Abstract

A many-flux (discrete ordinate) radiative transfer calculation procedure is described with the goal of making the mathematics easy to learn and use. The major approximation is the neglect of polarization. Emission within the scattering medium is not included, and the formulas are restricted to a scattering medium bounded by parallel planes. The boundary conditions allow for a variety of kinds of illumination, and the surface reflection coefficients at the boundaries of the scattering medium are accurately determined. A comparison is made with the two-flux (Kubelka-Munk) and four-flux calculation methods, and this leads to empirical expressions for the scattering and absorption coefficients in these simple theories, which make them give nearly the same results as exact theories. These empirical expressions provide a very simple method for estimating the absolute reflectance and transmittance of turbid media and greatly increase the utility of the two-flux and four-flux calculation methods. The two-flux equations give excellent results provided the absorption is small compared to scattering and the optical thickness is greater than 5. A comparison with experimental data taken with collimated illumination shows that the four-flux equations give good results at any optical thickness even if the absorption is strong.

© 1971 Optical Society of America

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References

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  1. The Symposium on the Interdisciplinary Aspects of Radiative Energy Transfer, J. Quant. Spectr. Radiative Transfer 8, 1 (1968).
  2. V. Kourganoff, Basic Methods in Transfer Problems (Dover, New York, 1963).
  3. R. Viskanta, R. J. Grosh, Appl. Mech. Rev. 17, 91 (1964).
  4. S. E. Orchard, J. Oil Colour Chem. Assoc. 51, 44 (1968).
  5. S. E. Orchard, J. Opt. Soc. Amer. 59, 1584 (1969).
    [CrossRef]
  6. I. A. Vasalos, “Effect of Separation Distance on the Optical Properties of Dense Dielectric Particle Suspensions,” Ph.D. thesis, MIT (August1969); H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, J. Heat Transfer 92, 285 (1970).
    [CrossRef]
  7. A. Schuster, Astrophys. J. 21, 1 (1905). Reprinted in D. H. Menzel, Selected Papers on the Transfer of Radiation (Dover, New York, 1966).
    [CrossRef]
  8. L. Silberstein, Phil. Mag. 4, 1291 (1927).
  9. P. Kubelka, F. Munk, Z. Tech. Phys. 12, 593 (1931).
  10. P. Kubelka, J. Opt. Soc. Amer. 38, 448, 1067 (1948).
    [CrossRef]
  11. D. B. Judd, G. Wyszecki, Colorin Business, Science and Industry, 2nd ed. (Wiley, New York, 1963).
  12. G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
    [CrossRef]
  13. G. C. Wick, Z. Phys. 120, 702 (1943).
  14. S. Chandrasekhar, Astrophys. J. 100, 76 (1944).
    [CrossRef]
  15. S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950) and (Dover, New York, 1960).
  16. H. C. Hottel, A. F. Sarofim, L. B. Evans, I. A. Vasalos, J. Heat Transfer 90, 56 (1968).
    [CrossRef]
  17. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965).
  18. International Business Machines, System/360 Scientific Subroutine Package. See, for example, manual H20-0205-3.
  19. See, for example, H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry, Vol. I (Van Nostrand, Princeton, 1956), p. 109.
  20. L. W. Richards, J. Paint Technol. 42, 276 (1970).
  21. H. M. Hsia, T. J. Love, J. Heat Transfer 89, 197 (1967).
    [CrossRef]
  22. R. G. Giovanelli, Opt. Acta 2, 153 (1955).
    [CrossRef]
  23. S. E. Orchard, Astrophys. J. 149, 665 (1967).
    [CrossRef]
  24. J. P. Kratohvil, C. Smart, J. Colloid Sci. 20, 875 (1965).
    [CrossRef]
  25. H. G. Völz, VIth FATIPEC Congress1962 (Verlag Chemie, Weinheim/Bergstr., 1962), p. 98.
  26. H. G. Völz, VIIth FATIPEC Congress1964 (Verlag Chemie, Weinheim/Bergstr., 1964), p. 194.
  27. J. K. Beasley, J. T. Atkins, F. W. Billmeyer, in the Proceedings of the Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967), p. 765.
  28. H. C. Hamaker, Philips Res. Rept. 2, 55 (1947).
  29. J. L. Saunderson, J. Opt. Soc. Amer. 32, 727 (1942).
    [CrossRef]
  30. W. R. Blevin, W. J. Brown, J. Opt. Soc. Amer. 52, 1250 (1962).
    [CrossRef]
  31. H. G. HechtModern Aspects of Reflectance Spectroscopy, W. W. Wendlandt, Ed. (Plenum, New York, 1968), p. 1.
    [CrossRef]
  32. W. R. Blevin, W. J. Brown, J. Opt. Soc. Amer. 51, 975 (1961).
    [CrossRef]
  33. See, for example, F. W. Sears, Optics (Addison-Wesley, Reading, Mass., 1949), pp. 169–176.
  34. J. W. T. Walsh, in the Illumination Research Technical Paper 2 by A. K. Taylor and C. J. W. Grieveson, Department of Scientific and Industrial Research (His Majesty’s Stationery Office, London, 1926).
  35. J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).
  36. S. Q. Duntley, J. Opt. Soc. Amer. 3261 (1942).
    [CrossRef]
  37. For a review of previous practice, see F. Kottler, J. Opt. Soc. Amer. 50, 483 (1960).
    [CrossRef]
  38. R. G. Giovanelli, Opt. Acta 3, 127 (1956).
    [CrossRef]
  39. J. T. Atkins, Absorption and Scattering of Light in Turbid Media, Ph.D. thesis, University of Delaware (1965); Dissertation Abstr. B27, 1844 (1966).
  40. J. T. Atkins, F. W. Billmeyer, Color Eng. 6, No. 3, 40 (May–June 1968).
  41. W. E. Craker, F. D. Robinson, J. Oil Colour Chem. Assoc. 50, 111 (1967).

1970

L. W. Richards, J. Paint Technol. 42, 276 (1970).

1969

S. E. Orchard, J. Opt. Soc. Amer. 59, 1584 (1969).
[CrossRef]

1968

The Symposium on the Interdisciplinary Aspects of Radiative Energy Transfer, J. Quant. Spectr. Radiative Transfer 8, 1 (1968).

S. E. Orchard, J. Oil Colour Chem. Assoc. 51, 44 (1968).

H. C. Hottel, A. F. Sarofim, L. B. Evans, I. A. Vasalos, J. Heat Transfer 90, 56 (1968).
[CrossRef]

J. T. Atkins, F. W. Billmeyer, Color Eng. 6, No. 3, 40 (May–June 1968).

1967

W. E. Craker, F. D. Robinson, J. Oil Colour Chem. Assoc. 50, 111 (1967).

H. M. Hsia, T. J. Love, J. Heat Transfer 89, 197 (1967).
[CrossRef]

S. E. Orchard, Astrophys. J. 149, 665 (1967).
[CrossRef]

1965

J. P. Kratohvil, C. Smart, J. Colloid Sci. 20, 875 (1965).
[CrossRef]

1964

R. Viskanta, R. J. Grosh, Appl. Mech. Rev. 17, 91 (1964).

1962

W. R. Blevin, W. J. Brown, J. Opt. Soc. Amer. 52, 1250 (1962).
[CrossRef]

1961

W. R. Blevin, W. J. Brown, J. Opt. Soc. Amer. 51, 975 (1961).
[CrossRef]

1960

For a review of previous practice, see F. Kottler, J. Opt. Soc. Amer. 50, 483 (1960).
[CrossRef]

1956

R. G. Giovanelli, Opt. Acta 3, 127 (1956).
[CrossRef]

1955

R. G. Giovanelli, Opt. Acta 2, 153 (1955).
[CrossRef]

1948

P. Kubelka, J. Opt. Soc. Amer. 38, 448, 1067 (1948).
[CrossRef]

1947

H. C. Hamaker, Philips Res. Rept. 2, 55 (1947).

1944

S. Chandrasekhar, Astrophys. J. 100, 76 (1944).
[CrossRef]

1943

G. C. Wick, Z. Phys. 120, 702 (1943).

1942

J. L. Saunderson, J. Opt. Soc. Amer. 32, 727 (1942).
[CrossRef]

S. Q. Duntley, J. Opt. Soc. Amer. 3261 (1942).
[CrossRef]

1931

J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).

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

1927

L. Silberstein, Phil. Mag. 4, 1291 (1927).

1905

A. Schuster, Astrophys. J. 21, 1 (1905). Reprinted in D. H. Menzel, Selected Papers on the Transfer of Radiation (Dover, New York, 1966).
[CrossRef]

Atkins, J. T.

J. T. Atkins, F. W. Billmeyer, Color Eng. 6, No. 3, 40 (May–June 1968).

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, in the Proceedings of the Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967), p. 765.

J. T. Atkins, Absorption and Scattering of Light in Turbid Media, Ph.D. thesis, University of Delaware (1965); Dissertation Abstr. B27, 1844 (1966).

Beasley, J. K.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, in the Proceedings of the Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967), p. 765.

Billmeyer, F. W.

J. T. Atkins, F. W. Billmeyer, Color Eng. 6, No. 3, 40 (May–June 1968).

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, in the Proceedings of the Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967), p. 765.

Blevin, W. R.

W. R. Blevin, W. J. Brown, J. Opt. Soc. Amer. 52, 1250 (1962).
[CrossRef]

W. R. Blevin, W. J. Brown, J. Opt. Soc. Amer. 51, 975 (1961).
[CrossRef]

Brown, W. J.

W. R. Blevin, W. J. Brown, J. Opt. Soc. Amer. 52, 1250 (1962).
[CrossRef]

W. R. Blevin, W. J. Brown, J. Opt. Soc. Amer. 51, 975 (1961).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Astrophys. J. 100, 76 (1944).
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950) and (Dover, New York, 1960).

Craker, W. E.

W. E. Craker, F. D. Robinson, J. Oil Colour Chem. Assoc. 50, 111 (1967).

Duntley, S. Q.

S. Q. Duntley, J. Opt. Soc. Amer. 3261 (1942).
[CrossRef]

Evans, L. B.

H. C. Hottel, A. F. Sarofim, L. B. Evans, I. A. Vasalos, J. Heat Transfer 90, 56 (1968).
[CrossRef]

Giovanelli, R. G.

R. G. Giovanelli, Opt. Acta 3, 127 (1956).
[CrossRef]

R. G. Giovanelli, Opt. Acta 2, 153 (1955).
[CrossRef]

Grosh, R. J.

R. Viskanta, R. J. Grosh, Appl. Mech. Rev. 17, 91 (1964).

Hamaker, H. C.

H. C. Hamaker, Philips Res. Rept. 2, 55 (1947).

Hecht, H. G.

H. G. HechtModern Aspects of Reflectance Spectroscopy, W. W. Wendlandt, Ed. (Plenum, New York, 1968), p. 1.
[CrossRef]

Hottel, H. C.

H. C. Hottel, A. F. Sarofim, L. B. Evans, I. A. Vasalos, J. Heat Transfer 90, 56 (1968).
[CrossRef]

Hsia, H. M.

H. M. Hsia, T. J. Love, J. Heat Transfer 89, 197 (1967).
[CrossRef]

Judd, D. B.

D. B. Judd, G. Wyszecki, Colorin Business, Science and Industry, 2nd ed. (Wiley, New York, 1963).

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
[CrossRef]

Kottler, F.

For a review of previous practice, see F. Kottler, J. Opt. Soc. Amer. 50, 483 (1960).
[CrossRef]

Kourganoff, V.

V. Kourganoff, Basic Methods in Transfer Problems (Dover, New York, 1963).

Kratohvil, J. P.

J. P. Kratohvil, C. Smart, J. Colloid Sci. 20, 875 (1965).
[CrossRef]

Kubelka, P.

P. Kubelka, J. Opt. Soc. Amer. 38, 448, 1067 (1948).
[CrossRef]

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

Love, T. J.

H. M. Hsia, T. J. Love, J. Heat Transfer 89, 197 (1967).
[CrossRef]

Margenau, H.

See, for example, H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry, Vol. I (Van Nostrand, Princeton, 1956), p. 109.

Munk, F.

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

Murphy, G. M.

See, for example, H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry, Vol. I (Van Nostrand, Princeton, 1956), p. 109.

Orchard, S. E.

S. E. Orchard, J. Opt. Soc. Amer. 59, 1584 (1969).
[CrossRef]

S. E. Orchard, J. Oil Colour Chem. Assoc. 51, 44 (1968).

S. E. Orchard, Astrophys. J. 149, 665 (1967).
[CrossRef]

Richards, L. W.

L. W. Richards, J. Paint Technol. 42, 276 (1970).

Robinson, F. D.

W. E. Craker, F. D. Robinson, J. Oil Colour Chem. Assoc. 50, 111 (1967).

Ryde, J. W.

J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).

Sarofim, A. F.

H. C. Hottel, A. F. Sarofim, L. B. Evans, I. A. Vasalos, J. Heat Transfer 90, 56 (1968).
[CrossRef]

Saunderson, J. L.

J. L. Saunderson, J. Opt. Soc. Amer. 32, 727 (1942).
[CrossRef]

Schuster, A.

A. Schuster, Astrophys. J. 21, 1 (1905). Reprinted in D. H. Menzel, Selected Papers on the Transfer of Radiation (Dover, New York, 1966).
[CrossRef]

Sears, F. W.

See, for example, F. W. Sears, Optics (Addison-Wesley, Reading, Mass., 1949), pp. 169–176.

Silberstein, L.

L. Silberstein, Phil. Mag. 4, 1291 (1927).

Smart, C.

J. P. Kratohvil, C. Smart, J. Colloid Sci. 20, 875 (1965).
[CrossRef]

Vasalos, I. A.

H. C. Hottel, A. F. Sarofim, L. B. Evans, I. A. Vasalos, J. Heat Transfer 90, 56 (1968).
[CrossRef]

I. A. Vasalos, “Effect of Separation Distance on the Optical Properties of Dense Dielectric Particle Suspensions,” Ph.D. thesis, MIT (August1969); H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, J. Heat Transfer 92, 285 (1970).
[CrossRef]

Viskanta, R.

R. Viskanta, R. J. Grosh, Appl. Mech. Rev. 17, 91 (1964).

Völz, H. G.

H. G. Völz, VIth FATIPEC Congress1962 (Verlag Chemie, Weinheim/Bergstr., 1962), p. 98.

H. G. Völz, VIIth FATIPEC Congress1964 (Verlag Chemie, Weinheim/Bergstr., 1964), p. 194.

Walsh, J. W. T.

J. W. T. Walsh, in the Illumination Research Technical Paper 2 by A. K. Taylor and C. J. W. Grieveson, Department of Scientific and Industrial Research (His Majesty’s Stationery Office, London, 1926).

Wick, G. C.

G. C. Wick, Z. Phys. 120, 702 (1943).

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965).

Wyszecki, G.

D. B. Judd, G. Wyszecki, Colorin Business, Science and Industry, 2nd ed. (Wiley, New York, 1963).

Appl. Mech. Rev.

R. Viskanta, R. J. Grosh, Appl. Mech. Rev. 17, 91 (1964).

Astrophys. J.

A. Schuster, Astrophys. J. 21, 1 (1905). Reprinted in D. H. Menzel, Selected Papers on the Transfer of Radiation (Dover, New York, 1966).
[CrossRef]

S. Chandrasekhar, Astrophys. J. 100, 76 (1944).
[CrossRef]

S. E. Orchard, Astrophys. J. 149, 665 (1967).
[CrossRef]

Color Eng.

J. T. Atkins, F. W. Billmeyer, Color Eng. 6, No. 3, 40 (May–June 1968).

J. Colloid Sci.

J. P. Kratohvil, C. Smart, J. Colloid Sci. 20, 875 (1965).
[CrossRef]

J. Heat Transfer

H. M. Hsia, T. J. Love, J. Heat Transfer 89, 197 (1967).
[CrossRef]

H. C. Hottel, A. F. Sarofim, L. B. Evans, I. A. Vasalos, J. Heat Transfer 90, 56 (1968).
[CrossRef]

J. Oil Colour Chem. Assoc.

S. E. Orchard, J. Oil Colour Chem. Assoc. 51, 44 (1968).

W. E. Craker, F. D. Robinson, J. Oil Colour Chem. Assoc. 50, 111 (1967).

J. Opt. Soc. Amer.

S. E. Orchard, J. Opt. Soc. Amer. 59, 1584 (1969).
[CrossRef]

P. Kubelka, J. Opt. Soc. Amer. 38, 448, 1067 (1948).
[CrossRef]

J. L. Saunderson, J. Opt. Soc. Amer. 32, 727 (1942).
[CrossRef]

W. R. Blevin, W. J. Brown, J. Opt. Soc. Amer. 52, 1250 (1962).
[CrossRef]

W. R. Blevin, W. J. Brown, J. Opt. Soc. Amer. 51, 975 (1961).
[CrossRef]

S. Q. Duntley, J. Opt. Soc. Amer. 3261 (1942).
[CrossRef]

For a review of previous practice, see F. Kottler, J. Opt. Soc. Amer. 50, 483 (1960).
[CrossRef]

J. Paint Technol.

L. W. Richards, J. Paint Technol. 42, 276 (1970).

J. Quant. Spectr. Radiative Transfer

The Symposium on the Interdisciplinary Aspects of Radiative Energy Transfer, J. Quant. Spectr. Radiative Transfer 8, 1 (1968).

Opt. Acta

R. G. Giovanelli, Opt. Acta 2, 153 (1955).
[CrossRef]

R. G. Giovanelli, Opt. Acta 3, 127 (1956).
[CrossRef]

Phil. Mag.

L. Silberstein, Phil. Mag. 4, 1291 (1927).

Philips Res. Rept.

H. C. Hamaker, Philips Res. Rept. 2, 55 (1947).

Proc. Roy. Soc. (London)

J. W. Ryde, Proc. Roy. Soc. (London) A131, 451 (1931).

Z. Phys.

G. C. Wick, Z. Phys. 120, 702 (1943).

Z. Tech. Phys.

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

Other

V. Kourganoff, Basic Methods in Transfer Problems (Dover, New York, 1963).

I. A. Vasalos, “Effect of Separation Distance on the Optical Properties of Dense Dielectric Particle Suspensions,” Ph.D. thesis, MIT (August1969); H. C. Hottel, A. F. Sarofim, I. A. Vasalos, W. H. Dalzell, J. Heat Transfer 92, 285 (1970).
[CrossRef]

D. B. Judd, G. Wyszecki, Colorin Business, Science and Industry, 2nd ed. (Wiley, New York, 1963).

G. Kortüm, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Clarendon Press, Oxford, 1950) and (Dover, New York, 1960).

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965).

International Business Machines, System/360 Scientific Subroutine Package. See, for example, manual H20-0205-3.

See, for example, H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry, Vol. I (Van Nostrand, Princeton, 1956), p. 109.

H. G. Völz, VIth FATIPEC Congress1962 (Verlag Chemie, Weinheim/Bergstr., 1962), p. 98.

H. G. Völz, VIIth FATIPEC Congress1964 (Verlag Chemie, Weinheim/Bergstr., 1964), p. 194.

J. K. Beasley, J. T. Atkins, F. W. Billmeyer, in the Proceedings of the Second Interdisciplinary Conference on Electromagnetic Scattering, R. L. Rowell, R. S. Stein, Eds. (Gordon and Breach, New York, 1967), p. 765.

J. T. Atkins, Absorption and Scattering of Light in Turbid Media, Ph.D. thesis, University of Delaware (1965); Dissertation Abstr. B27, 1844 (1966).

See, for example, F. W. Sears, Optics (Addison-Wesley, Reading, Mass., 1949), pp. 169–176.

J. W. T. Walsh, in the Illumination Research Technical Paper 2 by A. K. Taylor and C. J. W. Grieveson, Department of Scientific and Industrial Research (His Majesty’s Stationery Office, London, 1926).

H. G. HechtModern Aspects of Reflectance Spectroscopy, W. W. Wendlandt, Ed. (Plenum, New York, 1968), p. 1.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

The division of the directions in space into channels. The channel divisions shown here are those in Table I.

Fig. 2
Fig. 2

The geometry of scattering from one infinitesimal solid angle into another.

Fig. 3
Fig. 3

A schematic diagram representing the light flux that is specularly reflected from the backing.

Fig. 4
Fig. 4

A schematic diagram showing the light flux γ that is diffusely reflected from the backing. The flux α appears in Fig. 3 and has not been diffused.

Fig. 5
Fig. 5

A schematic diagram representing the reflection of light from channel 2 into channel 3 at the back surface and the backing.

Fig. 6
Fig. 6

The ratio of the transmittances calculated by the two-flux and many-flux theories as a function of k/s. The numbers beside each curve give sX, the optical thickness for scattering only; m is the ratio of the indices of refraction at the boundaries of the scattering medium, and the phase function for the pigment is given in column B of Table III.

Fig. 7
Fig. 7

The ratio of the reflectances calculated by the two-flux and many-flux theories as a function of k/s. The top frames give the reflectance of a medium with nothing behind it, and middle and bottom frames the reflectance of a medium in contact with a perfect absorber and a white with a reflectance of 0.94, respectively. The definition of m and the phase functions are as in Fig. 6.

Tables (6)

Tables Icon

Table I Channel Divisions for Many-Flux Calculations with Twenty-Two Channelsa

Tables Icon

Table II Comparison of Reflectance Calculated Using the Many-Flux Equations with Those Reported by Orchard23 for Three Different Phase Functionsa

Tables Icon

Table III Legendre Polynomial Coefficients for the Phase Functions.a

Tables Icon

Table IV Channel Divisions for Many-Flux Calculations with Twenty-six Channelsa

Tables Icon

Table V Values of S and K Required to Make the Two-Flux Calculation Give the Same Results As the Many-Flux Calculation

Tables Icon

Table VI Comparison of Four-Flux Calculations with the Data of Atkinsa

Equations (91)

Equations on this page are rendered with MathJax. Learn more.

F = I cos θ .
F = I cos θ d ω 2 π ,
d F i d x = j = 1 n S i j F j ,             i n / 2 ,
- d F i d x = j = 1 n S i j F j ,             i > n / 2 ,
S i j = S i j when i n / 2 ,
S i j = - S i j when i > n / 2 ,
( S - E d d x ) F = 0.
F i = j = 1 n A i j C j e λ j x ,             i = 1 , 2 , , n ,
j = 1 n S i j A j k - λ k A i k = 0 ,             i = 1 , 2 , , n .
F = I cos θ d ν d ω d a ,
d F = - s F d l
d x = d l cos θ ,
d F = - ( s F d x / cos θ ) .
d ( d F ) = - s F d x cos θ · p ( cos ψ ) d ω 4 π .
4 π p ( cos ψ ) d ω 4 π = 1.
d ω = 2 π sin ψ d ψ ,
1 2 - 1 + 1 p ( cos ψ ) d ( cos ψ ) = 1.
d ( d F ) = - s F d x cos θ j · p ( cos ψ ) 4 π · ω i d ϕ i 2 π .
d F = - s F ω i d x 8 π 2 cos θ j 0 2 π p ( cos x ) d ϕ i ,
cos ψ = cos θ i cos θ j + sin θ i sin θ j cos ( ϕ i - ϕ j ) .
p ( cos ψ ) = l = 0 L a l P l ( cos ψ ) .
a l = ( l + 1 2 ) - 1 1 p ( cos ψ ) P l ( cos ψ ) d ( cos ψ ) .
0 2 π p ( cos ψ ) d ϕ i = l = 0 L a l 0 2 π P l ( cos ψ ) d ϕ i = 2 π l = 0 L a l P l ( cos θ i ) P l ( cos θ j ) .
d F = - ω i s F d x 4 π cos θ j l = 0 L a l P l ( cos θ i ) P l ( cos θ j ) .
S i j i j = ω i 4 π cos θ j l = 0 L a l P l ( cos θ i ) P l ( cos θ j ) .
S j j = - k / s cos θ j - m = 1 m j n S m j .
F i = D i + r i F n + 1 - i ;             i = 1 , 2 , , n / 2 ;             x = 0.
F i = k = 1 n / 2 R i k F k ;             i = n 2 + 1 , n 2 + 2 , , n ;             x = X ,
k = 1 n A i j C j = D i + r i j = 1 n A m j C j , or j = 1 n C j ( A i j - r i A m j ) = D i ;             i = 1 , 2 , , n / 2 ;             m = n + 1 - i
k = 1 n C j ( A i j - k = 1 n / 2 R i k A k j ) e λ j X = 0 ; i = n / 2 + 1 , n / 2 + 2 , , n
R i k = R s + R d ω i cos θ i / m = n / 2 + 1 n ω m cos θ m ,
R i k = R d ω i cos θ i / m = n / 2 + 1 n ω m cos θ m .
α = ( 1 - r k ) F k + r i α R s ,
α = ( 1 - r k ) F k 1 - r i R s .
( 1 - r i ) R s ( 1 - r k ) F k 1 - r i R s = R s ( 1 - r k ) 2 1 - r k R s F k ,
R F = 1 - i = 1 n / 2 D i .
γ = α R d + R F γ ( R s + R d ) .
γ = α R d 1 - R F ( R s + R d ) ,
D i γ = D i R d ( 1 - r k ) F k [ 1 - R F ( R s + R d ) ] ( 1 - r k R s ) .
R i k = r k + R s ( 1 - r k ) 2 1 - r k R s + D i R d ( 1 - r k ) [ 1 - R F ( R s + R d ) ] ( 1 - r k R s ) , for i + k = n + 1 ,
R i k = D i R d ( 1 - r k ) [ 1 - R F ( R s + R d ) ] ( 1 - r k R s ) , for i + k n + 1 ,
and             R i k = 1 for i + k = n + 1 R i k = 0 for i + k n + 1
k = n + 1 - i , m = n + 1 - j ,
S i j = - S k m .
TS T - 1 = - S ,
T - 1 = T .
S - E λ = 0.
F 1 = collimated flux in the positive direction . F 2 = diffuse flux in the positive direction , F 3 = diffuse flux in the negative direction , F 4 = collimated flux in the negative direction .
S 12 = S 13 = S 14 = S 41 = S 42 = S 43 = 0.
S 21 = S 34 = S 1 , S 31 = S 24 = S 2 , S 23 = S 32 = S .
d F 1 / d x = - ( k + S 1 + S 2 ) F 1 , d F 2 / d x = S 1 F 1 - ( K + S ) F 2 + S F 3 + S 2 F 4 , - ( d F 3 / d x ) = S 2 F 1 + S F 2 - ( K + S ) F 3 + S 1 F 4 , - ( d F 4 / d x ) = - ( k + S 1 + S 2 ) F 4 .
F 1 = C 1 e - λ x , F 2 = C 1 A 1 e - λ x + C 2 ( 1 + β ) e - σ x + C 3 ( 1 - β ) e σ x + C 4 A 2 e λ x , F 3 = C 1 A 2 e - λ x + C 2 ( 1 - β ) e - σ x + C 3 ( 1 + β ) e σ x + C 4 A 1 e λ x , F 4 = C 4 e λ x ,
λ = k + S 1 + S 2 ,
σ = [ K ( K + 2 S ) ] 1 2 ,
A 1 = S S 2 + ( K + S + λ ) S 1 σ 2 - λ 2 ,
A 2 = S S 1 + ( K + S - λ ) S 2 σ 2 - λ 2 ,
β = [ K / ( K + 2 S ) ] . 1 2
R 13 = R 42 = 0.
F 1 = f ( 1 - R c ) + R 14 F 4 ,
F 2 = ( 1 - f ) ( 1 - R e ) + R 23 F 3
F 3 = R 31 F 1 + R 32 F 2 ,
F 4 = R 41 F 1
C 1 = f ( 1 - R c ) 1 - R 14 R 41 e - 2 λ X ,
C 4 = R 41 C 1 e - 2 λ X = f ( 1 - R c ) R 41 e - 2 λ X 1 - R 14 R 41 e - 2 λ X .
C 2 [ 1 + β - R 23 ( 1 - β ) ] + C 3 [ 1 - β - R 23 ( 1 + β ) ] = ( 1 - f ) ( 1 - R e ) + C 1 ( R 23 A 2 - A 1 ) + C 4 ( R 23 A 1 - A 2 ) ,
C 2 [ 1 - β - R 32 ( 1 + β ) ] e - σ X + C 3 [ 1 + β - R 32 ( 1 - β ) ] e σ X = C 1 [ R 31 + R 32 A 1 + R 32 R 41 A 2 - A 2 - R 41 A 1 ] e - λ X ,
R 14 = R c , R 31 = R d , R 32 = R d + R s , R 41 = R s , R 23 = r 23 ,
R 14 = R c .
R 41 = R c + ( 1 - R c ) 2 1 - R c R s R s .
R 31 = ( 1 - R c ) ( 1 - R e ) [ 1 - R e ( R s + R d ) ] ( 1 - R c R s ) R d .
R 32 = r 32 + ( 1 - r 32 ) ( 1 - R e ) 1 - R e ( R s + R d ) ( R s + R d ) .
K 2 k .
S = s - 1 1 g ( cos ψ ) p ( cos ψ ) d ( cos ψ ) ,
g ( cos ψ ) = l = 0 M b l P l ( cos ψ ) ,
S = s l = 0 M 2 a l b l 2 l + 1 .
S = s ( 0.7502 a 0 - 0.2496 a 1 - 0.0010 a 2 + 0.0001 a 3 ) .
S s ( 3 a 0 - a 1 ) / 4.
g ( cos ψ ) 3 8 P 0 ( cos ψ ) - 3 8 P 1 ( cos ψ ) = 3 8 ( 1 - cos ψ ) .
r 32 0.619 = 0.968 - ( 0.742 ± 0.058 ) k / s ,
r 23 / 0.619 = 1.005 + ( 1.065 ± 0.093 ) k / s + ( 0.070 ± 0.003 ) ( 1 - R 32 ) / ( S X ) ,
S 1 = s 2 0 1 p ( cos ψ ) d ( cos ψ ) ,
S 2 = s 2 - 1 0 p ( cos ψ ) d ( cos ψ ) ,
S 1 + S 2 = s .
0 1 P l ( cos ψ ) d ( cos ψ ) = 0 ;             l = 2 , 4 , 6 , = ( - 1 ) ( l - 1 ) / 2 ( 1 · 3 · 5 l ) 2 l ( l + 1 ) l ! ;             l = 1 , 3 , 5 ,
S 1 = s 2 ( 1 + l = 1 , 3 , 5 , m a l ( - 1 ) ( l - 1 ) / 2 ( 1 · 3 · 5 l ) 2 l ( l + 1 ) l ! ) = s 2 ( 1 + a 1 2 - a 3 8 + a 5 16 - 5 a 7 128 + 7 a 9 256 - ) .
transmitted collimated flux = ( 1 - R c ) F 1 ( X ) ,
reflected collimated flux = f R c + ( 1 - R c ) F 4 ( 0 ) ,
transmitted diffuse flux = ( 1 - r 32 ) F 2 ( X ) ,
reflected diffuse flux = ( 1 - f ) R e + ( 1 - R 23 ) F 3 ( 0 ) ,
a 0 = 1.0000 , a 7 = 0.05092 , a 1 = 1.5560 , a 8 = 0.06530 , a 2 = 1.5493 , a 9 = 0.00416 , a 3 = 1.0429 , a 10 = 0.01336 , a 4 = 0.7195 , a 11 = - 0.00054 , a 5 = 0.3095 , a 12 = 0.00082 , a 6 = 0.24234 , a 13 = 0.00004.
f = 1 , X = measured thickness of each disk in microns , R s = R d = 0 , R c = 0.040 , R e is not used , k = 7.28 ( weight fraction of red dye ) μ - 1 , K = 2 k , s = 4.71 ( weight fraction of rutile ) μ - 1 , S = 0.3610 s , S 1 = 0.8326 s , S 2 = 0.1674 s .

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