brillouin scattering and its application in...

14Reviews in Mineralogy & GeochemistryVol. 78 pp. 543-603, 2014Copyright © Mineralogical Society of America

1529-6466/14/0078-0014$10.00 http://dx.doi.org/10.2138/rmg.2014.78.14

Brillouin Scattering and its Application in Geosciences

Sergio Speziale, Hauke MarquardtDeutsches GeoForschungsZentrum GFZ

Telegrafenberg, 14473 Potsdam, [email protected] [email protected]

Thomas S. DuffyPrinceton University

Princeton, New Jersey 08544, [email protected]

INTRODUCTION

Brillouin spectroscopy is an optical technique that allows one to determine the directional dependence of acoustic velocities in minerals and materials subject to a wide range of environmental conditions. It is based on the inelastic scattering of light by spontaneous collective motions of particles in a material in the frequency range between 10−2 to 10 GHz.

Brillouin spectroscopy is used to determine acoustic velocities and elastic properties of a number of crystalline solids, glasses, and liquids. It is most commonly performed on transparent single crystals where the complete elastic tensor of the sample material can be derived. However, Brillouin spectra can be also measured from opaque materials, from which partial or complete information on the elastic tensor can be determined. It is a very flexible technique with many possible areas of application in research disciplines from condensed matter physics to biophysics to materials sciences to geophysics.

Brillouin scattering can be performed on very small samples and it can be easily combined with the diamond anvil cell and carried out at high pressures and temperatures (see reviews by Grimsditch and Polian 1989 and Eremets 1996). This makes this technique the method of choice to study the elastic properties of deep Earth materials, relevant to construct a mineralogical model of the interior of our planet that is consistent with the constraints from seismology.

Several of the candidate minerals of the Earth’s interior are not stable at ambient conditions, and only recently has there been substantial progress in their synthesis. Unfortunately, those deep earth minerals that can be metastably preserved at ambient pressure and temperature are only available as single crystals with sizes of the order of several tens of microns at most. However, crystals of this size are large enough for Brillouin scattering to be performed. In addition, more sophisticated methods for the characterization, manipulation and preparation of such precious samples are becoming available. These developments allow one to significantly broaden the accessible pressure range of Brillouin scattering experiments.

A challenging application of Brillouin scattering in geophysics is to characterize the elastic properties of materials of the lowermost mantle that are synthesized as microcrystalline aggregates in the diamond anvil cell and cannot be quenched to ambient conditions. New results regarding the average shear velocities of Mg-silicate perovskite and post-perovskite have been recently obtained by Brillouin scattering. These results open new opportunities of developing a systematic understanding of pressure and compositional effects on the elastic properties of the rocks of the lowermost mantle based on experimental data.

544 Speziale, Marquardt, Duffy

In this chapter, we will discuss some important results obtained in geosciences by Brillouin scattering. We will focus particularly on the measurements performed at elevated pressure and/or temperature, and their implications for our understanding of the Earth’s interior. As with any experimental technique Brillouin scattering has several limitations. We will also discuss some of the potential solutions to these technical limitations.

HISTORICAL BACKGROUND

The fundamental problem of light scattering attracted much interest among physicists more than a century ago. This interest was initially directed towards the intensity of the scattered light. Among the pioneering early developments were the work by Tyndall (1868) and Rayleigh (1899) regarding the scattering of light by particles suspended in a homogeneous fluid. Expressions for the intensity of molecular light scattering from fluids were initially proposed by Smoluchowski (1908) and Einstein (1910). The theory was then improved by Landau and Placzek (1934) and a formulation based on a generalized hydrodynamic theory was outlined by Mountain (1966b).

Brillouin (1922) and Mandelstam (1926) independently investigated the problem of light scattered by density fluctuations associated with acoustic waves in a homogeneous medium. The theory proposed by both Brillouin and Mandelstam was that light scattered by thermally excited acoustic waves in a medium should be shifted in frequency with respect to the incident light by an amount equal to the frequency of the scattering fluctuations. The kinematics of the scattering is described by the Bragg equation and the energy shift by the Doppler effect caused by the thermal acoustic waves propagating across the medium. This phenomenon, called Brillouin scattering (or Brillouin-Mandelstam scattering) could be detected spectroscopically in experiments in which light was scattered by acoustic waves with wavelength in the range of 100 nm. Brillouin scattering was first observed experimentally by Gross (1930b). However, with the conventional light sources available before the development of lasers, the measurement of Brillouin scattering was limited to compressed gases and fluids plus a handful of solids (see Krishnan 1971 for a review). The advent of laser sources in the 1960s made it possible to perform measurements on a wider range of solids, liquids and gases.

Since then, Brillouin scattering has been used to investigate elastic properties and high-frequency viscous behavior of crystalline and amorphous solids (Shapiro et al. 1966) liquids, molecular solids and glass-forming compounds, with a special interest in polymers (Patterson 1983; Krüger 1989). The advancement of the technology of Fabry-Perot interferometers then opened new fields of application of this technique, from the study of ferroelectric compounds and their behavior at phase transitions (Jiang and Kojima 2000; Ahart et al. 2010; Marquardt et al. 2013b), to the investigation of surface and interface vibrational and magnetic excitations of bulk materials and of thin films and multilayers (Wittkowski et al. 2002; Milano and Grimsditch 2010). Additional recent applications of Brillouin scattering range from the study of viscoelastic behavior of delicate biomaterials (Speziale et al. 2003) to the investigation of the elastic response of nanocomposites (Li Bassi et al. 2004), the characterization and design of phononic materials (Cheng et al. 2006) to the recent experimental detection of Bose-Einstein condensation of quasiparticles as an effect of external energy pumping (Demokritov et al. 2006).

The application of Brillouin scattering to the study of Earth materials dates back to the very early days of the technique. The first Brillouin measurements were performed on quartz by Gross (1930a) and on gypsum by Raman and Venkateswaran (1938). Starting in the 1970s a significant contribution to the knowledge of the anisotropic elasticity of geophysically important minerals at ambient conditions has been furnished by the groups led by D.J. Weidner at Stony Brook and J.D. Bass at the University of Illinois (e.g., Weidner et al. 1975, 1982; Weidner and Carleton 1977; Vaughan and Weidner 1978; Bass and Weidner 1984; Bass 1989; Yeganeh-Haeri

Brillouin Scattering 545

et al. 1992; Xia et al. 1998; Sinogeikin and Bass 2000; Sinogeikin et al. 2004b; Jackson et al. 2005; Stixrude and Jeanloz 2009). An important advance in the field of high-pressure Brillouin scattering was the study of the elastic tensor of NaCl to 3.5 GPa by Whitfield et al. (1976). In this work, the Brillouin frequency shift was measured in a symmetric scattering geometry that simplifies the determination of acoustic velocity from the measured frequency shift by making its conversion independent of the refractive index of the sample. The new experimental set-up opened the opportunity to investigate the elasticity of upper mantle and transition-zone minerals at the pressure conditions of the Earth’s interior as well as to study of simple molecular solids of relevance in planetary science (Polian and Grimsditch 1984; Duffy et al. 1995; Shimizu et al. 1996; Zha et al. 1998; Sinogeikin and Bass 2000; Sinogeikin et al. 2003; Jiang et al. 2004a; Speziale and Duffy 2004; Murakami et al. 2007a,b; Mao et al. 2008b; Marquardt et al. 2009b).

PHYSICAL PRINCIPLES OF THE BRILLOUIN EFFECT

Linear or thermal Brillouin scattering is inelastic scattering of light by spontaneous thermal fluctuations/excitations (in the wavelength range between 10−7 and 10−6 m) in a material. In a quantum view, such fluctuations correspond to acoustic phonons at the Brillouin zone center (i.e., small wavevector or long wavelength). Phonons, together with other excitations (magnons, plasmons and others) contribute to fluctuations of the dielectric tensor of the material that are responsible for the scattering of light in directions different from that of the incident light propagation (Nizzoli and Sandercock 1990).

In the classical description of light scattering, the electromagnetic field acts on charged particles in the medium that, in turn, act as independent scatterers (Rayleigh 1899). The charged particles accelerated in the scattering volume radiate light. However, the theory encountered a serious difficulty in explaining the observations in dense media: the oscillating electromagnetic field polarizes the medium, but different scatterers that are closer to one another than the wavelength of light each scatter out of phase, thereby creating destructive interference in all but the forward scattering direction. The solution to this problem in explaining the experimentally observed scattering in all directions is the presence of spatially correlated heterogeneities (fluctuations) of the dielectric constant of a material at finite temperatures due to the presence of translational and rotational motion of particles (Smoluchowski 1908; Einstein 1910).

In the classical picture, if we consider a medium in thermodynamic equilibrium, thermal motions of its atoms (particles) will determine fluctuations of their number density as a function of position and time (Fleury 1970; Pecora 1972; Dil 1982):

( ) ( ), , (1)N r t N N r t= + D

Where N is the average number density, DN(r,t) is its fluctuating part, r is position and t is time.

Incident light interacts with the particles acting as dipoles. The interaction results in a Doppler shift of the scattered light. The wavevector and frequency of the scattered light depend on those of the incident light and on the wavevectors and frequencies that are present in the density fluctuations. We can express the density fluctuations as 3D Fourier series:

( )( )

( ) ( )34

1, , exp (2)

2N r t dq d N q i q r t

+∞ +∞

−∞ −∞ D = ΩD Ω ⋅ −Ω π ∫ ∫

where q is the wavevector, Ω is frequency and DN(q,Ω) is the fluctuating part of the number density as a function of wavevector and frequency. The fluctuations are superpositions of phase (sinusoidal) gratings with wavelength Λ:

2(3)

q

πΛ =


and phase velocity ν:

ν =Ωq

( )4

The scattering from the phase gratings is analogous to Bragg’s reflection picture of X-ray scat-tering, and the wavevector and frequency of the scattered light in the medium are:

k k qs i= ± ( )5

(6)s iω = ω ±Ω

where ki, ks, ωi and ωs are the incident and scattered light wavevectors and frequencies respec-tively (Cummins and Schoen 1972).

Brillouin scattering takes place in the hypersonic frequency region (107 s−1 < Ω < 1012 s−1) of the vibrational energy spectrum of the medium. The dispersion q = q(Ω) is determined by the (visco)elastic properties of the medium.

The frequencies of the incident and scattered light are approximately equal (ks ≈ ki) and based on the scattering geometry (the angle between incident and scattered wave normals) only one Fourier component of the fluctuation is involved in the scattering. Its wavenumber is (Cummins and Schoen 1972):

2 sin (7)2

iq kθ =

where θ is the scattering angle (Fig. 1).

The wavelength of the scattering inho-mogeneity extends over many atoms and the phase velocity corresponds to the hypersonic acoustic velocity. The spectrum of the scat-tered light includes a doublet at frequencies (Fig. 2):

ω ω νθ

s i ik= ±

2

28sin ( )

The analysis of Brillouin spectra allows us to access information about four features characteristic of Brillouin scattering: (a) spectral character that allows us to determine information about the acoustic velocity and (visco)elastic coefficients, (b) intensity that allows us to quantify the coupling coefficient between fluctuations and the electromagnetic field, (c) line width that is related to the lifetime of the interaction between fluctuations and light (giving information

Figure 1. Schematic diagram of the geometry of Brillouin scattering. Symbols are explained in the text.

Figure 2. Idealized Brillouin scattering spectrum from an elastically anisotropic medium. The spec-trum contains an unshifted line, corresponding to Rayleigh scattering, and three doublets (symmetric with respect to the unshifted line) due to Brillouin scattering from the three polarizations of the scat-tering wave. Vs1 is the fast quasi-shear acoustic mode; Vs2 is the slow quasi-shear acoustic mode and Vp is the quasi-longitudinal mode.


about viscous properties), and (d) polarization that is related to anisotropic properties of the medium at the microscopic level.

The coupling of light with fluctuations in a medium is given by the fluctuating part of the dielectric tensor as a function of position and time

( ) ( ), , (9)ij ij ijr t r tε = ε + Dε

where εij is the average dielectric tensor and Dεij(r,t) is its fluctuating part. In the simple case of a monoatomic isotropic dielectric medium, the dielectric tensor is of the form εij = (ε + Dε)δij (where δij is the Kronecker delta) and the coupling with density fluctuations is expressed by (Fleury 1970; Dil 1982):

( ) ( ), , (10)r t r t∂ε

ε + Dε = ε + Dρ∂ρ

This is related to density fluctuations Dρ. The strength of the coupling (∂ε/∂ρ) can be expressed based on the Lorentz-Lorenz relation:

( )2

20 21 (11)

3

nn

∂ε ε += − ∂ρ ρ

where ε0 is the permittivity of free space and n is the refractive index.

The macroscopic electric polarization

( ) ( ) ( ) ( ), , ( , ) , , (12)r t N r t p r t Np r t r t= = + DP P

(where p(r,t) is the dipole moment and DP(r,t) is an excess polarization due to fluctuations) caused by the incident electromagnetic field is connected with the microscopic atomic dipole moment (and the density fluctuations):

( ) ( ) ( ) ( ), , , , (13)iis s i s i s ik q e k k

∂ε ωD ω = Dε ω Ω = Dρ − ω −ω

∂ρP

where ki is the wavevector and ei is the polarization of the incident electromagnetic field usually represented as a monochromatic wave.

The details of the coupling and the basic theory of Brillouin scattering in fluids and solids are summarized in the next two sections. A far more detailed presentation of these topics can be found elsewhere (Fabelinskii 1968; Fleury 1970; Cummins and Schoen 1972; Dil 1982). However, it is important here to highlight that in systems that are more complex than monoatomic fluids the second rank dielectric tensor is not diagonal having non-zero off-diagonal coefficients.

Brillouin scattering in fluids

The calculation of the intensity and the spectral characteristics of Brillouin scattering in fluids is based on the determination of the appropriate value of the fluctuating part of the dielectric tensor, Dε, and its coupling with fluctuations of pressure, density, temperature or entropy. In absence of a complete dynamical theory of fluids, the problem has been approached at three different levels of theory: (a) thermodynamic, (b) hydrodynamic and (c) microscopic. Here we will give a very brief account of each of them. More detailed information can be found by the interested reader in several reviews and in specialized studies (Rytov 1958; Benedek and Greytak 1965; Mountain 1966b; Wang 1986).

(a) The thermodynamic approach considers that the fluctuating part of the dielectric tensor can be characterized by its dependence on two parameters, pressure and entropy:


∆ ∆ ∆εε ε

=∂∂

+

∂∂

P

PS

SS P

( )14

Where P is pressure and S is entropy.

The total intensity of the scattered electromagnetic field derived by the fluctuation- dissipation theorem (e.g., Landau and Lifshitz 1969) is proportional to the mean square fluctuations of ε that are related to fluctuations in temperature and density (Fleury 1970; Cummins and Schoen 1972):

2(15)sI ∝< Dε >

By simple thermodynamic identities we can express 2

< Dε > as

2 2 22

0 0

(16)B B

S PS P

k T k T

P V T C V

∂ε ∂ε < Dε >= + ∂ β ∂ ρ

where

∂∂

=

∂∂

< ( ) >

ερ

εT

k T

C V SS

P

B

P P

2 2

0

22

∆

and kB is Boltzmann’s constant, βS is the isentropic compressibility, CP is the specific heat at constant pressure, and V0 is the ambient-pressure volume.

The two terms on the right side of (16) represent two spectral components of the scattered light: (I) the Brillouin scattered light caused by isentropic pressure fluctuations (i.e., the acoustic vibrations) producing a doublet with frequencies ωB = ωi ± Ω (where ωB is the frequency of the Brillouin spectral feature, ωi and Ω are the frequencies of incident light and scattering vibration, see also Eqn. 6 and Fig. 2) and (II) a component caused by isobaric (non-propagating) entropy fluctuations. The second component is not shifted in frequency, ωC = ωi, where ωC indicates that this spectral feature is not shifted in frequency.

The ratio of the intensities of the two components has been the subject of intense theoretical investigation. One important result is the famous Landau-Placzek ratio (Landau and Placzek 1934):

2 2

(17)C P V T S

B V SP S

I C C

I T P C

−∂ε ∂ε − β −β = ≈ = ∂ ∂ β

(where IC, IB are the intensities of the central feature and the shifted doublet, CP, CV are the specific heat at constant volume and at constant pressure, βS is the isentropic compressibility, and βT is the isothermal compressibility) based on simple thermodynamic identities and on the simplifying assumption that (∂ε/∂T)ρ ≈ 0 (where the subscript ρ indicates constant density).

Due to the fact that the Brillouin components are related to hypersonic acoustic waves while the central unshifted feature is due to slow (isothermal) fluctuations, the parameters in the Landau-Placzek ratio should take into account the possibility of dispersion in the hyper-sonic frequency range. A revised form of the ratio was presented by Cummins and Gammon (1965) as:

( )( )

(18)T SC ST

B S HS

I

I

β −β≈

β

where the indices ST and HS refer to static and hypersonic.


(b) The hydrodynamic approximation represents a more complete description of the spec-trum. It includes the dynamics of the fluctuations that affect the spectral line-shapes which cannot be captured by a thermodynamic (equilibrium) approach. In this case, the decay of the spontaneous fluctuations is described with the classical equations describing macroscopic re-laxation processes. One example of this approach is the model presented by Mountain (1966a) in which the dynamics of the fluctuations is expressed by the linearized equations of hydrody-namics. The spectrum is of the form:

( )( ) ( ) ( ) ( )

2 2 2

2 22 2 22 2

2, (19)P V T V

V T P

C C D q C q qS q

C D C q q q q

− Γ Γ ω ∝ + + + ω Γ + ω+ ν Γ − ω+ ν

where DT is the thermal diffusivity, and the terms in brackets are the central and the two shifted Lorentzian peaks whose full width at half maximum is equal to Γq2, where Γ is the acoustic absorption coefficient.

This model correctly describes the frequency content of the spectrum, the intensity ratio of the spectral features (similarly to the Landau-Placzek ratio) and their linewidth. The addition of phenomenological parameters was introduced by Mountain (1966a) to describe spectral features due to the coupling of internal (molecular) and external degrees of freedom (whole molecule translations) in real liquids.

Phenomenological models based on the hydrodynamic approximation can also tackle the problem of modeling the depolarized scattering from molecular liquids with intrinsic molecular anisotropy (scattering involving 90° change of the polarization with respect to the incident light) that cannot be accounted for by a thermodynamic theory because they are related to the presence of non-zero off-diagonal coefficients in the dielectric fluctuation tensor Dεij. Detailed theories of Brillouin scattering of anisotropic molecular liquids have been developed (e.g., Rytov 1970; Wang 1986).

(c) Models based on microscopic theories describe the relaxation of the fluctuations of physical properties of the medium by means of stochastic differential equations with a random driving force. The microscopic foundation of the generalized relaxation equations is based on the work of Zwanzig (1961) and Mori (1965). Microscopic theories of the Brillouin spectra of fluids have been developed in several studies often in order to interpret the depolarized spectra of viscous fluids (e.g., Kivelson and Madden 1980) and of supercooled glass-forming liquids.

Brillouin scattering in solids

In the case of solids, any atomic displacement uij(r,t) affects the local polarization field. The excess polarization for a non-piezoelectric solid can be expressed as (Dil 1982):

( ) ( ) ( ) ( ) ( )10

1, , , , (20)

2i i

i s s ij i j im i jn i mnkl kl s i s i jk q e p u k k e− D ω = Dε ω Ω = − ε ε ω ε ω − ω −ω P

Where pijkl is the photoelastic tensor and ukl(q,Ω) is the frequency and wavevector Fourier transform of the displacement gradient ukl(r,t) = ∂uk/∂xl. All the remaining quantities have already been introduced above. In case of optically isotropic solids and of static homogeneous deformations, the fourth rank tensor pmnkl has the symmetry of the Laue symmetry class of the solid (e.g., Nye 1985).

Nelson and Lax (1971) have developed a microscopic theory of the photoelastic interaction that includes also coupling of rotations to the scattering process. The fluctuating part of the dielectric tensor is expressed as a function of the strain tensor Sij and also of the rotation tensor Rij (the anti-symmetric part of the displacement gradient). In strongly birefringent materials, the Pockels photoelastic tensor is symmetric upon interchange of m and n but it is not symmetric


upon exchange of the indices k and l. In this case, the dependence on strain and rotation (the symmetric and anti-symmetric parts of uij) is treated by explicitly separating the symmetric and asymmetric contributions.

∆ Ω Ωε ω ε ε ω ε ωij i ji

im i jn i mnkl kl mn kl kq e p S q p R, , , [ ] [( ) = − ( ) ( ) ( )−0

1ll j

iq e] , ( )Ω( ) 21

where the brackets indicate antisymmetry upon interchange of indices, Skl = 1/2(∂uk/∂xl + ∂ul/∂xk) is the strain tensor and Rkl = 1/2(∂uk/∂xl − ∂ul/∂xk) is the rotation tensor.

In a thermodynamic approach, Dεij for an isotropic solid can be decomposed into a density and a thermal component. The microscopic theory of the effect of temperature fluctuations on the polarizability of solids is treated by Wehner and Klein (1972). Central non-shifted spectral features (see discussion of Eqn. 16) have actually been observed in Si, diamond, TiO2, SrTiO3, ZnSe (Lyons and Fleury 1976; Hehlen et al. 1995; Stoddart and Comins 2000; Koreeda et al. 2006). Some of these spectral features have been interpreted as the effect of two-phonon difference scattering (Anderson et al. 1984; Stoddart and Comins 2000; Koreeda et al. 2006).

In the long wavelength limit, the equation of motion of the atomic displacement field can be treated as that of an elastic continuum (Cummins and Schoen 1972; Every 2001):

( ) ( )2 2

2

, ,(22)i k

ijklj l

u r t u r tc

t x x

∂ ∂ρ =

∂ ∂ ∂

where cijkl is the stiffness tensor. The derivation of the equation of motion for discrete atomic displacements starts from the expansion of the interatomic potential energy in a power series, which is truncated to the second order term in the case of the harmonic approximation, or to higher orders if we want to take into account damping of the elastic constants.

Equation (22) expresses the dependence of the components of the atomic displacement ui from the equilibrium position on the spatial derivatives of the displacement field by means of the elastic constants. For plane wave solutions of the form ui(q,ω) = ui

0 expi (qr – ωt) we obtain by substitution in (22):

( ) ( )2 0 , 0 (23)ijkl j l ik kc q q u q−ρω δ ω =

where qj are the components of the scattering wavevector, where δik is the Kronecker delta, and uk

0 is the polarization.

This equation admits non-trivial solutions that satisfy the secular equation:

2 0 (24)ijkl j l ikc q q −ρω δ =

The secular equation has three solutions per each direction of q, which correspond to the three phonon branches. The solutions Ωα(q,cijkl) = ((cijkl/ρ)1/2)αq = ναq (where the superscript α = 1, 2, 3 identifies the three solutions and να is the phase velocity) for high symmetry q directions have displacement vectors parallel and perpendicular to q and correspond to one pure longitudinal and two pure shear modes. In a general direction of an anisotropic medium the displacement vectors are mixed and the vibrations are composed of one quasi-longitudinal and two quasi-shear modes (Fig. 2).

The spectral intensity, Is, can be derived from the excess dipole moment. In thermal equilibrium it is proportional (as in the case of liquids) to the mean square of the fluctuating part of the dielectric tensor:

22 2( ) ( ) (25)s isI e e∝ ⋅ < Dε > ⋅

where es and ei are the polarizations of the incident and scattered light.


The intensity of the αth acoustic mode with velocity να for a direction q can be expressed as (Cummins and Schoen 1972):

( )2

2

24 2T (26)

2

B ss i s i

is

k T V nI I e e

nr

α

α

π = ⋅ ⋅ λ ρ ν

where Ii is the intensity of the incident beam, V is the scattering volume, ns and ni are the refrac-tive indices of the medium in the incident and scattered directions, λs is the wavelength (in vac-uum) of the scattered light, r is the distance of observation from the scattering volume and Tα is a tensor that describes the distortion of the dielectric tensor associated with the acoustic mode α.

Tabulated expressions for the tensors Tα are given in Cummins and Schoen (1972) and Vacher and Boyer (1972) for several high symmetry directions in different crystal systems.

BRILLOUIN SPECTROSCOPY

Brillouin scattering measurements are performed in a range of experimental setups and use different scattering geometries. However, the basic features of the experiments are similar across the different setups. In the following subsections, we will introduce the basic design of Brillouin scattering experiments. We will then describe some details of the setups required for specific applications and address limitations and common problems encountered in performing measurements at extreme conditions.

Basic experimental setup

The basic design of a Brillouin scattering experiment is illustrated in Figure 3. Monochro-matic light is focused with a lens L1 on the sample, thereby defining a scattering volume V and an incident external wavevector, ki*. Scattered light in a selected direction is collected by the lens L2 that defines the external scattered wavevector, ks*, parallel to the direction of observa-tion, the external scattering angle, θ* (these quantities are, in general, different from ki, ks and θ in the scattering material as defined in the section “Physical Principles of the Brillouin Ef-fect”), and a solid angle δΩ around the direction of observation; the collimated light from L2

Figure 3. Schematic outline of a typical Brillouin scattering experiment. L: lens; δΩ: solid angle around the direction of observation determined by the numerical aperture of the collecting lens; ki: incident wavevec-tor in the sample; ks: scattered wavevector in the sample; θ: scattering angle inside the sample; ki*: incident wavevector outside the sample environment; ks*: scattered wavevector outside the sample environment; θ*: external scattering angle (determined by the directions of the incident and scattered wavevectors outside the sample environment); PMT: photomultiplier tube.


is focused with a lens L3 on the entrance of the spectrometer. After passing through the spec-trometer, the light is detected by a photomultiplier tube or a solid state detector and recorded by a multichannel analyzer.

Light source

The light source for Brillouin scattering experiments has to be strictly monochromatic (i.e., single longitudinal mode, generally by means of an etalon in the laser resonator) to prevent ambiguities in the spectrum that would make it impossible to assign the observed peaks. Gas or solid-state lasers are both commonly used in current Brillouin scattering system designs. Due to the λ−4 dependence of the intensity of scattered light, green and blue lasers are the most commonly adopted. However, systems designed for opaque semiconductors use near infrared laser sources (Gehrsitz et al. 1997). The power of the source radiation rarely exceeds 100 mW except for experiments performed on samples at extreme conditions (i.e., very small samples compressed in the diamond anvil cell).

Scattering geometry

In order to determine acoustic velocities from Brillouin spectra we must know the value of the scattering wavevector and the scattering angle in the medium (Eqns. 4-7). In order to define the scattering angle within the sample, we have to account for the paths of the incident and scattered light outside the sample, and we have to precisely know the characteristics of the sample (or the sample container) such that we can calculate the internal scattering angle by applying Snell’s law. By constraining the external light path and by knowing the refractive index of the medium we can determine the value of the wavevector of the scattering fluctuation and, from the measurement of its frequency, also determine its phase velocity.

The geometry of the light path outside the sample can be easily set to better than 10−2 degrees. Similar levels of precision can be achieved by accurate machining and polishing of the external surface of the sample (or sample assemblage). The final accuracy at which the internal scattering geometry can be constrained is limited by the uncertainty in the refractive indices of sample (and container). This becomes the limiting factor especially in high-pressure experiments, where both the refractive index of highly stressed diamond windows and that of the sample under pressure are poorly known.

Different scattering geometries are used to simplify the determination of the internal scattering angle (Fig. 4). Among the most commonly used geometries is the 90° normal geometry (90N) in which the sample incidence and scattering faces are 90° from each other, and perpendicular to the incident and scattered light paths. In this geometry, the effect of refraction at the samples surfaces is removed by symmetry and the wavevector of the scattering vibration is at 45° from both the incident and scattered wavevectors (Fig. 4a). The magnitude of the scattering wavevector is q = n(2)1/2/λ0, where n is the refractive index and λ0 is the wavelength of the incident light in vacuum. The most useful scattering geometry for high-pressure applications is the forward symmetric geometry (Fig. 4b). In this case, the incident and scattered sample interfaces are parallel to each other and perpendicular to the bisector of the external scattering angle set by the arrangement of the focusing and collecting optics. This geometry does not require knowledge of the refractive index of the sample and allows one to determine the orientation of the scattering wavevector if just one orientation on the sample face is known. By rotating the sample around an axis perpendicular to the plane of the sample, one can select any scattering orientation within the samples’ surface plane. The scattering wavevector is q = 2 sin (θ*/2)/λ0 where θ* is the external scattering angle as defined by the axes of the focusing and collecting lenses (see also Fig. 3). Two additional examples of scattering geometries are presented in Figure 4. 180° (backscattering) geometry (Fig. 4c) gives only information about the quasi-longitudinal mode (in this geometry the elasto-optic coupling with the shear modes vanishes). The magnitude of the scattering wavevector is q = 2n/λ0.


It can be used in combination with the symmetric forward scattering geometry to determine the refractive index of an isotropic medium (such as a fluid) at the laser frequency. A tilted backscattering geometry (Fig. 4d) is used to measure surface Brillouin scattering from opaque materials. In this case, the magnitude of the scattering wavevector (of the surface vibration) is q

= 2 sin φ/λ0, where φ is the angle between the incident (and scattered) wavevector and the surface normal. This scattering geometry will be further discussed in the section “Measurements of surface Brillouin scattering on opaque materials and thin-films.”

The spectrometer

Due to the very small frequency shifts of the thermal acoustic phonons measured in Brillouin scattering, Fabry-Perot interferometers are used to separate the spectral components of the light instead of grating monochromators. Brillouin spectral lines are extremely close to the unshifted source frequency (the measured frequency shift Dω is generally limited to the range 10−2 < Dω < 10 cm−1) and the spectral resolution required for such measurements is of the order of 10−3 cm−1. Grating monochromators are rarely capable of reaching frequency regions closer than 102 cm−1 to the source frequency and their resolution is in the range of 10−1 cm−1.

A Fabry-Perot interferometer consists of two flat mirror plates with faces very parallel to each other separated by a distance L. An incident beam is subject to multiple reflections at the opposing mirrors’ surfaces. Light of wavelength λ is transmitted only if it satisfies the interference condition:

2 cos (27)m nLλ = α

where m is an integer, n is the refractive index of the medium between the two mirrors of the interferometer, L is the mirror spacing and α is the incidence angle (normally 0°). The free spectral range, Dk, of the interferometer is defined as the frequency interval (expressed usually in wavenumbers) corresponding to the difference between two wavelengths that are simultane-ously transmitted by adjacent interference orders: Dk = 1/Dλ = 1/(2nL) for λ << L.

Figure 4. Schematic diagrams of different scattering geometries used in Brillouin spectroscopy. (a) 90° normal geometry; (b) forward symmetric geometry (here refraction is not taken into account and θ* is the external scattering angle (see text); (c) 180° backscattering geometry; (d) tilted backscattering geometry for surface Brillouin scattering measurements. Symbols are explained in the text.


The instrumental function of a single Fabry-Perot interferometer is determined by the convolution of three factors that depend on (a) the non-perfect reflectivity of the mirrors, (b) their non-perfect parallelism (and surface imperfections), and (c) the finite range of angles incident on the Fabry-Perot (which causes a finite transmitted wavelength bandwidth). The ratio of transmitted to incident light (transmission function) can be expressed by the Airy function (e.g., Jacquinot 1960):

( ) 22

2

, , (28)2

1 4 sin cos

TA L

F nLλ α =

π + α π λ

where T is the maximum (peak) transmitted intensity and F is the effective finesse, defined as the ratio of the instrumental linewidth divided by the free spectral range. The finesse determines the maximum number of features that can be resolved within a given free spectral range. The effective finesse is strongly affected by the imperfections of the mirror surfaces. If the surfaces of the two mirrors match each other within λ/m, the effective finesse will not exceed m/2. For high quality mirror surfaces, the effective finesse is F ≈ 102. Increasing the finesse can only be achieved at the cost of a substantial loss of throughput. The spectral contrast, C = 1 + 4F2/π2, defined as the ratio of maximum and minimum transmission, is always below 104. The throughput of the Fabry-Perot is controlled by geometric parameters of the external optical system (lenses, focal lenses, mirrors, aperture, maximum angular spread of the beam) that determine the minimum achievable free spectral range, which for a single Fabry-Perot interferometer is in the order of 0.5×10−2 cm−1. A thorough discussion of the Fabry-Perot interometer and its applications can be found in Jacquinot (1960) and Hernandez (1988).

In the standard Brillouin scattering experiment, the Fabry-Perot interferometer is used as a variable narrow band-pass filter by scanning the selected free spectral range. This can be performed by one of two main methods: (a) by varying the refractive index, n, of the medium between the mirrors, or (b) by scanning the mirror separation, L. Method (a) is implemented by pressurizing a gas medium in which the interferometer is immersed inside a sealed vessel (Durvasula and Gammon 1978); method (b) is the most commonly used in modern designs of spectrometers for Brillouin scattering, and it is implemented using systems that drive the mirror motion by piezoelectric transducers and achieve rapid and linear scan rates. In the following we will focus only on (b).

The main limitations in the use of a single Fabry-Perot interferometer as a Brillouin scat-tering spectrometer is due to low spectral contrast and by overlapping of transmission from different interference orders. A single Fabry-Perot interferometer does not have a sufficient contrast to measure Brillouin scattered light in presence of very intense elastic scattering from the surface of a material (this is true especially for opaque materials). Multi-passing a single Fabry-Perot or using multiple interferometers operated in series both increase the spectral con-trast as Cn = (C)n (where Cn is the overall contrast and C is the contrast of the single Fabry-Perot interferometer). However, as systems based on multiple interferometers in series suffer from imperfect synchronization of the interferometers scans, multi-passing a single interferometer is a solution that guarantees higher stability and performance. Studies on the design and per-formance of multi-pass Fabry-Perot systems for Brillouin scattering spectrometry show that 5-passes through the same interferometer increase the contrast C from 104 to 109-1010 (Sander-cock 1971; Hillebrands 1999) while maintaining a finesse that is comparable with that of a sin-gle-pass system (50-100). However there is an appreciable reduction of the throughput (Fig. 5).

At a constant effective spectral finesse, the spectral resolution is increased by reducing the free spectral range. This allows one to resolve low-frequency spectral features generally related to shear acoustic modes. The disadvantage of reducing the free spectral range below twice the highest frequency Brillouin peak is that adjacent orders of interference of the Brillouin compo-


nents can partially overlap. The identification and attribution of the observed spectral features is then ambiguous.

Alternatively, the overall free spectral range can be expanded by combining (in series) interferometers of different mirror spacings Li (where the index i refers to the ith interferometer in series). In case of a “tandem” design and incidence angle α = 0 (see Eqn. 27), the first inter-ferometer transmits wavelengths that satisfy λ1 = 2nL1/m1, the second interferometer transmits wavelengths that satisfy λ2 = 2nL2/m2. The tandem system will transmit only when both the conditions are simultaneously satisfied, that is when λ1 = λ2. This condition effectively in-creases the free spectral range by a factor of few tens (Fig. 6). In order to effectively use such combinations of interferometers, their scanning needs to be precisely synchronized such that the increments of the mirror distances for the first and second interferometer, δL1 and δL2, sat-isfy the condition:

1 1

2 2

(29)L L

L L

δ=

δ

Achieving both synchronization and constant increment of the mirrors’ distances is technically challenging when the interferometers are scanned separately (Dil et al. 1981). One ingenious design of a tandem multi-pass interferometer has been developed by Sandercock (1978, 1980) (Fig. 5). In this system, two Fabry-Perot interferometers are scanned using a single stage actuated by a highly linear piezo-transducer. The axis of the second interferometer is oriented at an angle β with respect to the first one. A fixed mirror scanned together with the two interferometers reflects light transmitted by the first interferometer to the second one. The mirror distance of the second interferometer is such that L2 = L1cosβ. This condition is maintained across the whole scan, such that Equation (29) is strictly satisfied at all times

Figure 5. Multi-pass Fabry-Perot interferometer in the “tandem” implementation by Sandercock (1980), Symbols are explained in the text.


during operation of the tandem interferometer. This design is by far the most widely used and it has allowed the application of Brillouin spectroscopy to various types of weakly scattering samples, such as samples in the diamond anvil cell, and systems characterized by the presence of intense elastically scattered signal that could otherwise mask Brillouin scattering features (such as opaque materials). Problems connected to the long-term alignment stability of the interferometers are reduced by controlling and stabilizing the environmental temperature within ±2 °C over a 24 h period. In addition, the implementation of feed-back alignment stabilization control (that systematically adjusts the fine mirrors’ tilting to ensure their best parallelism) and of dynamic vibration controls -to filter environmental vibrations (in the frequency range between 1 and 102 Hz) that can affect the efficiency of the interferometer- extends the stability of the system allowing up to 48-h continuous acquisition times (Lindsay et al. 1981; Sandercock 1987; Hillebrands 1999).

Detectors

The selection of the detector for a Brillouin scattering spectrometer represents a compro-mise between high efficiency in the energy range of interest and good signal-to-noise ratio. In experiments performed on large samples producing intense signals, the use of photomultiplier tubes guarantees the best signal-to-noise ratio due to extremely low dark count (down to less than 5 counts/s) and large dynamic range even though the quantum efficiency is only of the order of 20%. In case of small, weakly scattering samples, such as those in the diamond anvil cell, the high quantum efficiency of avalanche photodiodes (up to 70%) and moderately low dark count (about 25 counts/s) substantially reduces collection times, and can thus mitigate problems of long-term stability of the interferometers’ alignment and the natural drift of the source frequen-cy. However, due to the lower dark current, photomultiplier tubes can still be the best choice in case of materials that produce extremely weak Brillouin signal in the presence of a comparably strong elastic scattering from surface imperfections, such as in case of opaque materials.

Figure 6. Transmission spectrum (as a function of frequency) of the two interferometers and the total trans-mission of a tandem Fabry-Perot system. The use of a second Fabry-Perot interferometer in series with the first but with offset mirror distance suppresses the signal of the first interferometer at any position when the condition L1/m1 = L2/m2 is not satisfied (see text for explanations). This effectively increases the free spectral range of the spectrometer. ω is the Brillouin scattering frequency shift.


The essential components of a setup for Brillouin scattering at high pressure in the diamond anvil cell are sketched in Figure 7. This experimental configuration allows one to determine acoustic velocities with a precision of ∼ 0.5% and accuracy better than 1% (measured on standard materials). The best achievable uncertainty on the recovered single crystal elastic constants is better than 1% (Speziale and Duffy 2002).

Measurements on transparent materials

The most common use of Brillouin scattering is to perform measurements on optically transparent materials. In this case, the scattering geometry can be selected from nearly forward scattering to 180° back scattering. The region of the first Brillouin zone that can be accessed goes from the zone center (q → 0 corresponding to θ → 0; see Eqn. 7) to q = 2nk0 (where k0 is the incident light wavenumber in vacuum and n is the refractive index), assuming an optically isotropic material. Brillouin scattering in optically anisotropic materials is described by generalizing Equation (7) (Krishnan 1971):

( )2 20 2 cos (30)i s i sq k n n n n= + − θ

where ni, ns are the refractive indices in the directions of the incident and scattered light. Thus, for a general (external) orientation of the incident light, there are 12 doublets in the Brillouin scattering spectrum, one for each acoustic mode for each combination of the refractive indices. The application of Equation (30) allows the correct determination of the scattering wavevector. However, for typical birefringences and with the spectral resolution of the available spectrometers, no measurements have been reported that resolved components due to birefringence effects even in materials with exceptionally strong birefringence such as calcite (Chen et al. 2001).

The orientation of the scattering phonon in experiments performed on single crystals (that is, acoustically anisotropic materials) requires a precise knowledge of the orientation of the sample with respect to the experimental reference system. In the simple case of an optically isotropic material, it is important to determine the effect of refraction at the interface between the sample and the surrounding medium. This requires that the external surface of the sample

Figure 7. Schematic drawing of the setup for Brillouin scattering in the diamond anvil cell. BS: beamsplit-ter; L: lens; M: mirror.


(or sample assemblage) is shaped such that it has flat faces in the direction of the incident and scattered light. In case of irregularly shaped (small) samples, one solution is to immerse the sample in a fluid of matching refractive index (with a tolerance of the order of ≈ 3-5%; Weidner et al. 1975; Gleason et al. 2009) in a container with a regular shape appropriate for the scattering geometry selected in that experiment. This set-up allows one to exactly know the scattering wavevector once the coefficients of the optical indices of the sample material (at the wavelength of the light source) are known. The internal scattering angle θ is determined by applying Snell’s law to the external surfaces of the container (Vaughan and Weidner 1978).

Among the different scattering geometries, the platelet geometry (Fig. 4b) has the advantage that it does not require any knowledge of the refractive index of the sample. Due to the parallelism of the interfaces, the application of Snell’s law for both entrance and final exit surface allows one to determine the internal scattering wavevector by only using the “external” scattering angle θ* and the refractive index of the surrounding medium (in case of air nair ≅ 1). In case of an optically isotropic sample material, Equation (7) transforms into q = 2k0sin(θ*/2), where k0 is the incident light wavevector in vacuum.

The measurement of Brillouin scattering is facilitated when the scattering volume is free of large dislocations and fractures and the surfaces are optically clear. In fact, intense elastic scattering from flaws and inclusions can increase the spectrum background to the level of masking the weak Brillouin spectral features. Thus, optical-grade crystals are typically required. The minimum size of a single-crystal sample for Brillouin scattering measurements can be as small as the focal volume determined by the system optics. Samples smaller than 50 µm in diameter and as thin as 15 µm are sufficient for Brillouin scattering measurements in the most common experimental setups. The majority of geophysically relevant minerals of the Earth’s mantle are optically transparent even when they contain amounts of iron (~10 at%) in substitution of magnesium (Keppler and Smyth 2005; Keppler et al. 2007, 2008). This makes it possible to use Brillouin scattering to study their elastic properties to the relevant conditions of the deep Earth.

Measurements of surface Brillouin scattering on opaque materials and thin films

Measurements of Brillouin scattering from opaque materials have been a subject of interest starting from the 1970s (Sandercock 1972; Comins 2001). Due to the sample’s optical opacity, the scattering process is limited to a very shallow region near the surface. For this reason, the technique is generally referred as surface Brillouin scattering (e.g., Comins et al. 2000). Surface Brillouin scattering is in general not observable in transparent materials, where bulk effects dominate the scattered intensity. Brillouin scattering is a viable method to investigate acoustic wave velocities in opaque solids including metals (e.g., Mendik et al. 1992; Zhang et al. 2001; Graczykowski et al. 2011) and metallic liquids (Visser et al. 1984) and of metals at high pressures (Crowhurst et al. 1999) and it represents a potential method to study the elastic properties of candidate planetary core materials at extreme conditions.

The dominant scattering mechanism depends on the opacity of the material. In an isotropic solid with complex refractive index n = η + iκ and κ < η (semi-opaque material), elasto-optic coupling will be the dominant process. In case of materials with κ ≈ η, due to its small penetration, light will be mostly scattered by surface ripples that are caused by phonons and that propagate in thermal equilibrium. This second effect does not involve coupling of light with phonon-induced modulations of the dielectric constant of the sample material. The coexistence of the two mechanisms in the same system produces additional spectral features due to degeneracies of surface excitations with bulk ones. Additional spectral features can arise from scattering by interfacial waves between a thin opaque layer and a substrate. These occur when the thickness of the layer is comparable or smaller than the wavelength of the scattering acoustic mode (Sandercock 1982; Every 2002).


In the simple case of an opaque solid in backscattering geometry (θ = 180°) with incident (and scattered) wavevectors oriented perpendicular to the sample’s surface (Fig. 4c), the scatter-ing volume is reduced to a depth of the same order of magnitude as the wavelength of the vibra-tions involved in the process as a result of the strong light absorption. This relaxes the wavevec-tor conservation requirement expressed by Equation (5). The interaction of the incident light with the phonons is not limited to the frequency Ω but extends to a finite range around the value determined by Equation (6). This causes a broadening of the Brillouin peak (Sandercock 1982).

Experimental results (e.g., Sandercock 1972) show that the Brillouin peaks from backscat-tering measurements performed on opaque solids show asymmetry and a shift in frequency, with respect to the calculated scattering phonon frequency Ω. These can be determined by a thorough analysis including the effect of coherent reflection of the phonon at the surface (Der-visch and Loudon 1976). The calculated Brillouin lineshape is then asymmetric because q and –q contribute to both Stokes and anti-Stokes processes while this is not the case in transparent materials (each contributes only to one of the two processes). In addition to the asymmetric broadening, the maximum of the calculated lineshape is shifted in frequency from Ω = 2ηk0ν to Ω = 2(η2 + κ2)1/2k0ν, where η and κ are the real and imaginary part of the complex refractive index of the scattering solid n = η + iκ, k0 is the wavevector of the incident light in vacuum and ν is the acoustic velocity of the opaque solid (Sandercock 1982; Nizzoli and Sandercock 1990).

In the more general case in which Brillouin scattering experiments are performed in ge-ometries with oblique incidence on the sample surface (Fig. 4d), the scattering process involves excitations whose wavevectors have non-zero component on the surface plane. In the simple case of backscattering at an oblique incidence angle φ (defined as the angle of the incident and scattered wavevectors with respect to the normal to the surface), the relationship between the wavevector of the scattering surface mode and those of the incident and scattered light will be

|| 02 sin (30)iq k= φ

where q

is the surface wavevector, k0i is the incident wavevector (assumed equal to the scat-tered wavevector k0s; k0i ≅ k0s). The additional index 0 has been added here to specify that the wavevectors are considered outside the sample and there is not any dependence on the refractive index of the sample material (this is not the case for Eqn. (7) used for bulk phonons).

In the isotropic case, the general features of the surface modes involved in Brillouin scattering from opaque materials, thin-films and multilayered materials can be identified by comparison with the simple case of transverse resonance in an isotropic plate of thickness H (see Auld 1973 for the details). For the case of shear waves with polarization on the surface of the plate (SH), the solution that fulfils the required boundary conditions and symmetry of the plate forms a continuum with a lower cutoff at ω = πνt/H (where νt is the bulk shear velocity). The SH waves in the “free” plate correspond to a limiting case of the Love waves (Love 1911), which are defined for thin plates on a semi-infinite substrate. For shear waves with a general polarization, a longitudinal (L) in addition to a shear vertical (SV) component can be produced by reflection at the plate boundaries. Four plane wave solutions satisfy the boundary conditions, and the frequency spectrum will be characterized by three regimes: (a) ω/q < νt (where q is the plate normal mode wavevector and νt is the shear velocity of the bulk sample) in which all the transverse and longitudinal modes are dissipative (their q are imaginary) and localized at the surface and are related to the Rayleigh mode of a semi-infinite solid; (b) νt < ω/q < νl (where νl is the bulk longitudinal velocity of the plate) where the solutions are combinations of transverse and longitudinal modes and they are called Lamb waves; (c) ω/q > νl where all the solutions are real and they are bulk normal modes. In the case of a semi-infinite medium (corresponding to the limit Hq → ∞) the low order solutions (a) give rise to the Rayleigh surface mode and the SH and the Lamb wave solutions form continua that are indistinguishable from bulk shear modes. Only the Lamb waves have a longitudinal component localized at the surface. In case of


anisotropic extended opaque media, surface Brillouin scattering spectra present peaks arising from both pure surface transverse modes (surface acoustic waves, SAW) or their combinations with transverse bulk normal modes of the material (pseudo-SAW; e.g., Every 2002).

The most important features detected by surface Brillouin scattering from anisotropic thin films supported by thick substrates and multilayered composites can be summarized in two basic cases depending on the relative magnitudes of the acoustic velocity of the film and the substrate. (a) In the case of a supported thin layer of lower acoustic velocity than the semi-infi-nite substrate in the low frequency regime (ω/q < νt

layer) it is possible to observe a SAW whose velocity approaches (at large thin-film thickness) the bulk Rayleigh wave velocity; Lamb waves and Sezawa waves (localized waves supported by the film layer whose displacement field de-cays towards the substrate) are observed in the frequency regime νs

layer < ω/q < νtsubstrate. Finally,

at frequencies such that ω/q > νtsubstrate, combinations of Sezawa waves with bulk modes of the

substrate can be observed. (b) In case of a supported thin layer with higher acoustic velocity than the substrate at very low layer thickness, the Rayleigh wave of the substrate dominates the spectrum. It transforms into a SAW reaching the bulk shear wave threshold of the substrate at increasing thickness. At still increasing thickness, the SAW of the layer becomes detectable and it merges the Rayleigh wave velocity at the limit at which the thin layer becomes effectively a semi-infinite layer. In the case of thick semi-transparent layers with appropriate matching elas-tic and elasto-optic properties of both layer and substrate, localized interfacial waves (Stoneley waves) can be also observed (Every 2002; Li Bassi et al. 2004). It is noteworthy to remember that Brillouin scattering from SH waves can take place only by the elasto-optic coupling mecha-nism because they do not contribute to surface ripples (for a theoretical analysis of the associ-ated spectral intensity see Albuquerque et al. 1980). The detection of Brillouin scattering from SH waves is limited to semi-transparent materials (e.g., Albuquerque et al. 1980; Zhang et al. 2000). In order to detect Brillouin scattering from SH waves in metals, it is necessary to take advantage of the penetration through the interface between metal films and transparent sub-strates of the displacement field associated with SH waves of the substrate (Bell et al. 1987b).

A theoretical interpretation of surface Brillouin scattering spectral intensity has been the subject of several studies from different groups of researchers. Two different approaches have been used: (a) the elastodynamic Green’s function approach and (b) the direct calculation of the surface scattered electromagnetic field intensity.

(a) As an example of the first approach, for a layer supported by a substrate (Every 2002), the appropriate surface elastodynamic Green’s function (in the frequency domain) is G3,3(k; x = 0; ω), which describes the (surface-)normal component of displacement of the free surface (x = 0) of a supported layer or a semi-infinite solid in response to periodic forces acting on the same surface and having spatial frequency k

and time frequency ω. The imaginary part of the function, in the limit of ω << kBT/ (where kB is Boltzmann’s constant and = h/2π is the reduced Plank’s constant), can be related to the surface displacement power spectrum that is proportional to the scattering efficiency and it gives information about the relevant surface modes that contribute to the measured Brillouin signal (Zhang et al. 1998b; Beghi et al. 2004):

( ) ( ) 3,3 ||Im ; 0; (31)T

I D G k xω = = ωω

where I(ω) is the scattering efficiency and D is a factor depending on the properties of the sample, the scattering geometry, and the frequency and polarization of the incident light.

The Green’s function appropriate for the system under study is calculated based on the density and elastic constants of the material (or materials in case of supported thin films) of interest. If the model includes the surface modified bulk fluctuations of the dielectric constant of the material (elasto-optic contribution to the scattering) the calculated scattering efficiency will include terms describing the elasto-optic coupling (Pockel’s coefficients) of the scattering


materials (Loudon 1978). The prediction of the frequencies (and amplitudes) of the spectral features allows one to put constraints on the elastic constants and the Pockel’s coefficients of the scattering medium by least-square fits of dispersion curves of the surface waves, which are obtained as a function of the momentum transfer by changing the incidence angle φ (see Eqn. 30) or as a function of orientation by rotating the sample around the axis normal to the scattering surface. Single-crystal elastic constants of metals and semiconductors have been successfully determined with this approach (e.g., Mendik et al. 1992; Tlali et al. 2004). The comparison of this approach with the direct inversion of closed-form expressions of the solutions of the secular equation for surface waves in special directions (Every et al. 2001) and a recent Monte Carlo method (Wittkowski et al. 2004) has produced consistent results in case of tungsten carbide films (Wittkowski et al. 2006). A thorough analysis of the stability of the solutions and of the quality of the elastic constants retrieved from surface acoustic wave velocities is presented by Sklar et al. (1995).

(b) In the second approach, the electromagnetic scattered field in the bulk system is calculated and it is specialized for the case of surface scattering. The scattering cross-section is then calculated from the Poynting vector of the scattered field. This approach allows a more direct identification of the modes that contribute to the scattering cross-section (see discussion in Marvin et al. 1980a). In this approach, as well as in the one based on the elastodynamic Green’s function, the inclusion of both ripple and elasto-optic contributions allows one to interpret the experimental spectra and the dispersion curves both in terms of elastic constants and of photo-elastic coefficients of the investigated materials.

Several studies develop the theory for scattering in isotropic and anisotropic semi-infinite media and in supported or unsupported plates (Rowell and Stegeman 1978; Bortolani et al. 1983). The comparison with available experimental results confirms the prevalence of the ripple scattering mechanism in the highly opaque systems (Zhang et al. 1998a). Experimental studies performed with semiconductor layers supported by metals, have confirmed the importance of adding the elasto-optic contribution to fully interpret the observed spectral features (Sandercock 1978; Loudon and Sandercock 1980).

Finally, theoretical studies have been carried out to predict the intensity of surface Brillouin scattering in dielectric and metallic liquids (Dil and Brody 1976; Albuquerque 1983) and the results have been utilized to interpret existing experimental results on alcohols, mercury and gallium (Dil and Brody 1976; Visser et al. 1984).

The limited penetration of light in strongly opaque materials strongly limits Brillouin scat-tering as a technique to determine the elements of their elastic tensor. The use of surface Bril-louin scattering from surface ripples involves additional experimental problems. The quality of the flatness and polishing of the sample surface is extremely strict. In addition, a critical aspect of surface Brillouin scattering experimental design is precisely determining the scattering ge-ometry, especially in experiments performed in “tilted” backscattering geometry. This is due to the limited constraint on the scattering wavevector (see discussion above about the coherent reflection at the surface). In order to map the dispersion curve of the surface modes, the numeri-cal aperture of the collecting optical system has to be minimized, compatibly with the need for a sufficient light throughput. The effect of a finite numerical aperture on the collected spectra is that of integrating both across wavevector and orientation space. This broadens and shifts the positions of the spectral features as a function of the experimental geometry and of the proper-ties of the material. Decreasing the numerical aperture of the collecting lens reduces the width of the solid angle, δΩ, about the scattering wavevector and increases the resolution of the wave-vector. A careful quantitative evaluation of the effect of adding spatial filtering and slits to the basic experimental design on the accuracy of the measurements and of the recovered properties of the material is presented in Gigault and Dutcher (1998) and in Stoddart et al. (1998).


Brillouin scattering at ambient or near-ambient conditions

Most Brillouin scattering studies are performed at ambient conditions and used to determine the acoustic velocities and the full elastic tensors of single crystals and polycrystalline samples. Compared with other methods for elasticity determination, Brillouin spectroscopy is especially useful for small samples (even down to a few tens of microns in size) (Sinogeikin et al. 2004b). Examples include rare natural samples that do not grow to a large size, crystals synthesized at very high pressures, and technological materials with heterogeneities at very small scales (Beghi et al. 2002a). The development of the theoretical interpretation of Brillouin scattering from surfaces and interfaces as discussed in the preceeding subsection has opened a whole range of applications to the study of the behavior of thin films and nano-layered materials (Nizzoli and Sandercock 1990; Comins et al. 2000).

Brillouin spectroscopy is also useful as a non-contact method to investigate the elastic properties of delicate organic and biomaterials that cannot sustain large mechanical loads (Tao et al. 1988; Speziale et al. 2003). It is a method to investigate viscoelastic behavior of materials by analyzing the peaks’ broadening associated with acoustic attenuation in the GHz frequency regime in polymers and other soft materials as a function of temperature (up to few hundred degrees K; Adshead and Lindsay 1982; Takagi et al. 2007). Finally, the analysis of the intensity of Brillouin scattering spectra is a method to obtain quantitative information on the photo-elastic tensor of materials (e.g., Nelson et al. 1972; Vacher and Boyer 1972; Grimsditch and Ramdas 1976; Wallnöfer et al. 1994; Mielcarek et al. 2008).

The potential of Brillouin scattering in Geosciences was recognized by Anderson et al. (1969), the first experimental realization was due to Weidner et al. (1975) and nowadays this technique is one of the few methods of choice to investigate the elastic properties of the minerals of the Earth mantle. In the following parts of this section, we briefly summarize the most relevant (in our opinion) results and technical advancements in the different fields of application of Brillouin scattering at ambient conditions.

Condensed matter physics. The interest in the dynamics of molecular systems such as noble gases and simple fluids and solids has driven early Brillouin scattering research about acoustic wave velocity and acoustic attenuation in the GHz frequency range as a function of temperature down to few K (Palin et al. 1971). The measurements at extremely low temperatures required cryogenic vessels optimized for Brillouin scattering (Pike et al. 1970; Grigoriantz and Clouter 1998), usually performed in 90° scattering geometry (Fig. 4a). The development of multi-pass Fabry-Perot spectrometers has improved the quality of the data especially regarding the central features in the depolarized spectra in fluids and the generally weak shear modes of soft solids (Patterson 1976; Vacher and Pelous 1976; Gammon et al. 1983).

Brillouin scattering is a well-established technique for the study of the mechanical proper-ties of polymers as a function of composition or temperature at ambient pressure (Patterson 1983; Krüger 1989). Brillouin spectra of polymers and other viscoelastic organic materials allow researchers to obtain information about (a) acoustic velocities and elastic moduli (Krüger et al. 1986), and (b) acoustic attenuation in the GHz frequency regime from the analysis of the peak linewidths (Levelut et al. 1996). Additional bulk properties can be inferred by the analysis of the peak-shape of the central (unshifted) features of the spectrum, the Rayleigh peak (related to thermal diffusivity) and the Mountain peak, an additional broad central feature that is re-lated to viscous relaxation strengths of intramolecular vibrational degrees of freedom (Patterson 1983; Fioretto et al. 1999).

Brillouin scattering studies of liquid-like polymers as temperature decreases across the melting temperature and the glass transition show that: (a) the Brillouin linewidth increases in the high-temperature regime in which the dissipative processes have a timescale that is shorter than the inverse Brillouin frequency, (b) the linewidth decreases in the lower temperature re-


gime where the relaxation timescales are longer compared to the inverse Brillouin frequency and the measured acoustic velocity increases above the zero-frequency (thermodynamic) value corresponding to the relaxed elastic modulus (e.g., Patterson 1983). The quantitative analysis of the temperature dependence of Brillouin linewidth in terms of fluctuations relaxation time and longitudinal viscosity is generally based on the hydrodynamic approximation (Mountain 1966a). Tests of different functions to describe the temperature dependent Brillouin results are presented by Brodin et al. (2002).

Important additional information on the dynamics of the molecules in fluid polymers comes from Brillouin spectra measured by selecting the incident light polarization and the scattered light polarization such that they are perpendicular to each other (depolarized spectra; e.g., Dil 1982). Depolarized spectra present an additional low-frequency doublet, not visible in the polarized spectra (i.e., incident and scattered light polarizations are parallel), that is related to coupling between rotations and translations of the molecules in viscous molecular systems (Wang 1980). With decreasing temperature, these additional features progressively narrow down and eventually collapse in an unresolved single line as a consequence of the viscoelastic behavior of the material, up to the appearance of two transverse acoustic modes (Brillouin peaks) at even lower temperature in the solid state (Patterson 1976). New macroscopic and microscopic theoretical analyses of scattering from coupling of molecular rotation-translation dynamics of anisotropic molecular fluids have been recently tested by new experimental observations (Zhang et al. 2004). Recently, Brillouin scattering has been used to monitor the time-progress of the photo-induced polymerization of binary mixtures of monomer organic compounds as a function of their composition and of temperature (Ziobrowski et al. 2011).

Brillouin spectroscopy is being traditionally applied to the study of acoustic dissipation by analyzing the peak linewidth and important information has been obtained from the central (unshifted) peaks. This type of analysis, initially restricted to fluids (O’Connor and Schlupf 1967; Lucas et al. 1970; Zhao and Vanderwal 1997), has since then become very important also in the field of glass-forming materials (Loheider et al. 1990; Monaco et al. 1998; Rat et al. 2005) and of crystalline solids near phase transitions (Mroz et al. 1989; Jiang and Kojima 2000). Indeed, Brillouin scattering at ambient conditions or at low to moderate temperatures (100-300 K) has proved to be a method of choice in the study of ferroelectric phase transitions where the transition is associated with a soft acoustic mode and with additional mode coupling with central features related to a frequency-dependent shear viscosity (e.g., Ahart et al. 2007, 2009, 2011; Ohta et al. 2011; Marquardt et al. 2013b). Detailed studies of the quasi-elastic (unshifted) components of the Brillouin scattering spectra observed in few solids including Si and SrTiO3 at conditions far from phase transitions, show the presence of two different components each characterized by different linewidth dependencies on frequency, temperature and wavevector. These features are interpreted either as two-phonon difference scattering or diffusive entropy fluctuations (see for instance Koreeda et al. 2006).

The experimental and theoretical developments in the analysis of surface and interface Brillouin scattering have accompanied a specific interest in the study of the ripple scattering elasto-optic coupling in bulk strongly absorbing solids (Sandercock 1972, 1982; Sathish et al. 1991; Carlotti et al. 1992a; Every 2002). Surface Brillouin scattering has been used to investigate the elastic tensor of metallic and opaque solids from polycrystals and single crystals both as bulk samples, thin films and thin interlayered composites (superlattices), especially designed to enhance the intensity of waveguide shear and longitudinal acoustic modes (Mock and Guntherodt 1984; Bhadra et al. 1989; Chirita et al. 2001; Kotane et al. 2011). Experiments on supported thin layers and superlattices have often allowed the determination of anomalies in the elastic properties of films as a function of thickness, of reaction effects at the free surfaces and of stresses at the interface with the substrate due to large structural mismatches and microtextures induced by the thin film deposition techniques (Hillebrands et al. 1985; Pang et al. 1999; Wittkowski et al. 1999, 2002; de Bernabé et al. 2001).


Brillouin light scattering from bulk and surface collective fluctuations of electronic spins (magnons) has been experimentally observed together with phonon scattering since the advent of multi-pass tandem Fabry-Perot interferometers (Sandercock and Wettling 1973). Due to the non-linear dependence of frequency on wavevector, the study of the dispersion curve of the acoustic magnons near the Brillouin zone center is of interest together with their spatial dispersion (Borovik-Romanov and Kreines 1982; Wittkowski et al. 2002). Brillouin scattering has since then been used to investigate magnetic properties of dielectric and metallic ferromagnetic, antiferromagnetic and ferrimagnetic materials. Scattering from bulk acoustic magnons dominates in transparent and semi-transparent materials (Sandercock 1974; Jantz et al. 1976), while the negligible penetration depth of light in metallic materials makes the intensities of Brillouin scattering from surface and bulk magnons comparable (Grünberg and Metawe 1977; Grimsditch et al. 1980). Both spin dynamics and the elastic constants have been simultaneously determined for semitransparent materials (e.g., FeBO3; Jantz et al. 1976). Surface Brillouin scattering of single crystal antiferromagnetic NiO has helped to elucidate the details of the magnetic structure that exhibits eight antiferromagnetic resonance modes which are interpreted by a model of four interpenetrating antiferromagnetic sublattices (Grimsditch et al. 1994a; Milano et al. 2004; Milano and Grimsditch 2010). These results confirm the richness of information that Brillouin spectroscopy can give about the magnetic properties of materials. Theoretical models of light scattering from magnons have explained the characteristic large difference of the intensity of the Stokes and anti-Stokes magnon peaks and the asymmetric peak lineshape of surface magnon peaks (see review by Borovik-Romanov and Kreines 1982).

Surface Brillouin scattering has become a technique of choice to study the magnetic structure and properties of magnons in magnetic films, multilayers and composites with nanoscale magnetic patterns. This research has impact in the fundamental understanding of the spin dynamics in nanoscale magnetic systems with potential applications to storage and signal processing technologies (Grimsditch et al. 1979, 2001; Hicken et al. 1995; Carlotti and Gubbiotti 2002; Crew et al. 2004, 2005; Perzlmaier et al. 2005).

Brillouin scattering from spin waves has recently found application to fundamental topics in quantum thermodynamics. Measurements of Brillouin scattering in an yttrium iron garnet (YIG) thin film subject to microwave pumping at ambient temperature has given the first experimental proof of the existence of Bose-Einstein condensation transition of quasi-equilibrium quasiparticles (magnons in this case) produced in a solid as an effect of external energy pumping (Demokritov et al. 2006).

Materials science. Brillouin scattering is an ideal method for systematic studies of the mechanical properties of technological materials as a function of chemical substitutions, for instance studies of single-crystal elasticity of yttrium aluminum garnets (YAG, Y3Al5O12) as a function of Yb-Y substitution (Marquardt et al. 2009a). Beghi et al. (2000) in a study of Er-Y substitution in YAG used Brillouin scattering as a function of temperature in combination with photoluminescence spectroscopy to evaluate optimal levels of doping for optical communication applications. Brillouin scattering is also used to study bulk and surface elastic properties of materials with very high hardness such as natural and polycrystalline synthetic diamond (Grimsditch and Ramdas 1976; Jiang et al. 1991; Vogelgesang et al. 1996; Zouboulis et al. 1998; Krüger et al. 2000), diamond-like amorphous and nanocrystalline carbon (Djemia et al. 2001; Beghi et al. 2002b), carbides (Kamitani et al. 1997; Zhang et al. 1998a; Wittkowski et al. 2006; Zhuralev et al. 2013) and nitrides (Grimsditch et al. 1994b; Polian et al. 1996; Zinin et al. 2002; Tkachev et al. 2003; Manghnani et al. 2005; Zhang et al. 2011).

The study of the elastic and mechanical properties of a wide range of new single and multilayered materials is facilitated by the use of Brillouin scattering of thin films (thinner than 1 µm). Micro- and nano-multilayered composites represent a wide range of materials with important technological applications whose properties are strictly linked to their structure


and thickness (e.g., Alexopoulos and O’Sullivan 1990). The knowledge of the mechanical properties of such materials is necessary for their complete characterization and helps to direct the improvement of their design. Both supported and unsupported thin layers can be measured by Brillouin scattering (Bhadra et al. 1989; Pang et al. 1999; Zhang et al. 2004). The measurements performed on thin films generally combine surface and bulk Brillouin spectral features in transparent materials (Djemia et al. 2001). The measurement of Brillouin scattering has made the determination of the full elastic tensor of films possible (Carlotti et al. 1995; Wittkowski et al. 2004). In case of opaque thin films, the measurement of the spatial and wavenumber dispersion of surface and interfacial acoustic modes as a function of film thickness makes it possible to determine subsets or the full set of elastic constants of the film especially in the case in which the supporting substrate has higher elastic stiffnesses than the film material (Sumanya et al. 2007). Brillouin scattering has found successful application in the study of the effective elastic moduli of superlattices (composite multilayers with individual layers thinner than the wavelength of light) whose elastic properties are equivalent to those of a corresponding homogeneous medium (Bell et al. 1987a; Carlotti et al. 1992b).

Brillouin scattering has been extensively used to investigate the mechanical properties of nanomaterials and nanocomposites. Both acoustic velocity and acoustic attenuation have been determined by analyzing bulk Brillouin scattering and surface scattering (Li Bassi et al. 2004). Brillouin peaks broadening and central unshifted spectral features have been investigated in heterogeneous nanocomposites and represent a tool to investigate the lengthscale of the heterogeneities (Beghi et al. 2002a). The study on composite materials based on crystalline nanoparticles dispersed in oligomeric matrices has evidenced the presence of distinct enhancement of the mechanical properties even at very low nanoparticle concentrations (Eschbach et al. 2007). The systematic study of monodispersed hard nanospheres colloids in a fluid matrix of very different refractive index has also allowed the investigation of the behavior of modes propagating across both media and of localized interfacial modes as a function of increasing nanoparticle concentration. While the frequencies of the first type of excitations increase with increasing particle fraction, those of the second type soften with increasing particle fraction (Kriegs et al. 2004).

Recently, Brillouin scattering has become the tool for testing and designing phononic materials with band gaps in the hypersonic regime (106 < ω < 1011 Hz). The presence of a band gap in the hypersonic region of the phonon dispersion curve in synthetic face-centered colloidal crystals based on ordered assemblies of polystyrene nanoparticles infiltrated by fluids was observed by Cheng et al. (2006). The extension of phononic crystals to the hypersound regime and the progress in tuning the elastic impedance mismatch between their components promises to expand the potential for technological applications of these materials as acousto-optical devices (Sato et al. 2010). Brillouin scattering measurements of acoustic velocities of heterogeneous nanoscale composite materials allows testing of the different models to describe the effective elastic properties of highly heterogeneous media. In the case of nanoporous 4H-SiC, Devaty et al. (2010) verified the correctness of the effective medium model in the case of arrays of columnar 20 nm diameter pores in quasi-hexagonal arrangement.

Biomaterials. The study of the elastic properties of delicate macromolecular biomaterials supplements the structural characterization of their molecular architecture. It gives important information about the strength of the intermolecular interactions in close-packing molecular arrangements that can be compared with the arrangement of the same molecules in their natural fluid environment. Brillouin scattering is especially useful in the case of solid anisotropic biomaterials because it is a non-contact technique that allows for a systematic study of the directional dependence of the elastic properties and their sensitivity to changes of environmental conditions (such as temperature or relative humidity). The fields of application of Brillouin scattering to biomaterials range from the study of structural materials like tendon and bone


tissues, with implications for the stability of bone implants (Cusack and Miller 1979; Sakamoto et al. 2008; Mathieu et al. 2011) to the study of lipids, important in cell-membranes (by bulk and surface scattering measurements; LePesant et al. 1978; Chen et al. 1991; Manglkammer and Krüger 2005), eye lens (Randall and Vaughan 1982), and DNA and proteins in the form of molecular solids (Lee et al. 1987, 1993; Speziale et al. 2003). Brillouin scattering studies of the temperature effect on the elastic properties of tetragonal lysozyme have shown the presence of an unknown transition (e.g., Sakamoto et al. 2008; Svanidze et al. 2009). New time dependent measurements of dehydration of lysozyme crystals of tetragonal and monoclinic symmetry have been investigated by Hashimoto et al. (2008). Finally, Brillouin scattering has been used to probe the thermally induced denaturation of molecular lysozyme in its natural state in aqueous solutions (Svanidze et al. 2011). In the case of bone tissues and solid protein crystals, the comparison of Brillouin scattering in the GHz frequency regime with MHz ultrasonics and static bending measurements allows us to better understand the viscoelastic behavior of these materials (Speziale et al. 2003; Gautieri et al. 2012).

Determination of Pockel’s coefficients

In addition to elastic and magnetic properties and attenuation, Brillouin scattering allows determination of the photoelastic constants of materials (see the section “Physical Principles of the Brillouin Effect”). When the density, elastic constants and refractive index of the sample material are known, measurements of intensity performed in selected high-symmetry crystallographic directions while varying the incident and scattered light polarization can be used to determine the photo-elastic tensor. A substantial advancement in the interpretation of Brillouin peak intensities in anisotropic materials was the theoretical work by Nelson and Lax (1971) who first postulated and demonstrated the existence of non-zero antisymmetric elements in the elasto-optic tensor due to existence of coupling of rotations (instead of strains) with the light scattering process. Nelson et al. (1972) developed a generalized formula to analyze the intensity of Brillouin scattering in terms of the photo-elastic coupling constants valid for all the crystal symmetries based on a new Green’s radiation function in anisotropic media. Cummins and Schoen (1972) and Vacher and Boyer (1972) presented a large number of closed-form expressions to detetermine the values of the elements of the tensors Tα (see Eqn. 26) for a large number of crystallographic directions in the different symmetry classes.

While in transparent media, the photoelastic constants have been determined for many materials (crystalline and amorphous solids) from bulk phonons (Grimsditch and Ramdas 1976; Heiman et al. 1979; Schroeder 1980; Rand and Stoicheff 1982; Trzaskowska et al. 2010) constraining the full tensor in opaque materials is a much more difficult task due to the weakness of the Brillouin scattering by elasto-optic coupling and the larger relative intensity of ripple scattering (which is insensitive to the elasto-optic coupling). Subsets of the photoelastic tensor can still be constrained in semi-opaque materials by investigating the dispersion of surface waves and by fitting the spectral intensities (when the elastic constants are known) with the use of the elasto-optic constants as free parameters (Marvin et al. 1980b; Bassoli et al. 1986). This procedure is aided by the coexistence in the spectra of contributions from both the elasto-optic and the ripple scattering mechanism.

The determination of the photoelastic constants is restricted to measurements performed at ambient pressure because the measurements in the diamond anvil cell suffer from the effect of partial depolarization of light in the diamonds culets due to the large stress gradient to which they are subject when the cell is in operation.

Brillouin scattering at extreme conditions

There has been much interest in understanding the effects of pressure and temperature on the elastic properties of solids. As a non-contact probe, Brillouin scattering is ideally suited for studies at non-ambient conditions where the sample material is surrounded by a heater or con-


tained in a pressurizing apparatus. In addition, Brillouin scattering of single crystals determines the elastic anisotropy at extreme conditions. This makes it an invaluable source of information for interpreting seismological models of the very deep Earth and for geophysical modeling. Due to their transparency, diamond anvil cells are the optimal high-pressure device to perform Brillouin scattering at extreme conditions. The geometry of Brillouin scattering in a diamond anvil cell is drawn in Figure 8.

Brillouin scattering at high temperature. Use of Brillouin scattering at high temperatures introduces some experimental challenges. The intensity of Brillouin scattering is linearly proportional to temperature while the thermal background radiation (blackbody radiation) from the sample increases with temperature to the fourth power. As a result, at very high temperatures, the Brillouin peaks may be masked by blackbody radiation.

However, Brillouin measurements (using a green or blue light source) of both melts and solid oxides have been reported at temperatures as high as ~2300 K using resistive heating (Zouboulis and Grimsditch 1991b; Askarpour et al. 1993; Zaug et al. 1993; Vo-Thanh et al. 1996; Polian et al. 2002) and as high as ~2500 K (Sinogeikin et al. 2004a, 2005) using laser heating. However, there are only few very high-temperature Brillouin data and the (material-dependent) temperature limits for Brillouin scattering experiments in the diamond anvil cell are still to be explored.

Metal wires or foils (e.g., tungsten, platinum, rhodium) surrounding the sample are most commonly used as resistive heating element (e.g., Zouboulis and Grimsditch 1991a,b; Xu and Manghnani 1992; Askarpour et al. 1993; Krüger et al. 1998; Ko et al. 2008). Alternatively, a thin graphite foil can be employed (e.g., Bucaro and Dardy 1974). The sample is contained in an inert gas atmosphere to avoid chemical reactions. In order to minimize possible temperature gradients, the measurements are sometimes performed placing the samples in ovens with small surface windows for optic access (preferentially) in backscattering and 90° scattering geometry (Harley et al. 1978; Mjwara et al. 1991). The temperature at the sample position is usually measured with a thermocouple. Additional thermocouples whose junctions are positioned at different distance from the heating element, can detect the presence of temperature gradients.

Figure 8. Schematic drawing of the geometry for Brillouin scattering experiments at high pressure in the diamond anvil cell. These experiments are performed in forward symmetric scattering geometry.


The advent of platelet forward symmetric scattering geometry has aggravated the issue of temperature gradients. Due to the large scattering angles used in these experiments, furnaces with large optic access are required for these experiments (Xu and Manghnani 1992; Sinogeikin et al. 2000a). In case of resistive heating at high pressure, the space constraints imposed by the diamond anvil cell are very strict, and special miniature heaters are designed to fit in the small space available in the high-pressure cell (Fig. 9).

In general, continuous wave near infrared (IR) lasers are used as energy source in laser heating techniques. A simplified sketch of the geometry of double-side laser heating in a diamond anvil cell is shown in Figure 10. The two most popular choices are CO2 laser, with a wavelength of 10.6 µm, or solid-state lasers (e.g., YAG, YLF, fibre laser), with wavelength ~1 µm. CO2 laser radiation is strongly absorbed by the majority of optically transparent samples. It has

Figure 9. Schematic drawing of a high-temperature resistive-heated diamond anvil cell. On the left: the detail of the micro-heater, positioned inside the diamond cell piston, surrounding the sample chamber of a BX-90 DAC (Kantor et al. 2012). On the right: a custom designed water-cooling jacket for the BX-90 DAC. The cooling jacket has lateral windows that are optimized to combine Brillouin scattering and X-ray diffrac-tion in radial geometry (Drawing by R. Schultz, GFZ).

Figure 10. Schematic diagram of the geometry of laser heating of a sample in the DAC. The sample is heated from both sides. The sample has to be thermally insulated from the diamond anvils. The insulator can also act as a pressure transmitting medium.


been shown that by using CO2 lasers it is possible to perform Brillouin scattering experiments of oxides to temperatures in excess of 2000 K (Sinogeikin et al. 2004a, 2005). One important limitation in the use of CO2 laser heating is its poor laser stability compared with solid-state IR lasers. In addition, the 10.6 µm radiation is absorbed by optic glasses and the setup of the whole laser path requires the use of reflective optics.

Shorter wavelength solid-state lasers, however, are easier to handle and the recent develop-ments in fiber lasers makes them a very versatile and a promising tool for laser heating experi-ments (e.g., Dubrovinsky et al. 2009; Shen et al. 2010). However, ~1 µm wavelength lasers are weakly (or not) absorbed by optically transparent materials. This represents a serious limitation for their use in Brillouin scattering measurements in forward scattering geometry, which require sample materials that are transparent to the Brillouin probing laser and is the one that offers the most complete access to the full single-crystal elastic tensor. Customized sample preparation techniques are required to overcome this challenge (e.g., Marquardt and Marquardt 2012).

In laser heating experiments, temperature is usually determined by fitting the measured thermal radiation spectrum of the sample to a greybody curve. Here, the intensity of thermal emission depends on the surface conditions of the heated material and it has been documented that well-polished transparent plates, such as single-crystal samples used for Brillouin spectroscopy, show too low emissivity (e.g., Gardon 1956) for temperature to be determined at moderate temperatures (Sinogeikin et al. 2004a, 2005). In these cases, alternative ways of measuring temperature have to be used, such as measuring the intensity ratio of Stokes and anti-Stokes Raman peaks from Raman scattering. Thermal gradients are also a serious general problem in the laser-heating approach at both ambient pressure (e.g., Rantala and Levoska 1989; Shen and Zhang 2001) and in the diamond-anvil cell (Panero and Jeanloz 2001; Kiefer and Duffy 2005; Kavner and Nugent 2008).

Brillouin scattering at high pressure. Brillouin spectroscopy at high pressure has opened new opportunities to investigate anisotropic single-crystal elastic properties of materials espe-cially for geophysical applications. Measurements were conducted initially in 90° scattering geometry in gas pressure vessels adapted for Brillouin spectroscopy (e.g., Asenbaum et al. 1986). A major advance was the introduction of the diamond anvil cell in combination with the symmetric forward scattering geometry (see Fig. 4b) (Whitfield et al. 1976). However, the requirement of parallelism of the surfaces of a small sample makes it even more difficult to pre-pare ideal samples for this type of study. Deviations from parallelism of the interfaces in typical high-pressure Brillouin scattering experiments produce errors in the frequency shift measure-ments. The quantitative effect on the determination of acoustic velocity and elastic constants is analyzed by Zha et al. (1996), Sinogeikin and Bass (2000) and Zhuralev et al. (2013).

Starting from the 1980s, high-pressure Brillouin spectroscopy has been used both in the study of minerals (see the “Applications of Brillouin Spectroscopy in Geosciences” section) and for the study of the elastic properties of solid noble gases (Grimsditch et al. 1986; Polian and Grimsditch 1986; Shimizu et al. 2001; Zha et al. 2004), hydrogen (Zha et al. 1993; Duffy et al. 1994) and other simple molecular solids (Kiefte et al. 1985; Lee et al. 1986; Gagnon et al. 1990; Shimizu et al. 1994, 1995, 1996; Chen et al. 2010; Ahart et al. 2011). The determination of the full single-crystal tensor of high-pressure phases synthesized in the diamond anvil cell by compressing a fluid was achieved by combining Brillouin scattering measurements with X-ray diffraction that determine both density and the orientation of the plate-like single crystal (or few crystals) produced in the sample chamber (Shimizu and Sasaki 1992). In fact, for high symmetry solids (mostly cubic and hexagonal), it is possible to determine the full elastic tensor by measuring the spatial dispersion of the acoustic phonon branches in one or two single-crystal platelets (with different orientations) by forward symmetric Brillouin scattering (Shimizu and Sasaki 1992). High-pressure low-temperature Brillouin scattering measurements have been performed on H2S (Murase et al. 2002).


Recently, Stevens et al. (2007) conducted a high-pressure study of different polymeric elastomers at high pressures in the diamond anvil cell. The measurement of the shear modulus at high pressure adds important new information about the properties of systems for which it is often difficult to detect the propagation of shear acoustic waves at ambient and low pressure. In addition, Brillouin scattering of polymers at high pressures under nonhydrostatic conditions can furnish new information on the effect of non-isotropic stress on the hypersonic anisotropy of polymers. Thus, the results already available from Brillouin scattering studies of anisotropy in polymer films subject to tensile stress at ambient pressure are extended to high-pressure regimes (Yoshida et al. 2001).

Over the past decades, several important contributions to aspects of geophysics, materials science and condensed matter physics have been made by high-pressure Brillouin scattering experiments (e.g., Shimizu and Sasaki 1992; Duffy et al. 1995; Marquardt et al. 2009c) and the achievable pressure limits have been subsequently pushed forward (e.g., Polian 2003). Cur-rently, the highest pressure reported is 81 GPa for single-crystal measurements (Marquardt et al. 2009b) and 172 GPa and 207 GPa for polycrystalline materials and glasses, respectively (Murakami et al. 2007a; Murakami and Bass 2010).

The combination of Brillouin scattering and X-ray diffraction for density determination makes it possible to perform pressure-volume equation of state measurements independent of a secondary pressure scale. In fact, by the definition of the isothermal bulk modulus KT = (∂P/∂lnρ)T (where P, T, ρ are pressure, temperature and density, respectively) it is possible to determine pressure by combining volume measurements by X-ray diffraction and simultaneous determination of KS = (∂P/∂lnρ)S = KT(1 + αγT) (where S, α and γ are entropy, volume thermal expansion coefficient and Grüneisen parameter, respectively) by Brillouin scattering measure-ments, if γ and α are known (Zha et al. 2000; Zhuralev et al. 2013).

The success of Brillouin scattering to determine elastic properties of materials at elevated pressure comes together with a continual improvement of this experimental technique to tackle many challenges. In order to maximize the optical throughput, sample containers with very wide optic access and collecting lenses of large numerical aperture are often used. Additional slits or apertures are added in order to reduce the amount of parasitic diffuse scattered light from parts of the sample container close to the sample. However, if not perfectly aligned, the apertures may asymmetrically “clip” the solid angle defined by the collecting part of the optical system and effectively change the orientation of the scattering wavevector, causing an artificial shift of the position of Brillouin feature for all the observed acoustic modes. The effect of this vignetting is not easily detected unless by simultaneously measuring standard materials of known acoustic velocity (Oliver et al. 1992; Sinogeikin and Bass 2000).

One of the biggest technical challenges of Brillouin scattering at high pressure is the study of phases synthesized at high pressure in the diamond anvil cell, which are unquenchable to am-bient conditions. These materials are generally polycrystalline and Brillouin scattering can be used to infer the pressure dependence of aggregate (average) elastic moduli of the polycrystals (e.g., Murakami et al. 2007a,b, 2009b, 2012; Kudo et al. 2012). The interpretation of the mea-sured high-pressure average acoustic velocity in some systems is still a subject of investigation (Speziale et al. 2013). Indeed, Brillouin scattering of polycrystalline or composite materials at ambient conditions and moderate high pressures has sometimes shown additional spectral features that have been the subject of intense research (Hernandez et al. 1996; Ahart et al. 2006; Gleason et al. 2009). The persistence of regions of finer material at the interfaces between larger grains or in the original porous spaces can give rise to additional spectral features from interfacial modes, and the “bulk modes” present evidence of dispersion both in frequency and in the volume fraction of the grains of the dispersed phase (Kriegs et al. 2004). The interpreta-tion of similar spectral features that could be detected in high-pressure measurements in the DAC is relevant to assess the reliability of measurements on materials that cannot be analyzed as single-crystals.


Important results have been recently obtained from the studies of nano-sized materials by Brillouin scattering at high pressures. However, the results from some nanomaterials differ from those obtained from their equivalent bulk counterparts (Gleason et al. 2011; Marquardt et al. 2011). The interpretation of these differences is still a matter of debate, and the possibility that grain-size reduction in the DAC can produce nano-sized powders requires special attention. The goal is to determine the pressure/grain-size effect (in addition to stress heterogeneities) on the effective elastic properties of polycrystalline samples of the candidate phases of the deep Earth.

Brillouin scattering at simultaneous high pressures and temperatures. Until now few Brillouin scattering measurements have been reported at combined high P-T-conditions (Matsuishi et al. 2002, 2003; Li et al. 2006; Sinogeikin et al. 2006; Goncharov et al. 2007; Asahara et al. 2010; Mao et al. 2012; Murakami et al. 2012; Lu et al. 2013).

In principle, both resistive heating and laser heating can be employed in the diamond-anvil cell to heat the sample at high pressure. However, due to the limited temperature-stability of diamond, the achievable temperature range is limited to less than 1000 K for resistive heating unless the exposed diamond surfaces are flushed with a reducing gas mixture or the whole cell is immersed in a reducing atmosphere in order to prevent diamond surface oxidation (Liermann et al. 2009).

The first results have emerged from Brillouin scattering at simultaneously high pressure (up to 65 GPa) and high-temperature (up to 900 K) using a resistive-heated diamond cell (Sinogeikin et al. 2006, 2007; Mao et al. 2012; Lu et al. 2013; Marquardt et al. 2013a). The use of a graphite heater may allow one to extend the accessible temperature range (Liermann et al. 2009). Li et al. (2006) used a YLF heating laser to study “hot ice” in the diamond-anvil cell up to ~22 GPa and ~1200 K. To couple the short wavelength laser (1064 nm) to the transparent sample, a gold film was deposited inside the sample chamber. Murakami et al. (2009a) reported first Brillouin scattering results on the sound wave velocities of polycrystalline MgO compresed to 49 GPa in a DAC and simultaneously laser-heated to ~2300 K using a CO2-laser. In a more recent study both MgO and MgSiO3 perovskite have been investigated at simultaneous high pressures and temperatures to a maximum of 91 GPa and 2700 K (Murakami et al. 2012).

High-pressure/moderately high-temperature (T ≤ 800 K) measurements of CO2, CO2-H2O mixtures and of aqueous fluids relevant for metamorphic processes in the uppermost mantle have been recently performed in externally-heated diamond anvil cells paying special attention to precise pressure determination (Giordano et al. 2006; Qin et al. 2010; Mantegazzi et al. 2012; Valenti et al. 2011; Sanchez-Valle et al. 2012).

ANALYSIS OF THE BRILLOUIN SPECTRA AND RECOVERY OF THE ELASTIC TENSOR

One of the most important applications of Brillouin scattering, and the most relevant in the Earth sciences, is determining acoustic velocities in Earth materials, and from acoustic velocities and density, constraining the full elastic tensor. The objective of this section is to introduce the general problem of extracting information about the anisotropic elastic properties of the sample material (ideally a single crystal) from the velocity determined by Brillouin scattering measurements. The relationship between the Brillouin frequency shift and acoustic velocity was given in Equation (8).

In the long wavelength limit, the acoustic phonons involved in the scattering process can be considered acoustic waves propagating in an elastic continuum. The phase velocity determined by Equation (8) is a solution of the equations of motion of the form of Equation (22). In order to describe the actual approach used to reduce the data and determine the elements of the elastic


stiffness tensor, we will briefly introduce the basic concepts of anisotropic linear elasticity. The choice of limiting this introduction to linear elasticity is related to the fact that in crystalline Earth materials linear elastic behavior is generally a valid approximation in the frequency range of the acoustic vibrations involved in Brillouin scattering. More complete treatments of anisotropic elasticity can be found in Auld (1973), Musgrave (1970) and a concise summary is given by Every (2001).

Linear elasticity of anisotropic solids

A solid subject to a distortion (strain, εij) deforms elastically when it instantaneously recovers its original form when the cause of the distortion (stress, σij) is removed. Linear elasticity is restricted to the case in which the relationship between the stress and the distortion is linear. This is expressed in three dimensions by Hooke’s law:

(33)ij ijkl klsε = σ

or inversely(34)ij ijkl klσ = ε

where sijkl and cijkl are the second-order elastic compliances and stiffnesses, respectively. The indices have values 1, 2, 3, and summation is carried out for repeated indices (Einstein’s sum-mation rule). All the quantities of Equations (34) are tensors. The infinitesimal strain is defined as the symmetric part of the displacement gradient tensor. In a Cartesian reference system (x1, x2, x3):

1(35)

2ji

ijj i

uu

x u

∂∂ε = + ∂ ∂

where u(x) is the displacement. The strain tensor is a symmetric second rank tensor with only six independent components. The diagonal elements represent changes in linear extension or compression; the off-diagonal coefficients represent the change of the angle between two directions (Fig. 11).

The (Cauchy) stress tensor σij (Fig. 12) is defined as:

( ) (36)i ij jT n n= σ

Figure 11. Graphical representation of the elements of the strain tensor in a two-dimensional representation (modified from Every 2001). (a) compressional strain; (b) shear strain. Symbols are explained in the text.


where Ti(n) is the traction vector relative to an infinitesimal surface δA with normal direction n (with component ni in the direction xi). The traction vector is defined as Ti(n) = limδA→0(δFi/δA), where δFi is the ith component of the force acting on δA. The definition of traction and stress tensor requires that σijnj = −σij(−nj) to prevent infinite tractions at the limit for δA → 0. Another important property of the stress tensor is that σij = σji to prevent that the forces applied to a vanishingly small volume of a material can produce finite torques.

From Hooke’s law (Eqn. 34) we see that the compliance and stiffness tensors are fourth rank tensors with 81 components. Due to the symme-tries of the stress and strain tensors the number of independent components reduces to 36. Compli-ance and stiffness tensors are reciprocal, cijklsklmn = δimδin (where δim and δin are Kronecker deltas). Voigt contracted notation exploits the invariance by

exchange of indices i,j and k,l where the first and second pair of indices i,j (k,l) transform into α (β) following the rules: i,j → α:11 → 1, 22 → 2, 33 → 3, 23 = 32 → 4, 13 = 31 → 5, 12 = 21 → 6. It is applied to the stress, strain, stiffness and compliance tensors. For instance: c1123 → c14. In the case of strain (εij) the coefficient 1/2 is necessary for α > 3. In the case of compliance (sijlk) the coefficient 1/2 is necessary if one of the contracted indices is larger than 3, a coefficient of 1/4 is necessary if both the contracted indices are larger than 3. The contracted notation simpli-fies the computations in many cases. The com-pliance and stiffness tensors are thus reduced to 6 × 6 matrices.

The number of independent non-zero ele-ments of the elastic tensor is also affected by crystal symmetry (Nye 1985). The elastic ten-sor of materials belonging to the same Laue symmetry classes will have the same number of independent elastic constants. Isotropic solids have only two independent constants c11 and c44 and the third non-zero constant c12 is defined as c12 = c11 − 2c44. Anisotropic solids have a num-ber of non-zero independent elastic constants that increases at the decreasing of their symme-try to a maximum of 21 in the triclinic system (Table 1).

Determining the elastic constants

We can now rewrite the wave equation for a linear elastic anisotropic solid in a Cartesian reference system:

2 2

2(37)i k

ijklj l

u uc

t x x

∂ ∂ρ =∂ ∂ ∂

Plane wave solutions of Equation (37) have the form ui = ui0 expi (qx − ωt) where ui

0 is the polarization. By substituting into (37) we obtain:

Figure 12. Sketch of the orientation and the elements of the (Cauchy) stress tensor rela-tive to the face, whose normal is parallel to x3, of a cubic element of a material.

Table 1. Number of non-zero, indepen-dent single crystal elastic constants as a function of the symmetry of a material.

Crystal System Elastic Moduli

Isotropic 2

Cubic 3

Hexagonal 5

Trigonal 6 or 7

Tetragonal 6 or 7

Orthorhombic 9

Monoclinic 13

Triclinic 21


( )2 0 0 (38)ijkl j l ik kc q q u−ρω δ =

We can substitute direction cosines n = q/|q| and phase velocity ν = ω/q in Equation (38) and we can solve for acoustic velocity by solving the secular equation:

2 0 (39)ijkl j l ikc n n v−ρ δ =

The quantity cijklnjnl is the Christoffel’s tensor Γij. Equation (39) is a cubic equation in ν2 that ad-mits three real solutions that can be expressed in closed form by using Cardan’s method (Every 1980). The polarization vector u0 associated with the three phase velocity solutions can be de-termined for each solution by substituting each solution ν in (cijklqjql − ρω2δik)µk

0 = 0 (Eqn. 38).

Starting from velocities ν (and polarizations u0) determined by Brillouin scattering mea-surements performed along a series of directions (Figs. 13, 14) with direction cosines n on a single-crystal of density ρ, the elastic constants cijkl can be determined by inverting Equations (39) using a least-square method. In case of measurements performed in general directions, the polarization of the scattering acoustic wave is not easily constrained. In the case of measure-ments inside a diamond anvil cell, the polarization is typically unknown because of depolariza-tion by the stressed diamond windows. In this case the inversion procedure needs additional steps to assign the correct polarizations to the velocities. In general, the fastest velocity is as-sociated with the quasi-longitudinal mode (that with a displacement that is the closest to the phonon propagation wavevector), the assignment of the two slower solutions is often helped by symmetry considerations, but fundamental ambiguities may remain.

Additional difficulties arise when the orientation of the single crystal is not well defined. This can be the case for high-pressure experiments where crystalline phases are synthesized in the diamond-anvil cell (Shimizu and Sasaki 1992). In situ X-ray diffraction is always the best method to pre-determine the orientation of the sample. However, in the case of high-symmetry single crystals measured in platelet geometry it is possible to simultaneously refine both the elastic constants and sample orientation. Shimizu and Sasaki (1992) have resolved the elas-tic tensor of H2S grown in the diamond anvil cell with this approach. Alternatively, Brillouin measurements can be used to refine the initial X-ray orientation of the sample, accounting for differences in optical and X-ray paths through the sample (e.g., Castagnede et al. 1992; Mao et al. 2008c). The ability to recover or refine a crystal’s orientation through Brillouin measure-ments is a function of the degree of elastic anisotropy, and so varies from material to material.

The large redundancy of phase velocity measurements (with respect to the number of independent constants to be determined) is one of the strengths of Brillouin scattering (Fig. 14), especially when different scattering geometries are used. Except for directions with degeneracies of the acoustic phonon branches or directions with vanishingly low photoelastic efficiency, it is possible to observe all three acoustic modes for each direction. In principle, it would be possible to determine the full elastic tensor of crystals of any symmetry with a relatively small number of measurements performed in general directions. However, subsets of moduli, especially in materials with low symmetry, are strongly correlated, and in addition limited datasets can strongly decrease our ability to recover the full tensor (Castagnede et al. 1992). In reality, the determination of the full tensor of triclinic minerals is limited to a handful of cases (Brown et al. 2006). A much larger number of full tensors of monoclinic crystals have been determined by Brillouin scattering at ambient pressure. Materials with symmetries higher than monoclinic do not present serious difficulties at ambient or high pressures. Once the full elastic tensor is determined, it is possible to calculate the directional dependence of the elastic response to arbitrary applied stress fields (within the elastic limits). One of the most important properties is the volume compressibility of a crystal, defined as the volume change due to an applied pressure:


(40)iiiikks

P

εβ = =

where εii is the volume compression, P is pressure defined as σkl = Pδkl (where δkl is the Kronecker delta). Compressibility is a key parameter to construct the pressure volume equation of state of a solid. The thermodynamic definition of compressibility:

ln(41)

d

dP

ρβ =

where ρ is density, allows one to determine the isothermal pressure—density equation of state if we measure compressibility as a function of pressure just by integrating Equation (41) (Shimizu et al. 1981). However, it is necessary to convert the compressibility determined by

Figure 13. A typical Brillouin spec-trum of stishovite at high pressure in the diamond anvil cell (modified from Jiang et al. 2009).

Figure 14. Experimental spatial dispersions of acoustic velocity in synthetic spinel, MgAl2O4 (Speziale, unpublished data). (a) Measure-ments at ambient conditions. (b) Measure-ments at 11 GPa of pressure. The symbols represent measured phase velocities and the curves best fit models based on the elastic ten-sor retrieved from the same data.


Brillouin scattering, which is defined under isentropic conditions, to isothermal conditions. The conversion is obtained noting that the ratio of the isothermal and isentropic compressibilities is the same as that of specific heat measured at constant pressure and at constant volume:

1 (42)T P

S V

CT

C

β= = + αγ

β

where βT is the isothermal compressibility, βS is the isentropic compressibility, CP is specific heat at constant pressure, CV is the specific heat at constant volume, α is the thermal expansion coefficient, γ is the Grüneisen parameter, and T is temperature. The correction from isentropic to isothermal conditions does not affect the value of the room-temperature compressibility by more than a few percent in oxides and silicates of relevance for the Earth’s interior except for conditions close to phase transitions. The difference is larger for soft solids and fluids, or at high temperatures.

Average elastic constants can also be calculated for aggregates of crystals once their orien-tation distribution is determined (e.g., Tomé 2000). This is necessary in order to determine the effective anisotropy of a “real” material. In geophysics, it is relevant to determine the change of seismic speed in different directions across a rock. Average elastic properties for orientation-ally isotropic aggregates of crystals of a single mineral phase or weighted mixtures of different minerals can be calculated following different approaches (Watt et al. 1976). Two moduli are of special interest in geophysics, the bulk modulus or incompressibility K, defined as the inverse of the compressibility β and the shear modulus or rigidity µ that represents the resistance to a change in shape caused by a pure shear stress. The two elastic moduli, in terms of the elastic tensor (in matrix form) of the isotropic medium are defined as:

11 12 11 1244

2(43)

3 2

c c c cK c

+ −= µ = =

These moduli are directly related to the average compressional and shear seismic velocities (νP and νS, respectively) by the relationships:

4 / 3(44)P S

Kv v

+ µ µ= =

ρ ρ

What is the information from Brillouin scattering that is relevant to Earth science?

Determining the elastic properties of the candidate materials of inner layers of the Earth is of fundamental importance in order to translate the information from seismology into a consistent model of the composition and structure of the Earth. Average bulk and shear elastic moduli of candidate materials of the deep Earth compared to the average seismic velocities from radially symmetric seismic models of the Earth, such as PREM (Dziewonski and Anderson 1981), AK135 (Kennett et al. 1995) or STW105 (Kustowski et al. 2008) are eventually converging toward a dominant unified view of the radial compositional structure of the Earth’s interior, even though uncertainties are still present about the existence of a compositional layering at the transition from the upper to the lower mantle corresponding to the 660 km discontinuity or at deeper levels in the lower mantle (see review by Anderson 2005).

The increasing amount and quality of information from deep-Earth seismology has added a new dimension to our view of the structure of the Earth’s interior. The presence of lateral heterogeneity is depicted at various scales by seismic tomography (Trampert and van der Hilst 2005). To determine the causes of the heterogeneity at transition zone level and in the D″ zone, the lowermost part of the mantle, we need more detailed information about the compositional dependence of the elastic properties of the single mineral phases that are present in the rocks at such great depths inside our planet. This requires improvement of our available techniques


to quantify the effect of pressure and temperature on single minerals, pushing their resolution to the level of detecting the fine effects of chemical exchange at a wide range of physical conditions.

Brillouin scattering is one of the few techniques that can be effectively combined with the diamond-anvil cell to be able to determine elastic properties at the pressures and temperatures of the lower mantle: from 24 GPa and ∼2000 K at its top (660-km seismic discontinuity) to 135 GPa and ∼2800 K at its base (the core-mantle boundary). In addition to seismic heterogeneity, deep Earth seismological studies detect down to the deepest mantle the signature of seismic anisotropy, both as azimuthal variation of seismic velocity, and as shear polarization anisotropy (see review by Nowacki et al. 2011). Observations of seismic anisotropy in the D″ layer are growing in number and they are convincingly indicating that anisotropy is not limited to the uppermost layers of the Earth, but it is present also in the lower mantle and in the solid inner core (Ishii and Dziewonski 2002; Garnero et al. 2004; Fouch and Rondenay 2006; Wookey and Kendall 2008).

The complete directional dependence of the elastic properties of minerals under conditions approaching those of the deep Earth is also very important for our understanding of the relationship between plastic deformation and seismic anisotropy in regions of the deep mantle of the Earth (or even in the solid inner core). In fact, the seismic anisotropy of aggregates can be determined only if the full elastic anisotropy of the mineral components is known. In this sense, techniques such as Brillouin scattering are the only ones that can offer experimental tests of the computational results based on density functional theory, like in the case of MgO at ultrahigh pressures (Karki et al. 1999; Murakami et al. 2009b).

APPLICATIONS OF BRILLOUIN SPECTROSCOPY IN GEOSCIENCES

The elastic properties of crustal and mantle minerals are of critical importance for interpreting radial and lateral seismic velocity variations and seismic anisotropy in the Earth’s interior. To properly understand seismic observations, it is necessary to understand the variation of elastic properties as a function of composition, structure, pressure, and temperature. Brillouin spectroscopy is one of the most established and reliable techniques for the study of the single crystal elastic anisotropy of minerals of the Earth’s interior, and a large number of studies have been conducted both at ambient conditions and at high pressures or temperatures.

Here, after summarizing the other main techniques that allow one to determine single crystal elasticity, we will focus particularly on the measurements performed at elevated pressure and/or temperature, and their implications for our understanding of the Earth’s interior.

Experimental techniques to determine the anisotropic elasticity of Earth materials

Brillouin scattering is one of a family of methods that are used to access the full single crystal anisotropy of Earth materials (Fig. 15, Table 2). Ultrasonic interferometry (UI) techniques from MHz to GHz frequencies have been used to determine the full elastic tensor of cubic minerals up to 8 GPa and 10 GPa respectively (Chen et al. 1998; Jacobsen and Smyth 2006). Resonant ultrasound spectroscopy (RUS) is the method of choice to determine the elastic tensor at high temperatures (up to 1800 K) at ambient pressure, with a level of precision up to one order of magnitude better than Brillouin scattering (Isaak and Ohno 2003). A recent development of RUS to higher frequencies (HFRUS) has made it possible to use this method to study very small specimens from high-pressure syntheses such as stishovite (Yoneda et al. 2012). Phonon imaging is an optical technique that has been recently applied to the diamond anvil cell to investigate the single-crystal elastic constants of Si (Decremps et al. 2010). A femtosecond laser pulse focused on one face of a plate-shaped sample is absorbed setting a local thermal stress near the surface. A spatially uniform angular distribution of acoustic wavevectors will


produce at the exit surface focused patterns related to the topology of the anisotropic group velocity surface. Fast detection of the patterns as a function of time allows one to determine group velocity spatial dispersion and recover the full elastic tensor. Momentum-resolved inelastic X-ray scattering (IXS) is a synchrotron-based technique that can be considered as an extension of Brillouin scattering to the THz frequency range. It can be used to determine acoustic velocities in selected single crystals directions up to very high pressure in the DAC by measuring energy-wavevector dispersion curves (Antonangeli et al. 2011).

Finally, another optical spectroscopy technique that furnishes information about elastic anisotropy of materials is impulsive stimulated light scattering or ISLS (Nelson and Fayer 1980). This technique is related to (spontaneous) Brillouin scattering, and is also easily combined with the DAC allowing one to determine the spatial dispersion of velocities on single crystals at high pressures and temperatures (Chai et al. 1997b; Crowhurst et al. 2008). In ISLS, excitations, such as acoustic phonons, are coherently stimulated by infrared laser pulses by means of electrostriction and optical absorption producing a set of three-dimensional time-dependent standing wave gratings. A secondary light probe is then elastically (Bragg) scattered by the gratings and the diffracted intensity is analyzed in the time domain (Yan and Nelson 1987a; Brown et al. 1989). A theoretical analysis of this method in comparison with frequency domain laser scattering techniques is presented by Yan and Nelson (1987a; 1987b). In general, both stimulated scattering and spontaneous Brillouin scattering experiments yield similar

Figure 15. Available elasticity data of important mantle materials measured at high-pressures and/or high-temperatures. The shaded area illustrates the range of expected P-T-conditions in the Earth’s transition zone and lower mantle. Open symbols: Measurements performed on polycrystalline samples; full symbols: single-crystal data. Note that many data points collected at either high-pressures or high-temperatures were measured on single-crystals (by BS, ISLS, RUS), whereas measurements at simultaneous high-pressures and high-temperatures are mostly performed on polycrystals (MHz-UI in a multi-anvil press). However, the cross denote first single-crystal data sets measured at both high-pressures and high-temperatures by Bril-louin scattering (Mao et al. 2012; Liu et al. 2013).


Tabl

e 2.

Max

imum

pre

ssur

e an

d te

mpe

ratu

re c

ondi

tions

of

elas

ticity

mea

sure

men

ts o

f m

antle

min

eral

s.

Mat

eria

lm

ax P

[G

Pa]

(m

etho

d)m

ax T

[K

] (m

etho

d)m

ax P

/T [

GP

a, K

] (m

etho

d)

Ppv

172

(pol

y-B

S) M

urak

ami e

t al.

(200

7a)

——

Pv96

(po

ly-B

S) M

urak

ami e

t al.

(200

7b)

—9,

873

(M

Hz-

UI)

Li a

nd Z

hang

(20

05),

Sine

lnik

ov e

t al.

(199

8b)

Fp81

(B

S) M

arqu

ardt

et a

l. (2

009b

),

130

(pol

y-B

S) M

urak

ami e

t al.

(200

9b)

2500

K (

BS)

Sin

ogei

kin

et a

l. (2

004a

)8,

160

0 (s

c-M

Hz-

UI)

Che

n et

al.

(199

8)

Ca-

Pv—

—10

, 107

3 (M

Hz-

UI)

Sin

elni

kov

et a

l. (1

998a

)

Rw

23 (

BS)

Wan

g et

al.

(200

6)87

3 (B

S) J

acks

on e

t al.

(200

0)18

, 167

3 (M

Hz-

UI)

Hig

o et

al.

(200

8)

Wad

14 (

BS)

Zha

et a

l. (1

997)

660

(RU

S) I

saak

et a

l. (2

007)

7, 8

73 (

MH

z-U

I) L

i et a

l. (1

998)

Mj

26 (

BS)

Mur

akam

i et a

l. (2

008)

1073

(B

S) S

inog

eiki

n an

d B

ass

(200

2b)

18, 1

673

(MH

z-U

I) I

rifu

ne e

t al.

(200

8)

Stis

h22

(B

S) J

iang

et a

l. (2

009)

, 45

(B

S) L

aksh

tano

v et

al.

(200

7)

Ol

16 (

BS)

Duf

fy e

t al.

(199

5), Z

ha e

t al.

(199

6)17

00 (

RU

S) I

saak

et a

l. (1

989)

8, 1

073

(MH

z-U

I) L

iu e

t al.

(200

5)

Px12

.5 (

ISL

S) C

hai e

t al.

(199

7b)

1073

(B

S) J

acks

on e

t al.

(200

7),

cpx

1300

(R

US)

Isa

ak e

t al.

(200

6)13

, 107

3 (M

Hz-

UI)

Kun

g et

al.

(200

5)

Grt

11 (

BS)

Jia

ng e

t al.

(200

4a,b

)20

(ISL

S) C

hai e

t al.

(199

7a),

(BS)

Liu

et a

l. (2

013)

1073

(B

S) S

inog

eiki

n an

d B

ass

(200

2b),

13

50 (

RU

S) I

saak

et a

l. (1

992)

20, 1

700

(MH

z-U

I) Z

ou e

t al.

(201

2)20

, 750

(B

S) L

iu e

t al.

(201

3)

Ppv:

Mg-

rich

pos

t-pe

rovs

kite

; Pv:

Mg-

rich

per

ovsk

ite; F

p: M

g-ri

ch (M

g,Fe

)O; C

a-pv

: CaS

iO3 p

erov

skite

; Rw

: rin

gwoo

dite

; Wad

: wad

sley

ite; M

j: m

ajor

itic

garn

et; S

tish:

stis

hovi

te; O

l: ol

ivin

e;

Px: p

yrox

ene;

Grt

: gar

net.

BS:

Bri

lloui

n sc

atte

ring

; M

Hz-

UI:

MH

z ul

tras

onic

int

erfe

rom

etry

; G

Hz-

UI:

GH

z ul

tras

onic

int

erfe

rom

etry

; R

US:

res

onan

t ul

tras

ound

spe

ctro

scop

y; I

SLS:

im

puls

ivel

y st

imul

ated

lig

ht

scat

teri

ng.


information, except for differences in the efficiency of the two techniques that depend on material properties and experimental constraints. The two methods are largely complementary, and there are in the literature some rare examples of coupling time- and frequency domain analysis approaches in a single combined setup (Maznev et al. 1996).

Detailed descriptions of the application of these different techniques can be found in sev-eral review articles (e.g., Bass 2007; Angel et al. 2009).

Lithosphere and upper mantle

Olivine, (Mg,Fe)2SiO4, is the dominant constituent (40-60% by volume) in upper mantle mineralogical models and its phase transformations are believed responsible for the 410- and 660-km seismic discontinuities (Ringwood and Major 1966; Duffy and Anderson 1989; Kat-sura and Ito 1989). Not surprisingly, the elastic properties of olivines at high pressure have received considerable attention in Brillouin scattering studies (e.g., Shimizu et al. 1982; Duffy et al. 1995; Zha et al. 1996, 1998; Speziale and Duffy 2004). At ambient conditions, there have also been studies of Ca- Ni- and Mn-olivines as well as germanate analogs (Weidner and Hamaya 1983; Bass et al. 1984; Peercy and Bass 1990; Lin and Chen 2011). Olivine single crystal elasticity has also been investigated by MHz UI (Kumazawa and Anderson 1969; Chen et al. 1996a), GHz UI (Chen et al. 1996b), RUS (Isaak et al. 1989; Isaak 1992) and ISLS (Zaug et al. 1993; Abramson et al. 1997).

Olivine (and its high-pressure polymorphs) can store significant quantities of water as hy-droxyl-containing point defects (Smyth 1987; Kohlstedt et al. 1996; Smyth et al. 2006). Even if present in small quantities in the mantle, hydrogen can strongly affect phase relations, rheologi-cal and transport properties, and seismic properties of the deep Earth (Mei and Kohlstedt 2000; Karato 2006; Manthilake et al. 2009; Mao et al. 2010). Knowledge of the elastic properties of hydrous olivine polymorphs is thus necessary to interpret seismic data in potentially hydrous re-gions of the mantle. The elastic properties of hydrous olivine have recently been measured both at ambient conditions (Jacobsen et al. 2008) and high pressure (Mao et al. 2010). The presence of 0.9 wt% H2O results in an increase in the pressure derivatives of the aggregate moduli. As a result, the bulk and shear moduli and hence the sound velocities of hydrous forsterite become greater than those of anhydrous forsterite at high pressures. At 14 GPa (~400 km depth), the compressional and shear velocities of hydrous forsterite are surprisingly 1-2% faster than those of anhydrous forsterite (Mao et al. 2010).

Garnets are important minerals widely found in metamorphic and igneous rocks of the Earth’s crust. Garnets occur in both peridotitic and eclogitic compositions in upper mantle as-semblages and are expected to be important components down to 660 km depth. As high-quality natural garnet crystals are widely available and have cubic symmetry with generally little elastic anisotropy, these minerals have been extensively studied to understand compositional effects on elastic properties. Silicate garnets in the pyralspite and ugrandite groups have also been subject of Brillouin scattering studies at ambient conditions to deep upper mantle pressures (Bass 1986, 1989; O’Neill et al. 1989, 1991, 1993; Conrad et al. 1999; Sinogeikin and Bass 2000; Jiang et al. 2004a,b). In a very recent study, Liu et al. (2013) have investigated single-crystal elastic properties of Fe-bearing pyrope to simultaneous high pressures and temperatures up to 20 GPa and 750 K.

The results of studies of olivines and garnets were used in solid-solution models of Fe-Mg substitution in olivine and Fe-Ca-Mg substitution in garnet by Speziale et al. (2005) to model the systematics of heterogeneity parameters, defined as (∂lnνp,s/∂Xi)P (where νp,s are compres-sional and shear velocity, Xi is the atomic fraction of the element in isomorphic substitution and P is pressure), for both minerals as a function of depth through the whole upper mantle. These parameters are of prime importance for interpreting seismic tomographic data in terms of chemical heterogeneity (Karato and Karki 2001).


Ortho- and clinopyroxenes are chain silicates that are present as abundant components in the crust and upper mantle rocks. Measurements on orthophyroxenes, (Mg,Fe)SiO3, have fo-cused mainly on constraining the effect of Mg-Fe substitution at ambient conditions (Weidner et al. 1978; Bass and Weidner 1984; Duffy and Vaughan 1988; Jackson et al. 1999) but other compositional and structural effects on elasticity have been studied as well by Brillouin scat-tering (Vaughan and Bass 1983; Perrillat et al. 2007). High temperature investigations of Mg-SiO3 orthoenstatite show evidence for elastic softening associated with the onset of a displacive phase transition (Jackson et al. 2004a, 2007). Implications of orthopyroxene elasticity data for interpretation of seismic data in the crust and lithosphere have been discussed by Wagner et al. (2008) and Reynard et al. (2010).

Due to their monoclinic symmetry, the study of the elastic properties of clinopyroxenes is more challenging. Nevertheless, there have been extensive measurements, mostly by Brillouin scattering, of the full set of elastic constants at ambient conditions (Levien et al. 1979; Vaughan and Bass 1983; Kandelin and Weidner 1988a,b; Bhagat et al. 1992; Sang et al. 2011) and also by RUS (Isaak and Ohno 2003; and at high temperature, Isaak et al. 2006) and by ISLS (Collins and Brown 1998).

A variety of other common crustal minerals have been investigated by Brillouin scatter-ing. These include quartz and its polymorphs (Weidner and Carleton 1977; Yeganeh-Haeri et al. 1992; Gregoryanz et al. 2000; Lakshtanov et al. 2007). A particularly notable feature is the negative bulk Poisson’s ratio for the SiO2 polymorph, cristobalite (Yeganeh-Haeri et al. 1992). Elastic constants by Brillouin scattering have also been reported for zeolites, natrolite, analcine and pollucite (Sanchez-Valle et al. 2005, 2010); cordierite (Toohill et al. 1999); glaucophane (Bezacier et al. 2010b), zoisite (Mao et al. 2007), diaspore (Jiang et al. 2008), alunite (Majzlan et al. 2006), phenacite (Yeganeh-Haeri and Weidner 1989), fluorite (Speziale and Duffy 2002), ettringite (Speziale et al. 2008a) and portlandite (Speziale et al. 2008b). Brillouin scattering of synthetic spinel (MgA2O4) has been measured at ambient pressure by Askarpour et al. (1993) to 1273 K. The phyllosilicates that have also been characterized by the technique are muscovite (Vaughan and Guggenheim 1986) and antigorite (Bezacier et al. 2010a, 2013).

Brillouin scattering of hydrous minerals has helped to improve our knowledge of the changes in elastic properties associated with the incorporation of hydrogen in the structure of oxides and silicates, with important impact on our understanding of deep water recycling in areas of active subduction. In addition to the nominally anhydrous olivine polymorphs, hy-drous minerals studied to date include: brucite (Xia et al. 1998; Jiang et al. 2006), chondrodite (Sinogeikin and Bass 1999), clinohumite (Fritzel and Bass 1997), lawsonite (Sinogeikin et al. 2000b; Schilling et al. 2003) as well as the dense hydrous magnesian silicate phase A and phase D (Sanchez-Valle et al. 2008; Rosa et al. 2012). In addition to hydrogen, the recycling and deep budget of carbon is of interest in studying the composition of the deep Earth. Thus carbonates have been also studied by Brillouin spectroscopy in order to identify their potential contribution to the overall elastic properties of the subducting oceanic lithosphere at shallow depth (Chen et al. 2001, 2006; Sanchez-Valle et al. 2011).

Brillouin scattering of silica-rich glasses and melts has been measured as a function of pressure and temperature in order to determine compressibility and viscosity of granitic melts as a function of composition and put constraints on their density as a function of depth (Tkachev et al. 2005a,b; Hushur et al. 2013).

Transition zone

The Earth’s transition zone, i.e., the region between 410 km and 660 km depth, is char-acterised by several major phase transitions. In particular, the structural changes that occur in (Mg,Fe)2SiO4 from olivine to wadsleyite to ringwoodite to a perovskite assemblage are of enormous interest to geophysicists. Comparison of the characteristics of the observed seismic


discontinuities such as depth, width and velocity contrast to laboratory data provides constraints on the bulk chemical composition, the thermal state and potentially also the water content of the transition zone (e.g., Cammarano and Romanowicz 2007; Frost 2008).

The complete set of elastic stiffness coefficients of a single crystal of Mg2SiO4 in the wadsleyite structure has been reported at ambient conditions (Sawamoto et al. 1984) and at high pressure (Zha et al. 1997). The effect of Fe/(Fe+Mg) ratio on the single-crystal elastic properties of wadsleyite and ringwoodite has been determined at ambient conditions (Sinogeikin et al. 1998) and for ringwoodite at high pressure (Sinogeikin et al. 2001). The other major component of the transition zone is majoritic garnet. The bulk elastic properties of single crystal and of polycrystalline majorite have been measured by Pacalo et al. (1992) and by Sinogeikin and Bass (2002a) respectively. The effect of chemical substitution has been investigated by Yeganeh-Haeri et al. (1990) and Sinogeikin and Bass (2002a) in the majorite-pyrope system, as well as by Pacalo et al. (1992) and Reichmann et al. (2002) in the majorite-jadeite system. Recently, single-crystal majorite was grown in a high-pressure/high-temperature synthesis and measured by Brillouin scattering to high pressures (Murakami et al. 2008).

As with olivine, there has been much interest in the incorporation of OH− defects in the crystal structures of wadsleyite and ringwoodite (Smyth 1987, 1994; McMillan et al. 1991; Kohlstedt et al. 1996). The single-crystal elasticity of hydrous wadsleyite and ringwoodite has been determined at room conditions (Inoue et al. 1998; Wang et al. 2003; Mao et al. 2008c) and at high pressure (Wang et al. 2006; Mao et al. 2008b, 2010, 2011) and also by GHz UI (Jacobsen et al. 2004; Jacobsen and Smyth 2006), and the results imply that OH− defects have a pronounced effect on the elasticity of the high-pressure olivine polymorphs. The available data raise the hope that it might be feasible to “map” hydration in the transition zone by combining mineral physics data with constraints from seismic tomography (Jacobsen and Smyth 2006; Shito et al. 2006; Li et al. 2011). Very recently, the elastic properties of hydrous ringwoodite have been measured to 16 GPa and 673 K (Mao et al. 2012).

The elastic properties of stishovite have been determined by Weidner et al. (1982) at ambient conditions. Lakshtanov et al. (2007) has investigated the stishovite to CaCl2-type phase transition by measuring Brillouin scattering along the [110] crystallographic direction. This displacive transition, which takes place at pressures of the lower mantle, is characterized by a substantial shear acoustic softening along [110], and its depth is strongly influenced by aluminium (and hydrogen) content in stishovite. Jiang et al. (2009) in a study of stishovite up to 22 GPa, observed signs of the incipient elastic softening already at pressures of the transition zone.

Based on the available data on transition zone minerals, it appears that the steep velocity gradients indicated by seismology are not consistent with a homogeneous mantle of pyrolitic composition (Cammarano and Romanowicz 2007). However, to draw final conclusions about the state of the transition zone, more input data from mineral physics, ideally determined at realistic pressure and temperature conditions, is needed. This includes data on elastic properties, but also an improved understanding of anelastic effects on the propagation behaviour of seismic waves (e.g., Stixrude and Jeanloz 2009).

Lower mantle

The two major components of the pyrolitic Earth’s lower mantle are Mg-rich silicate perovskite (with a simplified formula (Mg1−xFex)SiO3) and ferropericlase (Mg1−yFey)O. Brillouin scattering of Mg-rich ferropericlase (y = 0.013) has been measured up to 9 GPa (Reichmann et al. 2008). Ferropericlase with composition (Mg0.94Fe0.06)O has been measured to a maximum pressure of 20 GPa (Jackson et al. 2006). Recently, Brillouin scattering of single-crystal (Mg0.9Fe0.1)O ferropericlase has been measured to 81 GPa corresponding to 1900 km depth (Marquardt et al. 2009b). This study showed that ferropericlase is elastically very


anisotropic in the Earth’s lower mantle, particularly when Fe2+ is in low spin state (Badro et al. 2003). However, it is still debated if the spin crossover of Fe2+ is accompanied by a “softening” of compressional wave velocities based on measurements of other ferropericlase compositions by ISLS (Crowhurst et al. 2008) and by IXS (Antonangeli et al. 2011). Brillouin scattering is not currently able to resolve this issue, because of the overlap of the Brillouin signal from diamonds anvils in the DAC (Marquardt et al. 2009c). However, in principle, this limitation can be overcome (e.g., Zha et al. 2000) and Brillouin scattering is certainly a candidate tool to resolve this fundamental question, with all its consequences for our understanding of the state of the lower mantle (e.g., Cammarano et al. 2010).

The other major component of the lower mantle, Mg-rich perovskite, has only been measured as a single-crystal at room pressure (Yeganeh-Haeri et al. 1989; Yeganeh-Haeri 1994; Jackson et al. 2004b; Sinogeikin et al. 2004b). High-pressure measurements are restricted at present to polycrystalline iron-free perovskite (Jackson et al. 2005; Murakami et al. 2007b). Measuring single-crystal anisotropic elastic properties of iron-bearing perovskite at pressure/temperature-conditions relevant for the lower mantle is certainly among the biggest challenges in experimental mineral physics. Brillouin scattering of polycrystalline CaSiO3 perovskite, which represents 5 vol% of the bulk lower mantle, has been measured throughout the whole lower mantle pressure range (Kudo et al. 2012).

In addition to the minerals of the pyrolitic bulk mantle, also the aluminum-rich phase with composition Na0.4Mg0.6Al1.6Si0.4O4 and its higher pressure polymorph with Ca-ferrite phase, both relevant components of lithospheric rocks deeply subducted at lower mantle pressures, have been subject of a high-pressure study based on Brillouin scattering of polycrystalline samples up to 73 GPa (Dai et al. 2013).

In the Earth’s lowermost mantle, perovskite undergoes a structural phase transition to post-perovskite (Murakami et al. 2004; Oganov and Ono 2004; Tsuchiya et al. 2004). The single-crystal elastic properties of this phase are experimentally unknown, even though they are of enormous importance for interpreting seismic anisotropy in the lowermost mantle (e.g., Miyagi et al. 2010). Brillouin scattering has, however, been performed on polycrystalline Mg-endmember post-perovskite to a pressure of 172 GPa (Murakami et al. 2007a). It should be noted here that the potential effect of the electronic spin transition of iron on the elastic properties of both perovskite and post-perovskite is still unknown (Lin et al. 2013).

Silicate glasses are model systems for understanding the behaviour of silicate melts at deep mantle conditions. Glasses with compositions of interest for geophysics have been measured to lower mantle pressures up to more than 200 GPa (Zha et al. 1994; Murakami and Bass 2010, 2011; Sanchez-Valle and Bass 2010). These studies reveal interesting changes in the pressure derivative of shear velocities that might indicate amorphous silicate densification. Furthermore, these measurements document that the experimental limits of Brillouin scattering at extreme conditions are still not mapped out (Murakami and Bass 2011).

FRONTIERS

Elasticity under deep mantle conditions

Brillouin scattering is one of the few techniques capable of providing information about single-crystal elastic anisotropy, at pressure and temperature conditions relevant to the deep Earth interior. However, the vast majority of experimental work on materials of geophysical relevance has been limited to either high pressure or high temperature, and not simultaneously at the high P-T conditions of the interior of our planet (Fig. 15). This poses limitations on the mineral physics interpretation of seismological observations at present.


Indeed, it has been shown that the P-T-cross-derivatives, which are only accessible by mea-surements performed at simultaneously high-P and -T, cannot be disregarded (Chen et al. 1998). In addition, only recently the first single-crystal measurements were performed (at 300 K) at pressures expected for the Earth’s lower mantle (Zha et al. 2000; Marquardt et al. 2009b,c) or at 1 bar at temperatures relevant to the deep mantle (Jackson et al. 2000, 2007; Sinogeikin and Bass 2002b; Sinogeikin et al. 2004a). At present, there are no high-pressure or high-temperature Brillouin data on single crystals of MgSiO3 perovskite, the most abundant mineral phase of the Earth’s lower mantle. In this chapter, we provide some perspectives on future directions for Brillouin spectroscopy.

Extending the experimental pressure range for single-crystal measurements. Sample preparation for Brillouin scattering is challenging as the technique requires small, optical grade, parallel, and well-polished samples in specific crystal orientations. It has recently been demonstrated that focused ion beam (FIB) cutting and milling are well suited techniques to prepare single-crystal samples with dimensions of few tens of microns and precisely defined shapes (Marquardt and Marquardt 2012). These procedures result in little damage and thus allow for machining designed samples from materials that are brittle, metastable, or show a strong cleavage. In addition, FIB techniques allow for cutting more than one sample from a single fragment of material, for instance, platelets with different orientations from one single-crystal of less than 100 µm width.

Another serious limitation in Brillouin-scattering experiments extending to very high pressures (>30 GPa), or on materials with fast acoustic velocities, arises from the overlap of Brillouin peaks from the shear waves of the diamond anvils with those of the sample’s compressional wave. As a result, the compressional waves can only be measured in certain directions or even not at all. This reduces the ability to recover the full set of elastic tensor components. In the case of cubic materials, in situ X-ray diffraction data can be used to supplement the Brillouin data. Using the (isothermal) bulk modulus KT = (c11 + 2c12)/3 together with the results from Brillouin scattering from the shear acoustic modes, one can extract the three independent elastic constants, if appropriate corrections are made between the isothermal and adiabatic moduli (e.g., Marquardt et al. 2009c). Methods for spatial filtering and the use of cylindrical lenses can, in principle, be used to suppress the diamond signal (Zha et al. 1998, 2000) and further developments of these techniques are needed.

Extending Brillouin spectroscopy to simultaneous high pressures and high temperatures. A small number of Brillouin scattering measurements at combined high pressures and high temperatures have been performed in the diamond-anvil cell, using resistive (Goncharov et al. 2007; Jia et al. 2008; Mao et al. 2012; Lu et al. 2013) or laser heating (Li et al. 2006; Murakami et al. 2009a, 2012). The maximum temperature that has been reported in single-crystal Brillouin experiments at high pressures by using resistive heating is about 900 K (Sinogeikin et al. 2006). The temperature range that can be accessed by resistive heating might be extended using different heaters and enclosing the whole diamond-anvil cell in a sealed container with a controlled atmosphere (e.g., Liermann et al. 2009), but is limited due to graphitization of the diamonds that occurs at roughly 1200-1500 K.

Alternatively, infrared laser heating can be employed to heat the sample. Recent progresses in implementation of laser-heating techniques for single-crystal studies (Dubrovinsky et al. 2010) has applications to Brillouin spectroscopy. The solid-state lasers that are commonly employed for laser-heating in the diamond-anvil cell operate at wavelengths in the near infrared (~1 µm) where there is tradeoff between the optical transparency required for Brillouin spectroscopy and the absorption required for effective heating at these wavelengths. Silicates and oxides (including lower-mantle ferropericlase and Mg-perovskite) that contain some amount of iron (~10 at%) are perfectly suited for Brillouin spectroscopy (e.g., Marquardt et al. 2009c), while absorbing enough energy from a ~1 µm heating laser to allow for effective


heating to thousands of degrees (Dubrovinsky et al. 2009). The possibility of laser-heating the sample chamber by employing “internal furnaces,” i.e., adding some laser absorbing material to the sample chamber, has also been discussed (Goncharov and Crowhurst 2005; Li et al. 2006). Alternatively, CO2 lasers (wavelength ~10 µm) have been employed to heat transparent samples (Sinogeikin et al. 2004a; Murakami et al. 2009a).

Brillouin scattering combined with synchrotron X-ray diffraction. The combination of Brillouin spectroscopy and synchrotron X-ray diffraction is very useful for studies at high P-T, because the density of the sample can be directly determined under the same experimental con-ditions as the Brillouin data. X-ray diffraction data provide a precise control and knowledge of the sample orientation (in case of slight rotation of the single crystal by increasing pressure and temperature), which reduces the number of phonon directions necessary to constrain the elastic tensor and thus minimizes experimental collection time, especially when measurements are performed at extreme conditions. In addition, the X-ray data give information on crystal qual-ity, twinning, etc. In case of experiments performed on polycrystalline samples, other useful results potentially derived from X-ray measurements include the average grain size, texture, and stress distribution (Marquardt et al. 2011; Speziale et al. 2013). The combination of Brillouin scattering and synchrotron X-ray diffraction can be realized by having the Brillouin system “on-line” at a beamline. Such systems are installed at the Advanced Light Source (APS) Sector 13 (Sinogeikin et al. 2006) and SPring 8 beamline BL10XU (Murakami et al. 2009a). They allow for a simultaneous collection of acoustic velocity measurements and X-ray diffraction data. Figure 16 shows both a Brillouin spectrum collected on a single-crystal of (Mg0.9Fe0.1)O ferropericlase at APS Sector 13 and the corresponding integrated diffraction pattern at around 10 GPa and 700 K.

One possible drawback of such systems lies in the experimental time requirements of the two techniques. While it takes only several minutes to collect X-ray diffraction datasets in the DAC, especially if only the unit cell volume is to be determined, the typical collection time for one Brillouin spectrum is several tens of minutes, sometimes even hours. Additionally, multiple crystallographic directions have to be probed in single-crystal Brillouin spectroscopy to invert for the complete elastic tensor at a given pressure, especially if the symmetry of the

Figure 16. Brillouin scattering spectrum of a single crystal of (Mg0.9Fe0.1)O and corresponding X-ray dif-fraction pattern simultaneously collected at high pressure and temperature at Sector 13 of the APS (Mar-quardt, unpublished data).


material is not cubic or hexagonal. Thus, the use of “on-line” systems represents the optimal use of beamtime only when the collection of Brillouin spectra can be limited to one or a few directions as is the case for liquids, glasses, random polycrystals, and elastically isotropic or high-symmetry single crystals.

An alternative approach to combining Brillouin scattering and X-ray diffraction is the installation of a Brillouin system in close proximity to a synchrotron beamline. With this arrangement, the time-consuming acoustic velocity measurements of elastic anisotropy can be performed “off-line” and the measurements of X-ray diffraction are performed only once per P-T step by transferring the sample to the beamline at the same environmental conditions. Such a setup is now available at DESY-PETRA III Beamline P02.2 (Extreme Conditions Beamline ECB) (Speziale et al. 2013). This approach is most effective for high-pressure and temperature measurements of single-crystals, especially when the crystal symmetry is lower than cubic.

The combination of Brillouin scattering with synchrotron X-ray diffraction at high pressure in the DAC can supplement the velocity data with very important information that help to constrain the full elastic tensor when the Brillouin data alone are insufficient. X-ray diffraction of powders can supply cell volume (and hence density) and constrain the isothermal bulk modulus (Sinogeikin et al. 2006; Marquardt et al. 2009b; Murakami et al. 2009a). Single crystal diffraction can place tight constraints on the axial compressibilities which place independent constraints on subsets of the elastic constants. X-ray diffraction of powders in radial geometry can supply information about texture development, useful to evaluate the results of Brillouin scattering from polycrystalline samples (Marquardt et al. 2011; Speziale et al. 2013), and in the case of high-symmetry materials it can also place constraints on subsets of single crystal elastic constants (Speziale et al. 2006; Mao et al. 2008a) by applying anisotropic lattice strains theories (e.g., Funamori et al. 1994; Singh et al. 1998; Matthies et al. 2001).

Brillouin scattering of polycrystalline aggregates. Brillouin measurements on polycrystalline aggregates offer some advantages, especially for experiments at higher pressures and combined high P-T conditions. For isotropic samples such as glasses and random polycrystals, only one single Brillouin measurement is necessary to obtain the two elastic moduli that characterize such materials, although more measurements are typically performed in practice. Good agreement for bulk properties (shear and bulk modulus) between Brillouin scattering measurements performed on polycrystalline and single-crystal perovskite has been demonstrated at ambient conditions (Sinogeikin et al. 2004b). However, even for elastically isotropic materials, the anisotropy of the photoelastic tensor (that determines the intensity of Brillouin scattered light; see “Brillouin scattering in solids” section) could bias the experimentally determined average velocity with respect to the theoretical average from single-crystal elastic constants.

In the case of loosely packed powders, the spectra show a characteristic distribution of frequency shifts caused by the large variation in scattering geometries (Hernandez et al. 1996). Thorough investigations of Brillouin spectra from heterogeneous micro-aggregates have elucidated the details of their dispersion relation in the GHz regime, including fine details of localized eigenmodes of the particles of the dispersed components (Kriegs et al. 2004). In the case of predefined microstructure, Brillouin scattering has been successfully used to determine the effective elastic properties of the composite by applying effective medium models (e.g., Devaty et al. 2010). These promising results suggest that Brillouin scattering could be used to infer properties of unknown materials when they form microstructurally well-characterized aggregates with materials of known elastic properties at extreme conditions.

Recently, Brillouin experiments on packed powders have been performed to high pressures (Murakami et al. 2007a,b, 2009b; Gleason et al. 2011; Marquardt et al. 2011). Even though these results are promising, effort is still necessary to understand the quality and reliability of bulk


properties determined by Brillouin scattering from polycrystalline materials at high pressures. As the stress distribution in a diamond-anvil cell at high pressure will not be uniform, a major complication might arise from the development of a crystallographic-preferred orientation (CPO) in the polycrystalline sample. In this case, the measured aggregate velocities will be average distribution functions weighted by the grains’ orientation and could strongly deviate from the isotropic average velocities. Furthermore, the anisotropic deformation behavior of the constituents leads to intergranular stresses, which may bias the measured velocities. Additional care has to be taken to precisely characterise the average crystallite size since small grain size can have an impact on the measured bulk elastic properties (Marquardt et al. 2011).

Combining Brillouin scattering with other techniques to characterize elastic anisotropy at high pressures

The combination of single-crystal Brillouin scattering at extreme pressures and tempera-tures with other techniques that determine elastic properties of materials can allow one to put constraints on the full elastic tensor of materials even when the individual techniques alone cannot. One example of this approach is represented by the study of the effect of the Fe2+ spin crossover on the elastic properties of ferropericlase (Mg1−xFex)O (with x < 0.3) at lower mantle pressures. Brillouin scattering, X-ray diffraction, IXS and ISLS experiments on ferropericlase of similar compositions have furnished direct information on all the constants of the elastic tensor of this mineral, which is the second most abundant phase of the lower mantle (Lin et al. 2005; Crowhurst et al. 2008; Marquardt et al. 2009c; Antonangeli et al. 2011). This body of ex-perimental results, generally in mutual agreement, has placed strong constraints on the behavior of the shear elastic anisotropy of ferropericlase, which shows an increase across the spin cross-over. The results regarding the compressional constant c11 are still contradictory, but the combi-nation of the available results along with density functional theory computations (Wentzcovitch et al. 2009; Wu et al. 2013) is progressing in the direction of an unequivocal picture. Due to inherent limitations of the different techniques, it is the combination of multiple approaches that best places quantitative constraints on the elastic anisotropy at these extreme conditions.

Surface Brillouin scattering at extreme conditions

One of the biggest challenges in mineral physics is that of directly probing the elastic properties of the materials of the Earth’s core. Measurements of the elastic properties of polycrystalline hcp-Fe (the most probable stable polymorph in the inner core) have been measured to megabar pressures by IXS (Fiquet et al. 2001; Antonangeli et al. 2004, including elastic anisotropy) and by ISLS (Crowhurst et al. 2005). Measurements of Fe-Ni and Fe-Ni-Si alloy compositions have been performed at high pressures and temperatures (Kantor et al. 2007; Antonangeli et al. 2010). These experiments provide, up to now, the best constraints on shear and compressional velocities of solid Fe and Fe alloys at extreme pressures. A coherent picture requires additional experimental investigations at even higher pressures and at significantly higher temperatures.

Brillouin scattering can be one technique to determine acoustic velocities of both solid and liquid metals at extreme conditions in the DAC. Surface Brillouin scattering of liquid metals has been reported at ambient conditions (Dil and Brody 1976) and the theory has been developed to interpret the spectral features present in spectra collected from the interface between a transparent container and a metal liquid (Visser et al. 1984). In addition, surface Brillouin scattering of solid thin-film Au has been measured to a pressure of 4.8 GPa (Crowhurst et al. 1999). Advances in the design of Brillouin spectrometers (that is both the optic path and the interferometers) allow, in principle to detect the scattering from interfacial waves at the contact between liquid and solid iron as well as between diamond and a transparent pressure medium.


ACKNOWLEDGMENTS

We thank E. S. Zouboulis, Z. Geballe, H. J. Reichmann and an anonymous reviewer for their very useful comments, which helped us to improve the manuscript. H. M. was supported by DFG-grant SP 1216/3-1 within SPP1236.

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