Theory of acoustic resonance of Ising magnets

Physica B 182 (1992) North-Holland

PHYSICA 1

71-7X

Theory of acoustic resonance

of Ising magnets

G.O. Berim, A.R. Kessel and S.S. Lapuschkin Kazur~ Physical-Technical Received 7 August Revised manuscript

Institute of the Russian Academy of Sciences,

1991 received

13 February

Kazan 420029. Russia

1992

The microscopic theory of acoustic resonance is developed for the linear Ising magnet: the operator of acoustic wave interaction with the spin system. resonance frequencies and resonance lines integral intensities are found. The latter are exactly expressed via the exactly known spin correlation functions of a model. The analysis was done of the integral intensity dependence of the value of the magnetic field. exchange integrals and the temperature.

1. Introduction Theoretical investigation of the nonequilibrium properties of statistical mechanics models shows that there is a possibility to describe exactly some nonequilibrium properties of those physical systems whose Hamiltonian allows an exact description of equilibrium properties. An example of such a system is the Ising model. This model is characterized by the Hamiltonian H,, = -no,,

c Sf - c ho,(p)S;S;+, i j.6,

)

(1)

possessing strong exchange interaction and allowing an exact description. Here Sf is the zcomponent of a spin placed in a lattice site j with radius-vector rj, Z,, = hw,( p) is the exchange integral between the neighbouring spins separated by the lattice vector p, o, is the Zeeman frequency for the magnetic field H, which is parallel to the exchange interaction anisotropy axis. The various exactly solvable modifications of the Ising model may differ from each other by the dimension and symmetry of the lattice as well as the spin value and the amount of the coordinaCorrespondence to: Prof. A.R. Institute of the IJSSR, Academy 10/7. Kazan 420029. Russia. OY2l-4526/Y2/$05.00

0

Kessel. Physical-Technical of Sciences, Sibirsky Tract.

1992 - Elsevier

Science

Publishers

tion spheres whose spins are involved in the exchange interaction [l-3]. Despite the simplicity of the Hamiltonian (1) the low-dimensional Ising models were successfully used for the description of magnetic properties of such substances as K,Fe(CN,), CoClz .2H,O, Rb,CoF,, DyPO,, . . . [4,5]. As a rule, nonequilibrium properties of spinsystems are studied in four regimes which differ from each other in the ratio between the external time-dependent influence value and the intensity of relaxation processes in the system. In weak stationary oscillating fields the resonance phenomena are investigated and the resonance frequencies, resonance line intensities and -widths are measured. For the intermediate values of oscillating fields the so-called saturation phenomenon arises and interesting kinetic processes take place in the spin system. Their investigation gives the information about the spin systemthermostat interaction. In strong oscillating fields which are applied usually as short impulses, the dynamics of a spin system during time intervals smaller than the spin-lattice relaxation times is studied. The free induction and spin echo signal which arise after impulses give the information about internal spin-spin interactions and kinetic processes. Enumerated are three types of experiments

B.V. All rights

reserved

dealing with the resonance excitation of the spin system. The fourth regime corresponds to the nonresonant excitation of the system. In this case the absorption curves have the Debye form and give information about the spin-spin and the spin-lattice relaxation. In the cases when the external influence is the electromagnetic field the nonequilibirum properties of the Ising systems were studied in detail theoretically as well as experimentally [4-c)]. At the same time similar investigations for acoustic excitation were not carried out. The only exception is the experimental work [lO] dealing with the investigation of a two-dimensional Ising magnet in the fourth regime. However, it is known that acoustic methods which are developed at the present time for electronic and nuclear paramagnets [ 1 l-131 extend considerably the information obtained by magnetic resonance methods. In particular they give the possibility to determine the dynamic spin-phonon interaction parameters. The theory of acoustic resonance (AR) of the one-dimensional Ising magnets proposed in the present paper corresponds to the first regime. From the methodical point of view such a theory has to preceed the theoretical study of AR in other regimes and we hope also to attract experimentalists’ attention to this interesting problem.

2. The Hamiltonian

of the problem

In the resonance phenomenon theory it is customary to subdivide the total Hamiltonian of the physical system into three parts: H = H, + H, .

H,

=

H,,

(2)

+ H' .

where the operator H,, determines the set of infinitely narrow energy levels of the spin system (in our case it is the Hamiltonian ( 1)). the operator H’ determines the width of these levels and the operator H, corresponds to the interaction with the time-dependent external field. H,, is the main Hamiltonian and it enters into

the theory without any assumption. This circumstance (and the fact that operator H,, contains two-particles exchange interaction) leads the theory to be essentially a many-particles enc. The important problem of the spin operator evolution under the influence of the Hamiltonian H,, was solved before [6. W]: .~I (t) = exp(iH,,t/h)S,’ = z

cxp(-iH,,t/h)

Kj(~,, ).S; cxp( tiwC,r) .

(3)

of the Hamilwhere wCrarc the cigcnfrcqucncies tonian H,,. S,’ = S.) + is:. Rj(w,,) are the proj,ective operators having the following propertlcs:

In the particular spin : and nearest w II = W,, + cyWC.

cast of the linear model with neighbour interaction WC’have u = 0. -+ I .

R,,(w,,) = i( I ~ 4S;+,S; Rj(w I,)=:(l~2s,,,)(1~2ss

,) .

(4) ,).

The expressions (4) for the eigenfrequcncies and the projective operators will be used below for the determination of the spectrum and the integral intensities of the Ising model AR absorption.. The line width in the solid concentrated magnets is determined usually by the spin-spin intcractions, which arc formed by the magnetic diinteraction. H<_. pole. H,‘), and the exchange The latter is not contained in H,,. The usual condition is Hi, B H,; (the typical values arc H,‘)= 10 c K. HJ, = 1 K), so we will conside] below the non-Ising part of the exchange intcraction only: H,!, = 2 c [I‘( p,)S)S; j 4

+ I “( ,u,)S;S;]

,

(5)

where pi = rj - rk runs over the nearest ncighbours of the site j and /“( pj) are the exchange integrals.

G.O.

Berim el al. I Theory of acoustic resonance of Ising magnets

To determine the operator H, the following considerations first used by Al’tschuler [14] are convenient. Under the action of acoustic vibration with wave vector q and polarization e the radius vector of a particle initially placed at the point rj becomes rj(t) = rj + u(rj, t), where

u(rj, t) = eu,, cos wt sin( qr, + cp)

In this case the linear response theory [15,16] is valid and so the power -absorbed my the spinsystem is

P(o)

is the acoustic displacement. As all the spin-spin interactions H, depends on the distances pj = lrj - r,l between the spins which are many times displacement acoustic greater then any (u(rj, t) G pi). The Hamiltonian H, can be expanded in a series on u,,. It is customary to treat the linear term of this expansion as the spinacoustic interaction operator H, [ 141. Let us note that the only part of the operator H, which does not commutate with H,, is responsible for the resonance transitions in the H,, spin system spectrum. That is why we take into account only the linear term of the expansion of the operator H& over u(rj, t) and obtain the operator H, in the form

73

= :

1 dt cos wt( { F(t), F]) -r

tanh( phw)

3 (7)

where

pP=2k~,

{A,B)=;(AB+BA),

(A) = Wexp(-2PH,,)Al A(t)‘= exp(iH,tlh)A

lTr[exp(-2PHd1,

exp(-iH,t/h)

.

Let us represent the operator F in the form where F(o,) is the FourierF = C, F(w&, component of operator F: exp(iH,,t/h)F = T F(w,)

exp(-iH,t/h)

exp(--i+t) ,

H,=-Fcoswt, F = c c (G_(p,)[S;S; i pi

+ S,S,]

+ G+(p,)[Sj’S,

+ S;s:])

(6)

X sin( qrj + cp) , G,(Pj)

= u,,(eV~‘(Pj.J)(Pjq)

where wC are the eigenfrequencies of the Hamiltonian H,). Then expression (7) can be represented in the form [15] P(w) = c A(+)& 5

7

1’ = I” + 1’ where the parameters G,( yj) = G, do not depend on j due to the translation invariance of the system. The expressions (l), (5) and (6) represent all the Hamiltonians needed in the AR absorption theory.

The quantities A(w,) and g(w - OJ*) are the integral intensity and the form function for the resonance line at the eigen-frequency w5 of H,, respectively. When the heat motion of the lattice can be neglected these functions are [15]

A(+) = g tanhW+)Qc Q,

3. AR line form When following hold:

resonance phenomena inequalities for the

(8)

- ws).

=

3

({F(Q F(-w,)H

(9)

3

g(w - n,> = [2%M;]~“2 take place, the matrix elements

X exp[-(w where

- wE - M:)*/2M:]

,

M: and M: are the first and the second

central moments of the resonance frequency wi. They arc equal to hM;

= ({[F(w,).

h’M’: = -({[F(q).

H’(O)].

F(-w,)})

line

iv,

at the

1

x[K,(l.l)+K,(pI.-l)].

H’(O)]})/Q,

~- [fiM;f

.

where H’(0) is the Fourier component of the operator H’ at the frequency wi = 0. The formulae of this section arc the basis fol the microscopic AR theory for different Ising models. They allow to determine the main parameters of the AR line: to find the resonance frequencies and to express the integral intensities and the parameters of the resonance line form function via the equilibrium correlation functions of the Ising model. In the low-dimensional models (one-dimensional. two-dimensional without magnetic field) the correlation functions arc known exactly. This fact opens the perspective for a construction of the AR theory which is exact relatively to all model parameters.

4. The integral intensities linear Ising model

of the AR of the

more carefully the AR lines of model characterized by the

H,,=~hw,,CS,-hw~~S/SI,, i

to b’(w,,,,, ) - F,.,, . whcrc

E;, ,) = X G ! [ s , s , i , + .Y, s ; / ] /

H’(O)].

[F(-q),

Let us consider the linear Ising Hamiltonian

and they arc equal

i

In this case the integral intensities A(w<) can bc calculated exactly for any value of the Zecman and exchange frequencies and for any tempcrature. Using the formulae (3) and (4) one ca11 calculate the Fourier components F(-(w,) of the operator F determining the spin-sound interaction. It turns out that the non-zero components F(w,) correspond to the main Hamiltonian H,, cigenvalues w< - WA.<*= hw,, + NWC ( A = 0. et2; N = 0. t- 1) .

F. ,,(, = z: WS,‘.S,,,[K,(-1. b’0 * I =k!,S,S, u ’ /

l)+K,(1.

L , j , ,K(1 , 1 -1) +Ls,y,,fi,(-L

Here

C;‘, = G

fi,(y,. (If.;=

I)].

CM ‘I’,,

I)].

and the operator\

y2) = :( 1 + 2y,s;

,)( 1 -+ 7y,s;

?)

+-I)

have the meaning of the projective operators acting in the space of the spin states Vr+,: = of the x,,, is the eigenstatc 11, x,,, - where opcr2~tor S) (Six,,,, = “1, x,,, . lH1 = t i ). and (111,) denotes the set of N quantum numbers for aII spins of the chain. The operator K,( y, . y, ) projccts these states on the subspacc where spins placed at the; ~ 1 and ; + 3 sites possess the y, /2 and yJ2 OZ projections, respectively. Let- us suppose that the AR lines arc ~cll resolved. that is. the resonance line form function s(w ~ CC); ) falls off sharply as its argument goes away from zero. Then the negative frcquencics give negligible contribution into the absorbed power (8) and the strong AR alssorption corresponds to the following four frcquencics: f1,, = (w ‘.,I1 = h/

1

f4 = I% 4I 1= IwJ k f1, .

( 10)

f1*, = /co,+ ,I = pw,, i LrJ~~ Using the formula (8) we obtain the following expressions for the resonance line integral inten-

G.O.

sities at the frequencies A(fl,,)

= g

G:(l

Berim et al.

I Theory

of acoustic

resonance

of Ising magnets

(10):

+ F, - Ed- aI ) tanh( Pf&)

3

A(~r,)=~GS(1~2~+~,+~2~2~,+a,) x tanh( phfl,,) A(C)

= $

G’,(l-

, .

F, - .s3 + LY,)tanh( /3h@)

(11) We use here the following of the Ising mode1

correlation

functions

0

2

4

6

a

IO

0

2

4

6

8

IO

12 ~Wtine

u=2($) = sinh( ,Bhw,,) x

(sinh’( @ho,,) + exp(-2phw,)))“’

&m = 4( s;s;+,,,>

= crz + (1 - (TZ)frn ,

Ym = m;q+,nq+,n+,> = CT&,+ (1 - C?)(l

ltf)Z[,f’~

(12)

,

%I = 16(Sf~,SfSS+,,SS+,,+,)

Fig. 1. The temperature dependencies of the AR line integral intensities at the frequency w = O,, for different values of the parameter p = w,,lR, for the ferro- (a) and antiferromagnetic (b) exchange coupling. The values of p are shown at each curve.

= Ff + (1 -- g?)( 1 + f)‘zif’” z,, = tanh pfiwo

CJ-- 20 , f= v+z”

in ref. These exact expressions were obtained [17] for the case of the linear Ising mode1 with spin S = $ and nearest neighbour interaction. Some asymptotically exact solutions for more general Ising models were obtained in ref. [3].

5. The AR line integral intensity the parameters of the model

dependence

on

The exact expressions (11) for the AR line integral intensities are rather complicated functions of the Zeeman frequency, exchange integral and temperature. That is why we study them numerically. Figures l-4 show the typical temperature dependencies of the integral intensities of all AR lines for the cases of ferromagnetic (w, > 0) and antiferromagnetic (we < 0) exchange integrals

and some different values of the parameter p = (w,,/w,I. All curves are given in the same relative units but in different scales. As it can be seen such dependencies form two different classes. The curves of the first class (fig. 1) possess a more or less sharp maximum at some temperature T,,, and tend to zero in the low temperature region. The curves of the second class (see for example fig. 2) rise monotonically as T goes to zero and reach their maximum value at T=O. Let us consider the corresponding integral intensities. In the ferromagnetic case all temperature dependencies of the resonance line integral intensities at the frequencies a(,, C, and a2 are of the first class. The position of the maximum goes smoothly to high temperatures with growing

Fig. 7. The same for the AR line at frequency co = fl,

p or, which is the same, with growing magnetic field. This means that the T, measurements can be removed to higher temperatures which is more attractive from the experimental point of view. It is of interest also because the exchange integral can be at some conditions found through the experimental value of T,. Namely, when the condition sinh( pfiiw,,) < exp(-phw,) holds and w, > 0, temperature T,, of the AR line at frequency (2,) determines the exchange integral I by the equality I = 4kT,X. where X satisfies the 1 + X - 3X th X = 0. equation transcendental Using the solution X- 0.886 we obtain the following simple formula for the exchange integral value: I = 3.544kT,

.

(13)

The integral intensity temperature dependence of the AR line at frequency 0, is of the second class. All integral intensities have the same maxi-

mum value at T = 0 for any value of the magnetic field. However, the decreasing rate of them is different and it increases with decreasing of the parameter p. There is a simple physical explanation for the difference in behaviour of the first and second class curves at T-+0. It can be exemplified by considering the AR lines at frequencies R,, and f1, in a ferromagnetic chain. From the microscopic point of view the reason for the presence of the AR line at frequency O,, is the reorientation of two equally directed j and j + 1 spins due to the acoustic influence when the nearest neighbour spins (j -- I and j t 2) arc directed opposite to each other (fig. S(a)). When the temperature goes to zero the linear Ising chain comes to the ground state when all spins are directed along the external magnetic field. The statistical weight of the local spin contiguration shown in fig. S(a) reduces to zero in that case and the number of spins (j and j + I in fig. 5(a)) giving the contribution to the AR absorp-

G.O.

A(nJ,

Berim

Theory

et al

of acousticresonanceof king magnets

re1.m.

0.5 .

0

2

2

0

4

6

8

IO

12

4

6

a

IO

12

Fig. 3. The same for the AR line at frequency

(4 j

j-1

.i+i

I

w

w = 0,

(b) j+z

...

t

j+i .,, . . F: : j-i

j

.i+2

Fig. 5. The local spin configurations “creating” the AR lines at frequencies w = R,, (a) and w = R, (b) for the case (I), > 0.

tion at frequency 0,, falls down. Thus the intensity A(fl,,) disappears at T+ 0. The line intensity at frequency 0, is determined by the local spin configuration when the j - 1 and j + 2 spins are directed along the magnetic field (fig. 5(b)). The statistical weight of such a configuration in-

77

creases with decreasing T and it is maximum at the ground state. Thus the intensity A(Q) has its maximum at T-+0. The explanation of the integral intensity temperature behaviour both in the ferro- and antiferromagnetic chains at any other frequencies is quite similar. The integral intensity temperature dependence in the antiferromagnetic chain is quite analogous to that described above. There exist, however, some differences due to the magnetic order changing when the magnetic field crosses the value given by the equation wg = 0,. When wg > 0, (p > 1) the antiferromagnetic chain ground state coincides with that of the ferromagnetic chain where all spins are parallel. In the opposite case when w(, < R, (p < l), the neighbouring spins are antiparallel. Due to this the temperature dependence of the AR line integral intensity in the antiferromagnetic chain can be given by the curves of both classes depending on the value of the parameter p. It can be seen in the example of the AR line at frequency 0, (fig. 2) appearing due to the local spin configuration represented in fig. 5(b). Figure 2 shows that the integral intensity temperature depedence of this line at p < 1 corresponds to the curves of the first class, and the other one at p > 1 to the curves of the second class. In the exotic case w(, = 0, all local spin configurations giving rise to different resonance frequencies are present in the ground state of the antiferromagnetic chain. That is why the integral intensities of all lines do not turn to zero when T-0 (see figs. l-4). Let us discuss briefly the possibilities of experimental verification of the obtained theoretical results. First of all we note that those Ising magnets are most convenient for the investigation by AR methods which have small exchange integral (l/k < 1 K). Only one such one-dimensional magnet is known at present to our knowledge. It is K,Fe(CN,) with Z/k - -0.5 K [5]. Such an exchange interaction gives a principal possibility to observe the whole spectrum of AR and to determine independently the value of the exchange integral using the formulae (lo), (13) with high accuracy. In addition, the temperature dependencies of the spin correlation functions

(12) can be determined from AR line integral intensities and in this way the supposition about the Ising-like form of the exchange interaction may be checked. For larger values of I the observation of AR lines at frequencies f&, O,, is very hard due to the difficulty to obtain experimentally sound frcquencies of needed values. However. the AR line at frequency n,, can be observed even in this case and the value of I can be determined from the temperature dependence of the integral intensity. Some other information can be obtained from the AR line width investigation and the line width theory we hope to develop in future.

References

Ill

R.J. Baxter. Exactly Solved Models in Statistical Mechanics (Academic Press, New York. 1982). PI V.A. Dobrovol’sky and A.R. Kessel. Teor. Mat. Fiz. Ih (1973) 135. Preprint JINI Rl7-8820, Dubna I31S.S. Lapuschkin, (1975).

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