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Critical behavior of spin relaxation in K2CuF4
C R I T I C A L B E H A V I O R O F S P I N R E L A X A T I O N IN K2CuF 4 T. H A S H I M O T O , Y. K O J I M A and T. I K E G A M I Department of Applied Physics, Faculty of Science, Tokyo Institute of Technology,, Oh-Okayama, Meguro, Tokyo, Japan
From the observation of x(q = 0, ~0)we obtained X0and the relaxation time ~-of spin and determined these critical indices. They show the crossover phenomena near ~ = 10-2 and e ~ 7 x 10-2. It has been also shown that pseudo-ferromagnetic-ordered clusters, which have a life time longer than ~ 10-6 s, exist below e ~ 10-2 even in the paramagnetic region.
1. Introduction
3. Results and discussion
Recently, from the view point of statistical physics the dynamical characters near the 2nd order phase transition point (To) have become an important and interesting area of study. In order to clarify this problem we considered the ferromagnetic substances as a typical example and have investigated the critical spin dynamics focusing on the following subjects: (i) Whether the crossover p h e n o m e n o n of the critical index with change of the lattice and spin dimensionality can be observed or not. (ii) Whether Onsager's kinetic coefficient, the ratio of the static susceptibility X0 to the relaxation time ~" of spin, depends on temperature or not. (iii) H o w the dispersion mechanism changes close to T~. This behavior is closely related to an important characteristic of the clustering process of the ferromagnetically ordered cluster. In order to clarify the critical dynamics of the time dependent uniform m o d e of spins, we have investigated the high frequency susceptibility x ( q = 0, to) of a ferromagnet near T¢. The quasi-twodimensional Heisenberg ferromagnet K2CuF 4 [1] was chosen as a sample.
In the monodispersive relaxation process the frequency dependence of complex magnetic susceptibility x ( q = O, to) can be described by x ( q = O, to) = X' - ix" = X o / (1 + ito,),
where X' and X" are real and imaginary parts of x ( q = O, to) and X0 and T are the static magnetic susceptibility and the relaxation time of the spin system, respectively. In the analysis of Xo a n d , we used eq. (1). Fig. 1 shows x0(a) and l"(b) in the paramagnetic region at several frequencies plotted against the reduced temperature, ~ -- ( T - T c ) / T ¢ , on a l o g log scale. In the temperature range of e ~> 5 × .
.'.
,
,
,
,
"2"
= 0.85
(m "c
2. Experiments
The single crystal of K 2 C u F 4 was formed into a coaxial shape, the axis of which is parallel to the c-axis of the crystal. In measuring, in order to exclude the demagnetization field effect, this sample was placed on the shorted end of the coaxial cavity [2]. By the analysis of impedance change of the sample part measured by the R X meter of the Boonton Radio Company, we obtained x ( q -- O, to) over the frequency range of 1 to 200 M H z in the temperature range of 4.2 to 7 K.
(1)
5
"%_.
~ .
• ,3.o MH, 1
o
42.0
lo_.~.,,,
\.
MHz
'
--10-9
~o o
-
I 10 -3
,
,
, I,,,,t
, I 0 "2
,
, i ....
I
10 -I
F..
Fig. I. (a) Xo(" X'[I + (X"/X')2]) and (b) ~(= Ly"/~ox']), obtained at the several frequencies, plotted against the reduced temperature, e - ( T - Tc) / Tc, on a log-log scale.
Journal of Magnetism and Magnetic Materials 15-18 (1980) 1025-1026 ©North Holland
1025
1026
T. Hashimoto et al./ Critical spin relaxation in K2CuF4
10-3, X0 in all frequencies almost coincide with each other. This clearly verifies that in this temperature range the relaxation process is monodispersive and also that the relaxation process changes at e ~ 5 X 10 -2 from monodispersive to polydispersive. Gradual changes of the gradient of Xo near e ---- 1 0 - 2 and e ~, 7 x 1 0 - 2 m a y show existence of the crossover phenomena. According to Dupas and Renard [3] X0 parallel to the a-axis shows two kinds of crossovers near e ~ 1 0 - 2 and ~ 8 × 10 -2. The latter crossover was tried to explain by the change of the dimension of the spin system, the former remaining inexplicable. Above e - - 10 -2, our result agrees well with theirs. Below e = 10 -2, since the relaxation process is polydispersive, we obtain y = 1 from the tangent of X0 in the lowest frequency, 1.8 MHz, which is not in contradiction to their value. In order to see precisely the spin dynamics below the former crossover, we show the C o l e - C o l e plots of x ( q = 0, 6o) at several-reduced temperature in fig. 2. In the range e ~> 5 x 10 -3, this plot makes a pure single semi-circle which indicates the monodispersive spin relaxation process. On the contrary, in the range e ~< 5 × 10 -3, this plot is composed of two kinds of curves. One of them changes from a pure semi-circle above e ~ 5 x 10 -3 to an approximate semi-ellipse and its long axis increases considerably as temperature approaches To. The other kind of semi-circle appears close to T¢ and seems to enlarge gradually even in the ferromagnetic region. Taking into account the strong temperature dependence of the former near T~, we m a y conclude that the former is due to the critical fluctuation of the spin system. In the ferromagnetic region the former disappears rapidly, but the latter remains and enlarges. This result may be explained by a model, in which the pseudo-ferromagnetically-ordered clusters having the finite life time, longer than ~ 10 -6 s, exist close to T¢ even in the paramagnetic region.
2001 -
! I
~ 1.6 x l O -3 2.4 X l O -3 C 3.2 X 1 0 -3 D 4.8 x l O -3
9.5 ~.z~
~ ~°//_~---_. -"~'~
.3 E A6'4 ×10-3 /
-----------_.
184
Fig. 2. The cole-cole plots of x(q = 0, to) at the several-reduced temperature in the paramagnetic region. The e dependence of z shows clearly that the critical slowing down phenomenon occurs and is very similar to that of X0 except the latter crossover near e ~ 7 x 10 -2, which is masked with experimental error. The present paper contributes the first verification of existence of crossover in dynamical character near T c in a ferromagnet. Finally, we touch briefly Onsager's kinetic coefficient L ( = X0(/z). Recently, in the critical region some investig~ttors [4, 5] have obtained theoretically that L does not depend on temperature. On the contrary, the experimental results in K2CuF 4 and EuS [2] show that L diverges by the exponent (y - A) 0.2. Finding the mechanism of this discrepancy should be the subject for a future study. This work is partially supported by the Kurata Foundation.
References
[1] I. Yamada, J. Phys. Soc. J a p a n 28 (1970) 1585. [2] M. Shiino and T. Hashimoto, J. Phys. Soc. Japan 45 (1978) 22. [3] A. Dupas and J. P. Renard, J. de Phys. 37 (1976) C1-213. [4] G. B. Teitelbaum, Sov. Phys. JETP Lett. 21 (1975) 145. [5] M. Shiino, Thesis, Tokyo Institute of Technology, Japan (1979).
From the observation of x(q = 0, ~0)we obtained X0and the relaxation time ~-of spin and determined these critical indices. They show the crossover phenomena near ~ = 10-2 and e ~ 7 x 10-2. It has been also shown that pseudo-ferromagnetic-ordered clusters, which have a life time longer than ~ 10-6 s, exist below e ~ 10-2 even in the paramagnetic region.
1. Introduction
3. Results and discussion
Recently, from the view point of statistical physics the dynamical characters near the 2nd order phase transition point (To) have become an important and interesting area of study. In order to clarify this problem we considered the ferromagnetic substances as a typical example and have investigated the critical spin dynamics focusing on the following subjects: (i) Whether the crossover p h e n o m e n o n of the critical index with change of the lattice and spin dimensionality can be observed or not. (ii) Whether Onsager's kinetic coefficient, the ratio of the static susceptibility X0 to the relaxation time ~" of spin, depends on temperature or not. (iii) H o w the dispersion mechanism changes close to T~. This behavior is closely related to an important characteristic of the clustering process of the ferromagnetically ordered cluster. In order to clarify the critical dynamics of the time dependent uniform m o d e of spins, we have investigated the high frequency susceptibility x ( q = 0, to) of a ferromagnet near T¢. The quasi-twodimensional Heisenberg ferromagnet K2CuF 4 [1] was chosen as a sample.
In the monodispersive relaxation process the frequency dependence of complex magnetic susceptibility x ( q = O, to) can be described by x ( q = O, to) = X' - ix" = X o / (1 + ito,),
where X' and X" are real and imaginary parts of x ( q = O, to) and X0 and T are the static magnetic susceptibility and the relaxation time of the spin system, respectively. In the analysis of Xo a n d , we used eq. (1). Fig. 1 shows x0(a) and l"(b) in the paramagnetic region at several frequencies plotted against the reduced temperature, ~ -- ( T - T c ) / T ¢ , on a l o g log scale. In the temperature range of e ~> 5 × .
.'.
,
,
,
,
"2"
= 0.85
(m "c
2. Experiments
The single crystal of K 2 C u F 4 was formed into a coaxial shape, the axis of which is parallel to the c-axis of the crystal. In measuring, in order to exclude the demagnetization field effect, this sample was placed on the shorted end of the coaxial cavity [2]. By the analysis of impedance change of the sample part measured by the R X meter of the Boonton Radio Company, we obtained x ( q -- O, to) over the frequency range of 1 to 200 M H z in the temperature range of 4.2 to 7 K.
(1)
5
"%_.
~ .
• ,3.o MH, 1
o
42.0
lo_.~.,,,
\.
MHz
'
--10-9
~o o
-
I 10 -3
,
,
, I,,,,t
, I 0 "2
,
, i ....
I
10 -I
F..
Fig. I. (a) Xo(" X'[I + (X"/X')2]) and (b) ~(= Ly"/~ox']), obtained at the several frequencies, plotted against the reduced temperature, e - ( T - Tc) / Tc, on a log-log scale.
Journal of Magnetism and Magnetic Materials 15-18 (1980) 1025-1026 ©North Holland
1025
1026
T. Hashimoto et al./ Critical spin relaxation in K2CuF4
10-3, X0 in all frequencies almost coincide with each other. This clearly verifies that in this temperature range the relaxation process is monodispersive and also that the relaxation process changes at e ~ 5 X 10 -2 from monodispersive to polydispersive. Gradual changes of the gradient of Xo near e ---- 1 0 - 2 and e ~, 7 x 1 0 - 2 m a y show existence of the crossover phenomena. According to Dupas and Renard [3] X0 parallel to the a-axis shows two kinds of crossovers near e ~ 1 0 - 2 and ~ 8 × 10 -2. The latter crossover was tried to explain by the change of the dimension of the spin system, the former remaining inexplicable. Above e - - 10 -2, our result agrees well with theirs. Below e = 10 -2, since the relaxation process is polydispersive, we obtain y = 1 from the tangent of X0 in the lowest frequency, 1.8 MHz, which is not in contradiction to their value. In order to see precisely the spin dynamics below the former crossover, we show the C o l e - C o l e plots of x ( q = 0, 6o) at several-reduced temperature in fig. 2. In the range e ~> 5 x 10 -3, this plot makes a pure single semi-circle which indicates the monodispersive spin relaxation process. On the contrary, in the range e ~< 5 × 10 -3, this plot is composed of two kinds of curves. One of them changes from a pure semi-circle above e ~ 5 x 10 -3 to an approximate semi-ellipse and its long axis increases considerably as temperature approaches To. The other kind of semi-circle appears close to T¢ and seems to enlarge gradually even in the ferromagnetic region. Taking into account the strong temperature dependence of the former near T~, we m a y conclude that the former is due to the critical fluctuation of the spin system. In the ferromagnetic region the former disappears rapidly, but the latter remains and enlarges. This result may be explained by a model, in which the pseudo-ferromagnetically-ordered clusters having the finite life time, longer than ~ 10 -6 s, exist close to T¢ even in the paramagnetic region.
2001 -
! I
~ 1.6 x l O -3 2.4 X l O -3 C 3.2 X 1 0 -3 D 4.8 x l O -3
9.5 ~.z~
~ ~°//_~---_. -"~'~
.3 E A6'4 ×10-3 /
-----------_.
184
Fig. 2. The cole-cole plots of x(q = 0, to) at the several-reduced temperature in the paramagnetic region. The e dependence of z shows clearly that the critical slowing down phenomenon occurs and is very similar to that of X0 except the latter crossover near e ~ 7 x 10 -2, which is masked with experimental error. The present paper contributes the first verification of existence of crossover in dynamical character near T c in a ferromagnet. Finally, we touch briefly Onsager's kinetic coefficient L ( = X0(/z). Recently, in the critical region some investig~ttors [4, 5] have obtained theoretically that L does not depend on temperature. On the contrary, the experimental results in K2CuF 4 and EuS [2] show that L diverges by the exponent (y - A) 0.2. Finding the mechanism of this discrepancy should be the subject for a future study. This work is partially supported by the Kurata Foundation.
References
[1] I. Yamada, J. Phys. Soc. J a p a n 28 (1970) 1585. [2] M. Shiino and T. Hashimoto, J. Phys. Soc. Japan 45 (1978) 22. [3] A. Dupas and J. P. Renard, J. de Phys. 37 (1976) C1-213. [4] G. B. Teitelbaum, Sov. Phys. JETP Lett. 21 (1975) 145. [5] M. Shiino, Thesis, Tokyo Institute of Technology, Japan (1979).