Abstract

A simplified stochastic model for the fiber structure of paper is introduced. The packing density and optical thickness of the fiber network are derived analytically, and their dependence on fiber characteristics can be seen. We undertake a Monte Carlo simulation of light scattering that is based on geometrical optics, using a realization of the model, which gives packing densities and optical thicknesses well in accordance with those given by the stochastic model and the scattering quantities as functions of three angles.

© 2000 Optical Society of America

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References

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  1. E. K. O. Hellén, M. J. Alava, K. J. Niskanen, “Porous structure of thick fiber webs,” J. Appl. Phys. 81, 6425–6431 (1997).
    [CrossRef]
  2. K. J. Niskanen, M. J. Alava, “Planar random networks with flexible fibers,” Phys. Rev. Lett. 73, 3475–3478 (1994).
    [CrossRef] [PubMed]
  3. E. Seppälä, M. Alava, K. Niskanen, “Onko kuitukoostumuksella havaittava vaikutus laboratorioarkkien karheuteen?” Paper Timber 78, 446–449 (1996).
  4. M. J. Alava, V. I. Räisänen, R. M. Nieminen, “Elasticity and plasticity of fiber networks,” CSC (Centre for Scientific Computing in Finland) News 6(1), 9–11 (1994).
  5. M. E. J. Karttunen, K. J. Niskanen, K. Kaski, “Fracture in mesoscopic disordered systems,” Phys. Rev. B 49, 9453–9459 (1994).
    [CrossRef]
  6. V. Räisänen, M. Alava, R. Neiminen, K. Niskanen, “Computer studies of elasticity and plasticity of random fibre networks,” (Center for Scientific Computing, Tieteellinen, Laskenta OY, Helsinki, Finland, 1994).
  7. P. Kubelka, F. Munk, “Ein breitag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).
  8. W. E. Vargas, “Generalized four-flux radiative transfer model,” Appl. Opt. 37, 2615–2623 (1998).
    [CrossRef]
  9. W. E. Vargas, “Two-flux radiative transfer model under nonisotropic propagating diffuse radiation,” Appl. Opt. 38, 1077–1085 (1999).
    [CrossRef]
  10. J. Rahola, “Solution of dense systems of linear equations in electromagnetic scattering calculations,” Licentiate’s thesis (Helsinki University of Technology, Helsinki, Finland, 1994).
  11. K. Muinonen, K. Saarinen, “Light scattering by Gaussian random cylinders: ray optics approximation,” in Proceedingsof the First Workshop on Electromagnetic and Light Scattering—Theory and Applications (University of Bremen, Bremen, Germany, 1996), pp. 139–142.
  12. K. Muinonen, K. Saarinen, “Ray optics approximation for Gaussian random cylinders,” J. Quant. Spectrosc. Radiat. Transfer 64, 201–218 (2000).
    [CrossRef]
  13. K. Saarinen, ABB Corporatee Research Oy, P.O. Box 608, FIN-65101 Vaasa, Finland; e-mail: Kari.Saarinen@fi.abb.com (personal communications, 1997–1998).
  14. M. Leskelä, “Simulation of particle packing for modelling the light scattering characteristics of paper,” Ph.D. dissertation (Helsinki University of Technology, Helsinki, Finland, 1997).
  15. A. M. Scallan, J. Borch, “An interpretation on paper reflectance based upon morphology. I: initial considerations,” Tappi 55, 583–588 (1972).
  16. A. M. Scallan, J. Borch, “An interpretation on paper reflectance based upon morphology. II: general applicability,” Tappi 57, 143–147 (1974).
  17. H. Cramer, M. R. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).
  18. M. Deng, C. T. J. Dodson, Paper—an Engineered Stochastic Structure (Tappi, Atlanta, Ga., 1994).
  19. A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).
  20. J. A. Rice, Mathematical Statistics and Data Analysis (Brooks/Cole, Pacific Grove, Calif., 1988).
  21. D. Stoyan, W. S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, 2nd ed. (Wiley, New York, 1995).
  22. K. Lumme, J. Rahola, K. Muinonen, H. Volten, “Scattering by rough particles and stochastic aggregates,” in Proceedings of the Fourth International Congress on Optical Particle Sizing (Nürnberger Messe, GmbH, Nürnberg, Germany, 1995), pp. 583–593.
  23. K. Muinonen, K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Light scattering by randomly oriented crystals,” Appl. Opt. 28, 3051–3060 (1989).
    [CrossRef] [PubMed]
  24. K. Muinonen, “Scattering of light by crystals: a modified Kirchhoff approximation,” Appl. Opt. 28, 3044–3050 (1989).
    [CrossRef] [PubMed]
  25. K. Muinonen, “Light scattering by Gaussian random particles,” Earth Moon Planets 72, 339–342 (1996).
    [CrossRef]

2000 (1)

K. Muinonen, K. Saarinen, “Ray optics approximation for Gaussian random cylinders,” J. Quant. Spectrosc. Radiat. Transfer 64, 201–218 (2000).
[CrossRef]

1999 (1)

1998 (1)

1997 (1)

E. K. O. Hellén, M. J. Alava, K. J. Niskanen, “Porous structure of thick fiber webs,” J. Appl. Phys. 81, 6425–6431 (1997).
[CrossRef]

1996 (2)

E. Seppälä, M. Alava, K. Niskanen, “Onko kuitukoostumuksella havaittava vaikutus laboratorioarkkien karheuteen?” Paper Timber 78, 446–449 (1996).

K. Muinonen, “Light scattering by Gaussian random particles,” Earth Moon Planets 72, 339–342 (1996).
[CrossRef]

1994 (3)

M. J. Alava, V. I. Räisänen, R. M. Nieminen, “Elasticity and plasticity of fiber networks,” CSC (Centre for Scientific Computing in Finland) News 6(1), 9–11 (1994).

M. E. J. Karttunen, K. J. Niskanen, K. Kaski, “Fracture in mesoscopic disordered systems,” Phys. Rev. B 49, 9453–9459 (1994).
[CrossRef]

K. J. Niskanen, M. J. Alava, “Planar random networks with flexible fibers,” Phys. Rev. Lett. 73, 3475–3478 (1994).
[CrossRef] [PubMed]

1989 (2)

1974 (1)

A. M. Scallan, J. Borch, “An interpretation on paper reflectance based upon morphology. II: general applicability,” Tappi 57, 143–147 (1974).

1972 (1)

A. M. Scallan, J. Borch, “An interpretation on paper reflectance based upon morphology. I: initial considerations,” Tappi 55, 583–588 (1972).

1931 (1)

P. Kubelka, F. Munk, “Ein breitag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Alava, M.

E. Seppälä, M. Alava, K. Niskanen, “Onko kuitukoostumuksella havaittava vaikutus laboratorioarkkien karheuteen?” Paper Timber 78, 446–449 (1996).

V. Räisänen, M. Alava, R. Neiminen, K. Niskanen, “Computer studies of elasticity and plasticity of random fibre networks,” (Center for Scientific Computing, Tieteellinen, Laskenta OY, Helsinki, Finland, 1994).

Alava, M. J.

E. K. O. Hellén, M. J. Alava, K. J. Niskanen, “Porous structure of thick fiber webs,” J. Appl. Phys. 81, 6425–6431 (1997).
[CrossRef]

K. J. Niskanen, M. J. Alava, “Planar random networks with flexible fibers,” Phys. Rev. Lett. 73, 3475–3478 (1994).
[CrossRef] [PubMed]

M. J. Alava, V. I. Räisänen, R. M. Nieminen, “Elasticity and plasticity of fiber networks,” CSC (Centre for Scientific Computing in Finland) News 6(1), 9–11 (1994).

Borch, J.

A. M. Scallan, J. Borch, “An interpretation on paper reflectance based upon morphology. II: general applicability,” Tappi 57, 143–147 (1974).

A. M. Scallan, J. Borch, “An interpretation on paper reflectance based upon morphology. I: initial considerations,” Tappi 55, 583–588 (1972).

Cramer, H.

H. Cramer, M. R. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).

Deng, M.

M. Deng, C. T. J. Dodson, Paper—an Engineered Stochastic Structure (Tappi, Atlanta, Ga., 1994).

Dodson, C. T. J.

M. Deng, C. T. J. Dodson, Paper—an Engineered Stochastic Structure (Tappi, Atlanta, Ga., 1994).

Hellén, E. K. O.

E. K. O. Hellén, M. J. Alava, K. J. Niskanen, “Porous structure of thick fiber webs,” J. Appl. Phys. 81, 6425–6431 (1997).
[CrossRef]

Irvine, W. M.

Karttunen, M. E. J.

M. E. J. Karttunen, K. J. Niskanen, K. Kaski, “Fracture in mesoscopic disordered systems,” Phys. Rev. B 49, 9453–9459 (1994).
[CrossRef]

Kaski, K.

M. E. J. Karttunen, K. J. Niskanen, K. Kaski, “Fracture in mesoscopic disordered systems,” Phys. Rev. B 49, 9453–9459 (1994).
[CrossRef]

Kendall, W. S.

D. Stoyan, W. S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, 2nd ed. (Wiley, New York, 1995).

Kubelka, P.

P. Kubelka, F. Munk, “Ein breitag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Leadbetter, M. R.

H. Cramer, M. R. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).

Leskelä, M.

M. Leskelä, “Simulation of particle packing for modelling the light scattering characteristics of paper,” Ph.D. dissertation (Helsinki University of Technology, Helsinki, Finland, 1997).

Lumme, K.

K. Muinonen, K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Light scattering by randomly oriented crystals,” Appl. Opt. 28, 3051–3060 (1989).
[CrossRef] [PubMed]

K. Lumme, J. Rahola, K. Muinonen, H. Volten, “Scattering by rough particles and stochastic aggregates,” in Proceedings of the Fourth International Congress on Optical Particle Sizing (Nürnberger Messe, GmbH, Nürnberg, Germany, 1995), pp. 583–593.

Mecke, J.

D. Stoyan, W. S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, 2nd ed. (Wiley, New York, 1995).

Muinonen, K.

K. Muinonen, K. Saarinen, “Ray optics approximation for Gaussian random cylinders,” J. Quant. Spectrosc. Radiat. Transfer 64, 201–218 (2000).
[CrossRef]

K. Muinonen, “Light scattering by Gaussian random particles,” Earth Moon Planets 72, 339–342 (1996).
[CrossRef]

K. Muinonen, K. Lumme, J. I. Peltoniemi, W. M. Irvine, “Light scattering by randomly oriented crystals,” Appl. Opt. 28, 3051–3060 (1989).
[CrossRef] [PubMed]

K. Muinonen, “Scattering of light by crystals: a modified Kirchhoff approximation,” Appl. Opt. 28, 3044–3050 (1989).
[CrossRef] [PubMed]

K. Lumme, J. Rahola, K. Muinonen, H. Volten, “Scattering by rough particles and stochastic aggregates,” in Proceedings of the Fourth International Congress on Optical Particle Sizing (Nürnberger Messe, GmbH, Nürnberg, Germany, 1995), pp. 583–593.

K. Muinonen, K. Saarinen, “Light scattering by Gaussian random cylinders: ray optics approximation,” in Proceedingsof the First Workshop on Electromagnetic and Light Scattering—Theory and Applications (University of Bremen, Bremen, Germany, 1996), pp. 139–142.

Munk, F.

P. Kubelka, F. Munk, “Ein breitag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Neiminen, R.

V. Räisänen, M. Alava, R. Neiminen, K. Niskanen, “Computer studies of elasticity and plasticity of random fibre networks,” (Center for Scientific Computing, Tieteellinen, Laskenta OY, Helsinki, Finland, 1994).

Nieminen, R. M.

M. J. Alava, V. I. Räisänen, R. M. Nieminen, “Elasticity and plasticity of fiber networks,” CSC (Centre for Scientific Computing in Finland) News 6(1), 9–11 (1994).

Niskanen, K.

E. Seppälä, M. Alava, K. Niskanen, “Onko kuitukoostumuksella havaittava vaikutus laboratorioarkkien karheuteen?” Paper Timber 78, 446–449 (1996).

V. Räisänen, M. Alava, R. Neiminen, K. Niskanen, “Computer studies of elasticity and plasticity of random fibre networks,” (Center for Scientific Computing, Tieteellinen, Laskenta OY, Helsinki, Finland, 1994).

Niskanen, K. J.

E. K. O. Hellén, M. J. Alava, K. J. Niskanen, “Porous structure of thick fiber webs,” J. Appl. Phys. 81, 6425–6431 (1997).
[CrossRef]

K. J. Niskanen, M. J. Alava, “Planar random networks with flexible fibers,” Phys. Rev. Lett. 73, 3475–3478 (1994).
[CrossRef] [PubMed]

M. E. J. Karttunen, K. J. Niskanen, K. Kaski, “Fracture in mesoscopic disordered systems,” Phys. Rev. B 49, 9453–9459 (1994).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

Peltoniemi, J. I.

Rahola, J.

K. Lumme, J. Rahola, K. Muinonen, H. Volten, “Scattering by rough particles and stochastic aggregates,” in Proceedings of the Fourth International Congress on Optical Particle Sizing (Nürnberger Messe, GmbH, Nürnberg, Germany, 1995), pp. 583–593.

J. Rahola, “Solution of dense systems of linear equations in electromagnetic scattering calculations,” Licentiate’s thesis (Helsinki University of Technology, Helsinki, Finland, 1994).

Räisänen, V.

V. Räisänen, M. Alava, R. Neiminen, K. Niskanen, “Computer studies of elasticity and plasticity of random fibre networks,” (Center for Scientific Computing, Tieteellinen, Laskenta OY, Helsinki, Finland, 1994).

Räisänen, V. I.

M. J. Alava, V. I. Räisänen, R. M. Nieminen, “Elasticity and plasticity of fiber networks,” CSC (Centre for Scientific Computing in Finland) News 6(1), 9–11 (1994).

Rice, J. A.

J. A. Rice, Mathematical Statistics and Data Analysis (Brooks/Cole, Pacific Grove, Calif., 1988).

Saarinen, K.

K. Muinonen, K. Saarinen, “Ray optics approximation for Gaussian random cylinders,” J. Quant. Spectrosc. Radiat. Transfer 64, 201–218 (2000).
[CrossRef]

K. Saarinen, ABB Corporatee Research Oy, P.O. Box 608, FIN-65101 Vaasa, Finland; e-mail: Kari.Saarinen@fi.abb.com (personal communications, 1997–1998).

K. Muinonen, K. Saarinen, “Light scattering by Gaussian random cylinders: ray optics approximation,” in Proceedingsof the First Workshop on Electromagnetic and Light Scattering—Theory and Applications (University of Bremen, Bremen, Germany, 1996), pp. 139–142.

Scallan, A. M.

A. M. Scallan, J. Borch, “An interpretation on paper reflectance based upon morphology. II: general applicability,” Tappi 57, 143–147 (1974).

A. M. Scallan, J. Borch, “An interpretation on paper reflectance based upon morphology. I: initial considerations,” Tappi 55, 583–588 (1972).

Seppälä, E.

E. Seppälä, M. Alava, K. Niskanen, “Onko kuitukoostumuksella havaittava vaikutus laboratorioarkkien karheuteen?” Paper Timber 78, 446–449 (1996).

Stoyan, D.

D. Stoyan, W. S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, 2nd ed. (Wiley, New York, 1995).

Vargas, W. E.

Volten, H.

K. Lumme, J. Rahola, K. Muinonen, H. Volten, “Scattering by rough particles and stochastic aggregates,” in Proceedings of the Fourth International Congress on Optical Particle Sizing (Nürnberger Messe, GmbH, Nürnberg, Germany, 1995), pp. 583–593.

Appl. Opt. (4)

CSC (Centre for Scientific Computing in Finland) News (1)

M. J. Alava, V. I. Räisänen, R. M. Nieminen, “Elasticity and plasticity of fiber networks,” CSC (Centre for Scientific Computing in Finland) News 6(1), 9–11 (1994).

Earth Moon Planets (1)

K. Muinonen, “Light scattering by Gaussian random particles,” Earth Moon Planets 72, 339–342 (1996).
[CrossRef]

J. Appl. Phys. (1)

E. K. O. Hellén, M. J. Alava, K. J. Niskanen, “Porous structure of thick fiber webs,” J. Appl. Phys. 81, 6425–6431 (1997).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

K. Muinonen, K. Saarinen, “Ray optics approximation for Gaussian random cylinders,” J. Quant. Spectrosc. Radiat. Transfer 64, 201–218 (2000).
[CrossRef]

Paper Timber (1)

E. Seppälä, M. Alava, K. Niskanen, “Onko kuitukoostumuksella havaittava vaikutus laboratorioarkkien karheuteen?” Paper Timber 78, 446–449 (1996).

Phys. Rev. B (1)

M. E. J. Karttunen, K. J. Niskanen, K. Kaski, “Fracture in mesoscopic disordered systems,” Phys. Rev. B 49, 9453–9459 (1994).
[CrossRef]

Phys. Rev. Lett. (1)

K. J. Niskanen, M. J. Alava, “Planar random networks with flexible fibers,” Phys. Rev. Lett. 73, 3475–3478 (1994).
[CrossRef] [PubMed]

Tappi (2)

A. M. Scallan, J. Borch, “An interpretation on paper reflectance based upon morphology. I: initial considerations,” Tappi 55, 583–588 (1972).

A. M. Scallan, J. Borch, “An interpretation on paper reflectance based upon morphology. II: general applicability,” Tappi 57, 143–147 (1974).

Z. Tech. Phys. (Leipzig) (1)

P. Kubelka, F. Munk, “Ein breitag zur optik der farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–601 (1931).

Other (11)

K. Saarinen, ABB Corporatee Research Oy, P.O. Box 608, FIN-65101 Vaasa, Finland; e-mail: Kari.Saarinen@fi.abb.com (personal communications, 1997–1998).

M. Leskelä, “Simulation of particle packing for modelling the light scattering characteristics of paper,” Ph.D. dissertation (Helsinki University of Technology, Helsinki, Finland, 1997).

H. Cramer, M. R. Leadbetter, Stationary and Related Stochastic Processes (Wiley, New York, 1967).

M. Deng, C. T. J. Dodson, Paper—an Engineered Stochastic Structure (Tappi, Atlanta, Ga., 1994).

A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

J. A. Rice, Mathematical Statistics and Data Analysis (Brooks/Cole, Pacific Grove, Calif., 1988).

D. Stoyan, W. S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, 2nd ed. (Wiley, New York, 1995).

K. Lumme, J. Rahola, K. Muinonen, H. Volten, “Scattering by rough particles and stochastic aggregates,” in Proceedings of the Fourth International Congress on Optical Particle Sizing (Nürnberger Messe, GmbH, Nürnberg, Germany, 1995), pp. 583–593.

V. Räisänen, M. Alava, R. Neiminen, K. Niskanen, “Computer studies of elasticity and plasticity of random fibre networks,” (Center for Scientific Computing, Tieteellinen, Laskenta OY, Helsinki, Finland, 1994).

J. Rahola, “Solution of dense systems of linear equations in electromagnetic scattering calculations,” Licentiate’s thesis (Helsinki University of Technology, Helsinki, Finland, 1994).

K. Muinonen, K. Saarinen, “Light scattering by Gaussian random cylinders: ray optics approximation,” in Proceedingsof the First Workshop on Electromagnetic and Light Scattering—Theory and Applications (University of Bremen, Bremen, Germany, 1996), pp. 139–142.

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Figures (12)

Fig. 1
Fig. 1

Part of a simulated network of fibers with the following parameters: expected radius 0.05, σ a = 0.02, σ b = 0.05, σ e = 0.5, σ00 = 0, σ10 = 0.2, σ20 = 0.2, and others are [in (σ kl )] 0.02. Visualized by use of matlab.

Fig. 2
Fig. 2

Example of a single simulated fiber (parameters as in Fig. 1). Visualized by use of matlab.

Fig. 3
Fig. 3

Some real fibers in suspension. This example is only qualitatively comparable with Fig. 1 because the fibers here have not yet been compressed to a paper sheet. An exemplary fiber in the middle of the picture has the following parameters: length, 2.86 mm; curl, 9.3%; and brightness, 36.1%. Courtesy of UPM-Kymmene Corporation, Helsinki, Finland.

Fig. 4
Fig. 4

Cross section of dyed real paper. Magnification is 200. Courtesy of the Finnish Pulp and Paper Research Institute, Espoo, Finland.

Fig. 5
Fig. 5

Part of a simulated fiber network; the same relative mass as in Fig. 1 but with an expected radius of 0.10. Visualized by use of matlab.

Fig. 6
Fig. 6

Simulated observed, scattered relative intensity as a function of the spherical coordinate ϕ. A few averaged curves in an interval of θ are shown. The value θ = 0 would correspond to direct backscattering and θ = 180° to forward scattering. A part of the corresponding fiber web can be seen in Fig. 5.

Fig. 7
Fig. 7

Simulated observed, scattered relative intensity as a function of the spherical coordinate θ. A few averaged curves in an interval of ϕ are shown. A part of the corresponding fiber web can be seen in Fig. 5.

Fig. 8
Fig. 8

Simulated observed, scattered relative intensity as a function of the spherical coordinate ϕ. A few averaged curves in an interval of θ are shown. The value θ = 0 would correspond to direct backscattering and θ = 180 deg to forward scattering. A part of the corresponding fiber web can be seen in Fig. 6.

Fig. 9
Fig. 9

Simulated observed, scattered relative intensity as a function of the spherical coordinate θ. A few averaged curves in an interval of ϕ are shown. A part of the corresponding fiber web can be seen in Fig. 6.

Fig. 10
Fig. 10

Propagation probabilities in a network whose fibers have a constant radius and are inflexible. No scattering takes place. Y axis: probability of traveling a certain length from a randomly picked starting point in the network.

Fig. 11
Fig. 11

Propagation probabilities in a fiber network without scattering. Y axis: probability of traveling a certain length from a randomly picked starting point in the network.

Fig. 12
Fig. 12

Slanted propagation probabilities in a fiber network.

Equations (53)

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

dxtdt2+dytdt2+dztdt2=1
dγtdt=sin θ cos ϕ, sin θ sin ϕ, cos θ,
θt=a+b sin t+c cos t, ϕt=d+e sin t+f cos t,
aNπ/2, σa2,  bN0, σb2,  cN0, σc2,  dhd,  eN0, σe2,  fN0, σf2,
covxt1, yt2=xt1yt2-xt1yt2=sin θt1sin θt2×sin ϕt2cos ϕt1-sin θt1×cos ϕt1sin θt2sin ϕt2
sin θt1sin θt2=R3 sina+b sin t1+c cos t1sina+b sin t2+c cos t2ga, b, cdadbdc=-14R3(expi2a+bsin t1+sin t2+ccos t1+cos t2-expibsin t1-sin t2+ccos t1-cos t2-expib-sin t1+sin t2+c-cos t1+cos t2+exp-i2a+bsin t1+sin t2+ccos t1+cos t2)×ga, b, cdadbdc,
Fexp-12x-xTΣ-1x-xω=2π-n/2Rn exp-iω·x×exp-12x-xTΣ-1x-xdx=|Σ| exp-ix·ωexp-12 ωTΣω,
ga, b, c=12π3/2|Σa,b,c| exp-12a, b, c-π2, 0, 0Σa,b,c-1a, b, c-π2, 0, 0T,
Σa,b,c=σa2000σb2000σc2
sin θt1sin θt2=-14σaσbσc×Fexp-12a-π2, b, cΣa,b,c-1a-π2, b, cT-2, -sin t1-sin t2, -cos t1-cos t2-Fexp-12a-π2, b, cΣa,b,c-1a-π2, b, cT0, -sin t1+sin t2, -cos t1+cos t2-Fexp-12a-π2, b, cΣa,b,c-1a-π2, b, cT0, sin t1-sin t2, cos t1-cos t2+Fexp-12a-π2, b, cΣa,b,c-1a-π2, b, cT2, sin t1+sin t2, cos t1+cos t2=-14σaσbσc|Σ|exp-i π2-2exp-12-2, -sin t1-sin t2, -cos t1-cos t2Σ×-2, -sin t1-sin t2, -cos t1-cos t2T-exp0exp-120, -sin t1+sin t2, -cos t1+cos t2Σ0, -sin t1+sin t2, -cos t1+cos t2T-exp0exp-120, sin t1-sin t2, cos t1-cos t2Σ×0, sin t1-sin t2, cos t1-cos t2T+exp-i π2+2×exp-122, sin t1+sin t2, cos t1+cos t2Σ2, sin t1+sin t2, cos t1+cos t2T=12exp-124σa2+σb2sin t1+sin t22+σc2cos t1+cos t22+exp-12σb2sin t1-sin t22+σc2cos t1-cos t22.
covxt1, yt2=14exp-124σa2+σb2+σc2×sin t1+sin t22+σc2+σf2×cos t1+cos t22+exp-12σb2sin t1-sin t22+σc2cos t1-cos t22+σe2sin t1+sin t22+σf2cos t1+cos t22×0π sin 2thtdt.
xt=-π/2+π/2 χ0, txτdτ=-π/2+π/2 χ0, txτdτ=0t xtdτ,
xt1yt2=-π/2+π/2 χ0, t1xτdτ -π/2+π/2 χ0, t2yτdτ=-π/2+π/2-π/2+π/2 χ0, t1τ1χ0, t2×τ2covxτ1, yτ2dτ1dτ2=0t10t2 covxτ1, yτ2dτ1dτ2.
rt, ψ=c expst, ψ,
st, ψ=k,lZ dkl expikψ+lt,  dkl*=d-k-l.
st, ψ=a00+0<llmaxa0l cos lt+b0l sin lt+0<kkmax, -lmaxllmaxakl coskψ+lt+bkl sinkψ+lt
a00N0, σ002,  aklbklN0, 2σkl2.
covst1, ψ1, st2, ψ2=σ002+2 0<llmax σ0l2 cos lt1-t2+20<kkmax,-lmaxllmax σkl2×coskψ1-ψ2+lt1-t2,
σ2=expσ002+2 0<llmax σ0l2+2 0<kkmax,-lmaxllmax σkl2-1
c=a1+σ2
rt, ψ=a,  covrt1, ψ1, rt2, ψ2=a2(expcovst1, ψ1, st2, ψ2-1).
var rt, ψ=a2σ2.
r0θt, ϕt, ψ)=cos ϕ-sin0sin ϕcos ϕ0001cos θ0sin θ010-sin θ0cos θcos ψsin ψ0=cos θtcos ϕtcos ψ-sin ϕtsin ψcos θtsin ϕtcos ψ+cos ϕtsin ψ-sin θtcos ψ.
ut, ψ=γt+rt, ψ,
γt=0tdxτdτ, dyτdτ, dzτdτdτ.
PΦB=k=exp-xxkk!,
PΦB=0=exp-λνdB.
PBΦ==exp-λν3Φ0-B,
PΦq0+αp0|α0, p==exp-λWp
PaΦ=1-PΦa==1-exp-λν3Φ0-a=1-exp-λV.
exp-1/cos θi,
V=Vshell+Vright end+Vleft endlπ=l3π02π-π/2+π/2ut, ψ·nt, ψdtdψ+l3πγπ2·γπ2γπ202π0rπ/2,ψ rdrdψ+l3π-γ-π2·γ-π2γ-π202π0r-π/2,ψ rdrdψ,
nt, ψ=uψt, ψ×utt, ψ
γt·γt=xtxt+ytyt+ztzt=xt0t xτdτ+yt0t yτdτ+zt0t zτdτ
γt·γt=0txtxτ+ytyτ+ztzτdτ,
V=l3π02π-π/2+π/2ut, ψ·uψt, ψ×utt, ψdtdψ+l3 a2σ2+10+π/2xπ2xτ+yπ2yτ+zπ2zτdτ+-π/20x-π2xτ+y-π2yτ+z-π2zτdτ,
ut, ψ=xt+r cos θ cos ϕ cos ψ-r sin ϕ sin ψyt+r cos θ sin ϕ cos ψ+r cos ϕ sin ψzt-r sin θ cos ψ,  uψt, ψ=rψ cos θ cos ϕ cos ψ-rψ sin ϕ sin ψ-r cos θ cos ϕ sin ψ-r sin ϕ cos ψrψ cos θ sin ϕ cos ψ+rψ cos ϕ sin ψ-r cos θ sin ϕ sin ψ+r cos ϕ cos ψrψ sin θ cos ϕ+r sin θ sin ψ,  utt, ψ=xt+rt cos θ cos ϕ cos ψ-rt sin ϕ sin ψyt+rt cos θ sin ϕ cos ψ+rt cos ϕ sin ψzt-rt sin θ cos ψ+-rθ sin θ cos ϕ cos ψ-rϕ cos θ sin ϕ cos ψ-rϕ cos ϕ sin ψ-rθ sin θ sin ϕ cos ψ+rϕ cos θ cos ϕ cos ψ-rϕ sin ϕ sin ψ-rθ cos θ cos ψ.
u·uψ×ut=-r2xθ cos θ cos ϕ cos2 ψ+r2×xϕ sin θ sin ϕ sin2 ψ+r2×cos2 θ cos2 ϕ cos2 ψ+r2×sin2 ϕ sin2 ψ + -r2×yθ cos θ sin ϕ cos2 ψ-r2×yϕ sin θ cos ϕ sin2 ψ+r2×z cos θ sin2 ϕ cos2 ψ+r2×z cos θ cos2 ϕ sin2 ψ+r2×x sin θ cos ϕ sin2 ψ+r2×zθ sin θ cos2 ψ+r2×sin2 θ cos2 ψ.
V=Va=23 lπa2σ2+1+16 la2σ2+10+π/2exp-12σb2+σe2sin τ-12+σc2+σf2cos2 τ+exp-124σa2+σb2sin τ+12+σc2 cos2 τ+σe2sin τ-12+σf2 cos2 τ+exp-12σb2sin τ-12+σc2 cos2 τ-exp-124σa2+σb2sin τ+12+σc2 cos2 τdτ+-π/20exp-12σb2+σe2sin τ+12+σc2+σf2cos2 τ+exp-124σa2+σb2sin τ-12+σc2 cos2 τ+σe2sin τ+12+σf2 cos2 τ+exp-12σb2sin τ+12+σc2 cos2 τ-exp-124σa2+σb2sin τ-12+σc2 cos2 τdτ+16 la2σ2+1t=-π/2+π/2τ=0tσb2-σe2sin τ cos t+σf2-σc2cos τ sin t+σb2+σe2-σc2-σf2×sin t cos texp-124σa2+σb2sin τ+sin t2+σc2cos τ+cos t2+σe2sin τ-sin t2+σf2cos τ-cos t2+σc2+σf2cos τ sin t-σb2+σe2sin τ cos t+σb2+σe2-σc2-σf2sin t cos t×exp-12σb2+σe2sin τ-sin t2+σc2+σf2cos τ-cos t2+-σb2 sin τ cos t+σc2 cos τ sin t + σc2-σb2sin t cos texp-124σa2+σb2sin τ+sin t2+σc2cos τ+cos t2+σc2 cos τ sin t-σb2 sin τ cos t+σb2-σc2sin t cos texp-12σb2sin τ-sin t2+σc2cos τ-cos t2dtdτ.
Vlπa2σ2+1.
Vlπa2σ2+1-16 la2σ2+1×32 π-4σa2σe2+12 πσa2σf2+3532 π-103σb4+σe4+516 π-23σb2σc2+σc2σe2+σe2σf2+1916 π-103σb2σe2+516 πσb2σf2+332 πσc4+σf4+316 πσc2σf2.
Wp=pAp0+V,
qt, ψ=1σ2+1 expst, ψ
Ap0t=-π/2+π/2aqt, ψ++aqt, ψ-ds+-π/2+π/212 a2q2t, ψ+-q2t, ψ-2αtdt,
γt·γt+Δt=1-αtΔt2+OtΔt3
ds=lπ1-p0Tγt21/2dt.
Ap0-π/2+π/2 aqt, ψ++qt, ψ-lπ×1-p0Tγt21/2dt+12 a2-π/2+π/2q2t, ψ+-q2t, ψ-×2αt1/2 dt.
Ap02a lπ ησkl-π/2+π/2 1-p0Tγt21/2dt=2a lπ ησklμσa,  , σf, p0.
1-p0Tγt21/2=xt2+yt21/2=sin θt=exp-12σa2+σb2 sin2 t+σc2 cos2 t.
μσa, , σf, p0=-π/2+π/21-p0Tγt21/2dt=-π/2+π/2 1-x2dt-π/2+π/21-12 x2-18 x4dt,
μσa, , σf, p0=π-p0x28π+I2, 0+I0, 2J1+I2, 2J1-p0y28π+I2, 0-I0, 2J1-I2, 2J1-p0z24π-I2, 0-p0xp0y4I0, 2J2+I2, 2J2-p0x45129π+12I2, 0+3I8, 0+12I0, 2J1+16I2, 2J1+4I8, 2J1+3I0, 8J3+4I2, 8J3+I8, 8J3-p0x3p0y128×6I0, 2J2+8I2, 2J2+2I8, 2J2+3I0, 8J4+4I2, 8J4+I8, 8J4-3p0x2p0y2256×3π+4I2, 0+I8, 0-3I0, 8J3-4I2, 8J3-I8, 8J3-p0xp0y31286I0, 2J2-8I2, 2J2+2I8, 2J2-3I0, 8J4-4I2, 8J4-I8, 8J4-p0y45129π+12I2, 0+3I8, 0-12I0, 2J1-16I2, 2J1-4I8, 2J1+3I0, 8J3+4I2, 8J3+I8, 8J3-p0x2p0z264π-I8, 0+I0, 2J1-I8, 2J1-3p0xp0yp0z232I0, 2J2-I8, 2J2-3p0y2p0z264π-I8, 0-I0, 2J1+I8, 2J1-p0z4643π-4I2, 0+I8, 0,
Iα, β=-π/2+π/2 exp-ασ1t-βσ2tdt,  σ1t=σa2+σb2 sin2 t+σc2 cos2 t,  σ2t=σe2 sin2 t+σf2 cos2 t,  J1=0π cos 2thtdt,  J2=0π sin 2thtdt,  J3=0π cos 4thtdt,  J4=0π sin 4thtdt.
Wp=pAp0+V2pa lπ ησklμσa, , σf, p0+V.

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