Abstract

This paper demonstrates that the scattering cross section per unit length of randomly oriented linear chains of optically soft spheres asymptotically converges toward those of randomly oriented and infinitely long cylinders with volume-equivalent diameter as the number of spheres increases. The critical number of spheres necessary to approximate the linear chains of spheres as infinitely long cylinders decreased rapidly as the size parameter of an individual sphere increased from 0.01 to 10. On the other hand, their absorption cross section per unit length was identical to that of an infinitely long volume-equivalent cylinder for any number of spheres. However, this approximation does not apply to the angle-dependent normalized Stokes scattering matrix element ratios.

© 2013 Optical Society of America

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  46. J. F. Cornet and C. G. Dussap, “A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors,” Biotechnol. Progress 25, 424–435 (2009).
    [CrossRef]
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2013 (4)

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transfer 123, 17–22 (2013).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transfer 116, 169–183 (2013).
[CrossRef]

E. Lee, R.-L. Heng, and L. Pilon, “Spectral optical properties of selected photosynthetic microalgae producing biofuels,” J. Quant. Spectrosc. Radiat. Transfer 114, 122–135 (2013).
[CrossRef]

N. F. Bunkin, A. V. Shkirin, N. V. Suyazov, and A. V. Starosvetskiy, “Calculations of light scattering matrices for stochastic ensembles of nanosphere clusters,” J. Quant. Spectrosc. Radiat. Transfer 123, 23–29 (2013).
[CrossRef]

2012 (1)

2011 (2)

L. Pilon, H. Berberoğlu, and R. Kandilian, “Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae,” J. Quant. Spectrosc. Radiat. Transfer 112, 2639–2660 (2011).
[CrossRef]

D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transfer 112, 2182–2192 (2011).
[CrossRef]

2009 (1)

J. F. Cornet and C. G. Dussap, “A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors,” Biotechnol. Progress 25, 424–435 (2009).
[CrossRef]

2006 (1)

D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quant. Spectrosc. Radiat. Transfer 100, 237–249 (2006).
[CrossRef]

2005 (1)

L. Liu and M. I. Mishchenko, “Effects of aggregation on scattering and radiative properties of soot aerosols,” J. Geophys. Res. Atmos. 110, D11211 (2005).
[CrossRef]

2004 (1)

2002 (2)

2001 (1)

2000 (1)

J. R. Benemann, “Hydrogen production by microalgae,” J. Appl. Phycol. 12, 291–300 (2000).
[CrossRef]

1998 (2)

S. Manickavasagam and M. P. Mengüc, “Scattering-matrix elements of coated infinite-length cylinders,” Appl. Opt. 37, 2473–2482 (1998).
[CrossRef]

J. W. Hovenier and D. W. Mackowski, “Symmetry relations for forward and backward scattering by randomly oriented particles,” J. Quant. Spectrosc. Radiat. Transfer 60, 483–492 (1998).
[CrossRef]

1997 (1)

1996 (4)

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

P. Yang and K. N. Liou, “Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space,” J. Opt. Soc. Am. A 13, 2072–2085 (1996).
[CrossRef]

M. I. Mishchenko and D. W. Mackowski, “Electromagnetic scattering by randomly oriented bispheres: comparison of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
[CrossRef]

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

1995 (3)

M. I. Mishchenko, D. W. Mackowski, and L. D. Travis, “Scattering of light by bispheres with touching and separated components,” Appl. Opt. 34, 4589–4599 (1995).
[CrossRef]

M. I. Mishchenko and J. W. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett. 20, 1356–1358 (1995).
[CrossRef]

J.-F. Cornet, C. G. Dussap, J. B. Gross, C. Binois, and C. Lasseur, “A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors,” Chem. Eng. Sci. 50, 1489–1500 (1995).
[CrossRef]

1994 (4)

1992 (1)

H. A. Yousif and E. Boutros, “A FORTRAN code for the scattering of EM plane waves by an infinitely long cylinder at oblique incidence,” Comput. Phys. Commun. 69, 406–414 (1992).
[CrossRef]

1991 (2)

M. F. Iskander, H. Y. Chen, and J. E. Penner, “Resonance optical absorption by fractal agglomerates of smoke aerosols,” Atmos. Environ. A 25, 2563–2569 (1991).
[CrossRef]

R. A. West, “Optical properties of aggregate particles whose outer diameter is comparable to the wavelength,” Appl. Opt. 30, 5316–5324 (1991).
[CrossRef]

1990 (1)

S. C. Lee, “Scattering phase function for fibrous media,” Int. J. Heat Mass Transfer 33, 2183–2190 (1990).
[CrossRef]

1989 (1)

1988 (2)

S. C. Lee, “Radiation heat-transfer model for fibers oriented parallel to diffuse boundaries,” J. Thermophys. Heat Transfer 2, 303–308 (1988).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1986 (2)

S. C. Lee, “Radiative transfer through a fibrous medium: allowance for fiber orientation,” J. Quant. Spectrosc. Radiat. Transfer 36, 253–263 (1986).
[CrossRef]

J. W. Hovenier, H. C. van de Hulst, and C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

1980 (1)

T. W. Tong and C. L. Tien, “Analytical models for thermal radiation in fibrous insulations,” J. Build. Phys. 4, 27–44 (1980).
[CrossRef]

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1966 (1)

A. C. Lind and J. M. Greenberg, “Electromagnetic scattering by obliquely oriented cylinders,” J. Appl. Phys. 37, 3195–3203 (1966).
[CrossRef]

Bai, L.

Benemann, J. R.

J. R. Benemann, “Hydrogen production by microalgae,” J. Appl. Phycol. 12, 291–300 (2000).
[CrossRef]

Berberoglu, H.

L. Pilon, H. Berberoğlu, and R. Kandilian, “Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae,” J. Quant. Spectrosc. Radiat. Transfer 112, 2639–2660 (2011).
[CrossRef]

Bi, L.

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transfer 116, 169–183 (2013).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transfer 123, 17–22 (2013).
[CrossRef]

Binois, C.

J.-F. Cornet, C. G. Dussap, J. B. Gross, C. Binois, and C. Lasseur, “A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors,” Chem. Eng. Sci. 50, 1489–1500 (1995).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Boutros, E.

H. A. Yousif and E. Boutros, “A FORTRAN code for the scattering of EM plane waves by an infinitely long cylinder at oblique incidence,” Comput. Phys. Commun. 69, 406–414 (1992).
[CrossRef]

Bricaud, A.

Bunkin, N. F.

N. F. Bunkin, A. V. Shkirin, N. V. Suyazov, and A. V. Starosvetskiy, “Calculations of light scattering matrices for stochastic ensembles of nanosphere clusters,” J. Quant. Spectrosc. Radiat. Transfer 123, 23–29 (2013).
[CrossRef]

Chen, H. Y.

M. F. Iskander, H. Y. Chen, and J. E. Penner, “Resonance optical absorption by fractal agglomerates of smoke aerosols,” Atmos. Environ. A 25, 2563–2569 (1991).
[CrossRef]

M. F. Iskander, H. Y. Chen, and J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
[CrossRef]

Cornet, J. F.

J. F. Cornet and C. G. Dussap, “A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors,” Biotechnol. Progress 25, 424–435 (2009).
[CrossRef]

Cornet, J.-F.

J.-F. Cornet, C. G. Dussap, J. B. Gross, C. Binois, and C. Lasseur, “A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors,” Chem. Eng. Sci. 50, 1489–1500 (1995).
[CrossRef]

Draine, B. T.

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Draper, N. R.

N. R. Draper and H. Smith, Applied Regression Analysis, 3rd ed. (Wiley, 1998).

Dussap, C. G.

J. F. Cornet and C. G. Dussap, “A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors,” Biotechnol. Progress 25, 424–435 (2009).
[CrossRef]

J.-F. Cornet, C. G. Dussap, J. B. Gross, C. Binois, and C. Lasseur, “A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors,” Chem. Eng. Sci. 50, 1489–1500 (1995).
[CrossRef]

Flatau, P. J.

Fournier, G. R.

M. Jonasz and G. R. Fournier, Light Scattering by Particles in Water: Theoretical and Experimental Foundations (Academic, 2007).

Greenberg, J. M.

A. C. Lind and J. M. Greenberg, “Electromagnetic scattering by obliquely oriented cylinders,” J. Appl. Phys. 37, 3195–3203 (1966).
[CrossRef]

Gross, J. B.

J.-F. Cornet, C. G. Dussap, J. B. Gross, C. Binois, and C. Lasseur, “A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors,” Chem. Eng. Sci. 50, 1489–1500 (1995).
[CrossRef]

Heng, R.-L.

E. Lee, R.-L. Heng, and L. Pilon, “Spectral optical properties of selected photosynthetic microalgae producing biofuels,” J. Quant. Spectrosc. Radiat. Transfer 114, 122–135 (2013).
[CrossRef]

Hovenier, J. W.

J. W. Hovenier and D. W. Mackowski, “Symmetry relations for forward and backward scattering by randomly oriented particles,” J. Quant. Spectrosc. Radiat. Transfer 60, 483–492 (1998).
[CrossRef]

M. I. Mishchenko and J. W. Hovenier, “Depolarization of light backscattered by randomly oriented nonspherical particles,” Opt. Lett. 20, 1356–1358 (1995).
[CrossRef]

J. W. Hovenier, H. C. van de Hulst, and C. V. M. van der Mee, “Conditions for the elements of the scattering matrix,” Astron. Astrophys. 157, 301–310 (1986).

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).

Iskander, M. F.

M. F. Iskander, H. Y. Chen, and J. E. Penner, “Resonance optical absorption by fractal agglomerates of smoke aerosols,” Atmos. Environ. A 25, 2563–2569 (1991).
[CrossRef]

M. F. Iskander, H. Y. Chen, and J. E. Penner, “Optical scattering and absorption by branched chains of aerosols,” Appl. Opt. 28, 3083–3091 (1989).
[CrossRef]

Jonasz, M.

M. Jonasz and G. R. Fournier, Light Scattering by Particles in Water: Theoretical and Experimental Foundations (Academic, 2007).

Kahnert, F. M.

Kandilian, R.

L. Pilon, H. Berberoğlu, and R. Kandilian, “Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae,” J. Quant. Spectrosc. Radiat. Transfer 112, 2639–2660 (2011).
[CrossRef]

Kattawar, G. W.

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transfer 116, 169–183 (2013).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transfer 123, 17–22 (2013).
[CrossRef]

P. Yang, G. W. Kattawar, and W. J. Wiscombe, “Effect of particle asphericity on single-scattering parameters: comparison between platonic solids and spheres,” Appl. Opt. 43, 4427–4435 (2004).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light, and Other Electromagnetic Radiation (Academic, 1969).

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2001).

Lasseur, C.

J.-F. Cornet, C. G. Dussap, J. B. Gross, C. Binois, and C. Lasseur, “A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors,” Chem. Eng. Sci. 50, 1489–1500 (1995).
[CrossRef]

Lee, E.

E. Lee, R.-L. Heng, and L. Pilon, “Spectral optical properties of selected photosynthetic microalgae producing biofuels,” J. Quant. Spectrosc. Radiat. Transfer 114, 122–135 (2013).
[CrossRef]

Lee, S. C.

S. C. Lee, “Scattering phase function for fibrous media,” Int. J. Heat Mass Transfer 33, 2183–2190 (1990).
[CrossRef]

S. C. Lee, “Radiation heat-transfer model for fibers oriented parallel to diffuse boundaries,” J. Thermophys. Heat Transfer 2, 303–308 (1988).
[CrossRef]

S. C. Lee, “Radiative transfer through a fibrous medium: allowance for fiber orientation,” J. Quant. Spectrosc. Radiat. Transfer 36, 253–263 (1986).
[CrossRef]

Li, H.-Y.

Li, Z.-J.

Lind, A. C.

A. C. Lind and J. M. Greenberg, “Electromagnetic scattering by obliquely oriented cylinders,” J. Appl. Phys. 37, 3195–3203 (1966).
[CrossRef]

Liou, K. N.

Liu, L.

L. Liu and M. I. Mishchenko, “Effects of aggregation on scattering and radiative properties of soot aerosols,” J. Geophys. Res. Atmos. 110, D11211 (2005).
[CrossRef]

Mackowski, D. W.

D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transfer 112, 2182–2192 (2011).
[CrossRef]

D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quant. Spectrosc. Radiat. Transfer 100, 237–249 (2006).
[CrossRef]

D. W. Mackowski, “Discrete dipole moment method for calculation of the T matrix for nonspherical particles,” J. Opt. Soc. Am. A 19, 881–893 (2002).
[CrossRef]

J. W. Hovenier and D. W. Mackowski, “Symmetry relations for forward and backward scattering by randomly oriented particles,” J. Quant. Spectrosc. Radiat. Transfer 60, 483–492 (1998).
[CrossRef]

M. I. Mishchenko and D. W. Mackowski, “Electromagnetic scattering by randomly oriented bispheres: comparison of theory and experiment and benchmark calculations,” J. Quant. Spectrosc. Radiat. Transfer 55, 683–694 (1996).
[CrossRef]

D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical particles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55, 535–575 (1996).
[CrossRef]

M. I. Mishchenko, D. W. Mackowski, and L. D. Travis, “Scattering of light by bispheres with touching and separated components,” Appl. Opt. 34, 4589–4599 (1995).
[CrossRef]

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Appl. Opt. (8)

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J. Quant. Spectrosc. Radiat. Transfer (11)

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “A numerical combination of extended boundary condition method and invariant imbedding method applied to light scattering by large spheroids and cylinders,” J. Quant. Spectrosc. Radiat. Transfer 123, 17–22 (2013).
[CrossRef]

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[CrossRef]

D. W. Mackowski, “A simplified model to predict the effects of aggregation on the absorption properties of soot particles,” J. Quant. Spectrosc. Radiat. Transfer 100, 237–249 (2006).
[CrossRef]

D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transfer 112, 2182–2192 (2011).
[CrossRef]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transfer 116, 169–183 (2013).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Other (10)

M. I. Mishchenko, J. W. Hovenier, and L. D. Travis, Light Scattering by Nonspherical Particles (Academic, 2000).

M. Kerker, The Scattering of Light, and Other Electromagnetic Radiation (Academic, 1969).

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K. N. Liou, An Introduction to Atmospheric Radiation, 2nd ed. (Academic, 2002).

M. Jonasz and G. R. Fournier, Light Scattering by Particles in Water: Theoretical and Experimental Foundations (Academic, 2007).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2001).

M. T. Madigan and J. M. Martinko, Biology of Microorganisms (Pearson Prentice Hall, 2006).

L. J. Stal, “Cyanobacteria,” in Algae and Cyanobacteria in Extreme Environments, J. Seckbach, ed., Vol. 11 of Cellular Origin, Life in Extreme Habitats and Astrobiology (Springer, 2007), pp. 659–680.

M. F. Modest, Radiative Heat Transfer (Academic, 2003).

N. R. Draper and H. Smith, Applied Regression Analysis, 3rd ed. (Wiley, 1998).

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Figures (7)

Fig. 1.
Fig. 1.

Micrographs of filamentous cyanobacteria: (a), (b) Nostoc pondiforme, (c) Anabaena sp., and (d) Anabaena iyengari. Reproduced with permission from (a) Isao Inouye (University of Tsukuba), Mark Schneegurt (Wichita State University), and Cyanosite (www-cyanosite.bio.purdue.edu); (b) Prof. Ann Magnuson (Uppsala University); and (c), (d) Prof. Yuuji Tsukii (Hosei University, http://protist.i.hosei.ac.jp/).

Fig. 2.
Fig. 2.

Schematic and coordinate system associated with absorption and scattering of incident radiation at incident angle of ϕ by (a) linear chain of Ns spheres of diameter ds with complex index of refraction m2=n2+ik2 in a nonabsorbing medium of m1=n1, and (b) infinitely long cylinder of diameter dc [37].

Fig. 3.
Fig. 3.

Absorption cross section Cabs(m,χs) per unit length (in m2/m) for randomly oriented linear chains of monodisperse spheres as a function of the number of spheres Ns and for randomly oriented and infinitely long equivalent cylinders of diameter dc,eq,S and dc,eq,V for size parameter χs=0.01, 0.1, 1.0, and 10.0 and m=1.0165+i0.003.

Fig. 4.
Fig. 4.

Scattering cross section Csca(m,χs) per unit length (in m2/m) for randomly oriented linear chains of monodisperse spheres as a function of the number of spheres Ns and for randomly oriented and infinitely long equivalent cylinders of diameter dc,eq,S and dc,eq,V for size parameter χs=0.01, 0.1, 1.0, and 10.0 and m=1.0165+i0.003.

Fig. 5.
Fig. 5.

Critical number of spheres Ns,cr beyond which the scattering cross section Csca(m,χs) of randomly oriented linear chains of spheres can be approximated as that of a randomly oriented and infinitely long volume-equivalent cylinder as a function of size parameter χs.

Fig. 6.
Fig. 6.

Scattering phase functions F11(Θ) of randomly oriented long linear chains consisting of Ns,cr monodisperse spheres or diameter ds and randomly oriented and infinitely long volume-equivalent cylinders of diameter dc,eq,V as a function of scattering angle for χs=0.01, 0.1, 1.0, and 10.0 and m=1.0165+i0.003.

Fig. 7.
Fig. 7.

Scattering matrix element ratios (a) F12(Θ)/F11(Θ), (b) F22(Θ)/F11(Θ), (c) F33(Θ)/F11(Θ), (d) F44(Θ)/F11(Θ), and (e) F34(Θ)/F11(Θ) as a function of scattering angle Θ for randomly oriented linear chains of spheres consisting of Ns,cr monodisperse spheres of diameter ds and of infinitely long cylinders of volume-equivalent diameter as function of scattering angle for χs=0.01, 0.1, 1.0, and 10 and m=1.0165+i0.003.

Tables (2)

Tables Icon

Table 1. Comparison between Selected Scattering Properties of Randomly Oriented Long Linear Chains of Spheres with Size Parameter χs Equal to 0.01, 0.1, 1.0, and 10.0 and their Randomly Oriented and Infinitely Long Volume-Equivalent Cylindersa

Tables Icon

Table 2. Comparison of the Absorption and Scattering Cross Sections Cabs,s and Csca,s of Two Randomly Oriented Linear Chains of Spheres (a) with Representative Arbitrary Diameter Distribution and (b) with Monodisperse Spheres with the Corresponding Average Diameter d¯sa

Equations (12)

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

Isca(r,s^)=1r2[Z(Θ)]Iinc(r,s^i),
[F(Θ)]=4πCsca[Z(Θ)],
[F(Θ)]=[F11(Θ)F12(Θ)00F12(Θ)F22(Θ)0000F33(Θ)F34(Θ)00F34(Θ)F44(Θ)].
14π4πF11(Θ)dΩ=1,
g=14π4πF11(Θ)cosΘdΩ.
δL=[F11(180°)F22(180°)][F11(180°)+F22(180°)]
δC=[F11(180°)+F44(180°)][F11(180°)F44(180°)].
Cabs/sca,s=πds,eq,V24Qabs/sca,s,
Cext/sca,c(m,χc,ϕ)=2dcQext/sca,c(m,χc,ϕ),
Cabs/sca,c(m,χc)=0π/2Cabs/sca,c(m,χc,ϕ)cosϕdϕ.
Cabs/sca,s=Cabs/sca,sNsds.
Ns,cr=Kχsp,

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