	The thickness, in cm, of the layer of dust or powdered sample collected on a filter.
k	The Kubelka-Munk absorption coefficient, defined as the fraction of light absorbed per centimeter, in an infinitesimally thin layer of the powdered sample.
k′	The absorption coefficient of a material, as determined from the Beer-Lambert law, of the glass or minerals used to test the method described in this report.
M₁	Diffuse reflectance of a membrane filter lying on a black back-ground.
M₂	Diffuse reflectance of a membrane filter lying on a white back-ground.
M₃	Diffuse reflectance of a dust sample, collected on a membrane filter, lying on a black background.
M₄	Diffuse reflectance of a dust sample, collected on a membrane filter, lying on a white background.
R	Diffuse reflectance of a layer of powder lying on a background of reflectance R_g.
R_g	Diffuse reflectance of the background material that is supporting a layer of powdered sample.
R_o	Diffuse reflectance of a layer of powdered sample with a black background.
R_∞	Diffuse reflectance of a layer of powdered sample thick enough that no detectable light passes through it.
s	The Kubelka-Munk scattering coefficient, defined as the fraction of light scattered per centimeter in an infinitesimally thin layer of the powdered sample.
T	The diffuse transmittance of a layer of powder.

		Wavelength in Micrometers
		0.528	0.550	0.585	0.620
Sample 1	s*	343	333	339	323
	k	4.8	0.94	16.6	0.51
Sample 2	s	309	304	304	295
	k	4.3	0.65	15.2	0.36
Sample 3	s	328	332	330	316
	k	4.5	0.80	16.6	0.51
Sample 4	s	340	337	338	327
	k	4.7	0.88	16.9	0.62
Transmission measurement	k′	4.2	0.27	16.0	0.19

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