2. Introduction to BRILLOUIN Spectroscopy

SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume

2. Introduction to BRILLOUIN spectroscopy

BRILLOUIN spectroscopy is is a well-established optical spectroscopic technique to investigate mechanical as well as magnetic properties of matter (e.g. SANDERCOCK 1982, KRÜGER 1989, HILLEBRANDS 2005). The technique bases on the phenomenon of inelastic laser light scattering, for wwhhichich the incident photons are scattered by thermally driven elementary excitations in matter, like sound waves (acoustic phonons) or spin waves (magnons) (BRILLOUIN 1922, MANDELSTAM 1926, GROSS 1930, FABELINSKII 1968, CHU 1974, BERNE & PECORA 1976, DIL 19819822). Note that BRILLOUIN spectroscopy gives access to hypersonic waves and spin waves without exciting them explicitly by transducers as necessary e.g. for ultrasonic pulse-echo techniques. Since manifold introductory review articles and monographs on BRILLRILLOOUINUIN scattering are available (FABELINSKII 1968, CHU 1974, BERNE & PECORA 1976, DIL 1982, SANDERCOCK 1982, KRÜGER 1989, HILLEBRANDS 2005) and since the current article focuses on scanning BRILLOUIN microscopy, we give only a short introduction to the theortheoreeticaltical background of BRILLOUIN scattering. In a first reading, the sections marked with an asterisk* 2.1 (except figure 2.1 and the related explications) and 2.4 can be skimmed over.

2.1. Scattering of laser light by thermally excited sound waves* The tthheoryeory of inelastic scattering of visible light by thermally excited hypersonic waves travelling through transparent matter will be recapitulated within a statistical framework (CHU 1974, BERNE & PECORA 1976). Remember that indeed the continuum of sound wavwaveess within condensed matter can be correlated to spatio- temporal fluctuations of the mass density, or more generally, of the strain tensor components (AULD 1973). ConsiderConsider a monochromatic a monochromatic laser beam laser that beam can thatbe decomposedcan be decomposed into planar into waves,planar incidentwaves, incident on a di electricon a dielectric sample. sample. The electric The electric part of part each of incident each incident planar planar light wave light canwave be can described be described as: as: , (2.1)

89 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch.Arch. Sci.Sci. Nat.Nat. Phys.Phys. Math.Math. NSNS 46,46, SpecialSpecial volumevolume where denotes the wave vector in the medium, thethe position position vector vector in in the the sample’ssample’s coordinatecoordinate system,system, thethe wave’swave’s angularangular frequency,frequency, thethe electricelectric fieldfield amplitudeamplitude andand defines the direction of polarispolarisation.ation. For the sake of simplicity, we assumeassume that that within anisotropicanisotropic matter, matter, the the incident incident and and scattered scattered electromagnetic electromagnetic waves represent eigenstates of the electromagnetic field. By this means we exclude effectseffects relatedrelated toto birefringence.birefringence. TheThe spatiallyspatially andand ttemporallyemporally fluctuatingfluctuating dielectricdielectric properties of the sample are described by the dielectric tensor .. IfIf the the componentscomponents ofof thethe dielectricdielectric tensortensor atat opticaloptical frequenciesfrequencies fluctuatefluctuate onlyonly spatially,spatially, but not temporally, elastic light scatscatteringtering can occur. In that case, the scattered electromagneticelectromagnetic waveswaves generallygenerally propagatepropagate alongalong otherother directionsdirections thanthan thethe incidentincident one while the energy is conserved: andand ,, with denoting the scatteredscattered wave wave vector. vector. In In the the domain domain of materials science, typical candidates for elasticelastic light light scattering scattering processes processes are are polycrystalline polycrystalline materials materials and and nanocomposites nanocomposites possessing optical heterogeneities in the 100 nanometer to micrometer rangerange.. InelasticInelastic lightlight scatteringscattering occursoccurs ifif thethe componentscomponents of the dielectric tensor at optical frequenciesfrequencies do not only fluctuate spatially, but also temporally. In this context,context, two two aspects aspects need need consideration: consideration: the the spatially spatially and and tempo temporallyrally fluctuating fluctuating elementaryelementary excitations excitations of of the the sample sample that that couple couple to to its its optical optical properties properties and and the the strengthstrength ofof thisthis coupling.coupling. TheThe spatiospatio--temporaltemporal elementaryelementary excitationsexcitations ofof thethe samplesample cancan bebe divideddivided intointo relaxationalrelaxational fluctuations,fluctuations, likelike fluctuationsfluctuations ofof thethe entropy,entropy, andand propagating fluctuations, like fluctuationsfluctuations of the mass density (BERNE & PECORA 1976). The corresponding modes are either diffusive or propagating ones. Sound waves (also denoted as acoustic phonons) are generally induced by thermally excitedexcited fluctuations fluctuations of of the the strain strain tensor tensor components, components, which which degenerate degenerate in in ideal ideal liquidsliquids toto massmass densitydensity fluctuations.fluctuations. TheThe elastoelasto--optical coupling between the strain fluctuationsfluctuations and and the the optical optical fluctuations fluctuations is is described described by by the the Pockels Pockels tensor tensor ( (NYE 1972, VACHER & BOYER 1972).1972). ForFor mostmost materials,materials, thethe elastoelasto--opticaloptical coupling for shearshear strainstrain isis muchmuch weakerweaker thanthan thatthat forfor longitudinallongitudinal strainstrain ((NYE 1972, VACHER & BOYER 1972). The spatiallyThe and spatially temporally and fluctuating temporally dielectric fluctuating tensor dielectric at optical tensor frequencies at optical can befrequencies described can as: be described as:

109 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume

, (2.2)

with denoting the spatially and temporally averaged dielectric tensor and the fluctuating part of the dielectric tensor. TheThe transversely polarized phonons are responsible for the off-diagonal entries of . The directions of polarization of the incident planar light wave and of the chosen inelastically scattered light wave yield the components according to:

. (2.3) These dielectric fluctuations can be described by the spatio-temporal autocorrelation function of , with V being the scattering volume and the considered time

interval:

(2.4)

The dynamic structure factor of the scattered light component travelling in the -direction results from the spatio-tempotemporalral FOURIER transformation of the autocorrelation function of :

, (2.5)

with designating the acoustic phonon’sphonon’s wave vector and

(with being the angular frequencies of the involved scattered electric fields; see section 2.2) its angular frequency. The spectral power density , whichwhich is closely related to the dynamic structure factor, is experiexperimentallymentally accessible by BRILLOUIN spectroscopy. The factor depends on

the intensity of the incident light. As will be delineated below in a rough overview, the BRILLOUINRILLOUIN lines of the spectral power density are intimately related to the hypersonic properties of the sample, since these lines are due to dielectric fluctuations induced by thermal fluctuations of the strain tensor components (SOMMERFELD 1945, GRIMVALL 1986,1986, LANDAU & LIFSHITZ 1991). As describedAs describedin KONDEPUDI in KONDEPUDI & PRIGOGINE & P (1998),RIGOGIN theE (1998),thermally the excited thermally fluctuations excited fluctuationsof the strain oftensor the straincomponents tensor developcomponents according develop to accordingthe same lawsto the as same macroscopic laws as macroscopicdeformations deformations of small amplitude. of small Within amplitude. the frameWithin of the linear frame res ofponse linear theory, response the

1011 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial ScanningScanning Brillouin Brillouin microscopy microscopy Arch. Arch. Sci. Sci. Nat. Nat. Phys. Phys. Math. Math. NS NS 46, 46, Special Special volume volume

straintheory, the strain can be related can be to relate the thermallyd to the thermally excited excited fluctuating fluctuating elastic elastic force densityforce density via the via elastic the elasticsusceptibility susceptibility tensor tensor , according, according to: to: . . (2.6)(2.6) ConsideringConsidering the the storing storing andand dissipative dissipative parts parts of of the the elastic elastic interaction interaction by by the the symmetricsymmetric forth forth rank rank tensors tensors of of the the elastic elastic moduli moduli andand viscosities viscosities

(with(with k, k, l, l, m m,, n n = = 1, 1, 2, 2, 3), 3), thethe strainsstrains evolveevolve with with timetime likelike dampeddamped oscillators. oscillators. The The components components of of the the inverse inverse elastic elastic susceptibilities susceptibilities correspondcorrespond to to ( (LLANDAUANDAU & & L LIFSHITZIFSHITZ 1991, 1991, K KONDEPUDIONDEPUDI & & P PRIGOGINERIGOGINE 1998 1998):):

, , (2.7)(2.7) withwith denotingdenoting the the mass mass densitydensity and and thethe Kronecker Kronecker symbol. symbol. The The tensor tensor

can can be be diagonalised diagonalised in in dependence dependence of of and and , ,which which results results in: in:

(2.8) (2.8) withwith (2.9) (2.9) wherewhere characterizes characterizes the the polarization polarization state state of of the the sound sound wave wave and and denotesdenotes the the effective effective elastic elastic modulus modulus for for the the given given wave wave vector vector and and polarization. polarization. TheThe threethree complexcomplex eigenfrequencieseigenfrequencies correspondcorrespond toto thethe polespoles of of equation equation (2.9) (2.9) with: with: (2.10)(2.10) andand . . (2.11)(2.11) HereHere designatesdesignates the the acoustic acoustic damping damping and and thethe eigenfrequencieseigenfrequencies of of the the undamped undamped sound sound waves waves for for a a given given wave wave vector vector . . The The relatedrelated eigenvectors, eigenvectors, i.e. i.e. the the polarization polarization vectors vectors of of the the three three sound sound waves, waves, are are orthogonalorthogonal to to each each other. other. For For p=1, p=1, the the mode mode is is generally generally quasi quasi--longitudinallylongitudinally polarized:polarized: itsits polarizationpolarization vectorvector isis almostalmost collinearcollinear toto thethe directiondirection ofof propagpropagationation ((AAULDULD 1973).1973). TwoTwo quasiquasi--transverselytransversely polarizedpolarized modesmodes areare describeddescribed byby p=2p=2 andand p=3.p=3. For For elastically elastically isotropic isotropic solids, solids, all all modes modes are are purely purely longitudinally longitudinally or or

1211 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume

transversely polarized, i.e. the propagation and polarization vectorsvectors are either collinear or perpendicular (AULD 1973). Moreover, the two transversely polarized modes are degenerated for isotropic solids. These shear modes do even not propagate at all in ideal isotropic liquids, where the term ‘ideal’ implicates the absenceabsence of viscoelastic behaviour. In contrast, if the acoustic modes couple to molecular structural relaxations in a liquid, transverstransverselyely polarized sound modes can be detected at sufficiently high frequencies (CHU 1974, BERNE & PECORA 1976, KRÜGER 1989).

Fig. 2.1. BRILLOUIN spectrum of a homogeneous elastically anisotropic solid containing the BBrillouinRILLOUIN doublets doublets of of thethe quasiquasi--longitudinallylongitudinally (p=1(p=1 oror qL)qL) andand the two quasi-transversely (p=2, 3 or qT1, qT2) polarized sound modes. (In thethe following the notation ‘qL, qT’ will be used.) Angular frequency or , hypehypersonicrsonic attenuation: oror ,, height height of of S StokesTOKES or Anti-StokesSTOKES lines: lines: oror .. TheThe centralcentral lineline (suppressed in the figure) corresponds to the RAYLEIGH line, reflecting elastically scattered light and diffusive modes.

1213 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume

In the classical limit, the fluctuation-dissipation theorem (KONDEPUDI & PRIGOGINE 1998) relates the polarization-independent dynamic structure factor to the imaginary part of the susceptibility by:

. (2.12)

Note that the opto-acoustic coupling is not part of the so-defined dynamic structure factor, and therefore has to be accounted for by an appropriate prefactor. In the present rough overview, the relationship between the experimentally accessible spectral power density and the elastic susceptibilities, described by equation (2.12), permits us to sense that the angular frequencies and attenuations of the sound waves can be determined from the measured BRILLOUIN spectrum. A typical BRILLOUIN spectrum for a homogeneous anisotropic solid is given in figure 2.1. In the centre of the BRILLOUIN spectrum appears the RAYLEIGH line, corresponding to the elastically scattered light and the diffusive modes. The three BRILLOUIN doublets are positioned symmetrically to the RAYLEIGH line on the horizontal angular frequency axis . In order to calculate the intrinsic hypersonic attenuations the instrumental broadening has to be deconvolved from the measured BRILLOUIN spectrum. The angular frequency and attenuation correspond to the position and the half width at half maximum of the BRILLOUIN lines, respectively. The statistical error of the hypersonic angular frequency typically lies in the one-tenth of percent regime, whereas that of the attenuation is larger by roughly a factor of 10. An increasing half width at half maximum of the BRILLOUIN lines indicates a decreased lifetime of the considered phonon mode . The here described lifetime of the phonon (or temporal attenuation of a sound wave) and the spatial attenuation of a sound wave known from ultrasonic investigations are related by the sound velocity.

For a very small temporal hypersonic attenuation, i.e. , the angular frequency is not renormalized by the attenuation (see equation (2.11)). Several

14 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume

mechanisms are known to contribute to the hypersonic attenuation, like structural relaxation processes with their main relaxation frequencies in the GHz regime

(BERNE & PECORA 1976, KRÜGER 1989).

2.2. The kinematic approach

In the frame of a quantum-mechanical description, BRILLOUIN scattering results from inelastic scattering of light photons by acoustic phonons. Applying the law of conservation of energy to this scattering process allows one to relate the hypersonic angular frequency of the phonons to the angular frequencies and of the

incident and scattered light, respectively, according to:

. (2.13) The sign ‘+’ corresponds to the annihilation of phonons (Anti-STOKES scattering),

and the sign ‘-’ to the creation of phonons (STOKES scattering) (see figure 2.1 in section 2.1). The energy transfer between an incident light photon and an acoustic

phonon equals to about of the photon’s energy. Since typical frequencies of

visible light are about , the angular frequencies of the involved acoustic

phonons are usually in the GHz regime. Similarly the law of conservation of momentum relates the wave vector of the hypersonic wave to the wave vectors and of the incident and scattered light according to:

. (2.14)

A schematic representation of this law is given in figure 2.2. Using the inner scattering angle , the norm of the wave vector equals to:

. (2.15)

15 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume

Due to the very small energy transfer between phonons and photons, is a valid approximation leading to:

. (2.16)

If denotes the vacuum laser wavelength and n the sample’s refractive index, one obtains:

. (2.17)

Note that in BRILLOUIN scattering the acoustic wave vector is selected by adjusting the adequate scattering geometry, and that the phonon’s frequency is the usually complex response. As elucidated in section 2.1, in general three acoustic phonons, one quasi-longitudinally polarized and two quasi-transversely polarized are determined for one selected wave vector in an elastically homogeneous solid (AULD 1973).

Fig. 2.2. Schematic representation of momentum conservation during an inelastic light scattering process. Sample represented in blue; , : incident/scattered light wave vector within the sample; : wave vector of the acoustic phonon, and : inner scattering angle.

For a typical vacuum laser wavelength of , the phonon wavelength

equals to some hundred nanometres and is hence of the same order of magnitude as the laser wavelength. Thus, optical elastic scattering, like RAYLEIGH or MIE scattering (BORN & WOLF 1999) in a heterogeneous sample is in most cases accompanied by acoustic RAYLEIGH or MIE scattering (MORSE & INGARD 1968). For

16 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume

sufficiently small hypersonic attenuation ( ), the phase velocity of the quasi- longitudinally (qL) and quasi-transversely (qT) polarized acoustic modes depends on the phonon wavelength and frequency f according to:

. (2.18)

Or, using equation (2.17):

. (2.19)

Knowing in addition the sample’s mass density, equation (2.19) allows the calculation of the related elastic moduli (AULD 1973): . (2.20)

2.3. Classical scattering geometries

Three common scattering geometries, defining the -vector involved in the scattering process, are depicted in figure 2.3. In classical BRILLOUIN spectroscopy, a typical scattering volume is about . The 90N scattering geometry, shown in figure 2.3a, is a commonly used scattering geometry due to its easy alignment as , where denotes the outer scattering angle (i.e. the angle between the incident and scattered light beam outside the sample). Similarly easy to handle is the backscattering geometry depicted in figure 2.3b, with . The backscattering geometry is especially suitable in case of non-transparent samples as the properties within the scattering volume can often still be probed close to the sample’s surface. The major difference to the situation depicted in figure 2.3b is that

for a non-transparent sample the scattering volume is much more reduced in the x3- direction (see also section 3.2). Note that in the backscattering geometry for elastically isotropic samples only information about longitudinally polarized sound waves can be gathered because of symmetry reasons (VACHER & BOYER 1972). During the last years, the backscattering geometry has proven to be highly promising for studying structural processes in dependence of time for transparent

17 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial ScanningScanning BrillouinBrillouin microscopymicroscopy Arch.Arch. Sci.Sci. Nat.Nat. Phys.Phys. Math.Math. NSNS 46,46, SpecialSpecial volumevolume

materialsmaterials ( (PPHILIPPHILIPP et et al. al. 2009, 2009, 2011, 2011, SSANCTUARYANCTUARY etet al. al. 2010). 2010). Indeed, Indeed, the the large large scatteringscattering volume volume in in the the --directiondirection allows allows for for a a comparatively comparatively high high temporal temporal resolutionresolution asas BBRILLOUINRILLOUIN spectraspectra cancan bebe recordedrecorded fast.fast. EquationsEquations (2.17)(2.17) andand (2.18)(2.18) allowallow forfor calculating calculating the the phonon phonon wavelengths wavelengths and and sound sound velocities velocities for for both both scatteringscattering geometries geometries (using (using or or ,, respectively). respectively). The The related related values,values, bothboth dependentdependent onon thethe sample’ssample’s refractiverefractive index,index, areare indicatedindicated inin tabletable 2.1.2.1.

Fig.Fig. 2.3.2.3. RepresentationRepresentation ofof (a)(a) thethe 90N90N scatteringscattering geometry,geometry, (b)(b) thethe backscatteringbackscattering geometry,geometry, an andd (c) (c) the the scattering scattering geometry geometry (here: (here: outer outer scattering scattering angle angle ).). Sample Sample represented represented in in blue, blue, scattering scattering volume volume in in red. red. (1) (1) laser laser beam beam incidentincident onon thethe sample,sample, (2)(2) laserlaser beambeam leavingleaving thethe sample,sample, (3)(3) selectselecteded directiondirection ofof thethe inelastically inelastically scattered scattered light light emerging emerging from from the the scattering scattering volume. volume. ,, :: incidentincident and and scattered scattered wave wave vector, vector, :: wave wave vector vector of of the the acoustic acoustic phonon. phonon. :: samplesample coordinatecoordinate system.system.

The The scattering scattering geometry, geometry, with with being being strictly stri ctly between between 0° 0° and and 90°, 90°, isis specificallyspecifically adapted adapted to to many many experimental experimental challenges challenges (e.g. (e.g. KKRÜGERRÜGER etet al. al. 1978a, 1978a, 1981,1981, 1986, 1986, KKRÜGERRÜGER 1989). 1989). It It possesses possesses the the special special attribute attribute that that the the adjusted adjusted phononphonon wavewave vectorvector isis independentindependent ofof thethe sample’ssample’s refractiverefractive indexindex nn inin casecase ofof opticallyoptically isotropicisotropic samples.samples. Indeed,Indeed, applyingapplying SSNELLNELL’s’s lawlaw toto figurefigure 2.3c2.3c alallowslows forfor expressingexpressing thethe innerinner scatteringscattering angleangle inin dependencedependence ofof thethe outerouter scatteringscattering angleangle :: .. (2.21)(2.21)

1817 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume

Combining equations (2.19) and (2.21) then directly leads to the n-independent soundCombining velocity equations expression (2.19) for and the (2.21) then scattering directly geometry leads to the given n- inde in pendent table 2.1. Becausesound velocity of the usually expression small for birefringence the scattering of anisotropic geometry samples, given the in tableexpressions 2.1. indicatedBecause of in the table usually 2.1 aresmall also birefringence high quality of approximationsanisotropic samples, for thethe phononexpressions wave vectorindicated and inthe table sound 2.1 velocities are also of high optically quality anisotropic approximations samples, for thewith phonon absolute wave errors beingvector usually and the sound smaller velocities than 0.1% of optically (KRÜGER anisotropic et al. 1986).samples, Hence, with absolute this scattering errors geometrybeing usually is especially smaller useful than 0.1% if high (K precisionRÜGER et acoustic al. 1986). data Hence, are needed this scatteringfor samples ofgeometry unknown is especially optical useful properties. if high Forprecision instance acoustic it isdata frequently are needed employedfor samples for temperatureof unknown dependent optical properties.measurements For of instance samples itwith is an frequently unknown employedrefractive index for temperature dependent measurements of samples with an unknown refractive index evolution versus temperature (e.g. KRÜGER 1989, KRÜGER et al. 1990, 1994, evolution versus temperature (e.g. KRÜGER 1989, KRÜGER et al. 1990, 1994, JIMÉNEZ RIOBÓO et al. 1990, PHILIPP et al. 2008). As thoroughly described in JIMÉNEZ RIOBÓO et al. 1990, PHILIPP et al. 2008). As thoroughly described in chapter 3, this scattering geometry is often combined with angle-resolved chapter 3, this scattering geometry is often combined with angle-resolved BRILLOUIN spectroscopy to precisely determine the elastic tensor properties of anisotropicBRILLOUIN samples. spectroscopy Commonly, to precisely an outer determine scatt ering the elastic angle tensor of properties of anisotropic samples. Commonly, an outer scattering angle of is chosen anisotropic samples. Commonly, an outer scattering angle of is chosen because of the easy alignment. The corresponding scattering geometry is called the because of the easy alignment. The corresponding scattering geometry is called the 90A scattering geometry. 90A scattering geometry.

Table 2.1. Expressions for the phonon wavelength and the sound velocity Table 2.1. Expressions for the phonon wavelength and the sound velocity for different scattering geometries for optically isotropic samples. forfor differentdifferent scattering geometriesgeometries for for optically optically isotropic isotropic samples. samples. Scattering 90N backscattering 90A ScatteringScattering 90N90N backscatteringbackscattering 90A90A geometry geometrygeometry Phonon Phonon wavelength wavelengthwavelength

Sound velocity SoundSound velocityvelocity

19 19 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial ScanningScanning Brillouin Brillouin microscopy microscopy Arch. Arch. Sci. Sci. Nat. Nat. Phys. Phys. Math. Math. NS NS 46, 46, Special Special volume volume

2.4.2.4. Determination Determination of of elastic elastic tensor tensor components components for for anisotropicanisotropic matter*matter* ScanningScanning acoustic acoustic microscopy microscopy applied applied to to inhomogeneous inhomogeneous and and heterogeneous heterogeneous anisotropicanisotropic materials materials only only makes makes sense sense if if in in addition addition to to the the spatial spatial variations variations of of elasticelastic properties properties accompanying accompanying local local symmetry symmetry changes changes are are recorded. recorded. Otherwise Otherwise bothboth effects,effects, i.e.i.e. spatialspatial symmetrysymmetry changeschanges andand inherentinherent elasticelastic modulusmodulus variations,variations, cancan hardly hardly be be discriminated. discriminated. A A purposeful purposeful experimental experimental approach approach consists consists in in the the combinationcombination of of scanning scanning BBRILLOUINRILLOUIN microscopy microscopy with with angle angle--resresolvedolved BBRILLOUINRILLOUIN spectroscopy.spectroscopy. First First a a recapitulation recapitulation of of the the determination determination of of the the elastic elastic modulus modulus tensortensor is is proposed proposed before before proceeding proceeding to to this this combined combined experimental experimental technique technique in in the the nextnext section. section. As shownAs in the shown previous in thesections, previous in B RILLOUIN sections, spectroscopy in BRILLOUIN the adjustment spectroscopy of the the scatteringadjustment geometry of the scattering permits togeometry select a permits phonon to wave select vector a phonon with wave a vector desired directionwith a desired and norm. direction Angle and-resolved norm. Angle BRILLOUIN-resolved spectroscopy BRILLOUIN allowsspectroscopy a comparably allows a easycomparably access easy to the access elastic to the properties elastic properties in a given in scatta givenering scattering volume volume for many for differentlymany differently oriented oriented ’s within ’s withinone plate one-like plate sample-like sample (e.g. K RÜGER (e.g. K RÜGERet al. 1985,et al. 1986,1985, 1990, 1986, 1994, 1990, 2001). 1994, Furthermore, 2001). Furthermore, in a homogeneous in a homogeneousanisotropic material, anisotropic this techniquematerial, thisyields technique for a given yields phonon for a given wave phonon vector wave at vectorbest three at sound best three velocities sound (velocitiesAULD 1973). (AULD The 1973).term ‘at The best’ term means ‘at best’that themeans opto that-acoustic the opto coupling-acoustic coefficients coupling mustcoefficient be sos large must that be so sufficiently large that intense sufficiently phonon intense lines phonon appear lines in the appear BRILLOUIN in the spectrum.BRILLOUIN For spectrum. a given For the a related given phonon the related modes phononare orthogonally modes are polarized, orthogonally but thesepolarized, polarizations but these are polarizations usually neither are usually purely neither longitudinal purely norlongitudinal purely transversal. nor purely Thetransversal. evaluation The of evaluation a BRILLOUIN of aspectrum BRILLOUIN does spectrum not immediately does not leadimmediately to the material lead to- describingthe material elast-describingic susceptibility, elastic i.e. susceptibility, the fourth rank i.e. elastic the fourth modulus rank tensor. elastic modulus Fortensor. moderate or negligible acoustic losses this problem is solved by combining the constitutingFor C HRISTOFFEL moderate or equation negligible (A ULD acoustic 1973) losses with a this statistically problem representative is solved by datasetcombining the constitutingof measured CsoundHRISTOFFEL velocities equation (KRÜGER (AULD et al. 1973) 1986, with KRÜGER a statistically 1989). Therepresentative CHRISTOFFEL dataset equation combines of measured the measured sound velocities with (K RÜGERthe components et al. 1986, of theKRÜGER 1989). elastic The modulus CHRISTOFFEL matrix equation (V combinesOIGT notation; the measured AULD 1973): with the components of the elastic modulus matrix (VOIGT notation; AULD 1973):

2019 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume

. (2.22)

Here, denotes the sample’s mass density; the eigenvector , expressed in the symmetry coordinate system , gives the ‘particle velocity’, i.e. the oscillation velocity of a volume of the medium being small compared to the whole sample but large compared to the atomic scale. The particle velocity reflects the acoustic polarization of the acoustic mode related to the sound velocity . The -matrix in equation (2.22) is called the CHRISTOFFEL matrix and is defined for the lowest possible symmetry, the triclinic symmetry, by:

(2.23)

where the unit vector gives the propagation direction

of the acoustic wave, expressed in the symmetry coordinate system . The matrix representation of the fourth rank elastic modulus tensor for triclinic symmetry is given by:

. (2.24)

21 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial ScanningScanning Brillouin Brillouin microscopy microscopy Arch. Arch. Sci. Sci. Nat. Nat. Phys. Phys. Math. Math. NS NS 46, 46, Special Special volume volume

ForFor cubic cubic symmetry symmetry the the CCHRISTOFFELHRISTOFFEL matrix matrix significantly significantly simplifies simplifies and and is is expressedexpressed as as follo follows:ws:

. . (2.25)(2.25)

TheThe threethree--dimensionaldimensional soundsound velocityvelocity polarpolar diagramdiagram forfor crystalscrystals ofof cubiccubic symmetrysymmetry isis still still a a three three--sheetedsheeted hypersuperface, hypersuperface, similar similar to to the the lower lower symmetries symmetries ( (AAULDULD 1973, 1973, KKRÜGERRÜGER etet al.al. 1986,1986, KKRÜGERRÜGER 1989).1989). ThisThis isis inin concontrasttrast toto thethe degenerateddegenerated opticaloptical phasephase velocity velocity surface surface for for crystals crystals of of cubic cubic symmetry. symmetry.

Fig.Fig. 2.4. 2.4. Differently Differently oriented oriented symmetry symmetry coordinate coordinate systems systems with with respectrespect to to the the sample sample coordinate coordinate system system . .The The blue blue face face denotes denotes a a plane plane parallelparallel to to the the --plane.plane.

The combinationThe combination of the 90A of scattering the 90A geometry scattering with geometry angle- resolved with angle BRILLOUIN-resolved spectroscopyBRILLOUIN spectroscopyis especially suited is especially for determining suited for the determining elastic tensor the of elastic low-symmetry tensor of transparentlow-symmetry anis transparentotropic materials; anisotropic a materials; task which a task is hardlywhich is possible hardly possible by other by experimentalother experimental techniques techniques (e.g. K (e.g.RÜGER KRÜGER et al. et1990, al. 1990, 1994, 1994, 2001). 2001). A prerequisite A prerequisite for thisfor this method method is theis the availability availability of of plate plate-like-like samples samples of of the the same same material, material, eacheach havinghaving aa differentlydifferently orientedoriented symsymmetrymetry coordinatecoordinate systemsystem withwith respectrespect

2221 SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial SCANNING BRILLOUIN MICROSCOPY Arch. Sci. Nat. Phys. Math. NS 46, volume spécial Scanning Brillouin microscopy Arch. Sci. Nat. Phys. Math. NS 46, Special volume

to the sample coordinate system (see figure 2.4). As elucidated in the next chapter, rotating the sample plate around an axis being normal to the sample plane and confining the phonon wave vector within this plane yields the acoustic indicatrix of this sample cut (KRÜGER et al. 1985, 1986). The determination of the acoustic indicatrices for a suitable choice of different crystal cuts provides a representative set of elastic data sufficient to calculate, on the base of the CHRISTOFFEL equation, the components of the elastic modulus tensor. This will be exemplarily illustrated in chapter 4 for several crystalline materials.