Hydrodynamic time-correlation functions of a Heisenberg ferrofluid

Physica A 277 (2000) 389–404www.elsevier.com/locate/physa

Hydrodynamic time-correlation functionsof a Heisenberg ferro uid

I. Mrygloda;b , R. Folkb;∗, S. Dubykc , Yu. RudavskiicaInstitute for Condensed Matter Physics, National Academy of Scienses of Ukraine,

UA-290011 Lviv, UkrainebInstitute of Theoretical Physics, Linz University, A-4040 Linz, AustriacState University “Lvivska Politekhnika”, UA-29013 Lviv, Ukraine

Received 5 March 1999

Abstract

Using our previous results [Physica A 220 (1995) 325 and A 234 (1996) 129], where thegeneralized transport equations for a Heisenberg-like model of a ferro uid were obtained and thehydrodynamic collective mode spectrum has been studied, we derive the analytical expressionsfor hydrodynamic time-correlation functions of a Heisenberg model ferro uid in a homogeneousmagnetic �eld and in the limit of small wave numbers k and frequencies !. These results,being exact in the hydrodynamic limit, are presented in the form, where contributions from eachof the hydrodynamic modes are separated. It is shown that the sound excitations contribute tothe time-correlation function of ‘spin density–spin density’, and the weight of this contributiondepends on the value of an external magnetic �eld. On the other side, the spin di�usive modecontributes to the central line of the dynamic structure factor. Both of these functions maybe extracted from scattering experiments. The Landau–Placzek ratios for the dynamic structurefactor S(k; !) and the magnetic dynamic structure factor Sm(k; !) are calculated. c© 2000 ElsevierScience B.V. All rights reserved.

PACS: 05.60.+w; 51.10.+y; 75.50.Mm

Keywords: Magnetic liquids; Hydrodynamics; Liquid metals; Magnetic relaxation

One of the simplest models of a disordered continuous system exhibiting ferromag-netic behavior is a uid with an isotropic Heisenberg-like interaction between spin(internal) degrees of freedom. The theoretical study of static and dynamical propertiesfor this model is of special interest because of its simplicity and because it could be

∗ Corresponding author.E-mail address: [email protected] (I. Mryglod)

0378-4371/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(99)00496 -3

390 I. Mryglod et al. / Physica A 277 (2000) 389–404

considered as a test for any theory developed for an inhomogeneous uid by compar-ison with results of computer simulations. On the other hand, the Heisenberg modelferro uid is of interest in its own right. In the beginning of 1970s it was reported thatCo/P alloys [1] and Co/Au melts [2,3] have a tendency to form amorphous ferromag-nets and could be undercooled below the Curie temperature. While these systems mightbe more representative as examples of quenched spin uids [5], very recently it wasdemonstrated [4] that a Co80Pd20 melt can be undercooled below its Curie temperature.Hence, the �rst evidence of a ferromagnetic behavior in a liquid metal was obtainedunder conditions where the Heisenberg exchange interaction absolutely dominates overthe contribution of the magnetic dipole–dipole interaction.The di�erent phase diagrams of the continuum Heisenberg model, depending on the

ratio of the strength of the exchange interaction to the spin-independent interaction,have been established by Hemmer and Imbro [6] within mean �eld theory and morerecently in several papers [7–12] using methods of mean �eld theory, integral equa-tions and density functional theory. More quantitative results have been obtained fromMonte Carlo simulations [10,13–15] for the case when the spin-independent interactionis of the hard-sphere type. Using functional integration methods, the calculations ofthe free energy, the ‘liquid’ equation of state, and the spin-wave spectrum for liquidand amorphous magnets with quantum Heisenberg interparticle spin interactions, werecarried out by Vakarchuk et al. [16–18]. In these papers the in uence of magnetic in-teractions on the structure of the liquid was also studied, so that herein a new aspect inthe study of magnetic uids was initiated, namely, to investigate that speci�c behaviorof a ferro uid which is caused by the in uence of the ‘magnetic’ subsystem (see alsoRef. [8]). In particular, in Refs. [16–18] it has been shown that the position of criticalpoint ‘gas–liquid’ depends on the value of an external magnetic �eld. More recently,this problem has been studied in more detail in papers [19,20].Until recently, the theoretical description of magnetic liquid dynamics was based to

a great extent on phenomenological approaches (see, e.g. Refs. [21–26]). However,some of the results obtained in di�erent approaches were contradictory. For exam-ple, one may note that the expressions for sound velocity found within two maingroups of phenomenological theories, belonging either to the so-called ‘co-rotational’or ‘co-deformational’ models of a ferro uid, di�er even qualitatively [24]. Hence, itbecame unavoidable to study the hydrodynamic behavior using a systematic rigorousstatistical treatment of the problem. In this connection one may recall the statisticalmethods developed in the theory of spin relaxation. 1 However, it should be noted thatall results obtained within these theories were mainly for the dynamics of the ‘spin’subsystem, and the mutual in uence of both subsystems has not been studied in detail.In the previous papers of this series [29–31] the spectrum of hydrodynamic collec-

tive modes for an isotropic Heisenberg-like model of a ferro uid at constant external

1 The projection operator methods and cumulant expansions based on perturbation theory have been discussedby Yoon et al. [27]. The method of a master equation for the reduced spin probability density and linearresponse theory were also applied to the description of spin dynamics (see, e.g., Ref. [28]).

I. Mryglod et al. / Physica A 277 (2000) 389–404 391

magnetic �eld has been studied. In Ref. [29] we used a rigorous microscopic treatmentfor deriving the generalized transport equations and equations for the time-correlationfunctions. These equations has been then analyzed in the hydrodynamic limit [30] andexplicit expressions for the static correlation functions in relation to the well-knownthermodynamic quantities as well as expressions for the transport coe�cients have beenderived. The results were then used [30] for the calculation of the hydrodynamic col-lective mode spectrum. We emphasize that the investigation of small (k; !)-region isworthwhile because, �rstly, the expressions derived are asymptotically exact [31] inthe hydrodynamic limit, and, secondly, all the input parameters in such expressions arejust thermodynamic quantities and hydrodynamic transport coe�cients, so that theseresults have far more applications than just for Heisenberg ferro uids.The goal of this paper is to derive for a Heisenberg model ferro uid in a homoge-

neous magnetic �eld analytical expressions for all the hydrodynamic time-correlationfunctions (TCFs), constructed on the densities of conserved dynamic variables, and toanalyze these results with respect to the interplay between the ‘liquid’ and ‘magnetic’subsystems. We note that some of these functions are of particular interest both fortheory and experiment. From the theoretical point of view, having the analytical ex-pressions for hydrodynamic TCFs, one can specify more clearly the contributions fromeach of the hydrodynamic collective modes. From the other side, for interpretationof scattering experiments on uid-like systems of particles with localized spin mo-ments the explicit expressions for the ‘density–density’ TCF and the ‘spin density–spindensity’ TCF have to be known.The paper is organized as follows. In Section 2 we present the general framework

for the calculation and introduce generalized thermodynamic quantities and generalizedtransport coe�cients. The theory of the hydrodynamic TCFs is here reformulated asan eigenvalue problem for the hydrodynamic matrix. The analytical solutions for thehydrodynamic TCFs functions are presented in Section 3 together with a discussionand some concluding remarks.

1. General framework

Let us assume that the set of hydrodynamic variables Y(k) includes all the densitiesof additive conserved quantities

Y(k) = {Y 1(k); Y 1(k); : : : ; YM (k)}and M is the total number of the additive integrals of motion, so that each componentof Y(k) satis�es the motion equation

dYi(k)dt

= i LYi(k) = ikJi(k) ;

where k is the wave vector, iL denotes an Liouville operator, and Ji(k) is the micro-scopic ux corresponding to Yi(k). For the many-component dynamic variable Y(k)


one can de�ne the matrix F(k; t) of equilibrium time-correlation functions (TCFs)

F(k; t) = (�Y(k); exp {−iLt}�Y+(k)) = ‖Fij(k; t)‖with the elements Fij(k; t) given by

Fij(k; t) = (�Yi(k); exp {−iLt}�Yj(−k))

≡∫ 1

0d�Sp[�Yi(k)��0(exp {−iLt}�Yj(−k))�1−�0 ] ; (1)

where

�Yi(k) = Yi(k)− 〈Yi(k)〉; 〈: : :〉= Sp (: : :)�0and �0 is an equilibrium statistical operator. In these and the following expressions wemake use of the fact that for an isotropic system (the Heisenberg model ferro uid is infact an example of such a system [30]) the correlation functions depend only on wavenumber k. Note that the quantum TCFs de�ned by Eq. (1) are so-called ‘symmetrized’TCFs, the properties of which are completely identical to the properties of its classicalanalogy. Then, for the matrix of Laplace transforms F(k; z) of the hydrodynamic TCFsF(k; t),

F(k; z) =∫ ∞

0dt e−ztF(k; t); z = i!+ �; �→ +0 ;

one can write [32] (see also Refs. [33–35]) an explicit matrix equation in the form

{zI − i0(k) + �(k; z)}F(k; z) = F(k; 0) ; (2)

where

i0(k) = (iLY(k);�Y+(k)) (�Y(k);�Y

+(k))−1 (3)

and

�(k; z) =(I (k);

1

z + (1−P)iLI+(k)

)(�Y(k);�Y

+(k))−1 (4)

are the frequency matrix and the matrix of memory functions, respectively. The operatorP in Eq. (4) is the Mori projection operator de�ned by

P : : := (: : : ;�Y+(k)) (�Y(k);�Y

+(k))−1 (5)

and

I (k) = (1−P) iLY(k) = (1−P) ikJ(k)

is the column vector of the generalized microscopic current of Y(k). Note that for theconserved dynamic variables one has iLY(k) ∼ k and in consequence I (k) ∼ k.For a Heisenberg model ferro uid with the Hamiltonian [16,17,29]

H =N∑f=1

p2f2m

+12

∑f 6=l

V (rfl)− 12

∑f 6=l

J (rfl)SfSl − h∑f

Szf ; (6)


the set of conserved dynamic variables contains

n(k) =N∑f=1

exp(ikrf) ; (7)

p�(k) =N∑f=1

p�f exp(ikrf); (8)

�(k) = �L(k) + �S(k) =N∑f=1

p

2f

2m+12

N∑f′(f′ 6=f)

V (r� ′)

exp(ikrf)

−12

∑f 6=f′

J (r� ′)SfSf′ exp(ikrf) ; (9)

m(k) =N∑f=1

Szf exp(ikrf) ; (10)

being the densities of particles’ number, momentum and energy, and the magnetizationdensity, respectively. The operator �(k) in (9) is split into two parts in order to showthe separated contributions from the ‘liquid’ �L(k) and ‘magnetic’ �S(k) subsystems.In Ref. [30] it was shown that it is more convenient to use instead of the vari-

ables {n(k); p�(k); �(k); m(k)} the set of orthogonalized dynamic variable Y(k) ={n(k); p�(k); h(k); s(k)}, the static correlation functions of which obey the relations

(�Yi(k);�Yj(−k)) = �ij(�Yi(k);�Yi(−k)) ; (11)

where i; j = {n; p; h; s} and the new dynamic variables s(k), h(k) are de�ned by theexpressions

s(k) = (1−Pn(k)) m(k); h(k) = (1−Ps(k)−Pn(k)) �(k) : (12)

The operators Pn(k) and Ps(k) are the corresponding Mori-like projection operatorsgiven by

Pn(k) : : := (: : : ; n(−k))(n(k); n(−k))−1n(k) ;Ps(k) : : := (: : : ; s(−k))(s(k); s(−k))−1s(k) : (13)

The diagonal static correlation functions (�Yi(k);�Yi(−k)) can be related [30] to thegeneralized k-dependent thermodynamic quantities, namely,

(n(k); n(−k)) = NS(k) = N n��T;h(k)|k→0 → NkBTn�T;h ; (14)

(p�(k); p�(−k)) = N�� m� = ��NkBTm ; (15)

(s(k); s(−k)) = 1��T;n(k)|k→0 → 1

��T;n = NkBT ��T;n ; (16)

(h(k); h(−k)) = T�Cn;m(k)|k→0 → T

�Cn;m = NkBT 2cn;m ; (17)


where S(k), �T;n(k), and Cn;m(k) are the static structure factor, the generalized magneticsusceptibility (along the direction of the magnetic �eld) and the generalized speci�cheat, respectively. The function �T;h(k) is the generalized isothermal compressibility atconstant value of an external �eld h. All the other static correlation functions, whichlead to non-zero elements of the frequency matrix (3), can be expressed similarlyby generalized thermodynamic quantities (see Ref. [30]). In such a way in additionto functions (14)–(17) the new ones appear which are the generalized isothermalcompressibility �T;m(k) at constant magnetization m, the generalized thermal expansion�P;m(k) and the magnetostriction �T;P(k) coe�cients. In the limit k → 0 they arerelated to the corresponding thermodynamic derivatives [36]

�T;m(k)|k→0 = �T;m =− 1V

(@V@P

)T;m

; (18)

�P;m(k)|k→0 = �P;m =1V

(@V@T

)P;m

; (19)

�T;P(k)|k→0 = �T;P =1V

(@V@h

)P;T

: (20)

Hence, in the hydrodynamic limit the frequency matrix iH0 = i0|k→0 for the longitu-dinal variables Y(k) = {n(k); p(l)(k); h(k); s(k)} has the following structure [30]:

iH0 = ik

01m

0 0

1n�T;h

01cn;m

�P;mn�T;m

1��T;n

�T;Pn�T;h

0T�P;m��T;m

0 0

0�T;P��T;h

0 0

; (21)

where �= nm is the mass density. We see already in (21) that the thermal expansion�P;m and the magnetostriction �T;P describe the coupling of heat uctuations with theviscous and magnetic processes, respectively.The elements ’ij(k; z) of the memory functions matrix (4) are the generalized

(k; !)-dependent transport coe�cients Lij(k; i!). In more explicit form one has

’ij(k; z) = k2Lij(k; z)

V

�(�Yj(k);�Yj(−k)): (22)

In the hydrodynamic limit the generalized transport coe�cients Lij(k; z) reduce [30] tothe well-known kinetic coe�cients Lij, which could be presented in the form of theGreen–Kubo formula

Lij =�V

∫ ∞

0(�fi ; e

−iLt�fj) dt ; (23)


where fi are the non-orthogonal parts of the generalized uxes I (k) in the small wavenumbers limit,

I i(k)|k→0 ' ik fi = (1−P) ikJ(i)(0) :

The expressions for the uxes J(i)(0) were obtained in Ref. [29]. More explicit the

meaning of the non-zero coe�cients Lij is the following: Lpp = �l = (43�+ �) gives alongitudinal viscosity, where � and � are the bulk and shear viscosities, respectively;Lhh = T� describes the heat di�usion with the thermal conductivity coe�cient �; Lssis the spin di�usion coe�cient; and the new kinetic coe�cients Lsh and Lhs describethe dynamic interplay between liquid and magnetic subsystems (namely, between theheat and magnetic properties) and are known in the literature as the thermomagneticdi�usion coe�cients (see, e.g., Refs. [37,38]).The symmetry properties of Lij immediately follow from the properties of the uxes

fl and were discussed in the previous paper [30]. In particular, it was shown thatLsh(h) = −Lsh(−h) and Lhs(h) = Lsh(h), so that these cross coe�cients are equal tozero above the Curie point when there is no external magnetic �eld (h= 0).The general structure of the matrix of memory functions (4) in the hydrodynamic

limit, ’(k; z)|!;k→0 = ’H (k), has been discussed in Ref. [30] and can be written as

follows:

’H (k) = k2

0 0 0

0�l�

0 0

0 0�

ncn;m

Lshn ��T;n

0 0Lsh

ncn;mTLssn ��T;n

: (24)

We see that the non-zero elements of ’H (k) are proportional to k2, so that one has’H (k) = k2‖ ij‖. Hence, in the hydrodynamic limit (denoted by the superscript “H”)the matrix equation (2) for the Laplace transforms F

H(k; z) of hydrodynamic time

correlation functions FH (k; t) has the form

{zI + TH (k)}FH (k; z) = FH (k) ; (25)

where

TH (k) =−iH0 (k) + ’H (k) (26)

is the so-called hydrodynamic matrix, and the matrices iH0 (k) and ’H (k) are given

by (21) and (24), respectively. One has to mention that the matrix equation (25) isasymptotically exact in the hydrodynamic limit, and the hydrodynamic matrix TH (k)depends only on the wavenumber k, not on the frequency !.


The solution of the matrix equation (25) can simply be written in the analyticalform via the eigenvalues z�(k) and the eigenvectors X � = ‖X i;�‖ of the hydrodynamicmatrix TH (k),∑

j

THij (k)X j;� = z�(k)X i;� ; (27)

where i; j = n; p; s; h and the index � labels the di�erent eigenvalues. For elementsFHij (k; z) we get

FHij (k; z) =

∑�

G ij� (k)z + z�(k)

(28)

or

F Hij (k; t) =∑�

G ij� (k) exp{−z�(k)t} ; (29)

where the weight coe�cients G ij� (k) are de�ned by the general expression

G ij� (k) =∑l

X i;�X−1�; lFlj(k; 0) (30)

and the matrix X−1is the inverse of X = ‖X �‖. For the orthogonal dynamic variables

(see (11)) expression (30) has the most simple form

G ij� (k) = X i;�X−1�; jFjj(k; 0) = Gij� (k)Fjj(k) ; (31)

which will be used in the next presentation.Thus for the case of a Heisenberg magnetic uid the hydrodynamic time-correlation

functions FH (k; t) are presented as a sum of four exponential terms, and each term isconnected with the corresponding hydrodynamic collective mode z�(k). The amplitudeG ij� (k) describes the partial contribution of the mode z�(k) in FHij (k; t). In order toderive the explicit expressions for the hydrodynamic TCFs we have to solve Eq. (27)for eigenvalues and eigenvectors of the hydrodynamic matrix TH (k). One can use forthis purpose the matrix perturbation theory considering the wave number k as a smallparameter.

2. Results and discussion

The eigenvalues of the matrix TH (k) have been already calculated in our previouspaper [30]. We reproduce here the main results because they will be used later. Thefollowing solutions for the eigenvalues of the hydrodynamic matrix TH (k), each ofthem corresponds to a certain hydrodynamic mode, have been found:(a) Two complex-conjugated sound modes with the eigenvalues:

z± =±ivsk + Dsk2 ; (32)


where the sound velocity vs is given by the expression

v2s = m��T;m

=(@P@�

)S;m

(33)

with m = CP;m=Cn;m, and the damping coe�cient Ds is

Ds =12�l�+12( m − 1)�ncP;m

+12(1− �T )Lssn m ��T;n

+�P;m�T;PLshn2�T;h ��T;ncP;m

; (34)

where �T = �T;m=�T;h is the ratio of the isothermal compressibilities at constant mag-netization m and magnetic �eld h. Of course, �T → 1 when h → 0 above Curiepoint;(b) A hydrodynamic heat mode with the eigenvalue:

zh = Dhk2; Dh = b+√b2 − c ; (35)

where

b=12

�ncP;m

+12

Lssn m ��T;n

( m + �T − 1)− �P;m�T;PLshn2�T;h ��T;ncP;m

; (36)

c =�T

n2TcP;m ��T;h(�TLss − L2sh) : (37)

(c) A hydrodynamic spin diffusion mode with the eigenvalue:

zm = Dmk2; Dm = b−√b2 − c (38)

with the coe�cients b and c de�ned by Eqs. (36) and (37).It is evident from (32)–(38) that when the external �eld is zero the results known

for simple uids [33,34] and solid magnets [37,38], respectively, are reproduced:

Ds =12�0l�+12( 0 − 1)�0ncP;0

; Dh =�0

ncP;0; Dm =

L0ssn ��0T;n

; (39)

where the index ‘0’ means that all quantities should be calculated for h= 0.We present below the results found for the hydrodynamic TCFs not going into the

details of calculations. A few general remarks should be made herein to be appliedfor any �xed wave number k. The �rst remark is the following: It can be shown[39,40] that for the hydrodynamic set of dynamic variables, in Markovian approxima-tion for the memory functions (which becomes exact in the hydrodynamic limit), thehydrodynamic TCFs can be obtained in the form for which the zero-order moments infrequency and time spaces are reproduced explicitly. For the zero-order frequency mo-ment this statement follows immediately from Eqs. (29) and (31). For the zero-ordertime moment it is not so obvious, but can be easily proved (see Refs. [39,40]), whenthe k-dependent memory functions in Markovian approximation are taken in explicitform. The exception is the ‘density–density’ TCF for which the frequency momentsare explicitly reproduced up to the second-order. 2 The second remark is that for small

2 This is because the �rst-order time derivative of n(k)∼pl(k) is indeed already included into the hydrody-namic set of variables.


wave numbers, when the eigenvalues are given by expressions (32)–(38), we have torestrict our consideration to the two lower orders with respect to the small parameter k(see (31)) for the weight coe�cients Gij� (k). Combining both remarks it can be shownthat the second-order terms to the weight coe�cients (proportional to wave numberk) are only important for contributions due to the sound modes. Keeping this in mindlet us consider the results for the diagonal elements of the matrix of TCFs. For the‘density–density’ TCF one gets

FHnn(k; t)=Fnn(k) =∑�

Gnn� (k) exp{−z�(k)t} ; (40)

where �= {+;−; h; m} and

Gnn+ =�T2 m

{1− ik bnn

vs

}; Gnn− =

�T2 m

{1 + ik

bnnvs

};

Gnnh =�T

m(Dm − Dh)(�l�

− 2Ds −(1− m

�T

)Dm

);

Gnnm =�T

m(Dh − Dm)(�l�

− 2Ds −(1− m

�T

)Dh

): (41)

The parameter bnn has been calculated by two di�erent methods, namely, using thematrix perturbation theory and the frequency moments approach. Both methods givethe same result

bnn =�l�

− 3Ds ;

which coincides formally with the result known for simple uids (see Refs. [33,34]).The di�erence between both cases lies only in the microscopic expressions for �l and3Ds.The weight coe�cients of the other diagonal elements of the matrix of TCFs FH (k; t)

are listed below:(i) For the ‘current–current’ TCF FHpp(k; t) we get

Gpp± =

12

{1∓ ik bpp

vs

}; bpp = bnn + 2Ds; G

pph = Gppm = 0 : (42)

(ii) For the TCF FHhh(k; t), describing the temperature uctuations, one has

Ghh± =( m − 1)2 m

{1∓ ik bhh

vs

}; bhh = bnn + 2

�ncn;m

+ 2�T�T;P LshnT�P;m ��T;n

;

Ghhh =1

m(Dm − Dh)(−Dh + �TLssn ��T;n

);

Ghhm =1

m(Dm − Dh)(Dm − �TLss

n ��T;n

): (43)


(iii) The weight coe�cients of the ‘spin density–spin density’ TCF FHmm(k; t) aregiven by

Gmm± =12(1− �T ) m

{1∓ ik bmm

vs

}; bmm = bnn + 2

Lssn ��T;n

+ 2Lsh�P;m

�P;T �T ncn;m;

Gmmh =1

m(Dm − Dh)(��Tncn;m

− (�T + m − 1)Dh);

Gmmm =1

m(Dh − Dm)(��Tncn;m

− (�T + m − 1)Dm): (44)

The non-diagonal TCFs, describing the dynamic coupling between the di�erent orthog-onalized hydrodynamic variables, has also been calculated (see also Ref. [41]) withinthe same approximation. In order to save room we report here only some resultsand concentrate the following discussion on the physical meaning of the expressionsobtained.For the weight coe�cients of the function FHnh(k; t) we get

Gnh± =�P;m2cP;m

{1∓ ik bnh

vs

}; bnh = bnn +

�ncn;m

+�T�T;P LshnT�P;m ��T;n

;

Gnhh =�P;m

cP;m(Dm − Dh)(Dh − Lss

n ��T;n+�T�T;P LshnT�P;m ��T;n

);

Gnhm =�P;m

cP;m(Dh − Dm)(Dm − Lss

n ��T;n+�T�T;P LshnT�P;m ��T;n

): (45)

It is seen in this expression that all the weight coe�cients are proportional to thethermal expansion �P;m as it should be. Besides that the contributions from the soundmode compensate the contributions from the heat and spin modes, so that we have�nally FHnh(k; 0)=Fhh(k)=

∑� G

nh� (k)=0. Similar conclusions can be made for F

Hns (k; t),

where the weight coe�cients are given by

Gns± =�T�T;P2 m ��T;n

{1∓ ik bns

vs

}; bns = bnn +

Lssn ��T;n

+�P;mLsh

�Tncn;m�T;P;

Gnsh =�T�T;P

m ��T;n(Dm − Dh)(Dh − �

ncn;m+

�P;mLsh�Tncn;m�T;P

);

Gnsm =�T�T;P

m ��T;n(Dh − Dm)(Dm − �

ncn;m+

�P;mLsh�Tncn;m�T;P

): (46)

In this case all the weight coe�cients are proportional to the magnetostriction coe�-cient �T;P and we have once again a cancellation of the di�erent contributions, so that∑

� Gnh� (k)=0. For the functions F

Hhs (k; t) and F

Hsh (k; t) we found that the weight coef-

�cients are proportional to the product of the thermal expansion and magnetostrictioncoe�cients.


Finally, one has for the TCF FHpn(k; t)

Gpn± =∓�Tmvs

2 m

{1∓ ik bpn

vs

}; bpn = bnn + Ds ;

Gpnh = ikmDhG

nnh ; Gpnm = ikmDmG

nnm : (47)

It is easy to check that the zero-order frequency moment for the function FHpn(k; t) isalso satis�ed. Taking into account the remarks presented above, it is evident that theexpressions for the weight coe�cients of the TCFs FHpn(k; t) and F

Hpp(k; t) may be also

derived using the explicit relations

d2

dt2Fnn(k; t) =− k

2

m2Fpp(k; t);

ddtFnn(k; t) =

ikmFpn(k; t) ;

when we consider terms up to the second order with respect to small k.The hydrodynamic TCFs Fnn(k; t) and Fmm(k; t), describing the ‘particle density–

particle density’ and ‘spin density–spin density’ correlations, are of special interest tothe theory, because they can be recovered from scattering experiments. Namely, theirFourier transforms, being measured experimentally, are the dynamic structure factor

S(k; !) =1NFnn(k; !) =

1NRe Fnn(k; i!+ �)

and the magnetic dynamic structure factor

Sm(k; !) =1NFmm(k; !) =

1NRe Fmm(k; i!+ �) :

Using Eqs. (40), (41) and (44) one has for these functions

S(k; !)S(k)

=�T2 m

∑�=+;−

Dsk2 − �k(!=vs + �k)bnn(!+ �kvs)2 + (Dsk2)2

+∑�=h;m

Gnn�D�k2

!2 + (D�k2)2; (48)

Sm(k; !)Sm(k)

=(1− �T )2 m

∑�=+;−

Dsk2 − �k(!=vs + �k)bmm(!+ �kvs)2 + (Dsk2)2

+∑�=h;m

Gmm�D�k2

!2 + (D�k2)2: (49)

We see in (48) and (49) that all the hydrodynamic modes contribute to the dynamicstructure factors when an external homogeneous magnetic �eld h is applied. The sameresult could be found for h = 0, if the ferromagnetic phase is considered. For h = 0and above the Curie point the coupling between subsystems disappears and one hasLsh = �P;T = 0 and �T = 1, so that all the expressions given above can be signi�cantlysimpli�ed. In particular, in this case we �nd that the dynamic structure factor S(k; !)has formally the same structure as for a simple uid [33,34], and the magnetic dynamicstructure factor in the hydrodynamic limit is completely determined by the spin di�usionmode [37,38]. Such behavior for the dynamic structure factor S(k; !) is qualitatively


Fig. 1. Schematic representation of the dynamic structure factor S(k; !) at three di�erent values of external�eld h. Contributions increasing with h from the spin di�usion mode at h 6= 0 are shown by thin lines inthe left bottom corner.

demonstrated in Fig. 1. We see in this �gure that the height of side Brillouin peak(of the central peak) decreases (increases) when h becomes larger, and an additionalcontribution to the central line due to the spin di�usion mode appears for h 6= 0.Of special interest is the magnetic dynamic structure factor Sm(k; !), in which due

to the magnetostriction e�ect, side Brillouin peaks appear (see Fig. 2), so that theform of Sm(k; !) changes qualitatively. Considering the case of a small magnetic �eldit can be shown that the weight coe�cients of sound modes in (49) are proportionalto h2. Although this e�ect is small, it can be in principle studied experimentally in uids consisting of particles with a localized spin moment. It is seen in Fig. 2 that incomparison with the case of S(k; !) we have for the magnetic dynamic structure factorSm(k; !) the inverse situation concerning its behavior for di�erent values of magnetic�eld. The height of central line decreases and the height of Brillouin peak increaseswhen h becomes larger.The Landau–Placzek ratios [33] of the integrated intensity of the Rayleigh central

peak to those of the Brillouin side peaks both for S(k; !) and Sm(k; !) can be easilyfound from expressions (48) and (49). We get the following results:

IR2IB

= m − �T�T

;ImR2ImB

= m − 1 + �T�T − 1

for S(k; !) and Sm(k; !), respectively. As it was mentioned already the central peak inboth cases has two separate Lorentzian components from the heat and spin-di�usionmodes for h 6= 0. This is quite similar to the case of a binary mixture [42–45] wherethe heat and concentration modes both form the central line. These results can be used


Fig. 2. The magnetic dynamic structure factor Sm(k; !) at three di�erent values of external �eld h: schematicrepresentation. Contributions increasing with h from the heat di�usion mode at h 6= 0 are shown by thinlines in the left bottom corner.

for an interpretation of scattering experiments. It is also worth to mention that fromthe thermodynamic relation

�T =

{1 +

�2T;Pn ��T;n�T;h

}−1;

the magnetostriction coe�cient �T;P can be extracted not only from thermodynamicexperiments, but also from the scattering experiments. In particular one may pro-ceed as follows: the thermodynamic quantities �T;n and �T;h can be obtained from thelong-wavelength limit of the zero-order frequency moments of S(k; !) and Sm(k; !) (or,in other words, from the static structure factors S(k) and Sm(k)). The ratio �T may befound from the position of Brillouin peak and the Landau–Placzek ratio IR=2IB.In conclusion, we note that the results obtained in this paper are derived for the

hydrodynamic region, where all the input parameters of the theory are well-de�nedthermodynamic quantities and hydrodynamic transport coe�cients. Because of this,they can be used not only for a Heisenberg model ferro uid with Hamiltonian (6),but for the description of other systems as well. Two main demands only are im-portant: (i) spin interaction has to be isotropic in space of translational motions andto depend only on the distance between particles; (ii) the total spin density shouldbe a conserved quantity. In particular, expressions (48) and (49) are directly appli-cable to the theory of polar liquids with isotropic interactions. The hydrodynamicsof polar liquids and ferro uids becomes formally identical [24] if one notes the cor-respondence between ‘electric �eld’ and ‘magnetic �eld’ and between ‘polarization’


and ‘magnetization’. However, the presence of long-range dipolar forces in real polarliquids leads to anisotropic e�ects which are not present for the short-range interactionconsidered in this paper.

Acknowledgements

This study is supported in part by the Fonds z�ur F�orderung der wissenschaftlichenForschung under Project P 12422 TPH.

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