Abstract

A rapid tool for the characterization of submicron particles is light spectroscopy. Rayleigh-Debye-Gans and Mie theories provide light scattering solutions that can be evaluated within the time constants required for continuous real time monitoring applications, as in characterization of biological particles. A multiwavelength assessment of Rayleigh-Debye-Gans theory for spheres was conducted over the UV-Vis wavelength range where strict adherence to the limits of the theory at a single wavelength could not be met. Reported corrections to the refractive indices were developed to extend the range of application of the Rayleigh-Debye-Gans approximation. The results of this study show that there is considerable disagreement between Rayleigh-Debye-Gans and Mie theory across the UV-Vis spectrum.

© 2006 Optical Society of America

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References

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  1. C. E. Alupoaei, J. A. Olivares, and L. H. Garcia-Rubio, "Quantitative spectroscopy analysis of prokaryotic cells: vegetative cells and spores," Biosens. Bioelectron. 19,893-903 (2003).
    [CrossRef]
  2. A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
    [CrossRef]
  3. 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. Transf. 55, 535-575 (1996).
  4. A. L. Koch, B. R. Robertson, and D. K. Button, "Deduction of cell volume and mass from forward scatter intensity from bacteria analyzed by flow cytometry," J. Microbiol. Methods. 2, 40-61 (1996).
  5. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science Paper Series, New York, 1998).
    [CrossRef]
  6. M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic Press, New York, 1969).
  7. M. Hammer, D. Schweitzerk, B. Michel, E. Thamm, and A. Kolb, " Single scattering by red blood cells," Appl. Opt. 37, 7410-7418 (1998).
    [CrossRef]
  8. A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)
  9. B. P. Latimer, "Scattering by Ellipsoids of Revolution; A Comparison of Theoretical Methods," J. Colloid Interface Science 63, 310-316 (1977).
    [CrossRef]
  10. N. L. Veshkin, "Screening Hypochromism of Biological Macromolecules and Suspensions," J. Photochem. Photobiol. B,  3, 625-630 (1989).
    [CrossRef]
  11. N. L. Veshkin, "Screening Hypochromism of Molecular Aggregates and Biopolymers," J. Biol. Phys. 25, 339-354 (1999).
    [CrossRef]
  12. N. L. Veshkin, "Screening Hypochromism of Chromophores in Macromolecular Biostructures," Biophys. J. 44, 41-51 (1999)
  13. P. J. Wyatt, and D. T. Phillips, "Structure of single bacteria from light scattering," J. Theor. Biol. 37, 493-501 (1972).
    [CrossRef]
  14. L. H. Garcia-Rubio, Private Communication.
  15. W. J. Wiscombe, "Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes," NCAR/TN-140 + STR. National Center for Atmospheric Research, Boulder Colorado (1979).
  16. A. C. Garcia-LopezHybrid Model for Characterization of Submicron Particles using Mulitwavelength Spectroscopy, (University of South Florida, 2005).
  17. P. Latimer, A. Brunsting, B. E. Pyle and C. Moor, "Effects of asphericity on single particle scattering," Appl. Opt. 17, 3152-3158 (1978).
    [CrossRef] [PubMed]
  18. A. Nonoyama, Using multiwavelength UV-Visible spectroscopy for the characterization of red blood cells: an investigation of hypochromism. (University of South Florida, 2004).
  19. S. Narayanan, Aggregation and Structural Changes in Biological Systems: An Ultraviolet Visible Spectroscopic Approach for Analysis of Blood Cell Aggregation and Protein Conformation. (University of South Florida, 1999).
    [PubMed]

2003

C. E. Alupoaei, J. A. Olivares, and L. H. Garcia-Rubio, "Quantitative spectroscopy analysis of prokaryotic cells: vegetative cells and spores," Biosens. Bioelectron. 19,893-903 (2003).
[CrossRef]

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

1999

N. L. Veshkin, "Screening Hypochromism of Molecular Aggregates and Biopolymers," J. Biol. Phys. 25, 339-354 (1999).
[CrossRef]

N. L. Veshkin, "Screening Hypochromism of Chromophores in Macromolecular Biostructures," Biophys. J. 44, 41-51 (1999)

1998

1996

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. Transf. 55, 535-575 (1996).

A. L. Koch, B. R. Robertson, and D. K. Button, "Deduction of cell volume and mass from forward scatter intensity from bacteria analyzed by flow cytometry," J. Microbiol. Methods. 2, 40-61 (1996).

1989

N. L. Veshkin, "Screening Hypochromism of Biological Macromolecules and Suspensions," J. Photochem. Photobiol. B,  3, 625-630 (1989).
[CrossRef]

1978

1977

B. P. Latimer, "Scattering by Ellipsoids of Revolution; A Comparison of Theoretical Methods," J. Colloid Interface Science 63, 310-316 (1977).
[CrossRef]

1972

P. J. Wyatt, and D. T. Phillips, "Structure of single bacteria from light scattering," J. Theor. Biol. 37, 493-501 (1972).
[CrossRef]

Alfano, R. R.

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

Alimova, A.

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

Alupoaei, C. E.

C. E. Alupoaei, J. A. Olivares, and L. H. Garcia-Rubio, "Quantitative spectroscopy analysis of prokaryotic cells: vegetative cells and spores," Biosens. Bioelectron. 19,893-903 (2003).
[CrossRef]

Brunsting, A.

Button, D. K.

A. L. Koch, B. R. Robertson, and D. K. Button, "Deduction of cell volume and mass from forward scatter intensity from bacteria analyzed by flow cytometry," J. Microbiol. Methods. 2, 40-61 (1996).

Garcia-Rubio, L. H.

C. E. Alupoaei, J. A. Olivares, and L. H. Garcia-Rubio, "Quantitative spectroscopy analysis of prokaryotic cells: vegetative cells and spores," Biosens. Bioelectron. 19,893-903 (2003).
[CrossRef]

Hammer, M.

Katz, A.

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

Koch, A. L.

A. L. Koch, B. R. Robertson, and D. K. Button, "Deduction of cell volume and mass from forward scatter intensity from bacteria analyzed by flow cytometry," J. Microbiol. Methods. 2, 40-61 (1996).

Kolb, A.

Latimer, B. P.

B. P. Latimer, "Scattering by Ellipsoids of Revolution; A Comparison of Theoretical Methods," J. Colloid Interface Science 63, 310-316 (1977).
[CrossRef]

Latimer, P.

Mackowski, D. W.

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. Transf. 55, 535-575 (1996).

McCormick, S. A.

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

Michel, B.

Mishchenko, M. I.

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. Transf. 55, 535-575 (1996).

Moor, C.

Olivares, J. A.

C. E. Alupoaei, J. A. Olivares, and L. H. Garcia-Rubio, "Quantitative spectroscopy analysis of prokaryotic cells: vegetative cells and spores," Biosens. Bioelectron. 19,893-903 (2003).
[CrossRef]

Phillips, D. T.

P. J. Wyatt, and D. T. Phillips, "Structure of single bacteria from light scattering," J. Theor. Biol. 37, 493-501 (1972).
[CrossRef]

Pyle, B. E.

Robertson, B. R.

A. L. Koch, B. R. Robertson, and D. K. Button, "Deduction of cell volume and mass from forward scatter intensity from bacteria analyzed by flow cytometry," J. Microbiol. Methods. 2, 40-61 (1996).

Rosen, R. B.

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

Rudolf, E.

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

Savage, H. E.

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

Schweitzerk, D.

Shah, M. K.

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

Thamm, E.

Travis, L. D.

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. Transf. 55, 535-575 (1996).

Veshkin, N. L.

N. L. Veshkin, "Screening Hypochromism of Molecular Aggregates and Biopolymers," J. Biol. Phys. 25, 339-354 (1999).
[CrossRef]

N. L. Veshkin, "Screening Hypochromism of Chromophores in Macromolecular Biostructures," Biophys. J. 44, 41-51 (1999)

N. L. Veshkin, "Screening Hypochromism of Biological Macromolecules and Suspensions," J. Photochem. Photobiol. B,  3, 625-630 (1989).
[CrossRef]

Wyatt, P. J.

P. J. Wyatt, and D. T. Phillips, "Structure of single bacteria from light scattering," J. Theor. Biol. 37, 493-501 (1972).
[CrossRef]

Xu, M.

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

Appl. Opt.

Biophys. J.

N. L. Veshkin, "Screening Hypochromism of Chromophores in Macromolecular Biostructures," Biophys. J. 44, 41-51 (1999)

Biosens. Bioelectron.

C. E. Alupoaei, J. A. Olivares, and L. H. Garcia-Rubio, "Quantitative spectroscopy analysis of prokaryotic cells: vegetative cells and spores," Biosens. Bioelectron. 19,893-903 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "Bacteria size determination by elastic light scattering," IEEE J. Sel. Top. Quantum Electron. 9, 277-287 (2003).
[CrossRef]

J. Biol. Phys.

N. L. Veshkin, "Screening Hypochromism of Molecular Aggregates and Biopolymers," J. Biol. Phys. 25, 339-354 (1999).
[CrossRef]

J. Colloid Interface Science

B. P. Latimer, "Scattering by Ellipsoids of Revolution; A Comparison of Theoretical Methods," J. Colloid Interface Science 63, 310-316 (1977).
[CrossRef]

J. Microbiol. Methods.

A. L. Koch, B. R. Robertson, and D. K. Button, "Deduction of cell volume and mass from forward scatter intensity from bacteria analyzed by flow cytometry," J. Microbiol. Methods. 2, 40-61 (1996).

J. Photochem. Photobiol. B

N. L. Veshkin, "Screening Hypochromism of Biological Macromolecules and Suspensions," J. Photochem. Photobiol. B,  3, 625-630 (1989).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

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. Transf. 55, 535-575 (1996).

J. Theor. Biol.

P. J. Wyatt, and D. T. Phillips, "Structure of single bacteria from light scattering," J. Theor. Biol. 37, 493-501 (1972).
[CrossRef]

Proc. SPIE

A. Katz, A. Alimova, M. Xu, E. Rudolf, M. K. Shah, H. E. Savage, R. B. Rosen, S. A. McCormick, and R. R. Alfano, "In situ identification of bacteria size by light scattering," Proc. SPIE 4965, 7376 (2003)

Other

C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science Paper Series, New York, 1998).
[CrossRef]

M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic Press, New York, 1969).

L. H. Garcia-Rubio, Private Communication.

W. J. Wiscombe, "Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes," NCAR/TN-140 + STR. National Center for Atmospheric Research, Boulder Colorado (1979).

A. C. Garcia-LopezHybrid Model for Characterization of Submicron Particles using Mulitwavelength Spectroscopy, (University of South Florida, 2005).

A. Nonoyama, Using multiwavelength UV-Visible spectroscopy for the characterization of red blood cells: an investigation of hypochromism. (University of South Florida, 2004).

S. Narayanan, Aggregation and Structural Changes in Biological Systems: An Ultraviolet Visible Spectroscopic Approach for Analysis of Blood Cell Aggregation and Protein Conformation. (University of South Florida, 1999).
[PubMed]

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Figures (23)

Fig. 1.
Fig. 1.

The attenuation of light due to absorption through a particle shown schematically for Mie theory and RDG approximation.

Fig. 2.
Fig. 2.

Schematic of chromophoric groups per particle size per wavelength within a particle.

Fig. 3.
Fig. 3.

Comparison of the calculated transmission of Mie and Rayleigh-Debye-Gans for 25 nm AgBr spheres suspended in water.

Fig. 4.
Fig. 4.

Comparison of the calculated transmission of Mie and Rayleigh-Debye-Gans for 25 nm AgCl spheres suspended in water.

Fig. 5.
Fig. 5.

Comparison of the calculated transmission of Mie and Rayleigh-Debye-Gans for 50 nm AgBr spheres suspended in water. Spectral differences in RDG are visible below 400 nm wavelength compared to those is Fig. 1.

Fig. 6.
Fig. 6.

Comparison of the calculated transmission of Mie and Rayleigh-Debye-Gans for a 50 nm AgCl spheres suspended in water. The spectral differences in RDG are visible below 400 nm wavelength compared to those in Fig. 4.

Fig. 7.
Fig. 7.

Comparison of Mie and Rayleigh-Debye-Gans calculated spectra of 500 nm Soft Body spheres suspended in water. Notice that the spectrumcalculated with Rayleigh-Debye-Gans theory does not approximate Mie theory.

Fig. 8.
Fig. 8.

Calculated transmission of Mie and Rayleigh-Debye-Gans for 1 µm Soft Body spheres suspended in water. Shape of the spectra by both theories remains similar.

Fig. 9.
Fig. 9.

Transmission spectra calculated using Mie and Rayleigh-Debye-Gans for 5.5 µm Soft Body spheres suspended in water. Rayleigh-Debye-Gans spectra does not coincide with Mie theory at such a large particle size.

Fig. 10.
Fig. 10.

Calculated spectra of Mie and Rayleigh-Debye-Gans for 500 nm Hemoglobin spheres with no absorption (κ=0) suspended in water. Note: the shape of the spectra are similar.

Fig. 11.
Fig. 11.

Spectral differences between Mie and Rayleigh-Debye-Gans theory for 1 µm Hemoglobin spheres with no absorption (κ=0) suspended in water. At shorter wavelengths the spectra calculated by Mie theory flattens considerably.

Fig. 12.
Fig. 12.

Semi log plot spectra for 5.5 µm Hemoglobin spheres with no absorption (κ=0) calculated using Mie and Rayleigh-Debye-Gans theory. The limits of RDG have been met for particle size.

Fig. 13.
Fig. 13.

Close approximation of the calculated transmission of Rayleigh-Debye-Gans to Mie theory for 100 nm Hemoglobin spheres suspended in water.

Fig. 14.
Fig. 14.

Comparison of calculated transmission for Rayleigh-Debye-Gans to that of Mie theory shows the divergence of theories for 500 nm Hemoglobin spheres suspended in water. Note the difference of RDG compared to Fig. 10 where no absorption is present.

Fig. 15.
Fig. 15.

The calculated spectra for 1 µm Hemoglobin spheres suspended in water shows that Mie theory appears to flatten when compared to Rayleigh-Debye-Gans.

Fig. 16.
Fig. 16.

Intermediate levels hypochromicity using Veshkin correction for Rayleigh-Debye-Gans compared to no hypochromicity correction in Mie and Rayleigh-Debye-Gans for 1 µm Hemoglobin spheres suspended in water.

Fig. 17.
Fig. 17.

Hypochromicity of 0% and 100% calculated using Mie and Rayleigh-Debye-Gans theory for a 1 µm Hemoglobin spheres suspended in water. Note that at 100% hypochromicity Rayleigh-Debye-Gans does not achieve reduced difference to Mie theory.

Fig. 18.
Fig. 18.

Calculated transmission of Mie and Rayleigh-Debye-Gans for 1 µm Hemoglobin spheres with Veshkin Correction to k c and an n eff calculated through Kramers-Kronig Transform.

Fig. 19.
Fig. 19.

Optical properties of AgBr as function of wavelength

Fig. 20.
Fig. 20.

Optical properties of AgCl as function of wavelength

Fig. 21.
Fig. 21.

Optical properties of Soft bodies as function of wavelength

Fig. 22.
Fig. 22.

Optical properties of Hemoglobin as function of wavelength

Fig. 23.
Fig. 23.

Optical properties of Water as function of wavelength

Tables (1)

Tables Icon

Table 1. Simulation Parameters for Turbidity using Mie and Rayleigh-Debye-Gans Theory.

Equations (31)

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

τ ( λ ) = N p l Q ext = N p l ( Q sca + Q abs )
τ ( λ ) = N p ( 9 k 4 V 2 16 π m 2 1 m 2 + 2 2 0 π P ( θ ) ( 1 + cos 2 θ ) sin θ d θ + 3 k V Im ( m 2 1 m 2 + 2 ) )
P ( θ ) = 9 π J 3 2 2 ( u ) 2 u 3 , u = h a , a = radius
h = 4 π λ sin ( θ 2 )
N = n + i κ
m = N 1 N 0 = n 1 + i κ 1 n 0 + i κ 0
n ( ω ) 1 = 1 π P 0 Ω k ( Ω ) Ω 2 ω 2 d Ω
κ ( ω ) = 2 ω π P 0 n ( Ω , ) Ω 2 ω 2 d Ω
h = 100 % ε ε ̂ ε
E ̂ = s 2.3 q k ( 1 ( 1 2.3 E q s ) k )
ε ̂ m = s 2.3 q k N A [ 1 ( 1 2.3 E q s 6.022 E 4 ) k ]
ε m = ε M w V
k ( λ ) = v f λ d
P = 2.3 q E ˙ s
E ˙ = ε m V ( 1 E 8 ) 2 N A
ε ̂ m = N A s V 1 E 8 2 k E ˙ q [ 1 ( 1 p ) k ]
ε ̂ = ε ̂ m V M w
κ c = ε ̂ λ 4 π
Q abs = 4 k a Im ( m 2 1 m 2 + 2 ) 4 k a Im ( 2 3 ( m 1 ) ) = 8 3 k a κ
Q sca = ( k a ) 4 m 2 1 m 2 + 2 2 0 π f 2 ( θ ) ( 1 cos 2 θ ) sin θ d θ
Q sca ( k a ) 4 Λ 2 3 ( n 1 ) + i 2 3 κ 2 = 4 9 ( k a ) 4 Λ ( ( n 1 ) 2 + κ 2 )
Q ext = 8 3 ( k a ) κ + 4 9 ( k a ) 4 Λ ( ( n 1 ) 2 + κ 2 )
= 4 9 ( k a ) 4 Λ ( n 1 ) 2 + 4 9 ( k a ) 2 Λ κ 2 + 8 3 ( k a ) κ
n eff = Q ko q 3 + 1
κ eff = q 2 ± q 2 2 + 4 q 1 Q no 2 q 1
Q ko = Q Mie ( n , κ ) q 1 κ 2 q 2 κ
Q no = Q Mie ( n , κ ) q 3 ( n 1 ) 2
( n eff 1 ) = ( Q Mie ( n , κ ) Q RDG ( n , κ ) 2 q 3 ( n 1 ) 2 + 1 ) 1 2 ( n 1 )
κ eff = ( q 2 ± ( 2 q 1 κ + q 2 ) 2 + 4 q 1 ( Q Mie ( n , κ ) Q RDG ( n , κ ) ) 2 q 1 )
( 2 q 1 κ + q 2 ) 2 4 q 1 ( Q Mie ( n , κ ) Q RDG ( n , κ ) )
( 2 q 1 κ + q 2 ) 2 + 4 q 1 ( Q Mie ( n , κ ) Q RDG ( n , κ ) ) q 2

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