Abstract

A new model for thermal radiation from spherical microparticles is presented. It treats small particles as volume emitters instead of the usual surface emitters to take into account the effect of temperature distributions inside the particle on emission. The model describes the spectral thermal emission of the particle’s volume cells as the radiation of classical electric dipoles with an averaged orientation in an absorbing dielectric sphere. Within this dipole model of thermal radiation Kirchhoff’s law is rederived. For nonisothermal temperature distributions inside particles, the angular dependence and the magnitude of the emitted radiation are computed. To test the influence of the model on the heat transfer in microparticles, we compare computed temperature distributions inside volume- and surface-emitting particles.

© 1994 Optical Society of America

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References

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  1. H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
    [CrossRef]
  2. H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
    [CrossRef] [PubMed]
  3. S. C. Ching, H. M. Lai, and K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987).
    [CrossRef]
  4. S. C. Ching, H. M. Lai, and K. Young, “Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shift,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
    [CrossRef]
  5. H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
    [CrossRef]
  6. M. Kerker, P. J. McNulty, M. Sculley, H. Chew, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for incoherent optical processes,” J Opt. Soc. Am. 68, 1676–1686 (1978).
    [CrossRef]
  7. H. Chew, M. Sculley, M. Kerker, P. J. McNulty, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
    [CrossRef]
  8. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).
  9. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957). Jackson’s Xlm are identical to Edmond’s Yllm.
  10. S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
    [CrossRef]
  11. J. Gersten and A. Nitzan, “Radiative properties of solvated molecules in dielectric clusters and small particles,” J. Chem. Phys. 95, 686–699 (1991).
    [CrossRef]
  12. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 123–126.
  13. G. W. Kattawar and M. Eisner, “Radiation from a homogeneous isothermal sphere,” Appl. Opt. 9, 2685–2690 (1970).
    [CrossRef] [PubMed]
  14. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 134, Eq. (8).
  15. P. W. Dusel, M. Kerker, and D. D. Cooke, “Distribution of absorption centers within irradiated spheres,” J. Opt. Soc. Am. 69, 55–59 (1979).
    [CrossRef]
  16. W. M. Greene, R. E. Spjut, E. Bar-Ziv, A. F. Sarofim, and J. P. Longwell, “Photophoresis of irradiated spheres: absorption centers,” J. Opt. Soc. Am. B 2, 998–1004 (1985).
    [CrossRef]
  17. A. B. Pluchino, “Surface waves and the radiative properties of micron-sized particles,” Appl. Opt. 20, 2986–2992 (1981).
    [CrossRef] [PubMed]
  18. S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988).
    [CrossRef]

1991 (1)

J. Gersten and A. Nitzan, “Radiative properties of solvated molecules in dielectric clusters and small particles,” J. Chem. Phys. 95, 686–699 (1991).
[CrossRef]

1988 (1)

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

1987 (4)

S. C. Ching, H. M. Lai, and K. Young, “Dielectric microspheres as optical cavities: thermal spectrum and density of states,” J. Opt. Soc. Am. B 4, 1995–2003 (1987).
[CrossRef]

S. C. Ching, H. M. Lai, and K. Young, “Dielectric microspheres as optical cavities: Einstein A and B coefficients and level shift,” J. Opt. Soc. Am. B 4, 2004–2009 (1987).
[CrossRef]

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[CrossRef]

S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

1985 (1)

1981 (1)

1979 (1)

1978 (2)

M. Kerker, P. J. McNulty, M. Sculley, H. Chew, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for incoherent optical processes,” J Opt. Soc. Am. 68, 1676–1686 (1978).
[CrossRef]

H. Chew, M. Sculley, M. Kerker, P. J. McNulty, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
[CrossRef]

1976 (1)

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

1970 (1)

Arnold, S.

S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

Bar-Ziv, E.

Benner, R. E.

S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 123–126.

Chew, H.

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[CrossRef]

M. Kerker, P. J. McNulty, M. Sculley, H. Chew, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for incoherent optical processes,” J Opt. Soc. Am. 68, 1676–1686 (1978).
[CrossRef]

H. Chew, M. Sculley, M. Kerker, P. J. McNulty, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
[CrossRef]

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Ching, S. C.

Cooke, D. D.

Druger, S. D.

S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

Dusel, P. W.

Edmonds, A. R.

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957). Jackson’s Xlm are identical to Edmond’s Yllm.

Eisner, M.

Folan, L. M.

S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

Gersten, J.

J. Gersten and A. Nitzan, “Radiative properties of solvated molecules in dielectric clusters and small particles,” J. Chem. Phys. 95, 686–699 (1991).
[CrossRef]

Greene, W. M.

Hill, S. C.

S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 123–126.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

Kattawar, G. W.

Kerker, M.

P. W. Dusel, M. Kerker, and D. D. Cooke, “Distribution of absorption centers within irradiated spheres,” J. Opt. Soc. Am. 69, 55–59 (1979).
[CrossRef]

M. Kerker, P. J. McNulty, M. Sculley, H. Chew, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for incoherent optical processes,” J Opt. Soc. Am. 68, 1676–1686 (1978).
[CrossRef]

H. Chew, M. Sculley, M. Kerker, P. J. McNulty, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
[CrossRef]

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Lai, H. M.

Longwell, J. P.

McNulty, P. J.

H. Chew, M. Sculley, M. Kerker, P. J. McNulty, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
[CrossRef]

M. Kerker, P. J. McNulty, M. Sculley, H. Chew, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for incoherent optical processes,” J Opt. Soc. Am. 68, 1676–1686 (1978).
[CrossRef]

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Nitzan, A.

J. Gersten and A. Nitzan, “Radiative properties of solvated molecules in dielectric clusters and small particles,” J. Chem. Phys. 95, 686–699 (1991).
[CrossRef]

Pluchino, A. B.

Sarofim, A. F.

Sculley, M.

H. Chew, M. Sculley, M. Kerker, P. J. McNulty, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1978).
[CrossRef]

M. Kerker, P. J. McNulty, M. Sculley, H. Chew, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for incoherent optical processes,” J Opt. Soc. Am. 68, 1676–1686 (1978).
[CrossRef]

Spjut, R. E.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 134, Eq. (8).

Young, K.

Appl. Opt. (2)

J Opt. Soc. Am. (1)

M. Kerker, P. J. McNulty, M. Sculley, H. Chew, and D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: numerical results for incoherent optical processes,” J Opt. Soc. Am. 68, 1676–1686 (1978).
[CrossRef]

J. Chem. Phys. (3)

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[CrossRef]

S. D. Druger, S. Arnold, and L. M. Folan, “Theory of enhanced energy transfer between molecules embedded in spherical dielectric particles,” J. Chem. Phys. 87, 2649–2659 (1987).
[CrossRef]

J. Gersten and A. Nitzan, “Radiative properties of solvated molecules in dielectric clusters and small particles,” J. Chem. Phys. 95, 686–699 (1991).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (3)

Phys. Rev. A (2)

H. Chew, P. J. McNulty, and M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[CrossRef] [PubMed]

Other (5)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton U. Press, Princeton, N.J., 1957). Jackson’s Xlm are identical to Edmond’s Yllm.

S. C. Hill and R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber and R. K. Chang, eds. (World Scientific, Singapore, 1988).
[CrossRef]

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 123–126.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1966), p. 134, Eq. (8).

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Figures (10)

Fig. 1
Fig. 1

Spectral thermal power ep versus wave number k for three different radial positions of the radiating cell inside a spherical particle of radius a. The middle and the upper curves are shifted upward by one half and by one unit, respectively, to prevent overlap of the curves. Refractive index n1 = 1.5 + 0.001i, particle radius a = 10 μm, temperature T = 1000 K.

Fig. 2
Fig. 2

Spectral thermal power ep versus normalized position r/a for size parameter x = 14.768615 (solid curve) and x = 15.66 (dashed curve). Refractive index n1 = 1.5 + 0.001i, particle radius a = 10 μm, temperature T = 1000 K.

Fig. 3
Fig. 3

Spectral thermal power per solid angle dep/dΩ as a function of the angle of emission θ and the imaginary part of the refractive index Im(n1). The real part of the refractive index Re(n1) = 1.5, the size parameter x = 12.32, the radial position r = 0.8a, the particle radius a = 10 μm, and the temperature T = 1000 K.

Fig. 4
Fig. 4

Total thermal power per solid angle dep/dΩ versus the angle of emission θ for various radial positions r = 0.1, ν = 1, …, 10. The refractive index n1 = 1.5 + 0.01i, the particle radius a = 10 μm, and the temperature T = 400 K.

Fig. 5
Fig. 5

Degree of polarization of the total thermal emission versus the angle of emission θ for different radial positions r; same parameters as in Fig. 4.

Fig. 6
Fig. 6

Total thermal power solid angle dep/dΩ versus the angle of emission θ for temperatures T equal to 300, 700, 1100, 1500, and 1900 K. The refractive index n1 = 1.5 + 0.01i, the particle radius a = 10 μm, and the radial position r = 0.8a.

Fig. 7
Fig. 7

Degree of polarization of the total thermal emission versus the angle of emission θ for different temperatures T; same parameters as in Fig. 6.

Fig. 8
Fig. 8

Total thermal power Ep as a function of the normalized radial position r/a and the imaginary part of the refractive index Im(n1). The real part of the refractive index Re(n1) = 1.5, the radial position r = 0.8a, the particle radius a = 10 μm, and the temperature T = 1000 K.

Fig. 9
Fig. 9

Nonstationary temperature distributions inside of a volume- and surface-emitting glass particle as a result of radiation cooling. The time step Δt = 10 μs, the refractive index n = 1.5 + 0.01i, the particle radius a = 10 μm, the temperature of the environment Tenv = 300 K, the heat conductivity λ = 0.8 W/(m K), the specific heat cp = 8.4 × 102 J/(kg K), and the mass density ρ = 2.5 × 103 kg/m3.

Fig. 10
Fig. 10

Stationary temperature distributions inside of volume and surface emitters for a rectangular source function. The refractive index n1 = 1.5 + 0.01i, the particle radius a = 10 μm, the heat conductivity λ = 0.8 W/(m K), the source function S = 5.263 × 1011 W/m3 for 0.4ar ≤ 0.6a.

Equations (54)

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E 1 = E dip + E sca ,
B 1 = B dip + B sca ,
E dip ( r ) = l , m i c n 1 2 ω a E ( l , m ) × [ h l ( 1 ) ( k 1 r ) X l m ( r ) ] + a M ( l , m ) h l ( 1 ) ( k 1 r ) X l m ( r ) ,
B dip ( r ) = l , m a E ( l , m ) h l ( 1 ) ( k 1 r ) X l m ( r ) - i c ω a M ( l , m ) × [ h l ( 1 ) ( k 1 r ) X l m ( r ) ] ,
E sca ( r ) = l , m i c n 1 2 ω b E ( l , m ) × [ j l ( k 1 r ) X l m ( r ) ] + b M ( l , m ) j l ( k 1 r ) X l m ( r ) ,
B sca ( r ) = l , m b E ( l , m ) j l ( k 1 r ) X l m ( r ) - i c ω b M ( l , m ) × [ j l ( k 1 r ) X l m ( r ) ] ,
E 2 ( r ) = l , m i c n 2 2 ω c E ( l , m ) × [ h l ( 1 ) ( k 2 r ) X l m ( r ) ] + c M ( l , m ) h l ( 1 ) ( k 2 r ) X l m ( r ) ,
B 2 ( r ) = l , m c E ( l , m ) h l ( 1 ) ( k 2 r ) X l m ( r ) - i c ω c M ( l , m ) × [ h l ( 1 ) ( k 2 r ) X l m ( r ) ] .
a E ( l , m ) = p · V E ( l , m ) ,
a M ( l , m ) = p · V M ( l , m ) ,
V E ( l , m ) = 4 π k 1 2 μ 1 n 1 × [ j l ( k 1 r ) X l m * ( r ) ] ,
V M ( l , m ) = 4 π i k 1 3 μ 1 n 1 2 j l ( k 1 r ) X l m * ( r ) .
c E ( l , m ) = f E ( l ) a E ( l , m ) ,
c M ( l , m ) = f M ( l ) a M ( l , m ) ,
f E ( l ) = i n 2 2 / ( μ 1 k 1 a ) ( n 1 2 / μ 1 ) j l ( k 1 a ) [ k 2 a h l ( k 2 a ) ] - ( n 2 2 / μ 2 ) h l ( k 2 a ) [ k 1 a j l ( k 1 a ) ] ,
f M ( l ) = i μ 2 / ( k 1 a ) μ 1 j l ( k 1 a ) [ k 2 a h l ( k 2 a ) ] - μ 2 h l ( k 2 a ) [ k 1 a j l ( k 1 a ) ] .
d P d Ω = c 3 8 π μ 2 n 2 3 ω 2 | l , m ( - i ) l + 1 [ c E ( l , m ) X l m ( r ) + n 2 c M ( l , m ) e r × X l m ( r ) ] | 2 .
d P d Ω = c 3 8 π μ 2 n 2 3 ω 2 l , m , l , m ( - i ) l - l × { c E ( l , m ) c E * ( l , m ) X l m ( r ) · X l m * ( r ) + n 2 2 c M ( l , m ) c M * ( l , m ) [ e r × X l m ( r ) ] · [ e r × X l m * ( r ) ] + n 2 c E ( l , m ) c M * ( l , m ) X l m ( r ) · [ e r × X l m * ( r ) ] + n 2 c M ( l , m ) c E * ( l , m ) × [ e r × X l m ( r ) ] · X l m * ( r ) } ,
f ( p ) = 1 4 π 4 π f ( p ) d Ω p .
c E ( l , m ) c M * ( l , m ) = p 2 f E ( l ) f M * ( l ) V E ( l , m ) · V M * ( l , m ) .
d P d Ω = c 3 24 π μ 2 n 2 3 ω 2 p 2 × ν = 1 3 | l , m ( - i ) l + 1 [ f E ( l ) V E ν ( l , m ) X l m ( r ) + n 2 f M ( l ) V M ν ( l , m ) e r × X l m ( r ) ] | 2 ,
c E ν = f E ( l ) V E ν ( l , m ) ,
c M ν = f M ( l ) V M ν ( l , m )
P = 4 π d P d Ω d Ω = c 3 24 π μ 2 n 2 3 ω 2 p 2 l f E ( l ) 2 m V E ( l , m ) 2 + n 2 2 f M ( l ) 2 m V M ( l , m ) 2 .
m V E ( l , m ) 2 = 4 π k 1 6 μ 1 2 n 1 2 [ ( l + 1 ) j 1 - 1 ( k 1 r ) 2 + l j + 1 ( k 1 r ) 2 ] ,
m V M ( l , m ) 2 = 4 π k 1 6 μ 1 2 n 1 4 ( 2 l + 1 ) j l ( k 1 r ) 2 ,
P = 1 6 c k 4 μ 1 2 n 1 4 μ 2 n 2 3 p 2 × l [ ( l + 1 ) j l - 1 ( k 1 r ) 2 + l j i + 1 ( k 1 r ) 2 ] f E ( l ) 2 + n 1 2 n 1 2 ( 2 l + 1 ) j l ( k 1 r ) 2 f M ( l ) 2 ,
P sphere = 4 π 0 a r 2 P ( r ) d r .
0 a r 2 j l ( k 1 r ) 2 d r = a 3 2 Im [ k 1 a j l * ( k 1 a ) j l + 1 ( k 1 a ) ] Re ( k 1 a ) Im ( k 1 a )             for Im ( k 1 ) 0
P sphere = π a 3 c k 2 μ 1 2 n 1 4 μ 2 n 2 3 Re ( n 1 ) Im ( n 1 ) p 2 × l Im [ ( l + 1 ) k 1 a j l - 1 * ( k 1 a ) j l ( k 1 a ) + l k 1 a j l + 1 * ( k 1 a ) j l + 2 ( k 1 a ) ] f E ( l ) 2 + n 2 2 n 1 2 ( 2 l + 1 ) Im [ k 1 a j l * ( k 1 a ) j l + 1 ( k 1 a ) ] f M ( l ) 2 .
S = 2 k μ 2 μ 1 n 2 Re ( n 1 ) Im ( n 1 ) I inc S ^ ,
S ^ = E 1 2 E inc 2 ,
I inc = c 8 π n 2 μ 2 E inc 2 ,
4 π S ^ ( r ) d Ω = 2 π l n 2 2 n 1 2 [ ( l + 1 ) j l - 1 ( k 1 r ) 2 + l j l + 1 ( k 1 r ) 2 ] γ E ( l ) 2 + ( 2 l + 1 ) j 1 ( k 1 r ) 2 γ M ( l ) 2 .
γ E ( l ) = μ 1 n 1 3 μ 2 n 2 3 f E ( l ) ,
γ M ( l ) = μ 1 n 1 μ 2 n 2 f M ( l ) ,
I therm = n 2 2 1 π e b ( k , T ) ,
e b ( k , T ) = c 2 4 π 2 k 3 exp ( k c / k B T ) - 1
P abs = 8 π a 2 k μ 1 n 1 4 μ 2 n 2 3 e b ( k , T ) × l Im [ ( l + 1 ) k 1 a j l - 1 * ( k 1 a ) j l ( k 1 a ) + l k 1 a j l + 1 * ( k 1 a ) j l + 2 ( k 1 a ) ] f E ( l ) 2 + n 2 2 n 1 2 ( 2 l + 1 ) Im [ k 1 a j l * ( k 1 a ) j l + 1 ( k 1 a ) ] f M ( l ) 2 .
p 2 = 24 c k 3 Re ( n 1 ) Im ( n 1 ) μ 1 e b ( k , T ) .
e p ( r , k , T ) d k = 4 k μ 1 μ 2 n 1 4 Re ( n 1 ) Im ( n 1 ) n 2 3 e b ( k , T ) × l ( [ l + 1 ) j l - 1 ( k 1 r ) 2 + l j l + 1 ( k 1 r ) 2 ] × f E ( l ) 2 + n 2 2 n 1 2 ( 2 l + 1 ) j l ( k 1 r ) 2 × f M ( l ) 2 ) d k ,
d e p d Ω d k = 1 π Re ( n 1 ) Im ( n 1 ) μ 1 μ 2 n 2 3 1 k 5 e b ( k , T ) × ν = 1 3 | l , m ( - 1 ) l + 1 [ f E ( l ) V E ν ( l , m ) X l m ( r ) + n 2 f M ( l ) V M ν ( l , m ) e r × X l m ( r ) ] | 2 d k .
s 0 = E θ 2 + E ϕ 2 ,
s 1 = E θ 2 - E ϕ 2 ,
s 2 = 2 Re ( E θ * E ϕ ) ,
s 3 = 2 Im ( E θ * E ϕ ) .
P = 1 s 0 ( s 1 2 + s 2 2 + s 3 3 ) 1 / 2 .
E p ( r , T ) = 0 e p ( r , k , T ) d k
λ a 2 ( 2 T r 2 + 2 r T r ) + S ( r ) = ρ c p T t ,
T r ( 0 ) = 0 ,
- λ T r ( 1 ) = 0 Q abs ( k ) e s ( k , T ) d k ,
λ a 2 ( 2 T r 2 + 2 r T r ) + S ( r ) - E ( r , T ) = ρ c p T t ,
T r ( 0 ) = 0 ,
T r ( 1 ) = 0 ,

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