Sound attenuation in one-dimensional magnetic systems at finite temperatures

Volume 61A, number 6

PHYSICS LETTERS

13 June 1977

SOUND ATTENUATION IN ONE-DIMENSIONAL MAGNETIC SYSTEMS AT FINITE TEMPERATURES Katsuhiko NAGANO and Hisao OKAMOTO Department of Physics, Kyushu University, Fukuoka Received 20 April 1977 Sound attenuation constants are calculated for the one-dimensional ferro- and antiferromagnetic systems at finite temperatures. Their frequency dependences, in both cases, turn out to be ii Wk ~ ‘~cand 4if ~‘k~ where wc is a “cut-off” frequency related to the three-dimensional anisotropic interactions.

In previous papers [1, 2], we discussed anomalies in the sound attenuation constant and the change in the sound speed in one- and two-dimensional magnetic systems in the high-temperature limit. In this letter, we consider the finite-temperature case of the one-dimensional system by making use of Fisher’s exact result for the static spin correlation function [3]. As in the previous work, we introduce a model function for the time correlations of spins which coincides with the diffusion model in the long-wavelength limit. According to Tani and Mon’s theory of the ultrasonic attenuation in the magnetic system [4], the attenuation constant ak and the relative change in the sound speed (I~u)k/v of the sound wave with frequency wk and wave vector k are expressed in the form

=

~k ÷iWk(~)k/V =

I dt

(1)

e_~)kt(fk(t)f).
where u is the bare sound speed, Ak the normal mode of the sound related to the creation operator b~of phonon andfk the fluctuating force acting on the normal mode; byAk = i(2IV~k,.,~/Mu2)h/2bZ fk

(wkIMu)EqF(q,kq) (Sq0 ~i~°_q+Sq~S~_q),

(2)

F(q, k q) = iE —

8Ikl’(k [aJ(r)/ar] r=~)[eW~ ei(1c_~)~]

(3)

—

where J(r) is the exchange integral and & the vector connecting nearest neighbors and S (a =0, ±). By using the random phase approximation, we have for the integrand of (1)

1I2E =

N

2 E~{~F(q, k —q) + F(q, k _q)*I2(sqO(t)s~)(S~O_q(t)s~~+q) (fk(t)ft) = (WkIMV) + F(q, k q) j2 (S;(t)s:q> (Sj~_q(t) S~k+q>}. —

1S~exp[i(q-r1)],

(4)

Now we assume for the time-correlation functions of spins the hydrodynamic form [5—7]

(S(t)S*)

=

(IS~I2)exp(—I’t),

(a = 0, ±),

(5)

and discuss them on the basis of Mon’s statistical-mechanical theory of generalized Brownian motions [8]. If the reaction of the phonon system to the spin system can be ignored compared to the exchange interaction, and the motion of spins is determined by the isotropic Heisenberg interaction, Hex. =—Eq~.Q’~q_q r( ~(~0~0 q—q’ where J(q) = E 5J(&)• exp [i(q &)], then we obtain a self-consistent equation for ~

415

Volume 61A, number 6

2)Fk =

(IS~°I

PHYSICS LETTERS

13 June 1977

f dt(~(t)~*) (h2N)-1 Eq~J(q) J(k —q)12 (IS~I2)(S~ql2>I(Fq rk_q), =

+

—

(7)

where “q stands for 17, (a = 0, ±).For the static correlation function of spins, we use the exact formula for the Heisenberg chain of classical spins [3];

(

1s012>F(AF)_1(ls±12>F(AF)_S(S+ sinhg q 2 q 3 1) coshic~cosq~’

(8 )

—

where S is the magnitude of spins, K = —ln I cothK K~I is the inverse correlation length with K = S(S -l-1)J/kBT, J being the nearest-neighbor coupling constant of the isotropic Heisenberg interaction, q~is the component of q along the chain axis and the superscript F and AF are assigned to refer to the ferromagnetic (J> 0; the upper sign in the r.h.s.) and the antiferromagnetic case (J<0; the lower sign), respectively. Then we can solve eq. (7) self-consistently if we may set as —

—,

c’O

2~.

/

—

‘“q /lq\~Jq=Q / 2D~l cosq~ ,

9

where D is independent of q. Substituting (8) and (9) in (7), we obtain ,1/2

coshK ±1/2 0 L cosh~±1 J where D0 = (~2IJI/1i) [S(S + 1)/311/2 represents the diffusion constant in the high temperature limit. Using (2)’-’(5) and (8)~—(10)in (1), we find that 2 - gF(AF)(T). 3 cos4O - ~~3/2(1 i), ~ (AF)~7..~ = [JcS~S+l~~

DF(AF)=D

—

(10)

(11)

where J’ = I aJ(r)/arI 11~.~, c is the nearest-neighbor distance along the chain, 0 is the angle between k and the chain axis and +

1/4

2 [C~R~/2)] (12) gF(AF)(T) = [coth(K/2)]t gF(T) and gAF(r) both approach unity as the temperature is raised, and (11) reproduces the previous result [11*1, in the case e =0 of ref. [1]. At low tem~eratures,on the other hand, the difference between the ferro- and the antiferromagnetic case becomes clear and cb~(T)increases as T3 whereas Ø~~(T) vanishes as T3/2. In real materials, a different behaviour is expected since, for example, anisotropic interactions exist. These interactions do not conServe the components of the total spin and thus give rise to a “cut-off’ for the correlation of long wavelength modes due to the diffusive behaviour of spins [7, 101. In this respect, we have calculated the “cut-off’ frequencies ~F(AF) 1-~o~F0(AF)(a = 0, 1), or the inverse correlation times, by taking into account the three-dimensional dipolar interaction and the single-ion type anisotropy in a similar way as used in deriving (10). 31’2(c/a)6. w~AF)(T)and Thus we obtain w~(T) = 2 wF(AF)(T)~.2~I~ Wd(COd/DO)1”3 hF(AF)(T), (13) .

where a is the nearest-neighbor distance in a plane perpendicular to the chain axis, Wd = [6(gj.t~)2/c3 + D 5/3] X 2/h, g being the g-factor, 1B the Bohr magneton and D [S(S + 1)/3] l/ 5 the single-ion type anisotropy constant, and 1/6 . 2/3- [c~~ll/2)] (14) hF~~(T) [coth(sc/2)]~ ~ D 0 in the present paper corresponds to D/2 in ref. Li], and the factor S(S+ 1)/3 in eq. (12a) there should be corrected as 2. [S(S+i)/3]

416

Volume 61A, number 6

PHYSICS LETTERS

13

June 1977

Thus the effect of the anisotropic interactions replaces w~2(1 1) in (11) by —

\4W~[(iWk+ 2w~AF))_1~’2 + 2(kok+ 2~,F~F))—1/2J ~ ~-~k~ ~,F(AF)

2(l—i) ~w~

w~2(l—i)

(15.a)

if w~’~CA?k~’~,

(15.b)

if

(15.c)

W&1~~~wk,

where we have assumed a > c which, in turn, leads to WF~AF) WF~&F). In the case of actual observations, therefore, the Wk dependence (11) is valid only for higher frequencies, as shown by (l5.b) and (15.c)*2, and in the low frequency region (15.a), the conventional (4-dependence of ak = v~Re cbk is obtained and (L~v)k/v= ~ ImØ~vanishes. The latter would be one possible mechanism for the dependence in the experiment on the one-dimensional antiferromagnet CsNiC1 3 by Almond and Rayne [10, 11]. It may be noted in this connection that, as seen from (14), the low-temperature experiments in the anti-ferromagnetic case will be advantageous in order to fulfill the high-frequency condition in (15.b) and (15.c). We expect experiments to investigate these problems. A full note will be presented elsewhere. ~

-

(4

We would like to express our sincere gratitude to Professor H. Mon for helpful discussions and the continued encouragement throughout the work. 5/2 times as the frequency increases should be about three times with increasing frequencies (as shown in (15.b) and (15.c) in the present paper).

*2 The statement in ref. [1] that the coefficient ofa~increases by

it does by

corrected that

References [1] K.NaganoandH.Okamoto,J.Phys.Soc.Japan 39 (1975) 1619. [2] H. Okamoto and K. Nagano, J. Phys. Soc. Japan 40 (1976) 1783. [3] M.E. Fisher, Am. J. Phys. 32 (1964) 343; H. Tomita and H. Mashlyama, Prog. Theor. Phys. 48 (1972) 1133. [4] K. Tani and H. Mori, Prog. Theor. Phys. 39 (1968) 876. [5] L. Kadanoff and P.C. Martin, Ann. Phys. (N.Y.) 24 (1963) 419. [6] R.E. Diets et al., Phys. Rev. Letters 26 (1971) 1186. [7] M.J. Hennessy, C.D. McElwee and P.M. Richards, Phys. Rev. B7 (1973) 930. [8] H. Mori, Prog. Theor. Phys. 33 (1965) 423. [9] J.-P. Boucher, M. Ahnied-Bakheit, M. Nechtschein, M. Villa, G. Bonera and F. Borsa, Phys. Rev. B13 (1976) 4098.

[10] D.P. Almond and i.A. Rayne, Phys. Letters 55A (1975) 295. Lii] Private communications. We are thankful to Dr. D.P. Almond

for sending his experimental data before publication and also

for useful comments.

417