Abstract

This paper presents one approach to the absorption and scattering of light from aggregates of primary particles. The primary particles are sphere-like and small compared to the wavelength, whereas the aggregate can be large compared to the wavelength. This situation applies to when soot particles formed in flames are measured using methods based on laser light. The method presented in this work, called generalized Rayleigh–Debye–Gans, leads to closed-form expressions for the scattered intensity and the absorbed power of an ensemble of aggregates with random positions and orientations. The expressions ensure a fast and accurate numerical evaluation of the scattering and absorption from ensembles of aggregates. The numerical results are compared with the ones obtained from the T-matrix method and the discrete dipole approximation method.

© 2013 Optical Society of America

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  1. H. Bladh, J. Johnsson, N.-E. Olofsson, A. Bohlin, and P.-E. Bengtsson, “Optical soot characterization using two-color laser-induced incandescence (2c–lii) in the soot growth region of a premixed flat flame,” Proc. Combust. Inst. 33, 641–648 (2011).
    [CrossRef]
  2. T. W. A. Doicu and Y. A. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).
  3. C. M. Sorensen, “Light scattering by fractal aggregate: a review,” Aerosol Sci. Technol. 35, 648–687 (2001).
    [CrossRef]
  4. P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).
  5. B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
    [CrossRef]
  6. L. Liu, M. I. Mishchenko, and W. P. Amott, “A study of radiative properties of soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109, 2656–2663 (2008).
    [CrossRef]
  7. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  8. B. T. Draine and P. J. Flatau, “User guide to the discrete dipole approximation code DDSCAT 7.2,” (2012). http://arxiv.org/abs/1202.3424 .
  9. A. Karlsson, H. Bladh, and P.-E. Bengtsson, “Accurate method for predicting light scattering from soot aggregates with subparticles of arbitrary shape and structure,” J. Opt. Soc. Am. A 26, 1704–1713 (2009).
    [CrossRef]
  10. M. Z. A. V. Filipov and D. E. Rosner, “Fractal-like aggregates: relation between morphology and physical properties,” J. Colloid Interface Sci. 229, 261–273 (2000).
    [CrossRef]
  11. H. W. M. Lattuada and M. Morbidelli, “A simple model for the structure of fractal aggregates,” J. Colloid Interface Sci. 268, 106–120 (2003).
    [CrossRef]
  12. T. T. Charalampopoulos and J. D. Felske, “Refractive indices of soot particles deduced from in-situ laser light scattering measurements,” Combust. Flame 68, 283–294 (1987).
    [CrossRef]
  13. 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]
  14. J. Johnsson, H. Bladh, N.-E. Olofsson, and P.-E. Bengtsson, “Influence of soot aggregate structure on particle sizing using laser-induced incandescence: importance of bridging between primary particles,” Appl. Phys. B (to be published).
  15. F. Liu and G. Smallwood, “Effect of aggregation on the absorption cross-section of fractal soot aggregates and its impact on LII modelling,” J. Quant. Spectrosc. Radiat. Transfer 111, 302–308 (2010).
    [CrossRef]
  16. 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]
  17. 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]
  18. J. Gregory, “Monitoring particle aggregation processes,” Adv. Colloid Interface Sci. 147, 109–123 (2009).
    [CrossRef]

2011

H. Bladh, J. Johnsson, N.-E. Olofsson, A. Bohlin, and P.-E. Bengtsson, “Optical soot characterization using two-color laser-induced incandescence (2c–lii) in the soot growth region of a premixed flat flame,” Proc. Combust. Inst. 33, 641–648 (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]

2010

F. Liu and G. Smallwood, “Effect of aggregation on the absorption cross-section of fractal soot aggregates and its impact on LII modelling,” J. Quant. Spectrosc. Radiat. Transfer 111, 302–308 (2010).
[CrossRef]

2009

2008

L. Liu, M. I. Mishchenko, and W. P. Amott, “A study of radiative properties of soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109, 2656–2663 (2008).
[CrossRef]

2006

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]

2003

H. W. M. Lattuada and M. Morbidelli, “A simple model for the structure of fractal aggregates,” J. Colloid Interface Sci. 268, 106–120 (2003).
[CrossRef]

2001

C. M. Sorensen, “Light scattering by fractal aggregate: a review,” Aerosol Sci. Technol. 35, 648–687 (2001).
[CrossRef]

2000

M. Z. A. V. Filipov and D. E. Rosner, “Fractal-like aggregates: relation between morphology and physical properties,” J. Colloid Interface Sci. 229, 261–273 (2000).
[CrossRef]

1996

1994

1987

T. T. Charalampopoulos and J. D. Felske, “Refractive indices of soot particles deduced from in-situ laser light scattering measurements,” Combust. Flame 68, 283–294 (1987).
[CrossRef]

1973

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Amott, W. P.

L. Liu, M. I. Mishchenko, and W. P. Amott, “A study of radiative properties of soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109, 2656–2663 (2008).
[CrossRef]

Barber, P. W.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

Bengtsson, P.-E.

H. Bladh, J. Johnsson, N.-E. Olofsson, A. Bohlin, and P.-E. Bengtsson, “Optical soot characterization using two-color laser-induced incandescence (2c–lii) in the soot growth region of a premixed flat flame,” Proc. Combust. Inst. 33, 641–648 (2011).
[CrossRef]

A. Karlsson, H. Bladh, and P.-E. Bengtsson, “Accurate method for predicting light scattering from soot aggregates with subparticles of arbitrary shape and structure,” J. Opt. Soc. Am. A 26, 1704–1713 (2009).
[CrossRef]

J. Johnsson, H. Bladh, N.-E. Olofsson, and P.-E. Bengtsson, “Influence of soot aggregate structure on particle sizing using laser-induced incandescence: importance of bridging between primary particles,” Appl. Phys. B (to be published).

Bladh, H.

H. Bladh, J. Johnsson, N.-E. Olofsson, A. Bohlin, and P.-E. Bengtsson, “Optical soot characterization using two-color laser-induced incandescence (2c–lii) in the soot growth region of a premixed flat flame,” Proc. Combust. Inst. 33, 641–648 (2011).
[CrossRef]

A. Karlsson, H. Bladh, and P.-E. Bengtsson, “Accurate method for predicting light scattering from soot aggregates with subparticles of arbitrary shape and structure,” J. Opt. Soc. Am. A 26, 1704–1713 (2009).
[CrossRef]

J. Johnsson, H. Bladh, N.-E. Olofsson, and P.-E. Bengtsson, “Influence of soot aggregate structure on particle sizing using laser-induced incandescence: importance of bridging between primary particles,” Appl. Phys. B (to be published).

Bohlin, A.

H. Bladh, J. Johnsson, N.-E. Olofsson, A. Bohlin, and P.-E. Bengtsson, “Optical soot characterization using two-color laser-induced incandescence (2c–lii) in the soot growth region of a premixed flat flame,” Proc. Combust. Inst. 33, 641–648 (2011).
[CrossRef]

Charalampopoulos, T. T.

T. T. Charalampopoulos and J. D. Felske, “Refractive indices of soot particles deduced from in-situ laser light scattering measurements,” Combust. Flame 68, 283–294 (1987).
[CrossRef]

Doicu, T. W. A.

T. W. A. Doicu and Y. A. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).

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 and P. J. Flatau, “User guide to the discrete dipole approximation code DDSCAT 7.2,” (2012). http://arxiv.org/abs/1202.3424 .

Eremin, Y. A.

T. W. A. Doicu and Y. A. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).

Felske, J. D.

T. T. Charalampopoulos and J. D. Felske, “Refractive indices of soot particles deduced from in-situ laser light scattering measurements,” Combust. Flame 68, 283–294 (1987).
[CrossRef]

Filipov, M. Z. A. V.

M. Z. A. V. Filipov and D. E. Rosner, “Fractal-like aggregates: relation between morphology and physical properties,” J. Colloid Interface Sci. 229, 261–273 (2000).
[CrossRef]

Flatau, P. J.

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 and P. J. Flatau, “User guide to the discrete dipole approximation code DDSCAT 7.2,” (2012). http://arxiv.org/abs/1202.3424 .

Gregory, J.

J. Gregory, “Monitoring particle aggregation processes,” Adv. Colloid Interface Sci. 147, 109–123 (2009).
[CrossRef]

Hill, S. C.

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

Johnsson, J.

H. Bladh, J. Johnsson, N.-E. Olofsson, A. Bohlin, and P.-E. Bengtsson, “Optical soot characterization using two-color laser-induced incandescence (2c–lii) in the soot growth region of a premixed flat flame,” Proc. Combust. Inst. 33, 641–648 (2011).
[CrossRef]

J. Johnsson, H. Bladh, N.-E. Olofsson, and P.-E. Bengtsson, “Influence of soot aggregate structure on particle sizing using laser-induced incandescence: importance of bridging between primary particles,” Appl. Phys. B (to be published).

Karlsson, A.

Lattuada, H. W. M.

H. W. M. Lattuada and M. Morbidelli, “A simple model for the structure of fractal aggregates,” J. Colloid Interface Sci. 268, 106–120 (2003).
[CrossRef]

Liu, F.

F. Liu and G. Smallwood, “Effect of aggregation on the absorption cross-section of fractal soot aggregates and its impact on LII modelling,” J. Quant. Spectrosc. Radiat. Transfer 111, 302–308 (2010).
[CrossRef]

Liu, L.

L. Liu, M. I. Mishchenko, and W. P. Amott, “A study of radiative properties of soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109, 2656–2663 (2008).
[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 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]

Mishchenko, M. I.

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. Liu, M. I. Mishchenko, and W. P. Amott, “A study of radiative properties of soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109, 2656–2663 (2008).
[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]

Morbidelli, M.

H. W. M. Lattuada and M. Morbidelli, “A simple model for the structure of fractal aggregates,” J. Colloid Interface Sci. 268, 106–120 (2003).
[CrossRef]

Olofsson, N.-E.

H. Bladh, J. Johnsson, N.-E. Olofsson, A. Bohlin, and P.-E. Bengtsson, “Optical soot characterization using two-color laser-induced incandescence (2c–lii) in the soot growth region of a premixed flat flame,” Proc. Combust. Inst. 33, 641–648 (2011).
[CrossRef]

J. Johnsson, H. Bladh, N.-E. Olofsson, and P.-E. Bengtsson, “Influence of soot aggregate structure on particle sizing using laser-induced incandescence: importance of bridging between primary particles,” Appl. Phys. B (to be published).

Peterson, B.

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Rosner, D. E.

M. Z. A. V. Filipov and D. E. Rosner, “Fractal-like aggregates: relation between morphology and physical properties,” J. Colloid Interface Sci. 229, 261–273 (2000).
[CrossRef]

Smallwood, G.

F. Liu and G. Smallwood, “Effect of aggregation on the absorption cross-section of fractal soot aggregates and its impact on LII modelling,” J. Quant. Spectrosc. Radiat. Transfer 111, 302–308 (2010).
[CrossRef]

Sorensen, C. M.

C. M. Sorensen, “Light scattering by fractal aggregate: a review,” Aerosol Sci. Technol. 35, 648–687 (2001).
[CrossRef]

Ström, S.

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Adv. Colloid Interface Sci.

J. Gregory, “Monitoring particle aggregation processes,” Adv. Colloid Interface Sci. 147, 109–123 (2009).
[CrossRef]

Aerosol Sci. Technol.

C. M. Sorensen, “Light scattering by fractal aggregate: a review,” Aerosol Sci. Technol. 35, 648–687 (2001).
[CrossRef]

Combust. Flame

T. T. Charalampopoulos and J. D. Felske, “Refractive indices of soot particles deduced from in-situ laser light scattering measurements,” Combust. Flame 68, 283–294 (1987).
[CrossRef]

J. Colloid Interface Sci.

M. Z. A. V. Filipov and D. E. Rosner, “Fractal-like aggregates: relation between morphology and physical properties,” J. Colloid Interface Sci. 229, 261–273 (2000).
[CrossRef]

H. W. M. Lattuada and M. Morbidelli, “A simple model for the structure of fractal aggregates,” J. Colloid Interface Sci. 268, 106–120 (2003).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

L. Liu, M. I. Mishchenko, and W. P. Amott, “A study of radiative properties of soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 109, 2656–2663 (2008).
[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]

F. Liu and G. Smallwood, “Effect of aggregation on the absorption cross-section of fractal soot aggregates and its impact on LII modelling,” J. Quant. Spectrosc. Radiat. Transfer 111, 302–308 (2010).
[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]

Phys. Rev. D

B. Peterson and S. Ström, “T-matrix for electromagnetic scattering from an arbitrary number of scatterers and representations of E(3),” Phys. Rev. D 8, 3661–3678 (1973).
[CrossRef]

Proc. Combust. Inst.

H. Bladh, J. Johnsson, N.-E. Olofsson, A. Bohlin, and P.-E. Bengtsson, “Optical soot characterization using two-color laser-induced incandescence (2c–lii) in the soot growth region of a premixed flat flame,” Proc. Combust. Inst. 33, 641–648 (2011).
[CrossRef]

Other

T. W. A. Doicu and Y. A. Eremin, Light Scattering by Systems of Particles (Springer-Verlag, 2006).

J. Johnsson, H. Bladh, N.-E. Olofsson, and P.-E. Bengtsson, “Influence of soot aggregate structure on particle sizing using laser-induced incandescence: importance of bridging between primary particles,” Appl. Phys. B (to be published).

B. T. Draine and P. J. Flatau, “User guide to the discrete dipole approximation code DDSCAT 7.2,” (2012). http://arxiv.org/abs/1202.3424 .

P. W. Barber and S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, 1990).

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Figures (5)

Fig. 1.
Fig. 1.

Comparison between the averaged absorbed power for the RDG, G-RDG, DDA, and T-matrix methods for an ensemble of identical aggregates, with random orientation. Each aggregate consists of N nonoverlapping spheres with radius 10 nm, kf=2.3, and Df=1.8. The wavelength is 532 nm. All of the results are normalized with the absorbed power obtained by the T-matrix method. The results from the T-matrix method are considered to be very accurate.

Fig. 2.
Fig. 2.

Comparison between the averaged absorbed power for the RDG, G-RDG, and DDA methods for an ensemble of identical aggregates, with random orientation. Each aggregate consists of N overlapping spheres with radius 10 nm, kf=1.25, and Df=1.8; cf. Figure 3. The distance between the centers of two adjacent spheres is 13.3 nm. The wavelength is 532 nm. All of the results are normalized with the absorbed power obtained by the DDA method.

Fig. 3.
Fig. 3.

Aggregate with 20 overlapping spheres used in Figs. 2 and 5. The radius of the spheres is 10 nm, the distance between the centers of two adjacent spheres is 13.3 nm, kf=1.25, and Df=1.8.

Fig. 4.
Fig. 4.

Averaged normalized intensity for an ensemble of identical aggregates, with random orientation. Each aggregate consists 20 nonoverlapping spheres with kf=2.3 and Df=1.5. The wavelengths are 532 and 1064 nm. The scattering plane is perpendicular to e^i, and the angle is the polar angle θ between k^i and r^.

Fig. 5.
Fig. 5.

Averaged normalized intensity for an ensemble of identical aggregates, with random orientation. Each aggregate consists 20 overlapping spheres with kf=1.25 and Df=1.8. The wavelengths are 532 and 1064 nm, the radius of the spheres is 10 nm, and the distance between the centers of two adjacent spheres is 13.3 nm. The scattering plane is perpendicular to e^i, and the angle is the polar angle θ between k^i and r^.

Tables (1)

Tables Icon

Table 1. Comparison between the Averaged Absorbed Power for the T-matrix, RDG, G-RDG, and DDA Methods for an Ensemble of Identical Aggregates, with Random Orientationa

Equations (57)

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

Ei(r)=e^iE0eiki·r,
Es(r)=k2eikr4πε0rr^×(r^×VP(r)eiq·rdv),
q=k(k^ir^).
P(r)=ε0(εc(r)1)E(r)eiki·r,
S=12Re{Es(r)×Hs*(r)}=12η0|Es|2r^,
I(r)=|Es(r)|2E02.
r^=sinαcosβf^+sinαsinβk^i+cosαe^i,α^=cosαcosβf^+cosαsinβk^isinαe^i,β^=sinβf^+cosβk^i.
I=Ico+Icross,Ico=1E02|α^·Es|2,Icross=1E02|β^·Es|2.
Wa=12σ|ε0(εc1)|2V|P(r)|2dv=12ωIm{m2}ε0|m21|2V|P(r)|2dv,
N=kf(Rga)Df.
Rg=1Nn=1N|rnrmean|2,rmean=1Nn=1Nrn,
P(r)=P(r)(eixeiyeiz)E0.
pn=VnP(r)dv.
pn=pn(eixeiyeiz)E0.
P˜(r)=e^i13tr{P(r)}E0,
P˜(r)=13tr{P(r)}I,
P(r)=P˜(r)+ΔP(r).
P(r)=P˜(r)+ΔP(r),
p˜n=13tr{pn}I,
pn=VnP˜(r)dv,
pn=p˜n+Δpn,
p˜n=p˜n(eixeiyeiz)E0=e^i13tr{pn}E0.
pn=p˜n+Δpn,
Es(r)=k2eikr4πε0rr^×(r^×(n=1Npneiq·rn)),
Ico=k4(4πE0ε0r)2|α^n=1N·pneiq·rn|2,Icross=k4(4πE0ε0r)2|β^n=1N·pneiq·rn|2.
Ico(r)=Mk432π2ε02r2n=1N[29sin2(α)n=1N(tr{p˜n}tr{p˜n*}sin(|q|rnn)|q|rnn)+3+sin2α15tr{ΔpnΔpn*}]
Icross(r)=Mk4160π2ε02r2n=1Ntr{ΔpnΔpn*},
Wa=MσE026|ε0(εc1)|2Vatr{P(r)P*(r)}dv,
tr{ΔpnΔpn*}=|Δpn11|2+|Δpn22|2+|Δpn33|2+2(|Δpn12|2+|Δpn13|2+|Δpn23|2).
WRDG=9E02Vaε0ωIm{m2}2|m2+2|2,
ki=kk^i=k(sinθcosϕ,sinθsinϕ,cosθ),
e^i=(ex,ey,ez)=ξ^cosψ+η^sinψ.
η^=k^i×z^|k^i×z^|,ξ^=η˜×k^i.
I(r)=|Es(r)|2E02=1E02m=1M|Ems(r)|2,
I(r)=MI1(r)=ME028π202π02π0π|E1s(r)|2sinθdθdϕdψ.
E1s(r)=k2eikr4πε0rr^×(r^×(n=1N(p˜n+Δpn)eiq·rn)).
r^×(r^×p˜n)·r^×(r^×p˜n*)=19sin2(α)tr{p˜n}tr{p˜n*},
I(r)=MI1(r)=2Mk4(4π)4ε02r2[19sin2(α)n=1Nn=1Ntr{p˜n}tr{p˜n*}02π02π0πeiq·(rnrn)sinθdθdϕdψ+n=1N02π02π0π|r^×(r^×Δpn)|2sinθdθdϕdψ].
Ico(r)=2Mk4(4π)4ε02r2[sin2α9n=1Nn=1Ntr{p˜n}tr{p˜n*}02π02π0πeiq·(rnrn)sinθdθdϕdψ+n=1N02π02π0π|α^·Δpn|2sinθdθdϕdψ],Icross(r)=2Mk4(4π)4ε02r2n=1N02π02π0π|β^·Δpn|2sinθdθdϕdψ.
02π02π0πeiq·(rnrn)sinθdθdϕdψ=2π02π0πeiq·(rnrn)sinθ¯dθ¯dϕ¯.
02πeiAcos(ϕ¯+ϕ0)dϕ¯=2πJ0(A),
q·(rnrn)=|q|(ρnnsinθ¯cos(ϕ¯arctan(ynxn))+(znzn)cosθ¯),
02π02π0πeiq·(rnrn)sinθdθdϕdψ=4π20πJ0(|q|ρnnsinθ¯)ei|q|cosθ¯(znzn)sinθ¯dθ¯=8π201J0(|q|ρnn1x2)cos(|q|x(znzn))dx.
01cos(|q|x(znzn))dx=01J0(|q||znzn|1x2)dx=sin(|q||znzn|)|q||znzn|.
01J0(|q|ρnn1x2)cos(|q|x(znzn))dx=01J0(|q|rnn1x2)dx.
01J0(|q|ρnn1x2)cos(|q|x(znzn))dx=sin(|q|rnn)|q|rnn.
02π02π0π|α˜·Δpn|2sinθdθdϕdψ=8π215((1+2sin2α)(|Δpn11|2+|Δpn22|2+|Δpn33|2)(13sin2α)Re{Δpn11Δpn22*+Δpn11Δpn33*+Δpn22Δpn33*}+(3+sin2α)(|Δpn12|2+|Δpn13|2+|Δpn232|))
02π02π0π|β^·Δpn|2sinθdθdϕdψ=8π215(|Δpn11|2+|Δpn22|2+|Δpn33|2Re{Δpn11Δpn22*+Δpn11Δpn33*+Δpn22Δpn33*}+3(Δpn122+Δpn132+Δpn232)),
Ico(r)=Mk432π2ε02r2n=1N[29sin2(α)n=1N(tr{p˜n}tr{p˜n*}sin(|q|rnn)|q|rnn)+3+sin2α15tr{ΔpnΔpn*}]
Icross(r)=Mk4160π2ε02r2n=1Ntr{ΔpnΔpn*}.
Ico(r)=Mk432π2ε02r2[29sin2(α)VaVa(tr{P˜(r)}tr{P˜(r)*}sin(|q||rr|)|q||rr|)dvdv+3+sin2α15Vatr{ΔP(r)ΔP(r)*}dv]
Icross(r)=Mk4160π2ε02r2Vatr{ΔP(r)ΔP*(r)}dv.
Wa=M(4π)2σ|ε0(εc1)|202π02π0πVa|P1(r)|2dVsinθdθdϕdψ=Mσ2|ε0(εc1)|2(Va|P˜(r)|2dV+18π202π02π0πVa|ΔP(r)|2dVsinθdθdϕdψ),
Wa=M(4π)2σ|ε0(εc1)|202π02π0πVa|P1(r)|2dVsinθdθdϕdψ=MσE026|ε0(εc1)|2Vatr{P(r)P*(r)}dV.
I(r)=Ico(r)=N2Mk4(4π)3E02ε02r2(|p˜av|sinα)202π0πS(θ¯,ϕ¯)sinθ¯dθ¯dϕ¯=Mk4(|p˜av|sinα)2(4π)2E02ε02r2n=1Nn=1Nsin(|q|rnn)|q|rnn,
S(θ¯,ϕ¯)=1N2|n=1Neiq·rn|2.
Wa=MNAσ2Vsn|ε0(εc1)|2|pav|2,

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