Abstract

Kubelka and Munk absorption and scattering coefficients for a nonwhite pigment dispersion (tinter) can be obtained from reflectance measurements on opaque samples of a reference white base and two tinter/white mixtures. The precision of the coefficients depends on the strengths of the mixtures, and the conditions for maximum precision have been determined from a theoretical analysis of the errors caused by a constant error in the measurement of absolute reflectance. One sample should consist of tinter without admixture of white, while the other should have reflectances as far as possible between 25% and 75% throughout the spectral range of interest.

© 1965 Optical Society of America

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References

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  1. D. R. Duncan, J. Oil & Colour Chemists’ Assoc. 32, 296 (1949).
  2. O. L. Davies, ed., Statistical Methods in Research and Production (Oliver and Boyd, London, 1957), 3rd ed., p. 41.

1949 (1)

D. R. Duncan, J. Oil & Colour Chemists’ Assoc. 32, 296 (1949).

Duncan, D. R.

D. R. Duncan, J. Oil & Colour Chemists’ Assoc. 32, 296 (1949).

J. Oil & Colour Chemists’ Assoc. (1)

D. R. Duncan, J. Oil & Colour Chemists’ Assoc. 32, 296 (1949).

Other (1)

O. L. Davies, ed., Statistical Methods in Research and Production (Oliver and Boyd, London, 1957), 3rd ed., p. 41.

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Tables (4)

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Table I Precision of scattering coefficients (coarse grid) (maximum=1000).

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Table II Precision of scattering coefficients (fine grid for R2⩽20%) (maximum=1000).

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Table III Precision of absorption coefficients (coarse grid) (maximum=1000).

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Table IV Precision of absorption coefficients (fine grid for R2 ⩽ 20%) (maximum=1000).

Equations (16)

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R = θ + 1 - ( θ 2 + 2 θ ) 1 2 ,
θ = [ c K + ( 1 - c ) K 0 ] / [ c S + ( 1 - c ) S 0 ] .
( K - K 0 ) + θ 1 ( 1 - S ) = ( θ 1 - K 0 ) / c 1 , ( K - K 0 ) + θ 2 ( 1 - S ) = ( θ 2 - K 0 ) / c 2 .
λ = θ 2 / c 2 - θ 1 / c 1 θ 2 - θ 1 - K 0 ( 1 / c 2 - 1 / c 1 ) θ 2 - θ 1 , μ = 1 / c 2 - 1 / c 1 1 / θ 2 - 1 / θ 1 - K 0 ( 1 / θ 2 c 2 - 1 / θ 1 c 1 ) ( 1 / θ 2 - 1 / θ 1 ) .
Var ( λ ) [ λ / R 1 ] 2 · Var ( R 1 ) + [ λ / R 2 ] 2 · Var ( R 2 ) ,
Var ( R 1 ) = Var ( R 2 ) = Var ( R ) , say , giving Var ( λ ) / Var ( R ) = [ λ / R 1 ] 2 + [ λ / R 2 ] 2 ,
λ / R = ( λ / θ ) ( d θ / d R ) ,
[ d θ / d R ] 2 = ( θ 2 + 2 θ ) / R 2 .
λ / θ 1 = ( - 1 / c ) / ( θ 2 - θ 1 ) + ( θ 2 / c 2 - θ 1 / c 1 ) / ( θ 2 - θ 1 ) 2 ,
λ θ 1 = μ θ 1 ( θ 1 - θ 2 )             and             λ θ 2 = μ θ 2 ( θ 2 - θ 1 )
[ λ R 1 ] 2 = μ 2 θ 1 2 ( θ 1 - θ 2 ) 2 θ 1 2 + 2 θ 1 R 1 2 = μ 2 ( θ 1 - θ 2 ) 2 θ 1 + 2 θ 1 R 1 2 ,
Var ( λ ) μ 2 Var ( R ) = 1 ( θ 1 - θ 2 ) 2 ( θ 1 + 2 θ 1 R 1 2 + θ 2 + 2 θ 2 R 2 2 ) .
θ 1 - θ 2 = [ ( 1 - R 1 ) 2 / 2 R 1 ] - [ ( 1 - R 2 ) 2 / 2 R 2 ] = [ ( R 2 - R 1 ) ( 1 - R 1 R 2 ) ] / 2 R 1 R 2
( θ + 2 ) / θ = ( 1 + R ) 2 / ( 1 - R ) 2 ,
Var ( λ ) μ 2 Var ( R ) = 4 { [ R 1 ( 1 - R 1 ) ( 1 + R 2 ) ] 2 + [ R 2 ( 1 - R 2 ) ( 1 + R 1 ) ] 2 [ ( R 1 - R 2 ) ( 1 - R 1 ) ( 1 - R 2 ) ( 1 - R 1 R 2 ) ] 2 } .
Var ( μ ) μ 2 Var ( R ) = [ ( 1 - R 1 ) 3 ( 1 + R 2 ) ] 2 + [ ( 1 - R 2 ) 3 ( 1 + R 1 ) ] 2 [ ( R 1 - R 2 ) ( 1 - R 1 ) ( 1 - R 2 ) ( 1 - R 1 R 2 ) ] 2 .