Characterization of lead zirconate titanate piezoceramic using high frequency ultrasonic spectroscopy

Transcription

1 JOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 1 15 JUNE 1999 Characterization of lea zirconate titanate piezoceramic using high frequency ultrasonic spectroscopy Haifeng Wang, Wenhua Jiang, a) an Wenwu Cao b) Materials Research Laboratory, National Resource Center for Meical Ultrasonic Transucer Engineering, The Pennsylvania State University, University Park, Pennsylvania 1680 Receive 4 November 1998; accepte for publication 9 March 1999 Dope piezoceramic lea zircanate titanate has been characterize in the frequency range of 0 60 MHz using ultrasonic spectroscopy. Theoretical analyses were performe for the reflection an refraction of acoustic waves at the interface of water-piezoelectric ceramic. The incient irections of the wave were chosen to be appropriate for ultrasonic spectroscopy measurements. Shear wave spectrum was obtaine through moe conversion using a pair of longituinal transucers submerge in water. The phase velocity shows linear epenence on frequency while the attenuation may be escribe by a secon orer polynomial of frequency in the frequency range investigate. The Kramers Kronig relation between ultrasonic phase velocity an attenuation was compare to measure results American Institute of Physics. S I. INTRODUCTION Application of higher frequency broaban ultrasonic transucer results in the improvement of the axial an lateral resolutions in meical imaging. Design high frequency transucer requires better knowlege of material properties since the ultrasonic ispersion becomes important for frequencies above 50 MHz. The ispersions of velocity an attenuation may eform the acoustic pulse an cause inappropriate interpretation of the pulse acoustic signal. Hence, knowing the properties of the transucer materials at high frequencies is important for esigning high frequency transucers. Currently, there are very limite experimental ata available in the literature on high frequency properties of piezoelectric materials ue to many technical ifficulties. 1 Reporte in this article are results of velocity an attenuation for ope lea zirconate titanate PZT-5H in the frequency range of 0 60 MHz measure by using an ultrasonic spectroscopy metho. The experimental results were also use to verify the Kramers Kronig relations. The ultrasonic spectroscopy metho has been wiely use in the characterization of soli materials. With angular incience, moe conversion effect was use to investigate shear wave properties of porous materials an polymeric materials. 3 8 If such a metho is generalize to characterize piezoelectric materials, one nees to eal with the problem of plane wave propagation through an interface between an isotropic meium an an anisotropic piezoelectric material. Since the propagation velocity of waves varies with propagation irection in an anisotropic material, the refraction becomes complicate when the ultrasonic wave is obliquely incient onto the interface. Generally speaking, the refraction coefficient cannot be given in an explicit analytic form. For pole PZT ceramics the symmetry is m, which has the a Present aress: Institute of Acoustics an State Key Laboratory of Moern Acoustics, Nanjing University, Nanjing 10093, People s Republic of China. b Electronic mail: cao@math.psu.eu same number of inepenent physical constants as that of 6 mm symmetry, i.e., five inepenent elastic constants, three inepenent piezoelectric coefficients, an two inepenent ielectric permittivities. Because of the reasonably high symmetry, the situation may be simplifie in some special incient angles. The present article is ivie into two parts: first, we iscuss the refraction of a plane wave at the interface of water-pzt ceramic. Base on the theoretical analysis, an ultrasonic spectroscopy technique suitable for characterization of PZT ceramic was evelope an use to characterize PZT-5H samples. Secon, the measure results were compare with those preicte by the Kramers Kronig relations. II. THE REFLECTION AND REFRACTION OF A LONGITUDINAL WAVE AT THE WATER- PIEZOCERAMIC INTERFACE Base on the principle escribe by Rokhlin, 8 we have erive the reflection/refraction of acoustic wave at the interface of water an a PZT ceramic plate. The poling irection of a plate PZT sample is either perpenicular or parallel to the large surface of the plate. As shown in Fig. 1, the plate sample with the poling irection perpenicular to its large surface is referre as PZT z, whereas the plate with poling irection parallel to its large surface is referre as PZT x. A. Reflection/refraction of a longituinal wave at the interface of water an PZT z When a longituinal wave is incient upon the interface of water an PZT z, we choose the coorinate system to make the incient plane coincient with either the x-z or the y-z plane of PZT since the plane perpenicular to the z axis is acoustically isotropic for PZT with m symmetry. If the x-z plane is a wave propagation plane as shown in Fig. 1a, the incient longituinal wave in water can be expresse as U i U 0i exp jtk w x 1 sin i k w x 3 cos i i1,, /99/85(1)/8083/9/$ American Institute of Physics

2 8084 J. Appl. Phys., Vol. 85, No. 1, 15 June 1999 Wang, Jiang, an Cao 1 c 11m 1 c 44 m 3 1u 01 1 c 13c 44 m 1 m 3 u 03 1 e 31e 15 m 1 m 3 u 04 0, 3a 1 1 c 11c 1 m 1 c 44 m 3 1 u 0 0, 1 c 13c 44 m 1 m 3 u 01 1 e 15m 1 e 33 m 3 u 04 0, 1 c 44m 1 c 33 m 3 1u 03 3b 3c e 15 e 31 m 1 m 3 u 01 e 15 m 1 e 33 m 3 u 03 FIG. 1. a Incience of a wave in the x-z plane of a PZT sample. b Incience of a wave in the x-y plane of a PZT sample. or U i U 0i exp jtm 1 w x 1 m 3 w x 3, 1 where k w /, is soun velocity of water, i is incient angle m w 1 n 1 sin i an m w v 3 n 3 cos i. w The unit vector n represents the wave propagation irection. Similarly, the refractive waves propagating in the PZT sample are expresse as u i u 0i exp jtkn 1 x 1 kn 3 x 3 i1,,3,4, where u 4 stans for electric potential wave with the electric fiel given by E i u 4 /x i. For convention we rewrite Eq. as u i u 0i exp jtm 1 x 1 m 3 x 3 with m i n i /v i1 or3. Substituting the wave solutions into the equation of motion ü i T ij /x j an consiering the constitutive equation T ij c ijkl u k /x l e mij E m, the amplitue u 0i (i14) in Eq. will be governe by the following equations: 11 m 1 33 m 3 u 04 0, 3 where is the ensity, c ij are the elastic stiffness constants, e i are the piezoelectric stress constants, an ij are the ielectric constants for the pole PZT ceramic. Short notation for the elastic constants, c, an the piezoelectric coefficients, ẽ, have been use; their efinition coul be foun in Ref. 9. From Eq. 3 it is seen that u 0 representing the amplitue of a shear wave oes not couple with u 01 an u 03. Hence, this wave cannot be generate through moe conversion effect in this incience arrangement. In other wors, the particle isplacement of the refractive waves in the PZT ceramic has only u 1 an u 3 components. From Eq. 3 u 04 e 15e 31 m 1 m 3 u 01 e 15 m 1 e 33 m 3 u m m 3 Substituting Eq. 4 into Eqs. 3a an 3c gives 1u 01 1 c 11 m 1 c 44 m 3 e 31e 15 m 1 m 3 11 m 1 33 m 3 1 c 13 c 44 e 33m 3 e 15 m 1 e 31 e m 1 33 m 3 m 1 m 3 u 03 0, 5a 1 c 13 c 44 e 33m 3 e 15 m 1 e 31 e m m 1 33 m 1 m 3 u c 44 m 1 c 33 m 3 e 15m 1 e 33 m 3 11 m 1 33 m 3 1u b The propagation irections of the refractive waves allowe in the PZT ceramic are etermine by the coefficient eterminate of Eq. 5, i.e., 1 c 11 m 1 c 44 m 3 e 31e 15 m 1 m m 1 33 m 3 1 c 13 c 44 e 33m 3 e 15 m 1 e 31 e m 1 33 m 3 m 1 m 3 1 c 13 c 44 e 33m 3 e 15 m 1 e 31 e m 1 33 m 3 m 1 m 3 c 44 m 1 c 33 m 3 e 33m 3 e 15 m 1 11 m 1 33 m

3 J. Appl. Phys., Vol. 85, No. 1, 15 June 1999 Wang, Jiang, an Cao 8085 It is known that the phase matching conition at the interface, or the Snell s law, emans m incient reflective 1 m 1 m refractive 1,orm w 1 m PZT 1. Since m w 1 is known for a given incient wave, one can etermine m PZT 3 of refractive waves propagating in the PZT from Eq. 6. There are two solutions of m PZT 3 from Eq. 6 corresponing to two refracte waves in PZT. Substituting the two roots of Eq. 6 back into Eq. 5, two eigenvectors with two components can be obtaine, which provies the polarization irections of the two refracte waves. In more general cases, one of them is a quasilongituinal wave an the other is a quasishear wave. Thus, when a longituinal wave is incient from water upon PZT ceramic, there are incient an refracte waves in the water meium; both are longituinal waves with amplitues given by u 0 an u r, respectively. In the PZT ceramic there are refracte quasilongituinal an quasishear waves. Their propagation irection an polarization irection are etermine by Eqs. 5 an 6 to be m L, I L an m S, I S, respectively. Their amplitues u L an u S are relate to that of the incient wave u 0 through the bounary conitions at the interface of x 3 0 u 3 wateru 3 PZT, 7a T 13 0, T 33 watert 33 PZT. The above equations may be rewritten as u r cos l 3 L u L l 3 S u S u 0 cos, ru L su S 0, Z w u r pu L qu S Z w u 0, where Z w is the acoustic impeance of water an rc 44 m 3 L l 1 L m 1 l 3 L e 15 A L l 1 L B L l 3 L, sc 44 m 3 S l 1 S m 1 l 3 S e 15 A S l 1 S B S l 3 S, pc 13 m 1 l 1 L c 33 m 3 L l 1 L e 33 m 3 L A L l 1 L B L l 3 L, 7b 7c 8a 8b 8c 9a 9b 9c qc 13 m 1 l S 1 c 33 m S 3 l S 1 e 33 m S 3 A S l S 1 B S l S 3, 9 with A L,S e L,S 15e 31 m 1 m 3 11 m L,S L,S, B L,S e L,S 15m 1 e 33 m m 3 11 m L,S L,S m 3 From Eq. 8 the ratios of u r /u 0, u L /u 0, an u S /u 0, i.e. the reflection an refraction coefficients, can be etermine. Using this proceure, one can etermine the amplitue an propagation irection of the refractive waves. In ultrasonic spectroscopy technique, however, one can use slightly ifferent ways to solve the problem. As mentione above, m w 1 is known for a given incient wave, i.e., m w 1 n 1 sin i. From Snell s law it is given that m PZT 1 sin i / sin p /v p, where p an v p pl or S are the refractive angle an velocity of longituinal L or shear S wave. If the velocity v p has been etermine by spectroscopy metho, one can simply calculate p through Snell s law. Obviously, m PZT PZT 3 can be foun by the relation of m 3 cos p /v p. Using Eq. 6 the longituinal an shear velocities can be correlate to appropriate elastic constants. In what follows we iscuss three special cases to illustrate the proceure. Case 1. Normal incience In this case m water 1 0, therefore, m PZT 1 0, p 0 an or m 3 S 1 v S m 3 L 1 v L. 10a 10b Substituting these results into Eq. 5 results in the following simple relations: v S c 44 an v L c D 33 c 33 e c, which are the velocities of the shear wave an the stiffene longituinal wave, respectively. From Eq. 6, two eigenvectors are obtaine l S 1,0,0 an l L 0,0,1. 11a,b This means that one of the possible refracte waves is a pure shear wave with polarization irection along the x axis an another is a pure longituinal wave. Substituting these results into Eqs. 8 an 9, one can obtain the reflection an refraction ratios R u r u 0 Z LZ w Z L Z w, T u L Z L, u 0 Z L Z w u s 0. 1a 1b 1c Equation 1c implies that moe conversion oes not exist in normal incience, an reflection coefficient R an transmission coefficient T of the longituinal wave are the same as for an interface of water an an isotropic soli. In the expressions Eqs. 1a,b, Z L is the acoustic impeance of the longituinal wave in PZT. Case. Incience at the critical angle of the longituinal wave. When a longituinal wave is obliquely incient from water upon PZT z, moe conversion takes place. In PZT meium there are, in general, refractive quasilongituinal an quasishear waves. If the wave is incient at the critical angle of longituinal wave, the refractive longituinal wave becomes an evanescent wave. This means that m 3 L is equal to zero ( i critical angle) or complex ( i critical angle). The vector m S representing the propagation irection of the refractive quasishear wave can be etermine by the following expressions:

4 8086 J. Appl. Phys., Vol. 85, No. 1, 15 June 1999 Wang, Jiang, an Cao m S 1 sin i sin S, 13a v S m S 3 cos S. 13b v S If the velocity v S is etermine from the ultrasonic spectroscopy metho, S can be calculate from Eq. 13a, therefore m S is totally etermine. Knowing m S, the polarization irection can be etermine by Eq. 5 an the transmission coefficient can be calculate from Eq. 8. The velocity of the quasishear wave is relate to a combination of elastic constants of PZT by where an v S c*, c*c 11 sin S c 33 cos S c 44 p 1 p c 11 c 44 sin S c 44 c 33 cos S p 1 p c 13 c 44 p 3 sin S 1/ 14 15a p 1 e 31e 15 sin S cos S 11 sin S 33 cos S, 15b p e 33 cos S e 15 sin S 11 sin S 33 cos S, 15c p 3 e 33 cos S e 15 sin S e 31 e sin S 33 cos, 15 S are terms associate with piezoelectric coupling. S is the refractive angle of the quasishear wave in PZT ceramic. B. Reflection/refraction of a longituinal wave at the interface of water an PZT x Assuming the plate normal irection is along the x axis an the poling irection is parallel to the z irection of the PZT plate as shown in Fig. 1b, the incient plane of the ultrasonic wave has three inepenent orientations: 1 x-y plane; y-z plane, an 3 incient in a plane that rotates aroun the y axis at an arbitrary angle. Since the last case makes the problem more complicate, it will not be iscusse here. The secon option is actually equivalent to the Case of Sec. II A, therefore we only nee to iscuss the first option which we call Case 3, as escribe below. The plane perpenicular to the z axis of PZT is an acoustically isotropic plane as mentione above. When an incient wave is in the x-y plane, as shown in Fig. 1b, the reflection/refraction of a longituinal wave at the interface of water an PZT ceramic is the same as at the interface of water an an isotropic soli. In this case, the particle isplacement u of the refractive waves has only components in the x an y irections. When a wave is normally incient, the only refractive wave is a pure longituinal wave with the velocity given by v L c a When a wave is obliquely incient, one of the two refractive waves is a pure longituinal wave with the same velocity as Eq. 16a, another refractive wave is a pure shear wave with the velocity of v S 1/c 11c 1 c b This shear wave polarizes in the x-y plane. Other cases of reflection/refraction of a plane wave at the interface of PZT ceramic water can be iscusse by using the same proceure. However, pure moes o not exist for general cases. In summary, by using angular incience of a longituinal wave from water to two PZT plates we can etermine the elastic constants an their ispersion from ispersion of velocity an attenuation in the following acoustic moes: A. For a PZT z sample: 1 Longituinal wave propagating along the poling irection with the wave normal incient upon the plate. From the velocity ispersion of this wave, the frequency epenence of elastic constant c D 33 can be etermine. Quasishear wave propagating in the x-z plane with the wave obliquely incient at the critical angle of the longituinal wave. From its velocity ispersion one can etermine an elastic constant combination c* an its frequency epenence. B. For a PZT x sample: 1 Longituinal wave propagating perpenicular to the poling irection. From its velocity ispersion one can etermine the elastic constant c 11 an its frequency epenence. Shear wave with irections of propagation an polarization normal to the poling irection when the wave is incient at the critical angle of the longituinal wave. From the velocity ispersion of this wave one can etermine the elastic constant c 66 an its frequency epenence. III. PRINCIPLES OF MEASUREMENTS The basic principle of the ultrasonic spectroscopy metho use to etermine the velocity an attenuation of materials is shown in Fig.. A pair of aligne transmitting an receiving transucers are immerse in water with an ajustable separation between them. When the transmitting transucer is riven by an electric pulse signal, an acoustic broaban signal is prouce at x0, note as u(t). Its Fourier transform is ũ( f ). For a linear an causal acoustic system shown in Fig. a, the transfer function of the meium for the wave to propagate can be expresse as H f exp w f Lexp j fl/ f, 17 where L is the istance between the transmitting an receiving transucers, an w are the wave velocity an attenuation of water, respectively. When the meium is consiere

5 J. Appl. Phys., Vol. 85, No. 1, 15 June 1999 Wang, Jiang, an Cao where is the thickness of the sample, an v are attenuation coefficient an phase velocity in the sample, respectively, i an are the incient an the refractive angles of the wave at the interface, respectively, T is the total transmission coefficient which is equal to the prouct of the transmission coefficients of the wave from water to sample an from sample to water. Thus, the amplitue an phase spectra of the output signal for the system shown in Fig. b can be written as FIG.. Principle of ultrasonic spectroscopy technique: a without sample, b with sample place in a rotate position. to be ispersive an issipative, both w an are frequency epenent. The spectrum of the output signal from the receiving transucer is ũ w f ũ f exp w f L exp j fl/ f ũf, 18 where ũ( f ) is the transfer function of the receiving transucer. Thus, the amplitue an phase spectra of the output signal can be expresse as A w f ũ f exp w f Lũ f, 19a w f fl/ f u u, 19b where u an u are the phase angles of the two transucers, respectively. They will not enter the calculations below since the spectroscopic metho epens only on the relative phase shift. When the sample to be measure is inserte between the transmitting an receiving transucers, the transfer function of the system as shown in Fig. b becomes H S f Tũf exp w f L tgtg cos i i sin i exp f cos exp j f L sin i f f tgtg cos i i cos v f ũ f, A f Tũ f exp w f L tgtg cos i i an sin i exp f fl f, cos ũ cos i tgtg i sin i f 1a f vf cos u u. 1b From Eqs. 19 an 1 the phase velocity an attenuation coefficients in the sample are given by v sin i w f cos i, w cos i ln TA w A cos /. 3 If the values of A w, A, w, an can be measure, the attenuation an phase velocity ispersion of the sample can be etermine. When the wave is normally incient to the sample, i 0, the above equations give the velocity an attenuation of the longituinal wave in the case iscusse in Sec. II for the longituinal wave propagating in either x-z plane or x-y plane as shown in Figs. 1a an 1b, respectively an v L 1 w f 4 L w ln T LA w A, 5 where T L 4 0 v L 0 v L 4z wz L z L z w 6 an 0 an are the mass ensity of water an sample, respectively, an v L is the longituinal wave velocity in the sample. If the wave is obliquely incient at the critical angle of the longituinal wave shear wave will be generate through moe convention effect. The velocity an attenuation of the shear wave can be calculate by the following expressions:

6 8088 J. Appl. Phys., Vol. 85, No. 1, 15 June 1999 Wang, Jiang, an Cao FIG. 4. The variation of amplitue spectrum with propagation istance in water. v S sin i w f cos i, 7 S w cos S i ln T SA w A cos S, 8 where T S an S are transmission coefficient an refractive angle of the shear wave. The transmission coefficients for flui-isotropic soli interface are simply given by T 1 0 z Sn sin S z Ln cos S z Sn sin, 9a S z wn T tan i sin S FIG. 3. Experiment setup. 1 z Ln cos S z Sn sin S z wn z Ln cos S z Sn sin S z wn, 9b T S T 1 T, 9c where T 1 an T 1 stan for the transmission coefficients of a wave from water to PZT an from PZT to water, respectively; z wn z w /cos i, z Ln z L /cos L, z Sn z S /cos S, an v S is the shear wave velocity in the sample. The refractive angles L an S are calculate from Snell s law sin i sin L sin S, 30 v L v S where the incient angle i is controlle by a computerize rotating table in our experiments. For the quasishear wave propagating in the x-z plane of a PZT ceramic. The transmission coefficient can be calculate from Eq. 8. IV. EXPERIMENT RESULTS FOR PZT-5H AND DISCUSSIONS The experimental setup is shown in Fig. 3. A pair of transucers with a center frequency of 50 MHz an banwith of 80% were use. Without sample, the spectra of the output signal from the receiving transucer are shown in Fig. 4. Since the ultrasonic attenuation of water is about 6 B/cm at 50 MHz the high frequency components of the Fourier spectrum are ecaye when the istance between the transmitter an receiver increases, as inicate in Fig. 4. The interval between transmitter an receiver shoul be selecte in such a way that the high frequency components are preserve as much as possible, an at the same time leaving enough room for the sample to be rotate. In our experiments, the istance L is set at about 3 cm. The output waveform was sample by a igital oscilloscope Tektronix TDS 460A at a sampling rate of 10 Gs/s. The ata were transferre into computer via a general purpose interface bus GPIB interface. The total recoring length for a waveform was 500 points. The amplitue A w an the phase spectra w were obtaine through fast Fourier transform FFT of the output signal from the configuration in Fig. a. When the sample was put in place, the trigger elay time was ajuste so that the shifting of the waveform cause by putting in the sample can be compensate. Using the same proceure, the amplitue A an the phase of the output signal from the configuration Fig. b were obtaine. Note: the velocity calculation must take into account the trigger time elay, v L 1, 31a w f f v S sin i w f f cos i. 31b The attenuation of the longituinal an shear waves can be calculate from Eqs. 3 an 5, respectively. Here the attenuation of water was given by l f B/mm, f in MHz. The measure velocity an attenuation for piezoceramic PZT-5H are given in Figs It was observe that velocity ispersion exists for both the longituinal an shear waves in the frequency range of the measurement an it is nearly linear, but the attenuation exhibits nonlinear frequency epenence. The attenuation of the shear wave is an orer of magnitue higher than that of the longituinal wave.

7 J. Appl. Phys., Vol. 85, No. 1, 15 June 1999 Wang, Jiang, an Cao 8089 FIG. 5. Dispersion relationship of the velocity an attenuation for water. Usually, the wave number of a ecay wave is consiere to be a complex number. Therefore, if the frequency is taken as real, the velocity becomes complex an is associate with the complex elastic moulus in the following fashion: ṽ C jc C. 3 FIG. 7. The phase velocity ispersion measure by experiment an erive by Kramers Kronig relationship for wave propagating in the x-y plane of a PZT sample an the corresponing elastic constants. The longituinal an shear velocities were fitte to linear curves: v L e 6*f (Hz) an v S e7*f (Hz). In general, it is true that C/C1, we may write the above equation as ṽ C 1 j C C. The complex wave number can be expresse as 33 FIG. 6. The phase velocity ispersion measure by experiment an erive by Kramers Kronig relationship for a wave propagating in the x-z plane of a PZT sample an the corresponing elastic constants. FIG. 8. The attenuation measure by experiment an erive by Kramers Kronig relationship for a wave propagating in the x-z plane of a PZT sample. The longituinal an shear velocities were fitte to linear curves: v L e6*f (Hz) an v S e7*f (Hz).

8 8090 J. Appl. Phys., Vol. 85, No. 1, 15 June 1999 Wang, Jiang, an Cao k k j, 34 where k/v an is the attenuation coefficient. Thus, the real an imaginary parts of the elastic moulus can be erive from the phase velocity an attenuation measurements Cv, 35a tan C v C. 35b If a meium in which an acoustic wave propagates can be consiere as a linear an causal system, its attenuation, which is associate with the imaginary part of the elastic moulus, an velocity ispersion, which is associate with the real part of elastic moulus, are relate by the Kramers Kronig relations. The approximation forms of the nearly local relationships can be expresse as 10,11 vv 0 v 0 0, 36a v 0 v, 36b where 0 is the starting frequency at which v( 0 ) an ( 0 ) are known. In our experiment, 0 *0 MHz. The ultrasonic spectroscopy technique is inherently base on the assumption of linearity. Thus, the velocity ispersion an attenuation obtaine from the technique are expecte to satisfy the above Kramers Kronig relations. To verify the valiity of these relations, the measure attenuation for the PZT-5H samples was fitte as a polynomial of frequency an the phase velocity ispersion was erive by the first Kramers Kronig relation Eq. 36a, then these calculate results were compare to measure results. As shown in Figs. 6 an 7 that the agreement is quite acceptable. The maximum eviation is less than 0.6% an the tren is correct. We have also performe the reverse checking, i.e., using the measure velocity ispersion to calculate the ispersion of the attenuation base on the local approximation Eq. 36b. We foun that this relation is extremely sensitive to the curvature of the velocity ispersion. Higher orer polynomial fitting of the velocity curve gives unreasonable results. Because the velocity ispersion is fairly small, we ecie to use a linear approximation to these velocity ata, which create error less than the experimental uncertainty for the velocity fitting but gave goo agreement to the measure attenuation ispersion. The results are shown in Figs. 8 an 9 for the two ifferent cut PZT samples. Owing to this sensitivity, one shoul be really careful when using the secon Kramers Kronig relation Eq. 36b. One may have problems using this relation if the velocity ispersion has a ownwar curvature, such as the shear velocity measure in this work, since it will lea to a ecrease of the attenuation at higher frequencies. The measure attenuation seems to always increase with frequency. Our experience is that a linear fitting of the velocity ispersion coul provie a much better preiction of the attenuation ispersion using the Kramers FIG. 9. The attenuation measure by experiment an erive by Kramers Kronig relationship for a wave propagating in the x-y plane of a PZT sample. Kronig relations. This linear fitting, of course, must be piece wise since the relationship is not linear in general for all frequency ranges. V. CONCLUSION The ispersions of velocity an attenuation for piezoceramic PZT-5H were investigate by using ultrasonic spectroscopy at the frequency range of 0 60 MHz. In the investigate frequency range, velocity ispersion of 1 3 m/s per MHz was observe. The attenuation epens nonlinearly on frequency an the shear wave exhibite an orer of magnitue larger attenuation than the longituinal wave. We showe that the Kramers Kronig relations between velocity ispersion an attenuation ispersion are satisfie for the longituinal waves. However, for the shear waves, the agreement between experiments an theory was not satisfactory, inicating the nonlinear origin of the shear wave attenuation. Base on these results, we conclue that the Kramers Kronig relation may be safely applie to longituinal waves, which provies us with a convenient way to measure the velocity ispersion. It is ifficult to be accurate with such measurement using other available techniques since the ispersion is quite small. By using the Kramers Kronig relations, one can erive the velocity ispersion from the velocity measurement at one frequency plus the attenuation spectrum. ACKNOWLEDGMENTS This research was sponsore by the NIH uner Grant No. P41-RR A1 an the ONR uner Grant No. N

9 J. Appl. Phys., Vol. 85, No. 1, 15 June 1999 Wang, Jiang, an Cao A. R. Selfrige, IEEE Trans. Sonics Ultrason. SU-3, 3, M. O Donnell, E. T. Jaynes, an J. G. Miller, J. Acoust. Soc. Am. 63, W. Sachse an Y. H. Pao, J. Appl. Phys. 49, R. A. Kline, J. Acoust. Soc. Am. 76, S. Bauouin an B. Hosten, Ultrasonics 34, T. J. Plona, R. D Angelo, an D. L. Johnson, Proceeings of the IEEE 1990 Ultrasonics Symposium, 1990, p J. Wu, J. Acoust. Soc. Am. 99, S. I. Rokhlin, T. K. Ballan, an L. Aler, J. Acoust. Soc. Am. 79, J. F. Nye, in Physical Properties of Crystals Clarenon, Oxfor, M. O Donnell, E. T. Jaynes, an J. G. Miller, J. Acoust. Soc. Am. 69, D. Zellouf, Y. Jayet, N. Saint-Pierre, J. Tatibouet, an J. C. Baboux, J. Appl. Phys. 80,