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Classical behaviour of the soliton density in Rb2ZnCl4 studied by 87Rb NMR
Solid State Communications, Vol. 44, No. 6, pp. 885-887, 1982. Printed in Great Britain.
0038-1098/82/420885-03 $03.00/0 Pergamon Press Ltd.
CLASSICAL BEHAVIOUR OF THE SOLITON DENSITY IN Rb2ZnCI4 STUDIED BY STRb NMR E. Schneider Fachbereich Physik, Universitiit des Saarlandes, D-6600 Saarbriicken, W. Germany
(Received 18 June 1982 by P.H. Dederichs) The temperature dependence of the soliton density ns(T) which has been studied in Rb2ZnC14 by STRb NMR could be fitted to a classical formula using only one fit parameter. From our data we cannot confirm recent experimental results which yield a critical exponent 1/2 for the soliton density. The results are discussed with respect to the free energy where the soliton density acts as the order parameter. Moreover, couplings to other degrees of freedom, e.g. the elastic strains, are considered.
RECENTLY, SEVERAL experimental investigations have been done to verify McMillans concept [1 ] of phase solitons or "discommensurations" in structurally incommensurate systems. Particularly, high frequency spectroscopy has been shown to constitute a sensitive method for investigating such systems. So, e.g. NMR has been applied to the 2d-modulated system 2H-TaSe2 [2] and to the ld-modulated systems Rb2ZnC14 and Rb2ZnBr4 [3,4] and K2SeO4 was investigated by ESR [5, 6]. Quantitative experimental results, however, concerning the temperature dependence of the density of "narrow" solitons or domain walls are rather scarce. It is well known that the study of incommensurate systems by magnetic resonance techniques can yield informations about the local environments and structures of the nuclei [7]. Blinc etal. [3, 4] were the first to estimate the soliton density and its temperature dependence in Rb2ZnC14 and Rb2ZnBr4 by STRb NMR. In these works the soliton density ns(T ) is reported to vanish at T e with a critical exponent 1/2, thus demonstrating strong deviations from the classical result n s oc -- In-it [1, 8, 9]. The authors state that this new result can be understood by a certain theory [10]. The square root power law for the soliton density, however, recently has been repudiated confirming the classical result for d = 3 systems [11 ]. Moreover, the classical results have been demonstrated to hold in three dimensions akeady some time ago for a special model [12]. Employing phenomenological arguments the classical behaviour has also been derived recently [13]. We note that the authors of this work [13] also criticize the interpretation of previous NMR results [3, 4] concerning the critical behaviour of the soliton density. For completeness it should be mentioned that recent ESR measurements [14] on Mn ÷ doped Rb2ZnC14 crystals did not give clear evidence either for the square root power law or for the logarithmic behaviour. Thus, whereas the theories presently 885
available predict a classical logarithmic behaviour from the experimental facts so far no clear picture can be derived for the soliton density. In order to obtain more precise information on the temperature behaviour of the soliton density n8 in Rb2ZnCIa we started 87Rb (I = 3/2) NMR measurements. From our investigations of the 87Rb NMR satellite transitions -+ 1/2 ~÷ -+ 3/2 [15] in the incommensurate phase of Rb2ZnC14 it was not possible to extract the intensities of the commensurate lines from the incommensurate background for T >~ Tc because of the rather broad and noisy spectra. Therefore, we measured the temperature dependence of the 1/2 ,~ 1/2 central transition around Tc as it has been done previously by Blinc et al. [3]. Experimental details of our work are described elsewhere [15]. A typical spectrum of the STRb central transition in the incommensurate phase of Rb2ZnC14 at about T = Tc + 0.4 K is shown in Fig. 1. The crystal orientation was nearly the same as used in [16, 18]. The spectrum essentially is a superposition of the quasi-continuous line shape functions of the Rb(1) and Rb(2) nuclei [15]. The line shape in Fig. 1 is in qualitative accordance with the theoretically predicted lineshape in an incommensurate multisoliton lattice [17]. The two lines indicated by arrows in Fig. 1 are of special interest because on passing from the incommensurate (IC) to the ferroelectric (FE) phase they are growing from an incommensurate background finally coming up as narrow discrete lines in the FE phase. We measured the temperature dependence of these lines in a range of about 25 K around Te (Fig. 2). The intensity of one commensurate line is proportional to the number of nuclei with nearly the same electrical field gradient tensor (EFG). These nuclei "he" in regions which are essentially commensurate with the underlying lattice and which are separated from each other by phase solitons or domain wails [16, 17] (leading to the -
-
CLASSICAL BEHAVIOUR OF THE SOLITON DENSITY IN Rb2ZnC14
886
87Rb:VL:98.17
HHz
i0
o I Ho
~t(c,H o)=27 °
,
10 kHz
,
0S
~
Fig. 1. Typical spectrum of the second order shifted 87Rb central lines near To in Rb2ZnC14. The meaning of the two lines indicated by arrows is discussed in the text.
severa[ • • "temperature runs
Vol. 44, No. 6
• A . j ~ r ~ ~.
AA
C~-
06
N
~ 02 -90
"' Id:°
0,-
'1/'1I,
02
II -86
-82
-78
-Tk
-70
Temperature [C ] Fig. 3. Temperaturedependence at the soliton density in Rb2ZnCI4 determined for different temperature runs which all were performed at increasingtemperature. The solid curve is obtained by computing equation (3). Inset: Calculation of equation (5) for a first order phase transition (d > 0) with limit of stability T~ and for a continuous phase transition (d = 0) with phase transition temperature T e. -
87.7 a*/3, i.e. p = 3. Thus disregarding constant terms the free energy density can be written as [8, 9]
-
8/+./+
F(x) = 1/2(O¢a(x)/ax -- 60) 2 -- v cos pc~(x),
-82!, -
81.3
-
80/+
TI°C - 76.3
~
-
68.9
Fig. 2. Temperature dependence of the two lines indicated in Fig. 1 by arrows.
incommensurate background limited by edge singularities). Thus, we calculated the intensity above the background o f these two lines in Fig. 2, which is proportional to the density of the commensurate regions ne = 1 -- n 8. The soliton density n s as measured as a function of temperature for three temperature runs is shown in Fig. 3. The experimental facts will be discussed by theories [1,8, 9, 19] describing the lock-in transition of incommensurate systems with a wave vector qi = a*/p + 8o. For Rb2ZnCI4 the modulation is along one direction, say the x-direction, and the wave vector qi is close to
(1)
where @(x) is the phase o f the complex order parameter whose amplitude A is assumed to be independent of x because so far no experimental evidence for amplitude solitons is known and v = -- v o A . Minimizing the free energy F = (A2/L) foc F(x) dx where L is the length of the crystal leads to the static sine Gordon equation a2@/ax 2 = p v sin p@. One special solution is the soliton ~ x ) = (4/p) arctan exp (px/-vx), which describes a single domain wall separating two commensurate regions with @~ 0 and @~ 2rt]p. Feeding this result back into equation (1) and calculating the free energy for an assembly o f M solitons one finds [8, 9] F = (2K/Tr2)~[1 + 4 exp (--t¢/~)] --060,
(2)
where K = 21rx/-v and ~ = 2Mrr/pL denotes the average wave vector for an incommensurate configuration. Thus, is proportional to the number of solitons and acts therefore as the order parameter of the lock-in transition. Minimizing equation (2) with respect to ~ one finds the soliton density 0 = ns as a function of 60 (1 + K/ns)4 exp ( - K/ns) = (6o-- 2Klrr2)l(2K/Tr2).
(3)
For the critical value 2Kfir 2 = 8o a continuous phase transition occurs where n8 = 0; for 2K[lr 2 < 80 an incommensurate state is favoured with non-zero n s. The temperature enters the problem through the temperature dependence of the ordering amplitude A which is proportional to (Ti -- T ~ [3, 9, 15]. Consequently one can linearize the right-hand side of equation (3) and
Vol. 44, No. 6
CLASSICAL BEHAVIOUR OF THE SOLITON DENSITY IN Rb2ZnC14
approximate by t = (T-- Te)[Te. In a suitable temperature range sufficiently far above To, equation (3) can be approximated by the well known classical result [8] n s cx - - l n - l t .
(4)
For the overall temperature dependence, particularly close to To, however, one has to solve equation (3) numerically. This has been done by assuming the parameter K to vary weakly with temperature according to the nearly saturated value of the ordering amplitude. Thus, r may be approximated by a constant value. A fit of the experimental data gives r = 5.5 + 0.1. The solid curve in Fig. 3 demonstrates that n s can be well represented by equation (3). Obviously the measuring points for T <~ Tc do not fit so well to equation (3). This may have several reasons. On the one hand there exists an unavoidable temperature gradient of about 0.3-0.5 K over the sample volume which leads to a smeared out behaviour in the phase transition region. On the other hand, it is well known that in most of the ferroelectric crystals there may appear domains of distinct polarization below Te which cause internal strains. Principally this effect could be avoided if one could apply an electrical field along the ferroelectric b-axis on decreasing the temperature below Te. Because of the reasons given above from our experiments we cannot decide whether or not the phase transition is of first or second order. In the literature it is widely assumed that the phase transition in Rb2ZnC14 is of first order with a temperature hysteresis which is probably due to soliton pinning on defects [20]. Comparing our results to those in the literature [3, 4] it can be stated that, near Te, n 8 depends much steeper on the temperature than in the previous works. This is related to the fact that contrary to the previous results the points in Fig. 3 cannot be fitted by a power law with a critical exponent 1/2. Rather the validity of the classical equation (3) is shown. The differences between these results possibly may be related to different qualities of the samples. The good quality standard of the crystals used in the present experiments becomes obvious from the fact that in these samples the 87Rb NMR satellite lines could be detected in Rb2ZnC14 for the first time [15]. Equation (1) can be extended by taking into consideration other degrees of freedom [9]. To be specific, we consider the coupling to an elastic strain, because defects can cause internal strains. Thus, one has to add to equation (1) the coupling between the strain r~ and the displacement profile and the usual elastic energy thus resulting in the terms 2w0 a¢/Ox + 1/2Cot/2. Calculating as before the free energy and minimizing this quantity
887
with respect to 71equation (2) takes an additional term -- 2(w2/co)~!2, provided w 2 < 2raCo exp (-- r/fl)/(rtfla). This case is always fulfilled for the weak coupling limit w -~ 0 [9]. Proceeding as before the temperature dependence of the soliton density is given by (1 + r / n s ) 4 e x p ( - - g / n s ) - - d n s = (T--Te)/Tc,
(5)
where d = 2w2rr/(rCo) >~O. The additional term in equation (5) leads to a first order phase transition (see inset in Fig. 3). A coupling of the displacement field to a long wave coordinate, i.e. the electrical polarization, can be analyzed in a similar way [9], resulting in a relation of the form of equation (5) provided the number of solitons is small.
Acknowledgements - The author is indebted to J. Petersson for helpful and stimulating discussions, to A. K16pperpieper for crystal growth and to the Deutsche Forschungsgemeinschaft for supporting this work within the frame of the Sonderforschungsbereich "Ferroelektrika". REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
W.L. McMillan, Phys. Rev. BI4, 1496 (1976). B.H. Suits, S. Couturi6 & C.P. Slichter, Phys. Rev. Lett. 45,194 (1980). R. Blinc, V. Rutar, B. TopiC, F. Milia & I.P. Aleksandrova, Phys. Rev. Lett. 46, 1406 (1981). V. Rutar, F. Milia, B. Topi~, R. Blinc & Th. Rasing, Phys. Rev. B25,281 (1982). M. Fukui & R. Abe, Jpn. J. Appl. Phys. 20, L533 (1981). A.S. Chaves, R. Gazzinelli & R. Blinc, Solid State Commun. 37,123 (1981). R. Blinc, Physics Reports 79, 331 (1981). P. Bak, Solitons and Condensed Matter Physics (Edited by A.R. Bishop & T. Schneider), p. 216. Springer-Verlag, New York (1978). A.D. Bruce, R.A. Cowley & A.F. Murray, J. Phys. C l l , 3591 (1978). T. Nattermann, J. Phys. C13, L265 (1980). T. Nattermann, J. Phys. 43,631 (1982). M.E. Fisher & W. Selke, Phys. Rev. Lett. 44, 1502 (1980). M.E. Fisher & D.S. Fisher, Phys. Rev. B25, 3192 (1982). M. Pezeril & J.C. Fayet, J. Phys. Lett. 43, L267 (1982). E. Schneider & J. Petersson, Z. Phys. B46, 169 (1982). R. Blinc, Ferroelectries 35,117 (198 0 . A.S. Chaves, R. Blinc, J. Seliger & S. Zumer, J. Magn. Res. 46, 146 (1982). R. Blinc, F. Milia, V. Rutar & S. ~umer, Phys. Rev. Lett. 48, 47 (1982). T. Nattermann & S. Trimper, Z Phys. 1214, 1603 (1981). K. Hamano, Y. Ikeda, T. Fujimoto, K. Ema & S. Hirotsu, J. Phys. Soc. Japan 49, 2278 (1980).
0038-1098/82/420885-03 $03.00/0 Pergamon Press Ltd.
CLASSICAL BEHAVIOUR OF THE SOLITON DENSITY IN Rb2ZnCI4 STUDIED BY STRb NMR E. Schneider Fachbereich Physik, Universitiit des Saarlandes, D-6600 Saarbriicken, W. Germany
(Received 18 June 1982 by P.H. Dederichs) The temperature dependence of the soliton density ns(T) which has been studied in Rb2ZnC14 by STRb NMR could be fitted to a classical formula using only one fit parameter. From our data we cannot confirm recent experimental results which yield a critical exponent 1/2 for the soliton density. The results are discussed with respect to the free energy where the soliton density acts as the order parameter. Moreover, couplings to other degrees of freedom, e.g. the elastic strains, are considered.
RECENTLY, SEVERAL experimental investigations have been done to verify McMillans concept [1 ] of phase solitons or "discommensurations" in structurally incommensurate systems. Particularly, high frequency spectroscopy has been shown to constitute a sensitive method for investigating such systems. So, e.g. NMR has been applied to the 2d-modulated system 2H-TaSe2 [2] and to the ld-modulated systems Rb2ZnC14 and Rb2ZnBr4 [3,4] and K2SeO4 was investigated by ESR [5, 6]. Quantitative experimental results, however, concerning the temperature dependence of the density of "narrow" solitons or domain walls are rather scarce. It is well known that the study of incommensurate systems by magnetic resonance techniques can yield informations about the local environments and structures of the nuclei [7]. Blinc etal. [3, 4] were the first to estimate the soliton density and its temperature dependence in Rb2ZnC14 and Rb2ZnBr4 by STRb NMR. In these works the soliton density ns(T ) is reported to vanish at T e with a critical exponent 1/2, thus demonstrating strong deviations from the classical result n s oc -- In-it [1, 8, 9]. The authors state that this new result can be understood by a certain theory [10]. The square root power law for the soliton density, however, recently has been repudiated confirming the classical result for d = 3 systems [11 ]. Moreover, the classical results have been demonstrated to hold in three dimensions akeady some time ago for a special model [12]. Employing phenomenological arguments the classical behaviour has also been derived recently [13]. We note that the authors of this work [13] also criticize the interpretation of previous NMR results [3, 4] concerning the critical behaviour of the soliton density. For completeness it should be mentioned that recent ESR measurements [14] on Mn ÷ doped Rb2ZnC14 crystals did not give clear evidence either for the square root power law or for the logarithmic behaviour. Thus, whereas the theories presently 885
available predict a classical logarithmic behaviour from the experimental facts so far no clear picture can be derived for the soliton density. In order to obtain more precise information on the temperature behaviour of the soliton density n8 in Rb2ZnCIa we started 87Rb (I = 3/2) NMR measurements. From our investigations of the 87Rb NMR satellite transitions -+ 1/2 ~÷ -+ 3/2 [15] in the incommensurate phase of Rb2ZnC14 it was not possible to extract the intensities of the commensurate lines from the incommensurate background for T >~ Tc because of the rather broad and noisy spectra. Therefore, we measured the temperature dependence of the 1/2 ,~ 1/2 central transition around Tc as it has been done previously by Blinc et al. [3]. Experimental details of our work are described elsewhere [15]. A typical spectrum of the STRb central transition in the incommensurate phase of Rb2ZnC14 at about T = Tc + 0.4 K is shown in Fig. 1. The crystal orientation was nearly the same as used in [16, 18]. The spectrum essentially is a superposition of the quasi-continuous line shape functions of the Rb(1) and Rb(2) nuclei [15]. The line shape in Fig. 1 is in qualitative accordance with the theoretically predicted lineshape in an incommensurate multisoliton lattice [17]. The two lines indicated by arrows in Fig. 1 are of special interest because on passing from the incommensurate (IC) to the ferroelectric (FE) phase they are growing from an incommensurate background finally coming up as narrow discrete lines in the FE phase. We measured the temperature dependence of these lines in a range of about 25 K around Te (Fig. 2). The intensity of one commensurate line is proportional to the number of nuclei with nearly the same electrical field gradient tensor (EFG). These nuclei "he" in regions which are essentially commensurate with the underlying lattice and which are separated from each other by phase solitons or domain wails [16, 17] (leading to the -
-
CLASSICAL BEHAVIOUR OF THE SOLITON DENSITY IN Rb2ZnC14
886
87Rb:VL:98.17
HHz
i0
o I Ho
~t(c,H o)=27 °
,
10 kHz
,
0S
~
Fig. 1. Typical spectrum of the second order shifted 87Rb central lines near To in Rb2ZnC14. The meaning of the two lines indicated by arrows is discussed in the text.
severa[ • • "temperature runs
Vol. 44, No. 6
• A . j ~ r ~ ~.
AA
C~-
06
N
~ 02 -90
"' Id:°
0,-
'1/'1I,
02
II -86
-82
-78
-Tk
-70
Temperature [C ] Fig. 3. Temperaturedependence at the soliton density in Rb2ZnCI4 determined for different temperature runs which all were performed at increasingtemperature. The solid curve is obtained by computing equation (3). Inset: Calculation of equation (5) for a first order phase transition (d > 0) with limit of stability T~ and for a continuous phase transition (d = 0) with phase transition temperature T e. -
87.7 a*/3, i.e. p = 3. Thus disregarding constant terms the free energy density can be written as [8, 9]
-
8/+./+
F(x) = 1/2(O¢a(x)/ax -- 60) 2 -- v cos pc~(x),
-82!, -
81.3
-
80/+
TI°C - 76.3
~
-
68.9
Fig. 2. Temperature dependence of the two lines indicated in Fig. 1 by arrows.
incommensurate background limited by edge singularities). Thus, we calculated the intensity above the background o f these two lines in Fig. 2, which is proportional to the density of the commensurate regions ne = 1 -- n 8. The soliton density n s as measured as a function of temperature for three temperature runs is shown in Fig. 3. The experimental facts will be discussed by theories [1,8, 9, 19] describing the lock-in transition of incommensurate systems with a wave vector qi = a*/p + 8o. For Rb2ZnCI4 the modulation is along one direction, say the x-direction, and the wave vector qi is close to
(1)
where @(x) is the phase o f the complex order parameter whose amplitude A is assumed to be independent of x because so far no experimental evidence for amplitude solitons is known and v = -- v o A . Minimizing the free energy F = (A2/L) foc F(x) dx where L is the length of the crystal leads to the static sine Gordon equation a2@/ax 2 = p v sin p@. One special solution is the soliton ~ x ) = (4/p) arctan exp (px/-vx), which describes a single domain wall separating two commensurate regions with @~ 0 and @~ 2rt]p. Feeding this result back into equation (1) and calculating the free energy for an assembly o f M solitons one finds [8, 9] F = (2K/Tr2)~[1 + 4 exp (--t¢/~)] --060,
(2)
where K = 21rx/-v and ~ = 2Mrr/pL denotes the average wave vector for an incommensurate configuration. Thus, is proportional to the number of solitons and acts therefore as the order parameter of the lock-in transition. Minimizing equation (2) with respect to ~ one finds the soliton density 0 = ns as a function of 60 (1 + K/ns)4 exp ( - K/ns) = (6o-- 2Klrr2)l(2K/Tr2).
(3)
For the critical value 2Kfir 2 = 8o a continuous phase transition occurs where n8 = 0; for 2K[lr 2 < 80 an incommensurate state is favoured with non-zero n s. The temperature enters the problem through the temperature dependence of the ordering amplitude A which is proportional to (Ti -- T ~ [3, 9, 15]. Consequently one can linearize the right-hand side of equation (3) and
Vol. 44, No. 6
CLASSICAL BEHAVIOUR OF THE SOLITON DENSITY IN Rb2ZnC14
approximate by t = (T-- Te)[Te. In a suitable temperature range sufficiently far above To, equation (3) can be approximated by the well known classical result [8] n s cx - - l n - l t .
(4)
For the overall temperature dependence, particularly close to To, however, one has to solve equation (3) numerically. This has been done by assuming the parameter K to vary weakly with temperature according to the nearly saturated value of the ordering amplitude. Thus, r may be approximated by a constant value. A fit of the experimental data gives r = 5.5 + 0.1. The solid curve in Fig. 3 demonstrates that n s can be well represented by equation (3). Obviously the measuring points for T <~ Tc do not fit so well to equation (3). This may have several reasons. On the one hand there exists an unavoidable temperature gradient of about 0.3-0.5 K over the sample volume which leads to a smeared out behaviour in the phase transition region. On the other hand, it is well known that in most of the ferroelectric crystals there may appear domains of distinct polarization below Te which cause internal strains. Principally this effect could be avoided if one could apply an electrical field along the ferroelectric b-axis on decreasing the temperature below Te. Because of the reasons given above from our experiments we cannot decide whether or not the phase transition is of first or second order. In the literature it is widely assumed that the phase transition in Rb2ZnC14 is of first order with a temperature hysteresis which is probably due to soliton pinning on defects [20]. Comparing our results to those in the literature [3, 4] it can be stated that, near Te, n 8 depends much steeper on the temperature than in the previous works. This is related to the fact that contrary to the previous results the points in Fig. 3 cannot be fitted by a power law with a critical exponent 1/2. Rather the validity of the classical equation (3) is shown. The differences between these results possibly may be related to different qualities of the samples. The good quality standard of the crystals used in the present experiments becomes obvious from the fact that in these samples the 87Rb NMR satellite lines could be detected in Rb2ZnC14 for the first time [15]. Equation (1) can be extended by taking into consideration other degrees of freedom [9]. To be specific, we consider the coupling to an elastic strain, because defects can cause internal strains. Thus, one has to add to equation (1) the coupling between the strain r~ and the displacement profile and the usual elastic energy thus resulting in the terms 2w0 a¢/Ox + 1/2Cot/2. Calculating as before the free energy and minimizing this quantity
887
with respect to 71equation (2) takes an additional term -- 2(w2/co)~!2, provided w 2 < 2raCo exp (-- r/fl)/(rtfla). This case is always fulfilled for the weak coupling limit w -~ 0 [9]. Proceeding as before the temperature dependence of the soliton density is given by (1 + r / n s ) 4 e x p ( - - g / n s ) - - d n s = (T--Te)/Tc,
(5)
where d = 2w2rr/(rCo) >~O. The additional term in equation (5) leads to a first order phase transition (see inset in Fig. 3). A coupling of the displacement field to a long wave coordinate, i.e. the electrical polarization, can be analyzed in a similar way [9], resulting in a relation of the form of equation (5) provided the number of solitons is small.
Acknowledgements - The author is indebted to J. Petersson for helpful and stimulating discussions, to A. K16pperpieper for crystal growth and to the Deutsche Forschungsgemeinschaft for supporting this work within the frame of the Sonderforschungsbereich "Ferroelektrika". REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
W.L. McMillan, Phys. Rev. BI4, 1496 (1976). B.H. Suits, S. Couturi6 & C.P. Slichter, Phys. Rev. Lett. 45,194 (1980). R. Blinc, V. Rutar, B. TopiC, F. Milia & I.P. Aleksandrova, Phys. Rev. Lett. 46, 1406 (1981). V. Rutar, F. Milia, B. Topi~, R. Blinc & Th. Rasing, Phys. Rev. B25,281 (1982). M. Fukui & R. Abe, Jpn. J. Appl. Phys. 20, L533 (1981). A.S. Chaves, R. Gazzinelli & R. Blinc, Solid State Commun. 37,123 (1981). R. Blinc, Physics Reports 79, 331 (1981). P. Bak, Solitons and Condensed Matter Physics (Edited by A.R. Bishop & T. Schneider), p. 216. Springer-Verlag, New York (1978). A.D. Bruce, R.A. Cowley & A.F. Murray, J. Phys. C l l , 3591 (1978). T. Nattermann, J. Phys. C13, L265 (1980). T. Nattermann, J. Phys. 43,631 (1982). M.E. Fisher & W. Selke, Phys. Rev. Lett. 44, 1502 (1980). M.E. Fisher & D.S. Fisher, Phys. Rev. B25, 3192 (1982). M. Pezeril & J.C. Fayet, J. Phys. Lett. 43, L267 (1982). E. Schneider & J. Petersson, Z. Phys. B46, 169 (1982). R. Blinc, Ferroelectries 35,117 (198 0 . A.S. Chaves, R. Blinc, J. Seliger & S. Zumer, J. Magn. Res. 46, 146 (1982). R. Blinc, F. Milia, V. Rutar & S. ~umer, Phys. Rev. Lett. 48, 47 (1982). T. Nattermann & S. Trimper, Z Phys. 1214, 1603 (1981). K. Hamano, Y. Ikeda, T. Fujimoto, K. Ema & S. Hirotsu, J. Phys. Soc. Japan 49, 2278 (1980).