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Long-time heat release in crystalline ferroelectrics at low-temperatures
Solid State Communications, Vol. 59, No. 9, pp. 643-646, 1986. Printed in Great Britain.
0038-I098/86 $3.00 + .00 Pergamon Journals Ltd.
LONG-TIME HEAT RELEASE IN CRYSTALLINE FERROELECTRICS AT LOW-TEMPERATURES S. Sahling*, A. Sahlingt, B.S. Neganov and M. Kol~:~ Joint Institute for Nuclear Research, Dubna, USSR
(Received 25 March 1986 by E.A. Kaner) Calorimetric measurements were performed with KH2PO4 (KDP) and PbomsLao.0asZro.~sTio.3s (PLZT) at helium temperatures. No heat release was observed in KDP single crystal. Heat release in polycrystalline PLZT after cooling from TI (1.3 K < Tt ~< 292 K) to To = 1.3 K is very similar to that in amorphous metals and dielectrics. Experimental results disagree with the standard tunneling model. The observed heat release may be explained assuming the existence of a maximum energy Ef in the distribution function. The maximum relaxation time ¢max was found as a function of Tt. 1. INTRODUCTION
1 t,h
AS HAS BEEN SHOWN recently [ 1 - 5 ] , the long.time heat release is probably one of the characteristic features of amorphous materials. This heat release was observed in vitreous silica [1, 2], various organic materials [ 2 - 4 ] , and also in*'metallic glasses [5]. A question has arisen about the existence of crystalline substances with similar heat release. For instance, the typical of glasses T.term in the heat capacity of hydrogen-saturated crystalline N b - T i was observed [6] along with the long-time heat release [2]. Amorphous-like heat capacity and heat conductivity were also seen in various polycrystaUine ferroelectrics and their single crystals, too [ 7 - 1 0 ] . To check possible heat releas~iin this type of materials, we performed an exf)erimental study of two of them.
PLZT
o,s
0
20
30
40
50
Fig. 1. The time necessary for cooling the specimens from the temperature TI to the temperature To = 1.3 K. glue was less than 10 mg). Due to their low heat capacity, the cooling of the specimens from the initial temperature TI (1.3 K < Tt ~< 292 K) to the measuring temperature To = 1.3 K was fairly rapid (Fig. 1). Before each cooling, the specimen remained at T1 at least for ten hours. During the experiments, the heat capacities of both specimens were measured.
2. EXPERIMENTAL Two specimens were studied: polycrystaUine I'bo.9tsLao.oss. Zro.~Tio.as (transition temperature T¢ = 370 K, the mass m = 50.6 g) (PLZT), and a single crystal of KI-I2PO4 (T¢ ~ 123 K, m = 94.8 g) (KDP). The measurements were performed in a 4Hecalorimeter [11]. Copper foil (30Wn thick) with a Ge-thermometer, a heater and the contact for the heat switch was glued to the specimens (the amount of the
* On leave of absence from the Technical University, Dresden, GDR. t On leave of absence from the Central Institute for Solid State Physics and Material Research, Academy of Sciences of GDR, Dresden, GDR. On leave of absence from the Institute of Physics, Czechoslovak Academy of Sciences, Prague, Czechoslovakia.
10
3. RESULTS AND DISCUSSION The measured heat capacity is shown in Fig. 2. In agreement with the previous measurements [7], PLZT heat capacity contains a rather large term roughly proportional to the temperature at T < 3 K. Such term is much smaller in KDP. Assuming
c = a T + b T 3,
(1)
we have from our experiment for 1 K ~< T ~< 1.5 K, a = (3.1 + 0.2)/d gK-2 , b = (0.8 + 0.2) Id gK -4 for PLZT, and a = (0.5 + 0.1)/~JgK -2, b = (2.05 + 0.10)/dgK -4
643
644
CRYSTALLINE FERROELECTRICS AT LOW-TEMPERATURES ,
c/T3'/U J;g K4
Vol. 59, No. 9
0,2
Cl[lOmin], nWlg
'
0,2
1
2
/f
x- 0,085,
y - 0,85
0,1
2
5
z, K z T•-1,3
10
I
I
0
Fig. 2. Specific heat of the specimens. 1 : PLZT experimental data; 2: PLZT_, [7] ; 3: c / T a - - a l l / T 2 ;4: c / T a -a/T 2 ; 5: b + CEI/T ~, ,v: = 0.21 mJ/gK, OEt = 15.5 K; 6: b + C E 2 / T °, ~2 = 5 m J / g K , OE2 = 3 4 . 5 K ; 7: b +
CE3/T~,
Ota=48mJ/gK ,
0Ea=72K;
8:
b+(Z~= 1
CEi)/T ~ ;9: KDP experimental data.
for KDP. Maybe, because of a small value of a no heat release was observed in KDP. On the other hand, distinct long-time heat release was seen in PLZT. The specific heat release t~ in this material for various initial temperatures TI and To = 1.3 K is shown in Fig. 3. The results are very similar to those of the experiments with amorphous materials: (a) q(t) ~ t -1 for small t, (b) q(t) does not significantly depend on T1 for Ts > 20 K. Such behaviour in vitreous
10s
I
100
200
Fig. 4. Specific heat release in PLZT 10min after the cooling from TI to To = 1.3 K as a function of T~ -1.32 K~. Open circles - t n = 10h~ triangles-tt-t = 5 min. Curve 1: equation (7) with Pm = 1.3__"10as/jg, 12 = 1 (Tf-~oo). Curve 2: equation (7) w i t h P m = 1.3" 103S/jg, Tf = 7.5K, T b = O.
silica [2] and glassy metals [5] was shown to be explainable by the following assumption about the density-of-states of two-level systems: n(E, t) = no(t) (1 + exp ((E--Ef)/kl~Tb)),
~1,pW/g
Ne J--~-O,"°t
101
,L¢O"
,,
1
i 1 ,oTt'K .:
101
(2)
where E f and Tb are constant and no(t) =/~ln (4t/rm~a); man is the shortest relaxation time and/~ is the parameter of the standard tunneling assumption P(ZX, X) = L
TMOX, i l i l
&
(3)
where p(A, ~.) is the distribution function of two-well energy difference A and tunneling parameter h. Equation (2) results in the heat capacity c t = Or2k~/12)rPmIl (T/Tt, Tb) In (4t/train),
10
.
.
.
.
'
c/T,/u J/gK 2
".,, 10'
!
.
.
.
.
!
(4)
(
0 0
0
!
lO~
lO!
Fig. 3. Time dependence of the specific heat release in PLZT after cooling from T1 to To = 1.3K. o: tu = 10h, 1.49K~
~0
0
O 0
O0
0
,
5
T 2, K2 .
o
.
.
.
:
5
•
.
,
i
I
to
Fig. 5. Specific heat of the SBN single crystal [9]. Curve 1 : c / T = b T 2 + a l l (T/Tf, Tb) , a = 5.0 pJ gK-2, b = 0 . 6 / a J g K -4, T f = 5 . 5 K , Tb=O. Curve 2: b T °, b = 0.6pJ g K -4 .
CRYSTALLINE FERROELECTRICS AT LOW-TEMPERATURES
Vol. 59, No. 9
645
where
It (T/Tf, Tb)
----
(6/~?)
x 2 exp (--x)dx (1 - - e x p (--x)) 2 (1 + exp ((xr/T t - - 1 ) / ( T b / T f ) ) ) '
k s is the Boltzmann constant, /~m =/~[P (P is the density of the specimen). Corresponding heat release is To
q = j (dc/dt)dT = a(Tx, To)/t,
(6)
TI
where
rb)rg),
(7) r/rt
I2(T/Tf, Tb) = 2(Tt/T) 2 f
(T/Tt)I'
0
(T/Tt, Tb )d(T/Tt).
(8)
When we take /~m = 1.3"103S/jg, T f = E f / k B = 7.5K, and Tb = 0 , equation (6) corresponds to the experimental data for PLZT well (Fig. 4). These values are fairly close to those of amorphous metals (/~ra = 2.3" 10aT/jg, Tf = 20K, Tb = 0 for FeaoB14Si6 and/~m = 1.2- 1037[jg, Tf = 24K, Tb = 0 forCo69Fe4.sCr2Si2.sB22) s and vitreous silica (fire ~--4-5"1037/jg, Tf = 13 K, Tb = 0) [2, 5]. Using necessary for calculation vitreous silica parameters (sound velocities Vt, Vz and coupling constants 7t, 7t) for calculation of /Sra [14], we get for PLZT,/~m = 7.2" 10aS/Jg from our heat capacity data (a = 3.1 pJ gK-2), and /~m = 1.6" 103S/jg from thermal conductivity measurements [8]. Like in the case of vitreous silica [2,14], "heat capacity" firs is greater than /~m from thermal conductivity measurements probably due to the existence of "anomalous" two-level systems [14]. As in metallic glasses [5], relaxation at T > T f / I O is faster than according to the standard tunneling theory. For instance, 1.3K heat release after t H = 5 m i n at T H = 4 . 5 K is almost the same as after tH = 10h at the same TH (see Figs. 3 and 4). According to the phenomenological equation for the ratio of power release after a short excitation time at TH to that after a very long (tn ~ ' , TH = Ta) excitation time [5], r = q(TH, To, tH, t)/cl(Tx, To, t) = (1 + (t/tt4)f(Tn)/f(To))-',
(9)
where
f ( T ) = exp (--6T/Tt) ,
There is a significant difference between PLZT and amorphous solids: heat release for large t decreases faster in PLZT. This may be due to a lower value of the maximum relaxation time rmax in this material than in amorphous solids. It was shown [5] that the deviations from t -1 law for t ~ rmex can be described with
gl(t) = glotot -1 exp (--t/rmax(T1)).
a(Tl, To) = (ff2k~/24)Pm(I2(Tl Tf, Tb)T~ - I,(ro/rr,
(5)
(10)
we get r = 0 . 8 7 for PLZT (T t = 7 . 5 K ) , T = 4 . 5 K , To = 1.30K, tH = 5 min, t = 10min. This value is very close to the experimental r = 0.86 + 0.05.
(11)
There is excellent agreement between (11) and the experimental data; for sufficiently low TI (T1 < 5 K), rraax is proportional to Tl (inset in Fig. 3). For E > El, the density of two-level systems falls to zero, there are no states with longer 7" for T1 > Tf and rmLx is independent of Ta. PLZT is probably the first example ot amorphous or quasiamorphous materials, where it was possible to fLx the upper limit of the relaxation-time spectrum of two-level systems. Modifications of the distribution function imply modification of the formula for the heat capacity. The proportionality of ct to T (ct/T = a/a) according to (4) holds for T~< 0.15 Tf only, if we assume Tb = 0 and neglect weak temperature dependence through train. For higher T, ct/T decreases with temperature increasing: substances with small Tf are more convenient for checking this. In Fig. 2, "residual" heat capacities co/T 3 = c/T a all/T 2 (curve 3) and colT s = c/T 3 --a/T 2 (curve 4) are shown. For T > 1 K in PLZT as in many others ferroelectrics, an additional contribution to the heat capacity [9, 12, 13] exists, which can be described with the hellz of several Einstein phonon modes. Because of this contribution, it is difficult to choose between curves 3 and 4, though curve 3 agrees better with the analysis with three Einstein terms (curves 5, 6, 7) of the form 3
Co = bT 3 + ~ t~x~ expxi ( e x p x t - - 1) -2,
(123
i=1
where b = 0.8/aJ gK -4, ax = 0.21 mJgK, OE1 = 15.5K or2 = 5 m J g K , OE2 = 3 4 . 5 K , ct3 = 4 8 m J g K , OE3 = 72 K x~ = OBi/E. Rather complicated temperature dependence of the low-temperature anomaly of the heat capacity was observed in the ferroelectric single crystal SBN (Sro.61Bao.agNb206) [9], shown in Fig. 5; here Einstein frequencies correspond to higher temperatures (32, 73.5,151K). If a = 5.0#JgK -2 , b = 0 . 5 / / J g K - 4 , T f = 5.5 K, Tb = 0, equation (4) is in good agreement with the experimental data.
646
CRYSTALLINE FERROELECTRICS AT LOW-TEMPERATURES
The discovery of the "amorphous-like" heat release in ferroelectric PLZT opens new possibilities for experimental study of this phenomenon. Since glass.like thermal properties also occur in single crystals, they are very likely to be due to ferroelectric structure and could be influenced by the electric field. Acknowledgement - The authors are obliged to Dr. J. Schrieber, Prof. E. Hegenbarth, Dr. G. Pompe, Dr. G. Spoerl, Dr. J. Helming, and Dr. M. Mertig from the TeclmicalUniversity, Dresden for delivering the specimens and for valuable discussion. We thank Dr. V.N. Pokrovski (JINR) for useful remarks in the course of manuscript preparation. REFERENCES 1. 2. 3.
J. Zimmermarm & G. Weber, Phys. Lett. 46, 661 (1981). M. Schwark, F. Pobell, M. Kubota & R.M. Mueller, J. Low Temp. Phys. 58, ,71 (1985). D.S. Greywall,Phys. Rev. BI8, 2127 (1985).
4. 5.
Vol. 59,
J. Zimmermann, Cryogenics 24, 27 (1984). S. Sahling, A. Sahling, B.S. Neganov & M. Kol~, ICEC 11 (1986) and JINR E8~6-103, Dubna (1986). 6. K. Neumaier, H. Wipf, G. Cannelli & R. Cannelli, Phys. Rev. Lett 49, 1423 (1982). 7. J. Henning, P. Frach, E. Hegenbarth & V.J. Fritsberg, Phys. Status Solidi (a) 70, K7 (1982). 8. E. Fischer, W. Haessler, E. Hegenbarth & V.J. Fritsberg, Phys. Status Solidi (a} 66, K 169 (1981). 9. J. Henning, M. Mertig, R. Hath, G. Pompe, E. Hegenbarth & R. Schalge, Phys. Status Solidi (a) 73, K106 (1982). 1(). E. Fischer, W. Haessler & E. Hegenbarth, Phys. Status Solidi (a) 72, K169 (1982). 11. M. Kol~, B.S. Neganov & S. Sahling, J. Low Temp. Phys. 59, 547 (1985), JINR E-8-84~60, Dubna, (1984). 12. W.N. Lawless, Phys. Rev. BI4, 134 (1976). 13. B. Gerth, A. Sahling, G. Pompe, andE. Hegenbarth & B. Brezina, Phys. Status Solidi (a) 57, K153 (1980). 14. J.L. Black, Phys. Rev. B17, 2740 (1978).
0038-I098/86 $3.00 + .00 Pergamon Journals Ltd.
LONG-TIME HEAT RELEASE IN CRYSTALLINE FERROELECTRICS AT LOW-TEMPERATURES S. Sahling*, A. Sahlingt, B.S. Neganov and M. Kol~:~ Joint Institute for Nuclear Research, Dubna, USSR
(Received 25 March 1986 by E.A. Kaner) Calorimetric measurements were performed with KH2PO4 (KDP) and PbomsLao.0asZro.~sTio.3s (PLZT) at helium temperatures. No heat release was observed in KDP single crystal. Heat release in polycrystalline PLZT after cooling from TI (1.3 K < Tt ~< 292 K) to To = 1.3 K is very similar to that in amorphous metals and dielectrics. Experimental results disagree with the standard tunneling model. The observed heat release may be explained assuming the existence of a maximum energy Ef in the distribution function. The maximum relaxation time ¢max was found as a function of Tt. 1. INTRODUCTION
1 t,h
AS HAS BEEN SHOWN recently [ 1 - 5 ] , the long.time heat release is probably one of the characteristic features of amorphous materials. This heat release was observed in vitreous silica [1, 2], various organic materials [ 2 - 4 ] , and also in*'metallic glasses [5]. A question has arisen about the existence of crystalline substances with similar heat release. For instance, the typical of glasses T.term in the heat capacity of hydrogen-saturated crystalline N b - T i was observed [6] along with the long-time heat release [2]. Amorphous-like heat capacity and heat conductivity were also seen in various polycrystaUine ferroelectrics and their single crystals, too [ 7 - 1 0 ] . To check possible heat releas~iin this type of materials, we performed an exf)erimental study of two of them.
PLZT
o,s
0
20
30
40
50
Fig. 1. The time necessary for cooling the specimens from the temperature TI to the temperature To = 1.3 K. glue was less than 10 mg). Due to their low heat capacity, the cooling of the specimens from the initial temperature TI (1.3 K < Tt ~< 292 K) to the measuring temperature To = 1.3 K was fairly rapid (Fig. 1). Before each cooling, the specimen remained at T1 at least for ten hours. During the experiments, the heat capacities of both specimens were measured.
2. EXPERIMENTAL Two specimens were studied: polycrystaUine I'bo.9tsLao.oss. Zro.~Tio.as (transition temperature T¢ = 370 K, the mass m = 50.6 g) (PLZT), and a single crystal of KI-I2PO4 (T¢ ~ 123 K, m = 94.8 g) (KDP). The measurements were performed in a 4Hecalorimeter [11]. Copper foil (30Wn thick) with a Ge-thermometer, a heater and the contact for the heat switch was glued to the specimens (the amount of the
* On leave of absence from the Technical University, Dresden, GDR. t On leave of absence from the Central Institute for Solid State Physics and Material Research, Academy of Sciences of GDR, Dresden, GDR. On leave of absence from the Institute of Physics, Czechoslovak Academy of Sciences, Prague, Czechoslovakia.
10
3. RESULTS AND DISCUSSION The measured heat capacity is shown in Fig. 2. In agreement with the previous measurements [7], PLZT heat capacity contains a rather large term roughly proportional to the temperature at T < 3 K. Such term is much smaller in KDP. Assuming
c = a T + b T 3,
(1)
we have from our experiment for 1 K ~< T ~< 1.5 K, a = (3.1 + 0.2)/d gK-2 , b = (0.8 + 0.2) Id gK -4 for PLZT, and a = (0.5 + 0.1)/~JgK -2, b = (2.05 + 0.10)/dgK -4
643
644
CRYSTALLINE FERROELECTRICS AT LOW-TEMPERATURES ,
c/T3'/U J;g K4
Vol. 59, No. 9
0,2
Cl[lOmin], nWlg
'
0,2
1
2
/f
x- 0,085,
y - 0,85
0,1
2
5
z, K z T•-1,3
10
I
I
0
Fig. 2. Specific heat of the specimens. 1 : PLZT experimental data; 2: PLZT_, [7] ; 3: c / T a - - a l l / T 2 ;4: c / T a -a/T 2 ; 5: b + CEI/T ~, ,v: = 0.21 mJ/gK, OEt = 15.5 K; 6: b + C E 2 / T °, ~2 = 5 m J / g K , OE2 = 3 4 . 5 K ; 7: b +
CE3/T~,
Ota=48mJ/gK ,
0Ea=72K;
8:
b+(Z~= 1
CEi)/T ~ ;9: KDP experimental data.
for KDP. Maybe, because of a small value of a no heat release was observed in KDP. On the other hand, distinct long-time heat release was seen in PLZT. The specific heat release t~ in this material for various initial temperatures TI and To = 1.3 K is shown in Fig. 3. The results are very similar to those of the experiments with amorphous materials: (a) q(t) ~ t -1 for small t, (b) q(t) does not significantly depend on T1 for Ts > 20 K. Such behaviour in vitreous
10s
I
100
200
Fig. 4. Specific heat release in PLZT 10min after the cooling from TI to To = 1.3 K as a function of T~ -1.32 K~. Open circles - t n = 10h~ triangles-tt-t = 5 min. Curve 1: equation (7) with Pm = 1.3__"10as/jg, 12 = 1 (Tf-~oo). Curve 2: equation (7) w i t h P m = 1.3" 103S/jg, Tf = 7.5K, T b = O.
silica [2] and glassy metals [5] was shown to be explainable by the following assumption about the density-of-states of two-level systems: n(E, t) = no(t) (1 + exp ((E--Ef)/kl~Tb)),
~1,pW/g
Ne J--~-O,"°t
101
,L¢O"
,,
1
i 1 ,oTt'K .:
101
(2)
where E f and Tb are constant and no(t) =/~ln (4t/rm~a); man is the shortest relaxation time and/~ is the parameter of the standard tunneling assumption P(ZX, X) = L
TMOX, i l i l
&
(3)
where p(A, ~.) is the distribution function of two-well energy difference A and tunneling parameter h. Equation (2) results in the heat capacity c t = Or2k~/12)rPmIl (T/Tt, Tb) In (4t/train),
10
.
.
.
.
'
c/T,/u J/gK 2
".,, 10'
!
.
.
.
.
!
(4)
(
0 0
0
!
lO~
lO!
Fig. 3. Time dependence of the specific heat release in PLZT after cooling from T1 to To = 1.3K. o: tu = 10h, 1.49K~
~0
0
O 0
O0
0
,
5
T 2, K2 .
o
.
.
.
:
5
•
.
,
i
I
to
Fig. 5. Specific heat of the SBN single crystal [9]. Curve 1 : c / T = b T 2 + a l l (T/Tf, Tb) , a = 5.0 pJ gK-2, b = 0 . 6 / a J g K -4, T f = 5 . 5 K , Tb=O. Curve 2: b T °, b = 0.6pJ g K -4 .
CRYSTALLINE FERROELECTRICS AT LOW-TEMPERATURES
Vol. 59, No. 9
645
where
It (T/Tf, Tb)
----
(6/~?)
x 2 exp (--x)dx (1 - - e x p (--x)) 2 (1 + exp ((xr/T t - - 1 ) / ( T b / T f ) ) ) '
k s is the Boltzmann constant, /~m =/~[P (P is the density of the specimen). Corresponding heat release is To
q = j (dc/dt)dT = a(Tx, To)/t,
(6)
TI
where
rb)rg),
(7) r/rt
I2(T/Tf, Tb) = 2(Tt/T) 2 f
(T/Tt)I'
0
(T/Tt, Tb )d(T/Tt).
(8)
When we take /~m = 1.3"103S/jg, T f = E f / k B = 7.5K, and Tb = 0 , equation (6) corresponds to the experimental data for PLZT well (Fig. 4). These values are fairly close to those of amorphous metals (/~ra = 2.3" 10aT/jg, Tf = 20K, Tb = 0 for FeaoB14Si6 and/~m = 1.2- 1037[jg, Tf = 24K, Tb = 0 forCo69Fe4.sCr2Si2.sB22) s and vitreous silica (fire ~--4-5"1037/jg, Tf = 13 K, Tb = 0) [2, 5]. Using necessary for calculation vitreous silica parameters (sound velocities Vt, Vz and coupling constants 7t, 7t) for calculation of /Sra [14], we get for PLZT,/~m = 7.2" 10aS/Jg from our heat capacity data (a = 3.1 pJ gK-2), and /~m = 1.6" 103S/jg from thermal conductivity measurements [8]. Like in the case of vitreous silica [2,14], "heat capacity" firs is greater than /~m from thermal conductivity measurements probably due to the existence of "anomalous" two-level systems [14]. As in metallic glasses [5], relaxation at T > T f / I O is faster than according to the standard tunneling theory. For instance, 1.3K heat release after t H = 5 m i n at T H = 4 . 5 K is almost the same as after tH = 10h at the same TH (see Figs. 3 and 4). According to the phenomenological equation for the ratio of power release after a short excitation time at TH to that after a very long (tn ~ ' , TH = Ta) excitation time [5], r = q(TH, To, tH, t)/cl(Tx, To, t) = (1 + (t/tt4)f(Tn)/f(To))-',
(9)
where
f ( T ) = exp (--6T/Tt) ,
There is a significant difference between PLZT and amorphous solids: heat release for large t decreases faster in PLZT. This may be due to a lower value of the maximum relaxation time rmax in this material than in amorphous solids. It was shown [5] that the deviations from t -1 law for t ~ rmex can be described with
gl(t) = glotot -1 exp (--t/rmax(T1)).
a(Tl, To) = (ff2k~/24)Pm(I2(Tl Tf, Tb)T~ - I,(ro/rr,
(5)
(10)
we get r = 0 . 8 7 for PLZT (T t = 7 . 5 K ) , T = 4 . 5 K , To = 1.30K, tH = 5 min, t = 10min. This value is very close to the experimental r = 0.86 + 0.05.
(11)
There is excellent agreement between (11) and the experimental data; for sufficiently low TI (T1 < 5 K), rraax is proportional to Tl (inset in Fig. 3). For E > El, the density of two-level systems falls to zero, there are no states with longer 7" for T1 > Tf and rmLx is independent of Ta. PLZT is probably the first example ot amorphous or quasiamorphous materials, where it was possible to fLx the upper limit of the relaxation-time spectrum of two-level systems. Modifications of the distribution function imply modification of the formula for the heat capacity. The proportionality of ct to T (ct/T = a/a) according to (4) holds for T~< 0.15 Tf only, if we assume Tb = 0 and neglect weak temperature dependence through train. For higher T, ct/T decreases with temperature increasing: substances with small Tf are more convenient for checking this. In Fig. 2, "residual" heat capacities co/T 3 = c/T a all/T 2 (curve 3) and colT s = c/T 3 --a/T 2 (curve 4) are shown. For T > 1 K in PLZT as in many others ferroelectrics, an additional contribution to the heat capacity [9, 12, 13] exists, which can be described with the hellz of several Einstein phonon modes. Because of this contribution, it is difficult to choose between curves 3 and 4, though curve 3 agrees better with the analysis with three Einstein terms (curves 5, 6, 7) of the form 3
Co = bT 3 + ~ t~x~ expxi ( e x p x t - - 1) -2,
(123
i=1
where b = 0.8/aJ gK -4, ax = 0.21 mJgK, OE1 = 15.5K or2 = 5 m J g K , OE2 = 3 4 . 5 K , ct3 = 4 8 m J g K , OE3 = 72 K x~ = OBi/E. Rather complicated temperature dependence of the low-temperature anomaly of the heat capacity was observed in the ferroelectric single crystal SBN (Sro.61Bao.agNb206) [9], shown in Fig. 5; here Einstein frequencies correspond to higher temperatures (32, 73.5,151K). If a = 5.0#JgK -2 , b = 0 . 5 / / J g K - 4 , T f = 5.5 K, Tb = 0, equation (4) is in good agreement with the experimental data.
646
CRYSTALLINE FERROELECTRICS AT LOW-TEMPERATURES
The discovery of the "amorphous-like" heat release in ferroelectric PLZT opens new possibilities for experimental study of this phenomenon. Since glass.like thermal properties also occur in single crystals, they are very likely to be due to ferroelectric structure and could be influenced by the electric field. Acknowledgement - The authors are obliged to Dr. J. Schrieber, Prof. E. Hegenbarth, Dr. G. Pompe, Dr. G. Spoerl, Dr. J. Helming, and Dr. M. Mertig from the TeclmicalUniversity, Dresden for delivering the specimens and for valuable discussion. We thank Dr. V.N. Pokrovski (JINR) for useful remarks in the course of manuscript preparation. REFERENCES 1. 2. 3.
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