Ferroelectrics Physical Effects Part 6 pot

Ferroelectrics – Physical Effects 190 composed only of compounds with fluoroalkyl group, giving the induction of SmC* A phase. Compound 1 was dopped with compound 13, having terphenyl rigid core lateraly substituted by four fluorine atoms, Fig. 18. This examples is an exeption from the rule that induction of SmC* A phase can appear in mixtures of compounds differing in the polarity. The steric factors play also crucial role in this behaviour, because the induction of SmC* A phase was not found in mixtures of both nonchiral compounds. For the last system, it cannot be also excluded that the appearance of SmC* A phase could be a result of interactions between nonfluorinated and fluorinated rigid core. 4. Conclusion There are many systems in which the induced antiferroelectric phase can be observed. Compounds with polar group (1, 3-5) have virtual SmC* A phase. They have similar structure to compounds forming SmC* A phase by themselvs; e.g. the structure of compound 1 and 4 is similar to the structure of the first antiferroelectric compound MHPOBC; they have the same rigid core and chiral terminal chain. The placement of molecules of compounds with virtual SmC* A phase, which have a tendency to form the antiferroelectric phase but cannot fulfil all conditions, in a suitable matrix of less polar compounds causes the appearance of an antiferroelectric phase. The ability of more polar compounds with virtual SmC* A phase for induction of this phase decreases in the following order: compounds 1, 3, 4 and 5. Compounds with cyano group in terminal chain have smaller ability than compounds with fluoroalkyl group. Compounds with biphenylate core have bigger ability than compounds with benzoate core. The same is observed in group of less polar compounds with alkyl group in the terminal chain, i.e. the biphenylate core is more preferable, as well as the increase of the terminal chain causes the inrease of tendency to induce SmC* A phase. In systems of similar polarity nonadditive behaviuor can be observed when fluorinated part of the non-branched chain is short. The appearance of liquid crystalline phases is possible due to intermolecular interactions. Interactions between permanent dipoles play a crucial role but interactions between induced dipoles cannot be neglectible. The steric interactions are also important. The possibility of obtaining antiferroelectric phase from compounds forming SmC* phase or only SmA phase broadens the range of compounds useful for preparation of antiferroelectric mixtures for display application. 5. References Chandani, A.D.L., Górecka, E., Ouchi, Y., Takezoe, H. & Fukuda, A. (1989). Antiferroelectric Chiral Smectic Phases Responsible for the Trislable Switching in MHPOBC, Jpn.J.Appl.Phys., Vol.28, No.7, (July 1989), pp. L1265-L1268, ISSN 0021-4922 Czupryński, K., Skrzypek, K., Tykarska, M. & Piecek, W. (2007). Properties of induced antiferroelectric phase, Phase Transitions, Vol.80, No.6-7, (June 2007), pp. 735-744, ISSN 0141-1594 Dąbrowski, R. (2000). Liquid crystals with fluorinated terminal chains and antiferroelectric properties, Ferroelectrics, Vol.243, No.1, (May 2000), pp. 1-18, ISSN 0015-0193 The Induced Antiferroelectric Phase - Structural Correlations 191 Dąbrowski, R., Czupryński, K., Gąsowska, J., Otón, J., Quintana, X., Castillo, P. & Bennis, N. (2004). Broad temperature range antiferroelectric regular mixtures, Proceedings of SPIE, Vol.5565, (2004), pp. 66-71, ISSN 0277-786X Drzewiński, W., Czupryński, K., Dąbrowski, R. & Neubert, M. (1999). New antiferroelectric compounds containing partially fluorinated terminal chains. Synthesis and mesomorphic properties, Mol.Cryst.Liq.Cryst., Vol.328, (August 1999), pp. 401-410, ISSN 1058-725X Drzewiński, W., Dąbrowski, R. & Czupryński, K. (2002). Synthesis and mesomorphic properties of optically active (S)-(+)-4-(1-methylheptyloxycarbonyl)phenyl 4’- (fluoroalkanoyloxyalkoxy)biphenyl-4-carboxylates and 4’-(alkanoyloxyalkoxy) biphenyl-4-carboxylates, Polish J.Chem., Vol.76, No.2-3, (February 2002), 273-284, ISSN 0137-5083 Dziaduszek, J., Dąbrowski, R., Czupryński, K. & Bennis, N. (2006). , Ferroelectrics, Vol.343, No.1, (November 2006), pp. 3-9, ISSN 0015-0193 Fukuda, A., Takanishi, Y., Isozaki, T., Ishikawa, K. & Takezoe, H. (1994). Antiferroelectric chiral smectic liquid crystals, J.Mater.Chem., Vol.4, No.7, (January 1994), pp. 997- 1016, ISSN 0959-9428 Gauza, S., Czupryński, K., Dąbrowski, R., Kenig, K., Kuczyński, W. & Goc, F. (2000). Bicomponent systems with induced or enhanced antiferroelectric SmC A * phase, Mol.Cryst.Liq.Cryst., Vol.351, (November 2000), 287-296, ISSN 1058-725X Gauza, S., Dąbrowski, R., Czypryński, K., Drzewiński, W. & Kenig, K. (2002). Induced antiferroelectric smectic C A * phase. Structural correlations, Ferroelectrics, Vol.276, (January 2002), pp. 207-217, ISSN 0015-0193 Gąsowska, J. (2004). PhD thesis, WAT, Warsaw. Gąsowska, J., Dąbrowski, R., Drzewiński, W., Filipowicz, M., Przedmojski, J. & Kenig, K. (2004). Comparison of mesomorphic properties in chiral and achiral homologous series of high tilted ferroelectrics and antiferroelectrics, Ferroelectrics, Vol.309, No.1, (2004), pp. 83-93, ISSN 0015-0193 Kobayashi, I., Hashimoto, S., Suzuki, Y., Yajima, T., Kawauchi, S., Imase, T., Terada, M. & Mikami, K. (1999). Effects of Conformation of Diastereomer Liquid Crystals on the Preference of Antiferroelectricity, Mol.Cryst.Liq.Cryst.A, Vol.328, (August 1999), pp. 131-137, ISSN 1058-725X Kula, P. (2008). PhD thesis, WAT, Warsaw. Mandal, P.K., Jaishi, B.R., Haase, W., Dąbrowski, R., Tykarska, M. & Kula, P. (2006). Optical and dielectric studies on ferroelectric liquid crystal MHPO(13F)BC: Evidence of SmC*  phase presence, Phase Transitions, Vol.79, No.3, (March 2006), pp. 223-235, ISSN 0141-1594 Matsumoto, T., Fukuda, A., Johno, M., Motoyama, Y., Yuki, T., Seomun, S.S. & Yamashita M. (1999). A novel property caused by frustration between ferroelectricity and antiferroelectricity and its application to liquid crystal displays-frustoelectricity and V-shaped swiching, J.Mater.Chem., Vol.9, No.9, (1999), pp. 2051-2080 ISSN 0959- 9428 Skrzypek, K. & Tykarska, M. (2006). The induction of antiferroelectric phase in the bicomponent system comprising cyano compound, Ferroelectrics, Vol.343, No.1, (November 2006), pp. 177-180, ISSN 0015-0193 Ferroelectrics – Physical Effects 192 Skrzypek, K., Czupryński, K., Perkowski, P. & Tykarska, M. (2009). Dielectric and DSC Studies of the Bicomponent Systems with Induced Antiferroelectric Phase Comprising Cyano Terminated Compounds, Mol.Cryst.Liq.Cryst., Vol.502, (2009), 154-163, ISSN 1542-1406 Tykarska, M. & Skrzypek, K. (2006). The enhancement and induction of antiferroelectric phase, Ferroelectrics, Vol.343, No.1, (November 2006), pp. 193-200, ISSN 0015-0193 Tykarska, M., Dąbrowski, R., Przedmojski, J., Piecek, W., Skrzypek, K., Donnio, B. & Guillon, D. (2008). Physical properties of two systems with induced antiferroelectric phase, Liq.Cryst., Vol.35, No.9, (September 2008), pp. 1053-1059, ISSN 0267-8292 Tykarska, M., Skrzypek, K., Ścibior, E. & Samsel, A. (2006). Helical pitch in bicomponent mixtures with induced antiferroelectric phase, Mol.Cryst.Liq.Cryst., Vol.449, No.1, (July 2006), pp. 71-77, ISSN 1542-1406 Tykarska, M., Garbat, K. & Rejmer, W. (2011). The influence of the concentration of fluoro- and cyanoterminated compounds on the induction of antiferroelectric phase and on helical pitch, Mol.Cryst.Liq.Cryst., accepted for publication 2011. Part 2 Piezoelectrics Andriy Andrusyk Institute for Condensed Matter Physics Ukraine 1. Introduction For the last several years there has been significant progress in the development of new piezoelectric materials (relaxor ferroelectric single crystals (Park & Shrout, 1997), solid solutions with high transition temperature (Zhang et al., 2003), lead-free materials (Saito et al., 2004)) and in understanding of mechanisms of the piezoelectric coupling in ferroelectric piezoelectrics (Fu & Cohen, 2000; Guo et al., 2000). This progress was triggered in particular by the wide use of piezoelectric effect in a variety of devices (resonators, tactile sensors, bandpass filters, ceramic discriminators, SAW filters, piezoresponse force microscopes and others). What concerns theoretical study of the piezoelectric effect, significant efforts were made in first-principles calculations. Such calculations are possible for the ferroelectrics with a relatively simple structure, in particular for simple and complex perovskites (Bellaiche et al., 2000; Garcia & Vanderbilt, 1998). For compounds with a complex structure often only the research within the Landau theory is possible. The structure complexity justifies the application of semimicroscopic models considering only that characteristic feature of the microscopic structure which is crucial in explaining the ferroelectric transition or the piezoelectric effect. Such models are adequate for the crystal under study if they are able to explain the wide range of physical properties. In this chapter sodium potassium tartrate tetrahydrate NaKC 4 H 4 O 6 ·4H 2 O (Rochelle salt or RS) is studied on the base of the semimicroscopic Mitsui model. The microscopic mechanism of ferroelectric phase transitions in RS was the subject of numerous investigations. Studies based on x-ray diffraction data (Shiozaki et al., 1998) argued that these were the order-disorder motions of O9 and O10 groups, coupled with the displacive vibrations of O8 groups, which were responsible for the phase transitions in Rochelle salt as well as for the spontaneous polarization. Later it was confirmed by the inelastic neutron scattering data (Hlinka et al., 2001). Respective static displacements initiate the emergence of dipole moments in local structure units in ferroelectric phase. Such displacements can be interpreted also as changes in the population ratio of two equilibrium positions of sites in the paraelectric structure (revealed in the structure studies (Noda et al., 2000; Shiozaki et al., 2001)). The order-disorder pattern of phase transitions in RS forms the basis of the semimicroscopic Mitsui model (Mitsui, 1958). In this model the asymmetry of occupancy of double local atomic positions and compensation of electric dipole moments occurring in paraelectric phases were taken into account. Piezoelectric Effect in Rochelle Salt 9 2 Will-be-set-by-IN-TECH Recently (Levitskii et al., 2003) Mitsui model as applied to RS was extended by accounting of the piezoelectric coupling between the order parameter and strain ε 4 . Later this model was extended to the four sublattice type (Levitskii et al., 2009; Stasyuk & Velychko, 2005) that gives more thorough consideration of real RS structure. We performed our research of piezoelectric effect in Rochelle salt on the basis of the Mitsui-type model containing additional term of transverse field type responsible for dynamic flipping of structural elements (Levitskii, Andrusyk & Zachek, 2010; Levitskii, Zachek & Andrusyk, 2010). Originally, this term was added with the aim to describe resonant dielectric response which takes place in RS in submillimeter region. First, we provide characteristics of ferroelectric phase transitions in RS and experimental data for constants of physical properties of RS. Then, we present our study results (thermodynamic and dynamic characteristics) obtained within Mitsui model for RS. Specifically, we calculate permittivity of free and clamped crystals, calculate piezoelectric stress coefficient e 14 , elastic constant c E 44 . The key attention is given to investigation of the phenomenon of piezoelectric resonance. 2. Physical properties of Rochelle salt Rochelle salt (RS), NaKC 4 H 4 O 6 ·4H 2 O is the oldest and has been for a long time the only known ferroelectrics. RS has been the subject of numerous studies over the past 60 years. It is known for its remarkable ferroelectric state between two Curie points T c1 = 255 K and T c2 = 297 K (Jona & Shirane, 1965). Second order phase transitions occur at both Curie points. The crystalline structure of RS proved to be complex. It is orthorhombic (space group D 3 2 —P2 1 2 1 2) in the paraelectric phases and monoclinic (space group C 2 2 —P2 1 )inthe ferroelectric phase (Solans et al., 1997). Spontaneous polarization is directed along the a crystal axis; it is accompanied by a spontaneous shear strain ε 4 . There are four formula units (Z = 4; 112 atoms) in the unit cell of Rochelle salt in both ferroelectric and paraelectric phase. In recent study (Görbitz & Sagstuen, 2008) the complete Rochelle salt structure in paraelectric phase was described. Due to the symmetry of RS crystal structure some elements of material tensors are zeroes. In RS case material tensors in Voigt index notations are of the form presented below (Shuvalov, 1988). Elastic stiffnesses or elastic constants (c E ij =(∂σ i /∂ε j ) E ): (c E )= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ c 11 c 12 c 13 c 14 00 c 12 c 22 c 23 c 24 00 c 13 c 23 c 33 c 34 00 c 14 c 24 c 34 c 44 00 0000c 55 c 56 0000c 56 c 66 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (1) Coefficients of piezoelectric stress (e ij =(∂P i /∂ε j ) E = −(∂σ j /∂E i ) ε ): (e)= ⎛ ⎝ e 11 e 12 e 13 e 14 00 0000e 25 e 26 0000e 35 e 36 ⎞ ⎠ (2) 196 Ferroelectrics – Physical Effects Piezoelectric Effect in Rochelle salt 3 Dielectric permittivity (χ α ij =(∂P i /∂E j ) α ,whereα is σ or ε): (χ σ,ε )= ⎛ ⎝ χ 11 00 0 χ 22 χ 23 0 χ 23 χ 33 ⎞ ⎠ (3) It is necessary to make some comments about the notations. Superscripts to the matrices of physical properties indicate that a physical characteristic denoted by a superscript is constant or zero (for instance (c) with superscript E denotes matrix of elastic constants at constant electric field: (c E )). We omitted superscripts E and σ, ε for components of tensors (c E ) and (χ σ,ε ) respectively but keep them in mind. Notation (χ σ,ε ) denotes two different tensors: (χ σ ) and (χ ε ) which we will call tensors of dielectric permittivity at constant stress and strain respectively. We will also call them tensor of free crystal dielectric permittivity (zero stress is assumed) and of clamped crystal dielectric permittivity. Hereinafter coefficients equal to zero in paraelectric phases are presented in bold. Experimental data for physical constants are presented in Figs. 1, 2, and 3. 240 260 280 300 -0,2 -0,1 0,0 0,1 c 24 + c 34 c 34 c 24 c 14 T (K) c i4 E (10 10 Nm -2 ) 200 225 250 275 300 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 c 13 c 12 c 23 c 11 c 33 c 22 c ij E (10 10 Nm -2 ) T (K) 250 275 300 0,0 0,2 0,4 0,6 0,8 c 44 E (10 10 Nm -2 ) T (K) Fig. 1. Rochelle salt elastic constants. Solid lines for c ij (i, j = 1, 2, 3) are experimental data (Mason, 1950), ∗ correspond to c 44 (Yu. Serdobolskaya, 1996), ◦ correspond to c E 24 + c E 34 (Shiozaki et al., 2006). Lines for c E 14 , c E 24 , c E 34 are the results of theoretical calculations (Levitskii et al., 2005). 260 280 300 320 1 2 3 4 5 6 7 240 260 280 300 -0,4 -0,2 0,0 0,2 0,4 0,6 e 14 (Cm -2 ) T (K) e 1j (Cm -2 ) e 12 e 13 e 11 T (K) Fig. 2. Rochelle salt coefficients of piezoelectric stress. Points  are experimental data (Beige & Kühnel, 1984) for e 14 . Solid lines correspond to the results of theoretical calculations e 1j (j = 1, 2, 3) (Levitskii et al., 2005). 197 Piezoelectric Effect in Rochelle Salt 4 Will-be-set-by-IN-TECH 240 260 280 300 0,00 0,04 0,08 0,12 0,16 0,20 240 260 280 300 0,00 0,04 0,08 0,12 0,16 0,20 ε 1/χ 11 T (K) σ 1/χ 11 T (K) Fig. 3. Experimental data for clamped and free crystal inverse dielectric permittivity of Rochelle salt. 1/χ ε 11 :  (Sandy & Jones, 1968), ♦ (Mueller, 1935). 1/χ σ 11 : + (Yurin, 1965), • (Mason, 1939),  (Taylor et al., 1984). As Fig. 1 shows elastic constants c E ij (i, j = 1, 2, 3) do not change their behaviour in phase transition points, whereas constant c E 44 becomes zero at the transition point. Besides, c 44 is strongly dependent on T as c E ij (i, j = 1, 2, 3) are almost independent of T. Constants c E 14 , c E 24 , c E 34 , as it is expected on symmetry grounds, are equal to zero in paraelectric phases. There are no experiments for c E 14 and there are no experiments for c E 24 , c E 34 measured separately. There is only the experiment for c E 24 + c E 34 (Shiozaki et al., 2006). However, c E 14 , c E 24 ,andc E 34 were estimated theoretically (Levitskii et al., 2005) and the result of estimation is presented in Fig. 1. Correspondent result for c E 24 + c E 34 is compared to experimental data where good agreement was derived. Fig. 2 presents temperature dependencies of piezoelectric stress coefficients. As one can see coefficients e 11 , e 12 , e 13 differ from zero only inside ferroelectric phase. Coefficient e 14 has sharp (but finite) peak in the transition point, whereas coefficients e 11 , e 12 , e 13 do not have it. Free and clamped crystal longitudinal susceptibilities are presented in Fig. 3. Free crystal susceptibility has singularities in transition point, whereas clamped susceptibility remains finite. 3. Thermodynamic characteristics of Rochelle salt 3.1 Theoretical study of the Mitsui model We give consideration to a two-sublattice order-disorder type system with an asymmetric double-well potential. Hamiltonian of such system is referred to as the Mitsui Hamiltonian. We assume this system has essential piezoelectric coupling of the order parameter with component of strain tensor ε 4 which should be accounted. We suppose the polarization is directed along x-axes and arises due to the structural units ordering in the one of two possible equilibrium positions. Precisely this case occurs in RS and such modified Mitsui model was considered earlier (Levitskii et al., 2003). We complement this model with transverse field to take into account the possibility of dynamic ordering units flipping between two equilibrium 198 Ferroelectrics – Physical Effects [...]... value, real part and imaginary part of u (y, z) at ν = 105 Hz 1,0 1,0 1,0 0,8 0,8 0,8 0 ,6 0 ,6 0 ,6 0,4 0,4 0,4 0,2 0,2 0,2 0,2 0,4 2 0 ,6 2 0,8 1,0 0,2 1/2 0,4 0 ,6 0,8 1,0 0,2 0,4 0 ,6 0,8 1,0 Im[u] Re[u] (Re [u]+Im [u]) Fig 9 Distributions of absolute value, real part and imaginary part of u (y, z) at ν = 2 .6 × 105 Hz 1,0 1,0 1,0 0,8 0,8 0,8 0 ,6 0 ,6 0 ,6 0,4 0,4 0,4 0,2 0,2 0,2 0,2 0,4 2 0 ,6 2 0,8 1/2... 31 Ak = ⎜ ⎜ a41 ⎜ ⎜ ⎝ a51 a22 0 a32 0 a42 0 a52 − a35 a61 a62 − a 36 ⎞ 0 a15 a 16 ⎟ ˜ − Ω a 16 a15 ⎟ ⎟ 0 a35 a 36 ⎟ ⎟ ⎟ , δxk (ω ) = 0 a 36 a35 ⎟ ⎟ ⎟ − a 36 a55 a 56 ⎠ − a35 a 56 a55 ⎛ z δξ k (ω ) ⎞ ⎛ b1 ⎞ ⎜ z ⎟ ⎜ ⎟ ⎜ δσk (ω ) ⎟ ⎜ b2 ⎟ ⎜ y ⎟ ⎜ ⎟ ⎜ δξ (ω ) ⎟ ⎜ ξx ⎟ ⎜ k ⎟ ⎜ ⎟ ⎜ y ⎟, b = ⎜ x ⎟ ⎜ δσ (ω ) ⎟ ⎜σ ⎟ ⎜ k ⎟ ⎜ ⎟ ⎜ x ⎟ ⎜ ⎟ ⎝ δξ k (ω ) ⎠ ⎝ b5 ⎠ x (ω ) b6 δσk (42) Matrix Ak components: ˜ a11 = U1 + R+ G1 ,... without piezoelectric coupling (Žekš et al., 1971) 210 Ferroelectrics – Physical Effects Will-be-set-by-IN-TECH 16 10 10 -1 (10 s ) 8 -1 6 τ 4 2 240 260 280 300 320 T (K) Fig 6 Dependency of inverse relaxation times on temperature Solid line presents the result of calculation Points present experimental data: • (Müser & Potthaest, 1 967 ), (Sandy & Jones, 1 968 ), ◦ (Kołodziej, 1975), (Volkov et al., 1980) 4.2... unexplored In particular, the influence of ferroelectric phase transition on piezoelectric resonance remains open The study of this issue requires to consider the system Eq (64 ), which can be performed only numerically 218 Ferroelectrics – Physical Effects Will-be-set-by-IN-TECH 24 1,0 1,0 1,0 0,8 0,8 0,8 0 ,6 0 ,6 0 ,6 0,4 0,4 0,4 0,2 0,2 0,2 0,2 0,4 2 0 ,6 2 0,8 1/2 (Re [u]+Im [u]) 1,0 0,2 0,4 0 ,6 Re[u] 0,8... clamped crystal, considered in 2 16 Ferroelectrics – Physical Effects Will-be-set-by-IN-TECH 22 previous subsection Similar behaviour of dynamic permittivity of free crystal is observed in low-temperature paraelectric and ferroelectric phases 1000 ε'11 80 800 70 60 0 ε''11 60 50 400 40 200 30 0 20 -200 10 -400 10 0 0 10 2 10 4 10 6 10 8 10 10 ν, Hz 10 0 10 2 10 4 10 6 10 8 10 10 ν, Hz Fig 7 Dynamic free... GHz, Sov Phys Solid State 15(8): 167 2– 167 3 Saito, Y., Takao, H., Tani, T., Nonoyama, T., Takatori, K., Homma, T., Nagaya, T & Nakamura, M (2004) Lead-free piezoceramics, Nature 432: 84–87 Sandy, F & Jones, R V (1 968 ) Dielectric relaxation of Rochelle salt, Phys Rev 168 (2): 481–493 Shiozaki, Y., Nakamura, E & Mitsui, T (eds) (20 06) 67 A-1 NaKC4 H4 O6 ·4H2 O [F], Vol 36C of Landolt-Börnstein — Group III... obtained for the physical coefficients were used for study of piezoelectric resonance It should be noted that the ratio between the elastic constants of Rochelle salt allowed to reduce 217 23 Piezoelectric Rochelle salt Rochelle Salt Piezoelectric Effect in Effect in 1,0 1,0 1,0 0,8 0,8 0,8 0 ,6 0 ,6 0 ,6 0,4 0,4 0,4 0,2 0,2 0,2 0,2 0,4 2 0 ,6 2 0,8 1,0 0,2 1/2 0,4 0 ,6 0,8 1,0 0,2 0,4 0 ,6 0,8 1,0 Im[u] Re[u]... microscopical mechanism of the phase transition of Rochelle salt 204 Ferroelectrics – Physical Effects Will-be-set-by-IN-TECH 10 -3 E -4 -2 P1 (10 Cm ) 6 2,5 ε4 (10 ) 6 4 1,5 3 1,0 2 0,5 -2 8 5 2,0 9 c44 (10 Nm ) 1 0,0 255 270 285 300 T 4 2 0 0 (K) 255 ε 270 285 300 σ 1/χ 11 1/χ T (K) 250 275 -2 300 T (K) e14 (Cm ) 11 6 0,20 0, 16 0,15 0,12 4 0,08 3 5 0,10 2 0,04 0,05 1 0,00 240 255 270 285 300 315... compared to δε 4 and can be treated as zero Therefore, system Eq (66 ) and boundary conditions Eq (67 ) reduce to Δδε 4 + δε 4 | Σ = ρω 2 δε = 0 c44 4 e14 δE c44 1 (68 a) (68 b) Similarly, equation (68 ) can be obtained for ferroelectric phase It is noteworthy that in paraelectric phase in case of small c22 , c23 , c33 we would receive equation (68 ) again However, in this case strains δε 2 and δε 3 are not... imaginary part of dielectric permittivity, calculated theoretically (lines) at different temperatures T (K): (a) 235, (b) 245, (c) 265 , (d) 285, (e) 305, (f) 315 Points represent experimental data: (Sandy & Jones, 1 968 ), ◦ (Poplavko et al., 1974), + (Pereverzeva, 1974), (Deyda, 1 967 ), • (Akao & Sasaki, 1955), (Müser & Potthaest, 1 967 ), × (Kołodziej, 1975), (Volkov et al., 1980), (Sandy & Jones, 1 968 ), (Jäckle, . unit, A k = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ a 11 a 12 − ˜ Ω 0 a 15 a 16 a 21 a 22 0 − ˜ Ω a 16 a 15 a 31 a 32 00a 35 a 36 a 41 a 42 00a 36 a 35 a 51 a 52 −a 35 −a 36 a 55 a 56 a 61 a 62 −a 36 −a 35 a 56 a 55 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,. (c E ij =(∂σ i /∂ε j ) E ): (c E )= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ c 11 c 12 c 13 c 14 00 c 12 c 22 c 23 c 24 00 c 13 c 23 c 33 c 34 00 c 14 c 24 c 34 c 44 00 0000c 55 c 56 0000c 56 c 66 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (1) Coefficients of piezoelectric stress (e ij =(∂P i /∂ε j ) E = −(∂σ j /∂E i ) ε ): (e)= ⎛ ⎝ e 11 e 12 e 13 e 14 00 0000e 25 e 26 0000e 35 e 36 ⎞ ⎠ (2) 1 96 Ferroelectrics. Proceedings of SPIE, Vol.5 565 , (2004), pp. 66 -71, ISSN 0277-786X Drzewiński, W., Czupryński, K., Dąbrowski, R. & Neubert, M. (1999). New antiferroelectric compounds containing partially fluorinated