Hindawi Publishing CorporationJournal of CeramicsVolume 2013, Article ID 261914, 6 pageshttp://dx.doi.org/10.1155/2013/261914
Research ArticleEffect of Processing on Synthesis and Dielectric Behavior ofBismuth Sodium Titanate Ceramics
Vijayeta Pal,1 R. K. Dwivedi,1 and O. P. Thakur2
1 Department of Physics and Material Science & Engineering, Jaypee Institute of Information Technology, Noida 201307, India2 Electroceramics Group, Solid State Physics Laboratory, Defence Research and Development Organization (DRDO), Timarpur,Delhi 110054, India
Correspondence should be addressed to Vijayeta Pal; [email protected]
An effort has been made to synthesize polycrystalline (Bi1−𝑥𝑥La𝑥𝑥)0.5Na0.5TiO3 (abbreviated as BLNT) system with compositionsx = 0, 0.02, and 0.04 by novel semiwet technique. Preparation of A-site oxides of BLNT for composition x = 0 was optimizedusing two precursor solutions such as ethylene glycol and citric acid. e XRD patterns revealed that the sample prepared byethylene glycol precursor solution has single phase perovskite structure with a rhombohedral symmetry at RT as compared tothe sample prepared by citric acid. Ethylene glycol precursor has been found to play a signi�cant role in the crystallization, phasetransitions, and electrical properties. e studies on structure, phase transitions, and dielectric properties for all the samples havebeen carried out over the temperature range fromRT to 450∘C at 100 kHz frequency. It has been observed that two phase transitions(i) ferroelectric to antiferroelectric and (ii) antiferroelectric to paraelectric occur in all the samples. All samples exhibit a modi�edCurie-Weiss law above Tc. A linear �tting of the modi�ed Curie-Weiss law to the experimental data shows diffuse-type transition.e dielectric as well as ferroelectric properties of BLNT ceramics have been found to be improved with the substitution of Laelements.
1. Introduction
Lead oxide-based ceramics with perovskite structure havebeen the subject of attraction for high-performance sensors,actuators, transducers, and other applications, owing to theirsuperior dielectric, piezoelectric, and electromechanical cou-pling coefficients. Applications are restricted to temperaturerange −50 to 150∘C [1]. However, in recent years many �eldshave expressed the need for actuation and sensing which canbe used at higher temperatures (>400∘C) such as automotive,aerospace, and related industrial applications. On the otherhand, in most of the cases lead constitutes more than 60% ofthe composition of these piezoelectric devices. Lead, knownto be highly toxic and volatile, is released to the atmosphereduring sintering causing serious environmental and healthproblems. Another cause for concern is the disposal of theseproducts at the end of the life cycle. Considering all thesehealth concerns posed by lead, multinational governments
like the European Union have enacted laws that ban the useof lead in the manufacture of many industrial products [2].is has led to the replacement of lead (Pb) in the �eld ofpiezoelectric ceramics. A lot of research has been carriedout on lead-free piezoceramic products in the last ��y yearsbut in the last few years, the momentum has tremendouslyincreased, accounting for about 75% of all published worksin this �eld. Bi0.5Na0.5TiO3- (BNT-) and K0.5Na0.5NbO3-(KNN-) based materials are two main material systems withperovskite structure, which have been studied to �nd thesubstitute of PZT for lead free piezoelectric applications.Pure BNT was discovered by Smolenskii et al. [3] and isa ferroelectric having Bi3+ and Na+ complexes on the A-site of ABO3-type perovskite structure with a rhombohedralsymmetry. Because of a large remanent polarization (𝑃𝑃𝑟𝑟 =38 𝜇𝜇C/cm2) at room temperature, BNT ceramic is consideredas one of the promising candidates for lead free piezoelectricceramics [4]. However, the poling of pure BNT ceramic is
2 Journal of Ceramics
very di�cult due to its high coercive �eld (𝐸𝐸𝑐𝑐 = 73 kV/cm).So, the pure BNT ceramic usually exhibits weak piezoelectricproperties.erefore, a number of BNT-based ceramics wereprepared to improve the electrical properties of this materialby the convectional solid state method [5, 6]. Recently, alot of efforts have been made to prepare the material byvarious chemical methods, such as hydrothermal process[7], citrate method [8, 9], sol-gel, autocombustion [10], andstearic acid gel route [11]. In the present work, A-site oxidesof lead-free Bi0.5Na0.5TiO3 ceramics were optimized usingtwo precursor solutions such as ethylene glycol and citricacid at low calcination temperature (750∘C) and furthersome La doped BNTs (BLNTs) have been developed usingethylene glycols precursors by a novel semiwet technique andstructural, dielectric, and ferroelectric properties have beenstudied for both systems. To the best of our knowledge, BLNTsystem has been synthesized for the �rst time using semiwettechnique. is technique has been applied to make othersystems enhance its properties [12].
2. Experimental Procedures
A novel semiwet technique was used to prepare lead-freeceramic (Bi1−𝑥𝑥La𝑥𝑥)0.5Na0.5TiO3 (BLNT) system. ese com-positions were prepared using analytical-grade metal oxidesor nitrate powders of sigma Aldrich as raw materials suchas Bi2O3 (99%), La2O3 (99%), NaNO3 (99%), TiO2 (99.9%)citric acid, and ethylene glycol. In this method, A-site ofBLNT with composition 𝑥𝑥 = 0 was prepared by usingtwo different precursor solutions� the �rst one is citric acid,and� the second is ethylene glycol. In the �rst process, anappropriate amount of citric acid solution was added to thesolution of nitrates of A-site cations in a beaker (C/M ∼ 1 : 1).Aqueous ammonia in the solution form was added drop bydrop to adjust the pH value of the solution in the range of 6–8.e precursor solution was dehydrated by putting on heaterwith continuous stirring at 80∘C for 3 hrs to form a viscousgel which was placed in an oven at 150∘C for overnight tocombust the gel into ash powder. In the second process,stoichiometry amount of the solution of nitrates of A-sitecation was dissolved with the ethylene glycol (E/M ∼ 1 : 1)with continuous stirring for 30min to get homogeneouslymixed solution. e precursor solution was dehydrated bythe same conditions, which is discussed in the �rst methodto form a gel into ash powder. Both precursor solutionsare expected to distribute the cations atomically homoge-neously throughout the polymeric structure forming a stablepolymeric complex, which was combusted at appropriatetemperature (𝑇𝑇 ∼ 500∘C) in the form of ash powder.e ash,highly �ne, homogeneous, and highly reactive powder wasmixed with appropriate amounts of TiO2 powder thoroughlyin ethanol using mortal pestle for 2 hrs followed by solidstate route. ese powders were dried and calcined at 750∘C(2 hrs) for BNT-CA and BNT-EG and calcined at 850∘C(2 hrs) for BLNT samples. e calcined powder was mixedthoroughly with a polyvinyl alcohol (PVA) binder solutionand then pressed into the form of disk with 10mm diameter.All samples are in pellet form, kept in alumina boat, and
o
(11
0)
(11
3)
oo
o
(22
0)
o
(21
0)
o
(20
0)
o
(11
1)
o
(10
0)
Perovskite structure
20 30 40 50 60 70
Inte
nsi
ty (
a.u
.)
2θ (deg)
(a)
20 30 40 50 60 70
Inte
nsi
ty (
a.u
.)
(11
3)
o
o
o
o
o
oo (2
20
)
(21
0)
(20
0)
(11
1)
(11
0)
(10
0)
2θ (deg)
Bi2Ti2O7∗
∗∗∗∗∗∗∗∗∗
∗
∗∗
(b)
F 1: XRD patterns of (a) BNT-EG and (b) BNT-CA samples.
sintered at temperature 1150∘C for 2 hours. Two pellets ofeach composition were electroded with silver paint on boththe surfaces of the samples for the subsequent electricalmeasurements.e crystallite size of all the samples in BLNTsystem was calculated using Debye-Scherrer formula (𝐷𝐷 =0.89𝜆𝜆/𝜆𝜆 𝜆𝜆𝜆 𝜆𝜆𝐵𝐵) where 𝐷𝐷 is the average crystallite size, 𝜆𝜆is the wavelength of X-ray radiation, 𝜆𝜆 is the full width athalf maximum (FWHM), and 𝜆𝜆𝐵𝐵 the diffraction angle. ecorresponding values are reported in Table 1.
e crystalline structure of the sintered samples wasexamined using X-ray diffraction (XRD) analysis with Cu-K𝛼𝛼 radiation (DX-1000).e surface morphology of sinteredceramics was observed by scanning electron microscopy(SEM, model JEOL A 800). e dielectric constant 𝜀𝜀𝑟𝑟 anddielectric loss (tan 𝛿𝛿) of the ceramic samples at 100 kHz weremeasured as a function of temperature over the temperaturerange from room temperature to 450∘C using an LCRmeter (Hioki 3522). A conventional P-E loop tracer (MarineIndia), which is based on Sawyer-Tower circuit, was used tomeasure the polarization-electrical �eld (P-E) hysteresis loopat 50Hz.
3. Results and Discussion
In the present work, preparation of pure Bi0.5Na0.5TiO3 wasoptimized using two chemical precursors, citric acid andethylene glycol. ese samples are abbreviated as BNT-CAand BNT-EG (Figures 1(a) and 1(b)). It is observed fromthe XRD patterns that the sample prepared by ethyleneglycol precursor solution has shown better phase forma-tion, whereas BNT-CA has formed partially along with
Journal of Ceramics 3
T 1: Dielectric properties of all BLNT system, prepared using ethylene glycol precursor.
F 2: XRD patterns of BLNT system with compositions 𝑥𝑥 𝑎𝑥, 𝑥𝑥𝑥𝑥, and 0.04.
the presence of other phase identi�ed as Bi𝑥Ti𝑥O7 [13].e experimental density observed in BNT-EG sample was5𝑥77 gm/cm𝑥 which is 95% of theoretical density and theexperimental density observed in BNT-CA sample was5𝑥56 gm/cm𝑥 which is 91% of theoretical density. erefore,a typical (Bi1−𝑥𝑥La𝑥𝑥)𝑥𝑥5Na𝑥𝑥5TiO3 (BLNT) systemwith compo-sitions 𝑥𝑥 𝑎 𝑥𝑥𝑥𝑥 and 𝑥𝑥𝑥𝑥 was prepared by semiwet techniqueusing ethylene glycol precursor, calcined at 85𝑥∘C. e XRDpatterns of all the samples in BLNT system have shown singlephase formation with a rhombohedral symmetry in Figure 2.
Pure BNT and BLNT powders have rhombohedral sym-metry at room temperature. However, rhombohedral struc-ture is hard to distinguish due to the overlapping of peaks that
could be due to nearly cubic lattice parameter. Owing to smalldegree of rhombohedral distortion, diffraction lines wereindexed on the basis of pseudocubic unit cell [14] and latticeparameters of all the BLNT samples, prepared by ethyleneglycol, are calculated by using the Unit Cell program package[15]. ere is a variation in the lattice parameters becauseof the different sizes of ionic radius of La3+ (1.03Å) whichare close to Bi3+ ionic radius (0.96Å) and Na+ (0.99Å). evalues in the parentheses refer to the Shannon’s effective ionicradius with the coordination number of six taken from [16].
e microstructure of the pure BNT ceramics, preparedby semiwet using ethylene glycol and citric acid, sinteredat 115𝑥∘C is shown in Figure 3. Ethylene glycol precursorplays a signi�cant role in the grain growth and densi�cation.e microstructure of pure BNT-EG ceramics is denser,homogeneous, and uniform grains with grain size of 2.12𝜇𝜇mas compared to BNT-CA with grain size of 1.19𝜇𝜇m.
e dielectric measurements of electroded samples ofBNT-EG, BNT-CA, and BLNT samples were carried at1𝑥𝑥 kHz frequency over temperature range from room tem-perature to 𝑥5𝑥∘C and are shown in Figures 4(a), 4(b), 4(c),and 4(d), respectively.
Two dielectric anomalies in all the samples are shown attemperatures 𝑇𝑇1 and 𝑇𝑇𝑥, termed as “𝑇𝑇𝑑𝑑” and “𝑇𝑇𝑚𝑚” respec-tively, which corresponds to dielectric transitions from ferro-electric (FE) to anti-ferroelectric (AFE) and anti-ferroelectric(AFE) to paraelectric (PE), respectively. e correspondingpeaks also appear in tan 𝛿𝛿 versus 𝑇𝑇 plots. However, thelow value of dielectric constant of the sample, preparedusing citric acid (BNT-CA), may be due to formation ofother phases of Bi𝑥Ti𝑥O7, which hampers the one dielectricanomaly and relative value of dielectric constant in thissample. It has been observed that the 𝑇𝑇𝑑𝑑 and 𝑇𝑇𝑚𝑚 shi tohigher temperature with increasing the concentration ofLa3+. e partial replacement of A-site cation, with largerionic radius cations and larger amount, decreases the relativedisplacement of B-site cation with respect to the oxygen octa-hedral cage and hence the increase in transition temperatureis observed with substitution of La at A-site. It is obvious thatLa3+ (1.03Å) can occupy the A-site of Bi3+ (0.96Å) or Na+
(0.99Å). When La3+ occupies A-site of Na+, it will lead tocharge imbalance, which creates defects on A-site. In general,A-site (Bi3+ and Na+) substitution by La3+ in BNT ceramicscan be formulated in the following way:
La𝑥O3 + BNT⟶𝑥LaBi + 3Oo (1)
La𝑥O3 + BNT⟶𝑥La••Na + 𝑥𝑉𝑉
�Na + 3Oo (2)
4 Journal of Ceramics
(a) (b)
F 3: Microphotographs patterns of (a) BNT-EG and (b) BNT-CA samples.
800
1600
2400
3200 BNT-EG
0
0.3
0.6
0.9
100 kHz
100 200 300 400
T (◦C)
εr
Tan
δ
Tm
Td
(a)
400
800
1200
1600 BNT-CA
0
0.3
0.6
0.9
100 200 300 400
T (◦C)
εr
Tan
δ
(b)
100 200 300 400
1400
2100
2800
3500
0
0.2
0.4
0.6
0.8
T (◦C)
εr
Tan
δ
(Bi0.98La0.02)0.5Na0.5TiO3
(c)
3200
4000
4800
5600Tm
Td
100 200 300 4000
0.2
0.4
0.6
0.8
T (◦C)
εr
Tan
δ
(Bi0.96La0.04)0.5Na0.5TiO3
(d)
F 4: Variations of 𝜀𝜀𝑟𝑟 and tan 𝛿𝛿 with temperature for samples (a) BNT-EG, (b) BNT-CA, (c) BLNT with 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥, and (d) 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥.
−0.2 −0.1 0 0.1 0.2−8
−4
0
4
8
Applied field E (kV/cm)
Po
lari
zati
onP
(µC
/Cm
2)
Pr = 5.3
Ec = 0.12
(a)
−80 −40 0 40 80−20
−10
0
10
20
Applied field E (kV/cm)
Po
lari
zati
onP
(µC
/Cm
2)
Pr = 13.16
Ec = 25.02
(b)
F 5: e P-E hysteresis loops at RT (50Hz) (a) for BNT-CA and (b) BNT-EG.
Journal of Ceramics 5
T 2: Ferroelectric properties of all saturated BLNT system,prepared using ethylene glycol precursor.
When La3+ occupies Bi-site, as shown in (1), the substitu-tion of Bi3+ by La3+ may cause the slack of BLNT lattice.e lattice deformation can make the ferroelectric domainsreorientation more easily. It leads to the enhancement ofdielectric as well as ferroelectric properties. Additionally,La3+ can also occupy the A-site of Na+, as shown in (2).In this case, the valence of La3+ ion is higher than that ofNa+ ion. To maintain overall electrical neutrality, La3+ actsas a donor leading to some Na-site vacancies [𝑉𝑉�Na], whichcan relax the strain caused by reorientation of domains.erefore, the movement of the domains becomes easierand thus the electrical properties of the BLNT ceramics areimproved signi�cantly. us, the substitution of La3+ in BNTsystem has signi�cantly in�uenced the phase transition anddielectric behavior (Figures 4(c) and 4(d)). e highest valueof 𝜀𝜀𝑟𝑟 (∼3020) and lowest value of Tan 𝛿𝛿 (∼0.045) are obtainedwith the composition 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥 in the BLNT system at roomtemperature. e physical and dielectric properties of all theBLNT samples are tabulated at RT (1𝑥𝑥 kHz) in Table 1.
It shows the typical character of a ferroelectric behavioraround transition temperature because of diffused phasetransition. Dielectric constant exhibits strong frequencydependence above𝑇𝑇𝑑𝑑 and themaximumvalue of 𝜀𝜀𝑟𝑟 decreasesas frequency increases in the BNT-EG samples suggestingthat the ceramic is relaxor ferroelectric. e diffuseness inthe phase transition can be described by 1/𝜀𝜀𝑟𝑟 − 1/𝜀𝜀𝑟𝑟max 𝑥𝐶𝐶−𝛾𝛾(𝑇𝑇 − 𝑇𝑇𝑚𝑚)
𝛾𝛾 in relaxor ferroelectrics [17], where 𝜀𝜀𝑟𝑟max isthe maximum value of dielectric constant at 𝑇𝑇𝑚𝑚, 𝛾𝛾 is thedegree of diffuseness, and 𝐶𝐶 is the curie-like coefficient. 𝛾𝛾can have a value ranging from 1 for normal ferroelectric to 2for an ideal relaxor ferroelectric.is sample exhibits a linearrelationship. e value of the exponent “𝛾𝛾” was determinedby least-squared �tting experimental data to the equationwhich is in the range of 1.50 to 1.80.is con�rms the diffusephase transition in BLNT-EG system. e polarization-electrical �eld (P-E) hysteresis loop is shown in Figures 5(a)and 5(b). It has been found that the P-E loop of BNT usingcitric acid is round shaped, not well developed (saturated),Figure 4(a). BNT-CA sample gets breakdown with increasingelectric �eld.is may be due to (I) large leakage current and(II) due to insufficient annealing process. P-E loop for BNTsample prepared using ethylene glycol is well developed orsaturated, Figure 5(b).
is may be attributed to smaller grain size and relativelybetter density of the samples and con�rm that all the samplesare ferroelectric in nature.
e ferroelectric properties of all the BLNT system aretabulated at RT (50Hz) in Table 2.
4. Conclusion
In summary, ethylene glycol prepared samples have revealedbetter crystallization of pure phase for BNT-EG sample. XRDpatterns have revealed that the sample BNT-EG has singlephase perovskite structure with a rhombohedral symmetry atRT.e structural, phase transition, and electrical propertiesof all the samples were investigated. e Bismuth sodiumtitanate, prepared by ethylene glycol precursor, has not onlyshown excellent dielectric but also ferroelectric behavior.eBNT sample has high value of dielectric constant (𝜀𝜀𝑟𝑟 𝑥 7𝑥5),dielectric loss (Tan 𝛿𝛿 𝑥 𝑥𝑥𝑥𝑥), remnant polarization (𝑃𝑃𝑟𝑟 𝑥13𝑥16 𝜇𝜇C/Cm𝑥), and Coercive �eld (𝐸𝐸𝑐𝑐 𝑥 𝑥5𝑥𝑥6 kV/Cm) atroom temperature. e relatively highest value of dielectricconstant for BLNT with composition 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥 may beattributed to the La doping. Composition with La substitu-tion of 𝑥𝑥 𝑥 𝑥𝑥𝑥𝑥 has shown the highest value of dielectric(3𝑥𝑥𝑥) constant and low loss (𝑥𝑥𝑥𝑥5) as well as the highestvalue of remanent polarization (𝑃𝑃𝑟𝑟 𝑥 17𝑥7𝑥 𝜇𝜇C/Cm
𝑥). etransition temperature Tm is also maximum (385∘C) for thiscomposition which reveals that the material can be useful forhigh-temperature device applications.
Acknowledgment
One of the authorsMs. V. Pal is thankful to JIIT for providingteaching assistance ship and other research facilities to carryout her research work at JIIT, Noida (India).
References
[1] B. Jaffe, W. R. Cook Jr., and H. Jaffe, Piezoelectric Ceramics, vol.3, Academic Press, London, UK, 1971.
[2] M. Pecht, Y. Fukuda, and S. Rajagopal, “e impact of lead-free legislation exemptions on the electronics industry,” IEEETransactions on Electronics Packaging Manufacturing, vol. 27,no. 4, pp. 221–232, 2004.
[3] G. A. Smolenskii, V. A. Isupov, A. I. Agranovskaya, and N. N.Krainik, “New ferroelectrics of complex composition,” SovietPhysics, Solid State, vol. 2, pp. 2651–2654, 1961.
[4] T. Takenaka, K. I. Maruyama, and K. Sakata,“(Bi1/𝑥Na1/𝑥)TiO3-BaTiO3 system for lead-free piezoelectricceramics,” Japanese Journal of Applied Physics, Part 1, vol. 30,no. 9, pp. 2236–2239, 1991.
[5] A. Sasaki, T. Chiba, Y. Mamiya, and E. Otsuki, “Dielectricand piezoelectric properties of (Bi𝑥𝑥5Na𝑥𝑥5)TiO3-(Bi𝑥𝑥5K𝑥𝑥5)TiO3systems,” Japanese Journal of Applied Physics, Part 1, vol. 38, no.9, pp. 5564–5567, 1999.
[6] B. J. Chu, D. R. Chen, G. R. Li, and Q. R. Yin, “Electricalproperties of Na1/𝑥Bi1/𝑥TiO3-BaTiO3 ceramics,” Journal of theEuropean Ceramic Society, vol. 22, no. 13, pp. 2115–2121, 2002.
[7] P. Pookmanee, G. Rujijanagul, S. Ananta, R. B. Heimann,and S. Phanichphant, “Effect of sintering temperature onmicrostructure of hydrothermally prepared bismuth sodiumtitanate ceramics,” Journal of the European Ceramic Society, vol.24, no. 2, pp. 517–520, 2004.
[8] Q. Xu, X. L. Chen, W. Chen, B. H. Kim, S. L. Xu, and M. Chen,“Structure and electrical properties of (Na𝑥𝑥5Bi𝑥𝑥5)1−x Ba𝑥𝑥 TiO3ceramics made by a citrate method,” Journal of Electroceramics,vol. 21, no. 1–4, pp. 617–620, 2008.
6 Journal of Ceramics
[9] D. L. West and D. A. Payne, “Preparation of0.95Bi1/2Na1/2TiO3⋅0.05BaTiO3 ceramics by an aqueouscitrate-gel route,” Journal of the American Ceramic Society, vol.86, no. 1, pp. 192–194, 2003.
[10] J. G. Hou, Y. F. Qu, W. B. Ma, and D. Shan, “Synthesis andpiezoelectric properties of (Na0.5Bi0.5)0.94Ba0.06TiO3 ceramicsprepared by sol-gel auto-combustion method,” Journal of Mate-rials Science, vol. 42, no. 16, pp. 6787–6791, 2007.
[11] J. Hao, X. Wang, R. Chen, and L. Li, “Synthesis of(Bi0.5Na0.5)TiO3 nanocrystalline powders by stearic acidgel method,” Materials Chemistry and Physics, vol. 90, no. 2-3,pp. 282–285, 2005.
[12] A. P. Singh, S. K. Mishra, D. Pandey, C. D. Prasad, and R.Lal, “Low-temperature synthesis of chemically homogeneouslead zirconate titanate (PZT) powders by a semi-wet method,”Journal of Materials Science, vol. 28, no. 18, pp. 5050–5055,1993.
[13] K. Kitagawa, T. Toyoda, K. Kitagawa, and T. Yamamoto,“(Bi1/2Na1/2)TiO3 additive effect for improved piezoelectric andmechanical properties in PZT ceramics,” Journal of MaterialsScience, vol. 38, no. 10, pp. 2241–2245, 2003.
[14] E. Fukuchi, T. Kimura, T. Tani, T. Takeuch, and Y. Saito, “Effectof potassium concentration on the grain orientation in bismuthsodium potassium titanate,” Journal of the American CeramicSociety, vol. 85, no. 6, pp. 1461–1466, 2002.
[15] N. Chaiyo, A. Ruangphanit, R. Muanghlua, S. Niemcharoen, B.Boonchom, and N. Vittayakorn, “Synthesis of potassium nio-bate (KNbO3) nano-powder by amodi�ed solid-state reaction,”Journal of Materials Science, vol. 46, no. 6, pp. 1585–1590, 2011.
[16] R. D. Shannon, “Revised effective ionic radii and systematicstudies of interatomic distances in halides and chalcogenides,”Acta Crystallographica, vol. 32, pp. 751–767, 1976.
[17] K. Uchino and S. Nomura, “Critical exponents of the dielectricconstants in diffused-phase-transition crystals,” FerroelectricsLetters Section, vol. 44, no. 3, pp. 55–61, 1982.
