Abstract

A simplified method is described for computation of Kubelka–Munk coefficients from internal transmittance measurements of two specimens of the same material where one specimen is twice as thick as the other. Equations are derived for determining the scattering coefficient S, the absorption coefficient K, the Kubelka parameters a and b, the internal transmittance τi, the reflectance of an infinitely thick specimen ρ, and the reflectance of a specimen with an ideal black background ρ0. Graphs are given to estimate the range of error of ρ, S, and K as a function of estimated errors of measurement of transmittance.

© 1968 Optical Society of America

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References

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  1. P. Kubelka and F. Munk, Z. Tech. Physik 12, 593 (1931).
  2. F. A. Steele, Paper Trade J. 100, 37 (1935).
  3. P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948).
    [Crossref] [PubMed]
  4. D. B. Judd, J. Res. Natl. Bur. Std. (U. S.) 19, 287 (1937).
    [Crossref]
  5. D. B. Judd, Paper Trade J. 106, 5 (1938).

1948 (1)

1938 (1)

D. B. Judd, Paper Trade J. 106, 5 (1938).

1937 (1)

D. B. Judd, J. Res. Natl. Bur. Std. (U. S.) 19, 287 (1937).
[Crossref]

1935 (1)

F. A. Steele, Paper Trade J. 100, 37 (1935).

1931 (1)

P. Kubelka and F. Munk, Z. Tech. Physik 12, 593 (1931).

Judd, D. B.

D. B. Judd, Paper Trade J. 106, 5 (1938).

D. B. Judd, J. Res. Natl. Bur. Std. (U. S.) 19, 287 (1937).
[Crossref]

Kubelka, P.

P. Kubelka, J. Opt. Soc. Am. 38, 448 (1948).
[Crossref] [PubMed]

P. Kubelka and F. Munk, Z. Tech. Physik 12, 593 (1931).

Munk, F.

P. Kubelka and F. Munk, Z. Tech. Physik 12, 593 (1931).

Steele, F. A.

F. A. Steele, Paper Trade J. 100, 37 (1935).

J. Opt. Soc. Am. (1)

J. Res. Natl. Bur. Std. (U. S.) (1)

D. B. Judd, J. Res. Natl. Bur. Std. (U. S.) 19, 287 (1937).
[Crossref]

Paper Trade J. (2)

D. B. Judd, Paper Trade J. 106, 5 (1938).

F. A. Steele, Paper Trade J. 100, 37 (1935).

Z. Tech. Physik (1)

P. Kubelka and F. Munk, Z. Tech. Physik 12, 593 (1931).

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Figures (3)

Fig. 1
Fig. 1

Curves of constant ρ.

Fig. 2
Fig. 2

Curves of constant Sx1.

Fig. 3
Fig. 3

Curves of constant Kx1.

Tables (1)

Tables Icon

Table I Equations giving some of the interrelationships between α, β, a, b, S, K, τi, ρ0, and ρ.

Equations (38)

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τ i = b / [ a sinh ( b S x ) + b cosh ( b S x ) ] ,
a ( S + K ) / S
b ( a 2 1 ) 1 2 = ( 2 S K + K 2 ) 1 2 / S .
τ i , 1 = b / [ a sinh ( b S x 1 ) + b cosh ( b S x 1 ) ] τ i , 2 = b / [ a sinh ( b S x 2 ) + b cosh ( b S x 2 ) ] } ,
2 x 1 = x 2 ,
a sinh ( b S x 1 ) + b cosh ( b S x 1 ) = b / τ i , 1 ,
a sinh ( 2 b S x 1 ) + b cosh ( 2 b S x 1 ) = b / τ i , 2 .
sinh ( 2 z ) 2 sinh ( z ) cosh ( z )
cosh ( 2 z ) 2 cosh 2 ( z ) 1 ,
2 cosh ( b S x 1 ) [ a sinh ( b S x 1 ) + b cosh ( b S x 1 ) ] b = b / τ i , 2 ,
( 2 b / τ i , 1 ) cosh ( b S x 1 ) b = b / τ i , 2 ,
cosh ( b S x 1 ) = ( 1 2 ) τ i , 1 ( 1 + 1 / τ i , 2 ) .
cosh 2 ( b S x 1 ) = ( 1 4 ) τ i , 1 2 ( 1 + 1 / τ i , 2 ) 2 .
sinh ( 2 z ) 2 sinh ( z ) cosh ( z )
cosh ( 2 z ) cosh 2 ( z ) + sinh 2 ( z ) ,
2 a b sinh ( b S x 1 ) cosh ( b S x 1 ) + b 2 cosh 2 ( b S x 1 ) + b 2 sinh 2 ( b S x 1 ) = b 2 / τ i , 2 .
a 2 sinh 2 ( b S x 1 ) + b 2 cosh 2 ( b S x 1 ) + 2 a b sinh ( b S x 1 ) × cosh ( b S x 1 ) = b 2 / τ i , 1 2 .
b 2 [ ( 1 / τ i , 1 2 ) ( 1 / τ i , 2 ) ] = sinh 2 ( b S x 1 ) .
cosh 2 ( z ) sinh 2 ( z ) 1 ,
b 2 = τ i , 1 2 [ τ i , 1 2 ( 1 + τ i , 2 ) 2 4 τ i , 2 2 ] / 4 τ i , 2 ( τ i , 2 τ i , 1 2 ) .
S = ( 1 / b x 1 ) arccosh [ ( 1 2 ) τ i , 1 ( 1 + 1 / τ i , 2 ) ] ,
K = S [ ( b 2 + 1 ) 1 2 1 ] = S ( a 1 ) ,
a = ( b 2 + 1 ) 1 2 .
ρ 0 = ( b 2 + 1 ) 1 2 ( τ i 2 + b 2 ) 1 2 , = a ( τ i 2 + b 2 ) 1 2 , = 1 / [ a + b coth ( b S x ) ] ,
ρ = ( b 2 + 1 ) 1 2 b = a b ,
α cosh ( b S x 1 ) = ( 1 2 ) τ i , 1 ( 1 + 1 / τ i , 2 ) ,
β ( 1 / τ i , 1 2 1 / τ i , 2 ) 1 2 ,
b 2 β 2 sinh 2 ( b S x 1 ) = b 2 [ ( 1 / τ i , 1 2 ) ( 1 / τ i , 2 ) ] .
b 2 = ( α 2 1 ) / β 2 ,
b = ( α 2 1 ) 1 2 / β .
S = ( 1 / b x 1 ) arccosh α , = ( 1 / b x 1 ) arcsinh b β , = ( 1 / b x 1 ) ln [ α + ( α 2 1 ) 1 2 ] , = ( 1 / b x 1 ) ln [ b β + ( b 2 β 2 + 1 ) 1 2 ] , = ( 1 / b x 1 ) ln ( α + b β ) .
b 0 , S 0 , 0 < x < , x 1 < x 2 , 0 τ i 1 , τ i , 1 > τ i , 2 ,
2 x 1 = x 2 .
τ i , 1 2 τ i , 2 / ( τ i , 2 + 1 ) .
τ i , 1 ( τ i , 2 ) 1 2 .
2 τ i , 2 / ( τ i , 2 + 1 ) τ i , 1 ( τ i , 2 ) 1 2 ,
τ i , 2 / τ i , 1 = τ i , 1 ,
τ i , 2 / τ i , 1 = 1 / ( 2 τ i , 1 ) ,