Elastic, piezoelectric, and dielectric properties of 0.58 Pb(Mg 1/3 Nb 2/3 ) O 3 -0.42 PbTiO 3 single crystal Hu Cao, V. Hugo Schmidt, Rui Zhang, Wenwu Cao, and Haosu Luo Citation: Journal of Applied Physics 96, 549 (2004); doi: 10.1063/1.1712020 View online: http://dx.doi.org/10.1063/1.1712020 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/96/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Face shear piezoelectric properties of relaxor- PbTiO 3 single crystals Appl. Phys. Lett. 98, 182903 (2011); 10.1063/1.3584851 Elastic, dielectric, and piezoelectric constants of Pb ( In 1 / 2 Nb 1 / 2 ) O 3 – Pb ( Mg 1 / 3 Nb 2 / 3 ) O 3 – PbTiO 3 single crystal poled along [ 011 ] c Appl. Phys. Lett. 97, 032902 (2010); 10.1063/1.3466906 Comment on “Complete sets of elastic, dielectric, and piezoelectric properties of flux-grown [011]-poled Pb ( Mg 1 / 3 Nb 2 / 3 ) O 3 − ( 28 – 32 ) % PbTiO 3 single crystals” [Appl. Phys. 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IP: 153.90.170.56 On: Fri, 26 Feb 2016 19:47:52 http://scitation.aip.org/content/aip/journal/jap?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/296339739/x01/AIP-PT/JAP_ArticleDL_021716/APR_1640x440BannerAd11-15.jpg/434f71374e315a556e61414141774c75?x http://scitation.aip.org/search?value1=Hu+Cao&option1=author http://scitation.aip.org/search?value1=V.+Hugo+Schmidt&option1=author http://scitation.aip.org/search?value1=Rui+Zhang&option1=author http://scitation.aip.org/search?value1=Wenwu+Cao&option1=author http://scitation.aip.org/search?value1=Haosu+Luo&option1=author http://scitation.aip.org/content/aip/journal/jap?ver=pdfcov http://dx.doi.org/10.1063/1.1712020 http://scitation.aip.org/content/aip/journal/jap/96/1?ver=pdfcov http://scitation.aip.org/content/aip?ver=pdfcov http://scitation.aip.org/content/aip/journal/apl/98/18/10.1063/1.3584851?ver=pdfcov http://scitation.aip.org/content/aip/journal/apl/97/3/10.1063/1.3466906?ver=pdfcov http://scitation.aip.org/content/aip/journal/apl/97/3/10.1063/1.3466906?ver=pdfcov http://scitation.aip.org/content/aip/journal/apl/96/19/10.1063/1.3429603?ver=pdfcov http://scitation.aip.org/content/aip/journal/apl/96/19/10.1063/1.3429603?ver=pdfcov http://scitation.aip.org/content/aip/journal/apl/90/21/10.1063/1.2743393?ver=pdfcov http://scitation.aip.org/content/aip/journal/apl/90/21/10.1063/1.2743393?ver=pdfcov http://scitation.aip.org/content/aip/journal/jap/101/2/10.1063/1.2429724?ver=pdfcov http://scitation.aip.org/content/aip/journal/jap/101/2/10.1063/1.2429724?ver=pdfcov Elastic, piezoelectric, and dielectric properties of 0.58Pb„Mg1Õ3Nb2Õ3…O3-0.42PbTiO3 single crystal Hu Caoa) and V. Hugo Schmidt Department of Physics, Montana State University, Bozeman, Montana 59717 Rui Zhang and Wenwu Cao Material Research Institute, The Pennsylvania State University, University Park, Pennsylvania 16802 Haosu Luo Shanghai Institute of Ceramics, Chinese Academy of Sciences 215 Chengbei Road, Jiading, Shanghai 201800, People’s Republic of China ~Received 23 September 2003; accepted 29 February 2004! The elastic, piezoelectric, and dielectric constants of 0.58Pb~Mg1/3Nb2/3)O3-0.42PbTiO3 ~PMN-42%PT! were determined experimentally using combined resonance and ultrasonic methods. At room temperature the PMN-42%PT single crystal has tetragonal symmetry. The measured piezoelectric constant d33 is ;260310212 C/N. The electromechanical coefficients k15 , k31 , k33 , and kt are 0.80, 0.39, 0.78, and 0.62, respectively. From the measured material constants the orientational dependence of phase velocities and electromechanical coupling coefficients was calculated. The results showed the tetragonal crystal exhibits isotropy in the X-Y plane and k15 , k33 , and kt reach their maxima in @001#. © 2004 American Institute of Physics. @DOI: 10.1063/1.1712020# I. INTRODUCTION Relaxor ferroelectrics, such as Pb~Mg1/3Nb2/3)O3 ~PMN!, Pb~Sc1/3Nb2/3)O3 ~PSN!, and Pb~Zn1/3Nb2/3)O3 ~PZN! and their solid solutions with PbTiO3 have come into prominence due to their ultra-high dielectric and piezoelec- tric properties1–4 in contrast to the conventional piezoelectric ceramics. More recently, there is increased interest in the relaxor ferroelectric single crystals xPb~Mg1/3Nb2/3)O3-(1 2x)PbTiO3 ~PMN-PT!, which have a broad composition range and possess very high electromechanical coupling co- efficients, piezoelectric, and dielectric constants and field- induced strain response.5 For instance, PMN-33%PT single crystals, which have a rhombohedral phase near the morpho- tropic phase boundary ~MPB!, possess large piezoelectric (d33;2500 pC/N), dielectric (e r;5000– 5500), and electro- mechanical coupling coefficients (k33;94%). This system shows a promising potential of producing higher sensitivity ultrasonic transducers with superior broadband characteristics,6 large strain actuators, and other electrome- chanical devices. At present, most people focus their interests on the domain engineered single crystals with different compositions near or away from MPB. These available complete physical property data of 0.67Pb~Mg1/3Nb2/3)O3- 0.33PbTiO3 ,7 0.70Pb~Mg1/3Nb2/3)O3-0.30PbTiO3 ,8 0.955Pb~Zn1/3Nb2/3)O3-0.045PbTiO3 ,9 and 0.92Pb~Zn1/3Nb2/3)O3-0.08PbTiO3 ~Ref. 10! single crystals have been reported. However, more information is needed for us to understand and apply these materials. Therefore, the aim of this article is to provide a complete set of such data for the tetragonal 0.58Pb~Mg1/3Nb2/3)O3-0.42PbTiO3 single crystal. Practically, it is also very important to have a com- plete set of elastic, piezoelectric, and dielectric constants available. The data were obtained by using a combined method involving both pulse-echo and impedance resonance techniques. A PMN-42%PT crystal, where the PT content is a little away from the MPB, has a tetragonal phase. For the tetrag- onal symmetry there are a total of 11 independent electro- elastic constants: six elastic constants, three piezoelectric constants, and two dielectric constants to be determined.11 The dielectric constants e11 T and e33 T were measured from the low frequency capacitance using the parallel capacitor ap- proximation. The elastic compliance s11 E and the electrome- chanical coefficient k31 can be calculated from the resonance and antiresonance frequencies of the length-extensional mode of vibration bars. Similarly, s33 D and k33 can be calcu- lated from the longitudinal extensional mode and c33 D and kt calculated from the thickness-extensional mode. Another five elastic constants c11 E , c12 E , c44 E , c66 E , and c44 D were determined from the measured phase velocities of ultrasonic waves propagating along appropriate pure mode orientations. Using this measurement scheme, the only samples needed are those with the orientations of @001#/@010#/@001# and @001/@110#/@11̄0# . Thus, fewer samples are required when using this combined measurement technique.12 For each given wave propagation direction, the relation- ship between the phase velocity and associated material con- stants can be obtained by solving the Christoffel wave equations13 and these velocities can be measured using the pulse-echo technique.11 Other material constants can be de- rived from the piezoelectric constitutive equations and con- version formula.a!Electronic mail: caohu@hotmail.com JOURNAL OF APPLIED PHYSICS VOLUME 96, NUMBER 1 1 JULY 2004 5490021-8979/2004/96(1)/549/6/$22.00 © 2004 American Institute of Physics Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. IP: 153.90.170.56 On: Fri, 26 Feb 2016 19:47:52 http://dx.doi.org/10.1063/1.1712020 II. EXPERIMENTAL PROCEDURE The 0.58Pb~Mg1/3Nb2/3)O3-0.42PbTiO3 single crystals were grown by the modified Bridgman method.14 In the ferroelectric phase, the dipole in each cell of the PMN- 42%PT crystals is along one of the six ^001& directions. The as-grown crystals were cut along the pseudocubic crystallo- graphic face @001# determined by a x-ray diffractometer. The impedance resonance and ultrasonic pulse-echo techniques, described in the IEEE standards on piezoelectricity,11 are of- ten used to characterize material properties for piezoelectric materials.15,16 Because of segregation17 during the process of crystal growth, the PT content in different parts in the crystal boule varies. Usually, 5%–10% PT composition variation is common in a large crystal. This will greatly affect the con- sistency of the measured data set. The Curie temperature approximately reflects the PT content in samples. Therefore, in order to guarantee consistency, samples were cut from the same part of a crystal boule and Curie temperature of samples was limited in the range of 190 °C63 °C. Then, each sample, based on the requirement of different types of measurements, was cut and polished into a rectangular par- allelepiped shape with three pairs of parallel surfaces, where dimensions and geometries were determined by IEEE stan- dards. The final dimensions of the samples used for the ul- trasonic measurements were 53530.8 mm3. For the reso- nance vibration bars the aspect ratio of samples exceeded 5:1 in order to yield nearly pure resonance modes.11 Two sets of samples were prepared for a consistency check. Gold elec- trodes were sputtered onto parallel surfaces of samples for poling. An external electric field of 5–8 kV/cm was applied at room temperature in silicone oil along the @001# direction of cubic axes to pole these samples. A 15 MHz longitudinal wave transducer ~Ultra Labora- tories, Inc.! and a 20 kHz shear wave transducer ~Panamet- rics! were used for the pulse-echo measurements. The elec- tric pulses used to excite the transducer were generated by a Panametrics 200 MHz pulser/receiver, and the time of flight between echoes was measured by a Tektronix 460A digital oscilloscope. For the resonance measurements an HP 4194A impedance/gain-phase analyzer was employed to measure the resonance and anti-resonance frequencies of these reso- nators. Dielectric constants are determined by measuring ca- pacitances of @100# and @001# oriented parallel plates. The capacitance was measured at 1 kHz using a Stanford Re- search System SR715 LCR meter. The density of the samples was determined by applying the Archimedes principle. Dielectric properties as a function of temperature were determined using a multi-frequency LCR meter ~HP 4284A!. Polarization and strain hysteresis were plotted using a modi- fied Sawyer-Tower circuit driven by a lock-in amplifier ~Stanford Research System, Model SR830!. ~See Fig. 1.! III. RESULTS AND DISCUSSION The dielectric constants er and dielectric loss as a func- tion of temperature and frequency for @001#-oriented PMN- 42%PT on heating and cooling are presented in Fig. 2. The crystal exhibits very little frequency dispersion in the dielec- tric constants whether on heating or on cooling over the mea- sured temperature range. The er exhibits a sharp change near the transition temperature. Thus, PMN-42%PT is a normal FIG. 1. The six types of test samples shown were cut from the PMN-0.42PT bulk single crystal and used to determine the whole set of material proper- ties. FIG. 2. Temperature and frequency dependence of dielectric constant er and dielectric loss of @001#-oriented PMN-0.42PT on heating ~a! and cooling ~b!. 550 J. Appl. Phys., Vol. 96, No. 1, 1 July 2004 Cao et al. Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. IP: 153.90.170.56 On: Fri, 26 Feb 2016 19:47:52 ferroelectric and possesses a first-order phase transition from the tetragonal ferroelectric phase to the cubic paraelectric phase. There are marked rises in loss with increasing tempera- ture above Tc at 1 kHz and especially at 100 Hz that can be attributed to thermally activated electrical conductivity. These rises are superimposed on the decreasing loss with increasing temperature just above Tc that are associated with the ferroelectric transition. To minimize effects of this transition-associated loss, an analysis based on Fig. 2~b! was made only in the 225 to 250 °C range in which the transition- associated loss is small. If one assumes that all dielectric loss in this range is due to conductivity, the appropriate formulas are s85e0e9v; e950.01e8~ loss% !. Here, s8 is the real part of the conductivity, e9 is the lossy part of the permittivity, and v is the angular frequency. Upon plotting the conductivity found from the 100 Hz data against inverse temperature over the above temperature range, a straight line is obtained that gives a conductivity of 1.55 31028 S/m at 500 K ~227 °C! with an activation energy of 0.79 eV and a preexponential coefficient s051.44 S/m. Such conductivity is a common feature of perovskite crystals and ceramics.18 For instance, a SrTiO3 ~also perovskite! single crystal measured by Schmidt et al. showed conductivity at 500 K and 100 Hz of 4.1531027 S/m and comparable acti- vation energy, with a weak frequency dependence.18 The conductivity plot for PMN-42%PT at 1 kHz showed curva- ture because of the residual transition-associated loss, but at the highest temperature the curve approached the slope of the 100 Hz data, with conductivity 10% higher than at 100 Hz. From this result, it can be concluded that the frequency de- pendence of the conductivity is no stronger than f 0.05. Polarization hysteresis loops as a function of electric field, and strain versus electric field ~bipolar! behavior for the @001#-oriented PMN-42%PT crystal are plotted in Figs. 3~a! and 3~b!, respectively. The value of remnant polarization P r is ;42 mC/cm2 and the coercive electric field Ec is 5.3 kV/cm when the maximum applied electric field is 13 kV/ cm. Upon increasing the applied field the loop becomes ‘‘fat- ter’’ and Ec goes from 5.85 kV/cm for maximum field 16 kV/cm to 6.85 kV/cm for maximum field 20 kV/cm. How- ever, the remnant polarization has almost no change in this range of applied fields. As @001# is the polar direction for PMN-42%PT, complete poling results in a single domain state, and piezoelectric strain behavior might be expected to be hysteresis free for @001# poled PMN-42%PT crystals. However, Fig. 3~a! shows that the crystal leaves the single- domain state even before the field falls to zero, so some piezoelectric hysteresis is expected even for small fields. For large fields, as shown in Fig. 3~b!, the significant hysteresis together with high strain value ~0.08% at 13 kV/cm! indi- cates considerable domain reorientation under bias. Similar to the polarization loops, the strain loops become ‘‘fatter’’ with increasing maximum field. A complete set of elastic, piezoelectric, and dielectric constants of PMN-42%PT crystals is listed in Table I. Mate- rial constants marked with ‘‘*’’ were determined directly by measurement and the others are indirectly derived quantities. The relevant constants of PMN-30%PT and PMN-33%PT are also listed in Table I for comparison. It is seen that the difference in elastic stiffness constants is very small between PMN-30%PT and PMN-33%PT. The same behavior was ob- served in PZN-4.5%PT and PZN-8%PT crystals.10 However, the difference is obvious compared to the PMN-42%PT crys- tals. The PMN-30%PT and PMN-33%PT crystals have rhombohedral phases at room temperature, while the PMN- 42%PT has a tetragonal phase. Therefore, we can conclude that the structural difference has a significant influence on the elastic stiffness constants in the PMN-PT and PZN-PT systems. Similar phenomena were observed in the elastic compliance constants si j D . The relationships between the measured phase velocities and related elastic constants were derived from the Christof- fel wave equations, and these relationships and measured parameters are listed in Table II. For a crystal with 4 mm symmetry, sound velocities can be directly measured from eight independent pure modes. From each measurement, ei- ther one elastic constant or a linear combination of several elastic constants can be obtained as shown in Table II. It is seen that c44 D and c12 E can be determined by more than one measurement, which provides a control check. The differ- ence between the measured phase velocity vsi @100# in the sec- FIG. 3. Polarization ~a! and strain ~b! vs E-field ~bipolar! curves for PMN- 0.42PT crystals oriented along @001#. The maximum electric field is 13, 16, and 20 kV/cm, respectively. 551J. Appl. Phys., Vol. 96, No. 1, 1 July 2004 Cao et al. Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. IP: 153.90.170.56 On: Fri, 26 Feb 2016 19:47:52 ond cut and vsi @110# in the sixth cut in Fig. 1 is very small and the corresponding elastic stiffness constants c44 D are nearly equivalent, 7.9331010 and 8.1331010 N/m2, respectively. According to the relationship in Table II, c12 E can be sepa- rately determined by two equations and the corresponding calculated values of c12 E are 8.431010 and 8.831010 N/m2. The elastic stiffness constants c44 D and c44 E are calculated from the measured phase velocities. The shear coupling coefficient 0.80 of k15 can be calculated by k15 2 512c44 E /c44 D . In order to test the consistency, the frequency constants 1987 m/s of 2t f r ~t is the thickness of sample and f the resonance fre- quency of sample! and 3901 m/s of 2t f a in the thickness shear mode were measured and 0.798 of k15 can be calcu- lated by k15 2 5p f r/2f a tan$p(fa2fr)/2f a%. The elastic stiff- ness constants c44 D calculated from the ultrasonic method (c44 D ;8.1331010 N/m2) and from the impedance (c44 D ;7.78 31010 N/m2) method differ slightly. Similarly, c33 D 516.8 31010 N/m2 from the ultrasonic measurement and 17.1 31010 N/m2 from the resonance measurement. Also, a reso- nance bar with its long dimension in @110# and thickness in @001# was made to verify the consistency. From the measured frequency constants 4182 m/s of 2l f r ~l is the length of sample! and 4428 m/s of 2l f a in length-extension mode along @110# k31^110& of 0.365 can be calculated by (k31 2 21)/k31 2 5tan$(p/2)( f a / f r)%/(p/2)( f a / f r) and the corre- sponding elastic compliance constant s11^110& E is 7.06 310212 m2/N. We have easily gotten s11 E , s66 E , c11 E , c12 E , and c66 E . There are two sets of data to calculate s12 E . One is s11^110& E 5(s11 E 1s12 E )/21s66 E /4 and the other is s12 E 5s11 E 21/(c11 E 2c12 E ). Here we test the consistency by the combi- nation of ultrasonic and impedance techniques again. First, 21.56310212 m2/N for s12 E can be calculated from the elas- tic compliance constants in the resonance measurement. However, 21.67310212 m2/N of s12 E can be calculated from the stiffness constants in the ultrasonic method. The differ- ence of s12 E separately obtained by the two methods is small. Therefore, the combination of ultrasonic and impedance techniques provides an effective way to measure the material property constants. For the PMN-PT and PZN-PT systems the piezoelectric properties of rhombohedral crystals are much higher than those of tetragonal crystals. It is known that the piezoelectric constant d33 of PZN-PT and PMN-PT systems increases non- linearly and very drastically near the MPB. When the PT content in crystals is much higher than that near the MPB, the value of d33 is smaller. For example, the piezoelectric constants d33 ~measured for small fields! of PMN-33%PT and PMN-30%PT are much higher than 260310212 C/N of PMN-42%PT. This significant difference can be attributed to the engineered-domain configuration. For rhombohedral crystals the spontaneous polarization is along ^111&. There are eight possible polarization orientations for ^111& do- mains. When poling is done by applying an electric field along the @001# direction only four of the eight possible po- larization orientations remain, i.e., @111#, @ 1̄11# , @11̄1# , and @ 1̄1̄1# . Therefore, the components of all four polarization vectors along @001# are equal, so that the domain walls have no incentive to move under an external electric field along TABLE I. Measured and Derived Material Properties of PMN-42%PT Single Crystal Poled Along @001#. Elastic stiffness constants: ci j (1010 N/m2) PT ~%! c11 E* c12 E c13 E c33 E c44 E* c66 E* c11 D c12 D c13 D c33 D* c44 D* c66 D 42 17.51 8.51 8.3 10.5 2.85 8.0 17.7 8.7 7.18 16.99 8.05 8.0 33 11.5 10.3 10.2 10.3 6.9 6.6 11.7 10.5 9.0 17.4 7.7 6.6 30 11.7 10.3 10.1 10.8 7.1 6.6 11.8 10.4 9.5 17.4 7.8 6.6 Elastic compliance constants: si j (10212 m2/N) PT ~%! s11 E* s12 E s13 E s33 E s44 E s66 E s11 D s12 D s13 D s33 D* s44 D s66 D 42 9.43 21.68 26.13 19.21 35.09 12.5 8.02 23.10 22.08 7.64 12.42 12.5 33 69.0 211.1 255.7 119.6 14.5 15.2 44 234 24.1 11.1 13.0 15.2 30 52.0 218.9 231.1 67.7 14.0 15.2 39.7 231.2 24.7 10.8 12.9 15.2 Piezoelectric constants: ei j (C/m2) di j (10212 C/N) gi j (1023 Vm/N) hi j (108 V/m) PT ~%! e15 e31 e33 d15 d31 d33* g15 g31 g33 h15 h31 h33 42 37.50 22.1* 12.2 131 291 260 17.23 215.57 44.5 13.87 29.16 53.22 33 10.1 23.9 20.3 146 21330 2820 10.3 218.4 38.8 7.9 25.9 33.7 30 13.6 22.4 27.1 190 2921 1981 6.0 213.3 28.7 4.6 22.2 24.6 Dielectric constants: e(e0), b(1024/e0) electromechanical coupling constants PT ~%! e11 S e33 S e11 T* e33 T* b11 S b33 S b11 p b33 T k15* k31* k33* kt* 42 3054 259 8627 660 3.27 38.61 1.16 15.15 0.8 0.39 0.78 0.62 33 1434 680 1600 8200 7.0 14.7 6.3 1.2 0.32 0.59 0.94 0.64 30 3307 1242 3600 7800 3.0 8.0 2.8 1.3 0.29 0.49 0.92 0.62 ~a!Measured properties. ~b!Density: r58.103103 kg/m3 ~PMN-42%PT!, r58.063103 kg/m3 ~PMN-33%PT!,7 r58.043103 kg/m3 ~PMN-30%PT!.8 TABLE II. The relationships between measured phase velocities of ultrasonic waves and elastic constants in a PMN-42%PT crystal poled along @001#. v v l @001# vs @001# v l @100# vs' @100# vsi @100# v l @110# vs' @110# vsi @110# Velocity ~m/s! 4551 1830 4653 3138 3129 5110 2380 3168 rv25 c33 D c44 E c11 E c66 E c44 D (c11 E 1c12 E 12c66 E )/2 (c11 E 2c12 E )/2 c44 D 552 J. Appl. Phys., Vol. 96, No. 1, 1 July 2004 Cao et al. Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. IP: 153.90.170.56 On: Fri, 26 Feb 2016 19:47:52 @001#. However, the domains can change shape by both shearing and stretching, causing the piezoelectric constant to exhibit a high value. In contrast, the polarization is mostly along @001# in a @001# poled tetragonal crystal and the field can only cause stretching of the @001# domain, but not shear- ing, so the piezoelectric constant is lower. According to the electromechanical coupling coefficients listed in Table II, k33 of rhombohedral crystals is much larger than that of tetragonal crystals. However, kt is almost equal for rhombohedral and tetragonal crystals. The orientation de- pendence of the phase velocities and electromechanical cou- pling coefficients of PMN-42%PT single crystal based on the measured material constants in Table I is plotted in Figs. 4 and 5. Figure 4 provides the directional dependence of phase velocities for sound waves propagating in the ~a! X-Y, ~b! Y-Z, and ~c! @110#-Z planes. The calculated results reveal that the velocities of the longitudinal waves do not change with orientation as much as those of the shear waves. As FIG. 4. Orientational dependence of the longitudinal velocity v l and the two shear velocities vs1 and vs2 in ~a! the @100#-@010# plane, ~b! the @100#-@001# plane, and ~c! the @110#-@001# plane. FIG. 5. Orientational dependence of the coupling coefficients k15 , k33 , kt , and k31 , in ~a! the @100#-@010# plane, ~b! the @100#-@001# plane, and ~c! the @110#-@001# plane. 553J. Appl. Phys., Vol. 96, No. 1, 1 July 2004 Cao et al. Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. IP: 153.90.170.56 On: Fri, 26 Feb 2016 19:47:52 shown in Fig. 4~a!, the shear wave propagation in the X-Y plane and polarized in the same plane has the strong orien- tational dependence. It has a maximum in @100# and a mini- mum in @110#, respectively. Figure 5 presents the orienta- tional dependence of the electromechanical coupling coefficients k15 , k31 , k33 , and kt . In the X-Y plane the coefficients k15 , k33 , and kt are isotropic and in the X-Z plane k15 , k33 , and kt reach maxima in @001#. When poling is done along @001# the switching of 90° domains causes the lattice elastic energy to increase, however, resulting in a slight depoling after removing the applied field. Statistically, the remaining domains along @100# or @010# have an equal probability, which means the tetragonal PMN-42% crystal is isotropic in the X-Y plane. Therefore, the electromechanical coefficients exhibit isotropy and the situation in the @110#- @001# plane is very similar to that in the @010#-@001# plane as shown in Figs. 5~b! and 5~c!. However, k31 has a slight change with orientation, which is possibly caused by the in- complete single domain state. IV. SUMMARY AND CONCLUSION The material properties of a 0.58Pb~Mg1/3Nb2/3)O3- 0.42PbTiO3 single crystal poled in the @100# direction of original cubic axes were measured using a combined method of ultrasonic and resonance techniques. A complete set of elastic, piezoelectric, and dielectric constants for tetragonal PMN-42%PT was obtained. It was confirmed that the tetrag- onal PMN-PT system has lower piezoelectric constants com- pared to the rhombohedral PMN-PT system. The piezoelec- tric constant d33 is 260310212 C/N and the electro- mechanical coefficient k33 is 0.78. Based on the measured material constants, orientational dependence of phase veloci- ties of ultrasonic waves propagating in the X-Y , Y -Z , and @110#-Z planes and of the electromechanical coupling coef- ficients has been analyzed. It was observed that the anisot- ropy of phase velocities is strong for the shear wave, whereas it is relatively weak for the longitudinal wave. For the elec- tromechanical coupling coefficients the crystal exhibits isot- ropy in the X-Y plane and k15 , k33 , and kt reach their maxima at @001#. However, because of the incomplete single domain status k31 has a slight variation with orientation. ACKNOWLEDGMENTS This research was sponsored by the Department of De- fense under Grant No. N00014-02-1-0657. 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