PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Attenuation mechanisms of high-frequency acoustic waves inpiezoelectric cubic crystals
Farkhad AKHMEDZHANOV1; Sirojiddin MIRZAEV1; Ulugbek SAIDVALIEV1
1

Institute of Ion-Plasma and Laser Technologies, Uzbekistan

ABSTRACT
Attenuation mechanisms of high frequency acoustic waves in piezoelectric bismuth silicate (BSO) andbismuth germanate (BGO) crystals has been studied in the frequency range of 0.41.6 GHz. It is shown thatthe main mechanisms of attenuation are the Akhiezer mechanism and the effect of dielectric losses due to thepiezoelectric coupling of deformation with polarization in piezoelectric crystals. To describe the attenuation
anisotropy of acoustic waves, real and imaginary elastic moduli of these crystals were determined. It isestablished that the most contribution of dielectric loss to the attenuation of acoustic waves is observed forthe piezoactive transversal waves propagating in the (100) plane in bismuth germanate crystals. The
dielectric loss can also affect the frequency dependence of the attenuation coefficient of piezoactive acousticwaves.
Keywords: Acoustic wave, Attenuation, Piezoelectric crystals

1. INTRODUCTION
The piezoelectric crystals of bismuth silicate Bi12SiO20 (BSO) and bismuth germanate Bi12GeO20(BGO) are widely used as an active medium in different optic and acoustooptical devices [1-2]. One of themain parameters of these devices is the attenuation coefficient of the acoustic waves, which can be calculatedusing the real and imaginary components of the complex elasticity tensor [3, 4]:  icijkl
 ,cijkl  cijkl(1)which enter into the expression of effective elastic constant:  iceff  cijkl
  j l  i  k  icijkl  j l  i  k ,ceff  ceff(2)

where j and i are the direction cosines of the wave vector and the displacement vector, respectively.
Equations (1) and (2) are valid for centrosymmetric dielectric crystals, in which the Akhiezer
mechanism is the main mechanism of attenuation of acoustic waves [1].
As is known, in crystals with piezoelectric properties (non-centrosymmetric crystals), piezoactive
acoustic waves are accompanied by a polarization wave of the electric field [1]. In piezoelectric crystals, atlarge coefficients of electromechanical coupling, the dispersion of the sound propagation speed and theakhmedzhanov.f@gmail.com

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contribution of dielectric losses to the attenuation of acoustic waves due to the phonon-dipole interaction
mechanism can be observed. As a result of this interaction, an additional attenuation of the acoustic wave
can be occurred.
A related system of equations of elastodynamics and electrostatics for a piezoelectric crystal,
describing the propagation of acoustic waves in a piezoelectric crystal, can be represented as [1, 3]:

Cijkl

 2uk
 2u
 2
 enij  2i
x j xexn x j
t(3) 2u k 2
  mn 
0xm xlxm xn(4)emkl 

where еmkl are the components of the piezoelectric coefficient tensor,  is the potential of an alternating
electric field arising during the propagation of an acoustic wave. ui are the components of the displacement
vector, тп = тп -iтп are the components of the complex tensor of dielectric constants.
Taking as the solution of the system of equations the equation of a plane damped wave: u r , t   u0  e t  ei  t  q r  ,(5)

we can get the well-known Christoffel equation and expression determining the attenuation of acoustic waves
in piezoelectric crystals:  
ceff

(6)

ceff

wherec'эффc'ijkl  j l  i k ""cэфф cijkl  j l  i k (enij   n j  i ) 2(7)

 0 rs r  s(enij   n j  i )  (emkl  m l  k ) 0 rs r s

 tg mn   m n

(8)

In equation (8), tgmn characterizes dielectric losses that occur in dielectrics due to polarization relaxation.

 for these crystals were measured in a
The real components of the elastic constant tensor cijkl
 which describe the attenuation of acoustic
number of papers [5]. As for the imaginary components cijkl
waves, they were practically not investigated. In the present paper, the detailed study of the complex tensor ofelastic constants of BSO and BGO crystals was carried out to determine the anisotropy of attenuation ofhigh-frequency acoustic waves at room temperature.

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2. EXPERIMENTAL METHODS
The crystals Bi12SiO20 and Bi12GeO20 were grown by the Czochralski technique on a Donets-1 setup.
The crystals were prepared in the form of a parallelepiped with sides 1684 mm. The faces of samples havebeen oriented with accuracy no worse than 1° along the <001>, <110> and <111> crystallographic directions.
All faces of the parallelepipeds were processed by mechanical polishing and polishing with optical accuracy.
The velocity and attenuation of acoustic waves were investigated with the help of Bragg diffraction oflight on acoustic waves in frequency range 0.4–1.6 GHz. Longitudinal and transverse acoustic waves wereexcited by piezoelectric transducers of quartz, accordingly of the X-cut or Y-cut. The helium-neon laser witha wavelength of 632.8 nm was used as a light source. The intensities of diffracted light were measured withthe help of photoelectric multiplier.
The acoustic wave velocity V was determined from the Bragg diffraction angle with the accuracyof 0.2% [1]:V 
2  sin  B ,(9)where  is the linear frequency of acoustic wave, B  the external Bragg angle.
The acoustic wave attenuation has been investigated for longitudinal and transverse acoustic waves
propagating along the investigated directions of these crystals. The attenuation coefficient was calculatedwith the accuracy ~ 5% by the formula:10g I1 I 2  .
2 L0(10)Here L0 is the length of the sample, I1 and I2 are the intensity of the light diffracted on the direct acoustic
wave and the intensity of the light diffracted on the wave, reflected from the free end of the sample,
respectively. These measurements were carried out at the same point of the sample along the wave
propagation direction.

3. RESULTS OF EXPERIMENTS AND DISCUSSION
The experimental values of the velocity and the attenuation of acoustic waves along the <001>, <110>and <111> directions were used to determine all the independent components of the complex tensor of elastic
constants cijkl, which define effective elastic constant in equation (2). The results of the calculations arepresented in table1.
For example anisotropy of the velocity and attenuation of acoustic waves in BGO and BSO crystals wasconsidered for the (001) plane, which is orthogonal to the symmetry axis of the second order. The direction
cosines of the wave vector of waves propagating in this plane are equal to 1=cos, 2=sin, 3=0. Then thecomponents (respectively real or imaginary) of the Green-Christoffel tensor can be written in the form [1, 4]:8214

Table 1  Effective elastic constants of germanate and bismuth silicate crystals ( = 1 GHz)q[100]

Expressionс𝑒𝑓𝑓 ,1010 Nm-2

107 Nm-2

𝑐𝑒𝑓𝑓

BGO

BSO

BGO

BSO

[100]

𝑐11

12.8

3.72

2.44

2.60

[001]

𝑐44

2.57

1.67

0.11

0.60

𝑐12

2.88

c11  c12  2c 44
2

10.4

3.36

2.14

2.80

4.96

2.32

0.42

1.17

2.85

1.76

0.40

1.90

9.98

3.29

2.41

3.29

4.15

2.12

0.32

1.05

[110]

[110]

с𝑒𝑓𝑓 ,c11  c122e142
c 44 tg *
 0

[110]

[001]

[111]

c11  2c12  4c 44 4e14233 0 1.60

tg*[111]c11  c12  c 44
3[110]

11=с11(cos)2+с44(sin)222=с44(cos)2+с11(sin)233=c4412=21= (с12+с44) cossin
13=31=23=32=0

In these expressions, the angle  is the angle between the direction of the wave vector and thesecond-order symmetry axis [100]. Components c are substituted real or imaginary, respectively. In
particular, the real part of the effective elastic constant will be written as: ceff

  22 1
11  22 ) 2  412
2 .
(1122(11)As a result, the phase velocities of the longitudinal and one of transverse waves polarized in the (001)plane of the BSO and BGO crystals are determined in terms of the real components of the elastic constants
tensor by the expression:  c66
 ) 2 cos 2 2  (c12  c66
 ) 2 sin 2 2c  c  (c112V1,2
 11 66.
2

(12)

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Relation (12) determines the values of the velocities of two waves, the polarization of which lies in the(001) plane. In this case, the [100] and [010] axes are equivalent. The expression for imaginary effective
elastic constant is obtained; if in expressions (12) change the real components of the elastic constants tensor
into suitable imaginary elastic constants.
Further the obtained values of effective elastic constants can be used for determination of theattenuation of longitudinal and transverse waves along any arbitrary crystallographic direction by theequation (6). The results of calculation are shown in Fig. 1 in the form of cross-section of characteristic
surfaces of attenuation of acoustic waves by plane (001).
It can be seen that in the considered crystallographic plane a relatively small anisotropy of theattenuation of acoustic waves is observed for the longitudinal acoustic waves. At the same time it is seenthat the strongest anisotropy of transversal waves is observed in BGO crystals. It can be explained by the
relatively large value of the dielectric loss tangent in bismuth germanate crystals.
abFigure 1  Anisotropy of the attenuation of quasilongitudinal (1), quasitransverse (3) and puretransverse (2) waves in BSO (a) and BGO (b) crystals.

4. CONCLUSIONSBoth real and imaginary elastic moduli of the noncentrosymmetric cubic crystals of germanate andbismuth silicate (point symmetry group 23) has been determined for the first time. By the example of thesecrystals it is shown that in piezoelectric crystals it is necessary to take into account the contribution ofdielectric losses to attenuation [8]. The calculations with help of equations (6), (7) and (8) have shown that
this contribution is most significant in bismuth germinate crystals and it is about 45%.
For slow transverse waves along [110] the imaginary effective elastic constant is written in thesimplest form:8216

""ceff
 c44


e142

 0tg(13)

Here: e14 is the piezoelectric coefficient,  – the dielectric constant, and tg – the dielectric loss tangent
Since in the field tgmn << 1 the dielectric losses, as a rule, grow in direct proportion to the frequency,
the considered mechanism of attenuation of acoustic waves gives, like the Akhiezer mechanism, a quadratic
dependence of attenuation on frequency.

REFERENCES
1.

Dieulesaint E., Royer D. Elastic waves in solids. Application for treatment of signals. 1982. М.: Nauka,342 p.

2.

Ganeev R.A., Ryasnyansky A.I., Tugushev R.I. et al. Study of nonlinear-optical characteristics ofphotorefractive BSO and BGO crystals. Quantum Electronics. 2004. V. 34. –No. 2. P. 156-160.

3.

Bajac I.L., McNab A., Richter J., Wilkinson C.D.W. Attenuation of acoustic waves in lithium niobate. J.

Acoust. Soc. Am. - 1981. - V. 69. - No. 3. – P. 689-695.

4.

Akhmedzhanov F.R., Lemanov V.V., Nasyrov A. Surfaces of acoustic attenuation in crystals. Letters

Journal Technical Physics (SU), 1980. V.6, № 10. P. 589- 592.
5.

Acoustic crystals. Reference book. Ed. M.P. Shaskolskaya (Moskow, Nauka, 1982).

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