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Porosity dependence of microwave dielectric properties of complex perovskite (Pb0.5Ca0.5)(Fe0.5Ta0.5)O3 ceramics
Materials Chemistry and Physics 79 (2003) 213–217
Porosity dependence of microwave dielectric properties of complex perovskite (Pb0.5Ca0.5)(Fe0.5Ta0.5)O3 ceramics Eung Soo Kim a,∗ , Heung Soo Park b , Ki Hyun Yoon b a
Department of Materials Engineering, Kyonggi University, Suwon 442-760, South Korea b Department of Ceramic Engineering, Yonsei University, Seoul 120-749, South Korea
Abstract Dielectric constant (K) and loss quality (Qf) of the complex perovskite (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 with different porosities were evaluated by the dielectric mixing rule. The observed ionic polarizabilities and loss qualities of the specimens were obtained from the dielectric constant and loss measured by the post resonant method. The theoretical ionic polarizabilities were calculated from the molecular additive rule and Clausius–Mosotti equation, and the intrinsic loss qualities were calculated from the far-infrared (Far-IR) reflectivity spectra in the range of 50–4000 cm−1 by Kramers–Kronig analysis and classical oscillator model. For the specimens with porosity, the ionic polarizabilities modified by Maxwell’s equation were more close to the theoretical values rather than those modified by Wiener’s equation, and the predicted loss quality obtained from intrinsic ones and Maxwell’s equation was agreed with the observed ones. © 2002 Elsevier Science B.V. All rights reserved. Keywords: (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 ; Porosity; Ionic polarizabilities; Far-infrared reflectivity spectra; Maxwell’s equation
1. Introduction With recent development in communication technology, it has been required to the good microwave dielectric materials with respect to the dielectric constant (K), loss quality (Qf), and temperature coefficient of resonant frequency (TCF). Several types of microwave dielectric materials have been studied to meet the requirements of the microwave applications. It is necessary for improved microwave materials that the intrinsic properties of materials should be studied to control and design the dielectric properties of materials. Recently, some reports have been tried to get the intrinsic properties of materials by the infrared reflectivity spectra [1] and calculation of theoretical polarizability [2]. However, there are discrepancies between the intrinsic properties obtained from the extrinsic methods and the extrinsic ones due to the grain, grain boundary, and pore. Therefore, the effects of porosity on the dielectric properties should be considered to evaluate the intrinsic dielectric properties, which in turn, can be available to predict the dielectric properties of ceramics with pores. In this study, the microwave dielectric properties of (Pb0.5 Ca0.5 ) (Fe0.5 Ta0.5 )O3 ceramics were investigated as a function of the content of porosity. The modification of theoretical ∗ Corresponding author. Tel.: +82-31-249-9764; fax: +82-31-249-9775. E-mail address: [email protected] (E.S. Kim).
polarizability and the prediction of dielectric loss were also discussed.
2. Experimental (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 ceramics were prepared by the conventional mixed oxide method using high purity oxides (>99.9%) via the columbite route. Raw materials were ball milled and calcined at 900 ◦ C for 3 h and sintered at 1100–1250 ◦ C for 30 min. To inhibit the loss of PbO and decomposition during sintering, the specimens were buried in powder of the same composition and placed into a platinum crucible [3]. X-ray powder diffraction analysis was used to determine the crystalline structure and lattice parameters. The density and porosity of the specimens were obtained by immersion technique. The dielectric constant and Qf-value of the specimens were measured by Hakki and Coleman’s method [4] at 7 GHz. The reflectivity spectra were measured using a Fourier transform infrared spectrometer (Model DA-8.12, Bomen Inc., Canada) from 50 to 4000 cm−1 . The reflectivity spectra were obtained as the relative intensity to the reflectivity of a gold mirror. The spectra were recorded at the resolution of 4 cm−1 . The incident angle of the radiation was 7◦ . The reflectivity spectra were evaluated by the Kramers–Kronig analysis [5] and classical oscillator model.
0254-0584/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 2 ) 0 0 2 6 0 - 2
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E.S. Kim et al. / Materials Chemistry and Physics 79 (2003) 213–217
Fig. 1. Porosity of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimens sintered at different temperatures.
3. Results and discussion The powder XRD patterns of the (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimens indicated that a single perovskite phase of cubic structure was obtained for the specimens sintered at 1100–1250 ◦ C for 30 min. As the lattice parameter was not changed remarkably with sintering temperature, the average lattice parameter and molar volume were 3.946 Å and 61.46 Å3 , respectively. Fig. 1 shows the dependence of porosity on the sintering temperature. The porosity of the specimens drastically decreased with sintering temperature, and showed a minimum value of 3.8% for the specimens sintered above 1200 ◦ C. Dielectric constant and Qf-value increased with sintering temperature and K of 82, Qf of 6700 GHz were obtained for the specimens sintered at 1200 ◦ C for 30 min, as confirmed in Table 1. Bosman and Havinga [6] reported that the theoretical dielectric constant could be obtained from the measured dielectric constant by the modification of porosity, and suggested the experimental equation of dielectric constant and porosity, as the Wiener’s equation in Eq. (1) Kw = Kmea. (1 + 1.5 P )
(1)
where Kw , Kmea. and P were theoretical dielectric constant, measured dielectric constant, and porosity, respectively.
Also, the dielectric constant of polycrystalline could be evaluated by the Maxwell’s equation in Eqs. (2)–(5), assuming the mixture of dielectrics and spherical pores with 3-0 connectivity. 3Vf (K1 − K2 ) (2) K m = K2 1 + K1 + 2K2 − Vf (K1 − K2 ) where Km , K2 , K1 were the dielectric constant of mixture, the matrix dielectrics, and pores, respectively. Vf was the volume fraction of dispersed phase. As K2 K1 (= 1), Eq. (3) could be obtained from the rearrangement of Eq. (2). 2 + Vf − 3Vf K m = K2 (3) 2 + Vf Table 1 Porosity, dielectric constant and Qf of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimen sintered at different temperatures Sintering temperature (◦ C)
Porosity (%)
Dielectric constant (K)
Qf (GHz)
1100 1150 1200 1250
15.2 6.8 3.8 3.8
67.9 79.6 84.0 83.9
5681 6153 6652 6685
E.S. Kim et al. / Materials Chemistry and Physics 79 (2003) 213–217
With neglecting of Vf2 , which were corresponded to square of the porosity, 4 − 2Vf − 4Vf + 4Vf2 ∼ 2 − 3Vf K Km = K2 (4) = 2 2 4 − Vf2 where Km , K2 were corresponded to Kmea. of the measured dielectric constant, and KM of theoretical dielectric constant, respectively. 2 (5) KM = Kmea. 2 − 3P where KM , Kmea. and P were the theoretical dielectric constant, the measured dielectric constant and porosity, respectively. On the other hand, the observed and theoretical dielectric polarizabilites were calculated by the Clausius–Mosotti equation and additivity rule of dielectric polarizabilities in Eqs. (6) and (7), respectively [2]. αmea. =
Vm (K − 1) b(K + 2)
(6)
215
αtheo. (PCFT) = (0.5αPb + 0.5αCa ) + (0.5αFe + 0.5αTa ) + 3αO
(7)
where α mea. and α theo. are the observed and the theoretical polarizabilities, α i (i: Pb, Ca, Fe, Ta, O) is the ionic polarizability of ion i, K is the measured dielectric constant, b is the 4π /3 and Vm the molar volume, respectively. For the specimens sintered at 1200 ◦ C, the observed dielectric polarizability calculated from Eq. (6) showed 14.090 Å, while the theoretical dielectric polarizability calculated from Eq. (7) showed 14.410 Å, and the relative deviation ((α mea. −α theo. )/100αmea. ) was −2.91%. However, the theoretical dielectric polarizabilities modified Eqs. (1) and (5) were 14.187 Å (deviation: −1.57%) and 14.189 Å (deviation: −1.56%), respectively, as shown in Fig. 2. These results were agreed with the report of Shannon [2], of which deviations were 0.5–1.5% for the ionic polarizabilities obtained from additivity rule. Also, the relative deviations modified by Maxwell’s equation, Eq. (6) were not changed with porosity, while those modified by Wiener’s equation, Eq. (1) were only valid for the specimens with lower
Fig. 2. Dielectric constant and relative deviations of ionic polarizability of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimen sintered at different temperatures.
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Fig. 3. Infrared reflectivity spectra of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimen sintered at 1250 ◦ C for 30 min.
porosity than 4%. Therefore, the dielectric constant of the specimens with a single phase could be predicted by Eq. (5) from the ionic polarizabilities of composing elements and porosity. Far-infrared (Far-IR) reflectivity spectra of the (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimens sintered at 1250 ◦ C for 30 min were taken to calculate the intrinsic dielectric loss at microwave frequencies. The spectra fitted by the 10 resonant modes calculated using reflectivity spectra were well fitted with the measured ones as shown in Fig. 3 and Table 2. The dispersion parameters of the specimens in Table 2 were determined by the Kramers–Kronig analysis and classical oscillator model. The calculated Qf-values were higher than the measured ones by Hakki and Coleman’s method, Table 2 Dispersion parameters, measured and calculated dielectric properties of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 ceramics obtained from the best fit to the reflectivity data Dispersion parameters J
ωj (cm−1 )
1 2 3 4 5 6 7 8 9 10
70 93 125 162 210 252 293 577 610 727
Dielectric properties K Qf ε ∞ = 5.16.
γ j (cm−1 ) 34 61 74 66 45 70 36 48 47 140 Measured 83 6700
εj 26.0 8.0 17.3 5.7 19.0 1.0 1.0 1.2 0.05 0.08
Fig. 4. The Qf of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimens sintered at different temperatures.
which was due to the extrinsic effect such as grain size and porosity. Assuming the mixture of dielectrics and spherical pore with 3-0 connectivity, the measured loss quality also depended on the porosity as well as the intrinsic loss of materials, and Eq. (6) could be modified for loss quality, as in Eq. (8). 2 − 3P IR Qfpred. = Qftheo. (8) 2 With the intrinsic loss obtained from Far-IR reflectivity and porosity, the predicted loss qualities by Eq. (8) was showed in Fig. 4, comparing to the measured ones by Hakki and Coleman’s method. The predicted loss qualities were consistent with the measured ones. In general, Qf-values do not follow the dielectric mixing rule for the specimens with two more phases, which is due to the dependence of Qf-value on the microstructure resulted from the reactivity and sinterability of each phase. Based on the dielectric mixing rule with 3-0 connectivity, however, the Qf-values could be predicted by Eq. (8) for the single phase specimens with porosity.
tan δ j (×10−4 ) 4.0540 0.7793 2.4199 0.3678 0.3924 0.0223 0.0057 0.0040 0.0006 0.0006 Calculated 85 7111
4. Conclusion Assuming the mixture of dielectrics and spherical pore with 3-0 connectivity, the dielectric constant (K) and loss quality (Qf ) of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 with different porosities were evaluated by the dielectric mixing rule. For the specimens with porosity, the ionc polarizabilities modified by Maxwell’s equation were more close to the theoretical values rather than those modified by Wiener’s equation, and the predicted loss qualities obtained from intrinsic ones and Maxwell’s equation were agreed with the observed ones.
E.S. Kim et al. / Materials Chemistry and Physics 79 (2003) 213–217
Acknowledgements This work was supported by the Brain Korea 21 project. References [1] W.S. Kim, K.H. Yoon, E.S. Kim, J. Am. Ceram. Soc. 83 (9) (2000) 2327–2329.
217
[2] R.D. Shannon, J. Appl. Phys. 73 (1) (1993) 348–366. [3] M.A. Akbas, P.K. Devies, J. Mater. Res. 12 (9B) (1997) 2617– 2622. [4] B.W. Hakki, P.D. Coleman, IRE Trans. MTT 8 (1960) 402– 410. [5] W.G. Spitzer, R.C. Miller, D.A. Kleinman, L.E. Howarth, Phys. Rev. 126 (5) (1962) 1710. [6] A.J. Bosman, E.E. Havinga, Phys. Rev. 129 (4) (1963) 1593– 1600.
Porosity dependence of microwave dielectric properties of complex perovskite (Pb0.5Ca0.5)(Fe0.5Ta0.5)O3 ceramics Eung Soo Kim a,∗ , Heung Soo Park b , Ki Hyun Yoon b a
Department of Materials Engineering, Kyonggi University, Suwon 442-760, South Korea b Department of Ceramic Engineering, Yonsei University, Seoul 120-749, South Korea
Abstract Dielectric constant (K) and loss quality (Qf) of the complex perovskite (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 with different porosities were evaluated by the dielectric mixing rule. The observed ionic polarizabilities and loss qualities of the specimens were obtained from the dielectric constant and loss measured by the post resonant method. The theoretical ionic polarizabilities were calculated from the molecular additive rule and Clausius–Mosotti equation, and the intrinsic loss qualities were calculated from the far-infrared (Far-IR) reflectivity spectra in the range of 50–4000 cm−1 by Kramers–Kronig analysis and classical oscillator model. For the specimens with porosity, the ionic polarizabilities modified by Maxwell’s equation were more close to the theoretical values rather than those modified by Wiener’s equation, and the predicted loss quality obtained from intrinsic ones and Maxwell’s equation was agreed with the observed ones. © 2002 Elsevier Science B.V. All rights reserved. Keywords: (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 ; Porosity; Ionic polarizabilities; Far-infrared reflectivity spectra; Maxwell’s equation
1. Introduction With recent development in communication technology, it has been required to the good microwave dielectric materials with respect to the dielectric constant (K), loss quality (Qf), and temperature coefficient of resonant frequency (TCF). Several types of microwave dielectric materials have been studied to meet the requirements of the microwave applications. It is necessary for improved microwave materials that the intrinsic properties of materials should be studied to control and design the dielectric properties of materials. Recently, some reports have been tried to get the intrinsic properties of materials by the infrared reflectivity spectra [1] and calculation of theoretical polarizability [2]. However, there are discrepancies between the intrinsic properties obtained from the extrinsic methods and the extrinsic ones due to the grain, grain boundary, and pore. Therefore, the effects of porosity on the dielectric properties should be considered to evaluate the intrinsic dielectric properties, which in turn, can be available to predict the dielectric properties of ceramics with pores. In this study, the microwave dielectric properties of (Pb0.5 Ca0.5 ) (Fe0.5 Ta0.5 )O3 ceramics were investigated as a function of the content of porosity. The modification of theoretical ∗ Corresponding author. Tel.: +82-31-249-9764; fax: +82-31-249-9775. E-mail address: [email protected] (E.S. Kim).
polarizability and the prediction of dielectric loss were also discussed.
2. Experimental (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 ceramics were prepared by the conventional mixed oxide method using high purity oxides (>99.9%) via the columbite route. Raw materials were ball milled and calcined at 900 ◦ C for 3 h and sintered at 1100–1250 ◦ C for 30 min. To inhibit the loss of PbO and decomposition during sintering, the specimens were buried in powder of the same composition and placed into a platinum crucible [3]. X-ray powder diffraction analysis was used to determine the crystalline structure and lattice parameters. The density and porosity of the specimens were obtained by immersion technique. The dielectric constant and Qf-value of the specimens were measured by Hakki and Coleman’s method [4] at 7 GHz. The reflectivity spectra were measured using a Fourier transform infrared spectrometer (Model DA-8.12, Bomen Inc., Canada) from 50 to 4000 cm−1 . The reflectivity spectra were obtained as the relative intensity to the reflectivity of a gold mirror. The spectra were recorded at the resolution of 4 cm−1 . The incident angle of the radiation was 7◦ . The reflectivity spectra were evaluated by the Kramers–Kronig analysis [5] and classical oscillator model.
0254-0584/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 2 ) 0 0 2 6 0 - 2
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E.S. Kim et al. / Materials Chemistry and Physics 79 (2003) 213–217
Fig. 1. Porosity of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimens sintered at different temperatures.
3. Results and discussion The powder XRD patterns of the (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimens indicated that a single perovskite phase of cubic structure was obtained for the specimens sintered at 1100–1250 ◦ C for 30 min. As the lattice parameter was not changed remarkably with sintering temperature, the average lattice parameter and molar volume were 3.946 Å and 61.46 Å3 , respectively. Fig. 1 shows the dependence of porosity on the sintering temperature. The porosity of the specimens drastically decreased with sintering temperature, and showed a minimum value of 3.8% for the specimens sintered above 1200 ◦ C. Dielectric constant and Qf-value increased with sintering temperature and K of 82, Qf of 6700 GHz were obtained for the specimens sintered at 1200 ◦ C for 30 min, as confirmed in Table 1. Bosman and Havinga [6] reported that the theoretical dielectric constant could be obtained from the measured dielectric constant by the modification of porosity, and suggested the experimental equation of dielectric constant and porosity, as the Wiener’s equation in Eq. (1) Kw = Kmea. (1 + 1.5 P )
(1)
where Kw , Kmea. and P were theoretical dielectric constant, measured dielectric constant, and porosity, respectively.
Also, the dielectric constant of polycrystalline could be evaluated by the Maxwell’s equation in Eqs. (2)–(5), assuming the mixture of dielectrics and spherical pores with 3-0 connectivity. 3Vf (K1 − K2 ) (2) K m = K2 1 + K1 + 2K2 − Vf (K1 − K2 ) where Km , K2 , K1 were the dielectric constant of mixture, the matrix dielectrics, and pores, respectively. Vf was the volume fraction of dispersed phase. As K2 K1 (= 1), Eq. (3) could be obtained from the rearrangement of Eq. (2). 2 + Vf − 3Vf K m = K2 (3) 2 + Vf Table 1 Porosity, dielectric constant and Qf of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimen sintered at different temperatures Sintering temperature (◦ C)
Porosity (%)
Dielectric constant (K)
Qf (GHz)
1100 1150 1200 1250
15.2 6.8 3.8 3.8
67.9 79.6 84.0 83.9
5681 6153 6652 6685
E.S. Kim et al. / Materials Chemistry and Physics 79 (2003) 213–217
With neglecting of Vf2 , which were corresponded to square of the porosity, 4 − 2Vf − 4Vf + 4Vf2 ∼ 2 − 3Vf K Km = K2 (4) = 2 2 4 − Vf2 where Km , K2 were corresponded to Kmea. of the measured dielectric constant, and KM of theoretical dielectric constant, respectively. 2 (5) KM = Kmea. 2 − 3P where KM , Kmea. and P were the theoretical dielectric constant, the measured dielectric constant and porosity, respectively. On the other hand, the observed and theoretical dielectric polarizabilites were calculated by the Clausius–Mosotti equation and additivity rule of dielectric polarizabilities in Eqs. (6) and (7), respectively [2]. αmea. =
Vm (K − 1) b(K + 2)
(6)
215
αtheo. (PCFT) = (0.5αPb + 0.5αCa ) + (0.5αFe + 0.5αTa ) + 3αO
(7)
where α mea. and α theo. are the observed and the theoretical polarizabilities, α i (i: Pb, Ca, Fe, Ta, O) is the ionic polarizability of ion i, K is the measured dielectric constant, b is the 4π /3 and Vm the molar volume, respectively. For the specimens sintered at 1200 ◦ C, the observed dielectric polarizability calculated from Eq. (6) showed 14.090 Å, while the theoretical dielectric polarizability calculated from Eq. (7) showed 14.410 Å, and the relative deviation ((α mea. −α theo. )/100αmea. ) was −2.91%. However, the theoretical dielectric polarizabilities modified Eqs. (1) and (5) were 14.187 Å (deviation: −1.57%) and 14.189 Å (deviation: −1.56%), respectively, as shown in Fig. 2. These results were agreed with the report of Shannon [2], of which deviations were 0.5–1.5% for the ionic polarizabilities obtained from additivity rule. Also, the relative deviations modified by Maxwell’s equation, Eq. (6) were not changed with porosity, while those modified by Wiener’s equation, Eq. (1) were only valid for the specimens with lower
Fig. 2. Dielectric constant and relative deviations of ionic polarizability of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimen sintered at different temperatures.
216
E.S. Kim et al. / Materials Chemistry and Physics 79 (2003) 213–217
Fig. 3. Infrared reflectivity spectra of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimen sintered at 1250 ◦ C for 30 min.
porosity than 4%. Therefore, the dielectric constant of the specimens with a single phase could be predicted by Eq. (5) from the ionic polarizabilities of composing elements and porosity. Far-infrared (Far-IR) reflectivity spectra of the (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimens sintered at 1250 ◦ C for 30 min were taken to calculate the intrinsic dielectric loss at microwave frequencies. The spectra fitted by the 10 resonant modes calculated using reflectivity spectra were well fitted with the measured ones as shown in Fig. 3 and Table 2. The dispersion parameters of the specimens in Table 2 were determined by the Kramers–Kronig analysis and classical oscillator model. The calculated Qf-values were higher than the measured ones by Hakki and Coleman’s method, Table 2 Dispersion parameters, measured and calculated dielectric properties of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 ceramics obtained from the best fit to the reflectivity data Dispersion parameters J
ωj (cm−1 )
1 2 3 4 5 6 7 8 9 10
70 93 125 162 210 252 293 577 610 727
Dielectric properties K Qf ε ∞ = 5.16.
γ j (cm−1 ) 34 61 74 66 45 70 36 48 47 140 Measured 83 6700
εj 26.0 8.0 17.3 5.7 19.0 1.0 1.0 1.2 0.05 0.08
Fig. 4. The Qf of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 specimens sintered at different temperatures.
which was due to the extrinsic effect such as grain size and porosity. Assuming the mixture of dielectrics and spherical pore with 3-0 connectivity, the measured loss quality also depended on the porosity as well as the intrinsic loss of materials, and Eq. (6) could be modified for loss quality, as in Eq. (8). 2 − 3P IR Qfpred. = Qftheo. (8) 2 With the intrinsic loss obtained from Far-IR reflectivity and porosity, the predicted loss qualities by Eq. (8) was showed in Fig. 4, comparing to the measured ones by Hakki and Coleman’s method. The predicted loss qualities were consistent with the measured ones. In general, Qf-values do not follow the dielectric mixing rule for the specimens with two more phases, which is due to the dependence of Qf-value on the microstructure resulted from the reactivity and sinterability of each phase. Based on the dielectric mixing rule with 3-0 connectivity, however, the Qf-values could be predicted by Eq. (8) for the single phase specimens with porosity.
tan δ j (×10−4 ) 4.0540 0.7793 2.4199 0.3678 0.3924 0.0223 0.0057 0.0040 0.0006 0.0006 Calculated 85 7111
4. Conclusion Assuming the mixture of dielectrics and spherical pore with 3-0 connectivity, the dielectric constant (K) and loss quality (Qf ) of (Pb0.5 Ca0.5 )(Fe0.5 Ta0.5 )O3 with different porosities were evaluated by the dielectric mixing rule. For the specimens with porosity, the ionc polarizabilities modified by Maxwell’s equation were more close to the theoretical values rather than those modified by Wiener’s equation, and the predicted loss qualities obtained from intrinsic ones and Maxwell’s equation were agreed with the observed ones.
E.S. Kim et al. / Materials Chemistry and Physics 79 (2003) 213–217
Acknowledgements This work was supported by the Brain Korea 21 project. References [1] W.S. Kim, K.H. Yoon, E.S. Kim, J. Am. Ceram. Soc. 83 (9) (2000) 2327–2329.
217
[2] R.D. Shannon, J. Appl. Phys. 73 (1) (1993) 348–366. [3] M.A. Akbas, P.K. Devies, J. Mater. Res. 12 (9B) (1997) 2617– 2622. [4] B.W. Hakki, P.D. Coleman, IRE Trans. MTT 8 (1960) 402– 410. [5] W.G. Spitzer, R.C. Miller, D.A. Kleinman, L.E. Howarth, Phys. Rev. 126 (5) (1962) 1710. [6] A.J. Bosman, E.E. Havinga, Phys. Rev. 129 (4) (1963) 1593– 1600.