Abstract

We report an investigation on the optical properties of three-dimensional nanoclusters (NCs) made by spherical constellations of metallic nanospheres arranged around a central dielectric sphere, which can be realized and assembled by current state-of-the-art nanochemistry techniques. This type of NCs supports collective plasmon modes among which the most relevant are those associated with the induced electric and magnetic resonances. Combining a single dipole approximation for each nanoparticle and the multipole spherical-wave expansion of the scattered field, we achieve an effective characterization of the optical response of individual NCs in terms of their scattering, absorption, and extinction efficiencies. By this approximate model we analyze a few sample NCs identifying the electric and magnetic resonance frequencies and their dependence on the size and number of the constituent nanoparticles. Furthermore, we discuss the effective electric and magnetic polarizabilities of the NCs, and their isotropic properties. A homogenization method based on an extension of the Maxwell Garnett model to account for interaction effects due to higher order multipoles in dense packed arrays is applied to a distribution of NCs showing the possibility of obtaining metamaterials with very large, small, and negative values of permittivity and permeability, and even negative index.

© 2011 OSA

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  1. V. Ponsinet, A. Aradian, P. Barois, and S. Ravaine, “Self-assembly and nanochemistry techniques towards the fabrication of metamaterials,” in Applications of Metamaterials, Ed. F. Capolino, CRC Press, Boca Raton, FL, 2009, Chap. 32.
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  3. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
    [CrossRef] [PubMed]
  4. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
    [CrossRef] [PubMed]
  5. Q. Xu, R. M. Rioux, M. D. Dickey, and G. M. Whitesides, “Nanoskiving: a new method to produce arrays of nanostructures,” Acc. Chem. Res. 41(12), 1566–1577 (2008).
    [CrossRef] [PubMed]
  6. H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94(6), 063901 (2005).
    [CrossRef] [PubMed]
  7. N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
    [CrossRef]
  8. V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005).
    [CrossRef]
  9. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
    [CrossRef]
  10. A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express 14(4), 1557–1567 (2006).
    [CrossRef] [PubMed]
  11. Y. A. Urzhumov, G. Shvets, J. A. Fan, F. Capasso, D. Brandl, and P. Nordlander, “Plasmonic nanoclusters: a path towards negative-index metafluids,” Opt. Express 15(21), 14129–14145 (2007).
    [CrossRef] [PubMed]
  12. C. R. Simovski and S. A. Tretyakov, “Model of isotropic resonant magnetism in the visible range based on core-shell clusters,” Phys. Rev. B 79(4), 045111 (2009).
    [CrossRef]
  13. A. Alù and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17(7), 5723–5730 (2009).
    [CrossRef] [PubMed]
  14. J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
    [CrossRef] [PubMed]
  15. C. F. Bohren, and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983).
  16. S. Steshenko, and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers” in Theory and Phenomena of Metamaterials, Ed. F. Capolino, (CRC Press, Boca Raton, FL, 2009), Chap. 8.
  17. J. E. Hansen, Spherical Near-Field Antenna Measurements (Peter Peregrinus Ltd, London, 1988).
  18. J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, “Self-assembled monolayers of thiolates on metals as a form of nanotechnology,” Chem. Rev. 105(4), 1103–1169 (2005).
    [CrossRef] [PubMed]
  19. J. Israelachvili, Intermolecular and surface forces, Academic Press (2007).
  20. S. Reculusa, C. Poncet-Legrand, S. Ravaine, C. Mingotaud, E. Duguet, and E. Bourgeat-Lami, “Synthesis of raspberry-like silica/polystyrene materials,” Chem. Mater. 14(5), 2354–2359 (2002).
    [CrossRef]
  21. J.-C. Taveau, D. Nguyen, A. Perro, S. Ravaine, E. Duguet, A. Brisson, and O. Lambert, “New insights into the nucleation and growth of PS nodules on silica nanoparticles by 3D cryo-electron tomography,” Soft Matter 4(2), 311–315 (2008).
    [CrossRef]
  22. B. W. Clare and D. L. Kepert, “The closest packing of equal circles on a sphere,” Proc. R. Soc. Lond. A Math. Phys. Sci. 405(1829), 329–344 (1986).
    [CrossRef]
  23. J. R. Edmundson, “The distribution of point charges on the surface of a sphere,” Acta Crystallogr. A 48(1), 60–69 (1992).
    [CrossRef]
  24. D. A. Kottwitz, “The densest packing of equal circles on a sphere,” Acta Crystallogr. A 47(3), 158–165 (1991).
    [CrossRef]
  25. A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
    [CrossRef]
  26. M. Wheeler, J. Aitchison, and M. Mojahedi, “Coupled magnetic dipole resonances in sub-wavelength dielectric particle clusters,” J. Opt. Soc. Am. B 27(5), 1083–1091 (2010).
    [CrossRef]
  27. D. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11(11), 2851–2861 (1994).
    [CrossRef]
  28. I. Romero, J. Aizpurua, G. W. Bryant, and F. J. García De Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14(21), 9988–9999 (2006).
    [CrossRef] [PubMed]
  29. K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” N. J. Phys. 7, 168 (2005).
    [CrossRef]
  30. O. N. Singh, and A. Lakhtakia, eds., Electromagnetic Waves in Unconventional Materials and Structures (John Wiley, New York, 2000).
  31. P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59(8), 2609 (1986).
    [CrossRef]
  32. A. Moroz and C. Sommers, “Photonic band gaps of three-dimensional face-centred cubic lattices,” J. Phys. Condens. Matter 11(4), 997–1008 (1999).
    [CrossRef]
  33. A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66(11), 115109 (2002).
    [CrossRef]
  34. V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequencies,” J. Phys. Condens. Matter 17(25), 3717–3734 (2005).
    [CrossRef] [PubMed]
  35. S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).

2010

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

M. Wheeler, J. Aitchison, and M. Mojahedi, “Coupled magnetic dipole resonances in sub-wavelength dielectric particle clusters,” J. Opt. Soc. Am. B 27(5), 1083–1091 (2010).
[CrossRef]

2009

A. Alù and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17(7), 5723–5730 (2009).
[CrossRef] [PubMed]

C. R. Simovski and S. A. Tretyakov, “Model of isotropic resonant magnetism in the visible range based on core-shell clusters,” Phys. Rev. B 79(4), 045111 (2009).
[CrossRef]

2008

J.-C. Taveau, D. Nguyen, A. Perro, S. Ravaine, E. Duguet, A. Brisson, and O. Lambert, “New insights into the nucleation and growth of PS nodules on silica nanoparticles by 3D cryo-electron tomography,” Soft Matter 4(2), 311–315 (2008).
[CrossRef]

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Q. Xu, R. M. Rioux, M. D. Dickey, and G. M. Whitesides, “Nanoskiving: a new method to produce arrays of nanostructures,” Acc. Chem. Res. 41(12), 1566–1577 (2008).
[CrossRef] [PubMed]

2007

2006

2005

J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, “Self-assembled monolayers of thiolates on metals as a form of nanotechnology,” Chem. Rev. 105(4), 1103–1169 (2005).
[CrossRef] [PubMed]

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94(6), 063901 (2005).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” N. J. Phys. 7, 168 (2005).
[CrossRef]

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequencies,” J. Phys. Condens. Matter 17(25), 3717–3734 (2005).
[CrossRef] [PubMed]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30(24), 3356–3358 (2005).
[CrossRef]

2003

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[CrossRef]

2002

S. Reculusa, C. Poncet-Legrand, S. Ravaine, C. Mingotaud, E. Duguet, and E. Bourgeat-Lami, “Synthesis of raspberry-like silica/polystyrene materials,” Chem. Mater. 14(5), 2354–2359 (2002).
[CrossRef]

A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66(11), 115109 (2002).
[CrossRef]

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

1999

A. Moroz and C. Sommers, “Photonic band gaps of three-dimensional face-centred cubic lattices,” J. Phys. Condens. Matter 11(4), 997–1008 (1999).
[CrossRef]

1994

1992

J. R. Edmundson, “The distribution of point charges on the surface of a sphere,” Acta Crystallogr. A 48(1), 60–69 (1992).
[CrossRef]

1991

D. A. Kottwitz, “The densest packing of equal circles on a sphere,” Acta Crystallogr. A 47(3), 158–165 (1991).
[CrossRef]

1986

B. W. Clare and D. L. Kepert, “The closest packing of equal circles on a sphere,” Proc. R. Soc. Lond. A Math. Phys. Sci. 405(1829), 329–344 (1986).
[CrossRef]

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59(8), 2609 (1986).
[CrossRef]

Aitchison, J.

Aizpurua, J.

Alù, A.

Aydin, K.

K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” N. J. Phys. 7, 168 (2005).
[CrossRef]

Bao, J.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Bao, K.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Bardhan, R.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Bourgeat-Lami, E.

S. Reculusa, C. Poncet-Legrand, S. Ravaine, C. Mingotaud, E. Duguet, and E. Bourgeat-Lami, “Synthesis of raspberry-like silica/polystyrene materials,” Chem. Mater. 14(5), 2354–2359 (2002).
[CrossRef]

Brandl, D.

Brisson, A.

J.-C. Taveau, D. Nguyen, A. Perro, S. Ravaine, E. Duguet, A. Brisson, and O. Lambert, “New insights into the nucleation and growth of PS nodules on silica nanoparticles by 3D cryo-electron tomography,” Soft Matter 4(2), 311–315 (2008).
[CrossRef]

Bryant, G. W.

Bulu, I.

K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” N. J. Phys. 7, 168 (2005).
[CrossRef]

Cai, W.

Capasso, F.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Y. A. Urzhumov, G. Shvets, J. A. Fan, F. Capasso, D. Brandl, and P. Nordlander, “Plasmonic nanoclusters: a path towards negative-index metafluids,” Opt. Express 15(21), 14129–14145 (2007).
[CrossRef] [PubMed]

Casse, B. D. F.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94(6), 063901 (2005).
[CrossRef] [PubMed]

Chettiar, U. K.

Clare, B. W.

B. W. Clare and D. L. Kepert, “The closest packing of equal circles on a sphere,” Proc. R. Soc. Lond. A Math. Phys. Sci. 405(1829), 329–344 (1986).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Dickey, M. D.

Q. Xu, R. M. Rioux, M. D. Dickey, and G. M. Whitesides, “Nanoskiving: a new method to produce arrays of nanostructures,” Acc. Chem. Res. 41(12), 1566–1577 (2008).
[CrossRef] [PubMed]

Dolling, G.

Drachev, V. P.

Duguet, E.

J.-C. Taveau, D. Nguyen, A. Perro, S. Ravaine, E. Duguet, A. Brisson, and O. Lambert, “New insights into the nucleation and growth of PS nodules on silica nanoparticles by 3D cryo-electron tomography,” Soft Matter 4(2), 311–315 (2008).
[CrossRef]

S. Reculusa, C. Poncet-Legrand, S. Ravaine, C. Mingotaud, E. Duguet, and E. Bourgeat-Lami, “Synthesis of raspberry-like silica/polystyrene materials,” Chem. Mater. 14(5), 2354–2359 (2002).
[CrossRef]

Edmundson, J. R.

J. R. Edmundson, “The distribution of point charges on the surface of a sphere,” Acta Crystallogr. A 48(1), 60–69 (1992).
[CrossRef]

Engheta, N.

Estroff, L. A.

J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, “Self-assembled monolayers of thiolates on metals as a form of nanotechnology,” Chem. Rev. 105(4), 1103–1169 (2005).
[CrossRef] [PubMed]

Fan, J. A.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Y. A. Urzhumov, G. Shvets, J. A. Fan, F. Capasso, D. Brandl, and P. Nordlander, “Plasmonic nanoclusters: a path towards negative-index metafluids,” Opt. Express 15(21), 14129–14145 (2007).
[CrossRef] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Fu, L.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

García De Abajo, F. J.

Giessen, H.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Guo, H.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Guven, K.

K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” N. J. Phys. 7, 168 (2005).
[CrossRef]

Halas, N. J.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[CrossRef]

Jandhyala, V.

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Kafesaki, M.

K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” N. J. Phys. 7, 168 (2005).
[CrossRef]

Kaiser, S.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Kepert, D. L.

B. W. Clare and D. L. Kepert, “The closest packing of equal circles on a sphere,” Proc. R. Soc. Lond. A Math. Phys. Sci. 405(1829), 329–344 (1986).
[CrossRef]

Kildishev, A. V.

Kottwitz, D. A.

D. A. Kottwitz, “The densest packing of equal circles on a sphere,” Acta Crystallogr. A 47(3), 158–165 (1991).
[CrossRef]

Kriebel, J. K.

J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, “Self-assembled monolayers of thiolates on metals as a form of nanotechnology,” Chem. Rev. 105(4), 1103–1169 (2005).
[CrossRef] [PubMed]

Kuga, Y.

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[CrossRef]

Lambert, O.

J.-C. Taveau, D. Nguyen, A. Perro, S. Ravaine, E. Duguet, A. Brisson, and O. Lambert, “New insights into the nucleation and growth of PS nodules on silica nanoparticles by 3D cryo-electron tomography,” Soft Matter 4(2), 311–315 (2008).
[CrossRef]

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Lee, S. W.

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[CrossRef]

Linden, S.

Liu, N.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Love, J. C.

J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, “Self-assembled monolayers of thiolates on metals as a form of nanotechnology,” Chem. Rev. 105(4), 1103–1169 (2005).
[CrossRef] [PubMed]

Mackowski, D.

Manoharan, V. N.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Mingotaud, C.

S. Reculusa, C. Poncet-Legrand, S. Ravaine, C. Mingotaud, E. Duguet, and E. Bourgeat-Lami, “Synthesis of raspberry-like silica/polystyrene materials,” Chem. Mater. 14(5), 2354–2359 (2002).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Mojahedi, M.

Moroz, A.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequencies,” J. Phys. Condens. Matter 17(25), 3717–3734 (2005).
[CrossRef] [PubMed]

A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66(11), 115109 (2002).
[CrossRef]

A. Moroz and C. Sommers, “Photonic band gaps of three-dimensional face-centred cubic lattices,” J. Phys. Condens. Matter 11(4), 997–1008 (1999).
[CrossRef]

Moser, H. O.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94(6), 063901 (2005).
[CrossRef] [PubMed]

Nguyen, D.

J.-C. Taveau, D. Nguyen, A. Perro, S. Ravaine, E. Duguet, A. Brisson, and O. Lambert, “New insights into the nucleation and growth of PS nodules on silica nanoparticles by 3D cryo-electron tomography,” Soft Matter 4(2), 311–315 (2008).
[CrossRef]

Nordlander, P.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Y. A. Urzhumov, G. Shvets, J. A. Fan, F. Capasso, D. Brandl, and P. Nordlander, “Plasmonic nanoclusters: a path towards negative-index metafluids,” Opt. Express 15(21), 14129–14145 (2007).
[CrossRef] [PubMed]

Nuzzo, R. G.

J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, “Self-assembled monolayers of thiolates on metals as a form of nanotechnology,” Chem. Rev. 105(4), 1103–1169 (2005).
[CrossRef] [PubMed]

Ozbay, E.

K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” N. J. Phys. 7, 168 (2005).
[CrossRef]

Pedersen, N. E.

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59(8), 2609 (1986).
[CrossRef]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Perro, A.

J.-C. Taveau, D. Nguyen, A. Perro, S. Ravaine, E. Duguet, A. Brisson, and O. Lambert, “New insights into the nucleation and growth of PS nodules on silica nanoparticles by 3D cryo-electron tomography,” Soft Matter 4(2), 311–315 (2008).
[CrossRef]

Poncet-Legrand, C.

S. Reculusa, C. Poncet-Legrand, S. Ravaine, C. Mingotaud, E. Duguet, and E. Bourgeat-Lami, “Synthesis of raspberry-like silica/polystyrene materials,” Chem. Mater. 14(5), 2354–2359 (2002).
[CrossRef]

Ravaine, S.

J.-C. Taveau, D. Nguyen, A. Perro, S. Ravaine, E. Duguet, A. Brisson, and O. Lambert, “New insights into the nucleation and growth of PS nodules on silica nanoparticles by 3D cryo-electron tomography,” Soft Matter 4(2), 311–315 (2008).
[CrossRef]

S. Reculusa, C. Poncet-Legrand, S. Ravaine, C. Mingotaud, E. Duguet, and E. Bourgeat-Lami, “Synthesis of raspberry-like silica/polystyrene materials,” Chem. Mater. 14(5), 2354–2359 (2002).
[CrossRef]

Reculusa, S.

S. Reculusa, C. Poncet-Legrand, S. Ravaine, C. Mingotaud, E. Duguet, and E. Bourgeat-Lami, “Synthesis of raspberry-like silica/polystyrene materials,” Chem. Mater. 14(5), 2354–2359 (2002).
[CrossRef]

Rioux, R. M.

Q. Xu, R. M. Rioux, M. D. Dickey, and G. M. Whitesides, “Nanoskiving: a new method to produce arrays of nanostructures,” Acc. Chem. Res. 41(12), 1566–1577 (2008).
[CrossRef] [PubMed]

Romero, I.

Salandrino, A.

Sarychev, A. K.

Saw, B. T.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94(6), 063901 (2005).
[CrossRef] [PubMed]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Schweizer, H.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Shalaev, V. M.

Shvets, G.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Y. A. Urzhumov, G. Shvets, J. A. Fan, F. Capasso, D. Brandl, and P. Nordlander, “Plasmonic nanoclusters: a path towards negative-index metafluids,” Opt. Express 15(21), 14129–14145 (2007).
[CrossRef] [PubMed]

Simovski, C. R.

C. R. Simovski and S. A. Tretyakov, “Model of isotropic resonant magnetism in the visible range based on core-shell clusters,” Phys. Rev. B 79(4), 045111 (2009).
[CrossRef]

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Sommers, C.

A. Moroz and C. Sommers, “Photonic band gaps of three-dimensional face-centred cubic lattices,” J. Phys. Condens. Matter 11(4), 997–1008 (1999).
[CrossRef]

Soukoulis, C. M.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32(1), 53–55 (2007).
[CrossRef]

K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” N. J. Phys. 7, 168 (2005).
[CrossRef]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Taveau, J.-C.

J.-C. Taveau, D. Nguyen, A. Perro, S. Ravaine, E. Duguet, A. Brisson, and O. Lambert, “New insights into the nucleation and growth of PS nodules on silica nanoparticles by 3D cryo-electron tomography,” Soft Matter 4(2), 311–315 (2008).
[CrossRef]

Tretyakov, S. A.

C. R. Simovski and S. A. Tretyakov, “Model of isotropic resonant magnetism in the visible range based on core-shell clusters,” Phys. Rev. B 79(4), 045111 (2009).
[CrossRef]

Urzhumov, Y. A.

Waterman, P. C.

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59(8), 2609 (1986).
[CrossRef]

Wegener, M.

Wheeler, M.

Whitesides, G. M.

Q. Xu, R. M. Rioux, M. D. Dickey, and G. M. Whitesides, “Nanoskiving: a new method to produce arrays of nanostructures,” Acc. Chem. Res. 41(12), 1566–1577 (2008).
[CrossRef] [PubMed]

J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, “Self-assembled monolayers of thiolates on metals as a form of nanotechnology,” Chem. Rev. 105(4), 1103–1169 (2005).
[CrossRef] [PubMed]

Wilhelmi, O.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94(6), 063901 (2005).
[CrossRef] [PubMed]

Wu, C.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Xu, Q.

Q. Xu, R. M. Rioux, M. D. Dickey, and G. M. Whitesides, “Nanoskiving: a new method to produce arrays of nanostructures,” Acc. Chem. Res. 41(12), 1566–1577 (2008).
[CrossRef] [PubMed]

Yannopapas, V.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequencies,” J. Phys. Condens. Matter 17(25), 3717–3734 (2005).
[CrossRef] [PubMed]

Yuan, H. K.

Zhang, X.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Acc. Chem. Res.

Q. Xu, R. M. Rioux, M. D. Dickey, and G. M. Whitesides, “Nanoskiving: a new method to produce arrays of nanostructures,” Acc. Chem. Res. 41(12), 1566–1577 (2008).
[CrossRef] [PubMed]

Acta Crystallogr. A

J. R. Edmundson, “The distribution of point charges on the surface of a sphere,” Acta Crystallogr. A 48(1), 60–69 (1992).
[CrossRef]

D. A. Kottwitz, “The densest packing of equal circles on a sphere,” Acta Crystallogr. A 47(3), 158–165 (1991).
[CrossRef]

Chem. Mater.

S. Reculusa, C. Poncet-Legrand, S. Ravaine, C. Mingotaud, E. Duguet, and E. Bourgeat-Lami, “Synthesis of raspberry-like silica/polystyrene materials,” Chem. Mater. 14(5), 2354–2359 (2002).
[CrossRef]

Chem. Rev.

J. C. Love, L. A. Estroff, J. K. Kriebel, R. G. Nuzzo, and G. M. Whitesides, “Self-assembled monolayers of thiolates on metals as a form of nanotechnology,” Chem. Rev. 105(4), 1103–1169 (2005).
[CrossRef] [PubMed]

IEEE Trans. Antenn. Propag.

A. Ishimaru, S. W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static Lorentz theory,” IEEE Trans. Antenn. Propag. 51(10), 2550–2557 (2003).
[CrossRef]

J. Appl. Phys.

P. C. Waterman and N. E. Pedersen, “Electromagnetic scattering by periodic arrays of particles,” J. Appl. Phys. 59(8), 2609 (1986).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. Condens. Matter

A. Moroz and C. Sommers, “Photonic band gaps of three-dimensional face-centred cubic lattices,” J. Phys. Condens. Matter 11(4), 997–1008 (1999).
[CrossRef]

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequencies,” J. Phys. Condens. Matter 17(25), 3717–3734 (2005).
[CrossRef] [PubMed]

N. J. Phys.

K. Aydin, I. Bulu, K. Guven, M. Kafesaki, C. M. Soukoulis, and E. Ozbay, “Investigation of magnetic resonances for different split-ring resonator parameters and designs,” N. J. Phys. 7, 168 (2005).
[CrossRef]

Nat. Mater.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

C. R. Simovski and S. A. Tretyakov, “Model of isotropic resonant magnetism in the visible range based on core-shell clusters,” Phys. Rev. B 79(4), 045111 (2009).
[CrossRef]

A. Moroz, “Metallo-dielectric diamond and zinc-blende photonic crystals,” Phys. Rev. B 66(11), 115109 (2002).
[CrossRef]

Phys. Rev. Lett.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rod-split-ring-resonator electromagnetic metamaterial,” Phys. Rev. Lett. 94(6), 063901 (2005).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci.

B. W. Clare and D. L. Kepert, “The closest packing of equal circles on a sphere,” Proc. R. Soc. Lond. A Math. Phys. Sci. 405(1829), 329–344 (1986).
[CrossRef]

Science

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[CrossRef] [PubMed]

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328(5982), 1135–1138 (2010).
[CrossRef] [PubMed]

Soft Matter

J.-C. Taveau, D. Nguyen, A. Perro, S. Ravaine, E. Duguet, A. Brisson, and O. Lambert, “New insights into the nucleation and growth of PS nodules on silica nanoparticles by 3D cryo-electron tomography,” Soft Matter 4(2), 311–315 (2008).
[CrossRef]

Other

J. Israelachvili, Intermolecular and surface forces, Academic Press (2007).

C. F. Bohren, and D. R. Huffman, Absorption and scattering of light by small particles (Wiley, New York, 1983).

S. Steshenko, and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers” in Theory and Phenomena of Metamaterials, Ed. F. Capolino, (CRC Press, Boca Raton, FL, 2009), Chap. 8.

J. E. Hansen, Spherical Near-Field Antenna Measurements (Peter Peregrinus Ltd, London, 1988).

V. Ponsinet, A. Aradian, P. Barois, and S. Ravaine, “Self-assembly and nanochemistry techniques towards the fabrication of metamaterials,” in Applications of Metamaterials, Ed. F. Capolino, CRC Press, Boca Raton, FL, 2009, Chap. 32.

O. N. Singh, and A. Lakhtakia, eds., Electromagnetic Waves in Unconventional Materials and Structures (John Wiley, New York, 2000).

S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).

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Figures (13)

Fig. 1
Fig. 1

Examples of NC geometries analyzed in this paper: (a) 4-element tetrahedral and (b) 12-element icosahedral NCs derived by locating nanospheres at the vertices of the respective Platonic solid; (c) 48-element NC with maximized minimum interparticle distance.

Fig. 2
Fig. 2

(a) Electric and (b) magnetic resonant modes for a sample NC.

Fig. 3
Fig. 3

Extinction, scattering, and absorption efficiencies of the tetrahedral NC type-A.

Fig. 4
Fig. 4

Extinction, scattering, and absorption efficiencies of the icosahedral NC type-B.

Fig. 5
Fig. 5

Extinction, scattering, and absorption efficiencies of the pseudo-regular NC type-C.

Fig. 6
Fig. 6

Polarizabilities of the tetrahedral NC type-A. (a) Electric, (b)-(c) magneto-electric, and (d) magnetic polarizabilities.

Fig. 7
Fig. 7

Polarizabilities of the icosahedral NC type-B. (a) Electric, (b)-(c) magneto-electric, and (d) magnetic polarizabilities.

Fig. 8
Fig. 8

Polarizabilities of the pseudo-regular NCtype-C. (a) Electric, (b)-(c) magneto-electric, and (d) magnetic polarizabilities.

Fig. 9
Fig. 9

Spherical wave spectral footprint (N = 5) calculated at 600 THz for NCs (a) type-A, (b) type-B, and (c) type-C. Colorbar unit is dB.

Fig. 10
Fig. 10

(a) Permittivity and (b) permeability of a close-packed 3D periodic fcc array of tetrahedral NCs type-A calculated by MG (black lines) and WP (red lines) formulas. Real and imaginary parts of the parameters are shown in solid and dashed lines, respectively.

Fig. 11
Fig. 11

As in Fig. 9, but for a close-packed 3D periodic fcc array of icosahedral NCs, type-B.

Fig. 12
Fig. 12

As in Fig. 9 or 10, but for close-packed 3D periodic arrays of pseudo-regular NCs type-C.

Fig. 13
Fig. 13

As in Fig. 9, but here the radius of silver nanospheres is 11 nm and the overall dimension of an individual tetrahedral cluster is D = 57.8 nm.

Equations (13)

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E ( r , θ , ϕ ) = k h ζ h s = 1 2 n = 1 N m = n n Q s m n ( 3 ) F s m n ( 3 ) ( r , θ , ϕ ) , H ( r , θ , ϕ ) = i k h ζ h s = 1 2 n = 1 N m = n n Q s m n ( 3 ) F 3 s , m , n ( 3 ) ( r , θ , ϕ ) , r 0 < r <
α = 6 i π ε 0 ε h k 3 m ψ 1 ( m k a ) ψ 1 ( k a ) ψ 1 ( k a ) ψ 1 ( m k a ) m ψ 1 ( m k a ) ξ 1 ( k a ) ξ 1 ( k a ) ψ 1 ( m k a ) ,
Q s m n ( 3 ) = ( 1 ) m + 1 V ( k h ζ h F s , m , n ( 1 ) J + i k h ζ h F 3 s , m , n ( 1 ) M ) d V .
J = i ω q = 1 N p p q ( r q ) δ ( r r q )
Q s m n ( 3 ) = i ω k h ζ h ( 1 ) m q = 1 N p F s , m , n ( 1 ) ( r q ) p q ( r q ) .
C e x t = ζ h 2 | E i | 2 Re [ s m n Q s m n ( 3 ) ( Q s m n i ) * ] , C s c a = ζ h 2 | E i | 2 s m n | Q s m n ( 3 ) | 2 ,
C a b s = C e x t C s c a
p e x = i c e ( Q 2 , 1 , 1 ( 3 ) Q 2 , 1 , 1 ( 3 ) ) , p e y = c e ( Q 2 , 1 , 1 ( 3 ) + Q 2 , 1 , 1 ( 3 ) ) , p e z = i 2 c e Q 201 ( 3 ) , p m x = c m ( Q 1 , 1 , 1 ( 3 ) Q 1 , 1 , 1 ( 3 ) ) , p m y = i c m ( Q 1 , 1 , 1 ( 3 ) + Q 1 , 1 , 1 ( 3 ) ) , p m z = 2 c m Q 101 ( 3 ) ,
Q x x = c q [ Q 202 ( 3 ) + 6 ( Q 222 ( 3 ) + Q 2 , 2 , 2 ( 3 ) ) ] , Q x y = i 6 c q ( Q 222 ( 3 ) Q 2 , 2 , 2 ( 3 ) ) , Q y y = c q [ Q 202 ( 3 ) 6 ( Q 222 ( 3 ) + Q 2 , 2 , 2 ( 3 ) ) ] , Q y z = 4 6 c q 14 + 15 ( Q 212 ( 3 ) + Q 2 , 1 , 2 ( 3 ) ) Q z z = c q Q 202 ( 3 ) , Q x z = 4 6 c q 14 + 15 ( Q 212 ( 3 ) Q 2 , 1 , 2 ( 3 ) )
[ p e p h ] = a ¯ [ E l o c H l o c ] = [ a ¯ e e a ¯ e m a ¯ m e a ¯ m m ] [ E l o c H l o c ] ,
ε r e f f = ε h ( 1 3 2 t 1 e f G e ) , μ r e f f = ( 1 3 2 t 1 h f G h )
G v = 1 + t 1 v f 2 t 1 v f [ 6 t 3 v σ 04 2 ( 3 f 4 π α ) 10 3 1 + 15 t 3 v σ 06 ( 3 f 4 π α ) 7 8 + 220 3 t 5 v σ 06 2 ( 3 f 4 π α ) 14 3 + 1120 33 t 7 v σ 08 2 ( 3 f 4 π α ) 6 ]
t 1 v = 3 i ( k h R e ) 3 S s 11 ; s 11 1 + S s 11 ; s 11 , t n v = 2 i ( 2 n ) ! ( 2 n + 1 ) ! ( n ! ) 2 ( 2 k h R e ) 2 n + 1 S s 1 n ; s 1 n , n = 3 , 5 , 7

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