Physica A, 2000, V. 277, No 3-4, p.389-404.

Hydrodynamic time correlation functions of
a Heisenberg model ferrofluid

I.Mryglod^1,2, R.Folk², S.Dubyk³, Yu.Rudavskii³

# Introduction
# General framework
# Results and discussion
# Bibliography

Short running title:
Hydrodynamic time correlation functions of a ferrofluid
PACS number: 05.60.+w, 51.10.+y, 75.50.Mm.

August 13, 2001

Abstract:

Using our previous results [Physica A 220 (1995) 325; Physica A 234 (1996) 129], where the generalized transport equations for a Heisenberg-like model of a ferrofluid were obtained and the hydrodynamic collective mode spectrum has been studied, we derive the analytical expressions for hydrodynamic time correlation functions of a Heisenberg mode ferrofluid in the limit of small wavenumbers k and frequencies $\omega$ . These results, being exact in the hydrodynamic limit, are presented in the form, where contributions from each of hydrodynamic modes are separated. It is shown that the sound excitations contribute to the time correlation function of `spin density-spin density', and the weight of this contribution depends on the value of an external magnetic field. On the other side, the spin diffusive mode contributes to the central line of the dynamic structure factor. Both of these functions may be extracted from scattering experiments. The Landau-Placzek ratios for the dynamic structure factor $S(k,\omega )$ and the magnetic dynamic structure factor $S_{m}(k,\omega)$ are calculated. Keywords: Magnetic liquids, hydrodynamics, liquid metals, magnetic relaxation.

Introduction

One of the simplest models of a disordered continuous system exhibiting ferromagnetic behavior is a fluid with an isotropic Heisenberg-like interaction between spin (internal) degrees of freedom. The theoretical study of static and dynamical properties for this model is of special interest because of its simplicity and because it could be considered as a test for any theory developed for an inhomogeneous fluid by comparison with results of computer simulations. On the other hand, the Heisenberg model ferrofluid is of interest in its own right. In the beginning of 70th it was reported that Co/P alloys [1] and Co/Au melts [2,3] have a tendency to form amorphous ferromagnets and could be undercooled below the Curie temperature. While these systems might be more representative as examples of quenched spin fluids [5], very recently it was demonstrated [4] that a Co₈₀Pd₂₀ melt can be undercooled below its Curie temperature. Hence, the first evidence of a ferromagnetic behavior in a liquid metal was obtained under conditions where the Heisenberg exchange interaction absolutely dominates over the contribution of the magnetic dipole-dipole interaction. The different 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 field theory and more recently in several papers [7,8,9,10,11,12] using methods of mean field theory, integral equations and density functional theory. More quantitative results have been obtained from Monte Carlo simulations [10,13,14,15] for the case when the spin-independent interaction is of the hard-sphere type. Using functional integration methods, the calculations of the free energy, the `liquid' equation of state, and the spin-wave spectrum for liquid and amorphous magnets with quantum Heisenberg interparticle spin interactions, were carried out by Vakarchuk, Rudavskii and Ponedilok [16,17,18]. In these papers the influence of magnetic interactions on the structure of liquid was also studied, so that herein a new aspect in the study of magnetic fluids was initiated, namely, to investigate that specific behavior of a ferrofluid which is caused by the influence of the `magnetic' subsystem (see also [8]). In particular, in Refs. [16,17,18] it has been shown that the position of critical point `gas-liquid' depends on the value of an external magnetic field. 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. [21,22,23,24,25,26]). However, some of the results obtained in different approaches were contradictory. For example, one may note that the expressions for sound velocity found within two main groups of phenomenological theories, belonging either to so-called `co-rotational' or `co-deformational' models of a ferrofluid, differ even qualitatively [24]. Hence, it became unavoidable to study the hydrodynamic behavior using a systematic rigorous statistical treatment of the problem. In this connection one may recall the statistical methods developed in the theory of spin-relaxation^*.

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

However, it should be noted that all results obtained within these theories were mainly for the dynamics of the `spin' subsystem, and the mutual influence of both subsystems has not been studied in detail. In the previous papers of this series [29,30,31] the spectrum of hydrodynamic collective modes for an isotropic Heisenberg-like model of a ferrofluid at constant external magnetic field has been studied. In Ref. [29] we used a rigorous microscopic treatment for deriving the generalized transport equations and equations for the time correlation functions. These equations has been then analysed in the hydrodynamic limit [30] and explicit expressions for the static correlation functions in relation to the well-known thermodynamic quantities as well as expressions for the transport coefficients have been derived. The results were then used [30] for the calculation of the hydrodynamic collective mode spectrum. We emphasize that the investigation of small ( $k,\omega$ )-region is worthwhile because, firstly, the expressions derived are asymptotically exact [31] in the hydrodynamic limit, and, secondly, all the input parameters in such expressions are just thermodynamic quantities and hydrodynamic transport coefficients, so that these results have a wider range of application than for Heisenberg ferrofluids only. The goal of this paper is to derive analytical expressions for all the hydrodynamic time correlation functions (TCFs), constructed on the conserved dynamic variables, and to analyze the results with respect to the interplay between the `liquid' and `magnetic' subsystems. We note that some of these functions are of particular interest both for theory and experiment. From the theoretical point of view, having the analytical expressions for hydrodynamic TCFs, one can specify more clearly the contributions from each of the hydrodynamic collective modes. for interpretations of scattering experiments on fluid-like systems of particles with localized spin moments the explicit expressions for the `density-density' TCF and the `spin density-spin density' 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 generalized transport coefficients. The theory of the hydrodynamic TCFs is here reformulated as an eigenvalue problem for the hydrodynamic matrix. The analytical solutions for the hydrodynamic TCFs functions are presented in Section 3 together with a discussion and some concluding remarks.

General framework

Let us assume that the set of hydrodynamic variables $\hat{\mbox{\boldmath$Y$}}(\mbox{\boldmath$k$})$ includes all the densities of additive conserved quantities,

$\begin{displaymath} \hat{\mbox{\boldmath $Y$}}(\mbox{\boldmath $k$})= \{ \hat{\... ...dmath $Y$}}_{\scriptscriptstyle M} (\mbox{\boldmath $k$}) \}, \end{displaymath}$

and M is the total number of the additive integrals of motion, so that each component of $\hat{\mbox{\boldmath$Y$}}(\mbox{\boldmath$k$})$ satisfies the motion equation

$\begin{displaymath} \frac{ d \hat{\mbox{\boldmath $Y$}}_i(\mbox{\boldmath $k$})... ...th $k$}} \hat{\mbox{\boldmath $J$}}_i(\mbox{\boldmath $k$}), \end{displaymath}$

where $\mbox{\boldmath$k$}$ is the wavevector, $i\hat{L}$ denotes an Liouville operator, and $\hat{\mbox{\boldmath$J$}}_i(\mbox{\boldmath$k$})$ is the microscopic flux corresponding to $\hat{\mbox{\boldmath$Y$}}_i(\mbox{\boldmath$k$})$ . For the many-component dynamic variable $\hat{\mbox{\boldmath$Y$}}(\mbox{\boldmath$k$})$ one can define the matrix ${\mbox{\boldmath$F$}}(k,t)$ of equilibrium time correlation functions (TCFs),

$\begin{displaymath} {\mbox{\boldmath $F$}}(k,t)= \Big(\Delta\hat{\mbox{\boldmat... ...th $k$}) \Big) = \Vert{\mbox{\boldmath $F$}}_{ij}(k,t) \Vert, \end{displaymath}$

with the elements ${\mbox{\boldmath$F$}}_{ij}(k,t)$ given by

$\displaystyle {\mbox{\boldmath$F$}}_{ij}(k,t)= \Big(\Delta\hat{\mbox{\boldmath$... ... \ \Delta\hat{\mbox{\boldmath$Y$}}_j(-\mbox{\boldmath$k$}) \Big) \qquad \qquad$

(1)

$\displaystyle \equiv \int \limits_{0}^{1} d\tau \ {\rm Sp} \left[ \Delta\hat{\m... ...hat{\mbox{\boldmath$Y$}}_j(-\mbox{\boldmath$k$}) \Big) \rho_0^{1-\tau} \right],$

where

$\begin{displaymath} \Delta\hat{\mbox{\boldmath $Y$}}_i(\mbox{\boldmath $k$})= \... ... \qquad \langle \ldots \rangle = {\rm Sp} \ (\ldots) \rho_0, \end{displaymath}$

and $\rho_0$ is an equilibrium statistical operator. In these and following expressions we make use of the fact that for an isotropic system (the Heisenberg model ferrofluid is in fact an example of such a system [30]) the correlation functions depend only on wavenumber k. Note that the quantum TCFs defined by Eq. (1) are so-called `symmetrized' TCFs, the properties of which are completely identical to the properties of its classical analogy. Then, for the matrix of Laplace transforms $\tilde{\mbox{\boldmath$F$}}(k,z)$ of the hydrodynamic TCFs ${\mbox{\boldmath$F$}}(k,t)$ ,

$\begin{displaymath} \tilde{\mbox{\boldmath $F$}}(k,z)= \int \limits_{0}^{\infty... ...}}(k,t), \qquad z= i\omega+\epsilon, \qquad \epsilon \to +0, \end{displaymath}$

one can write [32] (see also [33,34,35]) an explicit matrix equation in the form

$\begin{displaymath} \{z {\mbox{\boldmath$I$}} - i{\mbox{\boldmath$\Omega$}}_0(... ...tilde{\mbox{\boldmath$F$}}(k,z)= {\mbox{\boldmath$F$}}(k,0), \end{displaymath}$

(2)

where

$\begin{displaymath} i{\mbox{\boldmath$\Omega$}}_0 (k)= \left( i\hat L \hat{\mb... ...hat{\mbox{\boldmath$Y$}}^+(\mbox{\boldmath$k$}) \right)^{-1} \end{displaymath}$

(3)

and

$\begin{displaymath} {\mbox{\boldmath$\phi$}}(k,z)= \left(\hat{\mbox{\boldmath$... ...hat{\mbox{\boldmath$Y$}}^+(\mbox{\boldmath$k$}) \right)^{-1} \end{displaymath}$

(4)

are the frequency matrix and the matrix of memory functions, respectively. The operator ${\cal P}$ in Eq. (4) is the Mori projection operator defined by

$\begin{displaymath} {\cal P} \ldots = \left( \ldots, \Delta \hat{\mbox{\boldma... ...at{\mbox{\boldmath$Y$}}^+(\mbox{\boldmath$k$}) \right)^{-1}, \end{displaymath}$

(5)

and

$\begin{displaymath} \hat{\mbox{\boldmath $I$}}(\mbox{\boldmath $k$})=(1-{\cal P... ...dmath $k$}} \hat{\mbox{\boldmath $J$}} (\mbox{\boldmath $k$}) \end{displaymath}$

is the column-vector of the generalized microscopic current of $\hat{\mbox{\boldmath$Y$}}(\mbox{\boldmath$k$})$ . Note that for the conserved dynamic variables one has $i\hat L \hat{\mbox{\boldmath$Y$}}(\mbox{\boldmath$k$}) \sim k$ and in consequence $\hat{\mbox{\boldmath$I$}}(\mbox{\boldmath$k$}) \sim k$ . For a Heisenberg model ferrofluid with the Hamiltonian [16,17,29]

$\begin{displaymath} \hat{H} = \sum \limits _{f=1}^{N} \frac {{\mbox{\boldmath$... ...\mbox{\boldmath$S$}}_f{\mbox{\boldmath$S$}}_l- h \sum_f S_f^z \end{displaymath}$

(6)

the set of conserved dynamic variables contains

$\begin{displaymath} \hat{n}({\mbox{\boldmath$k$}}) =\sum \limits _{f=1}^N \exp(i{\mbox{\boldmath$k$}}{\mbox{\boldmath$r$}}_f), \end{displaymath}$

(7)

$\begin{displaymath} \hat{p}^{\alpha}({\mbox{\boldmath$k$}})=\sum \limits _{f=1... ...alpha} \exp (i{\mbox{\boldmath$k$}}{\mbox{\boldmath$r$}}_f), \end{displaymath}$

(8)

$\displaystyle \hat{\varepsilon}({\mbox{\boldmath$k$}})= \hat{\varepsilon}_{\scr... ...+ \hat{\varepsilon}_{\scriptscriptstyle S}({\mbox{\boldmath$k$}})= \hspace{8em}$

$\displaystyle = \sum \limits _{f=1}^N \{ \frac {p^2_f}{2m} +\frac {1}{2} \sum \... ...\mbox{\boldmath$S$}}_{f'} \exp (i{\mbox{\boldmath$k$}}{\mbox{\boldmath$r$}}_f),$

(9)

$\begin{displaymath} \hat{m}({\mbox{\boldmath$k$}})=\sum \limits _{f=1}^N \ S_f^z \exp (i{\mbox{\boldmath$k$}}{\mbox{\boldmath$r$}}_f), \end{displaymath}$

(10)

being the densities of particles' number, momentum and energy, and the magnetization density, respectively. The operator $\hat{\varepsilon}({\mbox{\boldmath$k$}})$ in (9) is split into two parts in order to show the separated contributions from the `liquid' $\hat{\varepsilon}_{\scriptscriptstyle L}({\mbox{\boldmath$k$}})$ and `magnetic' $\hat{\varepsilon}_{\scriptscriptstyle L}({\mbox{\boldmath$k$}})$ subsystems. In Ref. [30] it was shown that it is more convenient to use instead of the variables $\{ \hat{n}({\mbox{\boldmath$k$}}), \hat{p}^{\alpha}({\mbox{\boldmath$k$}}), \hat{\varepsilon}({\mbox{\boldmath$k$}}), \hat{m}({\mbox{\boldmath$k$}}) \}$ the set of orthogonalized dynamic variable $\hat{\mbox{\boldmath$Y$}}(\mbox{\boldmath$k$}) = \{ \hat{n}({\mbox{\boldmath$k... ...dmath$k$}}), \hat{h}({\mbox{\boldmath$k$}}), \hat{s}({\mbox{\boldmath$k$}}) \}$ , the static correlation functions of which obey the relations

$\begin{displaymath} \left( \Delta \hat{\mbox{\boldmath$Y$}}_{i} ({\mbox{\boldm... ...hat{\mbox{\boldmath$Y$}}_{i}(-{\mbox{\boldmath$k$}}) \right), \end{displaymath}$

(11)

where i,j= { n, p, h, s} and the new dynamic variables $\hat{s}({\mbox{\boldmath$k$}})$ , $\hat{h}({\mbox{\boldmath$k$}})$ are defined by the expressions

$\begin{displaymath} \hat{s}({\mbox{\boldmath$k$}})= \Big(1 - {\cal P}_n({\mbox... ...oldmath$k$}})\Big) \hat{\varepsilon}({\mbox{\boldmath$k$}}). \end{displaymath}$

(12)

The operators ${\cal P}_n({\mbox{\boldmath$k$}})$ and ${\cal P}_s({\mbox{\boldmath$k$}})$ are corresponding Mori-like projection operators given by

$\displaystyle {\cal P}_n({\mbox{\boldmath$k$}}) \ldots = \Big( \ldots, \hat{n}(... ...}), \hat{n}(-{\mbox{\boldmath$k$}})\Big)^{-1} \ \hat{n}({\mbox{\boldmath$k$}}),$

$\displaystyle {\cal P}_s({\mbox{\boldmath$k$}}) \ldots = \Big( \ldots, \hat{s}(... ...}), \hat{s}(-{\mbox{\boldmath$k$}})\Big)^{-1} \ \hat{s}({\mbox{\boldmath$k$}}).$

(13)

The diagonal static correlation functions $\left( \Delta \hat{\mbox{\boldmath$Y$}}_{i} ({\mbox{\boldmath$k$}}), \Delta \hat{\mbox{\boldmath$Y$}}_{i}(-{\mbox{\boldmath$k$}}) \right)$ can be related [30] to the generalized k-dependent thermodynamic quantities, namely,

$\begin{displaymath} (\hat{n}({\mbox{\boldmath$k$}}), \hat{n}(-{\mbox{\boldmath... ...\scriptscriptstyle B} T n \kappa_{{\scriptscriptstyle T},h}, \end{displaymath}$

(14)

$\begin{displaymath} (\hat{p}^{\alpha}({\mbox{\boldmath$k$}}), \hat{p}^{\beta}(... ... \delta_{\alpha \beta} N k_{\scriptscriptstyle B} T {\rm m}, \end{displaymath}$

(15)

$\begin{displaymath} (\hat{s}({\mbox{\boldmath$k$}}), \hat{s}(-{\mbox{\boldmath... ...scriptscriptstyle B} T \bar{\chi}_{{\scriptscriptstyle T},n}, \end{displaymath}$

(16)

$\begin{displaymath} (\hat{h}({\mbox{\boldmath$k$}}), \hat{h}(-{\mbox{\boldmath... ...ac{T}{\beta} C_{n,m}= N k_{\scriptscriptstyle B} T^2 c_{n,m}, \end{displaymath}$

(17)

where S(k), $\chi_{{\scriptscriptstyle T},n}(k)$ , and C_n,m(k) are the static structure factor, the generalized magnetic susceptibility (along the direction of the magnetic field) and the generalized specific heat, respectively. The function $\kappa_{{\scriptscriptstyle T},h}(k)$ is the generalized isothermal compressibility at constant value of an external field h. All the other static correlation functions, which lead to non-zero elements of the frequency matrix (3), can be expressed similarly by generalized thermodynamic quantities (see [30]). In such a way in addition to the functions (14)-(17) the new ones appear which are the generalized isothermal compressibility $\kappa_{{\scriptscriptstyle T},m}(k)$ at constant magnetization m, the generalized thermal expansion $\beta_{{\scriptscriptstyle P},m}(k)$ and the magnetostriction $\pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}(k)$ coefficients. In the limit $k \to 0$ they are related to the corresponding thermodynamic derivatives [36]

$\begin{displaymath} \kappa_{{\scriptscriptstyle T},m}(k)\vert _{k\to 0}= \kapp... ... \frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_{T,m}, \end{displaymath}$

(18)

$\begin{displaymath} \beta_{{\scriptscriptstyle P},m}(k)\vert _{k\to 0}= \beta_... ...frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_{P,m}, \end{displaymath}$

(19)

$\begin{displaymath} \pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}(k)\ver... ... \frac{1}{V}\left(\frac{\partial V}{\partial h}\right)_{P,T}. \end{displaymath}$

(20)

Hence, in the hydrodynamic limit the frequency matrix $i\Omega_0^{\scriptscriptstyle H}= i\Omega_0\vert _{k \to 0}$ for the longitudinal variables $\hat{\mbox{\boldmath$Y$}}(\mbox{\boldmath$k$}) = \{ \hat{n}({\mbox{\boldmath$k... ...dmath$k$}}), \hat{h}({\mbox{\boldmath$k$}}), \hat{s}({\mbox{\boldmath$k$}}) \}$ has the following structure [30]

$\begin{displaymath} i{\mbox{\boldmath$\Omega$}}_0^{\scriptscriptstyle H} = ik ... ...a_{{\scriptscriptstyle T},h}}} & 0 & 0 \end{array} \right), \end{displaymath}$

(21)

where $\rho =n {\rm m}$ is the mass density. We see already in (21) that the thermal expansion $\beta_{{\scriptscriptstyle P},m}$ and the magnetostriction $\pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}$ describe the coupling of heat fluctuations with the viscous and magnetic processes, respectively. The elements $\tilde{\mbox{\boldmath$\varphi$}}_{ij}(k,z)$ of the memory functions matrix (4) are the generalized ( $k,\omega$ )-dependent transport coefficients $\tilde L_{ij}(k,i\omega)$ . In more explicit form one has

$\begin{displaymath} \tilde{\mbox{\boldmath$\varphi$}}_{ij}(k,z)= k^2 \tilde L_... ... \hat{\mbox{\boldmath$Y$}}_{j}(-\mbox{\boldmath$k$})\right)}, \end{displaymath}$

(22)

In the hydrodynamic limit the generalized transport coefficients $\tilde L_{ij}(k,z)$ reduce [30] to the well-known kinetic coefficients L_ij, which could be presented in form of the Green-Kubo formula

$\begin{displaymath} L_{ij}= \frac{\beta}{V} \int \limits_0^{\infty} \left(\De... ...\hat L t} \Delta \hat{\mbox{\boldmath$f$}}_{j} \right) \ dt, \end{displaymath}$

(23)

where $\hat{\mbox{\boldmath$f$}}_i$ are the nonorthogonal parts of the generalized fluxes $\hat{\mbox{\boldmath$I$}}(\mbox{\boldmath$k$})$ in the small wavenumbers limit,

$\begin{displaymath} \hat{\mbox{\boldmath $I$}}_i(\mbox{\boldmath $k$})\vert _{k... ...}}_i= (1-{\cal P}) \ ik \hat{\mbox{\boldmath $J$}}^{(i)}(0). \end{displaymath}$

The expressions for the fluxes $\hat{\mbox{\boldmath$J$}}^{(i)}(0)$ were obtained in [29]. More explicit meaning of the non-zero coefficients L_ij is the following: $L_{pp}= \eta_l=(\frac{4}{3}\eta + \zeta)$ gives a longitudinal viscosity, where $\zeta$ and $\eta$ are the bulk and shear viscosities, respectively; $L_{hh}= T \lambda$ describes the heat diffusion with the thermal conductivity coefficient $\lambda$ ; L_ss is the spin diffusion coefficient; and the new kinetic coefficients L_sh and L_hs describe the dynamic interplay between liquid and magnetic subsystems (namely, between the heat and magnetic properties) and are known in the literature as the thermomagnetic diffusion coefficients (see, e.g., [37,38]). The symmetry properties of L_ij immediately follow from the properties of the fluxes $\hat{\mbox{\boldmath$f$}}_l$ and were discussed in the previous paper [30]. In particular, it was shown that L_sh(h)=-L_sh(-h) and L_hs(h)=L_sh(h), so that these cross coefficients are equal to zero above the Curie point when there is no external magnetic field (h=0). The general structure of the matrix of memory functions (4) in the hydrodynamic limit, $\tilde{\mbox{\boldmath$\varphi$}}(k,z)\vert _{\omega,k \to 0} = \tilde{\mbox{\boldmath$\varphi$}}^{\scriptscriptstyle H} (k),$ has been discussed in [30] and can be written as follows

$\begin{displaymath} \tilde{\mbox{\boldmath$\varphi$}}^{\scriptscriptstyle H} (... ...ar{\chi}_{{\scriptscriptstyle T},n}}} \end{array} \right). \end{displaymath}$

(24)

We see that the nonzero elements of $\tilde{\mbox{\boldmath$\varphi$}}^{\scriptscriptstyle H}(k)$ are proportional to k², so that one has $\tilde{\mbox{\boldmath$\varphi$}}^{\scriptscriptstyle H}(k) = k^2 \Vert \gamma_{ij} \Vert$ . Hence, in the hydrodynamic limit (denoted by the superscript "H") the matrix equation (2) for the Laplace transforms $\tilde{\mbox{\boldmath$F$}}^{\scriptscriptstyle H}(k,z)$ of hydrodynamic time correlation functions ${\mbox{\boldmath$F$}}^{\scriptscriptstyle H}(k,t)$ has the form

$\begin{displaymath} \{ z{\mbox{\boldmath$I$}} + {\mbox{\boldmath$T$}}^{\script... ...le H}(k,z)= {\mbox{\boldmath$F$}}^{\scriptscriptstyle H}(k), \end{displaymath}$

(25)

where

$\begin{displaymath} {\mbox{\boldmath$T$}}^{\scriptscriptstyle H}(k)= - i\,\mbo... ... \tilde{\mbox{\boldmath$\varphi$}}^{\scriptscriptstyle H}(k) \end{displaymath}$

(26)

is so-called hydrodynamic matrix, and the matrices $i\mbox{\boldmath$\Omega$}_0^{\scriptscriptstyle H}(k)$ and $\tilde{\mbox{\boldmath$\varphi$}}^{\scriptscriptstyle H}(k)$ are given by (21) and (24), respectively. One has to mention that the matrix equation (25) is asymptotically exact in the hydrodynamic limit, and the hydrodynamic matrix ${\mbox{\boldmath$T$}}^{\scriptscriptstyle H}(k)$ depends only on the wavenumber k, not on the frequency $\omega$ . The solution of the matrix equation (25) can be simply written in analytical form via the eigenvalues $z_{\alpha}(k)$ and the eigenvectors $\hat{X}_{\alpha}=\vert\vert\hat{X}_{i,\alpha}\vert\vert$ of the hydrodynamic matrix ${\mbox{\boldmath$T$}}^{\scriptscriptstyle H}(k)$ ,

$\begin{displaymath} \sum\limits_{j} \mbox{\boldmath$T$}^{\scriptscriptstyle H}_{ij}(k) \hat{X}_{j,\alpha} = z_{\alpha}(k) \hat{X}_{i,\alpha}, \end{displaymath}$

(27)

where i,j =n,p,s,h and the index $\alpha$ labels the different eigenvalues. For elements $\tilde{\mbox{\boldmath$F$}}^{\scriptscriptstyle H}_{ij} (k,z)$ we get

$\begin{displaymath} \tilde{\mbox{\boldmath$F$}}_{ij}^{\scriptscriptstyle H}(k,... ...\frac{\mbox{\boldmath$G$}_{\alpha}^{ij}(k)}{z+z_{\alpha}(k)}, \end{displaymath}$

(28)

$\begin{displaymath} \mbox{\boldmath$F$}_{ij}^{\scriptscriptstyle H}(k,t) = \su... ...ox{\boldmath$G$}_{\alpha}^{ij}(k) \exp \{-z_{\alpha}(k)t \}, \end{displaymath}$

(29)

where the weight coefficients $\mbox{\boldmath$G$}_{\alpha}^{ij}(k)$ are defined by the general expression

$\begin{displaymath} \mbox{\boldmath$G$}_{\alpha}^{ij}(k)= \sum \limits_{l} \ha... ...alpha} \hat{X}_{\alpha,l}^{-1} \mbox{\boldmath$F$}_{lj}(k,0) \end{displaymath}$

(30)

and the matrix $\hat{X}^{-1}$ is the inverse of $\hat{X}= \Vert\hat{X}_{\alpha}\Vert$ . For the orthogonal dynamic variables (see (11)) the expression (30) has the most simple form

$\begin{displaymath} \mbox{\boldmath$G$}_{\alpha}^{ij}(k)= \hat{X}_{i,\alpha} \... ...k,0) = {\cal G}_{\alpha}^{ij}(k) \mbox{\boldmath$F$}_{jj}(k), \end{displaymath}$

(31)

which will be used in the next presentation. Thus for the case of a Heisenberg magnetic fluid the hydrodynamic time correlation functions $\mbox{\boldmath$F$}^{\scriptscriptstyle H}(k,t)$ are presented as a sum of four exponential terms, and each term is connected with the corresponding hydrodynamic collective mode $z_{\alpha}(k)$ . The amplitude $\mbox{\boldmath$G$}_{\alpha}^{ij}(k)$ describes the partial contribution of the mode $z_{\alpha}(k)$ in $\mbox{\boldmath$F$}_{ij}^{\scriptscriptstyle H}(k,t)$ . In order to derive the explicit expressions for the hydrodynamic TCFs we have to solve the equation (27) for eigenvalues and eigenvectors of the hydrodynamic matrix ${\mbox{\boldmath$T$}}^{\scriptscriptstyle H}(k)$ . One can use for this purpose the matrix perturbation theory considering the wavenumber k as small parameter.

Results and discussion

The eigenvalues of the matrix ${\mbox{\boldmath$T$}}^{\scriptscriptstyle H}(k)$ have been already calculated in our previous paper [30]. We reproduce here the main results because they will be used later. The following solutions for the eigenvalues of the hydrodynamic matrix ${\mbox{\boldmath$T$}}^{\scriptscriptstyle H}(k)$ , each of them corresponds to a certain hydrodynamic mode, have been found:

a) two complex-conjugated sound modes with the eigenvalues

$\begin{displaymath} z_{\pm} = \pm iv_s k + D_s k^2 \end{displaymath}$

(32)

where the sound velocity v_s is given by the expression

$\begin{displaymath} v_s^2 = \frac{\gamma_m}{\rho \kappa_{{\scriptscriptstyle T},m}}= \left(\frac{\partial P}{\partial \rho}\right)_{S,m}, \end{displaymath}$

(33)

with $\gamma_m= C_{{\scriptscriptstyle P},m}/C_{n,m}$ , and the damping coefficient D_s is

$\displaystyle D_s = \frac12\frac{\eta_l}{\rho} +\frac12\frac{(\gamma_m-1)\lambd... ...iptstyle T},h}\bar\chi_{{\scriptscriptstyle T},n}c_{{\scriptscriptstyle P},m}},$

(34)

where $\delta_{\scriptscriptstyle T} = \kappa_{{\scriptscriptstyle T},m}/\kappa_{{\scriptscriptstyle T},h}$ is the ratio of the isothermal compressibilities at constant magnetization m and magnetic field h. Of course, $\delta_{\scriptscriptstyle T} \to 1$ when $h \to 0$ above Curie point;

b) a hydrodynamic heat mode with the eigenvalue

$\begin{displaymath} z_h = D_h k^2, \ \ D_h= b + \sqrt{b^2- c}, \end{displaymath}$

(35)

where

$\displaystyle b= \frac12\frac{\lambda}{nc_{{\scriptscriptstyle P},m}} +\frac12\... ...iptstyle T},h}\bar\chi_{{\scriptscriptstyle T},n}c_{{\scriptscriptstyle P},m}},$

(36)

$\displaystyle c =\frac{\delta_{\scriptscriptstyle T}}{n^2 T c_{{\scriptscriptstyle P},m}\bar\chi_{{\scriptscriptstyle T},h}} (\lambda T L_{ss}-L_{sh}^2);$

(37)

c) a hydrodynamic spin diffusion mode with the eigenvalue

$\begin{displaymath} z_{m} = D_m k^2, \ \ D_m = b - \sqrt{b^2- c}, \end{displaymath}$

(38)

with the coefficients b and c defined by Eqs. (36)-(37). It is evident form (32)-(38) that when the external field is zero the results known for simple fluids [33,34] and solid magnets [37,38], respectively, are reproduced:

$\begin{displaymath} D_s =\frac12\frac{\eta_l^0}{\rho} +\frac12\frac{(\gamma_0... ..._m = \frac{L_{ss}^0}{n\bar\chi_{{\scriptscriptstyle T},n}^0}, \end{displaymath}$

(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 no going into the details of calculations. A few general remarks should be done herein to be applied for any fixed wavenumber k. The first remark is the following: It can be shown [39,40] that for the hydrodynamic set of dynamic variables, in Markovian approximation for the memory functions (which becomes exact in the hydrodynamic limit), the hydrodynamic TCFs can be obtained in the form for which the zero order moments in frequency and time spaces are reproduced explicitly. For the zero order frequency moment this statement follows immediately from Eqs. (29) and (31). For the zero order time moment it is not so obvious, but can be easily proved (see [39,40]), when the k-dependent memory functions in Markovian approximation are taken in explicit form. The exception is the `density-density' TCF for which the frequency moments are explicitly reproduced up to the second order including^*.

^*This is because the first order time derivative of $\hat{n}(\mbox{\boldmath $k$}) \sim \hat{p}^l(\mbox{\boldmath $k$})$ is indeed already included into the hydrodynamic set of variables.

The second remark is that for small wavenumbers, when the eigenvalues are given by the expressions (32)-(38), we have to restrict our consideration to the two lower orders with respect to the small parameter k (see (31)) for the weight coefficients ${\cal G}_{\alpha}^{ij}(k)$ . Combining both remarks it can be shown that the second order terms to the weight coefficients (proportional to wavenumber k) are only important for contributions due to the sound modes. Keeping this in mind let us consider the results for the diagonal elements of the matrix of TCFs. For the `density-density' TCF one gets

$\begin{displaymath} \mbox{\boldmath$F$}_{nn}^{\scriptscriptstyle H}(k,t)/ \mbo... ...{\alpha} {\cal G}_{\alpha}^{nn}(k) \exp \{-z_{\alpha}(k)t \}, \end{displaymath}$

(40)

where $\alpha = \{+,-,h,m\}$ and

$\displaystyle {\cal G}^{nn}_{+}=\frac{\delta_{\scriptscriptstyle T}}{2\gamma_m}... ...{\scriptscriptstyle T}}{2\gamma_m} \left\{ 1 + ik \frac{b_{nn}}{v_s } \right\},$

$\displaystyle {\cal G}^{nn}_{h}=\frac{\delta_{\scriptscriptstyle T}}{\gamma_m(D... ..._l}{\rho}-2D_s -(1-\frac{\gamma_m}{\delta_{\scriptscriptstyle T}})D_{m}\right),$

(41)

$\displaystyle {\cal G}^{nn}_{m}=\frac{\delta_{\scriptscriptstyle T}}{\gamma_m(D... ..._l}{\rho}-2D_s -(1-\frac{\gamma_m}{\delta_{\scriptscriptstyle T}})D_{h}\right).$

The parameter b_nn has been calculated by two different methods, namely, using the matrix perturbation theory and the frequency moments approach. Both methods give the same result

$\begin{displaymath} b_{nn}= \frac{\eta_l}{\rho}- 3D_s, \end{displaymath}$

which coincides formally with the result known for simple fluids (see [33,34]). The difference between both cases lies only in the microscopic expressions for $\eta_l$ and 3D_s. The weight coefficients of the other diagonal elements of the matrix of TCFs $\mbox{\boldmath$F$}^{\scriptscriptstyle H}(k,t)$ are listed below:
(i) for the `current-current' TCF $\mbox{\boldmath$F$}_{pp}^{\scriptscriptstyle H}(k,t)$ we get

$\displaystyle {\cal G}^{pp}_{\pm}=\frac{1}{2}\left\{ 1 \mp ik \frac{b_{pp}}{v_s... ...ight\}, \quad b_{pp}=b_{nn}+2D_s, \quad {\cal G}^{pp}_{h}={\cal G}^{pp}_{m}= 0;$

(42)

(ii) for the TCF $\mbox{\boldmath$F$}_{hh}^{\scriptscriptstyle H}(k,t)$ , describing the temperature fluctuations, one has

$\displaystyle {\cal G}^{hh}_{\pm}=\frac{(\gamma_m-1)}{2\gamma_m} \left\{ 1\mp i... ..._{sh}} {nT\beta_{{\scriptscriptstyle P},m}\bar\chi_{{\scriptscriptstyle T},n}},$

$\displaystyle {\cal G}^{hh}_{h}=\frac{1}{\gamma_m (D_{m}-D_{h})} \left(-D_{h} +... ...lta_{\scriptscriptstyle T}L_{ss}}{n\bar\chi_{{\scriptscriptstyle T},n}}\right),$

(43)

$\displaystyle {\cal G}^{hh}_{m}=\frac{1}{\gamma_m (D_{m}-D_{h})} \left(D_{m} -\... ...lta_{\scriptscriptstyle T}L_{ss}}{n\bar\chi_{{\scriptscriptstyle T},n}}\right);$

(iii) the weight coefficients of the `spin density-spin density' TCF $\mbox{\boldmath$F$}_{mm}^{\scriptscriptstyle H}(k,t)$ are given by

$\displaystyle {\cal G}^{mm}_{\pm}=\frac{1}{2} \frac{(1-\delta_{\scriptscriptsty... ...tscriptstyle P},{\scriptscriptstyle T}} \delta_{\scriptscriptstyle T}nc_{n,m}},$

$\displaystyle {\cal G}^{mm}_{h}=\frac{1}{\gamma_m (D_{m}-D_{h})} \left(\frac{\l... ...iptstyle T}}{nc_{n,m}}-(\delta_{\scriptscriptstyle T}+\gamma_m-1)D_{h} \right),$

(44)

$\displaystyle {\cal G}^{mm}_{m}=\frac{1}{\gamma_m (D_{h}-D_{m})} \left(\frac{\l... ...iptstyle T}}{nc_{n,m}}-(\delta_{\scriptscriptstyle T}+\gamma_m-1)D_{m} \right).$

The nondiagonal TCFs, describing the dynamic coupling between the different orthogonalized hydrodynamic variables, has been also calculated (see also [41]) within the same approximation. In order to save room we report here only some results and concentrate the following discussion on the physical meaning of the expressions obtained. For the weight coefficients of the function $\mbox{\boldmath$F$}_{nh}^{\scriptscriptstyle H}(k,t)$ we get

$\displaystyle {\cal G}^{nh}_{\pm}=\frac{\beta_{{\scriptscriptstyle P},m}}{2c_{{... ..._{sh}}{nT\beta_{{\scriptscriptstyle P},m} \bar\chi_{{\scriptscriptstyle T},n}},$

$\displaystyle {\cal G}^{nh}_{h}=\frac{\beta_{{\scriptscriptstyle P},m}}{c_{{\sc... ...nT\beta_{{\scriptscriptstyle P},m} \bar\chi_{{\scriptscriptstyle T},n}}\right),$

(45)

$\displaystyle {\cal G}^{nh}_{m}=\frac{\beta_{{\scriptscriptstyle P},m}}{c_{{\sc... ...nT\beta_{{\scriptscriptstyle P},m} \bar\chi_{{\scriptscriptstyle T},n}}\right).$

It is seen in this expression that all the weight coefficients are proportional to the thermal expansion $\beta_{{\scriptscriptstyle P},m}$ as it should be. Besides that the contributions from the sound mode compensate the contributions from the heat and spin modes, so that we have finally $\mbox{\boldmath$F$}_{nh}^{\scriptscriptstyle H}(k,0)/ \mbox{\boldmath$F$}_{hh}(k) = \sum \limits_{\alpha} {\cal G}_{\alpha}^{nh}(k) =0.$ Similar conclusion can be made for $\mbox{\boldmath$F$}_{ns}^{\scriptscriptstyle H}(k,t)$ , where the weight coefficients are given by

$\displaystyle {\cal G}^{ns}_{\pm}=\frac{\delta_{\scriptscriptstyle T}\pi_{{\scr... ...ptscriptstyle T}n c_{n,m} \pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}},$

$\displaystyle {\cal G}^{ns}_{h}= \frac{\delta_{\scriptscriptstyle T}\pi_{{\scri... ...tstyle T}n c_{n,m} \pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}}\right),$

(46)

$\displaystyle {\cal G}^{ns}_{m}=\frac{\delta_{\scriptscriptstyle T}\pi_{{\scrip... ...tstyle T}n c_{n,m} \pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}}\right).$

In this case all the weight coefficients are proportional to the magnetostriction coefficient $\pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}$ and we have once again a cancellation of the different contributions, so that $\sum \limits_{\alpha} {\cal G}_{\alpha}^{nh}(k) =0$ . For the functions $\mbox{\boldmath$F$}_{hs}^{\scriptscriptstyle H}(k,t)$ and $\mbox{\boldmath$F$}_{sh}^{\scriptscriptstyle H}(k,t)$ we found that the weight coefficients are proportional to the product of the thermal expansion and magnetostriction coefficients. Finally, one has for the TCF $\mbox{\boldmath$F$}_{pn}^{\scriptscriptstyle H}(k,t)$

$\displaystyle {\cal G}^{pn}_{\pm}= \mp \frac{\delta_{\scriptscriptstyle T}{\rm ... ...gamma_m}\left\{ 1\mp ik \frac{b_{pn}}{v_s } \right\}, \quad b_{pn}=b_{nn}+ D_s,$

$\displaystyle {\cal G}^{pn}_{h}= i k{\rm m} D_h {\cal G}^{nn}_{h}, \quad {\cal G}^{pn}_{m}= i k{\rm m} D_m {\cal G}^{nn}_{m}.$

(47)

It is easily to check that the zero order frequency moment for the function $\mbox{\boldmath$F$}_{pn}^{\scriptscriptstyle H}(k,t)$ is also satisfied. Taking into account the remarks presented above, it is evident that the expressions for the weight coefficients of the TCFs $\mbox{\boldmath$F$}_{pn}^{\scriptscriptstyle H}(k,t)$ and $\mbox{\boldmath$F$}_{pp}^{\scriptscriptstyle H}(k,t)$ may be also derived using the explicit relations

$\begin{displaymath} \frac{d^2}{dt^2} \mbox{\boldmath $F$}_{nn}(k,t)= - \frac{k^... ...{nn}(k,t)= \frac{ik}{{\rm m}} \mbox{\boldmath $F$}_{pn}(k,t), \end{displaymath}$

when we consider terms up to the second order with respect to small k. The hydrodynamic TCFs ${\mbox{\boldmath$F$}}_{nn}(k,t)$ and ${\mbox{\boldmath$F$}}_{mm}(k,t)$ , describing the `particle density-particle density' and `spin density-spin density' correlations, are of special interest of the theory, because they can be recovered from scattering experiments. Namely, their Fourier transforms, being measured experimentally, are the dynamic structure factor

$\begin{displaymath} S(k,\omega) = \frac{1}{N}{\mbox{\boldmath $F$}}_{nn}(k,\ome... ...m Re} \ \tilde{\mbox{\boldmath $F$}}_{nn}(k,i\omega+\epsilon) \end{displaymath}$

and the magnetic dynamic structure factor

$\begin{displaymath} S_{m}(k,\omega)= \frac{1}{N}{\mbox{\boldmath $F$}}_{mm}(k,\... ... Re} \ \tilde{\mbox{\boldmath $F$}}_{mm}(k,i\omega+\epsilon). \end{displaymath}$

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

$\displaystyle \frac{S (k,\omega)}{S(k)} = \frac{\delta_{\scriptscriptstyle T}}{... ...2 - \alpha k(\omega/v_s + \alpha k) b_{nn}}{(\omega+\alpha kv_s)^2+(D_s k^2)^2}$

(48)

$\displaystyle + \sum_{\alpha=h,m} \ {\cal G}^{nn}_{\alpha} \frac{D_\alpha k^2} {\omega^2+(D_\alpha k^2)^2},$

$\displaystyle \frac{S_m(k,\omega)}{S_m(k)}= \frac{(1-\delta_{\scriptscriptstyle... ...2- \alpha k(\omega/v_s + \alpha k) b_{mm}}{(\omega+ \alpha kv_s)^2+(D_s k^2)^2}$

(49)

$\displaystyle + \sum_{\alpha=h,m} \ {\cal G}^{mm}_{\alpha} \frac{D_\alpha k^2}{\omega^2+(D_\alpha k^2)^2}.$

We see in (48) and (49) that all the hydrodynamic modes contribute to the dynamic structure factors when an external homogeneous magnetic field h is applied. The same result could be found for h=0, if the ferromagnetic phase is considered. For h=0 and above a Curie point the coupling between subsystems disappears and one has $L_{sh}= \pi_{{\scriptscriptstyle P},{\scriptscriptstyle T}}=0$ and $\delta_{\scriptscriptstyle T}=1$ , so that all the expressions given above can be significantly simplified. In particular, in this case we find that the dynamic structure factor $S(k,\omega )$ has formally the same structure as for a simple fluid [33,34], and the magnetic dynamic structure factor in the hydrodynamic limit is completely determined by the spin diffusion mode [37,38].

Figure: Schematic representation of the dynamic structure factor $S(k,\omega )$ at three different values of external field h. Contributions increasing with h from the spin diffusion mode at $h\neq 0$ are shown by thin lines in the left bottom corner.

$\begin{figure} \epsfxsize =130mm \centerline{\epsffile{snn.eps}} \end{figure}$

**Figure:** Schematic representation of the dynamic structure factor $S(k,\omega )$ at three different values of external field h. Contributions increasing with h from the spin diffusion mode at $h\neq 0$ are shown by thin lines in the left bottom corner.
$\begin{figure} \epsfxsize =130mm \centerline{\epsffile{snn.eps}} \end{figure}$

Such behavior for the dynamic structure factor $S(k,\omega )$ is qualitatively demonstrated in figure 1. We see in this figure that the height of side Brillouin peak (of the central peak) decreases (increases) when h becomes larger, and an additional contribution to the central line due to the spin diffusion mode appears for $h\neq 0$ . Of special interest is the magnetic dynamic structure factor $S_m(k,\omega )$ , in which due to the magnetostriction effect, side Brillouin peaks appear (see figure 2), so that the form of $S_m(k,\omega )$ changes qualitatively.

Figure: The magnetic dynamic structure factor $S_m(k,\omega )$ at three different values of external field h: schematic representation. Contributions increasing with h from the heat diffusion mode at $h\neq 0$ are shown by thin lines in the left bottom corner.

$\begin{figure} \epsfxsize =130mm \centerline{\epsffile{smm.eps}} \end{figure}$

Considering the case of a small magnetic field it can be shown that the weight coefficients of sound modes in (49) are proportional to h². Although this effect is small, it can be in principle studied experimentally in fluids consisting of particles with a localized spin moment. It is seen in figure 2 that in comparison with the case of $S(k,\omega )$ we have for the magnetic dynamic structure factor $S_m(k,\omega )$ the inverse situation concerning its behavior for different values of magnetic field. The height of central line decreases and the height of Brillouin peak increases when 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 $S(k,\omega )$ and $S_m(k,\omega )$ can be easily found from the expressions (48) and (49). We get the following results

$\begin{displaymath} \frac{I_{\scriptscriptstyle R}}{2I_{\scriptscriptstyle B}}=... ...elta_{\scriptscriptstyle T}}{\delta_{\scriptscriptstyle T}-1} \end{displaymath}$

for $S(k,\omega )$ and $S_m(k,\omega )$ , respectively. As it was mentioned already the central peak in both cases has two separate Lorentzian components from the heat and spin-diffusion modes for $h\neq 0$ . This is quite similar to the case of a binary mixture [42,43,44,45] where the heat and concentration modes both form the central line. These results can be used for an interpretation of scattering experiments. It is also worth to mention that from thermodynamic relation

$\begin{displaymath} \delta_{\scriptscriptstyle T} = \left\{1+ \frac{\pi_{{\scri... ...tstyle T},n}\kappa_{{\scriptscriptstyle T},h}} \right\}^{-1} \end{displaymath}$

the magnetostriction coefficient $\pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}$ can be extracted not only from thermodynamic experiments, but also from the scattering experiments. In particular one may proceed as follows: the thermodynamic quantities $\chi_{{\scriptscriptstyle T},n}$ and $\kappa_{{\scriptscriptstyle T},h}$ can be obtained from the long wave length limit of the zero-order frequency moments of $S(k,\omega )$ and $S_m(k,\omega )$ (or, in other words, from the static structure factors S(k) and S_m(k)). The ratio $\delta_{\scriptscriptstyle T}$ may be found from the position of Brillouin peak and the Landau-Placzek ratio $I_{\scriptscriptstyle R}/2I_{\scriptscriptstyle B}$ . 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-defined thermodynamic quantities and hydrodynamic transport coefficients. Because of this, they can be used not only for a Heisenberg model ferrofluid with the Hamiltonian (6), but for the description of other systems as well. Two main demands are important only:
(i) spin interaction has to be isotropic in space of translational motions and to depend only on the distance between particles;
(ii) the total spin density should be a conserved quantity.
In particular, expressions (48) and (49) are directly applicable to the theory of polar liquids with isotropic interactions. The hydrodynamics of polar liquids and ferrofluids becomes formally identical [24] if one notes the correspondence between `electric field' and `magnetic field' and between `polarization' and `magnetization'.

Acknowledgments. This study is supported in part by the Fonds fьr Fцrderung der wissenschaftlichen Forschung under Project P 12422 TPH.

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Hydrodynamic time correlation functions of a Heisenberg model ferrofluid

Footnotes:

¹Institute for Condensed Matter Physics, National Academy of Scienses of Ukraine, UA-290011 Lviv, Ukraine

²Institute of Theoretical Physics, Linz University, A-4040 Linz, Austria

³State University ``Lvivska Politekhnika'', UA-29013 Lviv, Ukraine

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записки.

$\displaystyle {\mbox{\boldmath$F$}}_{ij}(k,t)= \Big(\Delta\hat{\mbox{\boldmath$... ... \ \Delta\hat{\mbox{\boldmath$Y$}}_j(-\mbox{\boldmath$k$}) \Big) \qquad \qquad$			(1)
$\displaystyle \equiv \int \limits_{0}^{1} d\tau \ {\rm Sp} \left[ \Delta\hat{\m... ...hat{\mbox{\boldmath$Y$}}_j(-\mbox{\boldmath$k$}) \Big) \rho_0^{1-\tau} \right],$

$\displaystyle \hat{\varepsilon}({\mbox{\boldmath$k$}})= \hat{\varepsilon}_{\scr... ...+ \hat{\varepsilon}_{\scriptscriptstyle S}({\mbox{\boldmath$k$}})= \hspace{8em}$
$\displaystyle = \sum \limits _{f=1}^N \{ \frac {p^2_f}{2m} +\frac {1}{2} \sum \... ...\mbox{\boldmath$S$}}_{f'} \exp (i{\mbox{\boldmath$k$}}{\mbox{\boldmath$r$}}_f),$			(9)

$\displaystyle {\cal P}_n({\mbox{\boldmath$k$}}) \ldots = \Big( \ldots, \hat{n}(... ...}), \hat{n}(-{\mbox{\boldmath$k$}})\Big)^{-1} \ \hat{n}({\mbox{\boldmath$k$}}),$
$\displaystyle {\cal P}_s({\mbox{\boldmath$k$}}) \ldots = \Big( \ldots, \hat{s}(... ...}), \hat{s}(-{\mbox{\boldmath$k$}})\Big)^{-1} \ \hat{s}({\mbox{\boldmath$k$}}).$			(13)

$\displaystyle {\cal G}^{nn}_{+}=\frac{\delta_{\scriptscriptstyle T}}{2\gamma_m}... ...{\scriptscriptstyle T}}{2\gamma_m} \left\{ 1 + ik \frac{b_{nn}}{v_s } \right\},$
$\displaystyle {\cal G}^{nn}_{h}=\frac{\delta_{\scriptscriptstyle T}}{\gamma_m(D... ..._l}{\rho}-2D_s -(1-\frac{\gamma_m}{\delta_{\scriptscriptstyle T}})D_{m}\right),$			(41)
$\displaystyle {\cal G}^{nn}_{m}=\frac{\delta_{\scriptscriptstyle T}}{\gamma_m(D... ..._l}{\rho}-2D_s -(1-\frac{\gamma_m}{\delta_{\scriptscriptstyle T}})D_{h}\right).$

$\displaystyle {\cal G}^{hh}_{\pm}=\frac{(\gamma_m-1)}{2\gamma_m} \left\{ 1\mp i... ..._{sh}} {nT\beta_{{\scriptscriptstyle P},m}\bar\chi_{{\scriptscriptstyle T},n}},$
$\displaystyle {\cal G}^{hh}_{h}=\frac{1}{\gamma_m (D_{m}-D_{h})} \left(-D_{h} +... ...lta_{\scriptscriptstyle T}L_{ss}}{n\bar\chi_{{\scriptscriptstyle T},n}}\right),$			(43)
$\displaystyle {\cal G}^{hh}_{m}=\frac{1}{\gamma_m (D_{m}-D_{h})} \left(D_{m} -\... ...lta_{\scriptscriptstyle T}L_{ss}}{n\bar\chi_{{\scriptscriptstyle T},n}}\right);$

$\displaystyle {\cal G}^{mm}_{\pm}=\frac{1}{2} \frac{(1-\delta_{\scriptscriptsty... ...tscriptstyle P},{\scriptscriptstyle T}} \delta_{\scriptscriptstyle T}nc_{n,m}},$
$\displaystyle {\cal G}^{mm}_{h}=\frac{1}{\gamma_m (D_{m}-D_{h})} \left(\frac{\l... ...iptstyle T}}{nc_{n,m}}-(\delta_{\scriptscriptstyle T}+\gamma_m-1)D_{h} \right),$			(44)
$\displaystyle {\cal G}^{mm}_{m}=\frac{1}{\gamma_m (D_{h}-D_{m})} \left(\frac{\l... ...iptstyle T}}{nc_{n,m}}-(\delta_{\scriptscriptstyle T}+\gamma_m-1)D_{m} \right).$

$\displaystyle {\cal G}^{nh}_{\pm}=\frac{\beta_{{\scriptscriptstyle P},m}}{2c_{{... ..._{sh}}{nT\beta_{{\scriptscriptstyle P},m} \bar\chi_{{\scriptscriptstyle T},n}},$
$\displaystyle {\cal G}^{nh}_{h}=\frac{\beta_{{\scriptscriptstyle P},m}}{c_{{\sc... ...nT\beta_{{\scriptscriptstyle P},m} \bar\chi_{{\scriptscriptstyle T},n}}\right),$			(45)
$\displaystyle {\cal G}^{nh}_{m}=\frac{\beta_{{\scriptscriptstyle P},m}}{c_{{\sc... ...nT\beta_{{\scriptscriptstyle P},m} \bar\chi_{{\scriptscriptstyle T},n}}\right).$

$\displaystyle {\cal G}^{ns}_{\pm}=\frac{\delta_{\scriptscriptstyle T}\pi_{{\scr... ...ptscriptstyle T}n c_{n,m} \pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}},$
$\displaystyle {\cal G}^{ns}_{h}= \frac{\delta_{\scriptscriptstyle T}\pi_{{\scri... ...tstyle T}n c_{n,m} \pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}}\right),$			(46)
$\displaystyle {\cal G}^{ns}_{m}=\frac{\delta_{\scriptscriptstyle T}\pi_{{\scrip... ...tstyle T}n c_{n,m} \pi_{{\scriptscriptstyle T},{\scriptscriptstyle P}}}\right).$

$\displaystyle {\cal G}^{pn}_{\pm}= \mp \frac{\delta_{\scriptscriptstyle T}{\rm ... ...gamma_m}\left\{ 1\mp ik \frac{b_{pn}}{v_s } \right\}, \quad b_{pn}=b_{nn}+ D_s,$
$\displaystyle {\cal G}^{pn}_{h}= i k{\rm m} D_h {\cal G}^{nn}_{h}, \quad {\cal G}^{pn}_{m}= i k{\rm m} D_m {\cal G}^{nn}_{m}.$			(47)

$\displaystyle \frac{S (k,\omega)}{S(k)} = \frac{\delta_{\scriptscriptstyle T}}{... ...2 - \alpha k(\omega/v_s + \alpha k) b_{nn}}{(\omega+\alpha kv_s)^2+(D_s k^2)^2}$			(48)
$\displaystyle + \sum_{\alpha=h,m} \ {\cal G}^{nn}_{\alpha} \frac{D_\alpha k^2} {\omega^2+(D_\alpha k^2)^2},$

$\displaystyle \frac{S_m(k,\omega)}{S_m(k)}= \frac{(1-\delta_{\scriptscriptstyle... ...2- \alpha k(\omega/v_s + \alpha k) b_{mm}}{(\omega+ \alpha kv_s)^2+(D_s k^2)^2}$			(49)
$\displaystyle + \sum_{\alpha=h,m} \ {\cal G}^{mm}_{\alpha} \frac{D_\alpha k^2}{\omega^2+(D_\alpha k^2)^2}.$