Light-scattering spectra of supercooled molecular liquids

Light-scattering spectra of supercooled molecular liquids

T. Franosch,^1,*M. Fuchs,¹and A. Latz²

1Physik-Department, Technische Universita¨t Mu¨nchen, D-85747 Garching, Germany

2Institut fu¨r Physik, Johannes Gutenberg Universita¨t, Staudinger Weg 7, D-55099 Mainz, Germany 共Received 21 June 2000; published 25 May 2001兲

The light-scattering spectra of molecular liquids are derived within a generalized hydrodynamics. The wave-vector and scattering-angle dependencies are given in the most general case and the change of the spectral features from liquid to solidlike is discussed without phenomenological model assumptions for共gen- eral兲dielectric systems without long-ranged order. Exact microscopic expressions are derived for the frequency dependent transport kernels, generalized thermodynamic derivatives, and the background spectra.

DOI: 10.1103/PhysRevE.63.061209 PACS number共s兲: 64.70.Pf, 78.35.⫹c

I. INTRODUCTION

Light scattering is a powerful tool to study the dynamics of dense 共transparent兲materials 关1兴. The fluctuations of the dielectric tensor for wave-vector transfer q are measured, where the corresponding wavelength can be considered large compared to molecular length scales. In this case the theo- retical description of the light spectra simplifies as it suffices to determine the lowest orders in wave vector q only. The wave vector and scattering-angle dependence of the scatter- ing cross sections thus can be determined.

Focusing on low frequencies, the hydrodynamic approach can be used to calculate the spectral shapes. For the polarized-light-scattering spectra of gases and liquids, the as- sumption that dielectric fluctuations are dominantly caused by density fluctuations leads to the well-known Rayleigh- Brillouin spectra. The corresponding hydrodynamic depolar- ized spectra were first obtained within simplified models by Andersen and Pecora关2兴and Keyes and Kivelson关3兴, in the latter case, under the assumption that fluctuations in the ori- entations of the molecules cause the dielectric variations. As observed experimentally in molecular liquids, a negative central line appears called ‘‘Rytov dip’’ 关4,5兴.

The spectra, in particular the depolarized ones, change qualitatively if the relaxation times of the structural dynam- ics increase upon cooling the liquids. For the polarized spec- tra, Mountain introduced a frequency-dependent longitudinal viscosity in order to model the additionally appearing central line关6兴. In the depolarized-light-scattering spectra transverse sound peaks become visible as expected from hydrodynamic calculations for solids关7兴. Within phenomenological models including nonhydrodynamic variables, the changes of the spectra from liquidlike at high to solidlike at low tempera- tures could be explained 关8–13兴. However, assumptions about the included slow variables and about phenomenologi- cal kinetic equations coupling their time dependencies, were required. The introduction of memory functions, in most de- tail in the recent work of Dreyfus et al.关12,13兴, has relaxed the requirement to identify and include all slow variables but still uses phenomenological equations to couple rotational

and translational degrees of freedom.

Here, we clarify that generalized hydrodynamics and symmetry considerations suffice to explain the light- scattering spectra qualitatively as they change from liquid to solidlike. The results will be derived without any assump- tions about nonhydrodynamic variables共and their couplings兲 and will also not depend on specific light-scattering mecha- nisms nor molecular parameters like shape, dipole moment, polarizabilities nor chirality. The set of slow variables we consider are the standard slow hydrodynamic variables of liquids, density n(q), current density j(q), and temperature

⌰(q) 共connected to energy conservation兲, and the slow structural relaxation of supercooled liquids enters via a few memory functions. We thus provide a general framework for the analysis of light-scattering spectra in supercooled mo- lecular liquids, which we expect will prove useful either for phenomenological discussions using fit functions for the memory kernels—we will list all restrictions on these fit functions—or for the consideration of specific scattering mechanisms. Our central technique for simplifying the spec- tra consists in small-wave-vector expansions of the memory functions as should be appropriate for disordered systems.

Thus we adopt the idea of generalized hydrodynamics, which extends the regular hydrodynamic approach to larger fre- quencies. In detail, we use the one suggested by Go¨tze and Latz 关14兴as it provides a physically reasonable description of glassy systems. Finally, we also derive Green-Kubo for- mulas, which enable e.g., computer simulations, to determine the memory functions and thus the complete spectra directly.

Our assumptions pertain to the systems under consider- ation and can be tested experimentally: We consider only the lowest nontrivial orders in wave-vector transfer q, in order to find general results for the light scattering from amorphous, dielectric, macroscopically isotropic and optically inactive materials within the framework of linear response, classical statistical mechanics, and classical electromagnetism. The condition qaⰆ1, where a denotes either a typical molecular size, the average particle distance, or a collective correlation length, appears well satisfied for supercooled molecular liq- uids but excludes studies of critical phenomena. Electromag- netic retardation also can be neglected for␻Ⰶcq, where c is the speed of light.

The general formulas for the spectra and constitutive equations are presented in Sec. II. Section III lists the central

*Present address: Lyman Laboratory of Physics, Harvard Univer- sity, Cambridge, MA 02138.

PHYSICAL REVIEW E, VOLUME 63, 061209

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results that are discussed in Sec. IV. The results for the de- polarized spectra are compared to previous theoretical ap- proaches in Sec. V, and conclusions in Sec. VI summarize our results. More technical aspects are contained in Appen- dices A and B, and Appendix C outlines the application of our general results to specific light-scattering mechanisms.

II. GENERAL FORMULAS A. Dielectric fluctuations

In a light-scattering experiment a laser beam at frequency

␻i induces a polarization in a transparent sample, which starts to radiate. For a homogeneous sample the radiated waves interfere constructively only in the forward direction at the same frequency as the incident wave. However, the dielectric permeability fluctuates in space and time around its average and therefore additionally a共diffusive兲scattering oc- curs. In general, this scattering spectrum reflects the dynami- cal processes in the sample and depends on the frequency of the incident, ␻i, as well as the scattered wave,␻f. A sim- plification is possible if one considers only small frequency shifts, ␻⫽␻i⫺␻f with兩␻兩Ⰶ␻i. Then dynamical dielectric fluctuations „⑀i j(q,t)兩⑀kl(q)… determine the scattering cross sections completely 关7兴. Here the Kubo scalar product,

„A(t)兩B…⫽(1/k_BT)具␦^A(t)*␦^B典, is used, with T temperature and k_B Boltzmann’s constant.

The fluctuation␦⑀i j(q,t) has even time-reversal symme- try, is a symmetric tensor of second rank, and, as Fourier transform of a real quantity, fulfills⑀i j(⫺q,t)⫽⑀i j(q,t)*. In particular, the long-wavelength limit⑀i j(q→0,t) is real. Dif- ferent Cartesian components i, j ,k,l are picked out depend- ing on the polarization directions of the incoming and scat- tered light 关1兴; see Appendix A for more details. The dynamical evolution is given by the Liouvillian L via ⳵tA

⫽iLA. A Laplace transformation—convention f (z)

⫽i兰0⬁dt e^iztf (t) for Iz⬎0—thus leads to the problem to calculate, for q→0, the matrix elements of:

„⑀i j共q,z兲兩⑀kl共q兲…⫽

冉

^{⑀i j共q兲}

^冏

^{L⫺1 z}

^冏

^⑀kl共q兲

冊

^{. 共1兲}

The spectra at frequency␻ then are given by the imaginary part of„⑀i j(q,␻⫹i0)兩⑀kl(q)…denoted by„⑀i j(q,␻⁾兩⑀kl(q)…

⬙

^.

B. Generalized hydrodynamics

Light scattering measures large wavelength dielectric fluctuations. Even though qaⰆ1 can thus be assumed, the limit q→0 cannot be performed naively in Eq.共1兲. Because of density, momentum, and energy conservation, there are poles in the resolvent R(z)⫽(L⫺z)^⫺1, which shift to van- ishing frequency in this limit关15兴. Using the Zwanzig–Mori formalism, these hydrodynamic low-frequency features can be identified. One introduces the reduced resolvent

R

⬘

共z兲⫽Q 1

QLQ⫺zQ, 共2兲

where the projector Q⫽1⫺P projects perpendicular to the hydrodynamic modes:

P⫽兩n共q兲)共n共q兲兩

„n共q兲兩n共q兲… ⫹兩⌰共q兲)共⌰共q兲兩

„⌰共q兲兩⌰共q兲…⫹

兺

_i ^兩_„^jj^i共_i共^qq^兲兲兩^)共j^j_iⁱ共^共q^q兲…^兲兩. 共3兲 The standard hydrodynamic formulas are obtained in this approach if the reduced resolvent R

⬘

(z) is treated in a Mar- kovian approximation, replacing its matrix elements with frequency-independent transport coefficients 关15兴. General- ized hydrodynamics differs from this by retaining the fre- quency dependence of R

⬘

(z) but still neglecting its wave- vector dependence. This generalization is required for liquids at lower temperatures as the structural relaxation slows down strongly.

Following Ref. 关14兴 we identify the fluctuating tempera- ture ⌰(q) with the kinetic-energy fluctuations, e^K(q), that are orthogonal to the density fluctuations, c_V⁰⌰(q)

⫽Q_ne^K(q)⫽e^K(q)⫺n(q)(e^K兩n)/(n兩n). Here c_V⁰⫽3k_B/2 abbreviates the specific heat per particle of the kinetic de- grees of freedom, and Q_n is the projector orthogonal to the density. Conservation of the total energy e(q)⫽e^K(q)

⫹e^P(q) implies

c_V⁰L⌰共q兲⫽q j_e^L共q兲⫺q j^L共q兲共e兩n兲

共n兩n兲⫺LQe^P共q兲, 共4兲 where superscripts L indicate the longitudinal part, j^L

⫽q•j/q, e^P(q) is the potential energy, and j_e(q) the total- energy current. Note that since „⌰(q)兩e^P(q)…⫽0 one can replace Q_ne^P(q)⫽Qe^P(q). The hydrodynamic variables are orthogonal with normalizations: „n(q)兩n(q)…⫽NS(q)/k_BT,

„j_k(q)兩j_l(q)…⫽(N/m)␦kl and „⌰(q)兩⌰(q)…⫽NT/c_V⁰. Here, n is the average density of N molecules, m the molecular mass, and S(q) is the equilibrium center-of-mass structure factor;␦kl is Kronecker’s symbol. Throughout the following we will neglect wave-vector dependencies caused by mo- lecular length scales, and replace, e.g., the structure factor by its homogeneous limit given by the isothermal compressibil- ity␬T: S(q)⫽S(0)⫹O„(qa)²…with S(0)⫽nk_BT␬T.

Considering, in generalized hydrodynamics关16兴, the fluc- tuating temperature instead of energy fluctuations rests upon the experimental observation that the heat conduction of glasses and liquids is not drastically different. This aspect is discussed in Refs. 关14,17兴 where the generalized hydrody- namics also is tested by molecular-dynamics simulations.

This formulation of generalized hydrodynamics accounts straightforwardly for a frequency-dependent isochoric spe- cific heat.

C. Decomposition of dielectric fluctuations

The exact resolvent calculus sketched in the previous sec- tion thus provides a reformulation of Eq.共1兲, see Eq.共A14兲 in Ref.关14兴. The reduced dynamics R

⬘

(z) and the projector P, which projects onto the hydrodynamic variables, appear

„⑀i j共q兲兩R共z兲兩⑀kl共q兲…⫽„⑀i j共q兲兩R

⬘

共z兲兩⑀kl共q兲…⫹„⑀i j共q兲兩

⫻关1⫺R

⬘

共z兲L兴PR共z兲P

⫻关1⫺LR

⬘

共z兲兴兩⑀kl共q兲…. 共5兲 Thus the coupling of the dielectric fluctuations to the hydro- dynamic variables is found; explicitly it is given when writ- ing out PR(z) P in Eq. 共5兲. Additionally there is a back- ground spectrum, the first term on the right-hand side of Eq.

共5兲.

Since the hydrodynamic modes have been projected out 共generalized兲 hydrodynamics postulates that the limit q→0 in R

⬘

(z) can now be performed safely. This leads to the well-known result for Raman scattering关7兴: The background spectrum consists of scalar, and symmetric-traceless-tensor scattering

„⑀i j共q兲兩R

⬘

共z兲兩⑀kl共q兲…⫽S共z兲␦i j␦kl⫹T共z兲

冉

^{␦ik␦jl⫹␦il␦jk}

⫺2

3␦i j␦kl

冊

^{⫹O共q2兲. 共6兲}

Explicit expression for S(z) and T(z) can be obtained by choosing special linear combinations of the dielectric tensor.

Let s₀₀⫽关⑀xx⫹⑀y y⫹⑀zz兴/3 denote the long-wavelength limit of the scalar part and t₂₀⫽关2⑀zz⫺⑀xx⫺⑀y y兴/

冑

^{12 the}共helic- ity兲 ␯⫽0 component of the corresponding spherical tensor t_2␯ 共the prefactors are conventional兲. Then, S(z)

⫽(s₀₀兩R

⬘

^(z)兩s₀₀) and T(z)⫽(t₂₀兩R

⬘

^(z)兩t₂₀) are the scalar and the tensor correlations. Let us state here explicitly that we assume that the long-wavelength static correlations of the dielectric tensor are characterized by two numbers only and are independent of the direction of q→0.

Both spectral contributionsS(z),T(z) are autocorrelation functions of real variables with even time inversion symme- try. Their spectra thus are even and non–negative.

In order to simplify the discussion of the couplings to the hydrodynamic variables in Eq.共5兲, it is useful to consider the corresponding generalized constitutive equations 关14,15兴. These describe the temporal decay of the deviations in a variable, say in ␦⑀i j(q,t), produced by an adiabatic pertur- bation with external fields coupling to the conserved vari- ables, after the perturbations are switched off at time t⫽0:

具␦⑀i j共q,t兲典⫽

兺

_␣⁵

冋

^{具␦A␣共t兲典„A␣兩⑀i j共q兲…/共A␣兩A␣兲}

⫺

冕

0 t

d␶具␦^A␣共␶兲典„A_␣兩iLR

⬘

共t

⫺␶兲兩⑀i j共q兲…/共A_␣兩A_␣兲

册

^{, 共7兲}

where the共orthogonal兲hydrodynamic variables are abbrevi- ated: A₁⫽n(q),A₂⫽⌰(q), A₃⫽j_x(q),A₄⫽j_y(q), and A₅

⫽j_z(q). The special choice of the external perturbation con- sidered in the constitutive equation, Eq.共7兲, prepares a fluc- tuation in 具␦⑀i j(q,t)典, which decays slowly because it re-

quires decay of fluctuations of the conserved variables at large wavelengths, O(1/q). Thus the time evolution of 具␦⑀i j(q,t)典 ^{in Eq.共7兲}, is determined by time-dependent cou- plings to the generalized hydrodynamics of the distinct vari- ables. As seen via a Laplace transformation, the identical couplings appear in Eq.共5兲as in Eq.共7兲, expressing that via these couplings, close to equilibrium, dielectric fluctuations at long wavelengths acquire slow hydrodynamic compo- nents. Equation 共5兲 further identifies the nonhydrodynamic components contributing to the background. The first term on the right-hand side of Eq.共7兲describes static or instanta- neous couplings, whereas the second term describes dynamic couplings that need time to build up and may be character- ized by a finite response time. These will become important when approaching the glass transition upon cooling as then the structural relaxation times increase.

Density fluctuations are coupled to the dielectric fluctua- tions statically via the scalar scattering mechanism only, as follows from Eq. 共7兲 when inserting the projector P from Eq. 共3兲:

„n共q兲兩关1⫺LR

⬘

共z兲兴兩⑀i j共q兲…⫽共n兩s₀₀兲␦i j⫹O共q²兲. 共8兲 The dynamic coupling vanishes because Ln(q)⫽q_kj_k(q) is again a hydrodynamic variable. Furthermore the scalar den- sity cannot couple to the dielectric tensor fluctuations t_2␯ in the limit q→0.

Similar arguments hold for the coupling of the tempera- ture to the dielectric fluctuations. Since the kinetic energy is not conserved, there is a dynamic coupling in addition to the static one. In order to guarantee conservation of the total energy, Eq. 共4兲 is used. Observing (LQe^P兩R

⬘

^(z)⫽(Qe^P兩

⫹z(e^P兩R

⬘

(z) and rearranging terms one finds to lowest or- der in q

c_V⁰„⌰共q兲兩关1⫺LR

⬘

共z兲兴兩⑀i j共q兲…

⫽␦i j兵^{共Qne兩s00兲⫹z}„e^P兩R

⬘

共z兲兩s₀₀…其^{⫹O共q兲. 共9兲} Further couplings of order q can be ignored. The dynamic coupling in Eq. 共9兲 arises from the separation of the total- energy fluctuations into fast kinetic and 共possibly兲slow po- tential ones. Thus the simplified handling of the reduced re- solvent in this generalized hydrodynamics has to be paid by an additional frequency-dependent coupling to temperature fluctuations.

Finally for the coupling of dielectric fluctuations to cur- rent fluctuations

„j_k共q兲兩关1⫺LR

⬘

共z兲兴兩⑀i j共q兲…

⫽⫺

兺

_l ^{ql„␶kl共q兲兩R}

^⬘

^{共z兲兩⑀i j共q兲…/m, 共10兲}

one finds a purely dynamical coupling, as first recognized within so-called two variable models关2,3兴. A static coupling is excluded, since the currents j_kand the dielectric tensor⑀i j

have different time-reversal symmetry. The dynamic cou- pling to the stress tensor,␶kl, which appears because of mo-

mentum conservation,Lj_k(q)⫽兺lq_l␶kl(q)/m, can be evalu- ated in the long-wavelength limit:

„␶kl共q兲兩R

⬘

共z兲兩⑀i j共q兲…

⫽„p兩R

⬘

共z兲兩s₀₀…␦i j␦kl⫹„␶20兩R

⬘

共z兲兩t₂₀…

⫻

冋

^{␦ik␦jl⫹␦il␦jk⫺23␦i j␦kl}

册

^{⫹O共q2兲. 共11兲}

Here p⫽关␶xx⫹␶y y⫹␶zz兴/3 and ␶20⫽关2␶zz⫺␶xx⫺␶y y兴/

冑

¹²

denote the long-wavelength limits of the pressure and of the transversal stress tensor.

According to the basic assumption of共generalized兲hydro- dynamics 关14,15兴, one has to keep terms only to the order indicated, the remaining ones are assumed to be regular with respect to frequency z in the limit q→0.

III. RESULTS

From Eqs.共5兲to共11兲the light-scattering spectra follow if a scattering geometry is chosen and the appropriate tensor elements are calculated; see Appendix A for the used geom- etry. Polarizations vertical to V, or in the scattering plane H, are considered, where the standard abbreviation I_io denotes polarizations for incoming and outgoing light. The spectra depend on q, ␻ and scattering angle ␪^{. We find} I_HV(q,␪^,␻⁾⫽I_VH(q,␪^,␻) as predicted by Rayleigh’s reci- procity theorem 关1兴.

A. Total scattered intensities

The total scattered intensities共except for standard coeffi- cients关1兴兲can be obtained directly from Eq.共A4兲in Appen- dix A and consist of scalar and tensor scattering关7兴:

I_VV共q,␪兲⫽共s₀₀兩s₀₀兲⫹4

3共t₂₀兩t₂₀兲, 共12兲 I_VH共q,␪兲⫽共t₂₀兩t₂₀兲, 共13兲 I_HH共q,␪兲⫽cos²␪共s₀₀兩s₀₀兲⫹

冉

^{1⫹13cos2␪}

冊

^{共t20兩t20兲.}

共14兲 The intensities are wave-vector independent as the limit qa Ⰶ1 is considered in systems where all molecular correlations are short ranged. The conservation laws affect the spectral shapes only, i.e. cause low-lying hydrodynamic lines, but do not lead to long-ranged static correlations. Concurrently, the static couplings of the conserved variables to the dielectric fluctuations in Eqs. 共8兲 to共10兲 are q independent for small wave numbers. See Sec. IV, for why these results appear violated when considering the hydrodynamic limits.

B. Depolarized spectrum

Using the decomposition of the off-diagonal dielectric fluctuations, Eqs. 共7兲,共10兲, and共11兲, one can identify a dy- namic coefficient, a_VH(z), which describes the coupling to the generalized hydrodynamic variables:

a_VH共t兲⫽1

N„␶20兩R

⬘

共t兲兩t₂₀…. 共15兲 It is a generalized elasto–optic or Pockels’ constant familiar from light scattering in solids关7兴, and is a real and symmet- ric function of time, as the two tensor variables determining it are real with even time parity. Its Laplace transform a_VH(z), therefore has an even spectrum a_VH

⬙

⁽␻). This Pock- els’ constant describes the dynamic coupling of the trans- verse current into the dielectric fluctuations and the constitu- tive equation becomes

具␦⑀VH共q,t兲典⫽⫺iq cos␪

2

冕

^{0td␶ aVH共t⫺␶兲具␦jy共q,␶兲典.}

共16兲 From Eqs. 共5兲and共6兲follows the general result for the de- polarized spectrum:

„⑀VH共q,z兲兩⑀VH共q兲…⫽T共z兲⫹q²cos²␪

2a_VH共z兲²C_{j j}^T共q,z兲. 共17兲 Here C_{j j}^T(q,z)⫽„j^T(q,z)兩j^T(q)… denotes the correlation function of the transversal current fluctuations. The spectrum consists of a background arising from the symmetric scatter- ing in Eq. 共6兲, which commonly is discussed as a Raman line, and of couplings to the current fluctuations. Naively evaluating the depolarized spectrum at vanishing wave vec- tor would neglect this additional contribution. It is small, of order O(q²), but is characterized by a time scale that di- verges in the hydrodynamic limit q→0, and therefore domi- nates the low-frequency spectrum. The full current correla- tors appear in Eq. 共17兲, which stresses that no assumptions about translational-rotational coupling are required in order to derive Eq. 共17兲. Explicitly this has been shown by the derivation of Eq. 共17兲 for a liquid of spherical particles in Ref. 关19兴, which was tested in a simulation关20兴.

C. Polarized spectra

For the VV spectrum, Eq. 共11兲suggests to introduce the elasto-optical constant

a_VV共t兲⫽2

3a_VH共t兲⫺„p兩R

⬘

共t兲兩s₀₀…

N . 共18兲

It has identical properties as a_VH(t) since the tensor vari- ables entering its definition again are real with even time parity. Another time-dependent coupling function arises from the temperature fluctuations as described in Eq. 共9兲:

␰共z兲⫽␰⫹z共e^P兩R

⬘

共z兲兩s₀₀兲

NT , 共19兲

where the thermodynamic derivative ␰⫽(⳵^s00/⳵^T)n/n

⫽(Q_ne兩s₀₀)/(NT) is written, which contains the total energy perpendicular to density fluctuations 关14兴. Clearly, Eq. 共19兲 presents a generalized time- or frequency-dependent thermo- dynamic derivative. The scalar variables entering its time or

frequency-dependent term are real with even time parity, and

␰

⬙

⁽␻^)/␻ consequently is an even function of␻. The consti- tutive equation coupling the fluctuating hydrodynamic vari- ables into␦⑀VV(q,t), from Eq. 共7兲, becomes

具␦⑀VV共q,z兲典⫽共⳵^s00/⳵ⁿ兲T具␦ⁿ共q,z兲典⫹␰共z兲具␦⌰共q,z兲典

⫹a_VV共z兲q具␦^jL共q,z兲典^{, 共20兲} where the thermodynamic relation (n兩s₀₀)

⫽Nn␬T(⳵^s00/⳵ⁿ⁾T is used. Collecting the terms in Eq. 共5兲 one obtains when applying the mass-conservation law, which gives q²C^L_{j j}(q,z)⫽qzC_{n j}^L(q,z)⫽z²C_nn(q,z)⫹zNn␬T and qC_⌰^L_j(q,z)⫽zC_⌰n(q,z),

„⑀VV共q,z兲兩⑀VV共q兲…⫽S共z兲⫹4T共z兲/3⫹za_VV² 共z兲Nn␬T

⫹2共⳵^s00/⳵ⁿ兲Ta_VV共z兲Nn␬T

⫹关共⳵^s00/⳵ⁿ兲T⫹za_VV共z兲兴²C_nn共q,z兲

⫹␰共z兲²C_⌰⌰共q,z兲⫹2关共⳵^s00/⳵ⁿ兲T

⫹za_VV共z兲兴␰共z兲C_n⌰共q,z兲. 共21兲

Here C_nn(q,z) denotes the density-density correlation func- tion, C_⌰⌰(q,z) the temperature-temperature correlation function, etc; see Appendix B. Equation共21兲is our principal result for the VV spectrum. It extends the conventional hy- drodynamic spectra to arbitrary frequencies. The small–

wave–vector singularities are encoded in the generalized hy- drodynamic correlation functions C_␣␤(q,z), which are determined by the true resolvent R(z), and can thus e.g., be obtained from simulations.

From the information on the depolarized and the polarized spectrum also the I_HHspectrum can be obtained even though it is not a simple linear combination. The fluctuating variable coupling to the distinct variables for the HH scattering in the geometry of Appendix A is given by

具␦⑀HH共q,z兲典^⫽⫺cos␪ 共⳵^s00/⳵ⁿ兲T具␦ⁿ共q,z兲典^⫺cos␪ ␰共z兲

⫻具␦⌰共q,z兲典⫺关a_HH共z兲cos␪

⫺a_VH共z兲兴q具␦^jL共q,z兲典^{, 共22兲} where we abbreviated a_HH(z)⫽a_VV(z)⫺a_VH(z). The gen- eral spectrum in the HH geometry now reads

„⑀HH共q,z兲兩⑀HH共q兲…⫽S共z兲cos²␪⫹T共z兲

冉

^{1⫹31cos2␪}

冊

^{⫹z关aHH共z兲cos␪⫺aVH共z兲兴2Nn␬T⫹2⳵⳵sn00}

冊

T

cos␪关a_HH共z兲cos␪

⫺a_VH共z兲兴Nn␬T⫹

再冋

^⳵⳵sn00

^冊

^{T⫹zaHH共z兲}

册

^{cos␪⫺zaVH共z兲}

冎

^{2Cnn共q,z兲⫹␰共z兲2cos2␪C⌰⌰共q,z兲}

⫹2␰共z兲cos␪

再冋

^⳵⳵sn00

^冊

^{T⫹zaHH共z兲}

册

^{cos␪⫺zaVH共z兲}

冎

^{Cn⌰共q,z兲. 共23兲}

One notices that even for a general molecular fluid, trans- verse current fluctuations do not couple into the HH spectrum.

D. Generalized Green-Kubo relations

Ten frequency-dependent matrix elements built with the reduced resolvent R

⬘

(z) have been identified in the expres- sions for the light-scattering spectra in supercooled liquids.

Five generalized transport coefficients and thermodynamic derivatives are needed in order to describe the correlators of the hydrodynamic variables关14兴. They are the shear viscos- ity K_s(z), the thermal conductivity ␭(z), the dynamic spe- cific heat c_V(z), the dynamic tension coefficient ␤^{(z), and} the longitudinal stress relaxation kernel K_l(z) 共explicit ex- pressions are summarized in Appendix B兲. The remaining five frequency-dependent kernels encode the details of the light-scattering process: S(z), T(z), a_VH(z), a_VV(z), and

␰^(z).

These expressions involving reduced resolvents are very suitable for approximations, since they do not exhibit hydro- dynamic singularities. In order to determine them from other

theoretical approaches or from computer simulations, it is, however, more convenient to find a formulation in terms of correlation functions involving the full dynamics. For the considered cases at q⫽0, this is made possible by the con- servation laws that allow one to derive Green-Kubo relations for the memory functions or transport coefficients expressing them in terms of autocorrelation functions of the correspond- ing fluxes or time integrals thereof关15,16兴. From the identity 关14兴:

共X˜兩R

⬘

共z兲兩Y˜兲⫽共X˜兩R共z兲兩Y˜兲⫹„X˜兩关1⫺R

⬘

共z兲L兴

⫻PR共z兲P关LR

⬘

共z兲⫺1兴兩Y˜…, 共24兲 one observes that the reduced matrix elements can be rewrit- ten as full matrix elements and correlation functions of the hydrodynamic variables contained in PR(z) P with frequency-dependent coefficients. Since for q→0 the coeffi- cients involvingLn_q andLj_qvanish due to particle and mo- mentum conservation, only the temperature fluctuations con- tribute to the frequency dependence of the coefficients. To derive a generalized Green-Kubo relation, we only need vari- ables X˜⫽QX and Y˜⫽QY , respectively. Therefore all static

couplings X˜兩PR(z) P to the hydrodynamic variables in Eq. 共24兲will vanish too and the generalized Green-Kubo relation is given by

„X兩R

⬘

共z兲兩Y…⫽„QX兩R共z兲兩QY…⫹兵共X兩Qe^P兲⫹z„X兩R

⬘

共z兲兩e^P…其兵共Qe^P兩Y兲⫹z„e^P兩R

⬘

共z兲兩Y…其

NTzc_V共z兲 for q→0. 共25兲

Here we made use of Eqs.共4兲and共B3兲in the limit q→0 and of the identity R

⬘

^(z)LQ⫽Q⫹zR

⬘

(z). Rotational invariance implies that the second term in Eq.共25兲is nonzero only for scalar variables X,Y . Thus, e.g., the standard Green-Kubo relations for the shear viscosity, X⫽Y⫽␶20, and heat diffu- sion, X⫽Y⫽j_e^L, are found. Moreover, it follows that in the elasto-optic constant a_VH(z), Eq. 共15兲, and in the tensor background spectrum T(z), Eq. 共6兲, the reduced resolvent can be replaced with the full dynamics.

In order to obtain tractable expressions for the remaining kernels and make contact with the standard Kadanoff-Martin approach关16,18兴, we define another, the conventional, fluc- tuating temperature by c_VT˜ (q)⫽e(q)⫺n(q)(e兩n)/(n兩n), with normalization (T˜ (q)兩˜ (q))⫽T NT/c_V. Also, let Q˜ de- note the projector orthogonal to density, currents, and T˜ (q).

Choosing X⫽Y⫽e^P in Eq.共25兲one finds

„Qe^P共z兲兩Qe^P…

NT ⫽c_V共z兲⫺c_V

z ⫺关c_V共z兲⫺c_V⁰兴²

zc_V共z兲 . 共26兲 Since Qe^P⫽Q˜ e^P⫹(c_V⫺c_V⁰)T˜ the left-hand side implicitly contains hydrodynamic poles due to energy conservation.

However,

„Qe^P共z兲兩Qe^P…

NT ⫽„Q˜ e^P共z兲兩Q˜ e^P…

NT ⫺共c_V⫺c_V⁰兲²

zc_V , 共27兲 and, according to Kadanoff-Martin, the first term on the right-hand side is free of poles in the limit z→0 关18兴. Com- bining Eqs. 共26兲and共27兲one derives

c_V共z兲⫽c_V 共c_V⁰兲²

共c_V⁰兲²⫺zc_V„Q˜ e^P共z兲兩Q˜ e^P…/共NT兲. 共28兲 In the liquid phase the dynamic specific heat attains its ther- modynamic value c_V(z)→c_V for z→0. Equation共28兲dem- onstrates explicitly that the Go¨tze-Latz resolvent R

⬘

^{(z) in-} deed is devoid of all hydrodynamic singularities and is compatible with the conventional Kadanoff-Martin formal- ism. It differs in an explicit frequency dependence of c_V(z), which arises from the splitting of the conventional tempera- ture fluctuations T˜ (q,t) into fast, kinetic ones ⌰(q,t) and structural slow ones.

Similarly, substituting X⫽p and Y⫽e^P in Eq.共25兲yields

„Q p共z兲兩Qe^P…

NmT ⫽␤共z兲⫺␤

z ⫺关␤共z兲⫺␤⁰兴关c_V共z兲⫺c_V⁰兴 zc_V共z兲 ,

共29兲

where ␤⁰⫽( p兩Q_ne^K)/(NmT). Since Q p⫽Q˜ p

⫹m␤^{0Q˜ eP/c}V

0⫹m(␤⫺␤^0)T˜ , the left-hand side can be writ- ten as

1

NmT„Q p共z兲兩Qe^P…⫽ 1

NmT关Q˜ p共z兲兩Q˜ e^P兴

⫹ ␤⁰

NTc_V⁰„Q˜ e^P共z兲兩Q˜ e^P…

⫺共␤⫺␤⁰兲共c_V⫺c_V⁰兲

zc_V . 共30兲 Again the memory kernels appearing on the right-hand side are regular in the low-frequency limit. Collecting terms leads to the generalized Green-Kubo formula for the dynamic ten- sion coefficient

␤共z兲⫽␤^cV_c^共z兲

V ⫹c_V共z兲

c_V⁰ z„Q˜ p共z兲兩Q˜ e^P…

NmT . 共31兲 In a similar fashion the corresponding Green-Kubo for- mulas for the dynamic temperature coupling, the scalar back- ground spectrum, the remaining term contributing to a_VV(z), and the longitudinal stress-stress correlation function can be obtained:

␰共z兲⫽␰^cV_c^共z兲

V

⫹c_V共z兲

c_V⁰ z„Q˜ s00共z兲兩Q˜ e^P…

NT , 共32兲

S共z兲⫽NT␰共z兲² zc_V共z兲 ⫺NT

zc_V␰²⫹„Q˜ s₀₀共z兲兩Q˜ s₀₀…, 共33兲 共p兩R

⬘

共z兲兩s₀₀兲

N ⫽m␤共z兲T

zc_V共z兲 ␰共z兲⫺m␤^T

zc_V ␰⫹„Q˜ p共z兲兩Q˜ s00…

N ,

共34兲

K_l共z兲 nm ⫽ mT

zc_V共z兲␤共z兲²⫺mT

zc_V␤²⫹„Q˜␶^L共z兲兩Q˜␶^L… Nm . 共35兲 In particular, the last term on the right-hand side of Eq.共35兲 is related to the longitudinal viscosity i␩l⫽„Q˜␶^L(z

→0)兩Q˜␶^L…n/N. Therefore in the low-frequency limit where c_V(z)⫽c_V⫹izc_V

⬙

^{, and}␤^(z)⫽␤⫹iz␤

⬙

^{one finds}

K_l共z→0兲

nm ⫽⫺imT␤^2cV

⬙

c_V²⫹2imT␤␤

⬙

c_V⫹i ␩l

nm. 共36兲

To summarize, all ten frequency-dependent kernels can be expressed in terms of the full resolvent R(z), and therefore they can be obtained directly from molecular-dynamics simulations.

IV. DISCUSSION A. Depolarized spectra

The depolarized frequency-dependent spectrum, Eq.共17兲, shall be discussed in detail as it provides the most compact expression but also exhibits clear qualitative changes when supercooling the liquid. It consists of three frequency- dependent contributions, a background, the Pockels constant, and the transverse current correlator.

The current correlators can be taken from theories for the dynamics of the liquid under study or from computer simu- lations. Alternatively, the generalized hydrodynamic ap- proach shifts the problem of calculating the transverse cor- relator to the problem of calculating correlations of the transversal-stress tensor, namely, the frequency-dependent shear modulus, K_s(z)⫽„␶20兩R

⬘

^(z)兩␶20…n/N, where the resol- vent devoid of hydrodynamic fluctuations from Eq. 共2兲ap- pears again. The separation of the hydrodynamic poles from structural relaxation thus is achieved leading to关14兴:

C^T_{j j}共q,z兲⫽ ⫺共N/m兲

z⫹q²K_s共z兲/nm. 共37兲 The result from hydrodynamic theory for the depolarized light scattering from equilibrium molecular liquids can be obtained if the frequency dependence of memory functions built with the reduced resolvent R

⬘

(z) is neglected by using the Markovian low-frequency limit. Then the depolarized Pockels’ constant becomes purely imaginary:

a_VH共z兲→ia_VH

⬙

⫽i

冕

0

⬁

dt a_VH共t兲 for z→0. 共38兲 Therefore, and using the standard hydrodynamic result for the shear viscosity, K_s(z→0)→i␩s, one recovers the result for the depolarized spectrum first obtained within simple models in Refs.关2,3兴:

I_VH共q,␪^,␻兲hy. l.⫽T

⬙

共␻⫽0兲⫺q²共a_VH

⬙

兲²cos²␪ 2

⫻ Nq²␩s/共nm²兲

␻²⫹共q²␩s/nm兲². 共39兲 The transverse momentum diffusion cuts a central line of half-width q²␩s/(mn) and amplitude proportional to (a_VH

⬙

^cos␪^/2)2nN/␩s out of a flat background. Note, that within hydrodynamics the background is structureless, and that the spectrum is positive owing to an elementary Schwartz inequality, (t₂₀兩t₂₀)(␶20兩␶20)⭓(␶20兩t₂₀)².

The structural relaxation of liquids cooled down to and below the melting temperature, slows down strongly, and the frequency dependence of the memory functions with reduced

resolvent, can no longer be neglected; for a discussion of structural relaxation, see e.g., the review 关21兴. The result in Eq. 共17兲 can handle this situation, as the memory functions built with R

⬘

(z)—there are 3 in Eqs. 共17兲 and 共37兲—may either be modeled appropriately or can be taken from other theories or simulations. Within the generalized hydrody- namic approach, a glass or amorphous solid is obtained whenever a structural relaxation process, with time scale␶␣, is slow compared to the hydrodynamic frequencies. Assum- ing further that the dynamics in R

⬘

(t) at shorter times, de- noted by␶␤, can be neglected for this frequency range, then the memory functions in Eq.共17兲can be approximated by

K_s共z兲→⫺G_⬁

z ⫹i⌫s, a_VH共z兲→⫺a_VH^⬁

z , 共40兲 for 1/␶␣Ⰶ兩z兩Ⰶ1/␶␤. This is equivalent to time-independent values, K_s(t)⫽G_⬁ and a_VH(t)⫽a_VH^⬁ for ␶␤Ⰶt Ⰶ␶␣. Therefore, the poles in Eq. 共40兲 are called nonergodicity poles as they describe frozen–in, nonrelaxing components.

G_⬁⫽mnc_T² is the glassy shear modulus familiar from Max- well’s model and a_HV^⬁ is the Pockels’ constant共often denoted P44) quantifying the elasto–optic coupling in the glass 关7兴. Whereas G_⬁ and⌫s need to be positive, the sign of a_HV^⬁ is undetermined; a next-to-leading imaginary part in a_HVexists in principle but does not contribute to the spectrum in the hydrodynamic limit. Equations共17兲and共40兲predict for the hydrodynamic glass spectrum:

I_VH共q,␪^,␻兲hy. g⫽Tg

⬙

⫹

冉

^{q cos␪2aVH⬁}

冊

²

⫻ q²⌫s/共mn兲

共␻²⫺q²c_T²兲²⫹共␻^q2⌫s/nm兲². 共41兲 Two transverse phonon peaks characterized by the transverse sound velocity c_T, and a width ⬀q²⌫s, appear, which are described as damped harmonic oscillations. The background consists of a central line, which cannot be resolved and a structureless continuumT(z)⫽⫺T_⬁/z⫹iTg

⬙

^.

Note, that both hydrodynamic expressions, Eqs.共39兲and 共41兲, do not fulfill the sum rule for the total intensity, Eq.

共13兲, and imply wave-vector-dependent total scattered inten- sities. The reasons, of course, are the Markovian approxima- tions in Eqs. 共17兲and共40兲, which are restricted to describe the dynamics in the hydrodynamic range. Nontrivial spectra obtained in glasses on frequency scales characterized by ␶␤

关22兴also require more elaborate expressions for the memory functions in Eq.共17兲.

For temperatures around the liquid-to-glass crossover at T_c, neither the assumption ␶␣␻Ⰷ1, nor the estimate ␶␤␻ Ⰶ1 hold and the depolarized spectra exhibit anomalies关23兴. The mode coupling theory of the structural relaxation there suggests modeling the reduced resolvent as K_s(z)⬇⫺G_⬁关1

⫺(1⫺iz␶_␣^)⫺␤CD⫹(⫺izt₀)^a兴/z for T⬇T_c, and similar ex- pressions for the other two memory functions. Whereas the Cole-Davidson behavior 共the first two terms兲 is a 共rough兲

model of the␣process, and␤CD—as well as␶_␣—will differ for different resolvent matrix elements, the power law with exponent a describes the universal ‘‘critical’’ decay close to the transition. Here t₀is a microscopic time and the exponent a and the 共true, universal兲exponent b of the high-frequency von Schweidler wing of the ␣ ^{process, K}s(1/␶␣ⰆzⰆ1/␶␤)

⬃(⫺iz␶_␣^)⫺b/z, are related; see e.g., the review关21兴for fur- ther information.

B. Polarized spectra

The most prominent feature of the polarized spectrum, the Brillouin peaks, arise from propagating sound waves. Upon cooling the liquid, structural relaxation manifests itself pre- dominately by a gradual change of the sound velocity and the damping constant. Considering the enormous increase of the transport coefficients, e.g., the longitudinal viscosity that describes the damping in the liquid, this clearly points out the necessity to consider the frequency dependence of the reduced resolvent as done in generalized hydrodynamics.

Furthermore, the couplings a_VV(z),␰(z) as well as the back- ground spectra S(z),T(z) exhibit nontrivial z dependencies in the frequency regime of interest.

The structural relaxation of the Pockels’ constant a_VV(z) 共with numerically different constants兲 can be modeled as

given in Eqs.共38兲and共40兲and as described at the end of the previous section. The explicit factor z in the frequency- dependent part in ␰^{(z), Eq.} 共19兲, cancels a possible noner- godicity pole. The coupling to temperature fluctuations inter- polates smoothly between its low-frequency thermodynamic value (⳵^s00/⳵^T)n and a high-frequency coupling, ␰_⬁^{, char-} acteristic for a glass. A renormalization also appears in the effective coupling to the density fluctuations, which in Eq.

共21兲 is described by the Pockels’ constant P12⫽(⳵^s00/⳵ⁿ⁾T

⫹za_VV(z) and in Eq.共23兲by (P12cos␪⫺P44), respectively, where P44⫽za_VH(z). Note, that the frequency-dependent renormalization of (⳵^s00/⳵^T)n in Eq.共19兲 vanishes, if only the hydrodynamic fluctuations, density and temperature, con- tribute to the scalar scattering; then also a_VV(z)⫽²3a_HV(z).

In order to obtain the spectrum in the true hydrodynamic limit one substitutes the appropriate correlation functions—

see Appendix B—and replaces all memory functions with their low-frequency limits; only in c_V(z), ␤^{(z), and} ␰^(z) linear terms in z need to be kept as can be seen from Eqs.

共28兲, 共31兲, and 共32兲. There are the three familiar hydrody- namic resonances superimposed on the Raman background:

the Brillouin doublet of sound modes and the Rayleigh heat pole. The spectrum is obtained from determining the residues of these poles to lowest order in frequency and wave vector.

I_VV共q,␪^,␻兲hy .l.⫽„Q˜ s00共␻⫽0兲兩Q˜ s00…

⬙

^⫹4T

⬙

共␻⫽0兲/3⫹Nn␬TP12

2

冋

^{X␥Br共␻2⫺c2qc22兲q24⫹⌫共l␻q2⌫l兲2}

⫹X^R␥⫺1

␥

q²D_T

␻²⫹共q²D_T兲²

册

^{, 共42兲}

where the adiabatic sound velocity c⫽

冑

␥^/(mn␬T), the lon- gitudinal sound damping⌫l⫽D_T(␥⫺1)⫹␩l/(mn), and the heat diffusion constant D_T⫽␭

⬙

^/cP appear 共see Appendix B for details兲. The Pockels constant is given by the thermody- namic derivative, P12⫽⳵^s00/⳵ⁿ⁾T, and the flat background consists of scalar and tensor parts. Neglecting contributions from temperature fluctuations, X^R⫽X^Br⫽1, one regains the well-known result for light scattering from 共hydrodynamic兲 density fluctuations 关1,7,15兴. Then, the Landau-Placzek re- sult (␥⫺1) is recovered for the relative intensity of the Ray- leigh to the Brillouin lines, where␥⫽c_P/c_V denotes the ra- tio of the isobaric to isochoric heat capacity关1,7,15兴. In the general case, scattering from temperature fluctuations leads to 共presumably small兲 corrections: X^R⫽关1⫺(c₀²/␤⁾

⫻(␰^/P12)兴² and X^Br⫽关1⫹(␥⫺1)(c₀²/␤⁾⁽␰^/P12)兴².

For the glass, ␻␶_␣Ⰷ1, actually the identical formula holds where, however, the isothermal compressibility, the sound velocity and damping constants, as well as the cou- plings and the background—compare Eqs. 共39兲 and 共41兲— are renormalized. The high-frequency values of the memory functions appear in the formally identical definitions of c, D_T, and ⌫l, where, in the frequency window 1/␶␣Ⰶ␻

Ⰶ1/␶_␤, simple Markovian expressions like c_V(z)⫽c_V^⬁

⫹izc_V,g

⬙

are appropriate for ergodic matrix elements关c_V(z),

␤^(z),␰^{(z), and}␭(z)], and frozen-in components, leading to K_l(z)⫽⫺K_l^⬁/z⫹iK_l,g

⬙

, appear in K_l(z), a_VV(z), and in the Raman background lines. Thus, e.g., the expression for the Pockels constant becomes: P12⬁⫽⳵^s00/⳵ⁿ⁾T⫺a_VV^⬁ . A non- trivial renormalization appears in the isothermal sound ve- locity or equivalently the isothermal compressibility, because of the frozen structural relaxation in the longitudinal friction function:

共c_T^⬁兲²⫽ 1

mn␬T⬁⫽c₀²⫹K_l^⬁

mn, 共43兲

as first observed by Mountain关6兴and predicted from micro- scopic expressions by the mode-coupling theory 关21兴. The glass is less compressible than the corresponding liquid.

Thus the glass sound velocity is c^⬁⫽

冑

␥^⬁cT⬁

⫽

冑

^(cT⬁)²⫹mT(␤^⬁)2/cV⬁. Note that the reduction of the Bril- louin and Rayleigh intensities described by Eqs. 共42兲 and 共43兲 is caused by the appearance of frozen-in density fluc-