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Ultrasonic properties of Ag3SI
Solid State Ionics 40/41 North-Holland
( 1990) 205-208
ULTRASONIC Yoshitaka
PROPERTIES
MICHIHIRO,
OF Ag,SI
Tatsuo KANASHIRO,
Department of Physics, Faculty of Engineering,
Yutaka KISHIMOTO,
Takashi OHNO
Tokushima University, Tokushima 770. Japan
and Akira KOJIMA Osaka Institute of Technology Junior College, Osaka 535, Japan
The ultrasonic attenuation of Ag,SI has been measured for longitudinal waves at frequencies of 10, 15,30 and 50 MHz over the range from slightly above the liquid nitrogen temperature to room temperature with a standard pulse echo method. It is shown from an Arrhenius plot of the peak temperature for each frequency that the attenuation is owing to the relaxation phenomenon of the thermally activated process. A relaxation function taking account of the interactions between ions is employed in the treatment of the data. The activation energy is obtained for the hopping motion of the mobile ions and found in good agreement with the values given by other experiments. The estimation about the magnitude of the mutual coupling between mobile ions is tried.
1. Introduction It is well known that a class of silver compounds takes an averaged structure and exhibits high ionic conductivity [ 11. One of such a class is Ag,SI, which possesses three crystallographic phases [2]. In the a-phase ( TI 519 K), the crystal structure is bee with a disordered arrangement of I- and S2- ions like a CsZn type alloy, and Ag+ ions take a statistical distribution among interstitial sites. The P-phase ( 157 s TS 519 K) is a simple cubic structure with Iand S2- ions ordered in a CsCl type array, while Ag+ ions are still disordered. In the y-phase ( TS 157 K), the crystal is a rhombohedral structure and Ag+ ions are also ordered. The crystal of Ag,SI is superionic in the a- and P-phase, while the ionic conductivity is low in the y-phase. Physical properties of Ag,SI have been studied by various experimental methods including X-ray diffraction [ 2,3 1, specific heat [ 2,3 1, far infrared [ 45 1, neutron scattering [ 6,7 1, and ionic conductivity [ 8,9 1. There has been, however, rather a little amount of information about the ultrasonic properties [ lo121. We measured ultrasonic attenuation of Ag,SI in the a-quenched phase. 0167-2738/90/$03.50 ( North-Holland )
0 Elsevier Science Publishers
B.V.
Models of the ultrasonic attenuation in superionic conductors were proposed by Huberman and Martin [ 13 1. The microscopic view of those models was discussed by Ishii [ 14 1. We present here an analysis for the ultrasonic attenuation of Ag,SI employing a modified model of Huberman and Martin. The activation energy and other physical parameters are also discussed.
2. Experimental As the main experimental procedures are the same as described in the foregoing work [ 111, only the outline is presented here for convenience. A crystal of Ag,SI with mosaic structure was grown from an equimolar mixture of AgI and Ag,S with an improved vertical Bridgman method [ 151. The obtained crystal was cut into the form of an almost cylindrical rod approximately 7 mm in diameter and 4 mm long. Both the end faces of this sample were polished so as to be flat and parallel to each other and an X-cut quartz transducer was bonded to one of them.
Y. Michihiro et al. / Ultrasonic properties ofAg,SI
206
We measured the attenuation of longitudinal ultrasonic waves at frequencies of 10, 15, 30, 50 and 70 MHz with a standard pulse echo method. The measurements were done over the range about from 77 K to 280 K and the changing rate was fixed at 1 K/min or slower.
3. Results and discussion
of the scope. In the following we focus our attention on the region above 100 K. For the quenched sample the peak of the ultrasonic attenuation was seen at about 150 K- 180 K, for increasing the frequency the peak temperature became higher. An Arrhenius plot of the peak temperature for each frequency is drawn in fig. 2. This shows that the peak temperature obeys the relation exp(-E/k,T)
w=o,
The temperature dependence of the ultrasonic attenuation constant is shown in fig. la and b. Below about 157 K, no echo was observed for the annealed sample (a); this may be owing to the structural change at the p-y transition temperature [ 111. In contrast with the annealed sample (a), the echo was still observed below 157 K for the sample (b); this shows that the transition does not occur for the quenched sample. An anomalous attenuation was seen at the region near 80 K for the quenched sample. The measurement was not possible for the range lower than the liquid nitrogen temperature owing to the experimental limitations, so the discussion is out
,
(1)
where w is the angular frequency, T the absolute temperature, kB the Boltzmann constant and E and w, are constants. So the attenuation is owing to the relaxation phenomenon of the thermally activated process; the constant E is an activation energy for the hopping motion of the mobile ions and constant o, is a prefactor. From the least-square fitting of eq. ( 1) to the Arrhenius plot, the activation energy E is estimated to be 0.15 - 0.23 eV and the prefactor wi to be (l.O- 10) x 1013 Hz. The value of the activation energy is in good agreement with those obtained from other experiments [ 81. The value of the prefactor is reasonable despite the fact that the one ob-
60
00
-
40
-
2 0 \
E 0 \
z
40
MHZ 50
v
z 1 u
30
MHI 70 20
20
50
15
30
10
10
o-
0 100
200
T
Fig. 1. Temperature dependence sample [ 111 and (b ) quenched
300
100
300
T (Ki
(K)
of the ultrasonic sample.
200
attenuation
constant,
(Y,for the longitudinal
waves at various frequencies:
(a) annealed
201
Y. Michihiro et al. / Ultrasonic properties ofAgJI
where g and J are the parameters of the couplings between mobile ions and lattice ions and the mutual coupling between mobile ions. The term in square brackets takes a maximum when eq. (3) is fulfilled, with neglecting the terms of order ( kBT) - ‘. The hopping motion of mobile ions is treated as a thermally activated process, so the relaxation time, 7, is represented as
109
r=roexp(E/keT)
7
6 1000/T
(l/K)
Fig. 2. Arrhenius plot of the peak temperature for each angular frequency.
tained from the Arrhenius plot is somewhat ambiguous. The Debye-type relaxation is that an individual ion hops with a certain relaxation time independently of each other. The ultrasonic attenuation constant, a( T,w), in this model is given as a( T,w) =A,, where v is relaxation The term maximum
&[I
+7&l
2
(2)
the velocity of the ultrasonic wave, 7 the time and A,, the relaxation strength [ 16 1. in the square brackets passes through a when the condition
WT=l,
(3)
is satisfied and this describes a Debye peak. Following the manner of Huberman and Martin [ 13 ] but the mutual interactions between mobile ions, the ultrasonic attenuation was calculated. The ultrasonic attenuation constant, (Y( T,w), is written as the form,
[l-(l,k,TO;J]2+(wr)2
1’
(5)
where E is the activation energy for the hopping motion of mobile ions and z. the prefactor. The background attenuation owing to the diffraction and other effects is included in the measured value of the attenuation constant, (Y,and that is taken to be a constant (Ye.Thus the measured value of the attenuation constant may be expressed as
10’
5
,
(4)
a=a(T,w)+ao.
(6)
We tried the fittings of the Debye model, eqs. (2), (5) and (6), and the model taking account of the interactions between ions, eqs. (4), (5), and (6), to the data. The condition was subjected in the fittings that the value of the activation energy and the inverse of the prefactor were the above mentioned ones for the peak to be prescribed by eq. (3). The results are shown for 30 MHz in fig. 3b; open circles show the data and the dotted and solid lines indicate the Debye and the latter model, respectively. Good fitting was not obtained for the Debye model. The agreement of the latter model was good in the high temperature region but poor in the low temperature region. This may be due to the high temperature approximation employed in the model. For comparison, the fitting of that model to the annealed sample is presented in fig. 3a. In this case, data are rather in a high-temperature region and the model fits the data well. The activation energy is estimated to be 0.09 N 0.16 eV and the prefactor to be about 10 - I2 s. Ishii [ 13 ] discussed the model of Huberman and Martin from microscopic point of view and pointed out the importance of the interactions between ions. We discuss the interactions based on that model briefly in the high temperature region. From the values of the fitting parameters, the magnitude of the coupling parameter J relative to E is 0.03 - 0.6 for the quenched sample and 0.06 N 0.6 for the annealed
Y. Michihiro et al. / Ultrasonic properties of Ag,SI
208
40
20
I
I
100
200
I
300
T (K)
0
I 100
I
I
200
T
300
(K)
Fig. 3. Fitting of the Debye type model, eqs. (2) (5) and (6) and the model taking account of interactions, eqs. (4), (5) and (6) under the condition that the peak prescribed by eq. (3). Open circles show the data and the dotted and solid lines indicate the Debye and the latter model, respectively: (a) annealed sample and (b ) quenched sample.
sample. This shows that the interactions between ions fulfill an important role. Also from those values, it seems favorable that the interactions between mobile ions is repulsive. The detailed discussion on the interactions will be presented elsewhere.
References [l] J.B.BoyceandB.A.Huberman,Phys.Rep. 51 (1979) 189. [2] S. Hoshino, T. Sakuma and Y. Fujii, J. Phys. Sot. Japan 47 (1979) 1252. [3] S. Hoshino, T. Sakuma and Y. Fujii, J. Phys. Sot. Japan 45 (1978) 705. [4] P. Briiesch, H.U. Beyeler and S. Strassler, Phys. Rev. B 25 (1982) 541. [ 5 ] B. Gras and K. Funke, Solid State Ionics 2 ( 198 1) 34 1.
[ 61 S. Hoshino, T. Sakuma, H. Fujishita and K. Shibata, J. Phys. Soc.Japan52(1983)1261. [ 71 J.J. Didisheim, R.K. McMullan and B.J. Wuensch, Solid State Ionics 18/19 (1986) 1150. [8] e.g., A. Kojima, K. Tozaki, T. Ogawa, T. Takizawa and T. Kanashiro, J. Phys. Sot. Japan 57 ( 1988) 176. [9] Y. Michihiro, A. Ohtani, T. Ohno, T. Kanashiro and A. Kojima, Solid State Ionics 35 (1989) 337. [lo] A. Kojima and F. Akao, J. Phys. Sot. Japan 51 (1982) 2377. [ 111 T. Kanashiro, Y. Michihiro, J. Ozaki, T. Ohno and A. Kojima, J. Phys. Sot. Japan 56 (1987) 560. [ 121 Y. Michihiro, A. Kojima, T. Ohno and T. Kanashiro, J. Phys. Soc.Japan58(1989)533. [ 131 B.A. Huberman and R.M. Martin, Phys. Rev. B 13 ( 1976) 1498. [ 141 T. Ishii, Solid State Ionics 28-30 (1989) 67. [ 151 A. Kojima, S. Hoshino, Y. Noda, K. Fujii and T. Kanashiro, J. Cryst. Growth 94 (1989) 293. [ 16 ] N. Mikoshiba, Onpabussei (Sanseido, 1972) [in Japanese].
( 1990) 205-208
ULTRASONIC Yoshitaka
PROPERTIES
MICHIHIRO,
OF Ag,SI
Tatsuo KANASHIRO,
Department of Physics, Faculty of Engineering,
Yutaka KISHIMOTO,
Takashi OHNO
Tokushima University, Tokushima 770. Japan
and Akira KOJIMA Osaka Institute of Technology Junior College, Osaka 535, Japan
The ultrasonic attenuation of Ag,SI has been measured for longitudinal waves at frequencies of 10, 15,30 and 50 MHz over the range from slightly above the liquid nitrogen temperature to room temperature with a standard pulse echo method. It is shown from an Arrhenius plot of the peak temperature for each frequency that the attenuation is owing to the relaxation phenomenon of the thermally activated process. A relaxation function taking account of the interactions between ions is employed in the treatment of the data. The activation energy is obtained for the hopping motion of the mobile ions and found in good agreement with the values given by other experiments. The estimation about the magnitude of the mutual coupling between mobile ions is tried.
1. Introduction It is well known that a class of silver compounds takes an averaged structure and exhibits high ionic conductivity [ 11. One of such a class is Ag,SI, which possesses three crystallographic phases [2]. In the a-phase ( TI 519 K), the crystal structure is bee with a disordered arrangement of I- and S2- ions like a CsZn type alloy, and Ag+ ions take a statistical distribution among interstitial sites. The P-phase ( 157 s TS 519 K) is a simple cubic structure with Iand S2- ions ordered in a CsCl type array, while Ag+ ions are still disordered. In the y-phase ( TS 157 K), the crystal is a rhombohedral structure and Ag+ ions are also ordered. The crystal of Ag,SI is superionic in the a- and P-phase, while the ionic conductivity is low in the y-phase. Physical properties of Ag,SI have been studied by various experimental methods including X-ray diffraction [ 2,3 1, specific heat [ 2,3 1, far infrared [ 45 1, neutron scattering [ 6,7 1, and ionic conductivity [ 8,9 1. There has been, however, rather a little amount of information about the ultrasonic properties [ lo121. We measured ultrasonic attenuation of Ag,SI in the a-quenched phase. 0167-2738/90/$03.50 ( North-Holland )
0 Elsevier Science Publishers
B.V.
Models of the ultrasonic attenuation in superionic conductors were proposed by Huberman and Martin [ 13 1. The microscopic view of those models was discussed by Ishii [ 14 1. We present here an analysis for the ultrasonic attenuation of Ag,SI employing a modified model of Huberman and Martin. The activation energy and other physical parameters are also discussed.
2. Experimental As the main experimental procedures are the same as described in the foregoing work [ 111, only the outline is presented here for convenience. A crystal of Ag,SI with mosaic structure was grown from an equimolar mixture of AgI and Ag,S with an improved vertical Bridgman method [ 151. The obtained crystal was cut into the form of an almost cylindrical rod approximately 7 mm in diameter and 4 mm long. Both the end faces of this sample were polished so as to be flat and parallel to each other and an X-cut quartz transducer was bonded to one of them.
Y. Michihiro et al. / Ultrasonic properties ofAg,SI
206
We measured the attenuation of longitudinal ultrasonic waves at frequencies of 10, 15, 30, 50 and 70 MHz with a standard pulse echo method. The measurements were done over the range about from 77 K to 280 K and the changing rate was fixed at 1 K/min or slower.
3. Results and discussion
of the scope. In the following we focus our attention on the region above 100 K. For the quenched sample the peak of the ultrasonic attenuation was seen at about 150 K- 180 K, for increasing the frequency the peak temperature became higher. An Arrhenius plot of the peak temperature for each frequency is drawn in fig. 2. This shows that the peak temperature obeys the relation exp(-E/k,T)
w=o,
The temperature dependence of the ultrasonic attenuation constant is shown in fig. la and b. Below about 157 K, no echo was observed for the annealed sample (a); this may be owing to the structural change at the p-y transition temperature [ 111. In contrast with the annealed sample (a), the echo was still observed below 157 K for the sample (b); this shows that the transition does not occur for the quenched sample. An anomalous attenuation was seen at the region near 80 K for the quenched sample. The measurement was not possible for the range lower than the liquid nitrogen temperature owing to the experimental limitations, so the discussion is out
,
(1)
where w is the angular frequency, T the absolute temperature, kB the Boltzmann constant and E and w, are constants. So the attenuation is owing to the relaxation phenomenon of the thermally activated process; the constant E is an activation energy for the hopping motion of the mobile ions and constant o, is a prefactor. From the least-square fitting of eq. ( 1) to the Arrhenius plot, the activation energy E is estimated to be 0.15 - 0.23 eV and the prefactor wi to be (l.O- 10) x 1013 Hz. The value of the activation energy is in good agreement with those obtained from other experiments [ 81. The value of the prefactor is reasonable despite the fact that the one ob-
60
00
-
40
-
2 0 \
E 0 \
z
40
MHZ 50
v
z 1 u
30
MHI 70 20
20
50
15
30
10
10
o-
0 100
200
T
Fig. 1. Temperature dependence sample [ 111 and (b ) quenched
300
100
300
T (Ki
(K)
of the ultrasonic sample.
200
attenuation
constant,
(Y,for the longitudinal
waves at various frequencies:
(a) annealed
201
Y. Michihiro et al. / Ultrasonic properties ofAgJI
where g and J are the parameters of the couplings between mobile ions and lattice ions and the mutual coupling between mobile ions. The term in square brackets takes a maximum when eq. (3) is fulfilled, with neglecting the terms of order ( kBT) - ‘. The hopping motion of mobile ions is treated as a thermally activated process, so the relaxation time, 7, is represented as
109
r=roexp(E/keT)
7
6 1000/T
(l/K)
Fig. 2. Arrhenius plot of the peak temperature for each angular frequency.
tained from the Arrhenius plot is somewhat ambiguous. The Debye-type relaxation is that an individual ion hops with a certain relaxation time independently of each other. The ultrasonic attenuation constant, a( T,w), in this model is given as a( T,w) =A,, where v is relaxation The term maximum
&[I
+7&l
2
(2)
the velocity of the ultrasonic wave, 7 the time and A,, the relaxation strength [ 16 1. in the square brackets passes through a when the condition
WT=l,
(3)
is satisfied and this describes a Debye peak. Following the manner of Huberman and Martin [ 13 ] but the mutual interactions between mobile ions, the ultrasonic attenuation was calculated. The ultrasonic attenuation constant, (Y( T,w), is written as the form,
[l-(l,k,TO;J]2+(wr)2
1’
(5)
where E is the activation energy for the hopping motion of mobile ions and z. the prefactor. The background attenuation owing to the diffraction and other effects is included in the measured value of the attenuation constant, (Y,and that is taken to be a constant (Ye.Thus the measured value of the attenuation constant may be expressed as
10’
5
,
(4)
a=a(T,w)+ao.
(6)
We tried the fittings of the Debye model, eqs. (2), (5) and (6), and the model taking account of the interactions between ions, eqs. (4), (5), and (6), to the data. The condition was subjected in the fittings that the value of the activation energy and the inverse of the prefactor were the above mentioned ones for the peak to be prescribed by eq. (3). The results are shown for 30 MHz in fig. 3b; open circles show the data and the dotted and solid lines indicate the Debye and the latter model, respectively. Good fitting was not obtained for the Debye model. The agreement of the latter model was good in the high temperature region but poor in the low temperature region. This may be due to the high temperature approximation employed in the model. For comparison, the fitting of that model to the annealed sample is presented in fig. 3a. In this case, data are rather in a high-temperature region and the model fits the data well. The activation energy is estimated to be 0.09 N 0.16 eV and the prefactor to be about 10 - I2 s. Ishii [ 13 ] discussed the model of Huberman and Martin from microscopic point of view and pointed out the importance of the interactions between ions. We discuss the interactions based on that model briefly in the high temperature region. From the values of the fitting parameters, the magnitude of the coupling parameter J relative to E is 0.03 - 0.6 for the quenched sample and 0.06 N 0.6 for the annealed
Y. Michihiro et al. / Ultrasonic properties of Ag,SI
208
40
20
I
I
100
200
I
300
T (K)
0
I 100
I
I
200
T
300
(K)
Fig. 3. Fitting of the Debye type model, eqs. (2) (5) and (6) and the model taking account of interactions, eqs. (4), (5) and (6) under the condition that the peak prescribed by eq. (3). Open circles show the data and the dotted and solid lines indicate the Debye and the latter model, respectively: (a) annealed sample and (b ) quenched sample.
sample. This shows that the interactions between ions fulfill an important role. Also from those values, it seems favorable that the interactions between mobile ions is repulsive. The detailed discussion on the interactions will be presented elsewhere.
References [l] J.B.BoyceandB.A.Huberman,Phys.Rep. 51 (1979) 189. [2] S. Hoshino, T. Sakuma and Y. Fujii, J. Phys. Sot. Japan 47 (1979) 1252. [3] S. Hoshino, T. Sakuma and Y. Fujii, J. Phys. Sot. Japan 45 (1978) 705. [4] P. Briiesch, H.U. Beyeler and S. Strassler, Phys. Rev. B 25 (1982) 541. [ 5 ] B. Gras and K. Funke, Solid State Ionics 2 ( 198 1) 34 1.
[ 61 S. Hoshino, T. Sakuma, H. Fujishita and K. Shibata, J. Phys. Soc.Japan52(1983)1261. [ 71 J.J. Didisheim, R.K. McMullan and B.J. Wuensch, Solid State Ionics 18/19 (1986) 1150. [8] e.g., A. Kojima, K. Tozaki, T. Ogawa, T. Takizawa and T. Kanashiro, J. Phys. Sot. Japan 57 ( 1988) 176. [9] Y. Michihiro, A. Ohtani, T. Ohno, T. Kanashiro and A. Kojima, Solid State Ionics 35 (1989) 337. [lo] A. Kojima and F. Akao, J. Phys. Sot. Japan 51 (1982) 2377. [ 111 T. Kanashiro, Y. Michihiro, J. Ozaki, T. Ohno and A. Kojima, J. Phys. Sot. Japan 56 (1987) 560. [ 121 Y. Michihiro, A. Kojima, T. Ohno and T. Kanashiro, J. Phys. Soc.Japan58(1989)533. [ 131 B.A. Huberman and R.M. Martin, Phys. Rev. B 13 ( 1976) 1498. [ 141 T. Ishii, Solid State Ionics 28-30 (1989) 67. [ 151 A. Kojima, S. Hoshino, Y. Noda, K. Fujii and T. Kanashiro, J. Cryst. Growth 94 (1989) 293. [ 16 ] N. Mikoshiba, Onpabussei (Sanseido, 1972) [in Japanese].