Abstract

A formula is derived for the effective complex dielectric constant of a dielectric medium containing a cubical array of dielectric or metallic impurities. The formula is given to two levels of approximation; one based on the Clausius-Mosotti equation which assumes zero contribution to the polarization from dipoles within the Lorentz sphere. The second is a generalization of a more accurate calculation of Rayleigh. Specific expressions are given for the real dielectric constant and the effective conductivity (and loss tangent) both as a function of frequency and impurity content. Formulas are valid for wavelengths large compared with the dimensions of the impurities. Results are compared with experimental values of Kharadly and Jackson. The complex dielectric constant is also obtained from the more accurate dc formula for a uniform array of spheres embedded in a uniform dielectric as derived by Zuzovsky and Brenner and again from the dc formula of a random array as derived by Felderhof.

© 1984 Optical Society of America

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References

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  1. J. W. Rayleigh, “On the Influence of Obstacles Arranged in Rectangular Order upon the Properties of a Medium,” Philos. Mag. 34, 481 (1982).
  2. W. K. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), p. 33.
  3. J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1980).
  4. M. M. Z. Kharadly, W. Jackson, “The Properties of Artificial Dielectrics Comprising Arrays of Conducting Elements,” Proc. Inst. Elec. Engrs. 100, 199 (1953).
  5. M. Zuzovsky, H. Brenner, “Effective Conductivities of Composite Materials Composed of Cubic Arrangements of Spherical Particles Embedded in an Isotropic Matrix,” Z. Angew. Math. Phys. 28, 979 (1977).
    [CrossRef]
  6. B. U. Felderhof, “Bounds for the Effective Dielectric Constant of a Suspension of Uniform Spheres,” J. Phys. 15, 1731 (1982).
  7. H. Frohlich, Theory of Dielectrics (Oxford U.P., London, 1949).
  8. C. J. F. Böttcher, P. Bordewijk, Theory of Elec. Polarization (Elsevier, New York, 1978).
  9. I. Runge, “On the Electrical Conductivity of Metallic Aggregates,” Z. Tech. Phys. 6, 61 (1925).
  10. D. A. DeVries, “Het Warmtegeleidingsvermogen van grond,” Meded. Landbouwhogesch. Wageningen 52, 1 (1952).
  11. A. S. Sangani, A. Acrivos, “The Effective Conductivity of a Periodic Array of Spheres,” Proc. R. Soc. London Ser A 386, 263 (1983).
    [CrossRef]
  12. J. G. Kirkwood, “On the Theory of Dielectric Polarization,” J. Chem. Phys. 4, 592 (1936).
    [CrossRef]
  13. J. Yvon, Recherches sur la Theorie Cinetique des Liquides (Hermann, Paris, 19).
  14. D. A. G. Bruggeman, “Calculations of Various Physical Constants of Heterogeneous Substances: Part I. Dielectric Constants and Conductivity of Mixtures of Isotropic Substances,” Ann. Phys. 24, 636 (1935).
    [CrossRef]

1983 (1)

A. S. Sangani, A. Acrivos, “The Effective Conductivity of a Periodic Array of Spheres,” Proc. R. Soc. London Ser A 386, 263 (1983).
[CrossRef]

1982 (2)

J. W. Rayleigh, “On the Influence of Obstacles Arranged in Rectangular Order upon the Properties of a Medium,” Philos. Mag. 34, 481 (1982).

B. U. Felderhof, “Bounds for the Effective Dielectric Constant of a Suspension of Uniform Spheres,” J. Phys. 15, 1731 (1982).

1977 (1)

M. Zuzovsky, H. Brenner, “Effective Conductivities of Composite Materials Composed of Cubic Arrangements of Spherical Particles Embedded in an Isotropic Matrix,” Z. Angew. Math. Phys. 28, 979 (1977).
[CrossRef]

1953 (1)

M. M. Z. Kharadly, W. Jackson, “The Properties of Artificial Dielectrics Comprising Arrays of Conducting Elements,” Proc. Inst. Elec. Engrs. 100, 199 (1953).

1952 (1)

D. A. DeVries, “Het Warmtegeleidingsvermogen van grond,” Meded. Landbouwhogesch. Wageningen 52, 1 (1952).

1936 (1)

J. G. Kirkwood, “On the Theory of Dielectric Polarization,” J. Chem. Phys. 4, 592 (1936).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, “Calculations of Various Physical Constants of Heterogeneous Substances: Part I. Dielectric Constants and Conductivity of Mixtures of Isotropic Substances,” Ann. Phys. 24, 636 (1935).
[CrossRef]

1925 (1)

I. Runge, “On the Electrical Conductivity of Metallic Aggregates,” Z. Tech. Phys. 6, 61 (1925).

Acrivos, A.

A. S. Sangani, A. Acrivos, “The Effective Conductivity of a Periodic Array of Spheres,” Proc. R. Soc. London Ser A 386, 263 (1983).
[CrossRef]

Bordewijk, P.

C. J. F. Böttcher, P. Bordewijk, Theory of Elec. Polarization (Elsevier, New York, 1978).

Böttcher, C. J. F.

C. J. F. Böttcher, P. Bordewijk, Theory of Elec. Polarization (Elsevier, New York, 1978).

Brenner, H.

M. Zuzovsky, H. Brenner, “Effective Conductivities of Composite Materials Composed of Cubic Arrangements of Spherical Particles Embedded in an Isotropic Matrix,” Z. Angew. Math. Phys. 28, 979 (1977).
[CrossRef]

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Calculations of Various Physical Constants of Heterogeneous Substances: Part I. Dielectric Constants and Conductivity of Mixtures of Isotropic Substances,” Ann. Phys. 24, 636 (1935).
[CrossRef]

Christy, R. W.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1980).

DeVries, D. A.

D. A. DeVries, “Het Warmtegeleidingsvermogen van grond,” Meded. Landbouwhogesch. Wageningen 52, 1 (1952).

Felderhof, B. U.

B. U. Felderhof, “Bounds for the Effective Dielectric Constant of a Suspension of Uniform Spheres,” J. Phys. 15, 1731 (1982).

Frohlich, H.

H. Frohlich, Theory of Dielectrics (Oxford U.P., London, 1949).

Jackson, W.

M. M. Z. Kharadly, W. Jackson, “The Properties of Artificial Dielectrics Comprising Arrays of Conducting Elements,” Proc. Inst. Elec. Engrs. 100, 199 (1953).

Kharadly, M. M. Z.

M. M. Z. Kharadly, W. Jackson, “The Properties of Artificial Dielectrics Comprising Arrays of Conducting Elements,” Proc. Inst. Elec. Engrs. 100, 199 (1953).

Kirkwood, J. G.

J. G. Kirkwood, “On the Theory of Dielectric Polarization,” J. Chem. Phys. 4, 592 (1936).
[CrossRef]

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1980).

Panofsky, W. K.

W. K. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), p. 33.

Phillips, M.

W. K. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), p. 33.

Rayleigh, J. W.

J. W. Rayleigh, “On the Influence of Obstacles Arranged in Rectangular Order upon the Properties of a Medium,” Philos. Mag. 34, 481 (1982).

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1980).

Runge, I.

I. Runge, “On the Electrical Conductivity of Metallic Aggregates,” Z. Tech. Phys. 6, 61 (1925).

Sangani, A. S.

A. S. Sangani, A. Acrivos, “The Effective Conductivity of a Periodic Array of Spheres,” Proc. R. Soc. London Ser A 386, 263 (1983).
[CrossRef]

Yvon, J.

J. Yvon, Recherches sur la Theorie Cinetique des Liquides (Hermann, Paris, 19).

Zuzovsky, M.

M. Zuzovsky, H. Brenner, “Effective Conductivities of Composite Materials Composed of Cubic Arrangements of Spherical Particles Embedded in an Isotropic Matrix,” Z. Angew. Math. Phys. 28, 979 (1977).
[CrossRef]

Ann. Phys. (1)

D. A. G. Bruggeman, “Calculations of Various Physical Constants of Heterogeneous Substances: Part I. Dielectric Constants and Conductivity of Mixtures of Isotropic Substances,” Ann. Phys. 24, 636 (1935).
[CrossRef]

J. Chem. Phys. (1)

J. G. Kirkwood, “On the Theory of Dielectric Polarization,” J. Chem. Phys. 4, 592 (1936).
[CrossRef]

J. Phys. (1)

B. U. Felderhof, “Bounds for the Effective Dielectric Constant of a Suspension of Uniform Spheres,” J. Phys. 15, 1731 (1982).

Meded. Landbouwhogesch. Wageningen (1)

D. A. DeVries, “Het Warmtegeleidingsvermogen van grond,” Meded. Landbouwhogesch. Wageningen 52, 1 (1952).

Philos. Mag. (1)

J. W. Rayleigh, “On the Influence of Obstacles Arranged in Rectangular Order upon the Properties of a Medium,” Philos. Mag. 34, 481 (1982).

Proc. Inst. Elec. Engrs. (1)

M. M. Z. Kharadly, W. Jackson, “The Properties of Artificial Dielectrics Comprising Arrays of Conducting Elements,” Proc. Inst. Elec. Engrs. 100, 199 (1953).

Proc. R. Soc. London Ser A (1)

A. S. Sangani, A. Acrivos, “The Effective Conductivity of a Periodic Array of Spheres,” Proc. R. Soc. London Ser A 386, 263 (1983).
[CrossRef]

Z. Angew. Math. Phys. (1)

M. Zuzovsky, H. Brenner, “Effective Conductivities of Composite Materials Composed of Cubic Arrangements of Spherical Particles Embedded in an Isotropic Matrix,” Z. Angew. Math. Phys. 28, 979 (1977).
[CrossRef]

Z. Tech. Phys. (1)

I. Runge, “On the Electrical Conductivity of Metallic Aggregates,” Z. Tech. Phys. 6, 61 (1925).

Other (5)

J. Yvon, Recherches sur la Theorie Cinetique des Liquides (Hermann, Paris, 19).

W. K. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), p. 33.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1980).

H. Frohlich, Theory of Dielectrics (Oxford U.P., London, 1949).

C. J. F. Böttcher, P. Bordewijk, Theory of Elec. Polarization (Elsevier, New York, 1978).

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Figures (11)

Fig. 1
Fig. 1

Ratio of effective permittivity κe to permittivity κm of supporting dielectric medium as a function of frequency for a uniform array of conducting dielectric spheres embedded in a pure dielectric. The dotted curves are experimental values (Ref. 4). The solid curves are the theoretical based on the Clausius-Mosotti formulas, Eqs. (11b) and (11c); the Rayleigh formulas, Eqs. (23b) and (23c) and similarly for the Zuzovsky-Brenner formula. The volume density of spheres is 9.8%.

Fig. 2
Fig. 2

Same as Fig. 1 when the volume density of spheres is 27.5%.

Fig. 3
Fig. 3

Same as Fig. 1 when the volume density of spheres is 46.5%.

Fig. 4
Fig. 4

Percentage difference between values for (κe/κm)R, Eq. (23b), and (κe/κm)CM, Eq. (11b), as a function of frequency using the parameters of Kharadly and Jackson. Also shown is the same quantity when (κe/κm)ZB, based on the Zuzovsky-Brenner formula, replaces the Rayleigh value. The volume density of spheres is 9.8%.

Fig. 5
Fig. 5

Same as Fig. 4 when the volume density of spheres is 27.5%.

Fig. 6
Fig. 6

Same as Fig. 5 when the volume density of spheres is 46.5%.

Fig. 7
Fig. 7

Percentage difference between values for (tanδ)R, Eq. (23c), and (tanδ)CM, Eq. (11c), as a function of frequency using the parameters of Kharadly and Jackson. Also shown is the same quantity when (tanδ)ZB, based on the Zuzovsky-Brenner formula, replaces the Rayleigh value. The volume density of spheres is 9.8%.

Fig. 8
Fig. 8

Same as Fig. 7 when the volume density of spheres is 27.5%.

Fig. 9
Fig. 9

Same as Fig. 7 when the volume density of spheres is 46.5%.

Fig. 10
Fig. 10

Relative deviation (ɛeɛcm)/ɛcm of the effective value ɛe from the Clausius-Mosotti value ɛcm of the dielectric constant for a suspension of spheres as a function of fractional impurity content for β = (ɛsɛm)/(ɛs + 2ɛm) = 0.1. These curves are given by Felderhoff. Curve KY corresponds to the theory of Kirkwood and Yvon. Curve W corresponds to Felderhof’s modified version of the KY theory. Curve B corresponds to the self-consistent theory of Bruggeman, Curves U and L correspond to upper and lower bounds as given by Felderhof. We have added the curves R and ZB which are, respectively, Rayleigh’s formula, Eq. (24a), and Zuzovsky and Brenner’s formula, Eq. (24b), for a uniform array of spheres of equal radii.

Fig. 11
Fig. 11

Same as Fig. 6 with β = 0.3. Felderhof includes curves for β = −0.1 and β = −0.3 which we omit here since the main features are covered by the values 0.1 and 0.3.

Equations (84)

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ɛ m * = ɛ 0 κ m * = ɛ 0 ( κ m - j σ m ω ɛ 0 ) = - j ɛ 0 ω ( σ m ɛ 0 + j ω κ m ) ;
ɛ s * = ɛ o κ s * = ɛ o ( κ s - j σ s ω ɛ o ) = - j ɛ o ω ( σ s ɛ o + j ω κ s ) ,
ɛ e * = - j ɛ o ω ( σ e ω + j ω κ e ) .
P = n o ɛ m * α E eff
P = ɛ m * χ e * E o
E eff = E o + γ P ɛ m * ,
χ e * = n o α 1 - n o α / 3 ,
κ e * = ɛ e * ɛ m * = 1 - n o / ( - 1 α + n o 3 ) = 1 + n o α 1 - n o α / 3 = 1 + 2 n o α / 3 1 - n o α / 3 ,
ɛ e * ɛ m * = ( σ e ɛ o + j ω κ e ) / ( σ m ɛ o + j ω κ m )
ϕ = [ ( ɛ s * - ɛ m * ɛ s * + 2 ɛ m * ) a 3 r 3 - 1 ] E eff z ,
ϕ = p z / 4 π ɛ m * r 3 = α z E eff / 4 π r 3 ,
α = 4 π a 3 ɛ s * - ɛ m * ɛ s * + 2 ɛ m * ,
ɛ m * ɛ s * = ( σ m ɛ o + j ω κ m ) / ( σ ɛ o + j ω κ s ) .
α 4 π a 3 = ɛ s * - ɛ m * ɛ s * + 2 ɛ m * = σ + j ω κ σ + j ω κ = B ,
σ = ( σ s - σ m ) / ɛ o             κ = κ s - κ m ,
σ = ( σ s + 2 σ m ) / ɛ o             κ = κ s + 2 κ m .
ɛ e * ɛ m * = 1 + 2 f B 1 - f B = 1 + 3 f B 1 - f B ,
σ e ɛ o + j ω κ e = ( σ m ɛ o + j ω κ m ) 1 + 2 f B 1 - f B = σ m ɛ o [ ( σ + 2 f σ ) + j ω ( κ + 2 f κ ) ( σ - f σ ) + j ω ( κ - f κ ) ] + j ω κ 0 [ ( σ + 2 f σ ) + j ω ( κ + 2 f κ ) ( σ - f σ ) + j ω ( κ - f κ ) ] .
σ α = [ σ s ( 1 + 2 f ) + 2 σ m ( 1 - f ) ] ɛ o , κ α = κ s ( 1 + 2 f ) + 2 κ m ( 1 - f ) , σ α - σ β = 3 f ( σ s - σ m ) / ɛ o , σ β = [ σ s ( 1 - f ) + σ m ( 2 - f ) ] / ɛ o , κ β = κ s ( 1 - f ) + κ m ( 2 + f ) , κ α - κ β = 3 f ( κ s - κ m ) .
σ e ɛ o + j ω κ e = σ m ɛ o ( σ α + j ω κ a σ β + j ω κ β ) + j ω κ m ( σ α + j ω κ α σ β + j ω κ β ) .
( σ e σ m ) C M = σ α σ β + ω 2 [ κ α κ β - κ m ɛ o σ m ( σ β κ α - σ α κ β ) ] σ β 2 + ω 2 κ β 2 = 1 + σ β ( σ α - σ β ) + ω 2 [ κ β ( κ α - κ β ) - κ m ɛ o σ m ( σ β κ α - σ α κ β ) ] σ β 2 + ω 2 κ β 2 ,
( κ e κ m ) CM = σ m ɛ o κ m ( σ β κ α - σ α κ β ) + σ α σ β + ω 2 κ α κ β σ β 2 + ω 2 κ β 2 = 1 + σ m ɛ o κ m ( σ β κ α - σ α κ β ) + σ β ( σ α - σ β ) + ω 2 κ β ( κ α - κ β ) σ β 2 + ω 2 κ β 2 .
( tan δ ) CM = σ e ω ɛ o κ e .
σ β = σ s ( 1 - f ) / ɛ o ,             σ α = σ s ( 1 + 2 f ) / ɛ o , σ α - σ β = σ s 3 f / ɛ o ,             κ α - κ β = 3 f ( κ s - κ m ) , ( σ e ) CM = 9 f σ s w 2 D 1 2 + w 2 D 2 2 ,
( κ e κ m ) CM = 1 + 3 f [ D 1 + w 2 ( γ - 1 ) D 2 ] D 1 2 + w 2 D 2 2 ,
( tan δ ) CM = 9 f σ s w 2 D 1 2 + w 2 D 2 + 3 f [ D 1 + w 2 ( γ - 1 ) D 2 ] ,
D 1 = 1 - f ,             D 2 = γ ( 1 - f ) + 2 + f , w = ω ɛ o κ m / σ s ,             γ = κ s / κ m .
( σ e ) CM = 0 ,
( κ e κ m ) CM = 1 + κ α - κ β κ β = 1 + 3 f ( κ s - κ o ) κ s + 2 κ m - f ( κ s - κ o ) = 1 + 3 f B 1 - f B ,
J E o = ( 1 - 1 α β γ E o / 4 π B 1 ) j ω ɛ m * = j ω ɛ e * ,
J E o = j ω ɛ o ( κ e - j σ e ω ɛ o ) = j ω ɛ e * ;
β 3 E o 4 π B 1 = β 3 4 π a 3 [ 2 + ν * 1 - ν * + 2 ( a β ) 3 S 2 - 32 5 ( a β ) 10 ( 1 - ν * 4 / 3 + ν * ) S 4 2 + ] ;
β 3 E o 4 π B 1 = β 2 ( 2 + ν * ) 4 π a 3 ( 1 - ν * ) + 1 3 β 3 - 32 × ( 3.11 ) 2 5 × 4 π ( a β ) 7 ( 1 - ν * 4 / 3 + ν * ) .
β 3 E o 4 π B 1 = 1 n o ( - 1 α + n o 3 - n o c 3 ) = 1 3 f ( 2 + ν * 1 - ν * + f - f c ) ,
c = 32 5 ( 3 4 π ) 10 / 3 ( 3.11 ) 2 f 7 / 3 ( 1 - ν * 4 / 3 + ν * ) = 0.523 f 7 / 3 ( 1 - ν * 4 / 3 + ν * ) = 1.57 f 7 / 3 ( 1 - ν * 4 + 3 ν * ) ,
c = 1.31 f 7 / 3 ( 1 - ν * 4 / 3 + ν * ) = 1.31 × 3 f 7 / 3 ( 1 - ν * 4 + 3 ν * ) .
κ e * = ɛ e * ɛ m * = 1 - n o ( - 1 α + n o 3 - n o c 3 ) = 1 + n o α 1 - n o ( 1 - c ) α / 3 = 1 + 3 f B 1 - f B ( 1 - c ) = 1 + 2 f B + f B c 1 - f B + f B c ,
E eff = E o + γ P ɛ m * ,
κ e * 1 + n o α = 1 + n o 4 π a 3 B .
c = - 4.93 f 7 / 3 ɛ s * - ɛ m * 3 ɛ s * + 4 ɛ m * = - 4.93 f 7 / 3 × [ ( σ s - σ m ) + j ω ɛ 0 ( κ s - κ m ) ( 3 σ s + 4 σ m ) + j ω ɛ o ( 3 κ s + 4 κ m ) ] A ( x + j y ) ,
x = - [ ( σ s - σ m ) ( 3 σ s + 4 σ o ) + ω 2 ɛ o 2 ( κ s - κ m ) ( 3 κ s + 4 κ m ) ] ( 3 σ s + 4 σ m ) 2 + ω 2 ɛ o 2 ( 3 κ s + 4 κ m ) 2
y = 7 ( σ s κ m - σ m κ s ) ω / ɛ o ( 3 σ s + 4 σ m ) 2 + ω 2 ɛ o 2 ( 3 κ s + 4 κ m ) 2 ,             A = 4.93 f 7 / 3 .
σ e ɛ o + j ω κ e = ( σ m ɛ o + j ω κ m ) [ σ + 2 f σ + f A ( σ x - ω κ y ) + j [ ω ( κ + 2 f κ ) + f A ( ω κ x + σ y ) ] σ - f σ + f A ( σ x - ω κ y ) + j [ ω ( κ - f κ ) + f A ( ω κ x + σ y ) ] ] ,
σ e ɛ o + j ω κ e = ( σ m ɛ o + j ω κ m ) × σ + 2 f σ + f β o + j ω [ ( κ + 2 f κ ) + f γ o / ω ] σ - f σ + f β o + j ω [ ( κ - f κ ) + f γ o / ω ] ,
σ α = σ + 2 f σ + f β o , κ α = κ + 2 f κ + f γ o / ω , σ β = σ - f σ + f β o , κ β = κ - f κ + f γ o / ω ,
σ = ( σ s - σ m ) / ɛ o = σ s ( 1 - ν ) / ɛ o ; σ = ( σ 2 + 2 σ m ) / ɛ o = σ s ( 1 + 2 ν ) / ɛ o ; κ = κ s - κ m = κ m ( γ - 1 ) ; κ = ( κ s + 2 κ m ) = κ m ( γ + 2 ) ;
( σ e ɛ o + j ω κ e ) = ( σ m ɛ o + j ω κ m ) ( σ α + j ω κ α σ β + j ω κ β ) = ( σ m ɛ o + j ω κ m ) [ σ α σ β + ω 2 κ α κ β + j ω ( κ α σ β - κ β σ α ) ] σ β 2 + ω 2 κ β 2 .
( σ e σ m ) R = 1 + σ β ( σ α - σ β ) + ω 2 [ κ β ( κ α - κ β ) - κ o ɛ o ( κ α σ β - κ β σ α ) / σ o ] σ β 2 + ω 2 κ β 2 .
( κ e κ m ) R = 1 + σ β ( σ α - σ β ) + [ σ o ( κ α σ β - κ β σ α ) / ɛ o κ o ] + ω 2 κ β ( κ α - κ β ) σ β 2 + ω 2 κ β 2 .
σ α - σ β = 3 f σ ,             κ α - κ β = 3 f κ , κ α σ β - κ β σ α = 3 f ( σ κ - κ σ ) + 3 f 2 ( κ β o - σ γ o / ω ) ,
( σ e σ m ) R = 1 + 3 f σ σ β + ω 2 ( 3 f κ κ β - κ o ɛ o σ o ) [ 3 f ( σ κ - κ σ ) + 3 f 2 ( κ β 0 - σ γ o / ω ] σ β 2 + ω 2 κ β 2 ,
( κ e κ m ) R = 1 + 3 f σ σ β + σ o ɛ o κ o [ 3 f ( σ κ - κ σ ) + 3 f 2 ( κ β o - σ γ o / ω ) ] + ω 2 3 f κ κ β σ β 2 + ω 2 κ β 2 ,
σ α = σ s ɛ o [ 2 ν ( 1 - f ) + 1 + 2 f + f β o ɛ o / σ s ] , κ α = κ m [ 1 + 2 f ) γ + 2 ( 1 - f ) + f γ o / ω κ m ] , β o = A σ s ɛ o [ ( 1 - ν ) x - w ( γ - 1 ) y ] , σ β = σ s ɛ o [ ν ( 2 + f ) + 1 - f + f β o ɛ o / σ s ] , κ β = κ [ γ ( 1 - f ) + 2 + f + f γ o / ω κ m ] , γ o = A σ s ɛ o [ w ( γ - 1 ) x - ( 1 - ν ) y ] ,
σ = σ s / ɛ o ,             σ = σ s / ɛ o ,             σ α = σ s ( 1 + 2 f ) / ɛ o + f β o , σ β = σ s ( 1 - f ) + f β o / w , β o = A σ s ɛ o [ x - w ( γ - 1 ) y ] , γ o = σ s ɛ o A ( w x - y ) , x = - [ 3 + w 2 ( γ - 1 ) ( 3 γ + 4 ) ] 9 + w 2 ( 3 γ + 4 ) 2 ,             y = 7 w 9 + w 2 ( 3 γ + 4 ) 2 ,
D 0 = D 1 2 + w 2 D 2 , D 1 = ( 1 - f ) + f A [ x - w ( γ - 1 ) y ] , D 2 = [ γ ( 1 - f ) + 2 + f + f A ( x - y / w ) ] .
3 f σ σ β = ( σ s ɛ o ) 2 3 f D 1 and 3 f κ κ β = 3 f κ m 2 ( γ - 1 ) D 2 ,
σ κ - κ σ = - 3 σ s κ m / ɛ o and 3 f 2 ( κ β o - σ γ o / ω ) = 3 f 2 A κ m σ s ɛ o E ,
( σ e ) R = 3 σ s f w 2 ( 3 - f A E ) / D 0 ,
( κ e κ m ) R = ( ɛ e ɛ m ) R = 1 + 3 f [ D 1 + w 2 ( γ - 1 ) D 2 ] / D 0 ,
( tan δ ) R = σ e ω ɛ o κ e = 3 f w ( 3 - f A E ) D 0 + 3 f [ D 1 + w 2 ( γ - 1 ) D 2 .
κ e ( ɛ e ɛ o ) ZB = 1 - 3 f [ 2 + ν 1 - ν + f ( 1 - c ) ] - 1 = 1 + 3 f β 1 - f β ( 1 - c ) ,
β = ν - 1 ν + 2 ,             c = 16 a f 7 / 3 [ 4 + 3 ν 3 ( 1 - ν ) + 20 b f 7 / 3 ] - 1 ,             ν = ɛ s / ɛ m .
c ZB = g o [ 4 + 3 ν * 3 ( 1 - ν * ) + h o ] - 1 = g o 3 ( ɛ m * - ɛ s * ) 4 ɛ m * + 3 ɛ s * + 3 h o ( ɛ m * - ɛ s * ) ,
c ZB = 3 g ( ɛ m * - ɛ s * ) h ɛ m * + g ɛ s * ,
c ZB = 3 g o { [ ( σ m - σ o ) C / ɛ o + ω 2 ( κ m - κ s ) D ] + j ω [ C ( κ m - κ s ) - ( σ m - σ s ) D / ɛ o ] } / ( C 2 + ω 2 D 2 ) ,
c ZB A ( x + i y ) ,             A = 3 g o ,
x = [ - g + w 2 ( 1 - γ ) ( h + g γ ) ] / D ,             y = w [ g ( 1 - γ ) + ( h + g γ ) ] / D ,
ɛ R ɛ m = 1 + 3 f B 1 - f B ( 1 - c ) , c = - 3.93 f 7 / 3 3 B / ( 7 + 2 B ) .
ɛ ZB ɛ m = 1 - 3 f [ 2 + ν 1 - ν + f - 16 ( 0.2857 ) 2 f 10 / 3 ( 4 / 3 + γ ) / ( 1 - γ ) + 20 ( 0.02036 ) f 7 / 3 - 53.5 ( 0.02036 ) 2 ( 1 - γ ) f 14 / 3 ( 6 / 5 + γ ) + O ( f 6 ) ] - 1 .
ɛ W - ɛ m ɛ W + 2 ɛ m = f B D ,
B = ( ɛ s - ɛ m ) / ( ɛ s + 2 ɛ m ) , D = 1 + [ ( ɛ s - ɛ m ) / ɛ m ] 2 a = 1 + [ 3 B / ( 1 - B ) ] 2 a , a = K ( f ) / f , K ( f ) = 0.04682 f 2 - 0.05724 f 3 + 0.05072 f 4 .
ɛ W ɛ m = 1 + 3 1 f B - 1 + c W .
ɛ W ɛ m = 1 + 2 f B D 1 - f B D = 1 + 3 f B D 1 - f B D = 1 + 3 1 f B D - 1 = 1 + 3 1 f B - 1 + 1 f B D - 1 f B .
c W = 1 - D f B D = - 9 B a f [ ( 1 - B ) 2 + 9 B 2 a ] .
c W = - a f [ A C 1 + W 2 B C 2 + j W ( B C 1 - A C 2 ) C 1 2 + W 2 C 2 2 ] ,
A = [ ( 1 - ν ) ( 1 + 2 ν ) - w 2 ( γ - 1 ) ( γ + 2 ) ] , C 1 = { ν 2 + a ( 1 - ν ) 2 - w 2 [ 1 + a ( γ + 1 ) 2 ] } , B = γ ( ν + 2 ) + ( 1 - 4 ν ) C 2 = 2 ν + 2 a ( 1 - ν ) ( γ - 1 ) , ν = σ m / σ s ,             γ = ɛ s / ɛ m w = ω ɛ o κ e / σ s .
c W = A 0 ( x + i y ) , A W = K ( f ) / f 2 = 0.04682 - 0.05724 f + 0.05072 f 2 .
x = - A C 1 + w 2 B C 2 C 1 2 + w 2 C 2 2 ,             y = w ( B C 1 - A C 2 ) C 1 2 + w 2 C 2 2 .
A = 1 - w 2 ( γ - 1 ) ( γ + 2 ) ,             B = 2 γ + 1 , C 1 = a - w 2 [ 1 - a ( γ - 1 ) 2 ] ,             C 2 = 2 a ( γ - 1 ) .
x = - ( A C 1 + w 2 B C 2 ) / D W ,             y = - w ( B C 1 - A C 2 ) / D W ,
w = ω ɛ o κ m σ s = κ m 2 n s k s 1 / 120.
( σ e ) CM 8 f σ s ( κ m / 2 n s k s ) 2 ( 1 - f ) 2 , ( κ e κ m ) CM 1 + 3 f 1 - f = 1 + 2 f 1 - f .
x - 1 / 3 ,             y 7 w / 9 ,             E [ - ( γ - 2 ) / 3 + 7 / 9 ] , D 1 = 1 - f + f A / e ,             D 0 D 1 2 ,             A = 4.93 ,
( σ e ) R 3 f σ s ( κ m / 2 n s k s ) 2 [ 3 - f A E ] / ( 1 - f - f A / 3 ) 2 , ( κ e κ m ) R 1 + 3 f ( 1 - f - f A / 3 ) / ( 1 - f ) 2 .

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