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Thermal unpinning of the modulation wave near TI in incommensurate Rb2ZnCl4 detected by 35Cl NQR
Solid State Communications, Vol. 50, No. 11, pp. 1019-1021, 1984. Printed in Great Britain.
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
THERMAL UNPINNING OF THE MODULATION WAVE NEAR TI IN INCOMMENSURATE RbzZnC14 DETECTED BY 3sC1 NQR F. Milia Nuclear Research Center Demokritos, Aghia Paraskevi Attikis, Greece and R. Blinc and S. Zumer J. Stefan Institute, E. Kardelj University of Ljubljana, Ljubljana, Yugoslavia (Received 28 February 1984 by P. Wachter)
Thermal unpinning and floating of the incommensurate modulation wave produces in Rb2ZnCI4 close to the incommensurate-paraelectric transition T t a partial motional averaging of the splitting between the two edge singularities in the 3SCl nuclear quadrupole resonance spectrum. The mean square phase fluctuations are close to Tr proportional to ( T I - - T) -2# with/~ = 0.39 -+ 0.04.
1. INTRODUCTION
f(v) = 5 ( v - Vo).
IT HAS BEEN recently shown that thermal depinning of the incommensurate modulation wave takes place near the incommensurate (I)-paraelectric (P) transition TI in a number of incommensurate crystals [1-3]. It is the purpose of this note to investigate whether this effect can be detected via asc1 nuclear quadrupole resonance (NQR) spectroscopy in Rb2ZnC14. 2. THEORY In the plane-wave limit which is appropriate for the high temperature part of the I phase, the NQR lineshape can be in the simplest possible case - when the NQR frequency linearly varies with the incommensurate nuclear displacement v(x) = Vo + vl c o s ( k t x )
(1)
described by [4] const. f ( v ) = V v ~ - (v - Vo) ~
(2)
Here kz stands for the incommensurate wave vector and expression (2) represents a frequency distribution characterized by two edge singularities at v -- v0 = + vl. The frequency vl is proportional to the amplitude of the incommensurate order parameter: v ¢¢A o= (7"i -- T) a. The above result is correct for a static, pinned modulation wave. If phase fluctuations are large and fast as compared to vl, the NQR spectrum (2) will be completely motionally averaged out and one finds [ 1] just a single line at the unperturbed position Vo
(3)
If phase fluctuations are fast but of relatively small amplitude the motional averaging will be incomplete and one obtains [3] const. f ( v ) = X/v] e_aS _ (v -- Vo) 2 '
(4)
where 02 = <¢2> represents the mean square fluctuation in the phase of the modulation wave. Phase fluctuations a ¢ are related to fluctuations in the position of the incommensurate modulation wave Ax by: Ax = A ¢ / k I. If phase fluctuations are negligible, o 2 = <¢2>_+ 0, equation (2) is recovered, whereas in the limit of large phase fluctuations 02 = <¢2>_+ oo the incommensurate splitting between the two edge singularities Av = 2vi is motionaUy averaged out and expression (4) reduces to expression (3). In the intermediate case of small but non-zero phase fluctuations we have a partial motional averaging. The splitting between the two edge singularities is reduced by e -°2/2 as compared to the static case but the shape of the spectrum is not changed. The temperature dependence of the effective splitting Avett will be now given [31 by avet~ = 2vl e -°~/2 = (Tx -- T) ~ e -°2/=.
(5)
Since o 2 depends on temperature, the effect of the floating of the modulation wave will thus show up in a temperature dependence of the apparent critical exponent
[31 Aver~ o: (TI -- T)ge~L
1019
(6)
1020
THERMAL UNPINNING OF THE MODULATION WAVE
Vol. 50, No. 11
100
i0 ) AV [kHz]
o,2
[ rod ;)]
t
t
10-1
101
1001 0o
~
~
~
,
T]-T
m ,,ml
lff 2
101
3. RESULTS AND DISCUSSION The 3sC1 NQR spectra of Rb2ZnC14 have been investigated by Aleksandrova and coworkers [5-7] as well as Milia and coworkers [8, 9]. Aleksandrova and coworkers [7] found a critical exponent 1/2 whereas Milia and Rutar [9] report a critical exponent 0.36. We have reinvestigated the temperature dependence of the incommensurate splitting of the 8.35 MHz 3sc1 NQR line in Rb2ZnC14. The temperature dependence of the splitting between the two v+(a) and v_(a) edge singularities [9] is plotted against T1 -- T in a log-log scale in Fig. 1. In accordance with equation (5) the plot is not a straight line and the slope changes with temperature. In a temperature interval T~ -- T ~< 5 K the slope can be characterized by an apparent critical exponent /3e~~ ~ 0.53 -+ 0.03 whereas for 5 ~< Tx -- T<~ 3 0 K the critical exponent becomes/~etf =/3 = 0.35 -+ 0.02 in agreement with the value of/~ determined from the intensity of the X-ray satellites [10] and the d = 3, n = 2 Heisenberg model. Thus both above mentioned groups [7, 9] have been right but the slopes and values of the quoted apparent critical exponents refer to different temperature intervals. By subtracting in the log-log plot the observed splitting from the one extrapolated from the data "far" from Tx where phase fluctuations can be neglected and where/3ee t = ~ = 0.35 one obtains the temperature dependence of o 2, i.e. the temperature dependence of
,
100
,
......
101
--..- TvT [K]
[K]
Fig. 1. Log-log plot of the splitting between the two u.(a) and u_(a) edge singularities [9] in the 3sc1 NQR spectrum of Rb2ZnC14 vs TI -- T.
~,1
Fig. 2. Log-log plot of the mean square phase fluctuations o 2 = (q~) in Rb2ZnCI4 - as derived from Fig. 1 - vs T t - - T. the mean square phase fluctuations. The results are shown in Fig. 2 and agree rather well with the ones obtained from the 87Rb NMR data [3]. The magnitude of the root mean square phase fluctuations varies between 10 ° and 30 ° in the interval 1 <~ Tt -- T~< 4K. These fluctuations are thus too small to produce a complete motional averaging [ 1] but large enough to produce a partial reduction of the splitting between the two edge singularities. The log-log plot of o 2 vs T1 -- T (Fig. 2) yields a slope of 0.78 -+ 0.08. Since o z is expected [3] to be inversely proportional to the square of the amplitude of the modulation wave
o~ ~ y
(7"I- T) -~
(7)
the above value of the slope 0.79 corresponds to j3 = 0.39 -+ 0.04 in relatively good agreement with other experiments [10] and theory. REFERENCES 1. 2. 3. 4.
R. Blinc, D.C. Ailion, P. Prelov~ek & V. Rutar, Phys. Rev. Lett. 50, 67 (1983). J. Emery, S. Hubert & J.C. Fayet, Solid State Commun. (to be published). R. Blinc, F. Milia, B. TopiE & S. Zumer, Phys. B (to be published). See, for instance, R. Blinc, Phys. Reports 79, 331 (1981).
Vol. 50, No. 11 5. 6. 7.
THERMAL UNPINNING OF THE MODULATION WAVE
A.K. Moskalev, I.A. Belobrova, I.P. Aleksandrova, S. Sawada & T. Shiroishi, Phys. Status Solidi (a) 50, 157 (1978). I.P. Aleksandrova, Ferroelectrics 24, 135 (1980). I.P. Aleksandrova, A.K. Moskalev & I.P. Belobrova, J. Phys. Soc. Japan Suppl. B49, 86 (1980).
8. 9. 10.
1021
F. Milia, Phys. Lett. 70A, 218 (1979 ); Ferroelectrics 24, 151 (1980). F. Milia& V. Rutar, Phys. Rev. B23,6061(1981). S.R. Andrews & H. Mashiyama, J. Phys. C16, 4985 (1983).
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
THERMAL UNPINNING OF THE MODULATION WAVE NEAR TI IN INCOMMENSURATE RbzZnC14 DETECTED BY 3sC1 NQR F. Milia Nuclear Research Center Demokritos, Aghia Paraskevi Attikis, Greece and R. Blinc and S. Zumer J. Stefan Institute, E. Kardelj University of Ljubljana, Ljubljana, Yugoslavia (Received 28 February 1984 by P. Wachter)
Thermal unpinning and floating of the incommensurate modulation wave produces in Rb2ZnCI4 close to the incommensurate-paraelectric transition T t a partial motional averaging of the splitting between the two edge singularities in the 3SCl nuclear quadrupole resonance spectrum. The mean square phase fluctuations are close to Tr proportional to ( T I - - T) -2# with/~ = 0.39 -+ 0.04.
1. INTRODUCTION
f(v) = 5 ( v - Vo).
IT HAS BEEN recently shown that thermal depinning of the incommensurate modulation wave takes place near the incommensurate (I)-paraelectric (P) transition TI in a number of incommensurate crystals [1-3]. It is the purpose of this note to investigate whether this effect can be detected via asc1 nuclear quadrupole resonance (NQR) spectroscopy in Rb2ZnC14. 2. THEORY In the plane-wave limit which is appropriate for the high temperature part of the I phase, the NQR lineshape can be in the simplest possible case - when the NQR frequency linearly varies with the incommensurate nuclear displacement v(x) = Vo + vl c o s ( k t x )
(1)
described by [4] const. f ( v ) = V v ~ - (v - Vo) ~
(2)
Here kz stands for the incommensurate wave vector and expression (2) represents a frequency distribution characterized by two edge singularities at v -- v0 = + vl. The frequency vl is proportional to the amplitude of the incommensurate order parameter: v ¢¢A o= (7"i -- T) a. The above result is correct for a static, pinned modulation wave. If phase fluctuations are large and fast as compared to vl, the NQR spectrum (2) will be completely motionally averaged out and one finds [ 1] just a single line at the unperturbed position Vo
(3)
If phase fluctuations are fast but of relatively small amplitude the motional averaging will be incomplete and one obtains [3] const. f ( v ) = X/v] e_aS _ (v -- Vo) 2 '
(4)
where 02 = <¢2> represents the mean square fluctuation in the phase of the modulation wave. Phase fluctuations a ¢ are related to fluctuations in the position of the incommensurate modulation wave Ax by: Ax = A ¢ / k I. If phase fluctuations are negligible, o 2 = <¢2>_+ 0, equation (2) is recovered, whereas in the limit of large phase fluctuations 02 = <¢2>_+ oo the incommensurate splitting between the two edge singularities Av = 2vi is motionaUy averaged out and expression (4) reduces to expression (3). In the intermediate case of small but non-zero phase fluctuations we have a partial motional averaging. The splitting between the two edge singularities is reduced by e -°2/2 as compared to the static case but the shape of the spectrum is not changed. The temperature dependence of the effective splitting Avett will be now given [31 by avet~ = 2vl e -°~/2 = (Tx -- T) ~ e -°2/=.
(5)
Since o 2 depends on temperature, the effect of the floating of the modulation wave will thus show up in a temperature dependence of the apparent critical exponent
[31 Aver~ o: (TI -- T)ge~L
1019
(6)
1020
THERMAL UNPINNING OF THE MODULATION WAVE
Vol. 50, No. 11
100
i0 ) AV [kHz]
o,2
[ rod ;)]
t
t
10-1
101
1001 0o
~
~
~
,
T]-T
m ,,ml
lff 2
101
3. RESULTS AND DISCUSSION The 3sC1 NQR spectra of Rb2ZnC14 have been investigated by Aleksandrova and coworkers [5-7] as well as Milia and coworkers [8, 9]. Aleksandrova and coworkers [7] found a critical exponent 1/2 whereas Milia and Rutar [9] report a critical exponent 0.36. We have reinvestigated the temperature dependence of the incommensurate splitting of the 8.35 MHz 3sc1 NQR line in Rb2ZnC14. The temperature dependence of the splitting between the two v+(a) and v_(a) edge singularities [9] is plotted against T1 -- T in a log-log scale in Fig. 1. In accordance with equation (5) the plot is not a straight line and the slope changes with temperature. In a temperature interval T~ -- T ~< 5 K the slope can be characterized by an apparent critical exponent /3e~~ ~ 0.53 -+ 0.03 whereas for 5 ~< Tx -- T<~ 3 0 K the critical exponent becomes/~etf =/3 = 0.35 -+ 0.02 in agreement with the value of/~ determined from the intensity of the X-ray satellites [10] and the d = 3, n = 2 Heisenberg model. Thus both above mentioned groups [7, 9] have been right but the slopes and values of the quoted apparent critical exponents refer to different temperature intervals. By subtracting in the log-log plot the observed splitting from the one extrapolated from the data "far" from Tx where phase fluctuations can be neglected and where/3ee t = ~ = 0.35 one obtains the temperature dependence of o 2, i.e. the temperature dependence of
,
100
,
......
101
--..- TvT [K]
[K]
Fig. 1. Log-log plot of the splitting between the two u.(a) and u_(a) edge singularities [9] in the 3sc1 NQR spectrum of Rb2ZnC14 vs TI -- T.
~,1
Fig. 2. Log-log plot of the mean square phase fluctuations o 2 = (q~) in Rb2ZnCI4 - as derived from Fig. 1 - vs T t - - T. the mean square phase fluctuations. The results are shown in Fig. 2 and agree rather well with the ones obtained from the 87Rb NMR data [3]. The magnitude of the root mean square phase fluctuations varies between 10 ° and 30 ° in the interval 1 <~ Tt -- T~< 4K. These fluctuations are thus too small to produce a complete motional averaging [ 1] but large enough to produce a partial reduction of the splitting between the two edge singularities. The log-log plot of o 2 vs T1 -- T (Fig. 2) yields a slope of 0.78 -+ 0.08. Since o z is expected [3] to be inversely proportional to the square of the amplitude of the modulation wave
o~ ~ y
(7"I- T) -~
(7)
the above value of the slope 0.79 corresponds to j3 = 0.39 -+ 0.04 in relatively good agreement with other experiments [10] and theory. REFERENCES 1. 2. 3. 4.
R. Blinc, D.C. Ailion, P. Prelov~ek & V. Rutar, Phys. Rev. Lett. 50, 67 (1983). J. Emery, S. Hubert & J.C. Fayet, Solid State Commun. (to be published). R. Blinc, F. Milia, B. TopiE & S. Zumer, Phys. B (to be published). See, for instance, R. Blinc, Phys. Reports 79, 331 (1981).
Vol. 50, No. 11 5. 6. 7.
THERMAL UNPINNING OF THE MODULATION WAVE
A.K. Moskalev, I.A. Belobrova, I.P. Aleksandrova, S. Sawada & T. Shiroishi, Phys. Status Solidi (a) 50, 157 (1978). I.P. Aleksandrova, Ferroelectrics 24, 135 (1980). I.P. Aleksandrova, A.K. Moskalev & I.P. Belobrova, J. Phys. Soc. Japan Suppl. B49, 86 (1980).
8. 9. 10.
1021
F. Milia, Phys. Lett. 70A, 218 (1979 ); Ferroelectrics 24, 151 (1980). F. Milia& V. Rutar, Phys. Rev. B23,6061(1981). S.R. Andrews & H. Mashiyama, J. Phys. C16, 4985 (1983).