Abstract

We compute near-field radiative transfer between two spheres of unequal radii R1 and R2 such that R2 ≲ 40R1. For R2 = 40R1, the smallest gap to which we have been able to compute radiative transfer is d = 0.016R1. To accomplish these computations, we have had to modify existing methods for computing near-field radiative transfer between two spheres in the following ways: (1) exact calculations of coefficients of vector translation theorem are replaced by approximations valid for the limit dR1, and (2) recursion relations for a normalized form of translation coefficients are derived which enable us to replace computations of spherical Bessel and Hankel functions by computations of ratios of spherical Bessel or spherical Hankel functions. The results are then compared with the predictions of the modified proximity approximation.

© 2014 Optical Society of America

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  32. W. M. Hirsch, A. Kraft, M. T. Hirsch, J. Parisi, and A. Kittel, “Heat transfer in ultrahigh vacuum scanning thermal microscopy,” J. Vac. Sci. Technol. A 17(4), 1205–1210 (1999).
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  43. K. Sasihithlu and A. Narayanaswamy, “Proximity effects in radiative heat transfer,” Phys. Rev. B 83(16), 161406 (2011).
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  45. S. K. Lamoreaux, “Demonstration of the Casimir force in the 0.6 to 6μm range,” Phys. Rev. Lett. 78, 5–8 (1997).
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  46. H. Gies and K. Klingmüller, “Casimir effect for curved geometries: Proximity-force-approximation validity limits,” Phys. Rev. Lett. 96(22), 220401 (2006).
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2014 (1)

A. Narayanaswamy and Y. Zheng, “A Green’s function formalism of energy and momentum transfer in fluctuational electrodynamics,” J. Quant. Spectrosc. Radiat. Transfer 132, 12–21 (2014).
[CrossRef]

2013 (2)

L. Worbes, D. Hellmann, and A. Kittel, “Enhanced near-field heat flow of a monolayer dielectric island,” Phys. Rev. Lett. 110(13), 134302 (2013).
[CrossRef] [PubMed]

V. Golyk, M. Krüger, A. P. McCauley, and M. Kardar, “Small distance expansion for radiative heat transfer between curved objects,” Europhys. Lett. 101(3), 34002 (2013).
[CrossRef]

2012 (5)

O. Ilic, M. Jablan, J. D. Joannopoulos, I. Celanovic, and M. Soljačić, “Overcoming the black body limit in plasmonic and graphene near-field thermophotovoltaic systems,” Opt. Express 20(103), A366–A384 (2012).
[CrossRef] [PubMed]

P. J. van Zwol, L. Ranno, and J. Chevrier, “Tuning near field radiative heat flux through surface excitations with a metal insulator transition,” Phys. Rev. Lett. 108(23), 234301 (2012).
[CrossRef] [PubMed]

S. Shen, A. Mavrokefalos, P. Sambegoro, and G. Chen, “Nanoscale thermal radiation between two gold surfaces,” Appl. Phys. Lett. 100(23), 233114 (2012).
[CrossRef]

A. W. Rodriguez, M. T. H. Reid, and S. G. Johnson, “Fluctuating-surface-current formulation of radiative heat transfer for arbitrary geometries,” Phys. Rev. B 86(22), 220302 (2012).
[CrossRef]

B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, “Near-field radiative cooling of nanostructuresm,” Nano Lett. 12(9), 4546–4550 (2012).
[CrossRef] [PubMed]

2011 (6)

R. Messina and M. Antezza, “Scattering-matrix approach to Casimir-Lifshitz force and heat transfer out of thermal equilibrium between arbitrary bodies,” Phys. Rev. A 84(4), 042102 (2011).
[CrossRef]

C. Otey and S. Fan, “Numerically exact calculation of electromagnetic heat transfer between a dielectric sphere and plate,” Phys. Rev. B 84, 245431 (2011).
[CrossRef]

M. Krüger, T. Emig, and M. Kardar, “Nonequilibrium electromagnetic fluctuations: Heat transfer and interactions,” Phys. Rev. Lett. 106(21), 210404 (2011).
[CrossRef] [PubMed]

P. Ben-Abdallah, S. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett.,  107(11), 114301 (2011).
[CrossRef] [PubMed]

K. Sasihithlu and A. Narayanaswamy, “Convergence of vector spherical wave expansion method applied to near-field radiative transfer,” Opt. Express 19(S4), A772–A785 (2011).
[CrossRef] [PubMed]

K. Sasihithlu and A. Narayanaswamy, “Proximity effects in radiative heat transfer,” Phys. Rev. B 83(16), 161406 (2011).
[CrossRef]

2010 (1)

C. Otey, W.-T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
[CrossRef] [PubMed]

2009 (4)

W. A. Challener, C. Peng, A. V. Itagi, D. Karns, W. Peng, Y. Peng, X. Yang, X. Zhu, N. J. Gokemeijer, and Y. T. Hsia, “Heat-assisted magnetic recording by a near-field transducer with efficient optical energy transfer,” Nat. Photon. 3(4), 220–224 (2009).
[CrossRef]

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[CrossRef] [PubMed]

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photon. 3(9), 514–517 (2009).
[CrossRef]

S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33(13), 1203–1232 (2009).
[CrossRef]

2008 (4)

A. Narayanaswamy, S. Shen, and G. Chen, “Near-field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78(11), 115303 (2008).
[CrossRef]

U. M. B. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, “Fluctuation–dissipation: response theory in statistical physics,” Phys. Rep. 461(4), 111–195 (2008).
[CrossRef]

B. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Confinement of infrared radiation to nanometer scales through metallic slit arrays,” J. Quant. Spectrosc. Radiat. Transfer 109(4), 608–619 (2008).
[CrossRef]

A. Narayanaswamy and G. Chen, “Thermal near-field radiative transfer between two spheres,” Phys. Rev. B 77(7), 075125 (2008).
[CrossRef]

2006 (3)

M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. Appl. Phys. 100(6), 063704 (2006).
[CrossRef]

Y. D. Wilde, F. Formanek, R. Carminati, B. Gralak, P. A. Lemoine, K. Joulain, J. P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[CrossRef] [PubMed]

H. Gies and K. Klingmüller, “Casimir effect for curved geometries: Proximity-force-approximation validity limits,” Phys. Rev. Lett. 96(22), 220401 (2006).
[CrossRef] [PubMed]

2005 (3)

A. Kittel, W. Müller-Hirsch, J. Parisi, S. A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95(22), 224301 (2005).
[CrossRef] [PubMed]

E. J. Rothwell, “Computation of the logarithm of bessel functions of complex argument,” Commun. Numer. Methods Eng. 21(10), 597–605 (2005).
[CrossRef]

G. Domingues, S. Volz, K. Joulain, and J.-J. Greffet, “Heat transfer between two nanoparticles through near-field interaction,” Phys. Rev. Lett. 94(8), 085901 (2005).
[CrossRef] [PubMed]

2003 (1)

A. Narayanaswamy and G. Chen, “Surface modes for near field thermophotovoltaics,” Appl. Phys. Lett. 82, 3544–3546 (2003).
[CrossRef]

2002 (2)

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative transfer at nanometric distances,” Microscale Thermophys. Eng. 6, 209–222 (2002).
[CrossRef]

C. H. Park, H. A. Haus, and M. S. Weinberg, “Proximity-enhanced thermal radiation,” J. Phys. D: Appl. Phys 35(21), 2857–2863 (2002).
[CrossRef]

2001 (2)

A. I. Volokitin and B. N. J. Persson, “Radiative heat transfer between nanostructures,” Phys. Rev. B 63, 205404 (2001).
[CrossRef]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. 78, 2931–2933 (2001).
[CrossRef]

1999 (2)

J. B. Pendry, “Radiative exchange of heat between nanostructures,” J. Phys.: Condens. Matter 11, 6621 (1999).

W. M. Hirsch, A. Kraft, M. T. Hirsch, J. Parisi, and A. Kittel, “Heat transfer in ultrahigh vacuum scanning thermal microscopy,” J. Vac. Sci. Technol. A 17(4), 1205–1210 (1999).
[CrossRef]

1997 (1)

S. K. Lamoreaux, “Demonstration of the Casimir force in the 0.6 to 6μm range,” Phys. Rev. Lett. 78, 5–8 (1997).
[CrossRef]

1994 (1)

J. J. Loomis and H. J. Maris, “Theory of heat transfer by evanescent electromagnetic waves,” Phys. Rev. B 50, 18517–18524 (1994).
[CrossRef]

1993 (1)

W. C. Chew, “Efficient ways to compute the vector addition theorem,” J. Electromagn. Wave 7, 651–665 (1993).
[CrossRef]

1992 (1)

W. C. Chew, “Recurrence relations for three-dimensional scalar addition theorem,” J. Electromagn. Wave 6, 133–142 (1992).
[CrossRef]

1990 (2)

W. C. Chew, “A derivation of the vector addition theorem,” Microwave Opt. Tech. Lett. 3(7), 256–260 (1990).
[CrossRef]

W. C. Chew, “Derivation of the vector addition theorem,” Microwave Opt. Technol. Lett. 3, 256–260 (1990).
[CrossRef]

1980 (1)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68(2), 248–263 (1980).
[CrossRef]

1977 (1)

J. Błocki, J. Randrup, W. J. Światecki, and C. F. Tsang, “Proximity forces,” Ann. Phys. 105(2), 427–462 (1977).
[CrossRef]

1976 (1)

N. H. Juul, “Investigation of approximate methods for calculation of the diffuse radiation configuration view factor between two spheres,” Lett. Heat Mass Transfer 3(6), 513–521 (1976).
[CrossRef]

1971 (1)

D. Polder and M. V. Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303–3314 (1971).
[CrossRef]

1967 (1)

E. G. Cravalho, C. L. Tien, and R. P. Caren, “Effect of small spacings on radiative transfer between two dielectrics,” J. Heat Transfer 89, 351–358 (1967).
[CrossRef]

1966 (1)

R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys. 29(1), 255 (1966).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Antezza, M.

R. Messina and M. Antezza, “Scattering-matrix approach to Casimir-Lifshitz force and heat transfer out of thermal equilibrium between arbitrary bodies,” Phys. Rev. A 84(4), 042102 (2011).
[CrossRef]

Basu, S.

S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33(13), 1203–1232 (2009).
[CrossRef]

Bell, J. S.

L. Landau, E. M. Lifšic, J. B. Sykes, J. S. Bell, M. J. Kearsley, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1960).

Ben-Abdallah, P.

P. Ben-Abdallah, S. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett.,  107(11), 114301 (2011).
[CrossRef] [PubMed]

Biehs, S.

P. Ben-Abdallah, S. Biehs, and K. Joulain, “Many-body radiative heat transfer theory,” Phys. Rev. Lett.,  107(11), 114301 (2011).
[CrossRef] [PubMed]

Biehs, S. A.

A. Kittel, W. Müller-Hirsch, J. Parisi, S. A. Biehs, D. Reddig, and M. Holthaus, “Near-field heat transfer in a scanning thermal microscope,” Phys. Rev. Lett. 95(22), 224301 (2005).
[CrossRef] [PubMed]

Blocki, J.

J. Błocki, J. Randrup, W. J. Światecki, and C. F. Tsang, “Proximity forces,” Ann. Phys. 105(2), 427–462 (1977).
[CrossRef]

Bruning, J. H.

J. H. Bruning and Y. T. Lo, “Multiple scattering by spheres,” Antenna Laboratory Report No.69(5) (1969).

Caren, R. P.

E. G. Cravalho, C. L. Tien, and R. P. Caren, “Effect of small spacings on radiative transfer between two dielectrics,” J. Heat Transfer 89, 351–358 (1967).
[CrossRef]

Carminati, R.

Y. D. Wilde, F. Formanek, R. Carminati, B. Gralak, P. A. Lemoine, K. Joulain, J. P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[CrossRef] [PubMed]

M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. Appl. Phys. 100(6), 063704 (2006).
[CrossRef]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative transfer at nanometric distances,” Microscale Thermophys. Eng. 6, 209–222 (2002).
[CrossRef]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Nanoscale radiative heat transfer between a small particle and a plane surface,” Appl. Phys. Lett. 78, 2931–2933 (2001).
[CrossRef]

Celanovic, I.

Challener, W. A.

W. A. Challener, C. Peng, A. V. Itagi, D. Karns, W. Peng, Y. Peng, X. Yang, X. Zhu, N. J. Gokemeijer, and Y. T. Hsia, “Heat-assisted magnetic recording by a near-field transducer with efficient optical energy transfer,” Nat. Photon. 3(4), 220–224 (2009).
[CrossRef]

Chen, G.

S. Shen, A. Mavrokefalos, P. Sambegoro, and G. Chen, “Nanoscale thermal radiation between two gold surfaces,” Appl. Phys. Lett. 100(23), 233114 (2012).
[CrossRef]

S. Shen, A. Narayanaswamy, and G. Chen, “Surface phonon polaritons mediated energy transfer between nanoscale gaps,” Nano Lett. 9(8), 2909–2913 (2009).
[CrossRef] [PubMed]

A. Narayanaswamy and G. Chen, “Thermal near-field radiative transfer between two spheres,” Phys. Rev. B 77(7), 075125 (2008).
[CrossRef]

A. Narayanaswamy, S. Shen, and G. Chen, “Near-field radiative heat transfer between a sphere and a substrate,” Phys. Rev. B 78(11), 115303 (2008).
[CrossRef]

A. Narayanaswamy and G. Chen, “Surface modes for near field thermophotovoltaics,” Appl. Phys. Lett. 82, 3544–3546 (2003).
[CrossRef]

Chen, Y.

Y. D. Wilde, F. Formanek, R. Carminati, B. Gralak, P. A. Lemoine, K. Joulain, J. P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[CrossRef] [PubMed]

Chen, Y.-B.

B. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Confinement of infrared radiation to nanometer scales through metallic slit arrays,” J. Quant. Spectrosc. Radiat. Transfer 109(4), 608–619 (2008).
[CrossRef]

Chevrier, J.

P. J. van Zwol, L. Ranno, and J. Chevrier, “Tuning near field radiative heat flux through surface excitations with a metal insulator transition,” Phys. Rev. Lett. 108(23), 234301 (2012).
[CrossRef] [PubMed]

E. Rousseau, A. Siria, G. Jourdan, S. Volz, F. Comin, J. Chevrier, and J.-J. Greffet, “Radiative heat transfer at the nanoscale,” Nat. Photon. 3(9), 514–517 (2009).
[CrossRef]

Chew, W. C.

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Figures (7)

Fig. 1
Fig. 1

The configuration for this study consisting of two spheres of unequal radii R1 and R2 (labeled sphere ‘a’ and sphere ‘b’ respectively) and the relation between r, r′ and r″. The surface to surface gap between the two spheres d is given by d = r″R1R2

Fig. 2
Fig. 2

(a) Plot of ( z n + ν ( 3 ) ( k f r ) / z n + ν 2 ( 3 ) ( k f r )) as a function of n + ν for kfr″ = 33 (arbitrarily chosen). The point n + ν = 7kfr″, beyond which the one-term approximation has been adopted in our computations for calculating the translation coefficients, is marked in in the figure. (b) The error in spectral conductance at the surface phonon-polariton frequency (0.061 eV) when different values of (n + ν)/kfr″ are chosen as the criteria for employing the one-term approximation. From the plot, a criterion n + ν = 7kfr″ is observed to give an error of ≈ 0.02% in the spectral conductance. The spectral conductance has been computed for two spheres of size R1 = 13.7 μm and R2 = 40R1 with minimum gap d/R1 = 0.01.

Fig. 3
Fig. 3

Contour plots of the expression log 10 | z ν ( 1 ) ( k f R 1 ) z n + ν ( 3 ) ( k f r ) / z n ( 3 ) ( k f R 2 ) | as a function of n and ν for two spheres with successive radius ratios R2/R1 = (a) 1, (b) 3, (c) 10, and (d) 20 with R1 = 10 μm, and the minimum gap maintained at 50 nm for all the cases. The dashed-lines denotes the contour line for a value of −6 which is taken as the cutoff point below which values for the normalized vector translation coefficients are approximated to zero. The line of maximum (shown as dotted lines) given by Eq. (35) has been superimposed on these contour plots

Fig. 4
Fig. 4

(a) Application of MPA for the two spheres where the curved surfaces are approximated by a series of flat surfaces with varying gaps z. (b) the plot of spectral emissivity for a silica half-plane as a function of frequency in eV which is used in the form of MPA (Eq. (41)) to predict the far-field contribution to the conductance

Fig. 5
Fig. 5

(a) Plot of computed values of the total conductance (dotted) and the MPA (solid line)as a function of the non-dimensional gap d/R1 for two spheres with R2/R1 = 40. The study has been performed for R1 = 13.7μm and 2.5μm (b) The % error between the computed values and values from the MPA are plotted as a function of d/R1

Fig. 6
Fig. 6

(a) Comparison between the computed values of Gω at a surface phonon-polariton frequency (0.061 eV) (dotted) and G ω MPA (solid line) as a function of the non-dimensional gap d/R1 for two spheres of radius R1 = 2.5 μm, 13.7 μm and R2 = 40R1 (b) The % error between Gω and G ω MPA as a function of d/R1

Fig. 7
Fig. 7

(a) Comparison between the computed values of Gω at frequency 0.0801 eV (dotted) and G ω MPA (solid line) as a function of the non-dimensional gap d/R1 for two spheres of radius R1 = 2.5μm, 13.7μm and R2 = 40R1 (b) The % error between Gω and G ω MPA as a function of d/R1

Tables (1)

Tables Icon

Table 1 Table showing the error from the two approximations. Here, G ω Exact is the spectral conductance value without any approximations, G ω A 1 is the spectral conductance value obtained by using the one-term approximation and G ω A 2 is obtained by considering the dependence of normalized translation coefficients on the radius ratio of the two spheres. All conductances are in units of nW.K−1. The errors from these two approximations are denoted by ‘Error A1’ and ‘Error A2’ respectively (both in %). The values shown here are computed at the surface phonon-polariton frequency 0.061 eV for two cases (1) R2 = R1, and (2) R2 = 10R1. R1 is kept a constant at 13.7 μm.

Equations (57)

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× × P ( r ) k 2 P ( r ) = 0 ,
M n m ( j ) ( k r ) = z n ( j ) ( k r ) V n m ( 2 ) ( θ , ϕ ) ,
N n m ( j ) ( k r ) = ζ n j ( k r ) V n m ( 3 ) ( θ , ϕ ) + z n ( j ) ( k r ) k r n ( n + 1 ) V n m ( 1 ) ( θ , ϕ ) ,
V n m ( 1 ) ( θ , ϕ ) = r ^ Y n m ,
V n m ( 2 ) ( θ , ϕ ) = 1 n ( n + 1 ) ( ϕ ^ Y n m θ + θ ^ i m sin θ Y n m ) ,
V n m ( 3 ) ( θ , ϕ ) = 1 n ( n + 1 ) ( θ ^ Y n m θ + ϕ ^ i m sin θ Y n m ) .
ψ n m ( j ) ( k , r ) = z n ( j ) ( k r ) Y n m ( θ , ϕ ) ,
M n m ( j ) ( k , r ) = 1 n ( n + 1 ) × r ψ n m ( j ) ( k , r ) ,
N n m ( j ) ( k , r ) = 1 n ( n + 1 ) 1 k × × r ψ n m ( j ) ( k , r ) .
ψ n m ( k , r ) = ν = 1 m = ν ν ψ ν m ( k , r ) β ν m , n m ,
M n m ( j ) ( k r ) = ν = 1 m = ν ν [ M ν m ( j ) ( k r ) A ν m , n m + N ν m ( j ) ( k r ) B ν m , n m ] ,
N n m ( j ) ( k r ) = ν = 1 ν = m = ν ν [ N ν m ( j ) ( k r ) A ν m , n m + M ν m ( j ) ( k r ) B ν m , n m ]
A ν m , n m = ν ( ν + 1 ) n ( n + 1 ) { 2 π ν ( ν + 1 ) p = | ν n | ν + n [ n ( n + 1 ) + ν ( ν + 1 ) p ( p + 1 ) ] × i ν + p n ψ p , 0 ( k f r ) A ( m , n , m , ν , p ) } ,
B ν m , n , m = ν ( ν + 1 ) n ( n + 1 ) i m k f r ν ( ν + 1 ) p = | ν n | ν + n i ν + p n ψ p , 0 ( k f r ) A ( m , n , m , ν , p ) ,
A ( m , n , m , ν , p ) = ( 1 ) m ( 2 n + 1 ) ( 2 ν + 1 ) ( 2 p + 1 ) 4 π × ( n ν p 0 0 0 ) ( n ν p m m 0 ) ,
β ν m , n m ( k f r ) = p = | ν n | ν + n 4 π i ν + p n z p ( 3 ) ( k f r ) 2 p + 1 4 π A ( m , n , m , ν , p ) .
β ν m , n + 1 , m = 1 a n m + ( a n m β ν m , n 1 , m + a ν 1 , m + β ν 1 m , n m + a ν + 1 , m β ν + 1 m , n m ) ,
β ν n + 1 , n + 1 n + 1 = 1 b n n + ( b ν 1 , n + β ν 1 n , n n + b ν + 1 , n β ν + 1 n , n n ) .
A ν m , n m = ν ( ν + 1 ) n ( n + 1 ) { β ν m , n m + k f r ( ν + 1 ) ( ν + m + 1 ) ( ν m + 1 ) ( 2 ν + 1 ) ( 2 ν + 3 ) β ν + 1 , m , n m + k f r ν ( ν + m ) ( ν m ) ( 2 ν 1 ) ( 2 ν + 1 ) β ν 1 , m , n m } ,
B ν m , n , m i m k f r ν ( ν + 1 ) n ( n + 1 ) β ν m , n , m .
β ν 0 , 00 = ( 1 ) ν 2 ν + 1 z ν ( 3 ) ( k f r ) ,
z ν ( 3 ) ( k f r ) i 4 ( 2 ν + 1 ) k f r ( 2 ν + 1 e k f r ) ( ν + 1 / 2 ) ,
z ν ( 1 ) ( k f r ) 4 ( 2 ν + 1 ) k f r ( e k f r 2 ν + 1 ) ( ν + 1 / 2 ) .
β ν m , n m N = z ν ( 1 ) ( k f R 1 ) z n ( 3 ) ( k f R 2 ) β ν m , n m .
β ν m , n + 1 m N = 1 a n 0 + ( a n 0 z n + 1 ( 1 ) ( k f R 1 ) z n 1 ( 1 ) ( k f R 1 ) β ν m , n 1 m N + a ν 1 , 0 + z n + 1 ( 1 ) ( k f R 1 ) z n ( 1 ) ( k f R 1 ) z ν 1 ( 3 ) ( k f R 2 ) z ν ( 3 ) ( k f R 2 ) β ν 1 m , n m N + a ν 1 , 0 z n + 1 ( 1 ) ( k f R 1 ) z n ( 1 ) ( k f R 1 ) z ν + 1 ( 3 ) ( k f R 2 ) z ν ( 3 ) ( k f R 2 ) β ν + 1 m , n m N ) ,
β ν n + 1 , n + 1 n + 1 N = 1 b n n + ( b ν 1 , n + z n ( 3 ) ( k f R 2 ) z n + 1 ( 3 ) ( k f R 2 ) z ν ( 1 ) ( k f R 1 ) z ν 1 ( 1 ) ( k f R 1 ) β ν 1 n , n n N + b ν + 1 , n z n ( 3 ) ( k f R 2 ) z n + 1 ( 3 ) ( k f R 2 ) z ν ( 1 ) ( k f R 1 ) z ν + 1 ( 1 ) ( k f R 1 ) β ν + 1 n , n n ) .
β ν 0 , 00 N = z 0 ( 1 ) ( k f R 1 ) z ν ( 3 ) ( k f R 2 ) z ν ( 3 ) ( k f r ) ( 1 ) ν 2 ν + 1 .
β ν m , n m ( k f r ) = 4 π i 2 ν z n + ν ( 3 ) ( k f r ) 2 ( n + ν ) + 1 4 π A ( m , n , m , ν , n + ν ) .
ε z n + ν 2 ( 3 ) ( k f r ) z n + ν ( 3 ) ( k f r ) + z n + ν 2 ( 3 ) ( k f r ) .
z p + 2 ( 3 ) ( k f r ) z p ( 3 ) ( k f r ) ( 2 p + 5 k f r ) 2 .
ε ( k f r 2 ( n + ν ) 3 ) 2 ( k f r 2 ( n + ν ) 3 ) 2 + 1 .
β ν m , n + 1 m = β ν m , n m n + 1 n ( 2 n + 3 ) ( 2 n + 1 ) ( n + 1 ) 2 m 2 n + ν + 1 2 n + 2 ν + 1 z n + ν + 1 ( 3 ) ( k f r ) z n + ν ( 3 ) ( k f r ) .
z n + ν ( 3 ) ( k f r ) z ν ( 1 ) ( k f R 1 ) z n ( 3 ) ( k f R 2 ) .
z ν ( 1 ) ( k f R 1 ) exp [ ( ν + 1 / 2 ) 2 ( k f R 1 ) 2 ( ν + 1 / 2 ) cosh 1 [ ( ν + 1 / 2 ) k f R 1 ] ] 2 k f R 1 ( ν + 1 / 2 ) 2 ( k f R 1 ) 2 .
exp [ ν 2 ( k f R 1 ) 2 + n 2 ( k f R 2 ) 2 ( n + ν ) 2 ( k f R 1 + k f R 2 + δ ) 2 ] × exp [ ( n + ν ) cos 1 ( n + ν k f R 1 + k f R 2 + δ ) n cosh 1 ( n k f R 2 ) ν cosh 1 ( ν k f R 1 ) ] .
( 1 + n ν ) n ( 1 + ν n ) ν ( 1 + R 2 R 1 ) ν ( 1 + R 1 R 2 ) n .
n ν = R 2 R 1
exp [ δ ν 2 ( k f R 1 ) 2 ( ν 2 k f R 1 k f R 1 ) ] .
ν ( R 1 d ) 2 + ( k f R 1 ) 2 ,
n R 2 R 1 ( R 1 d ) 2 + ( k f R 1 ) 2 .
N conv = 2 R 2 d + e k f r 2 ,
G = lim T A T B P ( T A , T B ) | T A T B | ,
G MPA ( d , T ) = 0 R 1 h n f ( z ) 2 π r d r + G c ( d , T ) ,
G c ( d , T ) = 4 σ T 3 ( 4 π R 1 2 ) [ ( 1 ε ) / ε ] [ 1 + R 1 2 / R 2 2 ] + 1 / F 12 ( d ) ,
F 12 ( d ) = 1 2 ( 1 1 1 ( d / R 2 + R 1 / R 2 + 1 ) 2 ) .
G c ( d 1 , T ) G c ( d 2 , T ) = [ ( 1 ε ) / ε ] ( 1 + R 1 2 / R 2 2 ) + 1 / F 12 ( d 2 ) [ ( 1 ε ) / ε ] ( 1 + R 1 2 / R 2 2 ) + 1 / F 12 ( d 1 ) ,
G ω ( d ) ~ R 1 R 2 R 1 + R 2 1 d R 1 d ( for R 2 R 1 ) .
C n m l M + u ¯ n ( R 1 ) z n ( 1 ) ( k f R 1 ) z n ( 3 ) ( k f R 1 ) ν = ( m , 1 ) N max [ D ν m l M A ν m , n m ( k f r ) + D ν m l N B ν m , n m ( k f r ) ] = p N M δ N l ,
D n m l M + u ¯ n ( R 2 ) z n ( 1 ) ( k f R 2 ) z n ( 3 ) ( k f R 2 ) ν = ( m , 1 ) N max [ C ν m l M A ν m , n m ( + k f r ) + C ν m l N B ν m , n m ( + k f r ) ] = 0 ,
C n m l N + v ¯ n ( R 1 ) z n ( 1 ) ( k f R 1 ) z n ( 3 ) ( k f R 1 ) ν = ( m , 1 ) N max [ D ν m l M B ν m , n m ( k f r ) + D ν m l N A ν m , n m ( k f r ) ] = 0 ,
D n m l N + v ¯ n ( R 2 ) z n ( 1 ) ( k f R 2 ) z n ( 3 ) ( k f R 2 ) ν = ( m , 1 ) N max [ C ν m l M B ν m , n m ( + k f r ) + C ν m l N A ν m , n m ( + k f r ) ] = 0 ,
u ¯ n ( R 1 ) = ( k b z n + 1 ( 1 ) ( k b R 1 ) z n ( 1 ) ( k b R 1 ) k f z n + 1 ( 1 ) ( k f R 1 ) z n ( 1 ) ( k f R 1 ) k b z n + 1 ( 1 ) ( k b R 1 ) z n ( 1 ) ( k b R 1 ) k f z n + 1 ( 3 ) ( k f R 1 ) z n ( 3 ) ( k f R 1 ) ) ,
v ¯ n ( R 1 ) = ( n b z n + 1 ( 1 ) ( k f R 1 ) z n ( 1 ) ( k f R 1 ) z n + 1 ( 1 ) ( k b R 1 ) z n ( 1 ) ( k b R 1 ) + ( n + 1 ) ( 1 n b 2 ) k f R 1 n b n b z n + 1 ( 1 ) ( k f R 1 ) z n ( 1 ) ( k f R 1 ) z n + 1 ( 3 ) ( k b R 1 ) z n ( 3 ) ( k b R 1 ) + ( n + 1 ) ( 1 n b 2 ) k f R 1 n b ) .
( z l ( 1 ) ( k b R 2 ) z n ( 1 ) ( k f R 1 ) C n m l M ) + u ¯ n ( a ) z l ( 1 ) ( k b R 2 ) z l ( 1 ) ( k a R 1 ) × ν = ( m , 1 ) N max [ ( z l ( 1 ) ( k a R 1 ) z ν ( 1 ) ( k f R 2 ) D ν m l M ) ( z ν ( 1 ) ( k f R 2 ) z n ( 3 ) ( k f R 1 ) A ν m , n m ( k f D ) ) + ( z l ( 1 ) ( k a R 1 ) ζ ν ( 1 ) ( k f R 2 ) D ν m l N ) ( ζ ν ( 1 ) ( k f R 2 ) z n ( 3 ) ( k f R 1 ) B ν m , n m ( k f D ) ) ] = z l ( 1 ) ( k b R 2 ) z n ( 1 ) ( k f R 1 ) p N M δ N l ,
( z l ( 1 ) ( k a R 1 ) z n ( 1 ) ( k f R 2 ) D n m l M ) + u ¯ n ( R 2 ) z l ( 1 ) ( k a R 1 ) z l ( 1 ) ( k b R 2 ) × ν = ( m , 1 ) N max [ ( z l ( 1 ) ( k b R 2 ) z ν ( 1 ) ( k f R 1 ) C ν m l M ) ( z ν ( 1 ) ( k f R 1 ) z n ( 3 ) ( k f R 2 ) A ν m , n m ( + k f D ) ) + ( z l ( 1 ) ( k b R 2 ) ζ ν ( 1 ) ( k f R 1 ) C ν m l N ) ( ζ ν ( 1 ) ( k f R 1 ) z n ( 3 ) ( k f R 2 ) B ν m , n m ( + k f D ) ) ] = 0 ,
( z l ( 1 ) ( k b R 2 ) ζ n ( 1 ) ( k f R 1 ) C n m l M ) + v ¯ n ( R 1 ) z n ( 1 ) ( k f R 1 ) ζ n ( 1 ) ( k f R 1 ) z l ( 1 ) ( k b R 2 ) z l ( 1 ) ( k a R 1 ) × ν = ( m , 1 ) N max [ ( z l ( 1 ) ( k a R 1 ) z ν ( 1 ) ( k f R 2 ) D ν m l M ) ( z ν ( 1 ) ( k f R 2 ) z n ( 3 ) ( k f R 1 ) B ν m , n m ( k f D ) ) + ( z l ( 1 ) ( k a R 1 ) ζ ν ( 1 ) ( k f R 2 ) D ν m l N ) ( ζ ν ( 1 ) ( k f R 2 ) z n ( 3 ) ( k f R 1 ) A ν m , n m ( k f D ) ) ] = 0 ,
( z l ( 1 ) ( k a R 1 ) ζ n ( 1 ) ( k f R 2 ) D n m l M ) + v ¯ n ( R 2 ) z n ( 1 ) ( k f R 2 ) ζ n ( 1 ) ( k f R 2 ) z l ( 1 ) ( k a R 1 ) z l ( 1 ) ( k b R 2 ) × ν = ( m , 1 ) N max [ ( z l ( 1 ) ( k b R 2 ) z ν ( 1 ) ( k f R 1 ) C ν m l M ) ( z ν ( 1 ) ( k f R 1 ) z n ( 3 ) ( k f R 2 ) B ν m , n m ( + k f D ) ) + ( z l ( 1 ) ( k b R 2 ) ζ ν ( 1 ) ( k f R 1 ) C ν m l N ) ( ζ ν ( 1 ) ( k f R 1 ) z n ( 3 ) ( k f R 2 ) A ν m , n m ( + k f D ) ) ] = 0 .

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