Abstract

We perform a rigorous theoretical convergence analysis of the discrete dipole approximation (DDA). We prove that errors in any measured quantity are bounded by a sum of a linear term and a quadratic term in the size of a dipole d when the latter is in the range of DDA applicability. Moreover, the linear term is significantly smaller for cubically than for noncubically shaped scatterers. Therefore, for small d, errors for cubically shaped particles are much smaller than for noncubically shaped ones. The relative importance of the linear term decreases with increasing size; hence convergence of DDA for large enough scatterers is quadratic in the common range of d. Extensive numerical simulations are carried out for a wide range of d. Finally, we discuss a number of new developments in DDA and their consequences for convergence.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2006 (1)

2005 (2)

2004 (4)

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
[CrossRef]

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004).
[CrossRef]

K. F. Warnick and W. C. Chew, "Error analysis of the moment method," IEEE Antennas Propag. Mag. 46, 38-53 (2004).
[CrossRef]

2003 (1)

F. M. Kahnert, "Numerical methods in electromagnetic scattering theory," J. Quant. Spectrosc. Radiat. Transf. 79, 775-824 (2003).
[CrossRef]

2002 (1)

2000 (1)

J. Rahola, "On the eigenvalues of the volume integral operator of electromagnetic scattering," SIAM (Soc. Ind. Appl. Math) J. Sci. Comput. (USA) 21, 1740-1754 (2000).
[CrossRef]

1999 (3)

S. D. Druger and B. V. Bronk, "Internal and scattered electric fields in the discrete dipole approximation," J. Opt. Soc. Am. B 16, 2239-2246 (1999).
[CrossRef]

Y. L. Xu and B. A. S. Gustafson, "Comparison between multisphere light-scattering calculations: rigorous solution and discrete-dipole approximation," Astrophys. J. 513, 894-909 (1999).
[CrossRef]

N. B. Piller, "Coupled-dipole approximation for high permittivity materials," Opt. Commun. 160, 10-14 (1999).
[CrossRef]

1998 (3)

A. G. Hoekstra, J. Rahola, and P. M. A. Sloot, "Accuracy of internal fields in volume integral equation simulations of light scattering," Appl. Opt. 37, 8482-8497 (1998).
[CrossRef]

N. B. Piller and O. J. F. Martin, "Increasing the performance of the coupled-dipole approximation: a spectral approach," IEEE Trans. Antennas Propag. 46, 1126-1137 (1998).
[CrossRef]

A. G. Hoekstra, M. D. Grimminck, and P. M. A. Sloot, "Large scale simulations of elastic light scattering by a fast discrete dipole approximation," Int. J. Mod. Phys. C 9, 87-102 (1998).
[CrossRef]

1997 (2)

N. B. Piller, "Influence of the edge meshes on the accuracy of the coupled-dipole approximation," Opt. Lett. 22, 1674-1676 (1997).
[CrossRef]

G. C. Hsiao and R. E. Kleinman, "Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics," IEEE Trans. Antennas Propag. 45, 316-328 (1997).
[CrossRef]

1996 (1)

J. I. Peltoniemi, "Variational volume integral equation method for electromagnetic scattering by irregular grains," J. Quant. Spectrosc. Radiat. Transf. 55, 637-647 (1996).
[CrossRef]

1995 (2)

H. Okamoto, "Light scattering by clusters: the A1-term method," Opt. Rev. 2, 407-412 (1995).
[CrossRef]

K. F. Evans and G. L. Stephens, "Microwave radiative transfer through clouds composed of realistically shaped ice crystals. Part 1. Single scattering properties," J. Atmos. Sci. 52, 2041-2057 (1995).

1994 (1)

1993 (3)

F. Rouleau and P. G. Martin, "A new method to calculate the extinction properties of irregularly shaped particles," Astrophys. J. 414, 803-814 (1993).
[CrossRef]

B. T. Draine and J. J. Goodman, "Beyond Clausius-Miossotti—wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

A. Lakhtakia and G. W. Mulholland, "On two numerical techniques for light scattering by dielectric agglomerated structures," J. Res. Natl. Inst. Stand. Technol. 98, 699-716 (1993).

1992 (1)

A. Lakhtakia, "Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic-fields," Int. J. Mod. Phys. C 3, 583-603 (1992).
[CrossRef]

1991 (3)

1990 (1)

J. I. Hage and J. M. Greenberg, "A model for the optical properties of porous grains," Astrophys. J. 361, 251-259 (1990).
[CrossRef]

1988 (2)

G. H. Goedecke and S. G. O'Brien, "Scattering by irregular inhomogeneous particles via the digitized Green's function algorithm," Appl. Opt. 27, 2431-2438 (1988).
[CrossRef] [PubMed]

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

1980 (1)

A. D. Yanghjian, "Electric dyadic Green's function in the source region," Proc. IEEE 68, 248-263 (1980).

1973 (1)

E. M. Purcell and C. R. Pennypacker, "Scattering and adsorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Bohren, C. F.

Bronk, B. V.

Bryant, G. W.

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
[CrossRef]

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "Coupled dipole method with an exact long-wavelength limit and improved accuracy at finite frequencies," Opt. Lett. 27, 2118-2120 (2002).
[CrossRef]

Chaumet, P. C.

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
[CrossRef]

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "Coupled dipole method with an exact long-wavelength limit and improved accuracy at finite frequencies," Opt. Lett. 27, 2118-2120 (2002).
[CrossRef]

Chemyshev, A. V.

Chew, W. C.

K. F. Warnick and W. C. Chew, "Error analysis of the moment method," IEEE Antennas Propag. Mag. 46, 38-53 (2004).
[CrossRef]

Collinge, M. J.

Davis, C. P.

C. P. Davis and K. F. Warnick, "On the physical interpretation of the Sobolev norm in error estimation," Appl. Comput. Electromagn. Soc. J. 20, 144-150 (2005).

Draine, B. T.

M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004).
[CrossRef]

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 J. J. Goodman, "Beyond Clausius-Miossotti—wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

J. J. Goodman, B. T. Draine, and P. J. Flatau, "Application of fast-Fourier-transform techniques to the discrete-dipole approximation," Opt. Lett. 16, 1198-1200 (1991).
[CrossRef] [PubMed]

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

B. T. Draine, "The discrete dipole approximation for light scattering by irregular targets," in Light Scattering by Nonspherical Particles, Theory, Measurements, and Applications, M.I.Mishchenko, J.W.Hovenier, and L.D.Travis, eds. (Academic, 2000), pp. 131-145.

B. T. Draine and P. J. Flatau, "User guide for the discrete dipole approximation code DDSCAT 6.1," http://xxx.arxiv.org/abs/astro-ph/0409262 (2004).

D. Gutkowicz-Krusin and B. T. Draine, "Propagation of electromagnetic waves on a rectangular lattice of polarizable points," http://xxx.arxiv.org/abs/astro-ph/0403082 (2004).

Druger, S. D.

Dungey, C. E.

Evans, K. F.

K. F. Evans and G. L. Stephens, "Microwave radiative transfer through clouds composed of realistically shaped ice crystals. Part 1. Single scattering properties," J. Atmos. Sci. 52, 2041-2057 (1995).

Flatau, P. J.

Goedecke, G. H.

Goodman, J. J.

B. T. Draine and J. J. Goodman, "Beyond Clausius-Miossotti—wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

J. J. Goodman, B. T. Draine, and P. J. Flatau, "Application of fast-Fourier-transform techniques to the discrete-dipole approximation," Opt. Lett. 16, 1198-1200 (1991).
[CrossRef] [PubMed]

Greenberg, J. M.

J. I. Hage, J. M. Greenberg, and R. T. Wang, "Scattering from arbitrarily shaped particles: theory and experiment," Appl. Opt. 30, 1141-1152 (1991).
[CrossRef] [PubMed]

J. I. Hage and J. M. Greenberg, "A model for the optical properties of porous grains," Astrophys. J. 361, 251-259 (1990).
[CrossRef]

Grimminck, M. D.

A. G. Hoekstra, M. D. Grimminck, and P. M. A. Sloot, "Large scale simulations of elastic light scattering by a fast discrete dipole approximation," Int. J. Mod. Phys. C 9, 87-102 (1998).
[CrossRef]

Gustafson, B. A. S.

Y. L. Xu and B. A. S. Gustafson, "Comparison between multisphere light-scattering calculations: rigorous solution and discrete-dipole approximation," Astrophys. J. 513, 894-909 (1999).
[CrossRef]

Gutkowicz-Krusin, D.

D. Gutkowicz-Krusin and B. T. Draine, "Propagation of electromagnetic waves on a rectangular lattice of polarizable points," http://xxx.arxiv.org/abs/astro-ph/0403082 (2004).

Hage, J. I.

J. I. Hage, J. M. Greenberg, and R. T. Wang, "Scattering from arbitrarily shaped particles: theory and experiment," Appl. Opt. 30, 1141-1152 (1991).
[CrossRef] [PubMed]

J. I. Hage and J. M. Greenberg, "A model for the optical properties of porous grains," Astrophys. J. 361, 251-259 (1990).
[CrossRef]

Hoekstra, A. G.

Hsiao, G. C.

G. C. Hsiao and R. E. Kleinman, "Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics," IEEE Trans. Antennas Propag. 45, 316-328 (1997).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Kahnert, F. M.

F. M. Kahnert, "Numerical methods in electromagnetic scattering theory," J. Quant. Spectrosc. Radiat. Transf. 79, 775-824 (2003).
[CrossRef]

Kleinman, R. E.

G. C. Hsiao and R. E. Kleinman, "Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics," IEEE Trans. Antennas Propag. 45, 316-328 (1997).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia and G. W. Mulholland, "On two numerical techniques for light scattering by dielectric agglomerated structures," J. Res. Natl. Inst. Stand. Technol. 98, 699-716 (1993).

A. Lakhtakia, "Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic-fields," Int. J. Mod. Phys. C 3, 583-603 (1992).
[CrossRef]

Maltsev, V. P.

Martin, O. J. F.

N. B. Piller and O. J. F. Martin, "Increasing the performance of the coupled-dipole approximation: a spectral approach," IEEE Trans. Antennas Propag. 46, 1126-1137 (1998).
[CrossRef]

Martin, P. G.

F. Rouleau and P. G. Martin, "A new method to calculate the extinction properties of irregularly shaped particles," Astrophys. J. 414, 803-814 (1993).
[CrossRef]

Mulholland, G. W.

A. Lakhtakia and G. W. Mulholland, "On two numerical techniques for light scattering by dielectric agglomerated structures," J. Res. Natl. Inst. Stand. Technol. 98, 699-716 (1993).

O'Brien, S. G.

Okamoto, H.

H. Okamoto, "Light scattering by clusters: the A1-term method," Opt. Rev. 2, 407-412 (1995).
[CrossRef]

Peltoniemi, J. I.

J. I. Peltoniemi, "Variational volume integral equation method for electromagnetic scattering by irregular grains," J. Quant. Spectrosc. Radiat. Transf. 55, 637-647 (1996).
[CrossRef]

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, "Scattering and adsorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Piller, N. B.

N. B. Piller, "Coupled-dipole approximation for high permittivity materials," Opt. Commun. 160, 10-14 (1999).
[CrossRef]

N. B. Piller and O. J. F. Martin, "Increasing the performance of the coupled-dipole approximation: a spectral approach," IEEE Trans. Antennas Propag. 46, 1126-1137 (1998).
[CrossRef]

N. B. Piller, "Influence of the edge meshes on the accuracy of the coupled-dipole approximation," Opt. Lett. 22, 1674-1676 (1997).
[CrossRef]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, "Scattering and adsorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

Rahmani, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
[CrossRef]

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "Coupled dipole method with an exact long-wavelength limit and improved accuracy at finite frequencies," Opt. Lett. 27, 2118-2120 (2002).
[CrossRef]

Rahola, J.

J. Rahola, "On the eigenvalues of the volume integral operator of electromagnetic scattering," SIAM (Soc. Ind. Appl. Math) J. Sci. Comput. (USA) 21, 1740-1754 (2000).
[CrossRef]

A. G. Hoekstra, J. Rahola, and P. M. A. Sloot, "Accuracy of internal fields in volume integral equation simulations of light scattering," Appl. Opt. 37, 8482-8497 (1998).
[CrossRef]

Rouleau, F.

F. Rouleau and P. G. Martin, "A new method to calculate the extinction properties of irregularly shaped particles," Astrophys. J. 414, 803-814 (1993).
[CrossRef]

Semyanov, K. A.

Sentenac, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, "Coupled dipole method for scatterers with large permittivity," Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Sloot, P. M. A.

A. G. Hoekstra, M. D. Grimminck, and P. M. A. Sloot, "Large scale simulations of elastic light scattering by a fast discrete dipole approximation," Int. J. Mod. Phys. C 9, 87-102 (1998).
[CrossRef]

A. G. Hoekstra, J. Rahola, and P. M. A. Sloot, "Accuracy of internal fields in volume integral equation simulations of light scattering," Appl. Opt. 37, 8482-8497 (1998).
[CrossRef]

Stephens, G. L.

K. F. Evans and G. L. Stephens, "Microwave radiative transfer through clouds composed of realistically shaped ice crystals. Part 1. Single scattering properties," J. Atmos. Sci. 52, 2041-2057 (1995).

Tarasov, P. A.

Wang, R. T.

Warnick, K. F.

C. P. Davis and K. F. Warnick, "On the physical interpretation of the Sobolev norm in error estimation," Appl. Comput. Electromagn. Soc. J. 20, 144-150 (2005).

K. F. Warnick and W. C. Chew, "Error analysis of the moment method," IEEE Antennas Propag. Mag. 46, 38-53 (2004).
[CrossRef]

Xu, Y. L.

Y. L. Xu and B. A. S. Gustafson, "Comparison between multisphere light-scattering calculations: rigorous solution and discrete-dipole approximation," Astrophys. J. 513, 894-909 (1999).
[CrossRef]

Yanghjian, A. D.

A. D. Yanghjian, "Electric dyadic Green's function in the source region," Proc. IEEE 68, 248-263 (1980).

Yurkin, M. A.

Appl. Comput. Electromagn. Soc. J. (1)

C. P. Davis and K. F. Warnick, "On the physical interpretation of the Sobolev norm in error estimation," Appl. Comput. Electromagn. Soc. J. 20, 144-150 (2005).

Appl. Opt. (4)

Astrophys. J. (7)

Y. L. Xu and B. A. S. Gustafson, "Comparison between multisphere light-scattering calculations: rigorous solution and discrete-dipole approximation," Astrophys. J. 513, 894-909 (1999).
[CrossRef]

F. Rouleau and P. G. Martin, "A new method to calculate the extinction properties of irregularly shaped particles," Astrophys. J. 414, 803-814 (1993).
[CrossRef]

E. M. Purcell and C. R. Pennypacker, "Scattering and adsorption of light by nonspherical dielectric grains," Astrophys. J. 186, 705-714 (1973).
[CrossRef]

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

J. I. Hage and J. M. Greenberg, "A model for the optical properties of porous grains," Astrophys. J. 361, 251-259 (1990).
[CrossRef]

B. T. Draine and J. J. Goodman, "Beyond Clausius-Miossotti—wave propagation on a polarizable point lattice and the discrete dipole approximation," Astrophys. J. 405, 685-697 (1993).
[CrossRef]

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

K. F. Warnick and W. C. Chew, "Error analysis of the moment method," IEEE Antennas Propag. Mag. 46, 38-53 (2004).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

G. C. Hsiao and R. E. Kleinman, "Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics," IEEE Trans. Antennas Propag. 45, 316-328 (1997).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Partition of the scatterer’s volume into three regions relative to dipole i.

Fig. 2
Fig. 2

S 11 ( θ ) for all five test cases in logarithmic scale. The result for the k D = 3 sphere is multiplied by 10 for convenience.

Fig. 3
Fig. 3

Relative errors of S 11 at different angles θ and maximum over all θ versus y for (a) the k D = 8 cube, (b) the cubical discretization of the k D = 10 sphere. A log–log scale is used. A linear fit of maximum over θ errors is shown ( m = 1.5 ) .

Fig. 4
Fig. 4

Same as Fig. 3 but for (a) k D = 3 , (b) k D = 10 , and (c) k D = 30 spheres.

Fig. 5
Fig. 5

Relative errors of Q e x t versus y for all five test cases. A log–log scale is used. A linear fit through five finest discretizations of the k D = 3 sphere is shown.

Tables (1)

Tables Icon

Table 1 Exact Values of Q e x t for the Five Test Cases

Equations (100)

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E ( r ) = E i n c ( r ) + V \ V 0 d 3 r G ¯ ( r , r ) χ ( r ) E ( r ) + M ( V 0 , r ) L ¯ ( V 0 , r ) χ ( r ) E ( r ) ,
G ¯ ( r , r ) = ( k 2 I ¯ + ̂ ̂ ) g ( R ) = g ( R ) [ k 2 ( I ¯ R ̂ R ̂ R 2 ) 1 i k R R 2 ( I ¯ 3 R ̂ R ̂ R 2 ) ] ,
g ( R ) = exp ( i k R ) R .
M ( V 0 , r ) = V 0 d 3 r ( G ¯ ( r , r ) χ ( r ) E ( r ) G ¯ s ( r , r ) χ ( r ) E ( r ) ) ,
G ¯ s ( r , r ) = ̂ ̂ 1 R = 1 R 3 ( I ¯ 3 R ̂ R ̂ R 2 ) .
L ¯ ( V 0 , r ) = V 0 d 2 r n ̂ R ̂ R 3 ,
A ̃ E ̃ = E ̃ i n c ,
E ( r ) = E i n c ( r ) + j i V j d 3 r G ¯ ( r , r ) χ ( r ) E ( r ) + M ( V i , r ) L ¯ ( V i , r ) χ ( r ) E ( r ) .
E ( r ) = E ( r i ) = E i , χ ( r ) = χ ( r i ) = χ i for r V i .
E i = E i i n c + j i G ¯ i j V j χ j E j + ( M ¯ i L ¯ i ) χ i E i ,
M ¯ i = V i d 3 r ( G ¯ ( r i , r ) G ¯ s ( r i , r ) ) ,
G ¯ i j = 1 V j V j d 3 r G ¯ ( r i , r ) .
G ¯ i j ( 0 ) = G ¯ ( r i , r j ) .
L ¯ i = 4 π 3 I ¯ .
A ¯ d E d = E i n c , d ,
( A ̃ E ̃ ) ( r i ) = E ̃ i n c ( r i ) = E i i n c , d .
h i d = ( A ̃ E ̃ ) ( r i ) ( A ¯ d E 0 , d ) i ,
δ E d = E d E 0 , d = ( A ¯ d ) 1 h d .
E s c a ( r ) = exp ( i k r ) i k r F ( n ) ,
F ( n ) = i k 3 ( I ¯ n ̂ n ̂ ) i V i d 3 r exp ( i k r n ) χ ( r ) E ( r ) .
E i n c ( r ) = e 0 exp ( i k r ) ,
C s c a = 1 k 2 d Ω F ( n ) 2 ,
C e x t = 4 π k 2 Re ( F ( a ) e 0 ) ,
C a b s = 4 π k i V i d 3 r Im ( χ ( r ) ) E ( r ) 2 .
F ( n ) = i k 3 ( I ¯ n ̂ n ̂ ) i V i χ i E i d exp ( i k r i n ) ,
C a b s = 4 π k i V i Im ( χ i ) E i d 2 .
ϕ ̃ ( E ̃ ) = ϕ d ( E d ) + δ ϕ d ,
δ ϕ d = [ ϕ ̃ ( E ̃ ) ϕ d ( E 0 , d ) ] + [ ϕ d ( E 0 , d ) ϕ d ( E d ) ] .
N = γ 1 d 3 .
E ( r ) γ 2 , μ E ( r ) γ 3 , μ ν E ( r ) γ 4 , μ ν ρ E ( r ) γ 5 , μ ν ρ τ E ( r ) γ 6 for r V and μ , ν , ρ , τ .
G ¯ ( R ) c 6 R 3 , μ G ¯ ( R ) c 7 R 4 , μ ν G ¯ ( R ) c 8 R 5 , μ ν ρ G ¯ ( R ) c 9 R 6 , μ ν ρ τ G ¯ ( R ) c 10 R 7 for μ , ν , ρ , τ .
1 d 3 V c d 3 r f ( r ) f ( 0 ) c 11 d 2 max μ ν , r V c μ ν f ( r ) ,
1 d 3 V c d 3 r f ( r ) f ( 0 ) d 2 24 ( 2 f ( r ) ) r = 0 + c 12 d 4 max μ ν ρ τ , r V c μ ν ρ τ f ( r ) .
h i d = j i ( V j d 3 r G ¯ ( r i , r ) P ( r ) d 3 G ¯ i j ( 0 ) P j ) + M ( V i , r i ) ,
P ( r ) = χ ( r ) E ( r ) , P i = P ( r i ) .
P ( R ) = P ( 0 ) + ρ R ρ ( ρ P ) ( 0 ) + 1 2 ρ τ R ρ R τ ( ρ τ P ) ( r ̃ ( ρ , τ , R ) ) ,
M ( V i , r i ) = V i d 3 R ( G ¯ ( R ) G ¯ s ( R ) ) P i + 1 2 V i d 3 R G ¯ ( R ) ρ τ R ρ R τ ( ρ τ P ) ( r ̃ ( ρ , τ , R ) ) .
V i d 3 R ( G ¯ ( R ) G ¯ s ( R ) ) P i = 2 3 I ¯ P i V i d 3 R g ( R ) c 18 d 2 ,
V i d 3 R G ¯ ( R ) ρ τ R ρ R τ ( ρ τ P ) ( r ̃ ( ρ , τ , R ) ) 3 c 15 V i d 3 R G ¯ ( R ) R 2 c 19 d 2 .
M ( V i , r i ) c 20 d 2 .
n s ( l ) ( 2 l + 1 ) 3 ( 2 l 1 ) 3 c 21 l 2 .
l = 1 K ( i ) 1 j S l ( i ) ( V j d 3 r G ¯ ( r i , r ) P ( r ) d 3 G ¯ i j ( 0 ) P j ) .
1 2 j S l ( i ) ( V j d 3 r G ¯ ( r ) ( P ( r ) + P ( r ) ) d 3 G ¯ i j ( 0 ) ( P j + P j ) ) .
u ( r ) = 1 2 ( P ( r ) + P ( r ) ) P ( 0 ) ,
u ( r ) c 22 r 2 , μ u ( r ) c 23 r , μ ν u ( r ) c 24 for μ , ν .
j S l ( i ) ( V j d 3 r G ¯ ( r ) u ( r ) d 3 G ¯ i j ( 0 ) u j ) + j S l ( i ) ( V j d 3 r G ¯ ( r ) d 3 G ¯ i j ( 0 ) ) P i ,
max μ ν , r V j μ ν ( G ¯ ( r ) u ( r ) ) c 25 R i j 3
j S l ( i ) ( V j d 3 r G ¯ ( r ) u ( r ) d 3 G ¯ i j ( 0 ) u j ) j S l ( i ) c 26 d 5 R i j 3 c 27 d 2 l 1 ,
j S l ( i ) V j d 3 r G ¯ ( r ) = 2 3 I ¯ j S l ( i ) V j d 3 r g ( r ) ,
j S l ( i ) G ¯ i j ( 0 ) = 2 3 I ¯ j S l ( i ) g ( R i j ) .
j S l ( i ) ( V j d 3 r G ¯ ( r ) d 3 G ¯ i j ( 0 ) ) P i c 28 j S l ( i ) V j d 3 r g ( r ) d 3 g ( R i j ) c 29 d 4 l + c 30 d 2 l 3 ,
g ( R ) c 31 R 1 , μ ν ρ τ g ( R ) c 32 R 5 for μ , ν , ρ , τ .
l = 1 K ( i ) 1 j S l ( i ) ( V j d 3 r G ¯ ( r i , r ) P ( r ) d 3 G ¯ i j ( 0 ) P j ) ( c 33 + c 34 ln K ( i ) ) d 2 ,
j , R i j > 1 ( V j d 3 r G ¯ ( r i , r ) P ( r ) d 3 G ¯ i j ( 0 ) P j ) j , R i j > 1 c 35 d 5 N c 35 d 5 c 36 d 2 .
2 G ¯ ( r ) = G ¯ ( r )
2 ( G ¯ ( r ) P ( r ) ) r = R i j c 37 R i j 4 ,
max μ ν ρ τ , r V j μ ν ρ τ ( G ¯ ( r ) P ( r ) ) c 38 R i j 7 ,
j S l ( i ) ( V j d 3 r G ¯ ( r i , r ) P ( r ) d 3 G ¯ i j ( 0 ) P j ) c 39 d l 2 + c 40 l 5
l = K ( i ) K max j S l ( i ) ( V j d 3 r G ¯ ( r i , r ) P ( r ) d 3 G ¯ i j ( 0 ) P j ) c 41 d K 1 ( i ) + c 42 K 4 ( i ) .
h i d c 41 d K 1 ( i ) + c 42 K 4 ( i ) + ( c 43 + c 44 ln K ( i ) ) d 2 .
h d 1 = i = 1 N h i d ( c 43 + c 44 ln K max ) N d 2 + K = 1 K max n ( K ) ( c 41 d K 1 + c 42 K 4 ) ,
n ( K ) n ( 1 ) γ 12 N d ,
h d 1 N [ ( c 43 c 45 ln d ) d 2 + c 46 d ] .
δ E d 1 ( A ¯ d ) 1 1 h d 1 .
lim d 0 ( A ¯ d ) 1 1 = A ̃ 1 1 = γ 13 .
for d < d 0 ( A ¯ d ) 1 1 c 47 ,
ϕ ̃ ( E ̃ ) ϕ d ( E 0 , d ) i c 51 d 5 c 52 d 2 .
ϕ d ( E 0 , d ) ϕ d ( E d ) i c 53 d 3 δ E i d c 53 d 3 δ E d 1 ( c 54 c 55 ln d ) d 2 + c 56 d ,
δ ϕ d ( c 58 c 55 ln d ) d 2 + c 56 d .
δ ϕ y ( c 59 c 60 ln y ) y 2 + c 61 y .
h i d = j V ( V j d 3 r G ¯ ( r i , r ) P ( r ) d 3 G ¯ i j ( 0 ) P j ) ,
h i d = j V j i ( V j d 3 r G ¯ ( r i , r ) P ( r ) d 3 G ¯ i j ( 0 ) P j ) + M ( V i , r i ) ( L ¯ ( V i , r i ) 4 π 3 I ¯ ) χ i E i .
h i j s h = V j d 3 r G ¯ ( r i , r ) P ( r ) d 3 G ¯ i j ( 0 ) P j ,
h i i s h = M ( V i , r i ) ( L ¯ ( V i , r i ) 4 π 3 I ¯ ) χ i E i .
h i j s h { c 62 d 3 R i j 3 , R i j < 2 c 63 d 3 , R i j > 1 .
h i i s h c 64 ;
M ( V i , r i ) = V i d 3 r ( G ¯ ( r i , r ) G ¯ s ( r i , r ) ) P ( r ) + V i d 3 r G ¯ s ( r i , r ) ( P ( r ) P ( r i ) )
M ( V i , r i ) c 66 d .
P ( r ) P ( r i ) = ( P ( r ) P ( r ) ) + ( P ( r ) P ( r i ) )
M ( V i , r i ) c 67 d + c 68 V i d 3 r G ¯ s ( r i , r ) ,
L ¯ ( infinite plane , r i ) = R 2 d 2 r ρ ̂ ρ ̂ ρ ( ρ 2 + r 2 ) 3 2 = 2 π ρ ̂ ρ ̂ ρ 2 ,
L ¯ ( V i , r i ) c 69 ,
h d 1 i j V h i j s h + j V h i i s h j V ( l = 1 K max c 62 n s ( l ) l 3 + c 70 ) N d ( c 71 c 72 ln d ) ,
h d 1 N [ ( c 43 c 45 ln d ) d 2 + ( c 73 c 72 ln d ) d ] .
ϕ ̃ ( E ̃ ) ϕ d ( E 0 , d ) i c 51 d 5 + i V c 74 d 3 c 52 d 2 + c 75 d ,
ϕ d ( E 0 , d ) ϕ d ( E d ) c 53 d 3 δ E d 1 ( c 54 c 55 ln d ) d 2 + ( c 76 c 77 ln d ) d .
δ ϕ y ( c 59 c 60 ln y ) y 2 + ( c 78 c 79 ln y ) y .
M ( V i , r i ) = [ ( b 1 + b 2 m 2 + b 3 m 2 S ) d 2 + ( 2 3 ) i d 3 ] P i
V j d 3 r G ¯ ( r i , r ) P ( r ) d 3 G ¯ i j P j = V j d 3 r G ¯ ( r i , r ) ( P ( r ) P j ) c 80 V j d 3 r r 2 max μ , r V j μ G ¯ ( r i , r ) + c 81 V j d 3 r G ¯ ( r i , r ) r 2 c 82 d l 4 .
E i s = T ¯ i E i .
χ ¯ i e = ( V i p χ i p I ¯ + V i s χ i s T ¯ i ) d 3 ,
M ¯ ( V i , r ) = ( V i p d 3 r ( G ¯ ( r i , r ) G ¯ s ( r i , r ) ) χ i p + V i s d 3 r ( G ¯ ( r , r ) G ¯ s ( r i , r ) ) χ i s T ¯ i ) E i .
P ( r ) P i s c 83 min r V i s r r i ,
h i j s h = V j p d 3 r ( G ¯ ( r i , r ) P ( r ) G ¯ i j ( 0 ) P j p ) + V j s d 3 r ( G ¯ ( r i , r ) P ( r ) G ¯ i j ( 0 ) P j s ) .
h i j s h { c 84 d 4 R i j 4 , R i j < 2 c 85 d 4 , R i j > 1 .
h i i s h = ( M ( V i , r i ) L ¯ ( V i , r i ) P i p ) ( V i p d 3 r ( G ¯ ( r i , r ) G ¯ s ( r i , r ) ) P i p + V i s d 3 r ( G ¯ ( r , r ) G ¯ s ( r i , r ) ) P i s L ¯ ( V i , r i ) χ ¯ i e E i ) = V i p d 3 r G ( r i , r ) ( P ( r ) P i p ) + V i s d 3 r G ¯ ( r i , r ) ( P ( r ) P i s ) + V i s d 3 r G ¯ s ( r i , r ) ( P i s P i p ) + L ¯ ( V i , r i ) χ ¯ i e E i L ¯ ( V i , r i ) P i p .
h i i s h c 86 d + L ¯ ( V i p , r i ) P i p + L ¯ ( V i s , r i ) P i s L ¯ ( V i , r i ) χ ¯ i e E i ,
h d 1 j V ( l = 1 K max c 84 n s ( l ) l 4 + c 87 + c 88 d ) c 89 N d .
ϕ ̃ ( E ̃ ) ϕ d ( E 0 , d ) i c 51 d 5 + i V c 90 d 4 c 91 d 2 .
δ ϕ y ( c 92 c 93 ln y ) y 2 + c 94 y ,

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