INVESTIGATION OF VELOCITY SURFACE OF BULK ACOUSTIC
WAVES FOR PROTON-EXCHANGED LiNbO3 CRYSTAL
G. Chen, X. R. Zhang, J. C. Cheng
State Key Lab. of Modern Acoustics and Institute of Acoustics, Nanjing University
Nanjing 210093, China
ABSTRACT. We calculate the velocity surface of bulk acoustic waves propagating in the YZ-,
XY- and ZX- planes of proton-exchanged LiNbO3, based on the theory of acoustic wave in
anisotropic solid with piezoelectricity. The calculation results show that the velocity change due to
proton-exchange when the wave propagating along X-axis is larger than that along Z- axis for Y-cut
substrate, in spite of longitudinal or transverse wave. The curve of velocity surface for PE Z-cut
substrate is near to a circle, while that for pure Z-cut LiNbO3 is a hexagonal. The velocity surface
for PE X-cut LiNbO3 seems to be turned an angle from that for pure X-cut LiNbO3. The detail
results, discussions and comparison with the experiment will present in this paper.
INTRODUCTION
It is well know that LiNbO3 crystal undergoes ion exchange in acide media, replacing
some or all of the lithium ions with protons is called proton exchange (PE) [1]. When this
reaction is carried out in strong acide (for example, aqueous nitric acid), complete
exchange results, accompanied by a structural transformation from the rhombohedral
LiNbO3 structure to the cubic perovskite structure [2]. With weaker acids, partically
exchanged materials can be prepared with retain the LiNbO3 structure. Those materials are
of particular interest because they can be prepared as thin layers at the surface of LiNbO3
crystal without affecting their optical quality, and cause large optical and acoustic
refraction index increases which have been used in a number of optical devices and
surface acoustic waveguides [3-4]. Rice studied the properties of Li: xkxNbO3 as a function
of x, temperature, and stoiciometry of the LiNbO3 used for its preparation [5].
The acoustic properties of PE LiNbO3 used in surface acoustic waveguides were
measured by using the Brillouin Scattering method [6], acoustic microscopy method [7]
and the interdigital transducers (here after IDT) method [8-13]. Hinkov fabricated the first
time SAW waveguides on Y-cut LiNbO3 substractes by PE and studied the influence of the
source dilution, the annealing, or the Ti pre-diffusion on the SAW velocity, by using IDT
as SAW excitation source and optic probe as detector [4]. They reported that: The
exchange in pure acid always leads to a velocity decrease. It is larger for Y-X sample and
smaller for X-Z sample. The velocity change initially decreases with the annealing time
and then remains essentially unchanged. The absolute value of the velocity change
decreases with the dilution of the proton source. In contrast to the exchange in pure acid,
CP657, Review of Quantitative Nondestructive Evaluation Vol. 22, ed. by D. O. Thompson and D. E. Chimenti
© 2003 American Institute of Physics 0-7354-0117-9/03/S20.00
1277
the SAW velocity increases for Y-Z samples exchanged in a diluted source. We are
interesting in Hinkov et al [6] experimental results because they also measured velocity of
longitudinal (L-) wave propagating along the proton exchanged (PE) layer. The velocity of
SAW and L-wave propagating along PE LiNbO3 surface are measured by using Brillouin
scattering techniques at microwave frequencies (~ 1010 Hz) for the observation of
soundwaves whose acoustic wavelengths are considerably smaller than the exchange
depths of the samples, the wavelength of longitudinal wave >-LA=0.11 |um, of SAW
>-SAW=0.251 and 0.276 |um, but the depths of PE are near to 4 |um. We can regard that all of
the velocity decrease measured are contributed by the PE part of the sample. It is
convenient to deduce the bulk wave property of the PE sample. However, they only report
that the reduction of the effective elastic constants amounting to about 40% for the PE
LiNbO3, but they did not make theoretical calculation for comparison. And nobody study
on the property of bulk wave in PE LiNbO3. It is consideration on investigation the
velocity surface of bulk wave in surface of xy-, yz- and xz- planes for the PE LiNbO3, to
observed the change of the structure due to PE treatment.
Recently, we calculate the velocity surface of bulk acoustic waves propagating in the
YZ-, XY- and ZX- planes of proton-exchanged LiNbO3, based on the theory of acoustic
wave in anisotropic solid with piezoelectricity [15], and by using the elastic constants
evaluated by Biebl [16]. The calculation results, discussion are present in this paper.
BRIEF DESCRIPTION OF PREVIOUS RESULTS
The variation of the SAW velocity in the x-z plane of a LiNbO3 crystal with the angle
a, measured by Hinkov et al., is shown in Figure 1. Figure 2 shows the value of the
velocity of L-wave versus the depth of PE for a y-x LiNbO3. The depth is obtained by
measurement of the extraordinary refractive index as a function of penetration depth by
light of wavelength 514.5 nm being guided along the x-axis in proton exchanged region of
a y-cut LiNbO3. From Figure 1, we know that the velocity change of SAW is anisotropic.
The velocity change of SAW propagating along x-direction (17%) is larger than that along
z-direction. The effective anisotropy of the exchange is so larger as to even reverse the
sign of the acoustic birefringence of the crystal. The velocity change is caused by the
decreases of the elastic moduli. From Figure 2, We can get the velocity of the L-wave is
related with the depth of the PE layer. However there was no calculation about the velocity
surface of bulk wave propagating in the PE LiNbO3. Since the wavelength of SAW and
L-wave are much small than the PE depth, the PE layer is anisotropic.
THEORETICAL CALCULATIONS
We calculate the velocity surface of bulk acoustic waves propagating in the YZ-, XYand ZX- planes of proton-exchanged LiNbO3, based on the theory of acoustic wave in
anisotropic solid with piezoelectricity for pure LiNBO3 and piezoelectric effect
ignired[15].The elastic constants are taken from the constants evaluated by Biebl (c.f.
Table 1).
1278
3700 3600-
I^
3500-
CO
•s
3200-
1
3000-
""o
>
* for pure YX LiNbO3 *.SAW =0.25 1 (in
5500-,
for YX PE:LiNbO3, ^=0.11 urn
for YX H+LiNb03, X,SAW=0.25 1 jam
o
5400A
A
A
A
A
5300-
3400A
3300-
52000
31002900-
0
5100-
o
o
o
00°
x
4900-
z
2800-
. . . . . . . J ...
9700-
0
20
40
60
80
4800
0
100
1
2
Depth d
Angle a (deg.)
FIGURE 1 The variation of the SAW velocity
measured in the x-z plane of a LiNbO3 crystal with the
angle a. (\AW=0.251|um), f=14.4 GHz, Relative VS
change along X-direction is about -70 %., and is larger
than Z-axis about 6%.
3
4
5
(jam)
FIGURE 2 the value of the velocity of L- wave
versus the depth of PE for a Y_X LiNbO3
Relative
change -25
%.(6570-»4950 m/s when dPE=1.5 |um, f=59,73
GHz), the maximam change is observed at
dPE=1.5|um.
The Christoffel equation for anisotropic crystals appears in the form
(1)
where
is called a piezoelectrically stiffened elastic constant. The r? called the piezoelectrically
Christoffel matrix. Its elements are functions only of the plane propagation direction and
the stiffness constants of the medium.
TABLE 1 The elastic constants q. (109N m2) for LiNbO3 and HxLi1.xNbO3.
Material
Cu
C12
C13
LiNb03
203
53
HxLi1.xNbO3
150
39
C14
C33
75
9
75
0
1279
C*
C66
245
60
75
210
72
55
lx 0 0
0 ly 0
L 0 0 0 L 1,
0 0 lz
0 ly 0 lz o ix , and ikLj = -iklLj —» ik
0 lz l y
0 0 1 1 1 0
iz o i x
L, L
0
where lx=kx/k, ly=ky/k, lz=kz/k are the direction cosines of the propagation direction.
The elastic contants matrix for LiNbO3crystal (trigonal crystal classes 3m) is
C12
"Cn
C13
c12 Cn c13
c13 c13 c33
I °
0
0
0
0
0
0
C
14
-Ci4
0
044
0
0
0
0
0
0
0"
0
0
0
044
CM
CM
C
c 6 6 =-(c n
(2)
66_
The values of the elastic constants for LiNbO3 and PE LiNbO3 are taken from the Table 1
respectively. The Piezoelectric properties [er] of LiNbO3 crystal is
0
-e y2
0
0 0 e x5 -e y2 "
0
0 0 0 3.7 -2.5"
0 = -2.5 2.5 0 3.7 0
0
0 e x5 0
0.2
0.2
0
0
1.3
0
0
0
^z3 0
(3)
where ex6= -ey2. And we assume that the piezoelectric coefficients decrease neare to zero.
We calculate the variation of velocity of longitudinal (L-) and transverse (T-) waves
with azimuthal angle a (propagation direction) for y-cut (i.e., within xz-plane), x-cut
(yz-plane, and z-cut (xy-plane) of virgin and proton-exchanged (PE) crystal respectively.
The calculated results are shown in Figure 3 to Figure 9.
RESULTS AND DISCUSSIONS
Figure 3 shows the velocity of the transverse wave versus the angle a for y-cut LiNbO3,
and PE LiNbO3 respectively. From Figure 3, it can be seen that: The velocity of T-wave
decrease in x direction is 21%, but increase in z direction is 11%. For the PE LiNbO3, The
velocity in z direction is larger than that in x direction about 200 m/s. The tend of the
T-waves curves in Figure 3 near to that of SAW drawn in Figure 1, but the values of
T-wave are higher than that of the SAW, particularly for the velocity of T-wave
propagating along the z-direction.. Figure 4. shows the velocity surface of the L-wave for
y-cut LiNbO3, including piezoelectricity for pure LiNbO3, where the solid line is for
unchanged LiNbO3 and the dash line is for PE LiNbO3. From Figure 4, we can seen that
1280
the relative decrease of the L-wave in x- direction is larger than that in z-direction. It is
the
decrease of
L-wave in
in xx- direction
direction is
is larger
larger than
than that
that in
in z-direction.
z-direction. ItIt isis
the relative
relative
of the
the as
L-wave
also
the
samedecrease
as the results
SAW. The
velocity surface
of the transverse
for xz-plane is
also
the
same
as
the
results
as
SAW.
The
velocity
surface
of
the
transverse
for xz-plane
xz-planeisis
also
the
same
as
the
results
as
SAW.
The
velocity
surface
of
the
transverse
for
shown in Figure 5. It can be seen again that: the velocity change for the transverse waves
shown
5. It
again
that: the
the velocity change
change for
for the
the transverse
transverse waves
waves
shown in
in Figure
Figure
It can
can be
be seen
seen
again that:
propagating
along5.x-direction
is larger
than thatvelocity
along z-direction.
Figure 6 shows
the
propagating
along
x-direction
is
larger
than
that
along
z-direction.
Figure
6
shows
the
propagating
along
x-directionwaves
is larger
than LiNbO
that along
z-direction.
Figure
6
shows
the
.
We
find
that:
the
velocity
surface
is
velocity
surface
of transverse
for z-cut
3
We find
find that:
that: the
the velocity
velocity surface
surfaceisis
velocity
surface
of
transverse
waves
for
z-cut LiNbO
LiNbO33.. We
velocity
surface
of
transverse
waves
for
z-cut
changed from hexagon for unchanged LiNbO3 to circle for PE LiNbO3. Figure 7 shows the
to circle for
for PE
PE LiNbO
LiNbO33..Figure
Figure77shows
showsthe
the
changed
from
for
LiNbO33 to
changedsurface
from hexagon
hexagon
for unchanged
unchanged
can be seen
that: the velocity
surface is
velocity
of L- waves
for z-cut LiNbO
LiNbO3. Itcircle
. It can be
be seen
seen that:
that: the
the velocity
velocity surface
surface isis
velocity
of
waves
for
z-cut
velocity surface
surface
of LL-change
waves of
forL-wave
z-cut LiNbO
LiNbO
33. It can along
changed,
the relative
propagating
y-direction
is larger than that
changed,
the
relative
change
of L-wave
L-wave propagating
propagating along
along y-direction
y-direction isis larger
larger than
than that
changed,
the
relative
change
of
along x-direction, but not change can be seen, when the wave propagating alongthat
the
along
x-direction,
but
not
change
can
be
seen,
when
the
wave
propagating
along
the
along
x-direction,
but
not
change
can
be
seen,
when
the
wave
propagating
along
the
direction with a deviating angle from x-axis 15° -45°, 135° -165°, 195° -225°, 315° -345°.
direction
with
deviating
angle from
from x-axis 15°
-45°, 135°
-165°, 195°
195° -225°,
-225°, 315°
315° -345°.
-345°.
direction
with aathe
deviating
15° -45°,
135°LiNbO
-165°,
Figure
8 shows
velocityangle
surface ofx-axis
T- waves
for x-cut
3. It can be seen that: the
can be
be seen
seen that:
that: the
the
Figure
of TT- waves
for x-cut
x-cut LiNbO
LiNbO3.. ItIt can
Figure 88 shows
shows the
the velocity
velocity surface
surface of
waves for
direction of maximum velocity seems turn an angle of 60°, along3 inverse clock direction,
direction
turn an
an angle
angle of
of 60°,
60°, along
along inverse
inverse clock
clock direction,
direction,
direction of
of maximum
maximum velocity
velocity seems
seems turn
from the angle 30 ° for unchanged LiNbO3 to 90° for PE LiNbO3.
to 90°
90° for
for PE
PE LiNbO
LiNbO33..
from
LiNbO33 to
from the
the angle
angle 30
30 °° for
for unchanged
unchanged LiNbO
7000
7000
7000
6000
6000
6000
5000
5000
5000150
4000
150
4000
4000
3000
3000
3000
2000
2000:
2000
1000
1000
1000
0 180
0 180
180 Iji 4
10000
:
1000
1000
2000
2000
2000
3000
3000
3000
4000
210
40004000
210
5000
5000
6000
6000
7000
70007000
Velocity
T-wave
(m/s)
Velocity
ofof
T-wave
(m/s)
Velocity
of
T-wave
(m/s)
For Y-cut LiNbO
3
4800
For
For Y-cut
Y-cut LiNbO
LiNbC)3
4800 '*••
Qusi-T for pure crystal
Qusi-T
for
purecrystal
crystal
* • •9
• Qusi-T
Qusi-Tfor
for PE:
pure
crystal
4400
4400
PE: crystal
%
A Qusi-T
Qusi-T for
for PE:
crystal
%%
4000
4000
3600
3600
3200
3200 0
0o
X
X
X
X>
^V
4 ^
20
20
20
40
40
' 60
go
60
*""
80
so Z
80
Z
Z
Angle (degree)
Angle (degree)
(degree)
Angle
90X
90X
120
120
L-W pure LiNbO3
L-W pure LiNbO3 3
L-WpureLiNb0
L-W PE LiNbO3
L-W PE LiNbO3
L-WPELiNbO,
60
60
30
30
0 Z
O0 ZZ
330
330
240
300
240
240
300
300
270
270
270
FIGURE
FIGURE33 The
transverse wave
wave
FIGURE
The velocity
velocity of
of the
the transverse
versus
the
angle
α.
for
y-cut
LiNbO
3,, including
versus
the
angle
a.
for
y-cut
LiNbO
versus the angle α.
including
33 including
piezoelectricity
piezoelectricity for
piezoelectricity
for pure
pure and
and PE
PE LiNbO
LiNbO33
respectively.
respectively.
respectively.
90 XX
90
4000
40004000
3000
30003000
2000
20002000
120
120
60
60
30003000
3000
40004000
4000
3
of pure LiNbO
x T2
T2ofpureLiNb0
T2 of pure LiNbO333
of Pure LiNbO
o T1
TlofPureLiNb0
T1 of Pure LiNbO33
LiNbO33
* T2
T2of
ofPE:
PE:
T2
of
PE: LiNb0
LiNbO3
T1 of PE: LiNbO33
---TlofPE:LiNb0
T1 of PE: LiNbO
000
1000
10001000
20002000
2000
ZZ
330
330
240
240
30003000
3000
40004000
4000
300
300
270
270
270
3
60
60
30
30
150
150
180
180
0
0
X
X
330
330
210
210
240
240
300
300
270
270
forY-cut
Y-cutLiNbO
LiNbC)
for
for
Y-cut
LiNbO33
Qusi-T-w pureLiNbO
LiNbO
——— Qusi-T-w
Qusi-T-wpure
pure LiNbO3 33
Qusi-T-w PE LiNbO
. Qusi-T-w
Qusi-T-wPELiNbO
PE LiNbO3 3
120
120
4000
40004000
3000
30003000
2000
20002000
1000
10001000
0
0
210
210
3
Y 90
Y
Yg90
o
3
1000
10001000
000-180
180
1000
10001000
20002000
2000
FIGURE
The
velocity
surfaceof
ofthe
theL-wave.
L-wave.For
For
FIGURE
FIGURE 444 The
The velocity
velocity surface
surface
of
the
L-wave.
For
,
including
piezoelectricity
for
pure
y-cut
LiNbO
3
y-cut
LiNbO
,
including
piezoelectricity
for
pure
y-cut LiNbO33, including piezoelectricity for pure
LiNbO
where
the
solid line
line isis
is for
for unchanged
unchanged
LiNbO
LiNbO333... where
where the
the solid
solid
line
for
unchanged
and
the
dash
line
is
for
PE
LiNbO
.
LiNbO
3
3
LiNbO
andthe
thedash
dashline
lineisisfor
forPE
PELiNbO
LiNbO
LiNbO3 and
3. .
30
30
30
150
150
for Y-cut LiNbO3
for
forY-cut
Y-cutLiNbC)
LiNbO3
For
ForZ-cut
Z-cutLiNbO,
LiNbO3
For
Z-cut
LiNbO
3
FIGURE555The
Thevelocity
velocity surface
surface of
of
transverse
FIGURE
FIGURE
The
velocity
surface
of transverse
transverse waves
waves
fory-cut
y-cutLiNbO
LiNbO33,,, including
including piezoelectricity
for
for
pure
for
y-cut
LiNbO
piezoelectricity for
for pure
pure
3 including piezoelectricity
LiNbO3.3..
LiNbO
LiNbO
3
FIGURE
FIGURE 66 The
The velocity
velocity surface
transverse
surface of
of transverse
transverse
waves
including
waves for
for z-cut
z-cut LiNbO
including
LiNbO3,33, including
piezoelectricity
piezoelectricity for
forpure
pureLiNbO
LiNbO3.3.
1281
3
m/s
90
L-w pure LiNbO3
Y
Z
L-w PE LiNbO3
4000
120
60
3500
6000
3000
5000
L-w
pure
LiNbO
L-w
pure
LiNbO
33 2500
150
4000
30
150
Y
90
L-w
L-wPE
PELiNbCX
LiNbO3 2000
3000 m/s
1500
4
0
0
0
H
4000
120
60
200060001000
3500
3500:
6000
10005000500
3000
3000
5000
0 : 180 150
180
0 X
2500
04000
2500
4000:
30
150
500
2000
2000
100030003000
1500
1000
1500
200020002000
1000
1500
1000:
30001000
1000500
500:
2000
330
210
4000 00: 180210
0 X
25000
0 180
500
500
3000
50001000
1000
1000
3500
600020002000
1500240
300
1500:
4000
30003000
2000
2000:
270
210
For Z-cut LiNbO3 330
210
40004000
2500
2500
300050003000:
5000
3500
60006000 7 The velocity surface of L- waves for 3500
FIGURE
FIGURE
240
300
4000
4000
T-wave for pure crystal
T-wave for PE crystal
90
120
60
Z
90
120
T-wave
T-wavefor
forpure
purecrystal
crystal
30
T-wavef
T-wave or
forPE
PEcrystal
crystal
60
30
0
330
0
240
Y
Y
300
270
330
for X-cut LiNbO
3
8240The velocity surface
of T- waves for
300
270
270
ForZ-cutLiNbO
z-cut LiNbO3 including piezoelectricity
piezoelectricity
for pure
x-cut LiNbO3 including
For Z-cut LiNbO for pure
for
forX-cut
X-cutLiNbO,
LiNbO
LiNbO3.
LiNbO3.
FIGURE 7 The
The velocity
velocity surface
surface of
of LL- waves
waves for
for FIGURE
FIGURE 88 The
The velocity
velocity surface
surface of
of TT- waves
waves for
for
z-cut LiNbO33 including
including piezoelectricity
piezoelectricity for
for pure
pure x-cut
including piezoelectricity
piezoelectricity for
for pure
pure
x-cut LiNbO
LiNbO33 including
The velocity surface of T- waves for x-cut
LiNbO
3. is shown in Figure 9. We learn
LiNbCX
LiNbCL
LiNbO
LiNbO
3.
3.
3
3
that the velocity surface is changed near to a velocity surface of T-wave propagation in a
cube face
cubic crystal
classes
PE for
LiNbO
Theofvelocity
surface
of T- for
waves
x-cut
3. LiNbO
shown in
in Figure
Figure 9.9. We
We learn
learn
surface
x-cut
LiNbO33.. isis shown
that Rice
had reported
when the
PE exchange
is carried
thatWe
theremember
velocity surface
is changed
near
aa velocity
surface
of
propagation
in
surface
changed
near to
tothat
velocity
surface
of T-wave
T-wavereaction
propagation
in aa
face
cubiccomplete
crystal
for
LiNbO
outcube
in strong
exchange
results,
accompanied
by a structural transformation
33. .
face ofacid,
crystal classes
classes
for PE
PE
LiNbO
We rhombohedral
remember that LiNbO
Rice had
reported
when
the
exchange
reaction
carried
to the
cubic
perovskite
[2].
With
from the
had
reported that
that
when
the PE
PE
exchangestructure
reaction isis
carried
3 structure
out
in
strong
acid,
complete
exchange
results,
accompanied
by
a
structural
transformation
weaker acids, partially exchanged
materials
be prepared
retain
the LiNbO3
exchange
results, can
accompanied
by a with
structural
transformation
from the
rhombohedral
the
perovskite
structure
[2].inWith
weaker
3 structure
to cubic
the cubic
perovskite
structure
With
from
the
rhombohedral
LiNbO
structure.
We
think that inLiNbO
the
middle
acidtocase,
may
have mixed
structure
the[2].
PE
layer.
3 structure
acids,
materials
be prepared
with
retain
the
LiNbO
We3
3 structure.
acids, exchanged
partially
materials
can be
prepared
with
retain
the LiNbO
Forweaker
bulkpartially
exchange,
may exchanged
cause
the can
structure
change,
then
enable
the
velocity
surface
think
that
in
the
middle
acid
case,
may
have
mixed
structure
in
the
PE
layer.
For
bulk
structure.
We
think
that
in
the
middle
acid
case,
may
have
mixed
structure
in
the
PE
layer.
rotation.
exchange,
maydesign
cause wave-guide
the
change,
then
thethen
velocity
surface
For
bulk
exchange,
maystructure
cause the
structure
change,
enable
the rotation.
velocity surface
Thus
we can
by
control
theenable
PE processing
conditions.
Thus
we
can
design
wave-guide
by
control
the
PE
processing
conditions.
rotation.
Thus we can design wave-guide by control the PE processing conditions.
Qusi-T of pure crystal
Z
4000
40003000
3000- 150
4000
2000
20003000
1000
150
10002000180
0
01000
1000
10000 180
2000
20001000 210
3000
30002000
4000
210
4000
3000
4000
120
120
Qusi-T of
crystal
Qusi-T
of pure
PE crystal
90
Qusi-T of PE crystal
60
^^~~"~~T~~~>-^
Z
90
120
Qusi-T of pure crystal
Qusi-T of PE crystal
60
30
30
0
Y
0
Y
330
240
240 "^<C___[_
270
270
240
300
330
for
LiNbO33
for X-cut
X-cut LiNb0
300
FIGURE
9 The
velocity
including
piezoelectricityforfor
pure
LiNbO
FIGURE
9 The
velocitysurface
surfaceofofT-T-waves
wavesfor
for x-cut
x-cut LiNbO
LiNbOfor
pure
LiNbO
X-cut
LiNbO3piezoelectricity
33 including
3. 3.
270
FIGURE 9 The velocity surface of T- waves for x-cut LiNbO3 including piezoelectricity for pure LiNbO3.
1282
CONCLUSIONS
As mentioned above, some conclusions can be given. We calculate the velocity surface
of bulk acoustic waves, including the L- and T-waves, propagating in the YZ-, XY- and
ZX- planes of proton-exchanged and unexchanged LiNbO3, based on the theory of
acoustic wave in anisotropic solid with piezoelectricity. The calculation results show that
the value of velocity change due to proton-exchange when waves propagating along
X-axis is larger than that along Z- axis for Y-cut crystal, in spite of Longitudinal or
Transverse wave. The curve of velocity surface for PE Z-cut substrate is near to a circle,
while that for pure Z-cut LiNbO3 is a hexagonal. The velocity surface for PE X-cut LiNbO3
seems to be turned an angle from that for pure X-cut LiNbO3. And the value of L-wave
calculated is the same as the measured by Hinkov et al., when using the elastic constants
evaluated by Biebl and multiply 0.8.
ACKNOWLEDGMENTS
This work is supported by The State Key Lab. of Modern Acoustics, Nanjing
University, and Nanjing, China.
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