Characterization of lead zirconate titanate ... - Semantic Scholar

JOURNAL OF APPLIED PHYSICS

VOLUME 85, NUMBER 12

15 JUNE 1999

Characterization of lead zirconate titanate piezoceramic using high frequency ultrasonic spectroscopy Haifeng Wang, Wenhua Jiang,a) and Wenwu Caob) Materials Research Laboratory, National Resource Center for Medical Ultrasonic Transducer Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802

~Received 4 November 1998; accepted for publication 9 March 1999! Doped piezoceramic lead zircanate titanate has been characterized in the frequency range of 20–60 MHz using ultrasonic spectroscopy. Theoretical analyses were performed for the reflection and refraction of acoustic waves at the interface of water-piezoelectric ceramic. The incident directions of the wave were chosen to be appropriate for ultrasonic spectroscopy measurements. Shear wave spectrum was obtained through mode conversion using a pair of longitudinal transducers submerged in water. The phase velocity shows linear dependence on frequency while the attenuation may be described by a second order polynomial of frequency in the frequency range investigated. The Kramers–Kronig relation between ultrasonic phase velocity and attenuation was compared to measured results. © 1999 American Institute of Physics. @S0021-8979~99!00412-0#

I. INTRODUCTION

same number of independent physical constants as that of 6 mm symmetry, i.e., five independent elastic constants, three independent piezoelectric coefficients, and two independent dielectric permittivities. Because of the reasonably high symmetry, the situation may be simplified in some special incident angles. The present article is divided into two parts: first, we discuss the refraction of a plane wave at the interface of water-PZT ceramic. Based on the theoretical analysis, an ultrasonic spectroscopy technique suitable for characterization of PZT ceramic was developed and used to characterize PZT-5H samples. Second, the measured results were compared with those predicted by the Kramers–Kronig relations.

Application of higher frequency broadband ultrasonic transducer results in the improvement of the axial and lateral resolutions in medical imaging. Design high frequency transducer requires better knowledge of material properties since the ultrasonic dispersion becomes important for frequencies above 50 MHz. The dispersions of velocity and attenuation may deform the acoustic pulse and cause inappropriate interpretation of the pulse acoustic signal. Hence, knowing the properties of the transducer materials at high frequencies is important for designing high frequency transducers. Currently, there are very limited experimental data available in the literature on high frequency properties of piezoelectric materials due to many technical difficulties.1 Reported in this article are results of velocity and attenuation for doped lead zirconate titanate ~PZT-5H! in the frequency range of 20–60 MHz measured by using an ultrasonic spectroscopy method. The experimental results were also used to verify the Kramers–Kronig relations.2 The ultrasonic spectroscopy method has been widely used in the characterization of solid materials. With angular incidence, mode conversion effect was used to investigate shear wave properties of porous materials and polymeric materials.3–8 If such a method is generalized to characterize piezoelectric materials, one needs to deal with the problem of plane wave propagation through an interface between an isotropic medium and an anisotropic piezoelectric material. Since the propagation velocity of waves varies with propagation direction in an anisotropic material, the refraction becomes complicated when the ultrasonic wave is obliquely incident onto the interface. Generally speaking, the refraction coefficient cannot be given in an explicit analytic form. For poled PZT ceramics the symmetry is `m, which has the

II. THE REFLECTION AND REFRACTION OF A LONGITUDINAL WAVE AT THE WATERPIEZOCERAMIC INTERFACE

Based on the principle described by Rokhlin,8 we have derived the reflection/refraction of acoustic wave at the interface of water and a PZT ceramic plate. The poling direction of a plate PZT sample is either perpendicular or parallel to the large surface of the plate. As shown in Fig. 1, the plate sample with the poling direction perpendicular to its large surface is referred as PZTI z, whereas the plate with poling direction parallel to its large surface is referred as PZTI x. A. Reflection/refraction of a longitudinal wave at the interface of water and PZT O z

When a longitudinal wave is incident upon the interface of water and PZTI z, we choose the coordinate system to make the incident plane coincident with either the x-z or the y-z plane of PZT since the plane perpendicular 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. 1~a!, the incident longitudinal wave in water can be expressed as

a!

Present address: Institute of Acoustics and State Key Laboratory of Modern Acoustics, Nanjing University, Nanjing 210093, People’s Republic of China. b! Electronic mail: [email protected] 0021-8979/99/85(12)/8083/9/$15.00

U i 5U 0i exp@ j ~ v t2k w x 1 sin u i 2k w x 3 cos u i !# ~ i51,2,3 ! 8083

~1! © 1999 American Institute of Physics

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8084

J. Appl. Phys., Vol. 85, No. 12, 15 June 1999

Wang, Jiang, and Cao

F

1 1 ~ c m 2 1c m 2 21 ! u 011 ~ c 131c 44! m 1 m 3 u 03 r 11 1 44 3 r

HF

1 1 ~ e 311e 15! m 1 m 3 u 0450, r

~3a!

G J

1 1 ~ c 2c ! m 2 1c m 2 21 u 0250, r 2 11 12 1 44 3

F

~3b!

G

1 1 ~ c 131c 44! m 1 m 3 u 011 ~ c 44m 21 1c 33m 23 ! 21 u 03 r r 1 1 ~ e 15m 21 1e 33m 23 ! u 0450, r

~3c!

~ e 151e 31! m 1 m 3 u 011 ~ e 15m 21 1e 33m 23 ! u 03

2 ~ e 11m 21 1 e 33m 23 ! u 0450,

FIG. 1. ~a! Incidence of a wave in the x-z plane of a PZT sample. ~b! Incidence of a wave in the x-y plane of a PZT sample.

or U i 5U 0i exp@ j v ~ t2m w1 x 1 2m w3 x 3 !# ,

~18!

where k w 5 v / v w , v w is sound velocity of water, u i is incident angle m w1 5

n 1 sin u i 5 vw vw

and

m w3 5

n 3 cos u i 5 . vw vw

The unit vector n represents the wave propagation direction. Similarly, the refractive waves propagating in the PZT sample are expressed as u i 5u 0i exp@ j ~ v t2kn 1 x 1 2kn 3 x 3 !#

~ i51,2,3,4! ,

~2!

where u 4 stands for electric potential wave with the electric field given by E i 5 ] u 4 / ] x i . For convention we rewrite Eq. ~2! as u i 5u 0i exp@ j v ~ t2m 1 x 1 2m 3 x 3 !#

~28!

with m i 5n i / v ~i51 or 3!. Substituting the wave solutions into the equation of motion r u¨ i 5 ] T i j / ] x j and considering the constitutive equation T i j 5c i jkl ] u k / ] x l 2e mi j E m , the amplitude u 0i (i5124) in Eq. ~2! will be governed by the following equations:

U

S S

D

S S

~3d!

where r is the density, c i j are the elastic stiffness constants, e il are the piezoelectric stress constants, and e i j are the dielectric constants for the poled PZT ceramic. Short notation for the elastic constants, ˜c , and the piezoelectric coefficients, ˜e , have been used; their definition could be found in Ref. 9. From Eq. ~3! it is seen that u 02 representing the amplitude of a shear wave does not couple with u 01 and u 03 . Hence, this wave cannot be generated through mode conversion effect in this incidence arrangement. In other words, the particle displacement of the refractive waves in the PZT ceramic has only u 1 and u 3 components. From Eq. ~3d! u 045

~ e 151e 31! m 1 m 3 u 011 ~ e 15m 21 1e 33m 23 ! u 03

e 11m 21 1 e 33m 23

~4!

Substituting Eq. ~4! into Eqs. ~3a! and ~3c! gives

FS

D G

~ e 311e 15! 2 m 21 m 23 1 c 11m 21 1c 44m 23 1 21 u 01 r e 11m 21 1 e 33m 23

1

F

F

~ e 33m 23 1e 15m 21 !~ e 311e 15! 1 c 131c 441 r e 11m 21 1 e 33m 23

3m 1 m 3 u 0350,

G

G ~5a!

~ e 33m 23 1e 15m 21 !~ e 311e 15! 1 c 131c 441 m 1 m 3 u 01 r e 11m 21 1 e 33m 23

FS

D G

~ e 15m 21 1e 33m 23 ! 2 1 2 2 1 c m 1c m 1 21 u 0350. r 44 1 33 3 e 11m 21 1 e 33m 23 ~5b!

The propagation directions of the refractive waves allowed in the PZT ceramic are determined by the coefficient determinate of Eq. ~5!, i.e.,

D

U

~ e 311e 15! 2 m 21 m 23 1 c 11m 21 1c 44m 23 1 21 r e 11m 21 1 e 33m 23

~ e 33m 23 1e 15m 21 !~ e 311e 15! 1 c 131c 441 m 1m 3 r e 11m 21 1 e 33m 23

~ e 33m 23 1e 15m 21 !~ e 311e 15! 1 c 131c 441 m 1m 3 r e 11m 21 1 e 33m 23

~ e 33m 23 1e 15m 21 ! 2 1 c 44m 21 1c 33m 23 1 21 r e 11m 21 1 e 33m 23

D

.

D

50.

~6!




It is known that the phase matching condition at the interface, or the Snell’s law, demands m incident 5m reflective 1 1 refractive w PZT w 5m 1 , or m 1 5m 1 . Since m 1 is known for a given incident wave, one can determine m PZT of refractive waves 3 propagating in the PZT from Eq. ~6!. There are two solutions of m PZT from Eq. ~6! corre3 sponding to two refracted waves in PZT. Substituting the two roots of Eq. ~6! back into Eq. ~5!, two eigenvectors with two components can be obtained, which provides the polarization directions of the two refracted waves. In more general cases, one of them is a quasilongitudinal wave and the other is a quasishear wave. Thus, when a longitudinal wave is incident from water upon PZT ceramic, there are incident and refracted waves in the water medium; both are longitudinal waves with amplitudes given by u 0 and u r , respectively. In the PZT ceramic there are refracted quasilongitudinal and quasishear waves. Their propagation direction and polarization direction are determined by Eqs. ~5! and ~6! to be mL , IL and mS , IS , respectively. Their amplitudes u L and u S are related to that of the incident wave u 0 through the boundary conditions at the interface of x 3 50 u 3 ~ water! 5u 3 ~ PZT! ,

~7a!

T 1350,

~7b!

T 33~ water! 5T 33~ PZT! .

~7c!

u r cos u 1l L3 u L 1l S3 u S 5u 0

cos u ,

~8a!

ru L 1su S 50,

~8b!

Z w u r 2pu L 2qu S 52Z w u 0 ,

~8c!

where Z w is the acoustic impedance of water and r5c 44~ m L3 l L1 1m 1 l L3 ! 1e 15~ A L l L1 1B L l L3 ! ,

~9a!

s5c 44~ m S3 l S1 1m 1 l S3 ! 1e 15~ A S l S1 1B S l S3 ! ,

~9b!

p5c 13m 1 l L1 1c 33m L3 l L1 1e 33m L3 ~ A L l L1 1B L l L3 ! ,

~9c!

q5c 13m 1 l S1 1c 33m S3 l S1 1e 33m S3 ~ A S l S1 1B S l S3 ! ,

~9d!

with A L,S 5

~ e 151e 31! m 1 m L,S 3 L,S e 11m L,S 1 1 e 33m 3

,

B L,S 5

e 15m 1 1e 33m L,S 3 L,S e 11m L,S 1 1 e 33m 3

.

From Eq. ~8! the ratios of u r /u 0 , u L /u 0 , and u S /u 0 , i.e. the reflection and refraction coefficients, can be determined. Using this procedure, one can determine the amplitude and propagation direction of the refractive waves. In ultrasonic spectroscopy technique, however, one can use slightly different ways to solve the problem. As mentioned above, m w1 is known for a given incident wave, i.e., m w1 5

method, one can simply calculate u p through Snell’s law. Obviously, m PZT can be found by the relation of m PZT 3 3 5cos up /vp . Using Eq. ~6! the longitudinal and shear velocities can be correlated to appropriate elastic constants. In what follows we discuss three special cases to illustrate the procedure. Case 1. Normal incidence In this case m water 50, therefore, m PZT 1 1 50, u p 50 and m S3 5

1 vS

~10a!

or m L3 5

1 vL

n 1 sin u i 5 . vw vw

From Snell’s law it is given that m PZT 1 5sin ui /vw 5sin up /vp , where u p and v p ~p5L or S! are the refractive angle and velocity of longitudinal ~L! or shear ~S! wave. If the velocity v p has been determined by spectroscopy

~10b!

.

Substituting these results into Eq. ~5! results in the following simple relations:

v S5

A

c 44 r

and

v L5

A

cD 33

r

5

A

c 331

r

e 233

e 33

~10c,d!

which are the velocities of the shear wave and the stiffened longitudinal wave, respectively. From Eq. ~6!, two eigenvectors are obtained lS 5 @ 1,0,0 #

The above equations may be rewritten as

8085

and lL 5 @ 0,0,1 # .

~11a,b!

This means that one of the possible refracted waves is a pure shear wave with polarization direction along the x axis and another is a pure longitudinal wave. Substituting these results into Eqs. ~8! and ~9!, one can obtain the reflection and refraction ratios R5

u r Z L 2Z w 5 , u 0 Z L 1Z w

~12a!

T5

2Z L uL 5 , u 0 Z L 1Z w

~12b!

u s 50.

~12c!

Equation ~12c! implies that mode conversion does not exist in normal incidence, and reflection coefficient R and transmission coefficient T of the longitudinal wave are the same as for an interface of water and an isotropic solid. In the expressions Eqs. ~12a,b!, Z L is the acoustic impedance of the longitudinal wave in PZT. Case 2. Incidence at the critical angle of the longitudinal wave. When a longitudinal wave is obliquely incident from water upon PZTI z, mode conversion takes place. In PZT medium there are, in general, refractive quasilongitudinal and quasishear waves. If the wave is incident at the critical angle of longitudinal wave, the refractive longitudinal wave becomes an evanescent wave. This means that m L3 is equal to zero ( u i 5critical angle) or complex ( u i .critical angle). The vector mS representing the propagation direction of the refractive quasishear wave can be determined by the following expressions:


8086



m S1 5

sin u i sin u S 5 , vw vS

~13a!

m S3 5

cos u S . vS

~13b!

If the velocity v S is determined from the ultrasonic spectroscopy method, u S can be calculated from Eq. ~13a!, therefore mS is totally determined. Knowing mS , the polarization direction can be determined by Eq. ~5! and the transmission coefficient can be calculated from Eq. ~8!. The velocity of the quasishear wave is related to a combination of elastic constants of PZT by v S5

A*

c , r

~14!

where 2c * 5c 11 sin2 u S 1c 33 cos2 u S 1c 441p 1 1p 2 1 $ @~ c 112c 44! sin2 u S 1 ~ c 442c 33! cos2 u S 1p 1 2p 2 # 2 1 ~ c 131c 441p 3 ! 2 sin2 2 u S % 1/2

~15a!

and p 15

~ e 311e 15! 2 sin2 u S cos2 u S , e 11 sin2 u S 1 e 33 cos2 u S

~15b!

p 25

~ e 33 cos2 u S 1e 15 sin2 u S ! 2 , e 11 sin2 u S 1 e 33 cos2 u S

~15c!

~ e 33 cos2 u S 1e 15 sin2 u S !~ e 311e 15! , p 35 e 11 sin2 u S 1 e 33 cos2 u S

~15d!

are terms associated with piezoelectric coupling. u S is the refractive angle of the quasishear wave in PZT ceramic.

B. Reflection/refraction of a longitudinal wave at the interface of water and PZT O x

Assuming the plate normal direction is along the x axis and the poling direction is parallel to the z direction of the PZT plate as shown in Fig. ~1b!, the incident plane of the ultrasonic wave has three independent orientations: ~1! x-y plane; ~2! y-z plane, and ~3! incident in a plane that rotates around the y axis at an arbitrary angle. Since the last case makes the problem more complicated, it will not be discussed here. The second option is actually equivalent to the Case 2 of Sec. II A, therefore we only need to discuss the first option which we call Case 3, as described below. The plane perpendicular to the z axis of PZT is an acoustically isotropic plane as mentioned above. When an incident wave is in the x-y plane, as shown in Fig. 1~b!, the reflection/refraction of a longitudinal wave at the interface of water and PZT ceramic is the same as at the interface of water and an isotropic solid. In this case, the particle displacement u of the refractive waves has only components in the x and y directions. When a wave is normally incident, the only refractive wave is a pure longitudinal wave with the velocity given by

v L5

A

c 11 . r

~16a!

When a wave is obliquely incident, one of the two refractive waves is a pure longitudinal wave with the same velocity as Eq. ~16a!, another refractive wave is a pure shear wave with the velocity of v S5

A

1/2~ c 112c 12! 5 r

A

c 66 . r

~16b!

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 discussed by using the same procedure. However, pure modes do not exist for general cases. In summary, by using angular incidence of a longitudinal wave from water to two PZT plates we can determine the elastic constants and their dispersion from dispersion of velocity and attenuation in the following acoustic modes: ~A!. For a PZTI z sample: ~1! Longitudinal wave propagating along the poling direction with the wave normal incident upon the plate. From the velocity dispersion of this wave, the frequency dependence of elastic constant c D 33 can be determined. ~2! Quasishear wave propagating in the x-z plane with the wave obliquely incident at the critical angle of the longitudinal wave. From its velocity dispersion one can determine an elastic constant combination c * and its frequency dependence. ~B!. For a PZTI x sample: ~1! Longitudinal wave propagating perpendicular to the poling direction. From its velocity dispersion one can determine the elastic constant c 11 and its frequency dependence. ~2! Shear wave with directions of propagation and polarization normal to the poling direction when the wave is incident at the critical angle of the longitudinal wave. From the velocity dispersion of this wave one can determine the elastic constant c 66 and its frequency dependence. III. PRINCIPLES OF MEASUREMENTS

The basic principle of the ultrasonic spectroscopy method used to determine the velocity and attenuation of materials is shown in Fig. 2. A pair of aligned transmitting and receiving transducers are immersed in water with an adjustable separation between them. When the transmitting transducer is driven by an electric pulse signal, an acoustic broadband signal is produced at x50, noted as u(t). Its Fourier transform is ˜u ( f ). For a linear and causal acoustic system shown in Fig. 2~a!, the transfer function of the medium for the wave to propagate can be expressed as H ~ f ! 5exp@ 2 a w ~ f ! L # exp$ 2 j @ 2 p f L/ v w ~ f !# % ,

~17!

where L is the distance between the transmitting and receiving transducers, v w and a w are the wave velocity and attenuation of water, respectively. When the medium is considered




8087

tively, u i and u are the incident and the refractive angles of the wave at the interface, respectively, T is the total transmission coefficient which is equal to the product of the transmission coefficients of the wave from water to sample and from sample to water. Thus, the amplitude and phase spectra of the output signal for the system shown in Fig. 2~b! can be written as

F

H

A ~ f ! 5T u˜u ~ f ! u exp 2 a w ~ f ! L2 3sin u i and

GJ F

F

exp 2 a ~ f !

w 52 p f L2 1 FIG. 2. Principle of ultrasonic spectroscopy technique: ~a! without sample, ~b! with sample placed in a rotated position.

v5

˜u w ~ f ! 5u ˜ ~ f ! exp@ 2 a w ~ f ! L #

A w ~ f ! 5 u˜u ~ f ! u exp@ 2 a w ~ f ! L # u˜u 8 ~ f ! u ,

~19a!

w w ~ f ! 52 p f L/ v w ~ f ! 1 f u 1 f u 8 ,

~19b!

where f u and f u 8 are the phase angles of the two transducers, respectively. They will not enter the calculations below since the spectroscopic method depends only on the relative phase shift. When the sample to be measured is inserted between the transmitting and receiving transducers, the transfer function of the system as shown in Fig. 2~b! becomes

H

F

H S ~ f ! 5Tu ˜ ~ f ! exp 2 a w ~ f ! L2

GJ F H F S DY

3sin u i

exp 2 a ~ f !

d 2d ~ tg u 2tg u i ! cos u i

d cos u

3sin u i

GY

v w~ f !

~21b!

A

sin2 u i 1

F

vw ~ w 2 w w !v w

2p f d

S

1cos u i

D

G

TA w cos u /d. A

2

,

~22!

~23!

If the values of A w , A, w w , and w can be measured, the attenuation and phase velocity dispersion of the sample can be determined. When the wave is normally incident to the sample, u i 5 u 50, the above equations give the velocity and attenuation of the longitudinal wave in the case discussed in Sec. II for the longitudinal wave propagating in either x-z plane or x-y plane as shown in Figs. 1~a! and 1~b!, respectively v L5

vw ~ w 2 w w !v w 11 2p f d

and

S DY T LA w A

~24!

d,

~25!

where T L5

d 3exp 2 j 2 p f L2 2d ~ tg u 2tg u i ! cos u i d v w ~ f ! 12 p f cos u

~21a!

2p f d 1 f u1 f u . v~ f ! cos u

a L 5 a w 1ln

G

d 2d ~ tg u 2tg u i ! sin u i cos u i

a 5 a w cos~ u 2 u i ! 1 ln

~18!

where ˜u 8 ( f ) is the transfer function of the receiving transducer. Thus, the amplitude and phase spectra of the output signal can be expressed as

G

d u˜u 8 ~ f ! u , cos u

From Eqs. ~19! and ~21! the phase velocity and attenuation coefficients in the sample are given by

to be dispersive and dissipative, both a w and v w are frequency dependent. The spectrum of the output signal from the receiving transducer is 3exp@ 2 j ~ 2 p f L/ v w ~ f !#˜u 8 ~ f ! ,

d 2d ~ tg u 2tg u i ! cos u i

Y GJ v~ f !

˜u 8 ~ f ! , ~20!

where d is the thickness of the sample, a and v are attenuation coefficient and phase velocity in the sample, respec-

4 r 0v wr v L 4z w z L 5 ~ r 0 v w 1 r v L ! 2 ~ z L 1z w ! 2

~26!

and r 0 and r are the mass density of water and sample, respectively, and v L is the longitudinal wave velocity in the sample. If the wave is obliquely incident at the critical angle of the longitudinal wave shear wave will be generated through mode convention effect. The velocity and attenuation of the shear wave can be calculated by the following expressions:


8088



FIG. 4. The variation of amplitude spectrum with propagation distance in water. FIG. 3. Experiment setup.

v S5

A

sin2 u i 1

F

vw ~ w 2 w w !v w

2p f d

S

a S 5 a w cos~ u S 2 u i ! 1 ln

1cos u i

G

2

D Y

T SA w cos u S A

~27!

,

~28!

d,

where T S and u S are transmission coefficient and refractive angle of the shear wave. The transmission coefficients for fluid-isotropic solid interface are simply given by T 1 52 T 25

S D

r0 2z Sn sin~ 2 u S ! , 2 r z Ln cos ~ 2 u S ! 1z Sn sin2 ~ 2 u S ! 1z wn

S

~29a!

D

tan u i z Ln cos2 ~ 2 u S ! 2z Sn sin2 ~ 2 u S ! 1z wn 12 , 2 sin2 u S z Ln cos2 ~ 2 u S ! 1z Sn sin2 ~ 2 u S ! 1z wn ~29b!

T S 5T 1 T 2 ,

~29c!

where T 12 and T 21 stand for the transmission coefficients of a wave from water to PZT and from PZT to water, respectively; z wn 5z w /cos ui , z Ln 5z L /cos uL , z Sn 5z S /cos uS , and v S is the shear wave velocity in the sample. The refractive angles u L and u S are calculated from Snell’s law sin u i sin u L sin u S 5 5 , vw vL vS

~30!

where the incident angle u i is controlled by a computerized rotating table in our experiments. For the quasishear wave propagating in the x-z plane of a PZT ceramic. The transmission coefficient can be calculated from Eq. ~8!. IV. EXPERIMENT RESULTS FOR PZT-5H AND DISCUSSIONS

The experimental setup is shown in Fig. 3. A pair of transducers with a center frequency of 50 MHz and bandwidth of 80% were used. Without sample, the spectra of the output signal from the receiving transducer are shown in Fig. 4. Since the ultrasonic attenuation of water is about 6 dB/cm

at 50 MHz the high frequency components of the Fourier spectrum are decayed when the distance between the transmitter and receiver increases, as indicated in Fig. 4. The interval between transmitter and receiver should be selected in such a way that the high frequency components are preserved as much as possible, and at the same time leaving enough room for the sample to be rotated. In our experiments, the distance L is set at about 3 cm. The output waveform was sampled by a digital oscilloscope ~Tektronix TDS 460A! at a sampling rate of 10 Gs/s. The data were transferred into computer via a general purpose interface bus ~GPIB! interface. The total recording length for a waveform was 2500 points. The amplitude A w and the phase spectra w w were obtained through fast Fourier transform ~FFT! of the output signal from the configuration in Fig. 2~a!. When the sample was put in place, the trigger delay time was adjusted so that the shifting of the waveform caused by putting in the sample can be compensated. Using the same procedure, the amplitude A and the phase w of the output signal from the configuration Fig. 2~b! were obtained. @Note: the velocity calculation must take into account the trigger time delay, t# v L5

v S5

vw , ~ w 2 w w 12 p f t !v w 11 2p f d

A

sin2 u i 1

F

~31a!

vw ~ w 2 w w 12 p f t !v w

2p f d

1cos u i

G

2

. ~31b!

The attenuation of the longitudinal and shear waves can be calculated from Eqs. ~23! and ~25!, respectively. Here the attenuation of water was given by 0.00027l f2 ~dB/mm!, ~f in MHz!. The measured velocity and attenuation for piezoceramic PZT-5H are given in Figs. 6–9. It was observed that velocity dispersion exists for both the longitudinal and shear waves in the frequency range of the measurement and it is nearly linear, but the attenuation exhibits nonlinear frequency dependence. The attenuation of the shear wave is an order of magnitude higher than that of the longitudinal wave.




FIG. 5. Dispersion relationship of the velocity and attenuation for water.

Usually, the wave number of a decay wave is considered to be a complex number. Therefore, if the frequency is taken as real, the velocity becomes complex and is associated with the complex elastic modulus in the following fashion: ˜v 5

A A ˜ C 5 r

C1 jC 8 . r

~32!

8089

FIG. 7. The phase velocity dispersion measured by experiment and derived by Kramers–Kronig relationship for wave propagating in the x-y plane of a PZT sample and the corresponding elastic constants. The longitudinal and shear velocities were fitted to linear curves: v L 53744.411.00e 26 * f (Hz) and v S 51762.515.47e27 * f (Hz).

In general, it is true that u C 8 u / u C u !1, we may write the above equation as ˜v '

AS

D

C C8 11 j . r 2C

~33!

The complex wave number can be expressed as

FIG. 6. The phase velocity dispersion measured by experiment and derived by Kramers–Kronig relationship for a wave propagating in the x-z plane of a PZT sample and the corresponding elastic constants.

FIG. 8. The attenuation measured by experiment and derived by Kramers– Kronig relationship for a wave propagating in the x-z plane of a PZT sample. The longitudinal and shear velocities were fitted to linear curves: v L 54156.811.44e26 * f (Hz) and v S 5159718.67e27 * f (Hz).


8090


˜k 5k2 j a ,


~34!

where k5 v / v and a is the attenuation coefficient. Thus, the real and imaginary parts of the elastic modulus can be derived from the phase velocity and attenuation measurements C5 r v 2 , tan d 5

~35a!

C8 2av 5 . C v

~35b!

If a medium in which an acoustic wave propagates can be considered as a linear and causal system, its attenuation, which is associated with the imaginary part of the elastic modulus, and velocity dispersion, which is associated with the real part of elastic modulus, are related by the Kramers– Kronig relations. The approximation forms of the nearly local relationships can be expressed as10,11 v~ v ! 5 v~ v 0 ! 1

2 v 2~ v 0 ! p

pv 2 d v~ v ! a~ v !5 2 , 2v ~v0! dv

E

a~ v ! 2 dv, v0 v v

~36a!

~36b!

where v 0 is the starting frequency at which v ( v 0 ) and a ( v 0 ) are known. In our experiment, v 0 52 p * 20 MHz. The ultrasonic spectroscopy technique is inherently based on the assumption of linearity. Thus, the velocity dispersion and attenuation obtained from the technique are expected to satisfy the above Kramers–Kronig relations. To verify the validity of these relations, the measured attenuation for the PZT-5H samples was fitted as a polynomial of frequency and the phase velocity dispersion was derived by the first Kramers–Kronig relation Eq. ~36a!, then these calculated results were compared to measured results. As shown in Figs. 6 and 7 that the agreement is quite acceptable. The maximum deviation is less than 0.6% and the trend is correct. We have also performed the reverse checking, i.e., using the measured velocity dispersion to calculate the dispersion of the attenuation based on the local approximation Eq. ~36b!. We found that this relation is extremely sensitive to the curvature of the velocity dispersion. Higher order polynomial fitting of the velocity curve gives unreasonable results. Because the velocity dispersion is fairly small, we decided to use a linear approximation to these velocity data, which created error less than the experimental uncertainty for the velocity fitting but gave good agreement to the measured attenuation dispersion. The results are shown in Figs. 8 and 9 for the two different cut PZT samples. Owing to this sensitivity, one should be really careful when using the second Kramers–Kronig relation Eq. ~36b!. One may have problems using this relation if the velocity dispersion has a downward curvature, such as the shear velocity measured in this work, since it will lead to a decrease of the attenuation at higher frequencies. The measured attenuation seems to always increase with frequency. Our experience is that a linear fitting of the velocity dispersion could provide a much better prediction of the attenuation dispersion using the Kramers–

FIG. 9. The attenuation measured by experiment and derived 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 wised since the relationship is not linear in general for all frequency ranges. V. CONCLUSION

The dispersions of velocity and attenuation for piezoceramic PZT-5H were investigated by using ultrasonic spectroscopy at the frequency range of 20–60 MHz. In the investigated frequency range, velocity dispersion of 1–3 m/s per MHz was observed. The attenuation depends nonlinearly on frequency and the shear wave exhibited an order of magnitude larger attenuation than the longitudinal wave. We showed that the Kramers–Kronig relations between velocity dispersion and attenuation dispersion are satisfied for the longitudinal waves. However, for the shear waves, the agreement between experiments and theory was not satisfactory, indicating the nonlinear origin of the shear wave attenuation. Based on these results, we conclude that the Kramers– Kronig relation may be safely applied to longitudinal waves, which provides us with a convenient way to measure the velocity dispersion. It is difficult to be accurate with such measurement using other available techniques since the dispersion is quite small. By using the Kramers–Kronig relations, one can derive the velocity dispersion from the velocity measurement at one frequency plus the attenuation spectrum. ACKNOWLEDGMENTS

This research was sponsored by the NIH under Grant No. P41-RR11795-01 A1 and the ONR under Grant No. N00014-98-1-0527.


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