Abstract

We derive analytical solutions for the scattering of electromagnetic waves by a nanoparticle with nearly spherical shape and nonlocal dielectric function by using an extended Mie scattering theory with additional boundary conditions. A perturbation method is used to treat the correction due to deviation from the spherical shape. A surface characteristic function is introduced to describe the non-spherical surface profile of the nanoparticle, and it plays an important role in our analytical formulation. Complex surface plasmon modes are obtained. It is found that not only the transverse but also the longitudinal surface plasmon modes of the nanoparticle are excited due to the nonlocal effect. Our analytical formulation provides an alternative method for investigating the optical behaviors of the surface plasmon of nanoparticles with nearly spherical shape and nonlocal effect.

© 2010 Optical Society of America

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  1. C. Bohren and D. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  2. Y. G. Sun and Y. N. Xia, “Shape-controlled synthesis of gold and silver nanoparticles,” Science 298, 2176–2179 (2002).
    [CrossRef] [PubMed]
  3. J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
    [CrossRef]
  4. T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
    [CrossRef]
  5. J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001).
    [CrossRef]
  6. L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. V. Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005).
    [CrossRef] [PubMed]
  7. H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: A hybrid plasmonic nanostructure,” Nano Lett. 6, 827–832 (2006).
    [CrossRef] [PubMed]
  8. J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3, 485–491 (2003).
    [CrossRef]
  9. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
    [CrossRef]
  10. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).
  11. H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
    [CrossRef] [PubMed]
  12. E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004).
    [CrossRef] [PubMed]
  13. G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 330, 377–445 (1908).
    [CrossRef]
  14. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
    [PubMed]
  15. V. A. Erma, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969).
    [CrossRef]
  16. J. Vielma and P. T. Leung, “Nonlocal optical effects on the fluorescence and decay rates for admolecules at a metallic nanoparticle,” J. Chem. Phys. 126, 194704 (2007).
    [CrossRef] [PubMed]
  17. W. Ekardt and Z. Penzar, “Nonradiative lifetime of excited states near a small metal particle,” Phys. Rev. B 34, 8444–8448 (1986).
    [CrossRef]
  18. P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).
  19. G. Barton, “Some surface effects in the hydrodynamic model of metals,” Rep. Prog. Phys. 42, 963–1016 (1979).
    [CrossRef]
  20. J. Lindhard, “On the properties of gas of charged particles,” K. Dan. Fidensk. Selsk. Mat. Fys. Medd. 28, 1–57 (1954).
  21. Y. -C. Chang, , “Exact dynamical exchange-correlation kernel of a weakly inhomogenous electron gas,” Phys. Rev. Lett. 102, 113001 (2009).
    [CrossRef] [PubMed]
  22. G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys. 74, 601–659 (2002).
    [CrossRef]
  23. R. Fuchs and F. Claro, “Multipolar response of small metallic spheres: Nonlocal theory,” Phys. Rev. B 35, 3722–3727 (1987).
    [CrossRef]
  24. R. Rojas, F. Claro, and R. Fuchs, “Nonlocal response of a small coated sphere,” Phys. Rev. B 37, 6799–6807 (1988).
    [CrossRef]
  25. R. Chang and P. T. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006).
    [CrossRef]
  26. R. Chang and P. T. Leung, “Erratum: Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 75, 079901 (2006).
    [CrossRef]
  27. H. Y. Xie, H. Y. Chung, P. T. Leung, and D. P. Tsai, “Plasmonic enhancement of Förster energy transfer between two molecules in the vicinity of a metallic nanoparticle: Nonlocal optical effects,” Phys. Rev. B 80, 155448 (2009).
    [CrossRef]
  28. R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975).
    [CrossRef]
  29. V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys. Condens. Matter 20, 325211 (2008).
    [CrossRef]
  30. R. Schoonover, J. M. Rutherford, O. Keller, and P. S. Carney, “Nonlocal constitutive relations and the quasi-homogeneous approximation,” Phys. Lett. A 342, 363–367 (2005).
    [CrossRef]
  31. A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Ann. Phys. (N.Y.) 315, 352–418 (2005).
    [CrossRef]
  32. J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103, 097403 (2009).
    [CrossRef] [PubMed]

2009 (3)

Y. -C. Chang, , “Exact dynamical exchange-correlation kernel of a weakly inhomogenous electron gas,” Phys. Rev. Lett. 102, 113001 (2009).
[CrossRef] [PubMed]

H. Y. Xie, H. Y. Chung, P. T. Leung, and D. P. Tsai, “Plasmonic enhancement of Förster energy transfer between two molecules in the vicinity of a metallic nanoparticle: Nonlocal optical effects,” Phys. Rev. B 80, 155448 (2009).
[CrossRef]

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103, 097403 (2009).
[CrossRef] [PubMed]

2008 (2)

V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys. Condens. Matter 20, 325211 (2008).
[CrossRef]

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

2007 (1)

J. Vielma and P. T. Leung, “Nonlocal optical effects on the fluorescence and decay rates for admolecules at a metallic nanoparticle,” J. Chem. Phys. 126, 194704 (2007).
[CrossRef] [PubMed]

2006 (3)

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: A hybrid plasmonic nanostructure,” Nano Lett. 6, 827–832 (2006).
[CrossRef] [PubMed]

R. Chang and P. T. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006).
[CrossRef]

R. Chang and P. T. Leung, “Erratum: Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 75, 079901 (2006).
[CrossRef]

2005 (3)

L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. V. Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005).
[CrossRef] [PubMed]

R. Schoonover, J. M. Rutherford, O. Keller, and P. S. Carney, “Nonlocal constitutive relations and the quasi-homogeneous approximation,” Phys. Lett. A 342, 363–367 (2005).
[CrossRef]

A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Ann. Phys. (N.Y.) 315, 352–418 (2005).
[CrossRef]

2004 (1)

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004).
[CrossRef] [PubMed]

2003 (2)

J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3, 485–491 (2003).
[CrossRef]

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

2002 (3)

Y. G. Sun and Y. N. Xia, “Shape-controlled synthesis of gold and silver nanoparticles,” Science 298, 2176–2179 (2002).
[CrossRef] [PubMed]

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[CrossRef]

G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys. 74, 601–659 (2002).
[CrossRef]

2001 (1)

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001).
[CrossRef]

1998 (1)

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

1995 (1)

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

1992 (1)

P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).

1988 (1)

R. Rojas, F. Claro, and R. Fuchs, “Nonlocal response of a small coated sphere,” Phys. Rev. B 37, 6799–6807 (1988).
[CrossRef]

1987 (1)

R. Fuchs and F. Claro, “Multipolar response of small metallic spheres: Nonlocal theory,” Phys. Rev. B 35, 3722–3727 (1987).
[CrossRef]

1986 (1)

W. Ekardt and Z. Penzar, “Nonradiative lifetime of excited states near a small metal particle,” Phys. Rev. B 34, 8444–8448 (1986).
[CrossRef]

1983 (1)

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

1979 (1)

G. Barton, “Some surface effects in the hydrodynamic model of metals,” Rep. Prog. Phys. 42, 963–1016 (1979).
[CrossRef]

1975 (2)

S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14, 29–49 (1975).
[PubMed]

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975).
[CrossRef]

1969 (1)

V. A. Erma, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969).
[CrossRef]

1954 (1)

J. Lindhard, “On the properties of gas of charged particles,” K. Dan. Fidensk. Selsk. Mat. Fys. Medd. 28, 1–57 (1954).

1908 (1)

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 330, 377–445 (1908).
[CrossRef]

Asano, S.

Barbic, M.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[CrossRef]

Barton, G.

G. Barton, “Some surface effects in the hydrodynamic model of metals,” Rep. Prog. Phys. 42, 963–1016 (1979).
[CrossRef]

Bohren, C.

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

Brandl, D. W.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: A hybrid plasmonic nanostructure,” Nano Lett. 6, 827–832 (2006).
[CrossRef] [PubMed]

Carney, P. S.

R. Schoonover, J. M. Rutherford, O. Keller, and P. S. Carney, “Nonlocal constitutive relations and the quasi-homogeneous approximation,” Phys. Lett. A 342, 363–367 (2005).
[CrossRef]

Chang, R.

R. Chang and P. T. Leung, “Erratum: Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 75, 079901 (2006).
[CrossRef]

R. Chang and P. T. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006).
[CrossRef]

Chang, S. H.

L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. V. Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005).
[CrossRef] [PubMed]

Chang, Y. -C.

Y. -C. Chang, , “Exact dynamical exchange-correlation kernel of a weakly inhomogenous electron gas,” Phys. Rev. Lett. 102, 113001 (2009).
[CrossRef] [PubMed]

Chen, H.

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

Chung, H. Y.

H. Y. Xie, H. Y. Chung, P. T. Leung, and D. P. Tsai, “Plasmonic enhancement of Förster energy transfer between two molecules in the vicinity of a metallic nanoparticle: Nonlocal optical effects,” Phys. Rev. B 80, 155448 (2009).
[CrossRef]

Claro, F.

R. Rojas, F. Claro, and R. Fuchs, “Nonlocal response of a small coated sphere,” Phys. Rev. B 37, 6799–6807 (1988).
[CrossRef]

R. Fuchs and F. Claro, “Multipolar response of small metallic spheres: Nonlocal theory,” Phys. Rev. B 35, 3722–3727 (1987).
[CrossRef]

Coronado, E.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Duyne, R. P. V.

L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. V. Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005).
[CrossRef] [PubMed]

Ekardt, W.

W. Ekardt and Z. Penzar, “Nonradiative lifetime of excited states near a small metal particle,” Phys. Rev. B 34, 8444–8448 (1986).
[CrossRef]

Erma, V. A.

V. A. Erma, “Exact solution for the scattering of electromagnetic waves from bodies of arbitrary shape. III. Obstacles with arbitrary electromagnetic properties,” Phys. Rev. 179, 1238–1246 (1969).
[CrossRef]

Feldmann, J.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

Fuchs, R.

R. Rojas, F. Claro, and R. Fuchs, “Nonlocal response of a small coated sphere,” Phys. Rev. B 37, 6799–6807 (1988).
[CrossRef]

R. Fuchs and F. Claro, “Multipolar response of small metallic spheres: Nonlocal theory,” Phys. Rev. B 35, 3722–3727 (1987).
[CrossRef]

Gray, S. K.

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103, 097403 (2009).
[CrossRef] [PubMed]

Grosse, S.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

Halas, N. J.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: A hybrid plasmonic nanostructure,” Nano Lett. 6, 827–832 (2006).
[CrossRef] [PubMed]

Halevi, P.

P. Halevi, Spatial Dispersion in Solid and Plasmas (North-Holland, 1992).

Hao, E.

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004).
[CrossRef] [PubMed]

Huffman, D.

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

Keller, O.

R. Schoonover, J. M. Rutherford, O. Keller, and P. S. Carney, “Nonlocal constitutive relations and the quasi-homogeneous approximation,” Phys. Lett. A 342, 363–367 (2005).
[CrossRef]

Kelly, K. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Klar, T.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

Kottmann, J. P.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001).
[CrossRef]

Kou, X.

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

Kreibig, U.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer-Verlag, 1995).

Le, F.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: A hybrid plasmonic nanostructure,” Nano Lett. 6, 827–832 (2006).
[CrossRef] [PubMed]

Leung, P. T.

H. Y. Xie, H. Y. Chung, P. T. Leung, and D. P. Tsai, “Plasmonic enhancement of Förster energy transfer between two molecules in the vicinity of a metallic nanoparticle: Nonlocal optical effects,” Phys. Rev. B 80, 155448 (2009).
[CrossRef]

J. Vielma and P. T. Leung, “Nonlocal optical effects on the fluorescence and decay rates for admolecules at a metallic nanoparticle,” J. Chem. Phys. 126, 194704 (2007).
[CrossRef] [PubMed]

R. Chang and P. T. Leung, “Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 73, 125438 (2006).
[CrossRef]

R. Chang and P. T. Leung, “Erratum: Nonlocal effects on optical and molecular interactions with metallic nanoshells,” Phys. Rev. B 75, 079901 (2006).
[CrossRef]

Lindhard, J.

J. Lindhard, “On the properties of gas of charged particles,” K. Dan. Fidensk. Selsk. Mat. Fys. Medd. 28, 1–57 (1954).

Martin, O. J. F.

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001).
[CrossRef]

McMahon, J. M.

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103, 097403 (2009).
[CrossRef] [PubMed]

Mie, G.

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. (Leipzig) 330, 377–445 (1908).
[CrossRef]

Mock, J. J.

J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3, 485–491 (2003).
[CrossRef]

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[CrossRef]

Moroz, A.

A. Moroz, “A recursive transfer-matrix solution for a dipole radiating inside and outside a stratified sphere,” Ann. Phys. (N.Y.) 315, 352–418 (2005).
[CrossRef]

Ni, W.

H. Chen, X. Kou, Z. Yang, W. Ni, and J. Wang, “Shape- and size-dependent refractive index sensitivity of gold nanoparticles,” Langmuir 24, 5233–5237 (2008).
[CrossRef] [PubMed]

Nordlander, P.

H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “Nanorice: A hybrid plasmonic nanostructure,” Nano Lett. 6, 827–832 (2006).
[CrossRef] [PubMed]

Onida, G.

G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys. 74, 601–659 (2002).
[CrossRef]

Penzar, Z.

W. Ekardt and Z. Penzar, “Nonradiative lifetime of excited states near a small metal particle,” Phys. Rev. B 34, 8444–8448 (1986).
[CrossRef]

Perner, M.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

Reining, L.

G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys. 74, 601–659 (2002).
[CrossRef]

Rojas, R.

R. Rojas, F. Claro, and R. Fuchs, “Nonlocal response of a small coated sphere,” Phys. Rev. B 37, 6799–6807 (1988).
[CrossRef]

Rubio, A.

G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Rev. Mod. Phys. 74, 601–659 (2002).
[CrossRef]

Ruppin, R.

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975).
[CrossRef]

Rutherford, J. M.

R. Schoonover, J. M. Rutherford, O. Keller, and P. S. Carney, “Nonlocal constitutive relations and the quasi-homogeneous approximation,” Phys. Lett. A 342, 363–367 (2005).
[CrossRef]

Schatz, G. C.

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103, 097403 (2009).
[CrossRef] [PubMed]

L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. V. Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005).
[CrossRef] [PubMed]

E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. 120, 357–366 (2004).
[CrossRef] [PubMed]

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: The Influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Schoonover, R.

R. Schoonover, J. M. Rutherford, O. Keller, and P. S. Carney, “Nonlocal constitutive relations and the quasi-homogeneous approximation,” Phys. Lett. A 342, 363–367 (2005).
[CrossRef]

Schultz, D. A.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[CrossRef]

Schultz, S.

J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3, 485–491 (2003).
[CrossRef]

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001).
[CrossRef]

Sherry, L. J.

L. J. Sherry, S. H. Chang, G. C. Schatz, R. P. V. Duyne, B. J. Wiley, and Y. Xia, “Localized surface plasmon resonance spectroscopy of single silver nanocubes,” Nano Lett. 5, 2034–2038 (2005).
[CrossRef] [PubMed]

Smith, D. R.

J. J. Mock, D. R. Smith, and S. Schultz, “Local refractive index dependence of plasmon resonance spectra from individual nanoparticles,” Nano Lett. 3, 485–491 (2003).
[CrossRef]

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[CrossRef]

J. P. Kottmann, O. J. F. Martin, D. R. Smith, and S. Schultz, “Plasmon resonances of silver nanowires with a nonregular cross section,” Phys. Rev. B 64, 235402 (2001).
[CrossRef]

Spirkl, W.

T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

Sun, Y. G.

Y. G. Sun and Y. N. Xia, “Shape-controlled synthesis of gold and silver nanoparticles,” Science 298, 2176–2179 (2002).
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Figures (2)

Fig. 1
Fig. 1

Geometry of the scattering problem of a non-spherical particle. The dashed line denotes an unperturbed sphere ( r s = a ) , and the solid line describes the surface shape function [ r s = a ( 1 + η f ( θ , φ ) ) ] of the particle considered.

Fig. 2
Fig. 2

Schematic diagrams for the electric fields associated with (a) local and (b) nonlocal optical responses of a non-spherical particle. E i denotes the incident electric field, E s denotes the scattered electric field, E t denotes the transverse electric field, and E denotes the longitudinal electric field.

Equations (74)

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D ( r , ω ) = ε ( r , r , ω ) E ( r , ω ) d 3 r ,
D ( k , ω ) = ε ( k , ω ) E ( k , ω ) .
r s = a ( 1 + η f ( θ , φ ) ) ,
| η f ( θ , φ ) | 1.
M n m = z n ( ρ ) m n m ( θ , φ ) ,
N n m = z n ( ρ ) ρ n ( n + 1 ) o n m ( θ , φ ) e ̂ r + 1 ρ d [ ρ z n ( ρ ) ] d ρ n n m ( θ , φ ) ,
m n m ( θ , φ ) = e i m φ [ i m sin   θ P n m ( cos   θ ) e ̂ θ d P n m ( cos   θ ) d θ e ̂ φ ] ,
n n m ( θ , φ ) = e i m φ [ d P n m ( cos   θ ) d θ e ̂ θ + i m sin   θ P n m ( cos   θ ) e ̂ φ ] ,
o n m ( θ , φ ) = P n m ( cos   θ ) e i m φ .
d Ω [ m n m ( θ , φ ) m n m ( θ , φ ) n n m ( θ , φ ) n n m ( θ , φ ) o n m ( θ , φ ) o n m ( θ , φ ) n ( n + 1 ) ] = C n m δ n n δ m m ,
E s = n , m ( a n m M n m s + b n m N n m s ) ,
E t = n , m ( c n m M n m t + d n m N n m t ) ,
H s = n , m χ 1 ( a n m N n m s + b n m M n m s ) ,
H t = n , m χ 2 ( c n m N n m t + d n m M n m t ) ,
n ̂ × E 1 = n ̂ × E 2 r S ,
n ̂ × H 1 = n ̂ × H 2 r S ,
ρ 1 s E i t ( ρ 1 s ) + n , m [ a n m ρ 1 s h n ( 1 ) ( ρ 1 s ) m n m + b n m d ( ρ 1 s h n ( 1 ) ( ρ 1 s ) ) d ρ 1 s n n m ] + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) × { E i r ( ρ 1 s ) + n , m [ b n m h n ( 1 ) ( ρ 1 s ) ρ 1 s n ( n + 1 ) o n m ] } = κ n , m { c n m ρ 2 s j n ( ρ 2 s ) m n m + d n m d ( ρ 2 s j n ( ρ 2 s ) ) d ρ 2 s n n m } + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) n , m [ d n m j n ( ρ 2 s ) ρ 2 s n ( n + 1 ) o n m ] ,
ρ 1 s H i t ( ρ 1 s ) + n , m χ 1 [ a n m d ( ρ 1 s h n ( 1 ) ( ρ 1 s ) ) d ρ 1 s n n m + b n m ρ 1 s h n ( 1 ) ( ρ 1 s ) m n m ] + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) × { H i r ( ρ 1 s ) + n , m [ χ 1 a n m h n ( 1 ) ( ρ 1 s ) ρ 1 s n ( n + 1 ) o n m ] } = κ n , m χ 2 { c n m d ( ρ 2 s j n ( ρ 2 s ) ) d ρ 2 s n n m + d n m ρ 2 s j n ( ρ 2 s ) m n m } + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) n , m [ χ 2 c n m j n ( ρ 2 s ) ρ 2 s n ( n + 1 ) o n m ] ,
( a n m , b n m , c n m , d n m ) = j = 0 ( a n m j , b n m j , c n m j , d n m j ) η j ,
ρ 1 s h n ( 1 ) ( ρ 1 s ) = j = 0 η j α n j ρ 10 j f j j ! ,     α n j | d j [ ρ 10 h n ( 1 ) ( ρ ) ] d ρ j | ρ 10 ,
ρ 2 s j n ( ρ 2 s ) = j = 0 η j β n j ρ 20 j f j j ! ,     β n j | d j [ ρ j n ( ρ ) ] d ρ j | ρ 20 ,
h n ( 1 ) ( ρ 1 s ) ρ 1 s = j = 0 η j γ n j ρ 10 j f j j ! ,     γ n j | d j d ρ j [ h n ( 1 ) ( ρ ) ρ ] | ρ 10 ,
j n ( ρ 2 s ) ρ 2 s = j = 0 η j δ n j ρ 20 j f j j ! ,     δ n j | d j d ρ j [ j n ( ρ ) ρ ] | ρ 20 ,
ρ 1 s E i t ( ρ 1 s ) = j = 0 η j ( ρ E i t ) ( j ) ρ 10 j f j j ! ,
( ρ E i t ) ( j ) | j ( ρ E i t ( ρ ) ) ρ j | ρ 10 ,
E i r ( ρ 1 s ) = j = 0 η j ( E i r ) ( j ) ρ 10 j f j j ! ,
( E i r ) ( j ) j ρ j E i r ( ρ ) ρ 10 ,
ρ 1 s H i t ( ρ 1 s ) = j = 0 η j ( ρ H i t ) ( j ) ρ 10 j f j j ! ,     ( ρ H i t ) ( j ) | j ( ρ H i t ( ρ ) ) ρ j | ρ 10 ,
H i r ( ρ 1 s ) = j = 0 η j ( H i r ) ( j ) ρ 10 j f j j ! ,     ( H i r ) ( j ) j ρ j H i r ( ρ ) ρ 10 .
( α n 0 0 κ β n 0 0 χ 1 α n 1 0 κ χ 2 β n 1 0 0 α n 1 0 κ β n 1 0 χ 1 α n 0 0 κ χ 2 β n 0 ) ( a n m j b n m j c n m j d n m j ) = 1 C n m d Ω ( A j m n m B j n n m A j n n m B j m n m ) ,
A j = κ q = 0 j 1 n , m [ c n m q β n j q ρ 20 j q f j q ( j q ) ! m n m + d n m q β n j q + 1 ρ 20 j q f j q ( j q ) ! n n m ] + ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) q = 0 j 1 n , m [ d n m q δ n j 1 q ρ 20 j 1 q f j 1 q ( j 1 q ) ! n ( n + 1 ) o n m ] q = 0 j 1 n , m [ a n m q α n j q ρ 10 j q f j q ( j q ) ! m n m + b n m q α n j q + 1 ρ 10 j q f j q ( j q ) ! n n m ] ( ρ E i t ) ( j ) ρ 10 j f j j ! ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) [ ( E i r ) ( j 1 ) ρ 10 j f j 1 ( j 1 ) ! + q = 0 j 1 n , m b n m q γ n j 1 q ρ 10 j q f j 1 q ( j 1 q ) ! n ( n + 1 ) o n m ] ,
B j = κ q = 0 j 1 n , m χ 2 [ c n m q β n j q + 1 ρ 20 j q f j q ( j q ) ! n n m + d n m q β n j q ρ 20 j q f j q ( j q ) ! m n m ] + ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) q = 0 j 1 n , m χ 2 c n m q δ n j 1 q ρ 20 j 1 q f j 1 q ( j 1 q ) ! n ( n + 1 ) o n m ( ρ H i t ) ( j ) ρ 10 j f j j ! ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) [ ( H i r ) ( j 1 ) ρ 10 j f j 1 ( j 1 ) ! + χ 1 q = 0 j 1 n , m a n m q γ n j 1 q ρ 10 j q f j 1 q ( j 1 q ) ! n ( n + 1 ) o n m ] q = 0 j 1 n , m χ 1 [ a n m q α n j q + 1 ρ 10 j q f j q ( j q ) ! n n m + b n m q α n j q ρ 10 j q f j q ( j q ) ! m n m ] .
( a n m j b n m j c n m j d n m j ) = 1 C n m d Ω ( α n 0 0 κ β n 0 0 χ 1 α n 1 0 κ χ 2 β n 1 0 0 α n 1 0 κ β n 1 0 χ 1 α n 0 0 κ χ 2 β n 0 ) 1 ( A j m n m B j n n m A j n n m B j m n m ) ,
ε t ( k t , ω ) = c 2 ω 2 k t 2 ,
ε ( k , ω ) = 0 ,
L n m = k d z n ( ρ ) d ρ o n m ( θ , φ ) e ̂ r + k z n ( ρ ) ρ n n m ( θ , φ ) ,
E 2 = E t + E = n , m ( c n m M n m t + d n m N n m t + e n m L n m ) ,
n ̂ t E 1 r S = n ̂ t E 2 r S .
ρ 1 s E i t ( ρ 1 s ) + n , m [ a n m ρ 1 s h n ( 1 ) ( ρ 1 s ) m n m + b n m | d ( ρ h n ( 1 ) ( ρ ) ) d ρ | ρ 1 s n n m ] + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) × { E i r ( ρ 1 s ) + n , m [ b n m h n ( 1 ) ( ρ 1 s ) ρ 1 s n ( n + 1 ) o n m ] } = κ n , m { c n m ρ 2 s j n ( ρ 2 s ) m n m + [ d n m | d ( ρ j n ( ρ ) ) d ρ | ρ 2 s + e n m ρ 2 s j n ( ρ 3 s ) ] n n m } + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) × n , m [ d n m j n ( ρ 2 s ) ρ 2 s n ( n + 1 ) + e n m | d ( j n ( ρ ) ) d ρ | ρ 3 s k ] o n m ,
ρ 1 s H i t ( ρ 1 s ) + n , m χ 1 [ a n m | d ( ρ h n ( 1 ) ( ρ ) ) d ρ | ρ 1 s n n m + b n m ρ 1 s h n ( 1 ) ( ρ 1 s ) m n m ] + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) × { H i r ( ρ 1 s ) + n , m [ χ 1 a n m h n ( 1 ) ( ρ 1 s ) ρ 1 s n ( n + 1 ) o n m ] } = κ n , m χ 2 { c n m | d ( ρ j n ( ρ ) ) d ρ | ρ 2 s n n m + d n m ρ 2 s j n ( ρ 2 s ) m n m } + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) n , m [ χ 2 c n m j n ( ρ 2 s ) ρ 2 s n ( n + 1 ) o n m ] ,
ρ 1 s { E i r ( ρ 1 s ) + n , m [ b n m h n ( 1 ) ( ρ 1 s ) ρ 1 s n ( n + 1 ) o n m ] } η f θ ρ 10 { E i θ + n , m [ a n m h n ( 1 ) ( ρ 1 s ) u n m + b n m | d ( ρ h n ( 1 ) ( ρ ) ) ρ d ρ | ρ 1 s t n m ] } η sin   θ f φ ρ 10 { E i φ n , m [ a n m h n ( 1 ) ( ρ 1 s ) t n m b n m | d ( ρ h n ( 1 ) ( ρ ) ) ρ d ρ | ρ 1 s u n m ] } = n , m { ρ 1 s [ d n m j n ( ρ 2 s ) ρ 2 s n ( n + 1 ) + e n m | d ( j n ( ρ ) ) d ρ | ρ 3 s k ] o n m } η f θ ρ 10 [ c n m j n ( ρ 2 s ) u n m + ( d n m | d ( ρ j n ( ρ ) ) ρ d ρ | ρ 2 s + e n m j n ( ρ 3 s ) ρ 3 s k ) t n m ] { η sin   θ f φ ρ 10 [ c n m j n ( ρ 2 s ) t n m + ( d n m | d ( ρ j n ( ρ ) ) ρ d ρ | ρ 2 s + e n m j n ( ρ 3 s ) ρ 3 s k ) u n m ] } ,
t n m = d P n m ( cos θ ) d θ e i m φ and u n m = i m P n m ( cos θ ) sin θ e i m φ .
( a n m , b n m , c n m , d n m , e n m ) = j = 0 ( a n m j , b n m j , c n m j , d n m j , e n m j ) η j ,
j n ( ρ 3 s ) = j = 0 η j Δ n j ρ 30 j f j j ! ,     Δ n j d j d ρ j [ j n ( ρ ) ] ρ 30 ,
ρ 3 s j n ( ρ 3 s ) = j = 0 η j ξ n j ρ 30 j f j j ! ,     ξ n j d j d ρ j [ ρ 30 j n ( ρ ) ] ρ 30 ,
h n ( 1 ) ( ρ 1 s ) = j = 0 η j ζ n j ρ 10 j f j j ! ,     ζ n j d j d ρ j [ h 1 ( 1 ) ( ρ ) ] ρ 10 ,
j n ( ρ 2 s ) = j = 0 η j Ω n j ρ 20 j f j j ! ,     Ω n j d j d ρ j [ j n ( ρ ) ] ρ 20 ,
j n ( ρ 3 s ) ρ 3 s = j = 0 η j n j ρ 30 j f j j ! ,     n j | d j d ρ j [ j n ( ρ ) ρ ] | ρ 30 ,
ρ 1 s E i r ( ρ 1 s ) = j = 0 η j ( ρ E i r ) ( j ) ρ 10 j f j p ! ,
( ρ E i r ) ( j ) j ρ j ( ρ 10 E i r ( ρ ) ) ρ 10 ,
E i θ ( ρ 1 s ) = j = 0 η j ( E i θ ) ( j ) ρ 10 j f j p ! ,     ( E i θ ) ( j ) j ρ j E i θ ( ρ ) ρ 10 ,
E i φ ( ρ 1 s ) = j = 0 η j ( E i φ ) ( j ) ρ 10 j f j j ! ,     ( E i φ ) ( j ) j ρ j E i φ ( ρ ) ρ 10 ,
j = 0 η j ( ρ E i t ) ( j ) ρ 10 j f j j ! + j = 0 q = 0 j n , m η j [ a n m q α n j q ρ 10 j q f j q ( j q ) ! m n m + b n m q α n j q + 1 ρ 10 j q f j q ( j q ) ! n n m ] + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) j = 0 { η j [ ( E i r ) ( j ) ρ 10 j f j j ! + q = 0 j n , m b n m q γ n j q ρ 10 j q f j q ( j q ) ! n ( n + 1 ) o n m ] } = κ j = 0 q = 0 j n , m η j { c n m q β n j q ρ 20 j q f j q ( j q ) ! m n m + [ d n m q β n j q + 1 ρ 20 j q f j q ( j q ) ! + k 2 k e n m q ξ n j q ρ 30 j q f j q ( j q ) ! ] n n m } + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) j = 0 q = 0 j n , m η j [ d n m q δ n j q ρ 20 j q f j q ( j q ) ! n ( n + 1 ) + e n m q Δ n j q + 1 ρ 30 j q f j q ( j q ) ! ] o n m ,
j = 0 η j ( ρ H i t ) ( j ) ρ 10 j f j j ! + j = 0 q = 0 j n , m χ 1 η j [ a n m q α n j q + 1 ρ 10 j q f j q ( j q ) ! n n m + b n m q α n j q ρ 10 j q f j q ( j q ) ! m n m ] + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) × j = 0 { η j [ ( H i r ) ( j ) ρ 10 j f j j ! + χ 1 q = 0 j n , m a n m q γ n j q ρ 10 j q f j q ( j q ) ! n ( n + 1 ) o n m ] } = κ p = 0 q = 0 j n , m χ 2 η j [ c n m q β n j q + 1 ρ 20 j q f j q ( j q ) ! n n m + d n m q β n j q ρ 20 j q f j q ( j q ) ! m n m ] + η ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) j = 0 q = 0 j n , m χ 2 η j c n m q δ n j q ρ 20 j q f j q ( j q ) ! n ( n + 1 ) o n m ,
j = 0 η j ( ρ E i r ) ( j ) ρ 10 j f j j ! + j = 0 q = 0 j n , m η j b n m q ζ n j q ρ 10 j q f j q ( j q ) ! n ( n + 1 ) o n m η f θ ρ 10 [ j = 0 η j ( E i θ ) ( j ) ρ 10 j f j j ! + j = 0 q = 0 j n , m η j a n m q ζ n j q ρ 10 j q f j q ( j q ) ! u n m + j = 0 q = 0 j n , m η j b n m q ( γ n j q + ζ n j q + 1 ) ρ 10 j q f j q ( j q ) ! t n m ] η sin   θ f φ ρ 10 [ j = 0 η j ( E i φ ) ( j ) ρ 10 j f j j ! j = 0 q = 0 j n , m η j a n m q ζ n j q ρ 10 j q f j q ( j q ) ! t n m + j = 0 q = 0 j n , m η j b n m q ( γ n j q + ζ n j q + 1 ) ρ 10 j q f j q ( j q ) ! u n m ] = j = 0 q = 0 j n , m η j { k 1 k 2 d n m j Ω n j q ρ 20 j q f j q ( j q ) ! n ( n + 1 ) o n m + k 1 e n m q ( ξ n j q + 1 + Δ n j q ) ρ 30 j q f j q ( j q ) ! o n m η f θ ρ 10 [ c n m q Ω n j q ρ 20 j q f j q ( j q ) ! u n m + d n m q ( δ n j q + Ω n j q + 1 ) ρ 20 j q f j q ( j q ) ! t m n + e n m q n j q ρ 30 j q f j q ( j q ) ! k t n m ] η sin   θ f φ ρ 10 [ c n m q Ω n j q ρ 20 j q f j q ( j q ) ! t n m + d n m q ( δ n j q + Ω n j q + 1 ) ρ 20 j q f j q ( j q ) ! u n m + e n m q n j q ρ 30 j q f j q ( j q ) ! k u n m ] } .
( ρ E i t ) ( p ) ρ 10 p f p p ! + q = 0 p n , m [ a n m q α n p q ρ 10 p q f p q ( p q ) ! m n m + b n m q α n p q + 1 ρ 10 p q f p q ( p q ) ! n n m ] + ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) [ ( E i r ) ( p 1 ) ρ 10 p f p 1 ( p 1 ) ! + q = 0 p 1 n , m b n m q γ n p 1 q ρ 10 p q f p 1 q ( p 1 q ) ! n ( n + 1 ) o n m ] = κ q = 0 p n , m { c n m q β n p q ρ 20 p q f p q ( p q ) ! m n m + [ d n m q β n p q + 1 ρ 20 p q f p q ( p q ) ! + k 2 k e n m q ξ n p q ρ 30 p q f p q ( p q ) ! ] n n m } + ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) q = 0 p 1 n , m [ d n m q δ n p 1 q ρ 20 p 1 q f p 1 q ( p 1 q ) ! n ( n + 1 ) + e n m q Δ n p q ρ 30 p 1 q f p 1 q ( p 1 q ) ! ] o n m ,
( ρ H i t ) ( p ) ρ 10 p f p p ! + q = 0 p n , m χ 1 [ a n m q α n p q + 1 ρ 10 p q f p q ( p q ) ! n n m + b n m q α n p q ρ 10 p q f p q ( p q ) ! m n m ] + ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) [ ( H i r ) ( p 1 ) ρ 10 p f p 1 ( p 1 ) ! + χ 1 q = 0 p 1 n , m a n m q γ n p 1 q ρ 10 p q f p 1 q ( p 1 q ) ! n ( n + 1 ) o n m ] = κ q = 0 p n , m χ 2 [ c n m q β n p q + 1 ρ 20 p q f p q ( p q ) ! n n m + d n m q β n p q ρ 20 p q f p q ( p q ) ! m n m ] + ρ 10 ( f θ e ̂ θ + 1 sin   θ f φ e ̂ φ ) q = 0 p 1 n , m χ 2 c n m q δ n p 1 q ρ 20 p 1 q f p 1 q ( p 1 q ) ! n ( n + 1 ) o n m ,
( ρ E i r ) ( p ) ρ 10 p f p p ! + q = 0 p n , m b n m q ζ n p q ρ 10 p q f p q ( p q ) ! n ( n + 1 ) o n m f θ [ ( E i θ ) ( p 1 ) ρ 10 p f p 1 ( p 1 ) ! + q = 0 p 1 n , m a n m q ζ n p 1 q ρ 10 p q f p 1 q ( p 1 q ) ! u n m + q = 0 p 1 n , m b n m q ( γ n p 1 q + ζ n p q ) ρ 10 p q f p 1 q ( p 1 q ) ! t n m ] 1 sin   θ f φ [ ( E i φ ) ( p 1 ) ρ 10 p f p 1 ( p 1 ) ! q = 0 p 1 n , m a n m q ζ n p 1 q ρ 10 p q f p 1 q ( p 1 q ) ! t n m + q = 0 p 1 n , m b n m q ( γ n p 1 q + ζ n p q ) ρ 10 p q f p 1 q ( p 1 q ) ! u n m ] = q = 0 p n , m [ k 1 k 2 d n m q Ω n p q ρ 20 p q f p q ( p q ) ! n ( n + 1 ) + k 1 e n m q ( ξ n p q + 1 + Δ n p q ) ρ 30 p q f p q ( p q ) ! ] o n m f θ