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Nonlinear elastic coefficients of KDP
ELSEVIER
Physica B 219&220 (1996) 584 586
Nonlinear elastic coefficients of K D P U. Straube*, H.
Beige
Martin-Luther-Universitiit Halle-Wittenberg, Fachbereich Physik, Friedemann-Bach-Platz 6, 06108 Halle/Saale, Germany
Abstract A new apparatus for the measurement of velocity changes due to applied mechanical uniaxial stress working from room temperature down to liquid nitrogen temperatures is described. The experimental data obtained with this apparatus allow to calculate nonlinear elastic stiffnesses. Results on Potassium Dihydrogen Phosphate (KDP) are presented. No anomaly of the nonlinear coefficient c456 near the phase transition could be found contrary to the Landau theory but in agreement with former measurements of the nonlinear piezomodul e345.
Potassium dihydrogen phosphate (KDP) undergoes a first-order phase transition from a paraelectric hightemperature tetragonal phase with the point group 742m to a ferroelectric orthorhombic phase [1]. Many papers concerning linear elastic and electromechanic properties are published (e.g. Refs. [2 4]). Korobov [5], Petrakov et al. [6], Sysoev et al. [7] and Straube et al. I-8] measured nonlinear elastic and electromechanical values. An ultrasonic pulse method was modified to investigate the temperature dependence of the nonlinear elastic properties of KDP. The crystals were exposed to continuous variable uniaxial pressures. These pressures were produced by means of a piezoelectric actuator with a maximum dilatation of 20 gm at 1 kV voltage. The force was measured with a sensor made of a bridge of dilatation resistors on a steel plate exposed to this force. In this way the deformation of the steel plate could be found from the bridge voltage. Especially care had to be taken to compensate the different thermal expansion coefficients of the materials. This was achieved with a special construction of invar and steel elements. The force was about 30 N at a sample area of l cm 2.
* Corresponding author.
The ultrasound system consists of a stable RF generator that excites a quartz transducer of 20 MHz with short pulses. The transit time variation introduced by the low-frequency alternating uniaxial pressure is converted into an alternating voltage via the slope of a delayed RF echo cycle. The sampling oscilloscope used for this procedure is coupled to a computer allowing to lower the noise by summation of the values. Transit time variations of 10-6-10 7 can be detected if the attenuation is low. This system allows the automatization of the velocity measurement with a tracking procedure using a zero crossing of one HF oscillation in a definite echo pulse. Details will be published elsewhere [9]. Nonlinear coefficients are calculated following Brugger [10]. The relation for the uniaxial stress derivative (P0 W 2 ) , = 2 p o v g F + G
(1)
holds for uniaxial pressures perpendicular to the sound propagation direction. The material density is denoted by po, the initial sound velocity by vo and the natural wave speed W is calculated using the unstressed sound path length. F and G denote linear and nonlinear elasticcontributions. The following arrangements of sound propagation directions, polarizations and force directions were chosen
0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 8 1 9 - 5
U. Straube, H. Beige/Physica B 219&220 (1996) 584 586 for the first measurements in KDP. The sound direction was always parallel to the z-axis and the force was applied in [1 1 0J-direction (?-direction). I
longitudinal sound,
Expressions for the other cases can be found in the same way. The determination of the nonlinear coefficient c456 is possible combining the results of the cases II and Ill and taking into account 1/$66 = C66 and Po v~ = c44.
II
transverse sound, polarization perpendicular to the force,
2400'
III
transverse sound, polarization parallel to the force.
2395
F and G have, for these cases, the form: I:
585
,%,.
2390.
F = s~3,
(2)
G : (Sll ~- S12)C133 "q- S13C333,
(3)
E
%-,,,. 2385,
2380.
II:
%..
(4)
F=½(Slt +s12)-¼s~,6,
2375
O = [(SI1 -~- S12)(C144 -- C155)
120
I;o
I~o
+ 2S13C344 -- 2S~6 C4.56],
(6)
III: F = ½(sll + s12) + 1 s 6 6 ,
~;o
I;o
~o
i;o
1;0
TinK
(5)
Fig. 2. Transverse sound velocity in z-direction on temperature.
G = ½[(sl~ + s12)(c1,~4 + c155) 50-
(7)
-]- 2S13C344 -1- 2S66C4.56 ].
45-
Figs. 1 and 2 show the temperature dependence of the longitudinal velocity v3 and of the transverse velocity v4. The results are well known from literature and show the good sensitivity of the method. The temperature dependence of the relative velocity variation due to uniaxial stress Av/(v AT3,,) is shown for the cases I, II and III in Figs. 3, 4 and 5, respectively. The results are connected with the nonlinear coefficients via formula (1) with F and G selected from the cases I, II and III. We find for the case I, 2poV 2 [Av/(v3A T3;.)] = 2pOV~Sl3 + (Sll q- S12) C133 -~-"S13C333
(8)
't~ 13_
4035-
.=_
3025-
£
20151050
1~o
1;o TinK
1,~o
2;0
Fig. 3. Temperature dependence of the relative velocity variation of longitudinal ultrasound along the z-axis due to uniaxial stress in [1 1 0]-direction.
with Po v32 = c33 and the stress amplitude A T3.,.. 50-
5020-
'm
40 -
IJi!gl Ii 5010-
.,,
500O-
E
30-
q 20-
k
.
4990-
10-
4980lf02 4970-
I 140
I 160
I 180
TinK
1~o
1~0
1;o
~;o
2;0
Tin K
Fig. 1. Longitudinal sound velocity in z-direction on temperature.
Fig. 4. Temperature dependence of the relative velocity variation of transverse ultrasound along the z-axis due to uniaxial stress in [110]-direction; stress perpendicular to the polarization of the wave.
586
U. Straube, H. Beige/Physica B 219&220 (1996) 584-586 Tin K
TinK 12o
14o
16o
180
200
t20
o
0
140
160
180
200 i
-2-
#_
-lO -4-
-20 -
~/-
._¢
~
-6-
0 •~
-30 -
o~
-8
-10
" -40 -12 -
-5c
Fig. 5. Temperature dependence of the relative velocity variation of transverse ultrasound along the z-axis due to uniaxial stress in [1 1 0J-direction; stress parallel to the polarization of the wave, Squares and triangles from different measurement runs.
Fig. 6. Nonlinear elastic stiffness coefficient c456 from fitted measurement results.
Thus, we get the result
constant, too to give a critical index of - 1 for e345 as predicted from Landau theory. N o w we have found eperimentally this constant behaviour of c456. To find reproducible results the electric field dependence A v / ( v A E ) sould be proved in crystals cut from the same mother crystals like for the stress experiment. The described method is well suited to measure small velocity variations caused by an uniaxial force. The main task is now to enhance the reproducibility and the resolution of the experimental setup.
C456 =
2C44 { [Av4 +/(v4A T3~) - A v 4 , / ( v 4 A T3~.)] C 6 6
-- ½}-
(9)
The value Av4+ denotes the speed variation of case II and AV41 t the speed variation of case III. Although the results are noisy we made an approximation for the results of II and III. The best fits were achieved with an exponential function. The temperature dependence of the anomalous part of c66 follows a 1 / ( T - To) law, where Tc stands for the transition temperature. The temperature dependence of the nonlinear coefficient c456 is calculated from the known data and displayed in Fig. 6. The result shows a nearly constant value of c456 over the whole measured temperature region. The sharp decrease near the transition may be an error of fitting. In summary, the obtained temperature effects of the velocity via stress dependencies are compensated by the decrease of the coefficient c66 near the transition. Expanding the Landau theory to nonlinear coefficients and taking into consideration the symmetry changes as done by Beige [11], a 1/(T - Tc) law should be valid for c456 too. Perhaps the coupling coefficients are too small. The result coincides with the conclusion of another publication from Straube et al. [8]. The nonlinear piezomodul e345 is calculated from
We thank Mr. Sergej Fadeev from Voronesh (Russia) for helpful comments during the construction of the stress apparatus.
References [1] [2] [3] [4] [5] [6]
[7] e345 = d36 C 4 5 6
--
2po v4 [ d 3 6
--
Av/(vAE)].
(10)
The polarization lies in the [1 1 0]-direction. The effect A v / ( v A E ) introduced from the electrical field E in zdirection is strong and has a critical exponent of - 1. The linear piezomodul shows the same temperature behaviour. Since pov4 is nearly constant, c456 should be
[8] [9] [10] [11]
G. Busch and P. Scherrer, Naturwiss. 23 (1935) 737. W.P. Mason, Phys. Rev. 69 (1946) 173. B. Zwicker, Helv. Phys. Acta 19 (1946) 523. A. yon Arx and B. Bantle, Helv. Phys. Acta 17 (1944) 298. A.I. Korobov, Kandidatskaya Dissertatsiya, Moskva (1979). V.S. Petrakov, N.G. Sorokin, S.I. Chizhikov, M.P. Shaskolskaya and A.A. Blislanov, Izv. Akad. Nauk SSSR Ser. Fiz. 39 (1975) 974. A.M. Sysoev, M.P. Zaitseva, Yu.I. Kokorin and K.S. Aleksandrov, Ferroelectrics 71 (1987) 247. U. Straube, A.I. Korobov, Yu.A. Brashkin and O.Yu. Serdobolskaya, Acta Phys. Slov. 40 (1990) 64. U. Straube, P. Grau, S. Fadeew and H. Beige, Acta Phys. Slov., to be published. K. Brugger, J. Appl. Phys. 36 (1965) 768. H. Beige, Dissertation B, Halle (1980).
Physica B 219&220 (1996) 584 586
Nonlinear elastic coefficients of K D P U. Straube*, H.
Beige
Martin-Luther-Universitiit Halle-Wittenberg, Fachbereich Physik, Friedemann-Bach-Platz 6, 06108 Halle/Saale, Germany
Abstract A new apparatus for the measurement of velocity changes due to applied mechanical uniaxial stress working from room temperature down to liquid nitrogen temperatures is described. The experimental data obtained with this apparatus allow to calculate nonlinear elastic stiffnesses. Results on Potassium Dihydrogen Phosphate (KDP) are presented. No anomaly of the nonlinear coefficient c456 near the phase transition could be found contrary to the Landau theory but in agreement with former measurements of the nonlinear piezomodul e345.
Potassium dihydrogen phosphate (KDP) undergoes a first-order phase transition from a paraelectric hightemperature tetragonal phase with the point group 742m to a ferroelectric orthorhombic phase [1]. Many papers concerning linear elastic and electromechanic properties are published (e.g. Refs. [2 4]). Korobov [5], Petrakov et al. [6], Sysoev et al. [7] and Straube et al. I-8] measured nonlinear elastic and electromechanical values. An ultrasonic pulse method was modified to investigate the temperature dependence of the nonlinear elastic properties of KDP. The crystals were exposed to continuous variable uniaxial pressures. These pressures were produced by means of a piezoelectric actuator with a maximum dilatation of 20 gm at 1 kV voltage. The force was measured with a sensor made of a bridge of dilatation resistors on a steel plate exposed to this force. In this way the deformation of the steel plate could be found from the bridge voltage. Especially care had to be taken to compensate the different thermal expansion coefficients of the materials. This was achieved with a special construction of invar and steel elements. The force was about 30 N at a sample area of l cm 2.
* Corresponding author.
The ultrasound system consists of a stable RF generator that excites a quartz transducer of 20 MHz with short pulses. The transit time variation introduced by the low-frequency alternating uniaxial pressure is converted into an alternating voltage via the slope of a delayed RF echo cycle. The sampling oscilloscope used for this procedure is coupled to a computer allowing to lower the noise by summation of the values. Transit time variations of 10-6-10 7 can be detected if the attenuation is low. This system allows the automatization of the velocity measurement with a tracking procedure using a zero crossing of one HF oscillation in a definite echo pulse. Details will be published elsewhere [9]. Nonlinear coefficients are calculated following Brugger [10]. The relation for the uniaxial stress derivative (P0 W 2 ) , = 2 p o v g F + G
(1)
holds for uniaxial pressures perpendicular to the sound propagation direction. The material density is denoted by po, the initial sound velocity by vo and the natural wave speed W is calculated using the unstressed sound path length. F and G denote linear and nonlinear elasticcontributions. The following arrangements of sound propagation directions, polarizations and force directions were chosen
0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 8 1 9 - 5
U. Straube, H. Beige/Physica B 219&220 (1996) 584 586 for the first measurements in KDP. The sound direction was always parallel to the z-axis and the force was applied in [1 1 0J-direction (?-direction). I
longitudinal sound,
Expressions for the other cases can be found in the same way. The determination of the nonlinear coefficient c456 is possible combining the results of the cases II and Ill and taking into account 1/$66 = C66 and Po v~ = c44.
II
transverse sound, polarization perpendicular to the force,
2400'
III
transverse sound, polarization parallel to the force.
2395
F and G have, for these cases, the form: I:
585
,%,.
2390.
F = s~3,
(2)
G : (Sll ~- S12)C133 "q- S13C333,
(3)
E
%-,,,. 2385,
2380.
II:
%..
(4)
F=½(Slt +s12)-¼s~,6,
2375
O = [(SI1 -~- S12)(C144 -- C155)
120
I;o
I~o
+ 2S13C344 -- 2S~6 C4.56],
(6)
III: F = ½(sll + s12) + 1 s 6 6 ,
~;o
I;o
~o
i;o
1;0
TinK
(5)
Fig. 2. Transverse sound velocity in z-direction on temperature.
G = ½[(sl~ + s12)(c1,~4 + c155) 50-
(7)
-]- 2S13C344 -1- 2S66C4.56 ].
45-
Figs. 1 and 2 show the temperature dependence of the longitudinal velocity v3 and of the transverse velocity v4. The results are well known from literature and show the good sensitivity of the method. The temperature dependence of the relative velocity variation due to uniaxial stress Av/(v AT3,,) is shown for the cases I, II and III in Figs. 3, 4 and 5, respectively. The results are connected with the nonlinear coefficients via formula (1) with F and G selected from the cases I, II and III. We find for the case I, 2poV 2 [Av/(v3A T3;.)] = 2pOV~Sl3 + (Sll q- S12) C133 -~-"S13C333
(8)
't~ 13_
4035-
.=_
3025-
£
20151050
1~o
1;o TinK
1,~o
2;0
Fig. 3. Temperature dependence of the relative velocity variation of longitudinal ultrasound along the z-axis due to uniaxial stress in [1 1 0]-direction.
with Po v32 = c33 and the stress amplitude A T3.,.. 50-
5020-
'm
40 -
IJi!gl Ii 5010-
.,,
500O-
E
30-
q 20-
k
.
4990-
10-
4980lf02 4970-
I 140
I 160
I 180
TinK
1~o
1~0
1;o
~;o
2;0
Tin K
Fig. 1. Longitudinal sound velocity in z-direction on temperature.
Fig. 4. Temperature dependence of the relative velocity variation of transverse ultrasound along the z-axis due to uniaxial stress in [110]-direction; stress perpendicular to the polarization of the wave.
586
U. Straube, H. Beige/Physica B 219&220 (1996) 584-586 Tin K
TinK 12o
14o
16o
180
200
t20
o
0
140
160
180
200 i
-2-
#_
-lO -4-
-20 -
~/-
._¢
~
-6-
0 •~
-30 -
o~
-8
-10
" -40 -12 -
-5c
Fig. 5. Temperature dependence of the relative velocity variation of transverse ultrasound along the z-axis due to uniaxial stress in [1 1 0J-direction; stress parallel to the polarization of the wave, Squares and triangles from different measurement runs.
Fig. 6. Nonlinear elastic stiffness coefficient c456 from fitted measurement results.
Thus, we get the result
constant, too to give a critical index of - 1 for e345 as predicted from Landau theory. N o w we have found eperimentally this constant behaviour of c456. To find reproducible results the electric field dependence A v / ( v A E ) sould be proved in crystals cut from the same mother crystals like for the stress experiment. The described method is well suited to measure small velocity variations caused by an uniaxial force. The main task is now to enhance the reproducibility and the resolution of the experimental setup.
C456 =
2C44 { [Av4 +/(v4A T3~) - A v 4 , / ( v 4 A T3~.)] C 6 6
-- ½}-
(9)
The value Av4+ denotes the speed variation of case II and AV41 t the speed variation of case III. Although the results are noisy we made an approximation for the results of II and III. The best fits were achieved with an exponential function. The temperature dependence of the anomalous part of c66 follows a 1 / ( T - To) law, where Tc stands for the transition temperature. The temperature dependence of the nonlinear coefficient c456 is calculated from the known data and displayed in Fig. 6. The result shows a nearly constant value of c456 over the whole measured temperature region. The sharp decrease near the transition may be an error of fitting. In summary, the obtained temperature effects of the velocity via stress dependencies are compensated by the decrease of the coefficient c66 near the transition. Expanding the Landau theory to nonlinear coefficients and taking into consideration the symmetry changes as done by Beige [11], a 1/(T - Tc) law should be valid for c456 too. Perhaps the coupling coefficients are too small. The result coincides with the conclusion of another publication from Straube et al. [8]. The nonlinear piezomodul e345 is calculated from
We thank Mr. Sergej Fadeev from Voronesh (Russia) for helpful comments during the construction of the stress apparatus.
References [1] [2] [3] [4] [5] [6]
[7] e345 = d36 C 4 5 6
--
2po v4 [ d 3 6
--
Av/(vAE)].
(10)
The polarization lies in the [1 1 0]-direction. The effect A v / ( v A E ) introduced from the electrical field E in zdirection is strong and has a critical exponent of - 1. The linear piezomodul shows the same temperature behaviour. Since pov4 is nearly constant, c456 should be
[8] [9] [10] [11]
G. Busch and P. Scherrer, Naturwiss. 23 (1935) 737. W.P. Mason, Phys. Rev. 69 (1946) 173. B. Zwicker, Helv. Phys. Acta 19 (1946) 523. A. yon Arx and B. Bantle, Helv. Phys. Acta 17 (1944) 298. A.I. Korobov, Kandidatskaya Dissertatsiya, Moskva (1979). V.S. Petrakov, N.G. Sorokin, S.I. Chizhikov, M.P. Shaskolskaya and A.A. Blislanov, Izv. Akad. Nauk SSSR Ser. Fiz. 39 (1975) 974. A.M. Sysoev, M.P. Zaitseva, Yu.I. Kokorin and K.S. Aleksandrov, Ferroelectrics 71 (1987) 247. U. Straube, A.I. Korobov, Yu.A. Brashkin and O.Yu. Serdobolskaya, Acta Phys. Slov. 40 (1990) 64. U. Straube, P. Grau, S. Fadeew and H. Beige, Acta Phys. Slov., to be published. K. Brugger, J. Appl. Phys. 36 (1965) 768. H. Beige, Dissertation B, Halle (1980).