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Anomalous relaxation in low-dimensional magnets
Journal of Magnetism and Magnetic Materials 31-34 (1983) 643-644
643
ANOMALOUS RELAXATION IN LOW-DIMENSIONAL MAGNETS Hisao O K A M O T O , Takashi K A R A S U D A N I *, K a t s u h i k o N A G A N O a n d H a z i m e M O R I Department of Physics, Faculty of Science, Kyushu University," * Research Institute for Applied Mechanics, Kyushu University," Fukuoka 812, Japan
Anomalies of the angle-dependent EPR line spectrum and its satellites due to the long-time tails in one- and two-dimensional Heisenberg paramagnets are treated in the high-temperature limit by making use of a new mode-coupling theory. Its numerical results are compared with experimental data for TMMC and (C2HsNHa)2MnCI 4.
The electron paramagentic resonance (EPR) in oneand two-dimensional Heisenberg magnets exhibits the following anomalies in the high-temperature region: (1) The line shape is non-Lorentzian. (2) Satellite lines appear at multiples of ~0o where ,%(--gl~BH/h) is the Zeeman frequency. The anomalies of the EPR in d-dimensional magnets are due to a long-time t -d/2 tail in the time correlation functions which couple to hydrodynamic processes. In a one-dimensional magnet, i.e., TMMC, the main line and the 2 ~00 satellite line have been accounted for from this point of view by Lagendijk and Schoemaker in qualitative agreement with experiment [1,2]. Benner has also carried out a similar analysis in two-dimensional magnets [3]. They used a mode-decoupling approximation for the memory spectrum, which seems to be insufficient to clarify how the long-time tail is cut off so as to give finite intensities at resonances ,0 = _+mo0. Recently, the present authors have developed a mode-coupling continued-fraction representation of the absorption spectrum by taking account of all higherorder couplings to the spin diffusion process and shown that it gives a systematic method for investigating the EPR spectrum, including the satellite line and the cutoff of the long-time tail due to the dipole-dipole interaction [4,5]. We have also explored an extended theory, on the basis of our previous theory [4,5], in order to treat Heisenberg systems with the dipole-dipole interaction and other anisotropic interactions [6], whose Hamiltonian is given by H = H E + H D + H A + H K + Hz,
(1)
where HE, HD, HA, H K and H z are the exchange interaction, the dipole-dipole interaction, the uniaxial anisotropic exchange interaction, the single-ion anisotropy and the Zeeman term, respectively. Each term of (1) takes the form
H E = -2JEESioSj, i >j
H o = ~-~.E ( g21~/r~ )[ S / . ~ - (3/r, 2 )( rij. s )( rij.S j )], i >j
H~ = A Z E [ i>j
SiXS7 + SiYSf - 2 S : ~ ] ,
(2)
z 2 .K= K E[(s;)-s(s+l)/3], i
n z = g BE Sin. i
The normalized relaxation function X~(t) of the Fourier transform of the # component of the spin density S~ with wave vector k is defined by
--'~(t) =- ( S ~ ( t ) I S ~ ) / ( S ~ I S ~ ) ,
(# = +, 0).
where ( . . . [... ) denots the canonical average. By employing the new mode-coupling continued-fraction representation [4,5], the relaxation spectrum .~[ito], the Laplace transform of the normalized relaxation function, can be written as -0u[io~] = l / ( i • - i/u% + 4~,0 [i~ ]),
(3)
where q~l~,0[io~] is the memory spectrum whose recurrence formulae have been derived by the present authors [4-6], The EPR line spectrum is given by the real part of (3). If a set of parameters of (2) satisfies the following relations: ~'(3)hoao+A - K / 3 = 0 , Z(3)h~0 D - 2A + K / 3 = O,
(1 dim.), (2 dim.),
(4)
then, the effect of the long-time tail in the memory spectrum may vanish and q~,0[i0~] reduces to an ~-independent function ~,~(0), where ~ o = g2#2/c3h, and ~'(3) and Z(3) are Zeta functions which come from the dipole sum on the one-dimensional linear chain and the twodimensional square lattice, respectively. Thus, in these special cases, (3) reduces to the Lorentzian line spectrum: .7~[io~]= 1/[i0~-i/u%+~,~(0)], where 0 is the angle between the external magnetic field and the magnetic chain axis in the one dimension or the normal of the magnetic plane in the two dimension, respectively. Figs. 1 and 2 show the numerical results calculated from (3) for the one- and two-dimensional magnets, i.e., TMMC and (C 2H 5NH 3)2 MnCI 4, respectively. In these figures, the angular dependence of the peak-to-peak derivative line width, AHpp, and its peak-to-peak height of the main line and the 2o:0- and 3a~0-satellite lines of the spectral function calculated from (3) with A = K = 0
0 3 0 4 - 8 8 5 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d
644
H. Okamoto et at / Anomalous relaxation in low-dimensional magnets
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Fig. 2. The 0-dependence of the peak-to-peak derivative line width AHpP (solid line) and its peak-to-peak height (dashed • hne) of (a) the main line, and (b) the 2 w0-satellite line, for the two-dimensional magnet, (C 2H 5NH 3)2 MnCI 4- The dotts in (a) and (b) represent the X-band data of Benner [3]. height of the satellite lines to that of the main resonance line is about 1 : 1 0 - 3 : 1 0 -5 and 1 : 1 0 - 5 : 1 0 -1° in the one- and two-dimensional magnets, respectively. The comparison between the theoretical result calculated from (3) with (1) and (2) and experimental data for an anisotropic two-dimensional magnet, K2CuF4 [9], is being carried out and shall be presented elsewhere.
0 90
Fig. 1. The 0-dependence of the peak-to-peak derivative line width AHpp (solid line) and its peak-to-peak height (dashed line) of (a) the main line, (b) the 2~0-satellite line, and (c) the 3~o-satellite line for the one-dimensional magnet, TMMC. The dotts in (a) and (b) represent the X-band data of Yamada et al. [7] and Benner and Wiese [8], respectively. are shown and are also compared with experimental results observed in X band for T M M C [7,8] and (C2HsNHa)2MnC14 [3]. An adjustable parameter, diffusion constant, was determined by the experimental line width of the main line at 8 = 0 in the X band. Consequently, the ratio of the relative peak-to-peak
References [1] A. Lagendijk and D. Schoemaker, Phys. Rev. BI6 (1977) 47. [2] A. Lagendijk, Phys. Rev. B18 (1978) 1322. [3] H. Benner, Phys. Rev. BI8 (1978) 319. [4] H. Okamoto, K. Nagano, T. Karasudani and H. Moil, Progr. Theoret. Phys. 66 (1981) 53, 437. [5] T. Karasudani, H. Okamoto, K, Nagano and H. Moil, to be submitted. [6] H. Okamoto and 1-I. Moil, to be submitted. [7] Y. Natsume, F. Sasagawa, M. Toyoda and I. Yamada, J. Phys. Soc. Japan 48 (1980) 50. [8] H. Benner and J. Wiese, Physica 96B (1979) 216. [9] I. Yamada, I. Morishita and T. Tokuvama. nrer~rint.
643
ANOMALOUS RELAXATION IN LOW-DIMENSIONAL MAGNETS Hisao O K A M O T O , Takashi K A R A S U D A N I *, K a t s u h i k o N A G A N O a n d H a z i m e M O R I Department of Physics, Faculty of Science, Kyushu University," * Research Institute for Applied Mechanics, Kyushu University," Fukuoka 812, Japan
Anomalies of the angle-dependent EPR line spectrum and its satellites due to the long-time tails in one- and two-dimensional Heisenberg paramagnets are treated in the high-temperature limit by making use of a new mode-coupling theory. Its numerical results are compared with experimental data for TMMC and (C2HsNHa)2MnCI 4.
The electron paramagentic resonance (EPR) in oneand two-dimensional Heisenberg magnets exhibits the following anomalies in the high-temperature region: (1) The line shape is non-Lorentzian. (2) Satellite lines appear at multiples of ~0o where ,%(--gl~BH/h) is the Zeeman frequency. The anomalies of the EPR in d-dimensional magnets are due to a long-time t -d/2 tail in the time correlation functions which couple to hydrodynamic processes. In a one-dimensional magnet, i.e., TMMC, the main line and the 2 ~00 satellite line have been accounted for from this point of view by Lagendijk and Schoemaker in qualitative agreement with experiment [1,2]. Benner has also carried out a similar analysis in two-dimensional magnets [3]. They used a mode-decoupling approximation for the memory spectrum, which seems to be insufficient to clarify how the long-time tail is cut off so as to give finite intensities at resonances ,0 = _+mo0. Recently, the present authors have developed a mode-coupling continued-fraction representation of the absorption spectrum by taking account of all higherorder couplings to the spin diffusion process and shown that it gives a systematic method for investigating the EPR spectrum, including the satellite line and the cutoff of the long-time tail due to the dipole-dipole interaction [4,5]. We have also explored an extended theory, on the basis of our previous theory [4,5], in order to treat Heisenberg systems with the dipole-dipole interaction and other anisotropic interactions [6], whose Hamiltonian is given by H = H E + H D + H A + H K + Hz,
(1)
where HE, HD, HA, H K and H z are the exchange interaction, the dipole-dipole interaction, the uniaxial anisotropic exchange interaction, the single-ion anisotropy and the Zeeman term, respectively. Each term of (1) takes the form
H E = -2JEESioSj, i >j
H o = ~-~.E ( g21~/r~ )[ S / . ~ - (3/r, 2 )( rij. s )( rij.S j )], i >j
H~ = A Z E [ i>j
SiXS7 + SiYSf - 2 S : ~ ] ,
(2)
z 2 .K= K E[(s;)-s(s+l)/3], i
n z = g BE Sin. i
The normalized relaxation function X~(t) of the Fourier transform of the # component of the spin density S~ with wave vector k is defined by
--'~(t) =- ( S ~ ( t ) I S ~ ) / ( S ~ I S ~ ) ,
(# = +, 0).
where ( . . . [... ) denots the canonical average. By employing the new mode-coupling continued-fraction representation [4,5], the relaxation spectrum .~[ito], the Laplace transform of the normalized relaxation function, can be written as -0u[io~] = l / ( i • - i/u% + 4~,0 [i~ ]),
(3)
where q~l~,0[io~] is the memory spectrum whose recurrence formulae have been derived by the present authors [4-6], The EPR line spectrum is given by the real part of (3). If a set of parameters of (2) satisfies the following relations: ~'(3)hoao+A - K / 3 = 0 , Z(3)h~0 D - 2A + K / 3 = O,
(1 dim.), (2 dim.),
(4)
then, the effect of the long-time tail in the memory spectrum may vanish and q~,0[i0~] reduces to an ~-independent function ~,~(0), where ~ o = g2#2/c3h, and ~'(3) and Z(3) are Zeta functions which come from the dipole sum on the one-dimensional linear chain and the twodimensional square lattice, respectively. Thus, in these special cases, (3) reduces to the Lorentzian line spectrum: .7~[io~]= 1/[i0~-i/u%+~,~(0)], where 0 is the angle between the external magnetic field and the magnetic chain axis in the one dimension or the normal of the magnetic plane in the two dimension, respectively. Figs. 1 and 2 show the numerical results calculated from (3) for the one- and two-dimensional magnets, i.e., TMMC and (C 2H 5NH 3)2 MnCI 4, respectively. In these figures, the angular dependence of the peak-to-peak derivative line width, AHpp, and its peak-to-peak height of the main line and the 2o:0- and 3a~0-satellite lines of the spectral function calculated from (3) with A = K = 0
0 3 0 4 - 8 8 5 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 N o r t h - H o l l a n d
644
H. Okamoto et at / Anomalous relaxation in low-dimensional magnets
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A
50
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//f \'\
501 AHpp 80C
•
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Fig. 2. The 0-dependence of the peak-to-peak derivative line width AHpP (solid line) and its peak-to-peak height (dashed • hne) of (a) the main line, and (b) the 2 w0-satellite line, for the two-dimensional magnet, (C 2H 5NH 3)2 MnCI 4- The dotts in (a) and (b) represent the X-band data of Benner [3]. height of the satellite lines to that of the main resonance line is about 1 : 1 0 - 3 : 1 0 -5 and 1 : 1 0 - 5 : 1 0 -1° in the one- and two-dimensional magnets, respectively. The comparison between the theoretical result calculated from (3) with (1) and (2) and experimental data for an anisotropic two-dimensional magnet, K2CuF4 [9], is being carried out and shall be presented elsewhere.
0 90
Fig. 1. The 0-dependence of the peak-to-peak derivative line width AHpp (solid line) and its peak-to-peak height (dashed line) of (a) the main line, (b) the 2~0-satellite line, and (c) the 3~o-satellite line for the one-dimensional magnet, TMMC. The dotts in (a) and (b) represent the X-band data of Yamada et al. [7] and Benner and Wiese [8], respectively. are shown and are also compared with experimental results observed in X band for T M M C [7,8] and (C2HsNHa)2MnC14 [3]. An adjustable parameter, diffusion constant, was determined by the experimental line width of the main line at 8 = 0 in the X band. Consequently, the ratio of the relative peak-to-peak
References [1] A. Lagendijk and D. Schoemaker, Phys. Rev. BI6 (1977) 47. [2] A. Lagendijk, Phys. Rev. B18 (1978) 1322. [3] H. Benner, Phys. Rev. BI8 (1978) 319. [4] H. Okamoto, K. Nagano, T. Karasudani and H. Moil, Progr. Theoret. Phys. 66 (1981) 53, 437. [5] T. Karasudani, H. Okamoto, K, Nagano and H. Moil, to be submitted. [6] H. Okamoto and 1-I. Moil, to be submitted. [7] Y. Natsume, F. Sasagawa, M. Toyoda and I. Yamada, J. Phys. Soc. Japan 48 (1980) 50. [8] H. Benner and J. Wiese, Physica 96B (1979) 216. [9] I. Yamada, I. Morishita and T. Tokuvama. nrer~rint.