Abstract

Two dispersion models of disordered solids, one parameterizing density of states (PDOS) and the other parameterizing joint density of states (PJDOS), are presented. Using these models, not only the complex dielectric function of the materials, but also some information about their electronic structure can be obtained. The numerical integration is necessary in the PDOS model. If analytical expressions are required the presented PJDOS model is, for some materials, a suitable option still providing information about the electronic structure of the material. It is demonstrated that the PDOS model can be successfully applied to a wide variety of materials. In this paper, its application to diamond-like carbon (DLC), a-Si and SiO2-like materials are discussed in detail. Unlike the PDOS model, the presented PJDOS model represents a special case of parameterization that can be applied to limited types of materials, for example DLC, ultrananocrystalline diamond (UNCD) and SiO2-like.

© 2007 Optical Society of America

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References

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  1. A. R. Forouhi and I. Bloomer, "Optical dispersion relations for amorphous semiconductors and amorphous dielectrics," Phys. Rev. B 34, 7018-7026 (1986).
    [CrossRef]
  2. G. E. Jellison and F. A. Modine, "Parameterization of the optical functions of amorphous materials in the interband region," Appl. Phys. Lett. 69, 371-373 (1996).
    [CrossRef]
  3. A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. M. Deng, and G. Ganguly, "Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: Applications in thin film photovoltaics," J. Appl. Phys. 92, 2424-2436 (2002).
    [CrossRef]
  4. N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1971).
  5. J. Tauc, "Optical Properties of Non-Crystaline Solids," in Optical Properties of Solids, F. Abel`es, ed., pp. 277-313 (North-Holland, Amsterdam, 1972).
  6. S. Adachi, Optical Properties of Crystaline and Amorphous Semiconductors: Matrials and Fundamental Principles (Kluwer, Boston, 1999).
    [CrossRef] [PubMed]
  7. P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 2001).
  8. F. Wooten, Optical Properties of Solids (Academic Press, New York, 1972).
  9. D. Franta, I. Ohlýdal, M. Frumar, and J. Jedelský, "Expression of the Optical Constants of Chalcogenide Thin Films Using the New Parameterization Dispersion Model," Appl. Surf. Sci. 212-213, 116-121 (2003).
    [CrossRef]
  10. D. Franta, I. Ohlýdal, P. Klapetek, and P. Roca i Cabarrocas, "Complete Characterization of Rough Polymorphous Silicon Films by Atomic Force Microscopy and the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry," Thin Solid Films 455-456, 399-403 (2004).
    [CrossRef]
  11. L. Zajý¡cková, V . Bur¡sýková, D . Franta, A.  Bousquet, A.  Granier, A.  Goullet, and J. Bur¡sýk, "Comparative Study of Films Deposited from HMDSO/O2 in Continuous Wave and Pulsed rf Discharges," Plasma Process. Polym. 4, S287-S293 (2007).
    [CrossRef]
  12. D. Franta, L. Zajý¡cková, V. Bur¡sýková, and I. Ohlýdal, "New Dispersion Model of the Optical Constants of the DLC Films," Acta Phys. Slov.  53, 373-384 (2003).
  13. D. Franta, I. Ohlýdal, V. Burýková, and L. Zajý¡cková, "Optical properties of diamond-like carbon films containing SiOx," Diamond Relat. Mater. 12, 1532-1538 (2003).
    [CrossRef]
  14. D. Franta, I. Ohlýdal, V. Burýková, and L.  Zajý¡cková, "Optical Properties of Diamond-Like Carbon Films Containing SiOx Studied by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry," Thin Solid Films 455-456, 393-398 (2004).
    [CrossRef]
  15. D. Franta, V. Burýková, I. Ohlýdal, L. Zajý¡cková, and P. Stáhel, "Thermal stability of the optical properties of plasma deposited diamond-like carbon thin films," Diamond Relat. Mater. 14, 1795-1798 (2005).
    [CrossRef]
  16. D. Franta, V. Burýková, I. Ohlýdal, P. Stáhel, M. Ohlýdal, and D. Ne¡cas, "Correlation of thermal stability of the mechanical and optical properties of diamond-like carbon films," Diamond Relat. Mater. 16, 1331-1335 (2007).
    [CrossRef]
  17. D. Franta, V. Burýková, D. Ne¡cas, and L. Zajý¡cková, "Modeling of optical constants of diamond-like carbon," Diamond Relat. Mater. (submitted for publication).
  18. D. Franta, M. Hrdli¡cka, D. Ne¡cas, M. Frumar, I. Ohlýdal, and M. Pavli¡sta, "Optical characterization of phase changing Ge2Sb2Te5 chalcogenide films," Phys. Status Solidi A-Appl. Mat. (to be published).
  19. D. C. Ingram, J. A. Woollam, and G. Bu-Abbud, "Mass density and hydrogen concentration in diamond-like carbon films: proton recoil, rutherford backscattering and ellipsometric analysis," Thin Solid Films 137, 225-230 (1986).
    [CrossRef]
  20. F. Demichelis, C. F. Pirri, and A. Tagliaferro, "Evaluation of the [C(sp3)]/[C(sp2)] ratio in diamondlike films through the use of a complex dielectric constant," Phys. Rev. B 45, 14,364-14,370 (1992).
    [CrossRef]
  21. D. Wood and J. Tauc, "Weak Absorption Tails in Amorphous Semiconductors," Phys. Rev. B 5, 3144-3151 (1972).
    [CrossRef]
  22. D. Franta and I. Ohlýdal, "Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries," J. Mod. Opt. 45, 903-934 (1998).
    [CrossRef]
  23. L. Pajasov’a, "Optical properties of GeO2 in the ultraviolet region," Czech. J. Phys. 19, 1265-1270 (1969).
    [CrossRef]
  24. H. R. Philipp, "Silicon Dioxide (SiO2) (Glass)," in Handbook of Optical Constants of Solids, E. Palik, ed., vol. I, pp. 749-763 (Academic Press, New York, 1985).
    [CrossRef]
  25. L. A. J. Garvie, P. Rez, J. R. Alvarez, and P. R. Buseck, "Interband transitions of crystalline and amorphous SiO2: An electron energy-loss spectroscopy (EELS) study of the low-loss region," Solid State Commun. 106(5), 303-307 (1998).
    [CrossRef]
  26. D. Franta, L. Zaj ¡cková, M. Karásková, O. Ja¡sek, D. Ne¡cas, P. Klapetek, and M. Valtr, "Optical Characterization of Ultrananocrystalline Diamond Films," Diamond Relat. Mater. (submitted for publication).
  27. M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, M. Booth, and F. Rossi, GNU Scientific Library Reference Manual, 2nd ed. (Network Theory Limited, Bristol, 2005).
  28. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1964).

2007 (2)

L. Zajý¡cková, V . Bur¡sýková, D . Franta, A.  Bousquet, A.  Granier, A.  Goullet, and J. Bur¡sýk, "Comparative Study of Films Deposited from HMDSO/O2 in Continuous Wave and Pulsed rf Discharges," Plasma Process. Polym. 4, S287-S293 (2007).
[CrossRef]

D. Franta, V. Burýková, I. Ohlýdal, P. Stáhel, M. Ohlýdal, and D. Ne¡cas, "Correlation of thermal stability of the mechanical and optical properties of diamond-like carbon films," Diamond Relat. Mater. 16, 1331-1335 (2007).
[CrossRef]

2005 (1)

D. Franta, V. Burýková, I. Ohlýdal, L. Zajý¡cková, and P. Stáhel, "Thermal stability of the optical properties of plasma deposited diamond-like carbon thin films," Diamond Relat. Mater. 14, 1795-1798 (2005).
[CrossRef]

2004 (2)

D. Franta, I. Ohlýdal, P. Klapetek, and P. Roca i Cabarrocas, "Complete Characterization of Rough Polymorphous Silicon Films by Atomic Force Microscopy and the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry," Thin Solid Films 455-456, 399-403 (2004).
[CrossRef]

D. Franta, I. Ohlýdal, V. Burýková, and L.  Zajý¡cková, "Optical Properties of Diamond-Like Carbon Films Containing SiOx Studied by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry," Thin Solid Films 455-456, 393-398 (2004).
[CrossRef]

2003 (2)

D. Franta, I. Ohlýdal, V. Burýková, and L. Zajý¡cková, "Optical properties of diamond-like carbon films containing SiOx," Diamond Relat. Mater. 12, 1532-1538 (2003).
[CrossRef]

D. Franta, I. Ohlýdal, M. Frumar, and J. Jedelský, "Expression of the Optical Constants of Chalcogenide Thin Films Using the New Parameterization Dispersion Model," Appl. Surf. Sci. 212-213, 116-121 (2003).
[CrossRef]

2002 (1)

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. M. Deng, and G. Ganguly, "Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: Applications in thin film photovoltaics," J. Appl. Phys. 92, 2424-2436 (2002).
[CrossRef]

1998 (2)

L. A. J. Garvie, P. Rez, J. R. Alvarez, and P. R. Buseck, "Interband transitions of crystalline and amorphous SiO2: An electron energy-loss spectroscopy (EELS) study of the low-loss region," Solid State Commun. 106(5), 303-307 (1998).
[CrossRef]

D. Franta and I. Ohlýdal, "Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries," J. Mod. Opt. 45, 903-934 (1998).
[CrossRef]

1996 (1)

G. E. Jellison and F. A. Modine, "Parameterization of the optical functions of amorphous materials in the interband region," Appl. Phys. Lett. 69, 371-373 (1996).
[CrossRef]

1992 (1)

F. Demichelis, C. F. Pirri, and A. Tagliaferro, "Evaluation of the [C(sp3)]/[C(sp2)] ratio in diamondlike films through the use of a complex dielectric constant," Phys. Rev. B 45, 14,364-14,370 (1992).
[CrossRef]

1986 (2)

D. C. Ingram, J. A. Woollam, and G. Bu-Abbud, "Mass density and hydrogen concentration in diamond-like carbon films: proton recoil, rutherford backscattering and ellipsometric analysis," Thin Solid Films 137, 225-230 (1986).
[CrossRef]

A. R. Forouhi and I. Bloomer, "Optical dispersion relations for amorphous semiconductors and amorphous dielectrics," Phys. Rev. B 34, 7018-7026 (1986).
[CrossRef]

1972 (1)

D. Wood and J. Tauc, "Weak Absorption Tails in Amorphous Semiconductors," Phys. Rev. B 5, 3144-3151 (1972).
[CrossRef]

1969 (1)

L. Pajasov’a, "Optical properties of GeO2 in the ultraviolet region," Czech. J. Phys. 19, 1265-1270 (1969).
[CrossRef]

Acta Phys. Slov (1)

D. Franta, L. Zajý¡cková, V. Bur¡sýková, and I. Ohlýdal, "New Dispersion Model of the Optical Constants of the DLC Films," Acta Phys. Slov.  53, 373-384 (2003).

Appl. Phys. Lett. (1)

G. E. Jellison and F. A. Modine, "Parameterization of the optical functions of amorphous materials in the interband region," Appl. Phys. Lett. 69, 371-373 (1996).
[CrossRef]

Appl. Surf. Sci. (1)

D. Franta, I. Ohlýdal, M. Frumar, and J. Jedelský, "Expression of the Optical Constants of Chalcogenide Thin Films Using the New Parameterization Dispersion Model," Appl. Surf. Sci. 212-213, 116-121 (2003).
[CrossRef]

Czech. J. Phys. (1)

L. Pajasov’a, "Optical properties of GeO2 in the ultraviolet region," Czech. J. Phys. 19, 1265-1270 (1969).
[CrossRef]

Diamond Relat. Mater. (3)

D. Franta, I. Ohlýdal, V. Burýková, and L. Zajý¡cková, "Optical properties of diamond-like carbon films containing SiOx," Diamond Relat. Mater. 12, 1532-1538 (2003).
[CrossRef]

D. Franta, V. Burýková, I. Ohlýdal, L. Zajý¡cková, and P. Stáhel, "Thermal stability of the optical properties of plasma deposited diamond-like carbon thin films," Diamond Relat. Mater. 14, 1795-1798 (2005).
[CrossRef]

D. Franta, V. Burýková, I. Ohlýdal, P. Stáhel, M. Ohlýdal, and D. Ne¡cas, "Correlation of thermal stability of the mechanical and optical properties of diamond-like carbon films," Diamond Relat. Mater. 16, 1331-1335 (2007).
[CrossRef]

J. Appl. Phys. (1)

A. S. Ferlauto, G. M. Ferreira, J. M. Pearce, C. R. Wronski, R. W. Collins, X. M. Deng, and G. Ganguly, "Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: Applications in thin film photovoltaics," J. Appl. Phys. 92, 2424-2436 (2002).
[CrossRef]

J. Mod. Opt. (1)

D. Franta and I. Ohlýdal, "Ellipsometric Parameters and Reflectances of Thin Films with Slightly Rough Boundaries," J. Mod. Opt. 45, 903-934 (1998).
[CrossRef]

Phys. Rev. B (3)

A. R. Forouhi and I. Bloomer, "Optical dispersion relations for amorphous semiconductors and amorphous dielectrics," Phys. Rev. B 34, 7018-7026 (1986).
[CrossRef]

F. Demichelis, C. F. Pirri, and A. Tagliaferro, "Evaluation of the [C(sp3)]/[C(sp2)] ratio in diamondlike films through the use of a complex dielectric constant," Phys. Rev. B 45, 14,364-14,370 (1992).
[CrossRef]

D. Wood and J. Tauc, "Weak Absorption Tails in Amorphous Semiconductors," Phys. Rev. B 5, 3144-3151 (1972).
[CrossRef]

Phys. Status Solidi A-Appl. Mat. (1)

D. Franta, M. Hrdli¡cka, D. Ne¡cas, M. Frumar, I. Ohlýdal, and M. Pavli¡sta, "Optical characterization of phase changing Ge2Sb2Te5 chalcogenide films," Phys. Status Solidi A-Appl. Mat. (to be published).

Plasma Process. Polym. (1)

L. Zajý¡cková, V . Bur¡sýková, D . Franta, A.  Bousquet, A.  Granier, A.  Goullet, and J. Bur¡sýk, "Comparative Study of Films Deposited from HMDSO/O2 in Continuous Wave and Pulsed rf Discharges," Plasma Process. Polym. 4, S287-S293 (2007).
[CrossRef]

Solid State Commun. (1)

L. A. J. Garvie, P. Rez, J. R. Alvarez, and P. R. Buseck, "Interband transitions of crystalline and amorphous SiO2: An electron energy-loss spectroscopy (EELS) study of the low-loss region," Solid State Commun. 106(5), 303-307 (1998).
[CrossRef]

Thin Solid Films (3)

D. Franta, I. Ohlýdal, P. Klapetek, and P. Roca i Cabarrocas, "Complete Characterization of Rough Polymorphous Silicon Films by Atomic Force Microscopy and the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry," Thin Solid Films 455-456, 399-403 (2004).
[CrossRef]

D. C. Ingram, J. A. Woollam, and G. Bu-Abbud, "Mass density and hydrogen concentration in diamond-like carbon films: proton recoil, rutherford backscattering and ellipsometric analysis," Thin Solid Films 137, 225-230 (1986).
[CrossRef]

D. Franta, I. Ohlýdal, V. Burýková, and L.  Zajý¡cková, "Optical Properties of Diamond-Like Carbon Films Containing SiOx Studied by the Combined Method of Spectroscopic Ellipsometry and Spectroscopic Reflectometry," Thin Solid Films 455-456, 393-398 (2004).
[CrossRef]

Other (10)

D. Franta, V. Burýková, D. Ne¡cas, and L. Zajý¡cková, "Modeling of optical constants of diamond-like carbon," Diamond Relat. Mater. (submitted for publication).

N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1971).

J. Tauc, "Optical Properties of Non-Crystaline Solids," in Optical Properties of Solids, F. Abel`es, ed., pp. 277-313 (North-Holland, Amsterdam, 1972).

S. Adachi, Optical Properties of Crystaline and Amorphous Semiconductors: Matrials and Fundamental Principles (Kluwer, Boston, 1999).
[CrossRef] [PubMed]

P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 2001).

F. Wooten, Optical Properties of Solids (Academic Press, New York, 1972).

D. Franta, L. Zaj ¡cková, M. Karásková, O. Ja¡sek, D. Ne¡cas, P. Klapetek, and M. Valtr, "Optical Characterization of Ultrananocrystalline Diamond Films," Diamond Relat. Mater. (submitted for publication).

M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, M. Booth, and F. Rossi, GNU Scientific Library Reference Manual, 2nd ed. (Network Theory Limited, Bristol, 2005).

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with Formulas, Graphs, and Mathematical Tables (National Bureau of Standards, 1964).

H. R. Philipp, "Silicon Dioxide (SiO2) (Glass)," in Handbook of Optical Constants of Solids, E. Palik, ed., vol. I, pp. 749-763 (Academic Press, New York, 1985).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of interband and intraband electronic transitions. Shaded area depicts occupied electronic states according to the Fermi-Dirac statistics. Symbol E F denotes the Fermi energy.

Fig. 2.
Fig. 2.

Schematic diagram of electronic structure of three types of amorphous solids. Shaded area depicts occupied electronic states according to the Fermi–Dirac statistics. Symbol E F denotes the Fermi energy.

Fig. 3.
Fig. 3.

Unnormalized DOS of two DLC films calculated from the fitted parameters summarized in Table 1. Solid and dashed lines correspond to as deposited and annealed (510 °C) DLC films.

Fig. 4.
Fig. 4.

Real (top) and imaginary (bottom) part of dielectric function for all the studied DLC films.

Fig. 5.
Fig. 5.

Unnormalized DOS of a-Si film calculated from the fitted parameters summarized in Table 2.

Fig. 6.
Fig. 6.

Real (top) and imaginary (bottom) part of dielectric function for a-Si film and c-Si substrate.

Fig. 7.
Fig. 7.

Real (top) and imaginary (bottom) parts of dielectric function of SiO2-like film with the thickness of 1.13 µm deposited on c-Si substrate. Data tabulated for fused silica [24] are shown for comparison.

Tables (3)

Tables Icon

Table 1. Fitting parameters of PDOS model applied to the DLC films from [15] together with hydrogen atomic fractions X H determined by ERDA, the π-to-σ ratio α, the ratio β of the number of valence electrons in the film after and before annealing and sp3-to-sp2 ratio X C ( s p 3 ) X C ( s p 2 ) .

Tables Icon

Table 2. Fitting parameters of model applied to the typical a-Si:H film.

Tables Icon

Table 3. Fitting parameters of model applied to a SiO2-like film prepared similarly as in [11].

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

ε i ( E ) = ( e h m E ) 2 1 4 π ε 0 B 0 j , k p j k 2 f e ( S ) 𝒩 j ( S ) f h ( S + E ) 𝒩 k ( S + E ) d S ,
N j = 𝒩 j ( S ) d S .
𝒥 j k ( E ) = 𝒩 j ( S ) 𝒩 k ( S + E ) d S .
ε i ( E ) = ( e h m E ) 2 sgn ( E ) 4 π ε 0 B 0 j , k p j k 2 f e ( S ) 𝒩 j ( S ) f h ( S + E ) 𝒩 k ( S + E ) d S ,
ε r ( E ) = 1 + 1 π X ε i ( X ) X 2 E 2 d X = 1 + 2 π 0 X ε i ( X ) X 2 E 2 d X .
ε i ( E ) = sgn ( E ) E 2 j = π , σ E F = 0 N j ( S ) N j * ( S + E ) d S ,
N j ( S ) = e h p j j * 2 m π ε 0 B 0 𝒩 j ( S ) .
N j ( S ) d S = N j e h p j j * 2 m π ε 0 B 0 Q j ,
N j ( S ) = { 32 Q j S E g j 2 E h j 2 + S π ( E h j E g j ) 2 for E h j 2 < S < E g j 2 0 otherwise
N j * ( S ) = { 32 Q j S E g j 2 E h j 2 S π ( E h j E g j ) 2 for E g j 2 < S < E h j 2 0 otherwise .
ε i ( E ) = 1 E 2 j = π , σ ( 32 Q j π ( E h j E g j ) 2 ) 2 e j ( E ) ,
A j 32 Q j π ( E h j E g j ) 2 .
α = N π N σ = 1 κ Q π Q σ
N e = N π + N σ Q π + κ Q σ
β n e ( t ) n e ( 0 ) = d f ( t ) N e ( t ) d f ( 0 ) N e ( 0 ) = d f ( t ) ( Q π ( t ) + κ Q σ ( t ) ) d f ( 0 ) ( Q π ( 0 ) + κ Q σ ( 0 ) ) ,
β = n H ( t ) + 4 n C ( t ) n H ( 0 ) + 4 n C ( 0 ) = 4 n a ( t ) 3 n H ( t ) 4 n a ( 0 ) 3 n H ( 0 ) = n a ( t ) ( 4 3 X H ( t ) ) n a ( 0 ) ( 4 3 X H ( 0 ) ) =
n a ( t ) n C ( 0 ) ( 4 3 X H ( t ) ) n a ( 0 ) n C ( t ) ( 4 3 X H ( 0 ) ) = n a ( t ) n a ( 0 ) X C ( 0 ) ( 4 3 X H ( t ) ) n a ( 0 ) n a ( t ) X C ( t ) ( 4 3 X H ( 0 ) ) = ( 1 X H ( 0 ) ) ( 4 3 X H ( t ) ) ( 1 X H ( t ) ) ( 4 3 X H ( 0 ) )
X C ( sp 3 ) X C ( sp 2 ) = ( 1 3 α ) X H ( 1 2 α ) α ( 4 3 X H ) .
ε i ( E ) ( E E g ) 2 E 2 .
ε i ( E ) = sgn ( E ) E 2 E F = 0 [ N ξ ( S ) N ξ * ( S + E ) + 2 N λ ( S ) N ξ * ( S + E ) ] d S ,
N ξ ( S ) = { A ξ C ( S ) S E g 2 E h 2 + S for E h 2 < S < E g 2 0 otherwise ,
N ξ * ( S ) = { A ξ C ( S ) S E g 2 E h 2 S for E g 2 < S < E h 2 0 otherwise ,
C ( S ) = 1 + A 1 exp [ ( S E 1 2 ) 2 2 B 1 2 ] + A 2 exp [ ( S E 2 2 ) 2 2 B 2 2 ] .
N λ ( S ) = A λ exp ( S E g 2 E λ ) .
G ( A δ , i , B δ , i , E δ , i ) = A δ , i 2 π B δ , i { exp [ ( E E δ , i ) 2 2 B δ , i 2 ] exp [ ( E + E δ , i ) 2 2 B δ , i 2 ] } ,
ε i ( E ) = sgn ( E ) E 2 [ E F = 0 N ξ ( S ) N ξ * ( S + E ) d S + i G ( A δ , i , B δ , i , E δ , i ) + G ( A p , B p , E p ) ] ,
( e h m ) 2 p j j * 2 4 π ε 0 B 0 𝒥 j j * ( E ) = J j j * ( E ) .
0 J j j * ( E ) d E = Q j 2 .
ε i ( E ) = J ( E ) E 2 = 1 E 2 j J j j * ( E ) .
J j j * ( E ) ( E E g j ) 2 ( E E h j ) 2
ε i , j j * ( E ) = { 30 Q j 2 ( E E g j ) 2 ( E E h j ) 2 ( E h j E g j ) 5 E 2 for E g j < E < E h j 0 otherwise .
ε i , j j * ( E ) = 60 Q j 2 π ( E h j E g j ) 5 [ B ( E ) ln E + E h j E + E g j + C ( E ) ln E E h j E E g j D ( E ) ]
B ( E ) = Y ( E ) + X ( E ) 2 E 2 , C ( E ) = Y ( E ) X ( E ) 2 E 2 ,
D ( E ) = E g j 2 E h j 2 E 2 ln E h j E g j + 3 ( E h j 2 E g j 2 ) 2 ,
X ( E ) = 2 E [ E h j ( E g j 2 + E 2 ) + E g j ( E h j 2 + E 2 ) ] ,
Y ( E ) = E 2 ( E h j 2 + E g j 2 + 4 E g j E h j + E 2 ) + E g j 2 E h j 2 .
lim E E h j , E g j C ( E ) ln E E h j E E g j = 0 ,
lim E 0 ε r , j j * ( E ) = 60 Q j 2 π ( E h j E g j ) 5 [ ( E h j 2 + E g j 2 + 4 E g j E h j ) ln E h j E g j 3 ( E h j 2 E g j 2 ) 2 ]
lim E ε r , j j * ( E ) = 0 .
ε ̂ ( E ) = 1 + ε ̂ π π * ( E ) + ε ̂ σ σ * ( E ) .
X ε i ( X ) X 2 E 2 d X = 1 2 [ ε i ( X ) X + E + ε i ( X ) X E ] d X =
1 2 ε i ( X ) X + E d X + 1 2 ε i ( X ) X E d X = ε i ( X ) X E d X .
ε r ( E ) = 1 + 1 π ε i ( X ) X E d X .
X = xE 1 x , X = Ex and X = E x ,
ε r ( E ) = 1 + 1 π 0 1 ε i ( E x ) x ε i ( xE ) x ε i ( xE 1 x ) x ( 1 x ) d x = 1 + 1 π 0 1 I ( x ) d x ,
lim x 0 + I ( x ) = J ( 0 ) E ,
lim x 1 I ( x ) = 2 E ε i ( E ) + ε i ( E ) ,
e ( E ) = S min S max ( S E g 2 ) ( E h 2 + S ) ( E + S E g 2 ) ( E h 2 E S ) d S ,
S min = max ( E h 2 , E g 2 E ) and S max = min ( E g 2 , E h 2 E )
S = x E 2 ,
e ( E ) = m m ( M 2 x 2 ) ( m 2 x 2 ) d x .
m = min ( E 2 E g 2 , E h 2 E 2 ) and M = max ( E 2 E g 2 , E h 2 E 2 ) .
e ( E ) = 2 m 2 M 0 π 2 1 k 2 sin 2 t cos 2 t d t , where k = m M .
e ( E ) = 2 3 M 3 [ ( k 2 1 ) K ( k ) + ( k 2 + 1 ) E ( k ) ] ,
K ( k ) = 0 π 2 1 1 k 2 sin 2 t d t , E ( k ) = 0 π 2 1 k 2 sin 2 t d t .

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