Abstract

We propose a new formalism for computing the optical properties of small clusters of particles. It is a generalization of the coupled dipole–dipole particle-interaction model and allows one in principle to take into account all multipolar interactions in the long-wavelength limit. The method is illustrated by computations of the optical properties of N=6 particle clusters for different multipolar approximations. We examine the effect of separation between particles and compare the optical spectra with the discrete-dipole approximation and the generalized Mie theory.

© 2002 Optical Society of America

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  1. Ch. Girard, A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
    [CrossRef]
  2. Ph. Lambin, A. A. Lucas, J.-P. Vigneron, “Polarization waves and van der Waals cohesion of C60 fullerite,” Phys. Rev. B 46, 1794–1803 (1992).
    [CrossRef]
  3. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  4. V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
    [CrossRef]
  5. J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations,” Phys. Rev. B 25, 4204–4229 (1982).
    [CrossRef]
  6. M. Quinten, U. Kreibig, “Absorption and elastic scattering of light by particle aggregates,” Appl. Opt. 32, 6173–6182 (1993).
    [CrossRef] [PubMed]
  7. J. A. Sotelo, G. A. Niklasson, “Optical properties of quasifractal metal nanoparticle aggregates,” Nanostruct. Mater. 12, 135–138 (1999).
    [CrossRef]
  8. P. C. Waterman, “Numerical solution of electromagnetic scattering problems,” Phys. Rev. D 8, 3661–3678 (1973).
  9. F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B 30, 4989–4999 (1984).
    [CrossRef]
  10. F. Claro, R. Fuchs, “Optical absorption by clusters of small metallic spheres,” Phys. Rev. B 33, 7956–7960 (1986).
    [CrossRef]
  11. G. B. Smith, W. E. Vargas, G. A. Niklasson, J. A. Sotelo, A. V. Paley, A. V. Radchik, “Optical properties of a pair of spheres: comparison of different theories,” Opt. Commun. 115, 8–12 (1994).
    [CrossRef]
  12. A. V. Vagov, A. V. Radchik, G. B. Smith, “Optical response of arrays of spheres from the theory of hypercomplex variables,” Phys. Rev. Lett. 73, 1035–1038 (1994).
    [CrossRef] [PubMed]
  13. A. V. Radchik, A. V. Paley, G. B. Smith, “Quasistatic optical response of pairs of touching spheres with arbitrary dielectric permeability,” J. Appl. Phys. 73, 3446–3453 (1993).
    [CrossRef]
  14. L. G. Grechko, K. W. Whites, V. N. Pustovit, V. S. Lysenko, “Macroscopic dielectric response of the metallic particles embedded in host dielectric medium,” Microelectron. Reliability 40(4,5), 893–895 (2000).
    [CrossRef]
  15. M. Danos, L. C. Maximon, “Multipole matrix elements of the translation operator,” J. Math. Phys. 6, 766–778 (1965).
    [CrossRef]
  16. O. R. Crusan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  17. B. U. Felderhof, G. W. Ford, E. G. D. Cohen, “Two-particle cluster integral,” J. Stat. Phys. 28, 649–672 (1982).
    [CrossRef]
  18. M. Quinten, U. Kreibig, “Optical properties of aggregates of small metal particles,” Surf. Sci. 172, 557–577 (1986).
    [CrossRef]
  19. U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Vol. 25 of Material Science Series (Springer, Berlin, 1995).

2000

L. G. Grechko, K. W. Whites, V. N. Pustovit, V. S. Lysenko, “Macroscopic dielectric response of the metallic particles embedded in host dielectric medium,” Microelectron. Reliability 40(4,5), 893–895 (2000).
[CrossRef]

1999

J. A. Sotelo, G. A. Niklasson, “Optical properties of quasifractal metal nanoparticle aggregates,” Nanostruct. Mater. 12, 135–138 (1999).
[CrossRef]

1996

Ch. Girard, A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
[CrossRef]

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

1994

G. B. Smith, W. E. Vargas, G. A. Niklasson, J. A. Sotelo, A. V. Paley, A. V. Radchik, “Optical properties of a pair of spheres: comparison of different theories,” Opt. Commun. 115, 8–12 (1994).
[CrossRef]

A. V. Vagov, A. V. Radchik, G. B. Smith, “Optical response of arrays of spheres from the theory of hypercomplex variables,” Phys. Rev. Lett. 73, 1035–1038 (1994).
[CrossRef] [PubMed]

1993

A. V. Radchik, A. V. Paley, G. B. Smith, “Quasistatic optical response of pairs of touching spheres with arbitrary dielectric permeability,” J. Appl. Phys. 73, 3446–3453 (1993).
[CrossRef]

M. Quinten, U. Kreibig, “Absorption and elastic scattering of light by particle aggregates,” Appl. Opt. 32, 6173–6182 (1993).
[CrossRef] [PubMed]

1992

Ph. Lambin, A. A. Lucas, J.-P. Vigneron, “Polarization waves and van der Waals cohesion of C60 fullerite,” Phys. Rev. B 46, 1794–1803 (1992).
[CrossRef]

1988

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

1986

F. Claro, R. Fuchs, “Optical absorption by clusters of small metallic spheres,” Phys. Rev. B 33, 7956–7960 (1986).
[CrossRef]

M. Quinten, U. Kreibig, “Optical properties of aggregates of small metal particles,” Surf. Sci. 172, 557–577 (1986).
[CrossRef]

1984

F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B 30, 4989–4999 (1984).
[CrossRef]

1982

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

B. U. Felderhof, G. W. Ford, E. G. D. Cohen, “Two-particle cluster integral,” J. Stat. Phys. 28, 649–672 (1982).
[CrossRef]

1973

P. C. Waterman, “Numerical solution of electromagnetic scattering problems,” Phys. Rev. D 8, 3661–3678 (1973).

1965

M. Danos, L. C. Maximon, “Multipole matrix elements of the translation operator,” J. Math. Phys. 6, 766–778 (1965).
[CrossRef]

1962

O. R. Crusan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Armstrong, R.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

Ausloos, M.

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Claro, F.

F. Claro, R. Fuchs, “Optical absorption by clusters of small metallic spheres,” Phys. Rev. B 33, 7956–7960 (1986).
[CrossRef]

F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B 30, 4989–4999 (1984).
[CrossRef]

Cohen, E. G. D.

B. U. Felderhof, G. W. Ford, E. G. D. Cohen, “Two-particle cluster integral,” J. Stat. Phys. 28, 649–672 (1982).
[CrossRef]

Crusan, O. R.

O. R. Crusan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Danos, M.

M. Danos, L. C. Maximon, “Multipole matrix elements of the translation operator,” J. Math. Phys. 6, 766–778 (1965).
[CrossRef]

Dereux, A.

Ch. Girard, A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
[CrossRef]

Draine, B. T.

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

Felderhof, B. U.

B. U. Felderhof, G. W. Ford, E. G. D. Cohen, “Two-particle cluster integral,” J. Stat. Phys. 28, 649–672 (1982).
[CrossRef]

Ford, G. W.

B. U. Felderhof, G. W. Ford, E. G. D. Cohen, “Two-particle cluster integral,” J. Stat. Phys. 28, 649–672 (1982).
[CrossRef]

Fuchs, R.

F. Claro, R. Fuchs, “Optical absorption by clusters of small metallic spheres,” Phys. Rev. B 33, 7956–7960 (1986).
[CrossRef]

Gerardy, J. M.

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Girard, Ch.

Ch. Girard, A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
[CrossRef]

Grechko, L. G.

L. G. Grechko, K. W. Whites, V. N. Pustovit, V. S. Lysenko, “Macroscopic dielectric response of the metallic particles embedded in host dielectric medium,” Microelectron. Reliability 40(4,5), 893–895 (2000).
[CrossRef]

Kim, W.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

Kreibig, U.

M. Quinten, U. Kreibig, “Absorption and elastic scattering of light by particle aggregates,” Appl. Opt. 32, 6173–6182 (1993).
[CrossRef] [PubMed]

M. Quinten, U. Kreibig, “Optical properties of aggregates of small metal particles,” Surf. Sci. 172, 557–577 (1986).
[CrossRef]

U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Vol. 25 of Material Science Series (Springer, Berlin, 1995).

Lambin, Ph.

Ph. Lambin, A. A. Lucas, J.-P. Vigneron, “Polarization waves and van der Waals cohesion of C60 fullerite,” Phys. Rev. B 46, 1794–1803 (1992).
[CrossRef]

Lucas, A. A.

Ph. Lambin, A. A. Lucas, J.-P. Vigneron, “Polarization waves and van der Waals cohesion of C60 fullerite,” Phys. Rev. B 46, 1794–1803 (1992).
[CrossRef]

Lysenko, V. S.

L. G. Grechko, K. W. Whites, V. N. Pustovit, V. S. Lysenko, “Macroscopic dielectric response of the metallic particles embedded in host dielectric medium,” Microelectron. Reliability 40(4,5), 893–895 (2000).
[CrossRef]

Markel, V. A.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

Maximon, L. C.

M. Danos, L. C. Maximon, “Multipole matrix elements of the translation operator,” J. Math. Phys. 6, 766–778 (1965).
[CrossRef]

Niklasson, G. A.

J. A. Sotelo, G. A. Niklasson, “Optical properties of quasifractal metal nanoparticle aggregates,” Nanostruct. Mater. 12, 135–138 (1999).
[CrossRef]

G. B. Smith, W. E. Vargas, G. A. Niklasson, J. A. Sotelo, A. V. Paley, A. V. Radchik, “Optical properties of a pair of spheres: comparison of different theories,” Opt. Commun. 115, 8–12 (1994).
[CrossRef]

Paley, A. V.

G. B. Smith, W. E. Vargas, G. A. Niklasson, J. A. Sotelo, A. V. Paley, A. V. Radchik, “Optical properties of a pair of spheres: comparison of different theories,” Opt. Commun. 115, 8–12 (1994).
[CrossRef]

A. V. Radchik, A. V. Paley, G. B. Smith, “Quasistatic optical response of pairs of touching spheres with arbitrary dielectric permeability,” J. Appl. Phys. 73, 3446–3453 (1993).
[CrossRef]

Pustovit, V. N.

L. G. Grechko, K. W. Whites, V. N. Pustovit, V. S. Lysenko, “Macroscopic dielectric response of the metallic particles embedded in host dielectric medium,” Microelectron. Reliability 40(4,5), 893–895 (2000).
[CrossRef]

Quinten, M.

M. Quinten, U. Kreibig, “Absorption and elastic scattering of light by particle aggregates,” Appl. Opt. 32, 6173–6182 (1993).
[CrossRef] [PubMed]

M. Quinten, U. Kreibig, “Optical properties of aggregates of small metal particles,” Surf. Sci. 172, 557–577 (1986).
[CrossRef]

Radchik, A. V.

A. V. Vagov, A. V. Radchik, G. B. Smith, “Optical response of arrays of spheres from the theory of hypercomplex variables,” Phys. Rev. Lett. 73, 1035–1038 (1994).
[CrossRef] [PubMed]

G. B. Smith, W. E. Vargas, G. A. Niklasson, J. A. Sotelo, A. V. Paley, A. V. Radchik, “Optical properties of a pair of spheres: comparison of different theories,” Opt. Commun. 115, 8–12 (1994).
[CrossRef]

A. V. Radchik, A. V. Paley, G. B. Smith, “Quasistatic optical response of pairs of touching spheres with arbitrary dielectric permeability,” J. Appl. Phys. 73, 3446–3453 (1993).
[CrossRef]

Shalaev, V. M.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

Smith, G. B.

A. V. Vagov, A. V. Radchik, G. B. Smith, “Optical response of arrays of spheres from the theory of hypercomplex variables,” Phys. Rev. Lett. 73, 1035–1038 (1994).
[CrossRef] [PubMed]

G. B. Smith, W. E. Vargas, G. A. Niklasson, J. A. Sotelo, A. V. Paley, A. V. Radchik, “Optical properties of a pair of spheres: comparison of different theories,” Opt. Commun. 115, 8–12 (1994).
[CrossRef]

A. V. Radchik, A. V. Paley, G. B. Smith, “Quasistatic optical response of pairs of touching spheres with arbitrary dielectric permeability,” J. Appl. Phys. 73, 3446–3453 (1993).
[CrossRef]

Sotelo, J. A.

J. A. Sotelo, G. A. Niklasson, “Optical properties of quasifractal metal nanoparticle aggregates,” Nanostruct. Mater. 12, 135–138 (1999).
[CrossRef]

G. B. Smith, W. E. Vargas, G. A. Niklasson, J. A. Sotelo, A. V. Paley, A. V. Radchik, “Optical properties of a pair of spheres: comparison of different theories,” Opt. Commun. 115, 8–12 (1994).
[CrossRef]

Stechel, E. B.

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

Vagov, A. V.

A. V. Vagov, A. V. Radchik, G. B. Smith, “Optical response of arrays of spheres from the theory of hypercomplex variables,” Phys. Rev. Lett. 73, 1035–1038 (1994).
[CrossRef] [PubMed]

Vargas, W. E.

G. B. Smith, W. E. Vargas, G. A. Niklasson, J. A. Sotelo, A. V. Paley, A. V. Radchik, “Optical properties of a pair of spheres: comparison of different theories,” Opt. Commun. 115, 8–12 (1994).
[CrossRef]

Vigneron, J.-P.

Ph. Lambin, A. A. Lucas, J.-P. Vigneron, “Polarization waves and van der Waals cohesion of C60 fullerite,” Phys. Rev. B 46, 1794–1803 (1992).
[CrossRef]

Vollmer, M.

U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Vol. 25 of Material Science Series (Springer, Berlin, 1995).

Waterman, P. C.

P. C. Waterman, “Numerical solution of electromagnetic scattering problems,” Phys. Rev. D 8, 3661–3678 (1973).

Whites, K. W.

L. G. Grechko, K. W. Whites, V. N. Pustovit, V. S. Lysenko, “Macroscopic dielectric response of the metallic particles embedded in host dielectric medium,” Microelectron. Reliability 40(4,5), 893–895 (2000).
[CrossRef]

Appl. Opt.

Astrophys. J.

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

J. Appl. Phys.

A. V. Radchik, A. V. Paley, G. B. Smith, “Quasistatic optical response of pairs of touching spheres with arbitrary dielectric permeability,” J. Appl. Phys. 73, 3446–3453 (1993).
[CrossRef]

J. Math. Phys.

M. Danos, L. C. Maximon, “Multipole matrix elements of the translation operator,” J. Math. Phys. 6, 766–778 (1965).
[CrossRef]

J. Stat. Phys.

B. U. Felderhof, G. W. Ford, E. G. D. Cohen, “Two-particle cluster integral,” J. Stat. Phys. 28, 649–672 (1982).
[CrossRef]

Microelectron. Reliability

L. G. Grechko, K. W. Whites, V. N. Pustovit, V. S. Lysenko, “Macroscopic dielectric response of the metallic particles embedded in host dielectric medium,” Microelectron. Reliability 40(4,5), 893–895 (2000).
[CrossRef]

Nanostruct. Mater.

J. A. Sotelo, G. A. Niklasson, “Optical properties of quasifractal metal nanoparticle aggregates,” Nanostruct. Mater. 12, 135–138 (1999).
[CrossRef]

Opt. Commun.

G. B. Smith, W. E. Vargas, G. A. Niklasson, J. A. Sotelo, A. V. Paley, A. V. Radchik, “Optical properties of a pair of spheres: comparison of different theories,” Opt. Commun. 115, 8–12 (1994).
[CrossRef]

Phys. Rev. B

F. Claro, “Theory of resonant modes in particulate matter,” Phys. Rev. B 30, 4989–4999 (1984).
[CrossRef]

F. Claro, R. Fuchs, “Optical absorption by clusters of small metallic spheres,” Phys. Rev. B 33, 7956–7960 (1986).
[CrossRef]

V. A. Markel, V. M. Shalaev, E. B. Stechel, W. Kim, R. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

J. M. Gerardy, M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Ph. Lambin, A. A. Lucas, J.-P. Vigneron, “Polarization waves and van der Waals cohesion of C60 fullerite,” Phys. Rev. B 46, 1794–1803 (1992).
[CrossRef]

Phys. Rev. D

P. C. Waterman, “Numerical solution of electromagnetic scattering problems,” Phys. Rev. D 8, 3661–3678 (1973).

Phys. Rev. Lett.

A. V. Vagov, A. V. Radchik, G. B. Smith, “Optical response of arrays of spheres from the theory of hypercomplex variables,” Phys. Rev. Lett. 73, 1035–1038 (1994).
[CrossRef] [PubMed]

Q. Appl. Math.

O. R. Crusan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Rep. Prog. Phys.

Ch. Girard, A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
[CrossRef]

Surf. Sci.

M. Quinten, U. Kreibig, “Optical properties of aggregates of small metal particles,” Surf. Sci. 172, 557–577 (1986).
[CrossRef]

Other

U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Vol. 25 of Material Science Series (Springer, Berlin, 1995).

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Figures (4)

Fig. 1
Fig. 1

Projection of a cluster with six spheres (no sphere at the center). Each sphere has diameter 2r, and the distance between the centers of nearest-neighbor spheres is R.

Fig. 2
Fig. 2

Effective extinction cross section as a function of normalized frequency for the cluster of six spheres in case of dipole n=1 interaction between particles and spacing parameter σ=R/2a=1 for the DDA model [curve (a)], the Gerardy–Ausloos model [curve (b)], and the model presented in this paper [curve (c)].

Fig. 3
Fig. 3

Effective extinction cross section as a function of normalized frequency for the cluster of six spheres with use of different numbers of multipoles, as shown in the inset. The spacing parameter was set to (a) σ=R/2a=1.2 and (b) σ=1.

Fig. 4
Fig. 4

Effective extinction cross section as a function of normalized frequency for the cluster of six spheres for the model presented in this paper (curves A) and the Gerardy–Ausloos model (curves B) for (a) spacing parameter σ=R/2a=1.2 and n=4 order of multipoles and (b) spacing parameter σ=1 and n=7 order of multipoles.

Equations (31)

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

pi(1, 2,N)=pi(0)+jiNpij+
pi(1, 2,N)=(χ0+χc+)E0,
pij=p(i, j)-pi(0),
Φin(i)=-E0nmCnm(i)|R-Ri|nYnm(Rˆ-Rˆi),
Φout(i)=-E0nmdnm|R-Ri|nYnm(Rˆ-Rˆi)-E0nmBnm(i)|R-Ri|-n-1Ynm(Rˆ-Rˆi)-E0jinmBnm(j)|R-Rj|-n-1Ynm(Rˆ-Rˆj),
Ynm(θj, φj)Rjn+1=nm(-1)m+nKnmnmRijn+n+1RinYnm(θi, φi)×Yn+n,m-m*(θij, φij),
Knmnm=4π(2n+1)(n+n+m-m)!(n+n+m-m)!(2n+1)(2n+2n+1)(m+n)!(n-m)!(m+n)!(n-m)!1/2.
Φin(i)|Ri=ri=Φout(i)|Ri=ri,
i(Φin(i)mˆi)|Ri=ri=0(Φout(i)mˆi)|Ri=ri,
Bnm(i)χn(i)+nmBnm(j)(-1)m+nYn+n,m-m(Rˆj-Rˆi)Rijn+n+1Knmnm
=dnm,
Bnm(j)χn(j)+nmBnm(i)(-1)m+nYn+n,m-m(Rˆj-Rˆi)Rijn+n+1Knmnm
=dnm,
dnm=-δn12π3[2δm0uz+i(δm1+δm-1)uy+(δm-1-δm1)ux],
n=1Tnn(m)Xnm=δn1,n=1, 2,;m=-n,n,
Tnn(m)=r2n+1χnδnn-(-1)mrRijn+n+1n+nn+m,
Xnm=Bnmr-(n+2),
nm=n!m!(n-m)!.
Xnm=cof[T1n(m)]det[T(m)],
p(i, j)=-E0m=-11B1m(i)Y1m(Rˆi-Rˆj),ji.
pij=p(i, j)-pi(0)=r3W(ij)E0,
Wαγ(ij)=β11δαγ+uαuγ(β10-β11),
β11=X11-A1,β10=X10-A1,
X11=1A1-1+S3,X10=1A1-1-2S3,
X11=1(A1-1+S3)-3S8A2-1+4S5,
X10=1(A1-1-2S3)-9S8A2-1-6S5,
|p=r3Wˆ|E,
χc,αγi=34πj,eiα|ee¯|jγie¯|iαiα|ewer3.
χc=1/3N iTr(χc,αγi).
σa=σe=4πNk Im(χ0+χc),
i(ω)=1-ωp2ω(ω+iφ),

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