Abstract

A new technique for determining the optical properties of organic thin films is presented. A detailed evaluation of the accuracy of the determined optical constants has been performed, and the best combination of measured values yielding the smallest errors in the index of refraction for realistic experimental uncertainties has been found. The proposed method utilizes the fact that optical constants are smooth continuous functions, which reduces the possibility of encountering multiple solutions. The method consists of two steps. In the first step the optical constants at all wavelengths and the film thickness are determined. In the second step the thickness and the imaginary part of the index of refraction are kept fixed while we reevaluate the real part of the index of refraction by using a different objective function with improved sensitivity to the refractive index. After verifying that the proposed method is capable of an accurate estimation of optical constants, we determine the index of refraction data of vanadyl-phthalocyanine in the visible spectral range.

© 2000 Optical Society of America

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References

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  1. J. A. Dobrowolski, F. C. Ho, A. Waldorf, “Determination of optical constants of thin film coating materials based on inverse synthesis,” Appl. Opt. 22, 3191–3200 (1983).
    [CrossRef] [PubMed]
  2. R. M. Bueno, J. F. Trigo, J. M. Martinez-Duart, E. Elizalde, J. M. Sanz, “Study of the optical constants determination of thin films: dependence on theoretical assumptions,” J. Vac. Sci. Technol. A 13, 2378–2383 (1995).
    [CrossRef]
  3. S. V. Babu, M. David, R. C. Patel, “Two-step regression procedure for the optical characterization of thin films,” Appl. Opt. 30, 839–846 (1991).
    [CrossRef] [PubMed]
  4. W. R. Hunter, “Measurements of optical constants in the vacuum ultraviolet (VUV) spectral region,” in Handbook of Optical Constants of Solids I, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 69–88.
    [CrossRef]
  5. L. Ward, “The accuracy of photometric methods for determination of optical constants of thin absorbing films,” J. Phys. D. 15, 1361–1371 (1982).
    [CrossRef]
  6. L. Vriens, W. Rippens, “Optical constants of absorbing thin films on a substrate,” Appl. Opt. 22, 4105–4110 (1983).
    [CrossRef]
  7. A. Hjortsberg, “Determination of optical constants of absorbing materials using transmission and reflection of thin films on partially metallized substrates: analysis of the new (T, Rm) technique,” Appl. Opt. 20, 1254–1263 (1981).
    [CrossRef] [PubMed]
  8. A. H. M. Holtslag, P. M. L. O. Scholte, “Optical measurements of the refractive index, layer thickness, and volume changes of thin films,” Appl. Opt. 28, 5095–5104 (1989).
    [CrossRef] [PubMed]
  9. W. N. Hansen, “Optical characterization of thin films: theory,” J. Opt. Soc. Am. 63, 793–802 (1973).
    [CrossRef]
  10. J. M. del Pozo, L. Diaz, “Method for determination of optical constants of thin films: dependence on experimental uncertainties,” Appl. Opt. 31, 4474–4481 (1992).
    [CrossRef] [PubMed]
  11. E. Elizalde, J. M. Frigerio, J. Rivory, “Determination of thickness and optical constants of thin films from photometric and ellipsometric measurements,” Appl. Opt. 25, 4557–4561 (1986).
    [CrossRef] [PubMed]
  12. E. Elizalde, F. Rueda, “On the determination of optical constants n(λ) and α(λ) of thin supported films,” Thin Solid Films 122, 45–57 (1984).
    [CrossRef]
  13. O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
    [CrossRef]
  14. O. Stenzel, R. Petrich, “Flexible construction of error functions and their minimization: application to the calculation of optical constants of absorbing or scattering thin-film materials from spectrophotometric data,” J. Phys. D 28, 978–989 (1995).
    [CrossRef]
  15. G. Leveque, Y. Villachon-Renard, “Determination of optical constants of a thin film from reflectance spectra,” Appl. Opt. 29, 3207–3212 (1990).
    [CrossRef]
  16. A. B. Djurišić, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal-incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
    [CrossRef]
  17. A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance–probability–controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
    [CrossRef]
  18. T. Fritz, J. Hahn, H. Böttcher, “Determination of optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
    [CrossRef]
  19. J. Blohwitz, M. Pfeiffer, T. Fritz, K. Leo, “Low voltage organic light emitting diodes featuring doped phthalocyanine as hole transport layers,” Appl. Phys. Lett. 73, 729–731 (1998).
    [CrossRef]
  20. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1955), pp. 95–154.
  21. R. F. Potter, “Basic parameters for measuring optical properties,” in Handbook of Optical Constants of Solids I, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 11–34.
    [CrossRef]
  22. B. Harbecke, “Coherent and incoherent reflection and transmission of multilayer systems,” Appl. Phys. B 39, 165–170 (1986).
    [CrossRef]

1999

A. B. Djurišić, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal-incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
[CrossRef]

1998

J. Blohwitz, M. Pfeiffer, T. Fritz, K. Leo, “Low voltage organic light emitting diodes featuring doped phthalocyanine as hole transport layers,” Appl. Phys. Lett. 73, 729–731 (1998).
[CrossRef]

1997

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance–probability–controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

1995

R. M. Bueno, J. F. Trigo, J. M. Martinez-Duart, E. Elizalde, J. M. Sanz, “Study of the optical constants determination of thin films: dependence on theoretical assumptions,” J. Vac. Sci. Technol. A 13, 2378–2383 (1995).
[CrossRef]

O. Stenzel, R. Petrich, “Flexible construction of error functions and their minimization: application to the calculation of optical constants of absorbing or scattering thin-film materials from spectrophotometric data,” J. Phys. D 28, 978–989 (1995).
[CrossRef]

1992

J. M. del Pozo, L. Diaz, “Method for determination of optical constants of thin films: dependence on experimental uncertainties,” Appl. Opt. 31, 4474–4481 (1992).
[CrossRef] [PubMed]

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

1991

1990

1989

A. H. M. Holtslag, P. M. L. O. Scholte, “Optical measurements of the refractive index, layer thickness, and volume changes of thin films,” Appl. Opt. 28, 5095–5104 (1989).
[CrossRef] [PubMed]

T. Fritz, J. Hahn, H. Böttcher, “Determination of optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
[CrossRef]

1986

1984

E. Elizalde, F. Rueda, “On the determination of optical constants n(λ) and α(λ) of thin supported films,” Thin Solid Films 122, 45–57 (1984).
[CrossRef]

1983

1982

L. Ward, “The accuracy of photometric methods for determination of optical constants of thin absorbing films,” J. Phys. D. 15, 1361–1371 (1982).
[CrossRef]

1981

1973

Babu, S. V.

Blohwitz, J.

J. Blohwitz, M. Pfeiffer, T. Fritz, K. Leo, “Low voltage organic light emitting diodes featuring doped phthalocyanine as hole transport layers,” Appl. Phys. Lett. 73, 729–731 (1998).
[CrossRef]

Böttcher, H.

T. Fritz, J. Hahn, H. Böttcher, “Determination of optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
[CrossRef]

Bueno, R. M.

R. M. Bueno, J. F. Trigo, J. M. Martinez-Duart, E. Elizalde, J. M. Sanz, “Study of the optical constants determination of thin films: dependence on theoretical assumptions,” J. Vac. Sci. Technol. A 13, 2378–2383 (1995).
[CrossRef]

David, M.

del Pozo, J. M.

Diaz, L.

Djurišic, A. B.

A. B. Djurišić, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal-incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
[CrossRef]

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance–probability–controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

Dobrowolski, J. A.

Elazar, J. M.

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance–probability–controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

Elizalde, E.

R. M. Bueno, J. F. Trigo, J. M. Martinez-Duart, E. Elizalde, J. M. Sanz, “Study of the optical constants determination of thin films: dependence on theoretical assumptions,” J. Vac. Sci. Technol. A 13, 2378–2383 (1995).
[CrossRef]

E. Elizalde, J. M. Frigerio, J. Rivory, “Determination of thickness and optical constants of thin films from photometric and ellipsometric measurements,” Appl. Opt. 25, 4557–4561 (1986).
[CrossRef] [PubMed]

E. Elizalde, F. Rueda, “On the determination of optical constants n(λ) and α(λ) of thin supported films,” Thin Solid Films 122, 45–57 (1984).
[CrossRef]

Frigerio, J. M.

Fritz, T.

A. B. Djurišić, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal-incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
[CrossRef]

J. Blohwitz, M. Pfeiffer, T. Fritz, K. Leo, “Low voltage organic light emitting diodes featuring doped phthalocyanine as hole transport layers,” Appl. Phys. Lett. 73, 729–731 (1998).
[CrossRef]

T. Fritz, J. Hahn, H. Böttcher, “Determination of optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
[CrossRef]

Hahn, J.

T. Fritz, J. Hahn, H. Böttcher, “Determination of optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
[CrossRef]

Hansen, W. N.

Harbecke, B.

B. Harbecke, “Coherent and incoherent reflection and transmission of multilayer systems,” Appl. Phys. B 39, 165–170 (1986).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1955), pp. 95–154.

Hjortsberg, A.

Ho, F. C.

Holtslag, A. H. M.

Hopfe, V.

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Hunter, W. R.

W. R. Hunter, “Measurements of optical constants in the vacuum ultraviolet (VUV) spectral region,” in Handbook of Optical Constants of Solids I, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 69–88.
[CrossRef]

Leo, K.

A. B. Djurišić, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal-incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
[CrossRef]

J. Blohwitz, M. Pfeiffer, T. Fritz, K. Leo, “Low voltage organic light emitting diodes featuring doped phthalocyanine as hole transport layers,” Appl. Phys. Lett. 73, 729–731 (1998).
[CrossRef]

Leveque, G.

Martinez-Duart, J. M.

R. M. Bueno, J. F. Trigo, J. M. Martinez-Duart, E. Elizalde, J. M. Sanz, “Study of the optical constants determination of thin films: dependence on theoretical assumptions,” J. Vac. Sci. Technol. A 13, 2378–2383 (1995).
[CrossRef]

Patel, R. C.

Petrich, R.

O. Stenzel, R. Petrich, “Flexible construction of error functions and their minimization: application to the calculation of optical constants of absorbing or scattering thin-film materials from spectrophotometric data,” J. Phys. D 28, 978–989 (1995).
[CrossRef]

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Pfeiffer, M.

J. Blohwitz, M. Pfeiffer, T. Fritz, K. Leo, “Low voltage organic light emitting diodes featuring doped phthalocyanine as hole transport layers,” Appl. Phys. Lett. 73, 729–731 (1998).
[CrossRef]

Potter, R. F.

R. F. Potter, “Basic parameters for measuring optical properties,” in Handbook of Optical Constants of Solids I, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 11–34.
[CrossRef]

Rakic, A. D.

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance–probability–controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

Rippens, W.

Rivory, J.

Rueda, F.

E. Elizalde, F. Rueda, “On the determination of optical constants n(λ) and α(λ) of thin supported films,” Thin Solid Films 122, 45–57 (1984).
[CrossRef]

Sanz, J. M.

R. M. Bueno, J. F. Trigo, J. M. Martinez-Duart, E. Elizalde, J. M. Sanz, “Study of the optical constants determination of thin films: dependence on theoretical assumptions,” J. Vac. Sci. Technol. A 13, 2378–2383 (1995).
[CrossRef]

Scharff, W.

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Scholte, P. M. L. O.

Stenzel, O.

O. Stenzel, R. Petrich, “Flexible construction of error functions and their minimization: application to the calculation of optical constants of absorbing or scattering thin-film materials from spectrophotometric data,” J. Phys. D 28, 978–989 (1995).
[CrossRef]

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Tikhonravov, A.

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Trigo, J. F.

R. M. Bueno, J. F. Trigo, J. M. Martinez-Duart, E. Elizalde, J. M. Sanz, “Study of the optical constants determination of thin films: dependence on theoretical assumptions,” J. Vac. Sci. Technol. A 13, 2378–2383 (1995).
[CrossRef]

Villachon-Renard, Y.

Vriens, L.

Waldorf, A.

Ward, L.

L. Ward, “The accuracy of photometric methods for determination of optical constants of thin absorbing films,” J. Phys. D. 15, 1361–1371 (1982).
[CrossRef]

Appl. Opt.

A. Hjortsberg, “Determination of optical constants of absorbing materials using transmission and reflection of thin films on partially metallized substrates: analysis of the new (T, Rm) technique,” Appl. Opt. 20, 1254–1263 (1981).
[CrossRef] [PubMed]

J. A. Dobrowolski, F. C. Ho, A. Waldorf, “Determination of optical constants of thin film coating materials based on inverse synthesis,” Appl. Opt. 22, 3191–3200 (1983).
[CrossRef] [PubMed]

L. Vriens, W. Rippens, “Optical constants of absorbing thin films on a substrate,” Appl. Opt. 22, 4105–4110 (1983).
[CrossRef]

E. Elizalde, J. M. Frigerio, J. Rivory, “Determination of thickness and optical constants of thin films from photometric and ellipsometric measurements,” Appl. Opt. 25, 4557–4561 (1986).
[CrossRef] [PubMed]

A. H. M. Holtslag, P. M. L. O. Scholte, “Optical measurements of the refractive index, layer thickness, and volume changes of thin films,” Appl. Opt. 28, 5095–5104 (1989).
[CrossRef] [PubMed]

S. V. Babu, M. David, R. C. Patel, “Two-step regression procedure for the optical characterization of thin films,” Appl. Opt. 30, 839–846 (1991).
[CrossRef] [PubMed]

J. M. del Pozo, L. Diaz, “Method for determination of optical constants of thin films: dependence on experimental uncertainties,” Appl. Opt. 31, 4474–4481 (1992).
[CrossRef] [PubMed]

G. Leveque, Y. Villachon-Renard, “Determination of optical constants of a thin film from reflectance spectra,” Appl. Opt. 29, 3207–3212 (1990).
[CrossRef]

Appl. Phys. B

B. Harbecke, “Coherent and incoherent reflection and transmission of multilayer systems,” Appl. Phys. B 39, 165–170 (1986).
[CrossRef]

Appl. Phys. Lett.

J. Blohwitz, M. Pfeiffer, T. Fritz, K. Leo, “Low voltage organic light emitting diodes featuring doped phthalocyanine as hole transport layers,” Appl. Phys. Lett. 73, 729–731 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. D

O. Stenzel, R. Petrich, “Flexible construction of error functions and their minimization: application to the calculation of optical constants of absorbing or scattering thin-film materials from spectrophotometric data,” J. Phys. D 28, 978–989 (1995).
[CrossRef]

J. Phys. D.

L. Ward, “The accuracy of photometric methods for determination of optical constants of thin absorbing films,” J. Phys. D. 15, 1361–1371 (1982).
[CrossRef]

J. Vac. Sci. Technol. A

R. M. Bueno, J. F. Trigo, J. M. Martinez-Duart, E. Elizalde, J. M. Sanz, “Study of the optical constants determination of thin films: dependence on theoretical assumptions,” J. Vac. Sci. Technol. A 13, 2378–2383 (1995).
[CrossRef]

Opt. Commun.

A. B. Djurišić, T. Fritz, K. Leo, “Determination of optical constants of thin absorbing films from normal-incidence reflectance and transmittance measurements,” Opt. Commun. 166, 35–42 (1999).
[CrossRef]

Phys. Rev. E

A. B. Djurišić, A. D. Rakić, J. M. Elazar, “Modeling the optical constants of solids using acceptance–probability–controlled simulated annealing with adaptive move generation procedure,” Phys. Rev. E 55, 4797–4803 (1997).
[CrossRef]

Thin Solid Films

T. Fritz, J. Hahn, H. Böttcher, “Determination of optical constants of evaporated dye layers,” Thin Solid Films 170, 249–257 (1989).
[CrossRef]

E. Elizalde, F. Rueda, “On the determination of optical constants n(λ) and α(λ) of thin supported films,” Thin Solid Films 122, 45–57 (1984).
[CrossRef]

O. Stenzel, R. Petrich, W. Scharff, A. Tikhonravov, V. Hopfe, “A hybrid method for determination of optical thin film constants,” Thin Solid Films 207, 324–329 (1992).
[CrossRef]

Other

W. R. Hunter, “Measurements of optical constants in the vacuum ultraviolet (VUV) spectral region,” in Handbook of Optical Constants of Solids I, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 69–88.
[CrossRef]

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1955), pp. 95–154.

R. F. Potter, “Basic parameters for measuring optical properties,” in Handbook of Optical Constants of Solids I, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 11–34.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Error for the real part of the index of refraction estimated from R and T values.

Fig. 2
Fig. 2

Error for the imaginary part of the index of refraction estimated from R and T values.

Fig. 3
Fig. 3

Error for the imaginary part of the index of refraction estimated from R p 30 and T values.

Fig. 4
Fig. 4

Average error (averaged over wavelengths) for the real part of the index of refraction estimated by the least-squares method from (R, T, R p 30) overdetermined system.

Fig. 5
Fig. 5

Average error (averaged over wavelengths) for the real part of the index of refraction estimated by the least-squares method from R and T values.

Fig. 6
Fig. 6

Real and imaginary part of the index of refraction as a function of wavelength: circles, target data; solid curve, method (a) with the penalty function; dashed curve, method (a) without the penalty function; dot–dash curve, method (b) without the penalty function.

Fig. 7
Fig. 7

Reflectance and transmittance as a function of wavelength: circles, target data; solid curve, method (a) with the penalty function; dashed curve, method (a) without the penalty function.

Fig. 8
Fig. 8

Real and imaginary part of the index of refraction as a function of wavelength: circles, target data; solid curve, method (a); dashed curve, method (b); dot–dash curve, method (c).

Fig. 9
Fig. 9

Real and imaginary parts of the index of refraction as a function of wavelength for VOPc.

Equations (29)

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

Bn=βnν1νN wνdndν2dν,
wν=k-2ν,
CR=Cs0,1·Cl1·Cs1,2  ClM·CsM,M+1.
Cli=expiγizi00exp-iγizi,
Csi,i+1=1ti,i+11ri,i+1ri,i+11,
ζi=Z0μ0/γifor s polarization-Ni2/γiZ0for p polarization,
CR=1t0fabcd,
Rf=cc*aa*,
Tf=t0ft0f*aa*Re1/ζfRe1/ζ0.
T=TsTf exp-αs1-RsRf exp-2αs,
R=Rf+RsTfTf exp-2αs1-RsRf exp-2αs,
F=i=1i=NgRRλi-Rexptλi2+gTTλi-Texptλi2+Δi,
Δi=anλi-nλi-1kλi+kλi-12+bkλi-kλi-1nλi2+enλi-nλi-1λi-λi-122; kλi>0.1cnλi-nλi-12+dkλi-kλi-12+enλi-nλi-1λi-λi-122; kλi<0.1,
ΔX2=Xn Δn2+Xk Δk2=aΔn2+bΔk2,
ΔY2=Yn Δn2+Yk Δk2=cΔn2+dΔk2.
Δn2=dΔX2-bΔY2ad-bc,
Δk2=aΔY2-cΔX2ad-bc.
G=i=1NXni, ki, d, λi-Xiexpt2+Yni, ki, d, λi-Yiexpt2+Zni, ki, d, λi-Ziexpt2,
Gni=2X-XiexptXni+Y-YiexptYni+Z-ZiexptZni=0,
Gki=2X-XiexptXki+Y-YiexptYki+Z-ZiexptZki=0,
Gd=2 i=1NX-XiexptXd+Y-YiexptYd+Z-ZiexptZd=0.
Δni2=fXiexpt2ΔX2+fYiexpt2ΔY2+fZiexpt2ΔZ2.
H=X-XiexptXni+Y-YiexptYni+Z-ZiexptZni=0,
dH=HXiexpt dXiexpt+HYiexpt dYiexpt+HZiexpt dZiexpt+Hni dni=HxdXiexpt+HydYiexpt+HzdZiexpt+Hnidni=0,
fXiexpt=-HxHni=Xni1Xiexptni2+Yiexptni2+Ziexptni2+X-Xiexpt2Xni2+Y-Yiexpt2Yni2+Z-Ziexpt2Zni2.
Hni=Xiexptni2+Yiexptni2+Ziexptni2.
Δni=Xiexptni2ΔX2+Yiexptni2ΔY2+Ziexptni2ΔZ21/2Xiexptni2+Yiexptni2+Ziexptni2,
Δki=Xiexptki2ΔX2+Yiexptki2ΔY2+Ziexptki2ΔZ21/2Xiexptki2+Yiexptki2+Ziexptki2,
Δd=i=1NXiexptni2ΔX2+Yiexptni2ΔY2+Ziexptni2ΔZ21/2i=1NXiexptni2+Yiexptni2+Ziexptni2.

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