Low-frequency vibrational spectroscopy of glasses
JJanuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 1
Chapter 8Low-frequency vibrational spectroscopy of glasses
Benoit Ruffl´e, Marie Foret and Bernard Hehlen
Laboratoire Charles Coulomb (L2C), University of Montpellier,CNRS, Montpellier, Francebenoit.ruffl[email protected]
Atomic vibrations in perfect, slightly defective or mixed crystals areto a large extent well understood since many decades. Theoretical de-scriptions are thus in excellent agreement with the experiments. As aconsequence, phonon-related properties like specific heat, thermal con-ductivity or sound attenuation are also well explained in these solids.This is not yet the case in glasses where the lack of periodicity generatesenormous difficulties in theoretical treatments as well as in experimentsor in numerical simulations. Thanks to recent developments along allthese lines, comprehensive studies have emerged in the last decades andseveral decisive advances have been made. This chapter is thus devotedto a discussion of the nature of the vibrational properties in glasses withparticular emphasis on the low-frequency part of the vibrational densityof states, including the acoustic excitations, and of the experimentaltechniques used to their study.
1. Introduction1.1.
Thermal properties
The lack of periodicity in glasses implies that no reciprocal lattice can bedefined. As a consequence, the wavevector q is no longer a good quantumnumber for the description of the vibrational excitations and the phononstates cannot be characterized by dispersion curves ω ( q ). The quantitywhich can be safely used is the vibrational density of states g ( ω ) (vDOS),which is the distribution of the number of vibrational modes as a functionof the frequency ω . This quantity is of paramount importance as it alsomakes the connection between the microscopic and macroscopic aspectsof atomic vibrations. The isochoric heat capacity C v ( T ) is for example a r X i v : . [ c ond - m a t . d i s - nn ] J a n anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 2 B. Ruffl´e, M. Foret & B. Hehlen directly derived from g ( ω ).The second macroscopic quantity of great interest is the thermal con-ductivity which relates heat transport and propagative elastic waves indielectric crystals. Early measurements of these two thermal propertiesrapidly showed evidences of significant differences when compared to thecrystalline counterpart. It is then natural to start the chapter with a veryshort review of what is usually called the low temperature thermal anoma-lies of glasses (see Chapter 2 and 3 for an extended discussion of the latter).1.1.1. Heat capacity
The temperature dependence of the heat capacity of crystalline dielectricsis well understood. Well below the Debye temperature θ d , the isochoricheat capacity C v follows the Debye law, C v = C d T , where C d ∝ θ − d isgiven by the sound velocities and the atomic density. At higher T , C v showsa slower increase depending on the details of the atomic structure throughthe phonon dispersion relation, then flattens eventually to the classicalDulong–Petit limit at high temperature.Figure 1a shows the reduced heat capacity C p /T at low temperature ofthree SiO polymorphs: vitreous silica, α -quartz and α -cristobalite. Below5 K, C p /T of α -quartz follows the expected constant Debye value whereasthe bump at higher T is fully described by the curvature of the acousticphonon branches. According to their comparable densities and structure,it is however more appropriate to compare silica glass with α -cristobalite.For that crystal, the constant Debye value of C p /T holds to about 2 Kwhile the large increase at higher T , giving rise to a large bump around13 K, originates essentially from the flattening of the low-velocity trans-verse acoustic branch in the < > direction [1, 2]. Hence both crystallinepolymorphs nicely illustrate the general trends mentioned above with char-acteristic temperatures related to their local order and atomic packing. Itis also worth recalling that optic modes contribute to C p . The Raman ac-tive zone center mode around 50 cm − [3] of α -cristobalite is for exampleexpected to strengthen C p just above the maximum of C p /T .Conversely, the heat capacity of vitreous silica is much higher than theDebye expectation at 2 K and further never reaches it at low T , indicatinglow-frequency extra modes. At the lowest T , most of this excess originatesfrom tunneling states giving rise to an almost linear contribution C p ∝ T ,see Chapter 2. Subtracting this contribution to C p still leaves an excess overthe Debye prediction as illustrated by the circles and the dashed region in anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 3 Low-frequency vibrational spectroscopy of glasses a Temperature (K) × C p / T ( J m o l - K - ) α -cristobalite α -quartzvitreous silica 10 -2 -1 ∼ T ∼ T ∼ T -1 k c ⊥ c b Temperature (K) κ ( W m - K - ) Ω -4 λν (THz) ‘ ( n m ) Fig. 1. a) Low-temperature heat capacity of vitreous silica (dots [4–9]), α -cristobalite(squares [1, 4, 5]) and α -quartz (triangles [1, 4, 8]) plotted as C p /T against T . Thehorizontal lines are the corresponding constant Debye values (dashed, dotted and solidline, respectively). The circles show the heat capacity of vitreous silica lessened by thetwo-level systems (TLS) contribution [10]. b) Temperature dependence of the thermalconductivity of vitreous silica (dots [8, 11, 12]) and α -quartz (up and down trianglesfor the two principal directions [8, 13–15]). The square is a room T measurement on α -cristobalite [16]. Inset: Frequency dependence of the mean free path of vitreous silicain the dominant phonon approximation. Fig. 1a. At higher T , C p /T of vitreous silica displays a peak around 10 Kanalogous to that of α -cristobalite, in agreement with their similar structureand atomic density.Such common features indicated very early [1] that a peak in C p /T atlow T is not a peculiarity of glasses [17, 18] although it does not precludethat part of the vibrational spectrum responsible for this peak in glassesmay be very different in nature from that in the corresponding crystal.Oppositely to the case of α -cristobalite, it is for example not possible toreconcile the low- T part of the broad hump in C p /T of vitreous silica withthe known linear dispersion of sound waves up to at least 440 GHz [19], thelatter giving a constant Debye value up to ∼ . C p /T of glasses is directly related to the peakthat shows up in the reduced vibrational density of states g ( ω ) /ω (mostlynot in g ( ω ) itself) and originally referred to as the boson peak in Ramanspectroscopy, see Sec.4.2. anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 4 B. Ruffl´e, M. Foret & B. Hehlen
Thermal conductivity
The temperature dependence of the thermal conductivity κ of crystalline di-electrics is also well understood. Using the standard gas kinetic equation, κ is approximately described in these solids by κ = C v v(cid:96) , where C v isthe specific heat per volume of the acoustic phonons providing the thermaltransport, v their velocity of propagation, and (cid:96) their mean free path. Start-ing from ambient temperature, κ increases with decreasing temperatures as (cid:96) increases: phonons are scattered by anharmonic umklapp processes whichbecome less frequent as there are less and less phonons. At a certain point, (cid:96) reaches sample dimensions and κ decreases rapidly with the T depen-dence of C v . Figure 1b illustrates such an ideal behavior in the case of α -quartz in the c -axis direction. It also shows a room temperature mea-surement of the thermal conductivity of α -cristobalite. Interestingly, thevalue ∼ − K − is almost identical to that obtained perpendicularlyto the c -axis in α -quartz, suggesting a similar behavior.In striking contrast, dielectric glassy materials are known to exhibitmuch lower thermal conductivities with a markedly different temperaturedependence. As an example, κ of vitreous silica is plotted in Fig. 1b. Itfollows an approximate T law at very low T instead of the T dependenceexpected from the Debye approximation. This initial T rise originates frominteractions between phonons and tunneling states, reducing (cid:96) in glasses [10,20, 21]. At higher T , κ displays a remarkable plateau around 10 K, atemperature close to the hump in C p /T . It was early recognized that avery efficient phonon scattering mechanism was needed to generate such aplateau, at least with (cid:96) ∝ ω − [22], suggesting a Rayleigh-type scatteringmechanism arising from the glass disorder.Based on a dominant phonon approximation (cid:126) ω ∼ . k b T , the fre-quency dependence of the mean free path can be estimated from the above-mentioned kinetic equation of κ . The resulting (cid:96) ( ω ) for vitreous silica isplotted in the inset of Fig. 1b as a line. The mean free path exceeds1 µ m below 100 GHz, a value much larger than the acoustic wavelength λ a illustrating the propagative character of these acoustic phonons. Inthis frequency range, resonant relaxation by tunneling states dominatesthe acoustic attenuation at the corresponding temperatures, i.e., below 1 Kas seen on the top x-scale, and leads to (cid:96) ∝ ω − . At higher frequencies, (cid:96) drops rapidly then follows an approximate ω − trend. Around 1 THz, (cid:96) is comparable to λ a , indicating that the Ioffe–Regel criterion (cid:96) = λ a / (cid:96) is here the energy mean free path). It anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 5 Low-frequency vibrational spectroscopy of glasses corresponds to the propagation of the sound wave over a distance less thanhalf of its wavelength. Above the corresponding frequency ω ir , sound wavesdo not propagate and cannot transfer energy anymore. The wave vector isno longer a good quantum number and the notion of phonon becomes ill-defined. At higher temperatures, κ rises again and eventually saturates atvalues around ∼ − K − , following approximately C p ( T ). As the heattransport cannot be mediated anymore by propagating sound waves, it isgenerally admitted that κ is governed here by diffusion mechanisms [23, 24]. Vibrations in glasses
In many aspects, the vibrational spectrum of glasses is similar to that oftheir crystalline counterparts. However, the anomalous thermal propertiesdisplayed by the disordered solids indicate that some fundamental differ-ences must exist. Well-defined low-frequency sound waves also exist inglasses as their wavelength are much larger than the disorder length-scale.At higher frequencies, the rapid decrease of the acoustic phonon mean freepath related to the plateau in κ around 10 K seems however specific todisordered solids. Whether and how this peculiar scattering of the acous-tic phonons occurring at high frequencies is related to the excess abovethe Debye expectation in the low-temperature specific heat at temperaturebelow the maximum in C p /T remains a central and still unsettled ques-tion. It is further interesting to note that while similar humps do exist inthe low-temperature reduced specific heat both for vitreous silica and α -cristobalite, indicating similar vibrational density of states in that region,thermal transport mechanisms at the nanometer scale seem very differentin the two polymorphs.Section 2 briefly describes the ( ω, q ) range which must be addressed bythe experimental techniques as well the quantities probed using scatteringtechniques. Section 3 is devoted to the spectroscopy of acoustic phononswhereas Section 4 discusses the different experimental techniques aimingat measuring the low-frequency part of the vibrational density of states, inparticular in the boson peak region.
2. Inelastic spectroscopy2.1.
Dispersion diagram and experimental techniques
Figure 2 displays a ( ω , q ) map of the main experimental techniques thatcan probe vibrational excitations in disordered solids. The diagonal region anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 6 B. Ruffl´e, M. Foret & B. Hehlen -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 -4 -3 -2 -1 U S B L S I U V S P U A c o u s t i c m o d e s RS,IR
Optic modes q ir IXS INSWavevector (˚A − ) F r e q u e n c y ( T H z ) Wavelength (˚A) E n e r g y ( m e V ) Fig. 2. Dispersion diagram and ( ω , q ) windows of experimental techniques probing vi-brational excitations in disordered materials. Ultrasonics (US), Brillouin light scattering(BLS), Raman scattering (RS), infrared absorption (IR), inelastic scattering with ultra-violet radiation (IUVS), picosecond acoustics (PU), inelastic neutron (INS) and x-rayscattering (IXS). The horizontal dashed lines mark the domain of network optic vibra-tions in glasses while the diagonal band indicates approximately the domain of acousticexcitations, merging with the optic ones at high q . represents the linear dispersion relation for sound waves of velocity between1 km s − and 6 km s − . Brillouin light scattering (BLS), inelastic scatteringwith ultra-violet radiation (IUVS), x-rays (IXS) or neutron (INS) probean increasing well defined q range associated with an increasing ω rangeaccording to the linear dispersion law. Conversely, ultrasonics (US) is nota scattering experiment but covers specific frequency ranges. Accordingly,no scattering vector q is really defined for this technique. Picosecond ul-trasonics (PU) forms actually a family of optical techniques using pulsedlasers and belongs to the two categories depending on the setup. Concern-ing the optic modes, the ( ω , q ) ranges covered by scattering techniques thatcan probe these excitations and measure the vibrational density of statesare shown approximately as vertical regions in Fig. 2. In increasing q rangeorder one finds Raman scattering (RS), IXS and INS. Due to kinematicsconstraints, the ( ω , q ) range covered by INS is somewhat limited as com-pared to the one covered by IXS as shown by the approximate trapezoidalshape in Fig. 2. Additionally, there also exists non-scattering techniqueslike infra-red (IR) spectroscopy or nuclear inelastic scattering (NIS) whichgive informations on the vibrational density of states in a q region related anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 7 Low-frequency vibrational spectroscopy of glasses to the incoming radiation.Even if much of the dispersion diagram seems nowadays accessible forstudying acoustic excitations from the macroscopic length scale down tothe atomic one, it is far from being the case for most glasses. Moreover,it is often feasible for the longitudinal acoustic mode only. As mentionedin the preceding section, of peculiar interest is the mean free path of theacoustic phonons, a quantity often difficult, if not impossible, to obtainwith a useful accuracy. This is dramatically true for the transverse modes,in particular in the high q range above about 5 × − (cid:6) A − , approachingthe Ioffe–Regel crossover region, defined by q ir in Fig. 2. This is all themore important as the transverse modes largely dominate the Debye densityof states. Consequently, study of the longitudinal modes will give onlyincomplete informations.A crucial point revealed by Fig. 2 is that the acoustic linear dispersioncurves merge with the low frequency optic vibrations in the Ioffe–Regelcrossover region, where stands the boson peak feature around 1 THz. Thisexplains why, concomitantly to the study of the acoustic excitations, a verylarge effort of the scientific community has been devoted to the under-standing of the low-frequency part of the vibrational density of states ofdisordered materials. Inelastic scattering intensity
The incident radiation impinging on a material is inelastically scatteredwith the intensity I ( q , ω ) corresponding to the space and time Fouriertransforms of the autocorrelation function of the physical quantity A whichcouples to the incoming radiation: I ( q , ω ) ∝ F T (cid:26)(cid:90) V (cid:90) t A ( r , t ) A ( r + r (cid:48) , t + t (cid:48) ) d t d r (cid:27) = C ( A (cid:48) ) S ( q , ω ) , (1)where A (cid:48) is the derivative of A over the atomic displacements U of the modewhile the correlation function C ( A (cid:48) ) contains the selection rules. The latterdepend on the strength of the coupling of the incident radiation to a vi-bration and therefore modulate the intensity of its spectral shape S ( q , ω ).The physical quantity A is the electronic density ρ e for X-ray scatteringwhereas it stands for the coherence length b for neutron scattering and thedielectric susceptibility χ for light scattering. In the latter case, the dy-namical dielectric susceptibility χ (cid:48) is usually expressed in terms of the po-larizability tensor ¯¯ α (cid:48) (Raman scattering), the hyper-polarizability tensor ¯¯¯ β (cid:48) anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 8 B. Ruffl´e, M. Foret & B. Hehlen (hyper-Raman Scattering), and the photo-elastic tensor ¯¯¯¯ p (cid:48) (Brillouin scat-tering). All the atomic vibrations can be detected using neutron scattering,enabling the phonon dispersion curves to be measured over a large ( q , ω )range, however limited by the kinematic conditions and the instrumentalresolution. The normalized integral over all q values leads to the numberof vibrations per unit frequency, also called the neutron vibrational densityof states g ( ω ) (vDOS). The latter is however modulated by the coherentlength of the atoms and remains an approximated quantity, in particularfor polyatomic systems.As mentioned before, the vibrations are very well defined in crystals soare their spectral response functions. In that case, the group theory pro-vides the mathematical support to compute the selection rules containedin C ( A (cid:48) ) for all scattering experiments. The situation is again much morecomplicated in glasses owing to the structural disorder. Each elementaryunit of the network is connected to neighboring units, inducing strong struc-tural local distortions. The modes of a given symmetry may also involvedifferent number of neighboring units. These effects modulate C ( A (cid:48) ) and S ( q , ω ) from site to site inhibiting up to now the development of a generaltheory describing the vibrational spectra in glasses. Coherent and incoherent scattering
The relative spatial localization of extended waves in disordered media in-duces a loss of coherence of the scattering process due to the partly pointlessvibration wavevector Q . Acoustic or optic phonons in perfect crystals areplane waves with atomic displacements defined by: U ( r , t ) = u e ± i ( Qr − Ω t ) . (2)They are extended modes characterized by a spatial extension larger thanthe probed wavelength. The scattered radiation results from the interfer-ence integral over all the modes in the scattering volume V , which in caseof light scattering reads: E s = (cid:90) V δ P e − i k s r dV, (3)where δ P = (cid:15) χ (cid:48) U ( r , t ) E i is the modulation of the electrical polarizationinduced by the incident field E i = E e i ( k i r − ωt ) . Putting Eq. 2 in Eq. 3leads to E s ∝ E (cid:90) V χ (cid:48) u e i (( k i − k s ± Q ) r − ( ω i ∓ Ω) t ) d r , (4) anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 9 Low-frequency vibrational spectroscopy of glasses where r integrates over all orientations so that E s is zero unless k i − k s ± Q = 0, i.e., for momentum conservation. For a given scattering geometry,only the vibrations whose wavevector Q matches the one of the experiment Q = ± ( k i − k s ) = q will be active, denoting a coherent scattering process.On the extreme opposite side, the atomic displacements of fully local-ized atomic motions are defined by U ( t ) = u e ± i Ω t as Q is not a relevantquantum number. k s r is constant over the very small spatial extension ofthe mode, and if χ (cid:48) is also supposed to be constant over the volume V (gases and non-viscous liquids), Eq. 4 transforms into E s ∝ χ (cid:48) u E e − i ( ω i ∓ Ω) t , (5)showing that the vibration scatters at all q values, indicating incoherentscattering.For intermediate situations, i.e., damped plane waves or quasi-localizedwaves in disordered solids, the atomic displacements of these vibrationscan be modeled by U ( r , t ) = u e − r/(cid:96) c e ± i ( Qr − Ω t ) , where (cid:96) c is the coherencelength of the vibrational mode. For sufficiently small values of l c , the spatiallocalization of the mode induces a wavevector spectral broadening ∆ Q . Inthat case, the contribution at the spectral frequency ω is the sum overall modes of frequency Ω = ω having a wavevector spectral component Q matching the scattering wavevector q of the experiment. The aboveconclusion holds in glasses for optic modes as well as for short wavelengthacoustic waves, since the latter progressively transform into quasi-localizedvibrations at THz frequencies.Alternatively, high frequency acoustic modes in amorphous solids canbe treated by considering plane waves propagating in an inhomogeneouselastic medium [25] where χ (cid:48) is position-dependent. It leads to a spatialmodulation of the photoelastic tensor which now reads ¯¯¯¯ p (cid:48) ( r ). The product¯¯¯¯ p (cid:48) ( r ) U ( r , t ) in δ P yields to distorted acoustic waves whose coherence length (cid:96) c is limited by the local elastic inhomogeneities. The full treatment showsthat the light scattering spectrum consists of the usual Brillouin peaks, i.e.,coherent scattering, and a ω incoherent background induced by these localheterogeneities.
3. Spectroscopy of acoustic phonons3.1.
Attenuation and sound velocity in dielectric crystals
In perfect dielectric crystals, it is well-known that the dominant mechanismfor sound attenuation originates from the interaction between the acous- anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 10 B. Ruffl´e, M. Foret & B. Hehlen tic wave and high frequency phonons constituting the thermal bath [26].This interaction also leads to a significant variation of the sound velocity.Within the original Akhiezer relaxation model [27], it is assumed that thetemperature is sufficiently high for the bath to be appreciably populated.The thermal phonon mode frequencies Ω i in equilibrium are then perturbedby a sound wave of frequency Ω by γ i = ∂ ln Ω i /∂e , where e is the straindeveloped by the sound wave. As a rough estimate, the γ i can be rational-ized by using a mean-square average Gr¨uneisen parameter γ = γ i . Theperturbed thermal bath relaxes towards equilibrium with a characteristicthermal time τ th via anharmonic interactions and dissipates the energy ofthe sound wave with an internal friction Q − (Ω , T ) = A Ω τ th τ , (6)where A = γ C v T v/ ρv d [26]. In the latter equation, C v is the specificheat per unit volume, ρ is the mass density, and v d is the Debye velocity.The temperature dependence of the thermal phonon lifetime τ th canbe roughly estimated from κ using the kinetic equation κ = C v v d τ th .The corresponding values for crystal quartz are shown in Fig. 3a as a solidline. As τ th is typically short and further decreases rapidly with T , asignificant damping of the sound wave is expected at high frequencies only.In the case of quartz, (2 πτ th ) − (cid:39) τ th (cid:29) Q − with increasing T is essentiallydriven by the temperature dependence of C v , leading to A ∝ T . At highertemperature, both A and τ th evolve moderately so does Q − . In the latterregion Ω τ th (cid:28) Q − ∝ Ω. The acoustic damping α ∝ Ω is thusbest observed using high frequency techniques. Another estimate of τ th can be extracted from the measured Q − and Eq. 6 using a temperatureindependent Gr¨uneisein parameter γ (cid:39) τ th as a mean lifetime of the vibrational excitationsin the thermal bath.The frequency-dependent damping induces a frequency-dependent cor-rection to the sound velocity via the Kramers–Kronig transform δvv (Ω , T ) = − A τ , (7)where δv = v − v ∞ . In a crystal, one expects that v ∞ , the high frequencylimit of v , is constant in T . For Ω τ th (cid:28)
1, which is the case in ultrasonics for anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 11
Low-frequency vibrational spectroscopy of glasses -12 -11 -10 -9 -8 a α -quartzTemperature (K) T h e r m a l r e l a x a t i o n t i m e τ t h ( s ) b α -quartz1 GHZTemperature (K) I n t e r n a l f r i c t i o n × Q - − . − . − . − . − . c α -quartz35 MHZTemperature (K) R e l. v e l o c i t y c h a n g e s × δ v / v Fig. 3. a) Temperature dependence of the thermal relaxation time τ th in α -quartz.The line is calculated from the kinetic equation κ = C v v d τ th while the symbols areobtained from Q − of panel b) using Eq. 6 and γ (cid:39) α -quartz [28]. c) Fractional velocity changes of 35 MHzshear waves in BC-cut α -quartz [30]. The line is obtained using τ th ( T ) from the kineticequation and γ (cid:39) . crystal quartz, one simply has δv/v = − A/
2. Such a decrease of the soundvelocity with T is illustrated in Fig. 3b for 35 MHz shear waves in BC-cutquartz [30]. It compares well with the predicted temperature dependenceof δv/v computed from Eq. 7 and τ th from κ using γ (cid:39) . Attenuation and sound velocity in glasses
As mentioned before, sound waves are key properties to understand thelow-temperature thermodynamic anomalies of disordered systems. In spiteof its long history, the subject remains of much actuality and activity, par-ticularly at high frequencies in relation with nanoscale thermal transport.Other additional mechanisms do affect sound velocity and attenuation inglasses. The direction of the present paragraph is to review progress dur-ing the last decades in the understanding of the frequency and temperaturedependencies of acoustic properties in glasses and which experimental tech-nique can be used to that purpose. anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 12 B. Ruffl´e, M. Foret & B. Hehlen
Ultrasonics and mechanical techniques
At sonic and ultrasonic frequencies there are several well established meth-ods to measure the sound velocity v (Ω , T ) and the energy sound dampingconstant α (Ω , T ) related to the energy mean fee path (cid:96) = α − or to theinternal friction Q − = αv/ Ω. Figure 4 shows examples of internal frictionand velocity changes in dielectric glasses measured at ultrasonic frequenciesas a function of temperature measured in the late 1960’s [31]. Oppositelyto the case of crystalline quartz plotted in Fig. 3b the internal friction inthese glasses increases rapidly at low temperature, Q − ∝ T , evidencinga clear maximum which depends on the material. Moreover, the values of Q − observed here are about two orders of magnitude larger than in quartz,while the frequency is 50 times lower. It is clear that the internal frictionshown in Fig. 4a must originate from another damping mechanism ratherthan anharmonic interactions. a Temperature (K) I n t e r n a l f r i c t i o n × Q - GeO Zn(PO ) B O BeF SiO − − b Temperature (K) R e l. v e l o c i t y c h a n g e s × δ v / v Fig. 4. a) Internal friction and b) fractional velocity changes of 20 MHz longitudinalwaves in different glass formers, adapted from [31].
There is also marked differences in the low- T decay of the relativechanges in the sound velocity illustrated in Fig. 4b as compared to Fig. 3cfor quartz. According to the Kramers–Kronig relation with the internalfriction, the onset in δv/v is much slower in crystalline quartz than in theseglasses. Further, the velocity changes with T show a clear minimum forfour of the five glasses. This anomalous behavior is specific to tetrahedrallycoordinated glasses [31] and is related to progressive structural changeswith increasing T . In contrast, the velocity changes in B O at sufficiently anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 13 Low-frequency vibrational spectroscopy of glasses high T are similar to those observed in quartz. It was indeed soon iden-tified that anharmonicity was governing the velocity changes in glasses inthis temperature range [32, 33], except for the few that are tetrahedrallycoordinated.The ultrasonic absorption peaks observed in glasses at low- T are usu-ally described using a thermally activated structural relaxation process(TAR) represented by a broad distribution of asymmetric double-well po-tentials [20, 21, 34–36]. Owing to structural disorder, several atoms orgroups of atoms can occupy two or more equilibrium positions. In that pic-ture, their displacements correspond to these double well potentials. Fromsite to site, the heights V of the potential barrier and their asymmetries∆ are distributed according to a probability P (∆ , V ). The sound wavecouples to the defect site via a deformation potential γ = ∂ ∆ ∂e , breakingthe thermal equilibrium population which relaxes within the characteristictime τ = τ exp (cid:18) VT (cid:19) sech (cid:18) ∆2 T (cid:19) , (8)where 1 /τ is the attempt frequency. Integrating over all the thermallyactivated relaxations, the internal friction can be calculated following asimilar approach used to described anharmonicity leading to [29, 37] Q − tar = γ ρv T (cid:90) ∞−∞ d∆ (cid:90) ∞ d V P (∆ , V )sech ∆2 T (cid:18) Ω τ τ (cid:19) . (9)The most advanced analysis so far have been carried out on vitreoussilica in the early 2000s [29]. This is illustrated in Fig. 5a which shows thetemperature dependence of the internal friction at several representativesonic and ultrasonic frequencies. The peak position depends logarithmicallyon T [38], supporting thermal activated relaxation as the main source of theinternal friction. All the data set are simultaneously adjusted with Eq. 9using for ∆ and V independent gaussian distributions with cutoff ∆ c and V , respectively. The same set of parameters is also used to described thelow- T part of the sound velocity variations using the following equation, (cid:18) δvv (cid:19) tar = − γ ρv T (cid:90) ∞−∞ d∆ (cid:90) ∞ d V P (∆ , V )sech ∆2 T (cid:18)
11 + Ω τ (cid:19) , (10)which is the Kramers–Kronig transform of Eq. 9. Interestingly, the threedata set covering nearly five decades, corrected from this relaxational ef-fect, collapse on a single curve at low temperature revealing that the diparound 50 K originates from dynamical effects. Oppositely the anomalous anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 14 B. Ruffl´e, M. Foret & B. Hehlen continuous increase of the sound velocity at high temperatures is specificto the tetrahedral networks, thus of structural origin. v -SiO a Temperature (K) I n t e r n a l f r i c t i o n × Q -
207 MHz [33]20 MHz [39]180 kHz [40]11 . v -SiO b Temperature (K) R e l. v e l o c i t y c h a n g e s × δ v / v
35 GHz [29]6 . Fig. 5. a) Internal friction of v -SiO at sonic and ultrasonic frequencies as a functionof temperature taken from the literature [33, 37, 39, 40]. The lines show the adjustmentof the data to Eq. 9. b) Fractional velocity changes of v -SiO at sonic, ultrasonic andhypersonic frequencies [29, 33, 36] (solid symbols). The lines show the adjustment oflow- T part of the data to Eq. 10 whereas the smaller hollow symbols are the experimental δv/v corrected from the ( δv/v ) tar effect. The parameters used to draw the lines in Fig. 5 are V (cid:39)
660 K, ∆ c (cid:39)
80 K, τ (cid:39) . C = 1 . × − related to γ and the density of relaxing defects [29], the latter being es-timated at N (cid:39) . − . This phenomenological description of internalfriction at sonic and ultrasonic frequencies recently found a strong supportfrom atomic-scale modeling [41]. In that work, statistical properties of thetwo-level systems responsible for sound absorption above 10 K are obtainedfor a model of vitreous silica. The agreement with the above analysis isvery good, both qualitatively and quantitatively. Additionally, the micro-scopic nature of the relaxing defects is revealed. It involves the cooperativerotation of Si − O − Si bonds, see Fig. 6 in [41]. Similar results have been ob-tained later for some other glasses like GeO [42], B O [43, 44], where theboroxol rings were identified as the units subjected to thermally activatedrelaxations or binary borate glasses [45, 46]. anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 15 Low-frequency vibrational spectroscopy of glasses Brillouin light scattering
Increasing frequency up to the hypersonic regime, it was soon anticipatedthat anharmonicity should contribute significantly to the sound attenua-tion in glasses as it is the case in dielectric crystals (see Sect. 3.1) [33].In this frequency domain, typically in the range of several tens of GHz,Brillouin light scattering (BLS) is a method of choice for sufficiently trans-parent glasses. BLS deals with light inelastically scattered by the densityfluctuations exiting in the probed material which are related to the ther-mally activated acoustic modes. The scattering vector q is fixed by thegeometry, the laser wavelength λ and the refractive index n of the mate-rial, q = 4 πn sin( θ ) /λ , where θ is the angle between the incident and thescattered light directions. It also corresponds to the wavevector Q of theobserved acoustic mode Q = 2 π/λ a , where λ a is the acoustic wavelength.BLS technique measures a Brillouin shift frequency, which is actually thefrequency Ω of the probed sound wave. Ω relates to the acoustic velocity v through Ω = v/q . The linewidth Γ of the Brillouin spectrum quantifiesthe acoustic energy attenuation α at the frequency Ω or the inverse energymean free path (cid:96) -1 by α = (cid:96) -1 = Γ /v where Γ is expressed in angular fre-quencies, just like Ω. The internal friction is actually a direct output ofa BLS experiment as Q -1 = Γ / Ω. The ratio Ω / Γ is thus the quality fac-tor of the damped harmonic oscillator (DHO) used to adjust the Brillouinspectrum.Typical BLS experiments are nowadays carried out using a tandemmulti-pass plane Fabry–Perot interferometer (T-FPI) developed 40 yearsago by J. R. Sandercock [47]. This spectrometer is characterized by a highcontrast, better than 10 thanks to the six Fabry–Perot (FP) etalons inseries, and a high flexibility due to the tandem arrangement. It gives afrequency resolving power of about 2 × corresponding to a spectralbandwidth of 250 MHz (HWHM) for a Brillouin frequency shift of 30 MHz.Compared to the very abundant ultrasonic results, there exist relatively fewdamping data at frequencies above 1 GHz in glasses. This is essentially dueto the fact that the frequency resolving power of the T-FPI hardly meetsthe demanding needs for proper sound attenuation measurements in glasseswell below the glass transition temperature.This experimental difficulty can be overcome by using a Brillouin spec-trometer consisting in a tandem arrangement of a multi-pass plane FPetalon followed by a confocal one [48]. The plane FP acts here as a prefiltereffectively eliminating other spectral components and reducing consider- anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 16 B. Ruffl´e, M. Foret & B. Hehlen ably the elastic signal strength. It is maintained at a fixed spacing whichis dynamically adjusted to one of the Brillouin lines with the help of anelectro-optically modulated signal at the Brillouin frequency. The confocalFP is piezo-electrically scanned for the spectra analysis. Different confocalFPs can be used to adjust the spectrometer resolution which is rangingfrom 10 MHz to 100 MHz. In this way a high contrast (better than 10 )and a high-frequency resolving power (10 − ) is obtained.As far as sound damping is concerned, most BLS experiments relateactually to the longitudinal acoustic (LA) mode. This is due to the factthat experiments are generally performed in the backscattering geometryoffering several advantages: (i) higher Brillouin frequency shift; (ii) loweruncertainty of velocity measurements; (iii) lower geometrical broadening ofthe Brillouin line due to the finite aperture. Selection rules for Brillouinscattering indicate however that transverse acoustic modes are not activein the backscattering geometry for isotropic materials. One concludes thathigh-resolution BLS gives accurate and absolute measurements of Q -1 asboth the Brillouin frequency shift Ω and the Brillouin linewidth Γ can beobtained simultaneously.We pursue with the case of vitreous silica which is by far the most stud-ied system. Figure 6a shows the internal friction of v -SiO at hypersonicfrequencies as a function of the temperature. A maximum of the dampingcoefficient is observed around 150 K, similar to what is shown in Fig. 5a.However, Q -1 does not decrease rapidly to zero above that maximum butremains at a high level up to the highest temperatures, indicating the ap-pearance of another damping mechanism in this frequency range. FromEq. 9, we can anticipate that Q -1 tar cannot account for the entire Q -1 . Atthe peak position corresponding to ωτ (cid:39) Q -1 tar is indeed nearly inde-pendent of Ω. Second, anharmonicity is still in the regime Ω τ th (cid:28) Q -1 anh ∝ Ω. This is illustrated inFig. 6a where the dash-dotted line is the Q -1 tar values calculated with theparameters given in the preceding section for the TAR damping mechanismin v -SiO to which a small contribution from the incoherent scattering ofthe two-level systems is added ( Q -1 tls dotted line, see Chapter 4). The dif-ference, Q -1 anh , is the contribution due to the anharmonic coupling of theLA waves with the thermally excited vibrational modes. It clearly appearsthat at sufficiently high temperature and frequency, Q -1 anh takes over Q -1 tar .The analysis of the anharmonic damping has been done using Eq. 6 with A ( T ) = γ C v T v/ ρv d [29]. The amplitude of Q -1 tar determines the value ofthe Gr¨uneisen parameter γ = 3 .
8, the unique unknown parameter entering anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 17
Low-frequency vibrational spectroscopy of glasses A ( T ) while the temperature dependence of the thermal time τ th is extractedfrom the shape of Q -1 tar ( T ). The values of τ th are proportional to T -2 inthe approximate range 100 K to 300 K, evolving towards a T -1 behavior athigher temperatures in a similar way to those of α -quartz plotted in Fig. 3but three time shorter. It is important to stress again at that point thatcontrarily to the case of α -quartz, it is not possible to obtain τ th ( T ) in thattemperature range from the thermal conductivity. The kinetic equation κ = C v v(cid:96) is indeed valid only for propagating modes, a condition that ispresumably no more fulfilled beyond the plateau in κ ( T ) located around10 K in v -SiO . Finally, the validity of the analysis of the internal frictionsolely based on the TAR contribution at ultrasonic frequencies illustratedin Fig. 5 is verified a posteriori . Q -1 tls Q -1 tar Q -1 anh Q -1 tot v -SiO
34 GHz a Temperature (K) I n t e r n a l f r i c t i o n × Q - c Temperature (K) v ∞ ( k m s - )
34 GHz20 MHz6 .
35 kHz 34 GHz b × δ v / v Fig. 6. a) Internal friction of v -SiO at hypersonic frequencies as a function of temper-ature [29, 33, 49]. The solid line shows the adjustment of the curve to the sum of thethree damping processes Q -1 tot = Q -1 tls + Q -1 tar + Q -1 anh . b) Calculated fractional velocitychanges at sonic and hypersonic frequencies resulting from TAR and anharmonic pro-cesses. c) Unrelaxed velocities v ∞ after correction for TAR and anharmonic processesat four selected frequencies. The solid line is a Wachtman’s adjustment of the data. The effect of anharmonicity on the velocity dispersion can be calculatedfollowing Eq. 7. This is illustrated in Fig. 6b where the relative sound veloc-ity variations δv/v induced by anharmonicity are plotted for two differentfrequencies and compared to those due to the thermally activated relaxationprocesses. While ( δv/v ) tar strongly depends on the frequency below 100 K,( δv/v ) anh remains essentially unaffected by changing ν , in particular above100 K. At higher temperatures, the magnitude of ( δv/v ) anh increases al- anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 18 B. Ruffl´e, M. Foret & B. Hehlen most linearly with temperature. Linear decrease of the sound velocity haveindeed been observed in a large number of glasses [31, 32, 44, 50]. Fi-nally, Fig. 6c shows the unrelaxed values v ∞ of the frequency dependentmeasured sound velocity v that are corrected for the two damping pro-cesses, v ∞ = v − [ δv tar + δv anh ]. One observes that the four series of datacovering nearly five decades in frequency nicely collapse on a single curve v ∞ ( T ). The latter is not a constant but rather increases rapidly with T and can be adjusted to a Wachtman’s equation [51]. This is a peculiarity oftetrahedrally-coordinated glasses and associated to a progressive structuralchange occurring in the network when the temperature is raised [52].It is expected that the above-mentioned competing damping processesare not unique to vitreous silica. Even if accurate sound attenuation dataare scarce at hypersonic frequencies some attempts have been made in thatdirection. Figure 7 illustrates the case for two other systems. In Fig. 7a theinternal friction of a cesium borate glass at ultrasonic frequencies had beenadjusted to TAR alone following Eq. 9 [53]. Its extrapolation at hypersonicfrequencies, shown by the dotted line, largely fails to reproduce the Bril-louin light scattering data. Clearly, the BLS damping data must containan appreciable contribution from anharmonicity, shown by the dash-dottedline. The latter has been analyzed using Eq. 6 [45] leading to the determi-nation of τ th and of the mean Gr¨uneisen parameter for that glass as well.Figure 7b shows the case of vitreous germania, v -GeO , for which severalsound absorption experiments exist at low frequencies ensuring a fair es-timate of Q -1 tar . In a nutshell, an analysis similar to that done for v -SiO lead to comparable conclusions [42].This does not seem to be restricted to oxide glasses as illustrated bythe internal friction of poly(methyl methacrylate) (PMMA) obtained at ul-trasonic [54, 55] and hypersonic [48, 56] frequencies and plotted in Fig. 8a.It has been shown that the TAR model alone fails to explain consistentlythe internal friction in the Hz–GHz frequency domain above 10 K [60]. Theauthors concluded that an additional relaxational mechanism seems to ap-pear at high frequency, whose strength increases with T . They invoked aconstant loss spectrum ( Q -1 loss ∝ ω ) whose exact nature is not clear. Alter-natively, anharmonicity could explain the missing contribution to the sounddamping at high frequency, a possibility actually suggested by the authorsthemselves. Permanently densified glasses seems to show a much lower con-tribution from the TAR, likely related to the loss of the free volume [61]. Itoffers unique possibilities to study the sound attenuation originating fromanharmonicity at hypersonic frequencies. Fig. 8b illustrates this point with anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 19 Low-frequency vibrational spectroscopy of glasses Q -1 tar,us Q -1 tar,bls Q -1 anh (Cs O) . (B O ) . ( US, BLS) a Temperature (K) I n t e r n a l f r i c t i o n × Q - Q -1 tot Q -1 tar Q -1 anh Q -1 tls v -GeO ( BLS) b Temperature (K) I n t e r n a l f r i c t i o n × Q - Fig. 7. a) Internal friction of (Cs O) . (B O ) . at ultrasonic [53] and hyper-sonic [45] frequencies. The dotted line is the extrapolation at hypersonic frequenciesof the Q -1 tar solid line adjusted to the ultrasonic data. The dash-dotted line is the an-harmonic contribution to the damping resulting from the subtraction of Q -1 tar from theBLS data [45]. b) Internal friction of v -GeO at hypersonic frequencies. The solidline shows the adjustment of the BLS data to the sum of the three damping processes Q -1 tot = Q -1 tls + Q -1 tar + Q -1 anh [42]. PMMA ( US, BLS) a Temperature (K) × Q - glycerol ( BLS) d -SiO ( BLS) b Temperature (K) × Q - Fig. 8. b) Internal friction of PMMA at ultrasonic [54, 55] and hypersonic [48, 56]frequencies. b) Internal friction obtained using BLS in densified silica [57] and in glyc-erol [58, 59]. Note the different y-scale of the plots compared to Fig. 7. densified silica glass having the density of crystal quartz. In that case, theTAR are almost completely suppressed [61]. The internal friction measuredat GHz frequencies is indeed quite small as compared to that of normal sil-ica [57]. Hence, adjusting the measured Q -1 with Eq. 6 gives the solid linein Fig. 8b. The suppressing of Q -1 tar with densification seems also to hap-pen in vitreous boron oxide but accurate high resolution BLS data are still anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 20 B. Ruffl´e, M. Foret & B. Hehlen missing to properly estimate the anharmonicity in d -B O [44]. Finally,Fig. 72b also shows internal friction data for glycerol obtained using BLS.The damping coefficient is surprisingly low for a molecular glass and nowell defined peak, characteristic of TAR processes is observed. No attempthas been made yet to described these data using Eq. 6.3.2.3. Inelastic scattering of UV radiation
Brillouin scattering at shorter wavelength can be used to increase the Bril-louin shift frequency. This can be done using excitation with UV radia-tion [62, 63], as long as the samples are sufficiently transparent [64]. Thishas been attempted for example in glycerol, using the second harmonic ofa visible laser operating at 488 nm and the IUVS spectrometer located atthe Elettra synchrotron light source in Trieste, Italy [62]. In this operationmode, the total resolving power is 1 . × . This setup allows an increaseof Q by a factor of 2. Figure 9a shows the Brillouin linewidth at T = 150 K( T g = 187 K) extracted from the inelastic spectrum obtained with IUVSat (cid:39)
43 GHz. It is compared to the values obtained at lower frequenciesusing stimulated Brillouin gain spectroscopy (SBG) [65] and BLS [58, 59].The data show that, in this frequency range, the sound attenuation coeffi-cient closely follows a Ω dependence, as expected for anharmonicity, andin agreement with the temperature dependence discussed just before.Vitreous silica is known to have a rather high absorption edge at E i ≈ .
14 nm − . In that case, theIUVS spectrometer uses the UV beam produced by the synchrotron andthe total resolving power is 5 × . Figure 9b shows part of the first IUVSresults obtained at two wavelengths [62] or at two angles [63], plotted assolid symbols. The graph also shows high resolution BLS values obtainedwith visible light at three scattering angles [64]. The IUVS data remarkablyalign with the expected damping values calculated from Eqs. 6 and 9 withthe parameters used to described internal friction of vitreous silica at lowerfrequencies. One observes that the power law frequency exponent of Γ(Ω)is about 1.7 at 20 GHz, still slightly smaller than the quadratic behavior ofthe growing anharmonic contribution Γ anh .The understanding of the sound attenuation in vitreous silica and moregenerally in glasses was challenged by deep UV Brillouin scattering exper-iments, i.e., close to the absorption gap, claiming an unexpected dramaticincrease of the linewidth beyond 110 GHz in v -SiO [66]. Below this fre- anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 21 Low-frequency vibrational spectroscopy of glasses -2 -1 ∝ Ω a glycerolFrequency (GHz) L i n e w i d t h ( G H z ) SBG [65]BLS [58, 59]IUVS [62] 20 50 10010 -1 Γ tar + Γ anh Γ anh ∝ Ω b v -SiO Frequency (GHz) L i n e w i d t h ( G H z ) BLS [64]IUVS [62, 63]IUVS [66]IUVS [67]
Fig. 9. a) Frequency dependence of the Brillouin linewidth in glycerol using SBG [65],BLS [58, 59] and IUVS [62] at T = 150 K. The dashed line is the Γ ∝ Ω adjustmentof the data. b) Frequency dependence of the Brillouin linewidth in vitreous silica usingBLS [64] and IUVS [62, 63, 66, 67] at T = 300 K. The line is the theoretical expectationbased on Eqs. 6 and 9 used to describe the internal friction at lower frequency in v -SiO as explained in the preceding paragraphs. The dotted line is Γ anh only, showing aquadratic behavior. quency, these new data agreed very well with the preceding IUVS ones. Athigher frequencies, a much steeper increase was clearly observed that wasinterpreted as a crossover associated with elastic disorder. The latter washowever not detected by a second experiment carried out using a high gradesuperpolished silica sample up to about 125 GHz [67]. As we will see in thenext section devoted to picosecond ultrasonics, several other experimentscovering this frequency range also disproved the deep UV results. As aconsequence, only the deep UV Brillouin data which truly relate to acous-tic damping are plotted in Fig. 9b as hollow symbols. Brillouin scatteringclose to the absorption gap is clearly a delicate experiment. Approachingthe gap from below, the collected Brillouin signal considerably decreases,owing to rapidly increasing sample absorption of both the incident andthe scattered light. At first sight, the unexpected increase of the Brillouinlinewidth observed in the deep UV data above 110 GHz could be related tofinite-size effects induced by the dramatic increase of the imaginary part ofthe refractive index approaching the absorption gap [64, 68]. However, thiseffect is still negligible in the investigated region [67]. If not an artifact,the anomalous increase in damping could originate from growing refrac-tive index fluctuations close to the gap, producing an enhanced uncertainty anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 22 B. Ruffl´e, M. Foret & B. Hehlen broadening in Brillouin scattering [69].3.2.4.
Picosecond ultrasonics
There is, then, a limit set by the optical absorption edge to the properdetermination of acoustic attenuation in dielectric samples using Brillouinlight scattering. It corresponds to acoustic waves around 100 GHz in thefavorable case of vitreous silica. The samples remain then opaque to elec-tromagnetic radiation up to the x-ray region where they become againtransparent. Brillouin spectroscopy known as inelastic x-ray scattering al-lows then observation of sound properties in the THz frequency range, seeSec. 3.2.5.In the crucial frequency region between 50 GHz and 1 THz, there ex-ists but a single spectroscopy, known as picosecond ultrasonics (PU). Thisdelicate but very promising technique is essentially based on the genera-tion and the detection of GHz −− THz acoustic waves propagating througha thin glass film by ultrashort optical pulses, typically shorter than 1 ps.This optical technique has been pioneered in the 1980s [70] but first atten-uation measurements on a glass have been reported in the 1990s only [71].Generally, an ultrashort light pulse is focused onto a small area of a thinmetallic film deposited on the dielectric glass sample, which is itself oftena film deposited on a substrate. This light pulse, the pump, is absorbed bythe metallic layer which expands, generating a longitudinal strain pulse intothe glass. Typically, for a 12 nm thick aluminum film on vitreous silica, thelaunched acoustic pulse is a plane wave packet of about 20 nm spatial extentwhose Fourier spectrum peaks around 250 GHz [72]. The propagation ofthe acoustic pulse in the media and its reflection at the different interfacescan be monitored using a second time-delayed light pulse focused on thesame spot. The main part of this probe light is reflected at the fixed in-terfaces whereas a weaker component is reflected by the propagating strainpulse. All these reflections interfere leading to oscillations of the measuredreflectivity with the time delay due to the strain pulse bouncing back andforth in the sample.In the early work by Zhu et al . [71], the frequency dependence of thesound attenuation in vitreous silica layers was determined from the decayof successive echoes reaching the Al transducer after a round trip in theglass. A single-crystal tungsten substrate was used to produce numerousand strong echoes thanks to the large acoustic mismatch between v -SiO and tungsten. Based on the modification of the shape of the acoustic pulse anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 23 Low-frequency vibrational spectroscopy of glasses reflecting the frequency dependent attenuation coefficient of its spectralcomponents, this method allows accessing frequencies well above the Bril-louin frequency. These early data obtained at room temperature are plottedas stars in Fig. 10 showing an approximate quadratic behavior. It is in-teresting to note that extrapolating this Ω behavior (dashdotted line inFig. 10) to the THz region leads to values comparable to those reportedusing IXS (hollow square in Fig. 10) [73]. We will come back to this pointin the next section. Up to about 200 GHz, the picosecond ultrasonic dataroughly follow the expected frequency dependence shown by the dashedline discussed in the preceding section as the sum Γ tar + Γ anh . However,it became clear with time that the sound attenuation was systematicallyoverestimated by a factor between 2 and 3. It is now well admitted thatincomplete corrections for losses at the interfaces were the likely source oferrors. -1 ∝ Ω ∝ Ω v -SiO
300 KFrequency (GHz) L i n e w i d t h ( G H z ) BLS [64]IUVS [63]IUVS [67]PU [71]PU [75]PU [76]PU [77]IXS [73] Ω ir Fig. 10. Frequency dependence of the Brillouin linewidth in vitreous silica at roomtemperature using BLS [64], IUVS [62, 63, 67], different PU schemes [71, 74–77] andIXS [73]. The solid line is the sum Γ tar + Γ anh discussed in the preceding section plusthe expected Ω law leading to the Ioffe–Regel crossover at Ω ir . Those difficulties could be alleviated with a different scheme using theoscillations produced by the interference of the probe partly reflected at thesample surface with its reflection by the moving acoustic pulse. In that case,the probe light interacts with one of the Fourier components of the acousticpulse according to momentum conservation. This modus operandi is actu- anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 24 B. Ruffl´e, M. Foret & B. Hehlen ally the time-domain analog of the usual Brillouin light scattering and assuch suffers in principle from the same limits in the accessible frequencydomain [78]. If the dielectric sample film is deposited on a substrate withsignificantly higher refractive index and acoustic velocity, like crystallinesilicon, the oscillations detected in the substrate correspond then to muchhigher frequency acoustic waves. A study of the amplitude of these higherfrequency acoustic oscillations in the silicon substrate as a function of theglass film thickness allows then determining hypersound attenuation in theamorphous sample at a frequency beyond the usual BLS limit. This schemehas been used for silica films on crystalline silicon [74, 75] and the morerecent data covering the frequency range 165 GHz to 320 GHz are plottedin Fig. 10 as solid pentagons. They nicely superimpose with the theoreticalexpectations (dashed line) based on Eqs. 6 and 9 up to about 250 GHz in adomain where the anharmonic contribution Γ anh ∝ Ω becomes dominantas described in Fig. 9b. At higher frequencies, the latter tends to satu-rates while the measured sound attenuation still increases. This is in fairagreement with the solid line which takes into account the expected Ω lawleading to the Ioffe–Regel crossover at Ω ir . The latter is located around1 THz in vitreous silica as shown in Fig. 10 together with the lowest pointobtained using IXS [73]. This was not yet a clear observation of the Ω lawin that frequency range but thanks to the very high quality of the data theonset seems to have been detected here.Similar results have been obtained later on using a narrowband ultra-fast photoacoustic approach that allows covering more than a decade infrequency [76]. The attenuation coefficients, converted to linewidth, areshown in Fig. 10 as hollow disks. As expected for a multiband technique,the most precise values are obtained in the mid-range from 50 to 270 GHzwhile data in the frequency wings are rapidly approximate. It is particularlytrue for the data above 300 GHz which cannot allow the observation of theanticipated Ω law. However, the PU technique is continuously improvingand other works reported ultra-broadband PU data reaching 650 GHz in v -SiO [77], shown by stars in Fig. 10. In that case, high frequency Fouriercomponents in the acoustic pulse are obtained replacing the metallic trans-ducer with a piezoelectric nanolayer one. Unfortunately, the reported soundattenuation extracted from the decaying echoes is still clearly overestimatedas in [71]. Nevertheless, with a proper analysis of the interface contributionone can envision in the near future to bridge the gap between BLS andIXS so as to firmly detect the onset of the expected Ω law up to the IRcrossover at THz frequencies. anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 25 Low-frequency vibrational spectroscopy of glasses One must note however that most of the PU data focus on vitreoussilica due to the tremendous requirements for making decent glass filmshaving properties comparable to bulk samples. Nevertheless, similar mea-surements for other amorphous materials have been sparsely collected toinvestigate the universality of the attenuation at high frequency [79]. Inthe latter work, a general Γ ∼ Ω law is evidenced up to about 320 GHzfor the amorphous polymers PMMA, Polyethyl methacrylate (PEMA) andpolystyrene (PS) as well as for the metallig glass TiNi, all at room tem-perature. Concerning PMMA, the attenuation coefficient measured at lowfrequencies using BLS nicely extrapolate the obtained Ω law at ambient T as shown in Fig. 11a. This does not seem to be the case on the highfrequency side where a single measurement at about 600 GHz using IXSshows an attenuation coefficient about 70% larger [80], leaving room forthe observation of a Ω law in PMMA. Besides, the temperature depen-dence of the sound attenuation can be studied using PU as well. Both thetransient and steady-state heating of the sample by the laser pump must becontrolled, in particular at low- T where heat capacity is rapidly decreasing.A typical result is displayed in Fig. 11b for PMMA at 232 GHz [79]. Thephonon linewidth is continuously decreasing with decreasing temperature,in agreement with a possibly anharmonicity-dominated sound absorption.The temperature dependence seems however different from that obtainedusing BLS around 18 GHz as shown in Fig. 11b where the dashed line is themeasured BLS linewidth [48] extrapolated to 232 GHz with a Γ ∝ Ω law.This probably means that several competing sound damping mechanisms,each with its own frequency and temperature dependence, are present inPMMA as well. More recently, the temperature dependence of the soundabsorption up to about 200 GHz has been studied in vitreous silica showingfair agreement with the theoretical expectations based on Eqs. 6 and 9 [81].An attractive perspective given by the rapidly developing picosecond ul-trasonic technique is finally the possibility to observe the predicted Ω lawfor sound attenuation in a ( ω, T ) region where this process would dominateover relaxation mechanisms. The first direction is to push the technique to-wards higher frequencies using more complex transducers as acoustic pulsegenerator/detector. The second path is to sufficiently reduce the tempera-ture to suppress the relaxations, leaving the disorder-induced process dom-inate the acoustic attenuation. anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 26 B. Ruffl´e, M. Foret & B. Hehlen -1 ∝ Ω a PMMAFrequency (GHz) L i n e w i d t h ( G H z ) BLS [48]PU [79]IXS [80] 0 100 200 3000102030 b PMMATemperature (K) L i n e w i d t h ( G H z ) PU 232 GHz [79]Ω -extrap. BLS [48, 56] Fig. 11. a) Frequency dependence of the Brillouin linewidth in PMMA at room tem-perature using BLS [48], PU [79] and IXS [80]. The solid line is the Γ ∝ Ω best fitof the PU data. b) Temperature dependence of the Brillouin linewidth in PMMA at232 GHz [79] compared to Ω -extrapolated 18 GHz BLS data (dashed line). Inelastic neutron and x-ray scattering
Glass samples become again transparent in the soft x-ray region. Thelatter allows then the observation of sound properties in the THz rangeusing a Brillouin scattering technique known as inelastic x-ray scattering(IXS). Due to the large decrease of the wavelength compared to visiblelight, the experiment must be done at relatively small angles to achievethe lowest momentum transfer [82]. Hence, to observe Brillouin scatteringof THz acoustic excitations in amorphous materials, IXS experiments aremade in forward scattering in transmission while conventional BLS usesbackscattering to increase the sensitivity. The instrumental resolution ofthe IXS spectrometer is now around 1 meV (cid:39) .
25 THz while the lowestachievable q value is in practice about 1 nm − . On the other hand, inelasticneutron scattering (INS) is not a technique of choice for such experiments,at least in amorphous materials with rather high sound velocities. Closeto q = 0, it is indeed not possible to measure sound waves whose velocitiesare larger than those of the incident neutrons due to kinematic conditions.Amorphous selenium is one of the few examples where LA phonons aresufficiently slow to be accessible by INS [83]. Both techniques however sharethe same drawback which is their inability in probing transverse acousticmodes at low q . anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 27 Low-frequency vibrational spectroscopy of glasses a) Two competing scenario Measuring the acoustic modes in glasses in the ( ω, q ) region of the ex-pected IR crossover together with the search for a damping Γ ∝ Ω hasbeen a strongly debated issue emerging in the late nineties when IXS ex-periments became operational. The very first IXS experiments in glassesreported on glycerol and LiCl:6H O in the q region 2–8 nm − [82]. Themeasured spectra, which are proportional to the dynamical structure fac-tor S ( q, ω ), consist of a strong elastic peak plus a weaker LA Brillouindoublet modeled as a damped harmonic oscillator (DHO) both convolutedwith the instrumental resolution. In the classical approximation and ne-glecting Debye–Waller effects, S ( q, ω ) can be written as S ( q, ω ) = f q S ( q ) δ ( ω ) + (1 − f q ) S ( q ) 1 π Ω q Γ q (cid:2) ω − Ω q (cid:3) + [ ω Γ q ] (11)where S ( q ) is the structure factor and f q the elastic fraction of the scatteringat this q . Ω q is the Brillouin frequency shift, related to the phase velocityof sound waves v q = Ω q /q while Γ q is the full width at half maximum ofthe Brillouin peak, associated to the sound attenuation. Parameters withthe index q depend on the magnitude of the scattering vector q but not onthe frequency ω .The pioneering values obtained in glassy glycerol at T = 145 K are dis-played in Figs. 12a and 12b showing an almost linear dependence of Ω q with q and an approximate quadratic q dependence of Γ q . The linear dispersionwas corresponding to a sound velocity v la (cid:39) − , lower but closeto the known longitudinal sound velocity measured at much larger acous-tic wavelength. Very similar results were obtained for the second glass,LiCl:6H O, leading the authors to conclude that propagating sound waveswere existing at THz frequencies in those two glasses. This picture wasreinforced when first IXS data measured in the archetypal glass v -SiO also revealed linear dispersion with a longitudinal sound velocity compara-ble to the macroscopic one together with a quadratic q dependence of thelinewidths. The latter behavior was further shown to be reasonably con-sistent with the sound attenuation coefficient known at lower frequenciesat that time [71, 73] as illustrated in Fig. 12c where the solid line is theadjustment of the sole IXS data to a Γ q ∝ q α law with α (cid:39) S in ( q, ω ) function derived from an effective-medium ap- anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 28 B. Ruffl´e, M. Foret & B. Hehlen b glycerol 145 K q (nm -2 ) ¯ h Γ q ( m e V ) v la ∼ − a glycerol 145 K q (nm -1 ) ¯ h Ω q ( m e V ) − − − − − − ∼ q c v -SiO q (nm -1 ) ¯ h Γ q ( m e V ) IXS 1050 K [73]PU 300 K [71]BLS 300 K [73]
Fig. 12. Wavenumber dependence of a) the Brillouin frequency shift Ω q and b) theBrillouin linewidth Γ q in glycerol at 145 K measured using IXS [82]. The solid line in a)is the Ω q = v la q best fit giving a sound velocity v la (cid:39) − while the dashed linein b) is the Γ q ∝ q best fit. Adapted from [82]. c) Wavenumber dependence of theBrillouin linewidth Γ q in vitreous silica at 1050 K measured using IXS [73], PU [71] andBLS [73] at 300 K. The solid line is the Γ q ∝ q α best fit of the IXS data giving α (cid:39) . proximation (EMA) for the force constants of a percolating network thatreads [85]: S in ema ( q, ω ) = k b T q πmω q W (cid:48)(cid:48) ( ω )[ ω − q W (cid:48) ( ω )] + [ q W (cid:48)(cid:48) ( ω )] (12)Eq. 12 is a generalization of the second term in Eq. 11, appearing clearlywhen the second moment rule giving (1 − f q ) S ( q )Ω q = k b T q /m is used inthe later [86]: S in dho ( q, ω ) = k b T q πmω ω Γ q (cid:2) ω − Ω q (cid:3) + ω Γ q (13)In the EMA, the disorder-induced inhomogeneous force constants arereplaced by a complex, frequency-dependent, uniform function W = W (cid:48) − iW (cid:48)(cid:48) . W (cid:48) ( ω ) and W (cid:48)(cid:48) ( ω ) result in effective sound velocity c ema ( ω ) andscattering length (cid:96) ema which depend on the frequency [85] c ema ( ω ) = | W ( ω ) | Re (cid:112) W ( ω ) (cid:96) ema ( ω ) = 1 ω | W ( ω ) | Im (cid:112) W ( ω ) (14)The physical picture is that, as ω increases towards ω ir , the definition in q of the vibrations becomes less and less precise due to the disorder. Thus anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 29 Low-frequency vibrational spectroscopy of glasses vibrations of different frequencies have an appreciable Fourier componentat the scattering vector q . This leads to a dependence in frequency (ratherthan in wavevector, which is not well defined for the vibrations) of theparameters entering S in ( q, ω ). Although by themselves measurements of S ( q, ω ) cannot demonstrate phonon propagation or localization, it seemedrelevant to investigate the applicability of the two models.To illustrate the difference between Eqs 12 and 13, typical spectralshapes are shown in Fig. 13. The frequency scale is normalized by thehypothetical Ioffe–Regel frequency ω ir occurring when the linewidth Γ q reaches Ω q /π at the wavenumber q ir = ω ir /c where c is the constant low-frequency sound velocity. Figure 13a shows EMA spectral shapes at several q values around q ir obtained from a phonon-fracton crossover model [87]which was also used to describe IXS data in v -SiO [84]. In that casethe effective sound velocity c ema ( ω ) is c well below ω ir and increases as √ ω above while the sound damping coefficient Γ ema ( ω ) = c ema ( ω ) /(cid:96) ema ( ω )rapidly increases as ω below ω ir , then flattens to a linear frequency depen-dence above. Figure. 13b displays the corresponding DHO spectral shapesobtained using the best Ω q and Γ q values that adjust the EMA profiles. Atsmall q values, below q ir , the spectral shapes in Figs. 13a and 13b are quitesimilar showing a sharp Brillouin peak. The main difference arises at larger q values when the linewidth is large. The homogeneous DHO spectrumshows an increasing high intensity for ω tending to 0 compared to the peakintensity. Conversely, the EMA spectrum, which is intrinsically inhomoge-neous, shows an intensity going to zero together with ω . Furthermore, theEMA spectrum for q (cid:29) q ir becomes practically q independent, with a peakaround ω ir .The inset of Fig. 13b shows the q -dependence of the normalized Ω q andΓ q values. An almost linear dispersion of the former is observed even for q > q ir while the sound damping fairly compares to a quadratic behavior.These schematic graphs illustrate how contradictory conclusions could bedrawn from similar IXS experiments: one supporting the existence of prop-agating sound waves up to high q values [73, 82] characterized by lineardispersion and quadratic sound damping, the other one implying a strongdisorder-induced scattering of these excitations leading to a Ioffe–Regelcrossover [84]. However, the limited statistics and energy resolution of theearly IXS experiments clearly prevented any definite conclusions at thattime. anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 30 B. Ruffl´e, M. Foret & B. Hehlen a EMA function˜ ω S i n ( ˜ q , ˜ ω ) b DHO function˜ ω ˜ q = 0 . q = 0 . q = 1 . q = 1 . q = 1 . q = 2 .
00 0 1 2012 c ˜ q ˜ Ω q () , ˜ Γ q () Fig. 13. Spectral shapes of a) the EMA function (Eq. 12) at several q values around q ir [84], b) of the DHO function (Eq. 13) obtained using the best Ω q and Γ q values thatfit the EMA profiles. Inset: q -dependence of Ω q and Γ q . The line is Ω q = c q whereasthe dashed line is Γ q ∝ q . The scales are normalized by ω ir or q ir . b) Search for a quartic attenuation law The two competing views have been extensively discussed for manyyears, thereby including better experiments, progresses in numerical mod-eling and theoretical advances. On the experimental side, the linear dis-persion of the longitudinal acoustic excitation and the quadratic trend ofthe linewidth using IXS were reported for many different amorphous sys-tems [88–98]. However, spectroscopic evidences for the alternative sce-nario were slowly showing up [83, 99–101]. The latter were concomitantlystrengthened by numerical simulations, in particular of v -SiO models. IRcrossovers for acoustic excitations were observed both for LA and TA modesat ∼ q box-size limitation of the simulation.Similarly, a direct observation of the ω law using x-rays was still avery demanding experiment. From the thermal conductivity, it was indeed anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 31 Low-frequency vibrational spectroscopy of glasses anticipated that the expected ω ir hardly exceed 1–2 THz in most materials.Combined with the typical sound velocities found in amorphous solids, itleads to crossover wave vectors q ir in the range 1–2 nm − . This happenedto be the lowest accessible limit in IXS. Further, with a Lorentzian-likeinstrumental resolution profile of about 1 . ω ir ∼ . q ir ∼ . − [99, 101], as opposed to theestimates ω ir ∼ q ir ∼ − for v -SiO [84]. d -SiO is alsomore homogeneous so that the elastic intensity is significantly reduced.For these reasons, it was anticipated that the crucial region below andnear ω ir might be accessible to spectroscopy in d -SiO using existing IXScapabilities [99, 101]. Figure 14 shows typical inelastic spectra obtained at T = 565 K [104]. The measured spectra were fitted to a DHO plus a deltafunction for the elastic part, convoluted with the Lorentzian instrumentalfunction. In Fig. 14a the width of the Brillouin peak is mostly instrumental,while in 14e it is due in large part to a real broadening of the Brillouin signal.The values obtained for Ω q are reported in Fig.14g and show an almostlinear q -dependence as in other materials and in agreement with the dis-cussion of Fig. 13. The q dependence of the resulting Γ q values is shownin Fig.14h. Γ q increases very rapidly with q and is significantly above thevalues that can be extrapolated from the precise BLS measurement. Asdiscussed before for d -SiO , see Fig. 8, the sound attenuation at long wave-length originates from anharmonicity in this temperature range and it seemsreasonable to assume Γ q ∝ q in the sub-THz range as shown as the dashedline in Fig.14h. This homogeneous broadening contributes only to 10% ofthe linewidth measured using IXS at q = 1 .
37 nm − . Below about 2 nm − ,the four points can be well described with a Γ q ∝ q expression, giving thesolid line in Fig.14h. At larger q values, Γ q extracted from the DHO fitsseems to increase as q , in agreement with observations in other materials.It was strongly suggesting that a new damping mechanism should exist inbetween. Clearly, the latter must lead to a rapid increase of the linewidth,faster than q , as two different q laws could not even crossover.In addition, the DHO parameters fulfill the Ioffe–Regel criterion (Γ q =Ω q /π ) at q (cid:39) − , corresponding approximately to (cid:126) ω ir (cid:39) (cid:126) ω bp . Finally,another piece of the puzzle came from the shape of the inelastic spectra. It anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 32 B. Ruffl´e, M. Foret & B. Hehlen a Q = 1 .
17 nm − C o un t s ( /90 s ) b Q = 1 .
37 nm − c Q = 1 .
60 nm − C o un t s ( /90 s ) d Q = 1 .
80 nm − − −
10 0 10 2001020 e Q = 2 .
00 nm − Energy (meV) C o un t s ( /90 s ) − −
10 0 10 20 f Q = 4 .
20 nm − Energy (meV) v la = 6940 ms − g Scattering vector (nm − ) ¯ h Ω q ( m e V ) -1 -4 -3 -2 -1 ∝ q ∝ q h d -SiO Scattering vector (nm − ) ¯ h Γ q ( m e V ) IXS [104]BLS [57] q ir Fig. 14. a-f) IXS spectra of d -SiO at several q values and their DHO fits of the inelas-tic part after subtraction of a central peak freely adjusted in the fitting process [104].The background shown by the thin baselines is fixed to the detector noise, measuredindependently. g) Evolution of Ω q with q where the line extrapolates the BLS result. h)Evolution of Γ q with q where the dashed line extrapolates the quadratic behavior of thesound attenuation from BLS results [57]. The solid line is a Γ q ∝ q law interpolatingthe five lowest points. Adapted from [104]. seems clear from Fig.14 that with increasing q , the DHO profile demandsa larger and larger signal at small ω which is not measured, as alreadynoted in the preceding experiments in d -SiO [99, 101]. This is particularlyclear in Fig. 14f at q (cid:39) q ir , an averaged spectrum obtained after a weekof measurement. All these spectra were also adjusted to an EMA functionusing Eq. 12, giving similar results at low q values but much better fitsapproaching q ir and above, as expected. The sole usefulness of the DHOprofile was then to evidence the existence of a new sound damping mech-anism for the LA waves which approximately behaves as q or ω up to aIR crossover [104]. c) Fate of the high-frequency acoustic waves It was crucial to examine to what extent the observations reported for d -SiO were common to glasses. To that effect, a second glass, lithiumdiborate Li O − O , or LB2, was carefully studied using IXS upon ap-proaching the expected IR crossover at two temperatures. Again, the feel- anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 33 Low-frequency vibrational spectroscopy of glasses ing that the IR frequency should be related to the boson peak guided thechoice of a material with a boson peak located at a high energy to facilitatethe spectroscopy. Further, some preliminary results already suggested thatsuch a IR crossover should exist in LB2 [94]. The same procedure as theone previously described for analyzing d -SiO IXS data was applied to LB2.The parameters extracted from the DHO fits of the IXS data are shownin Fig. 15a-b [105]. They exhibit very little dependence on T , if any. Belowabout 2 nm − there is only a slight departure from linear dispersion asseen in Fig. 15a where the slope of the line is the BLS sound velocity at573 K [106]. There is however a very rapid increase of the full width Γ q in this q region, as shown in Fig. 15b. The latter is definitely faster thana q behavior and is consistent with a q law as can be inferred from thesolid and the dotted lines. Above about 2 nm − , Γ q increases slower withan exponent close to 2, in agreement with numerous reports. All theseresults are fully consistent with those which have been obtained previouslyfor densified silica, including the occurrence of the Ioffe–Regel crossover q ir below 2 nm − which corresponds to a IR frequency ω ir close to the frequencyof the maximum of the boson peak ω bp . The two series of points reportedin Fig. 15b at lower energy have been obtained using BLS analyzed witha high-resolution tandem spectrometer [106] for three different scatteringangles. They show that contrarily to the d -SiO case the sound attenuationdepends linearly on the frequency in the hypersonic range, pointing to TARprocesses as the dominant broadening mechanism in this ( ω, T ) region. Theanharmonic contribution should take over at higher frequencies.As discussed before, the experimental evidence for a disorder-induced q sound attenuation mechanism leading to a Ioffe–Regel crossover indi-cates that the DHO cannot be the adequate representation of the spectrallineshape approaching q ir . This is illustrated in Figs. 15c-h which compareinelastic IXS spectra just below and above q ir fitted to the DHO lineshapeon the left and to an EMA lineshape on the right [105, 107]. Well below q ir ,the experimentally selected scattering vector q matches acoustic modes ofwell defined Q number, the linewidth Γ q truly reflects an inverse lifetime.In this case, the DHO model gives an excellent representation of the spec-tral shape. In the opposite limit, where Q is ill-defined at a given modefrequency Ω, a spectrum S ( q, ω ) at constant Ω = ω just reflects an averagespatial profile of the excitation packets at ω [108]. Conversely, a spectrum S ( q, ω ) at constant q is then the sum of contributions of all the modes withΩ = ω at this particular scattering q value. This is inhomogeneous broad-ening. There is no basis for using the DHO in this case. The failure of the anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 34 B. Ruffl´e, M. Foret & B. Hehlen v la = 6800 ms − a Scattering vector (nm − ) ¯ h Ω q ( m e V ) -1 -4 -3 -2 -1 Ω Ω Ω b ¯ h Ω q (meV) ¯ h Γ q ( m e V ) IXS 573K [105]IXS 300K [105]IXS 700K [94]BLS 573K [105]BLS 300K [105] Ω ir Li O-2B O c Q = 1 .
77 nm − C o un t s ( /60 s ) d Q = 1 .
77 nm − e Q = 2 .
40 nm − C o un t s ( /60 s ) f Q = 2 .
40 nm − − −
10 0 10 200255075 g Q = 4 .
25 nm − Energy (meV) C o un t s ( /60 s ) − −
10 0 10 20 h Q = 4 .
25 nm − Energy (meV)DHO EMA
Fig. 15. a) Evolution of Ω q with q where the line extrapolates the BLS result for LB2glass. b) Evolution of Γ q with Ω q where the dashed line at low energy extrapolatesthe linear behavior of the sound attenuation from BLS results [106]. The solid line isa Γ q ∝ q law interpolating the lowest q points of IXS data [105]. The dotted lineis a quadratic behavior extrapolating IXS data obtained above the IR crossover. c-h)Comparison between IXS spectra adjusted to a DHO lineshape or to an EMA lineshapeshowing the failure of the former approaching q ir . Adapted from [105]. DHO above q ir is clearly illustrated in Figs. 15e,g and can be observed inFigs. 14e,f as well.The question of why such a rapid increase of the Brillouin linewidth wasnot detected in all the other glasses investigated so far was elucidated byconsidering the values of the ratio Γ q / Ω q obtained in these IXS experiments.Figure 16 shows values of that ratio vs Ω q for ten different glasses. For eachglass, the points were arbitrarily interpolated with a straight line in the IRcrossover region. Its intercept with the dashed horizontal line Γ q / Ω q = 1 /π gave an estimate for ω ir,la . One observes that only for LB2 and d -SiO are high-quality data available below the crossover. In the other cases,this region where the q was expected was simply not investigated at thattime. For all these glasses, the estimated ω ir,la values were clearly in thesame frequency range than the maximum of the boson peak ω bp , markedby the hatched region, and not a decade above [110]. All in all, after adecade of vivid scientific discussions, it appeared that a large part of thepicture originally proposed in [84] was experimentally verified. It must be anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 35 Low-frequency vibrational spectroscopy of glasses π Li O-2B O Γ q / Ω q Li O-4B O glycerol polybutadiene propylene carbonate π d -SiO ¯ h Ω q (meV) Γ q / Ω q v -SiO ¯ h Ω q (meV) orthoterphenyl¯ h Ω q (meV) selenium¯ h Ω q (meV) BeF ¯ h Ω q (meV) Fig. 16. Determination of the Ioffe–Regel energy (cid:126) ω ir (intercept with the dashed line)for different glasses and comparison with the energy region of the maximum of the bosonpeak (cid:126) ω bp (hatched region). References for the experimental points can be found in [105].The positions and ranges of the BPs are discussed in [107, 109]. noted that the latter observation made on the LA modes gave an incompleteview. The low-frequency region of the vibrational density of states is indeedlargely dominated by the transverse acoustic excitations, i.e. as 2 c l /c t .Even if no experimental evidence exists yet, it is largely admitted that ω bp is closed to ω ir,ta for all glasses while the ratio ω ir,la /ω ir,ta is system-dependent. The case of orthoterphenyl in Fig. 16 already illustrated thispoint with ω ir,la /ω bp > ω bp in metallicglasses [112–114]. d) Interaction of theory and experiment Concomitantly with the latter experimental evidences, theoretical ap-proaches were developed to describe vibrations in glasses. They can begrossly separated in two main categories. The first one, the so-called SoftPotential Model (SPM), assumes that there exists in glasses, non-acoustic,soft quasi-local vibrations (QLVs) at low frequencies [115]. The latter forman excess g loc of the vibrational density of states above the Debye contri-bution, g loc ( ω ) ∝ ω [116]. Their mutual interactions, mediated by theacoustic phonons, lead to the universal shape of the boson peak [117]. Inreturn, the acoustic phonons are strongly scattered by the QLVs leading anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 36 B. Ruffl´e, M. Foret & B. Hehlen to Γ ∝ ω up to a IR crossover at ω ir (cid:39) ω bp [10, 118, 119]. As the SPMis an extension of the standard tunneling model [20, 21], it offers a unifiedapproach and a comprehensive understanding of the thermo-mechanicalanomalies observed in glasses and discussed in Sec. 1.1. For more details,the reader is referred to Chapter 9.The second category considers harmonic elastic disorder. A self consis-tent effective medium approach based on fluctuating elastic constants hasbeen worked out in great detail and is now often termed Fluctuating Elas-ticity Theory (FET) [120, 121]. The elastic disorder induces an excess ofthe reduced vibrational density of states over the Debye theory, which isinterpreted as the boson peak located at a fraction of the Debye frequency.Similarly to the SPM results, the acoustic waves are strongly scattered inthe boson peak frequency range [23]. In this region, a crossover from anacoustic damping proportional to q at low frequency to q at high fre-quency is obtained, as observed experimentally [110]. For more details, thereader is referred to Chapter 10.Both theories give very similar predictions, for which increasing thedisorder leads to a downshift of the boson peak, eventually driving thesystem to an instability. The boson peak anomaly marks the crossoverfrom plane-wave like vibrational states to disorder-dominated states withincreasing frequency. The inelastic structure factor related to the scatteringof the acoustic modes takes the EMA form given by Eq. 12 with distinctself-energy functions W ( ω ) for the two theoretical treatments [23, 122],permitting quantitative comparisons with experiments and computationalsimulations. It must be stressed than conversely to the FET approach whichleads to the vDOS of the eigenmodes of the disordered system, the SPMstarts from the coexistence at low frequency of pure propagating acous-tic modes and boson peak modes with g exc ( ω ) which are not the eigen-modes [105, 107, 109, 123].Numerical simulation has become an increasing powerful tool to shedlight on the vibrational properties of glasses. It can provide very relevantphysical quantities that are not yet accessible to experiments and thus arecertainly useful for testing theories. On one hand, elastic heterogeneity ata typical length scale ξ (cid:39) − σ , where σ is the atomic scale, is welldocumented in glasses. It leads to the breakdown of the continuum elasticdescription and thus of the Debye approximation at ξ which corresponds tothe acoustic wavelength of frequencies close to ω ir [124–128]. On the otherhand, there is also substantial evidence from simulation that QLVs exist indisordered systems [24, 111, 129–131]. More recently, large-scaled simula- anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 37 Low-frequency vibrational spectroscopy of glasses tions gave new insights into the low-frequency continuum limit, evidencingthe ω law for the QLVs thus supporting the SPM approach [132–139].As far as experiment is concerned, first comparisons using the IXS Bril-louin linewidths of the LA modes in several glasses showed a much betteragreement of the experimental data with the SPM expectations [140]. Thestrength of the q behavior derived from FET fell systematically well be-low the observed widths. This can be partly attributed to the fact thatthe model used was not able to produce sufficiently intense boson peaksabove the Debye level. This was particularly true for vitreous silica [140].It was latter shown that in principle the agreement between experimentsand FET calculations could be improved. Introducing spatial correlationsin the elastic disorder [141], or replacing the original Gaussian distributionof the elastic constants by some non-Gaussian ones [142], leads indeed toa significant enhancement of the boson peak close to the instability. Highquality new IXS data were later obtained on the H-bonded glass glycerolwhich, as guessed from Fig. 16, was the next system to study. IXS spec-troscopy have been performed in great details across the expected crossover q ir (cid:39) . d -SiO and LB2were confirmed, extending the phenomenology to molecular glasses [143].Additionally, a marked softening of the sound velocity was evidenced be-low the IR crossover, thus exactly in the region of the rapid increase of theacoustic damping. A negative dispersion is actually the Kramers–Kronigcounterpart of the q behavior of the sound attenuation and as such wasa further evidence of its existence. Both the SPM and FET approachesproduce a softening of the apparent sound velocity for frequency below theboson peak [122, 144]. Similar IXS results were obtained afterward in sil-ica melt [145], glassy sorbitol [146, 147] and 2Na O − [148]. Furthercomparisons between the original FET model [23, 110] and IXS data forthe case of silica melt and glassy sorbitol concluded that FET was able toreproduce qualitatively the experimental data but not quantitatively all thefeatures [147, 149]. For more details, the reader is referred to Chapter 7.The relationship between the boson peak spectrum and the Debye den-sity of states, characteristic of the continuous elastic medium was also stud-ied. A successful scaling was supposed to promote the idea that vibrationalmodes forming the boson peak were only acoustic, as developed by the FET.Small changes of the boson peak spectrum following the evolution of theDebye vDOS were indeed observed in some glasses [150–152], while the De-bye scaling clearly failed in others [49, 153–155]. In the vitreous silica case,it was found that the temperature dependence of the boson peak spectrum anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 38 B. Ruffl´e, M. Foret & B. Hehlen rather scales to the sound velocity based on the unrelaxed bulk moduluswith scaling exponents in line with the SPM. All in all, the Debye scalingapproach was not really successful to discriminate between both theories.The importance of coordination number or fluctuations of the parti-cle contact number and applied stress on elastic properties of disorderedsystems has been underlined recently [156]. An effective medium theorybased on these two quantities has been developed which reproduced all thefeatures obtained using the FET approach including Eq. 12. Interestingly,this work emphasizes the relevance of the effective sound velocity c ema ( ω )and scattering length (cid:96) ema ( ω ) of Eqs. 14 as the physical quantities charac-terizing the total response of the system. Above the IR crossover, thesequantities differ significantly from those extracted from a simple analysisof the IXS data using Eq. 13 which leads to: c q ( ω ) = (cid:112) Re W ( ω ) (cid:96) q ( ω ) = 1 ω Re W ( ω )Im (cid:112) W ( ω ) (15)with c q = Ω q /q and (cid:96) q = c q / Γ q . In addition, analysis in term of constant-energy scans using Eq. 12 instead of usual constant-wavenumber scans usingEq. 13 provide the opportunity to easily include the diffuse Umklapp scat-tering contribution as illustrated by U. Buchenau [86]. Depending on thesystem, this contribution can also greatly affects the obtained parametervalues. Hence, a promising way to test the several theories is to directlyuse Eq. 12 on constant-energy spectra to find the frequency dependence ofthe complex self-energy W ( ω ), taking into account the quadratic Umklappscattering contribution.
4. Spectroscopy of the low-frequency region of the vDOS
This section aims at describing the nature of the modes constituting thelow frequency region of the vibrational density of states and to what ex-tend this information can be obtained from the experiments. For example,inelastic neutron scattering and inelastic x-ray scattering are sensitive toall vibrations but the q -dependence of the scattering allows separating dif-ferent type of excitations. The vibrational contrast in light experiments ismore pronounced, even though the well established crystalline or molecularselections rules are lifted out by the structural disorder. In glasses thoseare gathered in a coupling-to-light function, C ( ω ), whose expression (in-cluding C ( ω ) = 0 for inactive vibrations) depends on the mode-symmetry(atomic displacements) as well as on the experimental technique. Compar-ing the vibrational responses of infrared spectroscopy, Raman scattering, anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 39 Low-frequency vibrational spectroscopy of glasses and hyper-Raman scattering, can therefore be very informative. It is par-ticularly true in simple glasses where selections rules are more pronounced. Inelastic neutron scattering
One of the first experimental breakthrough in the understanding of theboson peak modes occurred in the 80’s using inelastic neutron scattering(INS). Atomic vibrations are all actives in INS and hence it is a technique ofchoice to access the density of vibrational states g ( ω ) of a material (vDOS).One drawback however is that it is impossible (except by doing a series ofexperiments with atomic isotopic substitutions) to normalize the experi-mental data by the scattering length of each type of atom constituting thematerial. This possibly distorts the experimental result when one wants tocompare it, e.g., with atomistic simulations. Within the Debye model, theboson peak is an excess of modes over the ω dependence of g ( ω ) at lowfrequency and hence it is observed in a g ( ω ) /ω plot, see Fig. 17a. Despitesthe above-mentioned limitations, at the frequencies of the boson peak, thespecific heat C p ( T ) calculated from the density of vibrational states com-pares well with the experimental data, at least in vitreous silica [9, 157].In the pioneering work of Buchenau et al . [9], a quantitative analysisof the low-frequency vDOS of v -SiO is performed by decomposing the in-elastic signal into two components, one proportional to q S ( q ) and anotherone to q e − W , where S ( q ) is the elastic structure factor and e − W theDebye–Waller factor. This implies two kinds of low-frequency vibrations:a coherent contribution corresponding to in-phase motions of neighboringatoms or molecular units whose inelastic structure factor reproduces S ( q )and hence the first sharp diffraction peak (FSDP), and a quasi-incoherent part corresponding to uncorrelated atomic motions with its own oscilla-tions as a function of the scattering wavevector q . Long wavelength acous-tic modes are collective excitations and belong to the first family, whilelocal vibrations (at least to the extent of a few structural units) or optic-like modes, to the second. These two contributions add to construct thetotal vDOS. In vitreous silica the quasi-incoherent excitations were inter-preted in terms of coupled librations of SiO units. A simplified schematicrepresentation is given in Fig. 17a. Interestingly, these motions also de-fine the soft mode of the β - α transition in quartz [160] and are known tobe instable in quartz and in cristobalite. As such, they naturally vibrateat low frequency [2]. Similar decompositions were made in v -B O [158]and polybutadiene [159], and the two contributions are well separated, as anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 40 B. Ruffl´e, M. Foret & B. Hehlen
SiO a B O c in-phasequasi-incoherent0 5 10 15024 GeO b Energy transfer (meV) × g ( E ) / E ( m e V − ) polybutadiene d Energy transfer (meV) in-phasequasi-incoherent
Fig. 17. bosons peak of (a) SiO and (b) GeO at different wave vectors [157] andschematic representation of coupled SiO librations in an undistorted structural modelcontaining five units [9]. bosons peak of (c) B O [158] and (d) polybutadiene [159]decomposed into in-phase and quasi-incoherent excitations. The dashed lines indicatethe Debye level. shown in Figs. 17c,d. Differently, they superimpose in silica, suggesting astrong hybridization of the modes. The possibility that librations of cou-pled BO triangles and B O rings participate to the boson peak of v -B O was proposed later [161], providing thereby a possible explanation for thequasi-incoherent contribution in boron oxide as well.A second campaign of INS experiments was performed about 10 yearsafter using the high flux of the reactor at the Institut Laue–Langevin atGrenoble. The high quality of the data allowed improving the conclusionsof the first campaign as well as to compare the situation in v -SiO and v -GeO (see Fig. 17) [157]. The spectroscopy allowed defining at each fre-quency ω , a function S ω ( q ) corresponding to the oscillations of the inelasticstructure factor over a purely incoherent scattering (Fig. 18). The appear-ance of an additional oscillation at the position of the FSDP ( ∼ . (cid:6) A − ),solely in the low-frequency response (3–5 meV), shows that in-phase atomicmotions contribute mostly at frequency close or below the BP maximum.For frequency above 10 meV, typically, S ω ( q ) only reproduces the oscilla- anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 41 Low-frequency vibrational spectroscopy of glasses tions around 3 and 5 (cid:6) A − characteristic of coupled librations of SiO orGeO tetrahedra (solid lines). The full analysis reveals that in vitreous sil-ica, about 25% of in-phase motions and 75% of quasi-incoherent excitationscontribute to the BP below its maximum (40% and 60%, respectively, invitreous germania). Above the maximum, the in-phase part decreases sodo the long wavelength excitations. a v -SiO Momentum transfer q (˚A − ) S ω ( q ) b v -GeO Momentum transfer q (˚A − ) Fig. 18. Oscillation function of the inelastic structure factor integrated over two fre-quency domains for a) v -SiO and b) v -GeO . The arrows mark the position of the firstsharp diffraction peak. (adapted from [157]). Among all these excitations, all but the long wavelength acoustic modescan be identified as boson peak modes, the former defining the Debye level(dashed lines in Figs. 17). In silica, germania, and boron oxide, coupledlibrations of rigid elementary units (SiO and GeO tetrahedra, BO tri-angles, boroxol rings) contribute partly to the excess of modes, but it isimportant to notice that the sole contribution of in-phase motions also ex-ceeds the Debye level (see, e.g., Fig. 17c). This shows that in addition toacoustic plane waves , the in-phase contribution contains modes that alsobuilt-up the reservoir of boson peak modes. When considering the con-nected networks of silica, germania, and boron oxide, it is natural thatlocal librations of elementary structural units (SiO tetrahedra, BO trian-gles, B O rings) induce slightly delocalized atomic translations, providingthereby in-phase-type contributions to the boson peak.Attempts to measure the dispersion curves of long wavelength acousticphonons in glasses with neutrons have also been made. The main issue inusing INS is that in addition to working in near forward scattering ( θ (cid:39) − ° with good instrumental resolution, the experiment must also fulfill anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 42 B. Ruffl´e, M. Foret & B. Hehlen drastic kinematic conditions (conservation of momentum and energy). Thelatter basically require that the probed sound velocities must be slower thanthe speed of the incident neutrons. Fulfilling all these conditions at onceis very challenging but still, experiments using the thermal neutrons of atriple axis spectrometer have been performed in vitreous selenium, a glasswith a low longitudinal sound velocity, v la (cid:39) − [83]. Spectra wererecorded at wavevectors as low as 0 . (cid:6) A − , as shown in Fig. 19a. The cutoffat high frequency is due to the kinematic conditions and limits the analysis.However, one clearly observes the narrow lineshapes of the longitudinalphonons at low wavevector which rapidly broaden with increasing q . Aboveabout 1 (cid:6) A − , the measured spectral shape starts mirroring the BP measuredby time-of-flight experiment, see Fig. 19b. This qualitatively demonstratesthe Ioffe–Regel crossover of LA phonons as discusses in Sec 3.2.5.An analysis of the q -dependence of the signal, similar to that performedin [9, 157], highlighted three spectral components: the Brillouin LA re-sponse at low q (dashed line in Fig. 19a), the quasi-incoherent or local-mode component (dot-dashed line in Fig. 19a,b), and the in-phase compo-nent developing at high q (gray region in Fig. 19b). The latter was calledacoustic umklapp scattering in [83] as it corresponds to q -conserving in-elastic processes occurring by exchanging part of the neutron momentumwith the peaks of the elastic structure factor. It is worth noting that trans-verse modes dominate in this scattering process and the analysis thereforeprovides a unique way for obtaining high-frequency spectroscopic informa-tion on TA modes in glasses. Similarly to INS in silica and germania, itsstructure factor mimics that of S ( q ), i.e., ∝ q S ( q ), and we recall that forthe quasi-incoherent part one simply has a signal ∝ q . LA and quasi-incoherent contributions as well as the total scattering response have beenfitted with EMA functions derived from the effective-medium approxima-tion (Eq. 12), and the in-phase component corresponds to the differencesignal. The analysis shows that as q increases, the quasi-incoherent contri-bution progressively overwhelms that of acoustic phonons. At high q , thein-phase component (umklapp scattering) adds to these local modes to con-struct the boson peak. One also observes in Fig. 19b that the in-phase con-tribution disappears around 3 to 5 meV revealing that TA phonons ceasesto propagate at very low frequency in v -Se. anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 43 Low-frequency vibrational spectroscopy of glasses − − q = 2 nm − − − Energy (meV) S ( q , ω ) e W ( e V − ) a v -SeTAS7 . − q = 20 nm − q = 16 nm − q = 12 nm − b v -SeTOFEnergy (meV) S ( q , ω ) e W / q ( n m / e V ) Fig. 19. INS triple axis experiment in vitreous selenium. LA plane-waves are seen atthe smallest q values. Dashed lines in a) and d) illustrate the Brillouin component alone.The solid lines represent the overall fits (see [83] for details). Note the unusual abscissascale, ω/q in meV.nm. Inelastic spectroscopies of light
In a light scattering experiment the dipole p induced by an incident electricfield E can be expanded in terms of E , p = α. E + 12 β : EE + ... (16)where α and β are the polarizability and hyper-polarizability tensors, re-spectively. The first order term gives the Raman signal whose spectrum I ( ω ) for a mode σ corresponds to the space and time Fourier transformof the modulation of α by the mode amplitude W σ , α (cid:48) σ = ( ∂α/∂W σ ) (seeSec. 2.2). Contrary to INS where all modes participate to the scattering,a vibration will be active or not in light spectroscopy depending on thesymmetry properties of the atomic displacements W σ (eigenvectors). InRaman, the density of vibrationnal states g σ ( ω ) of the mode σ is observedvia the coupling-to-light coefficient C rs σ so that I ( ω ) reads: I ( ω ) = C rs σ g σ ( ω ) n ( ω ) + 1 ω (17) anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 44 B. Ruffl´e, M. Foret & B. Hehlen where n ( ω ) is the Bose population factor and a sum over all the modes σ is implicit. As the boson peak appears in a g ( ω ) /ω plot it is worthwhiledefining the ”reduced intensity” I red ( ω ) which can also be related to theimaginary part of the Raman susceptibility χ ” I red ( ω ) = I ( ω ) ω ( n ( ω ) + 1) = C rs σ g σ ( ω ) ω ∝ χ ”( ω ) ω (18) C rs σ is known for every space groups and normal modes in crystals andmolecules. These selections rules are lifted in amorphous materials owing tothe structural disorder and C rs σ transforms into a frequency-dependent func-tion C rs σ ( ω ). However, for simple network glasses with elementary struc-tural units similar to these of the crystalline polymorphs, the vibrationalspectrum resembles to a smeared-out version of the crystal one. For exam-ple, the similarities between the Raman spectrum of vitreous silica and thatof cristobalite or α -quartz are striking as illustrated in Fig. 20 [162]. Onemajor difference, however, is the broad boson peak response appearing atlow frequency in all vitreous materials. The knowledge of C rsbp ( ω ) (we willcall it C rs ( ω ) for ease of writing) and its frequency dependence is thereforeof crucial importance for understanding the nature of the vibrations at BPfrequencies. Frequency ω (cm − ) R a m a n i n t e n s i t y ( a r b . un i t s ) v -SiO α -quartz α -cristobalite Fig. 20. Raman spectra of vitreous silica, α -quartz, and α -cristobalite [162]. Within classical expectations, C rs ( ω ) is a constant for local or quasi-local vibrations [163]. For acoustic waves propagating in a mechanically anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 45 Low-frequency vibrational spectroscopy of glasses inhomogeneous medium, Martin and Brenig [25] predicted a response com-posed by the usual sharp Brillouin peaks at GHz frequencies and a back-ground rising as ω originating from the incoherent contribution of the localheterogeneities, so that C rs ( ω ) ∝ ω . This points out a fundamental differ-ence between Raman scattering in crystals and glasses at low frequency. Inthe former the signal in null while in the latter the disorder opens up a newscattering channel for short wavelength acoustics phonons based on inco-herent processes (see Sec. 2.3). This density of states of acoustic phononsbuilts the low frequency part of the boson peak (typically up to ω BP ) towhich may adds a coupling with low-lying optic modes.These predictions have been the subject of extensive experimental andtheoretical investigations to unravel the nature of the boson peak modes.The comparison between the boson peak position ω bp in Raman and thespectral broadening and position of the first sharp diffraction peak mea-sured by x-ray diffraction has highlighted the close link between this vi-brational feature and the medium range order [164] (Figure 21a,b). Basedon these observations it was proposed that the position of the BP relatesto a characteristic length scale in the glass, typically in the nanometerrange. The microscopic origin of the nanometric heterogeneities, eitherelastic, structural, or dynamical, is still controversial, but it is likely thatall of these effects coexist with a glass specific weighting. An early quali-tative model developed by Duval et al . [165] proposed that the frequencyof the Raman BP is inversely proportional to the size of the nanodomains, ξ (cid:39) v/ω bp where v is the transverse or longitudinal acoustic velocity fortorsional or spherical modes, respectively. For most glasses ξ , v , and ω bp vary by a factor about two only, giving thus similar numbers which limitthe predictive power of the model.Since each spectroscopy is sensitive to particular atomic motions, an-other way to probe the boson peak modes is to extract the frequency de-pendence of its coupling-to-light coefficient. According to Eq. 18, C rs ( ω )is extracted from the comparison between the Raman intensity with g ( ω )obtained using either INS or specific heat. The results reveal that below ∼ ω bp the coupling coefficient exhibits a universal behavior C rs ( ω ) = A + Bω/ω bp [166] with A = 0 or 0 . ∼ ω bp no universal character is found and the BPshape becomes fully material-dependent. The progressive transition froma universal to a non-universal behavior of C rs ( ω ) correlates with the fre-quency of the Ioffe–Regel crossover, providing further evidence of the closelink between the strong scattering regime of acoustic branches and the bo- anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 46 B. Ruffl´e, M. Foret & B. Hehlen
Table 1. Frequency (in cm − ) of the boson peak maximum ω bp in a Ramanreduced spectra and in a g ( ω ) /ω plot in INS or specific heat measurementcompiled from [164, 166] and additional data (when indicated).Glass ω rs max ω ins max Glass ω rs max ω ins max (2) (3) O − (4) O − O O (1) S from [167], from [168], from [83], from [169] a ν/c (cm − ) I / ω ( n ( ω ) + )( a r b . un i t s ) b q (˚A − ) I ( q )( a r b . un i t s )
10 20 300102030 1234567 vt/cω max (˚A) π / ∆ Q ( ˚A ) d Scaled frequencies ν/ν bp C ( ν )( a . u . ) c C ( ν )( a . u . ) Fig. 21. a) boson peak and b) FSDP of different glasses (number 1 to 7 in Tab. 1).The inset in b) shows the correlation between characteristic lengths derived from theglass structure and vibrations [164]. c) Frequency dependence of the coupling coefficient C rs ( ω ) for glasses 2, 8–11 in Tab. 1, CKN (circles), and Ag O − O (triangles) vs ν/ν bp . The dotted line in c) is C rs ( ν ) ∝ ν/ν bp + 0 .
5. d) same as in c) for glasses 1, 13 inTab. 1, As O (triangles), and GeO (circles). The dotted line is C rs ( ν ) ∝ ν/ν bp [166]. son peak maximum.Until recently, infrared spectroscopy has been barely used to study theBP modes due to the technical difficulty of reaching low frequencies. Therecent development of THz light sources (far IR) in laboratories or of softx-rays spectrometer at synchrotrons enabled to overcome this issue. Time-domain or THz-spectroscopies (THz-TDS) are now mature for measuringthe absorption coefficient α ( ω ) in the far infrared, i.e., down to a few anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 47 Low-frequency vibrational spectroscopy of glasses wavenumbers [170]. Within the linear response theory α ( ω ) relates to thedensity of vibrational states g ( ω ) by α ( ω ) = C ir ( ω ) g ( ω ), where C ir ( ω ) isthe infrared coupling coefficient. Since the BP is seen in a g ( ω ) /ω plotone simply has α ( ω ) ω = C ir ( ω ) g ( ω ) ω (19)In glasses, Taraskin et al . [171] proposed a model where C ir ( ω ) = A + Bω β for frequencies below the Ioffe–Regel crossover. A and B are material-dependent constants related to long-range uncorrelated charge fluctuationsand correlated short-range (inter-atomic) charge fluctuations, respectively. β = 2 in the model of Ref. [171] but exponents as low as β = 1 has beenmeasured in molecular glasses, e.g., glucose [172]. This departure has beeninterpreted in terms of acousto-optic mode couplings and bending of theacoustic branches which are not taking into account in Taraskin and Elliottmodel.Hyper-Raman scattering (HRS) is a non-linear spectroscopy where twoincident photons scatter one photon after interaction with a vibration inthe medium. It corresponds to the first non-linear term in Eq. 16 and hencethe signal originates from the fluctuations of the hyper-polarizability tensor β [173]. β (cid:48) has different symmetry properties than α (cid:48) yielding to differentselection rules. For example, polar vibrations are always active in HRSwhatever the crystalline or molecular symmetry is. Conversely, there existsmodes that are active in HRS but inactive in both RS and IR, improperlynamed silent modes . The relation between the measured intensity I hrs andthe density of state g ( ω ) is similar to that of Eq. 17 and 18 replacing C rs σ ( ω )by C hr σ ( ω ), the latter mirroring the properties of the hyper-polarizabilitytensor.Figure 22 compiles the boson peaks obtained in vitreous silica and vit-reous borate glasses using neutron scattering, Raman scattering, hyper-Raman scattering, and THz-TDS spectroscopy. The different spectralshapes arise from the different frequency dependence of the coupling coeffi-cients combined to the sensitivity of each technique to specific vibrations. In v -B O , see Fig. 22a, the Raman and hyper-Raman BP are rather similarand coincide with the quasi-incoherent INS contribution shown in Fig.17c.The combined analysis revealed that the latter could correspond to localvibrations involving out-of-plane librations of rigid BO triangles and B O rings whose schematic representation is also shown in Fig. 22a. Within the D point group of these molecular units, this corresponds to E (cid:48)(cid:48) -symmetrymodes which are indeed active in both RS and HRS [161, 167]. THz-TDS anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 48 B. Ruffl´e, M. Foret & B. Hehlen a B O ν/c (cm − ) I r s / ν ( n ( ν ) + ) , g ( ν ) / ν , α / ν b Li O − O ν/c (cm − ) c SiO ν/c (cm − ) HRSRS, INS RSTHz HRS 90 VVHRS 90 VH RSHRS 180 VU INS
Fig. 22. a) boson peak of v -B O : HRS (solid line), RS (dashed line), out-of-phaseINS component (circles), and in-phase INS component (crosses) [161]. A schematicrepresentation of E (cid:48)(cid:48) -symmetry librations of boroxol rings is also shown. b) boson peakin a lithium borate glass with 14 mol % Li O content in RS and THz-TDS spectroscopy[174]. c) boson peak of v -SiO : HRS (symbols), RS (solid line), and INS (dashedline) [175]. in pure v -B O is very challenging, but data obtained in a lightly dopedlithium borate glass Li O − O [174] highlights a BP at significantlylower frequency than the RS one (see Fig. 22b), and suggests that the IRresponse rather associates with the in-phase INS contribution. In silica,the HRS BP has been associated to local or quasi-local libration motionsinvolving rigid SiO units [162, 175] introduced by Buchenau et al . (seeFig. 17c). Within the T d point group symmetry of SiO tetrahedra, thisvibration is inactive in both IR (thus non-polar) and RS, and only active inHRS ( silent mode of F -symmetry). This is likely the reason why RS andHRS responses are so different. The former has been ascribed to a leakageof these modes due to the structural disorder and dipolar-type contribu-tions [175]. Moreover, since the F -symmetry modes are non-polar, theydo not carry an electric charge and therefore do not participate to the localcontribution of charge fluctuations ( B term). This is consistent with theobserved constant behavior of C ir in vitreous silica [171]. Finally, the factthat the HRS-BP perfectly matches the INS one supports INS assumptionthat these modes play an important role in the excess of modes at lowfrequency in vitreous silica. Besides, libration motion of SiO tetrahedrais the soft mode of the structural α - β instability at 846 K in quartz [160].Its frequency extrapolates to ∼
36 cm − at T g ∼ g ( ω ) /ω measured by INS. Librations of rigidunits are also low-lying vibrations in β -cristobalite, but are located at the anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 49 Low-frequency vibrational spectroscopy of glasses zone boundary of a transverse acoustic branch. In the glass these modes arecharacterized by weak bondings (those creating the disorder). It is there-fore natural that they vibrate at low frequency. In a connected networkthey also hybridize with transverse and longitudinal acoustic phonons ofsimilar frequency. This source of scattering combines with the incoherentscattering of acoustic modes to built up the reservoir of boson peak modes.The loss of the Brillouin zone in the glass combined with the localization ofplane wave makes identification of these vibrations as acoustic- or optic-likeuseless. On heating, the inter-unit bonds gradually break and these libra-tion motions (vibrations) transform into rotation motions (relaxations) inthe liquid state. Nuclear inelastic scattering
Phonon spectra can be obtained with a M¨ossbauer radioisotope source byanalyzing the frequency dependence of its nuclear resonance fluorescencewith recoil. The principles were formulated soon after the discovery ofthe M¨ossbauer effect [177], but Nuclear inelastic scattering (NIS) setupshave emerged only in the 90’s with the development of synchrotron radia-tion offering highly monochromatic x-rays beams, i.e., compatible with thephonon energy spectra. Albeit the technique is limited to elements pos-sessing a M¨ossbauer isotope, this can turn into an important advantage forexample when the dynamics of the nuclei must be studied separately.In molecular glasses the probe were neutral ferrocene molecules with thecentral Fe nucleus as M¨ossbauer isotope [178]. In that case, neither rota-tions of the probe nor intramolecular modes of the glassy matrix are seen.At boson peak frequencies, the probe selects only few modes among thetotal vDOS, mostly displacements of the rigid probe driven by the transla-tional collective motions of the glass network. The coherent length of suchvibrations was estimated to be about 20 (cid:6)
A. The net difference between theNIS vDOS and the INS vDOS of toluene is shown in Figs. 23a,b. The latteris much more complex due to the participation of rotations and librationsof the methyl groups, as well as other local modes at high frequency. Thecomparison of the NIS and INS bosons peaks of toluene and dibutyl ph-thalate is also very informative (see Figs. 23c,d). The higher number ofvibrational states in the total vDOS (INS) as compared to the NIS vDOS(by a factor of 2) in dibutyl phthalate at and below the BP maximum wasexplained by modes with coherent lengths shorter than the 20 (cid:6)
A probed bythe technique, i.e., quasi-local or local vibrations. NIS was also performed anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 50 B. Ruffl´e, M. Foret & B. Hehlen . . a toluene - NIS g ( E )( m e V − ) . . b toluene - INSEnergy (meV) g ( E )( m e V − ) crystalglass 0 10 201e -5 -4 -3 -2 c tolueneEnergy (meV) g ( E ) / E ( m e V − ) NISINS 0 10 20 d dibutyl phthalateEnergy (meV) NISINS
Fig. 23. Density of states of toluene glass measured by a) nuclear inelastic scattering at22 K from ferrocene molecules and b) inelastic neutron scattering. The arrows indicateslocal optic modes not seen in the NIS vDOS. Reduced NIS- and INS-vDOS in c) tolueneand d) dibutyl phthalate in semi-log plots. Adapted from [178]. in a Na FeSi O glass with 95% enrichment of Fe isotope [151, 179]. Thestructural analysis has shown that iron is mostly in tetrahedral coordina-tion and therefore plays a similar role as silicon. A comparative analysis ofthe NIS vDOS of the glass and of the crystalline polymorph of NaFeSi O revealed that the number of vibrational states of the boson peak is similarto that of the Van–Hove singularity of the TA branch in the crystal. Italso suggested that the latter are redistributed at lower frequency in theglass and construct the boson peak. it is worth reminding however that inthese pioneered NIS campaigns, the M¨ossbauer isotope Fe, positioned atthe center of mass of the ferrocene molecule and of the FeO tetrahedra, isinsensitive to libration motions which are supposed to play an importantrole at the boson peak frequencies in silicates. Inelastic x-ray scattering
The density of vibrational states of pristine and densified silica glasses havebeen collected by inelastic x-ray scattering with setups equipped eitherwith a crystal-type or by a M¨ossbauer-type analyzer [18]. The data havebeen compared with those obtained in various crystalline SiO polymorphs.Calculating the absolute value of the DOSs, the authors have shown thatthe number of modes in excess relative to the Debye level is similar inthe glass and in the crystalline polymorph with a comparable density. Inaddition, the shape of g ( ω ) /ω in the glass (BP) appears as a smeared outversion of g ( ω ) /ω in the corresponding crystal which shows the Van–Hove anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 51 Low-frequency vibrational spectroscopy of glasses singularity. Despite these striking resemblances, it would be misleading toconclude that the BP simply identifies to the Van–Hove singularity of theacoustic branches in the corresponding crystal. The nature of the vibrationsunderlying the low frequency vDOS in glasses and crystals are indeed verydifferent. In the latter, the vibrations are plane waves (acoustic and optic)propagating in an ordered material while in the former the acoustic modesare scattered by the disorder. These differences produce the shadowedregion in C p ( T ) /T (cf. Fig. 1) and may appear as a detail when comparingthe total vDOS of crystals and glasses. It is however these specific propertieswhich lead to the scattering regime of phonons at the origin, e.g., of theplateau in the thermal conductivity, and to the boson peak observed ininelastic light spectroscopies. References [1] N. Bilir and W. A. Phillips, Phonons in SiO : The low-temperature heatcapacity of cristobalite, Phil. Mag. (1), 113–122 (1975).[2] M. T. Dove, M. J. Harris, A. C. Hannon, J. M. Parker, I. P. Swainson, andM. Gambhir, Floppy modes in crystalline and amorphous silicates, Phys.Rev. Lett. (6), 1070–1073 (1997).[3] V. N. Sigaev, E. N. Smelyanskaya, V. G. Plotnichenko, V. V. Koltashev,A. A. Volkov, and P. Pernice, Low-frequency band at 50 cm − in the ramanspectrum of cristobalite: identification of similar structural motifs in glassesand crystals of similar composition, J. Non-Cryst. Solids . (2), 141–146(1999).[4] R. Wietzel, The stability conditions of the glass and crystal phase of silicondioxide, Z. Anorg. Allg. Chem. , 71–95 (1921).[5] F. Simon, Analysis of specific heat capacity in low temperatures,
Annalender Physik . (11), 241–280 (1922).[6] R. B. Sosman, The Properties of Silica: An Introduction to the Propertiesof Substances in the Solid Non-conducting State . vol. 37,
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J. Phys. Chem. Solids . (1), 53–65 (1959).[8] R. C. Zeller and R. O. Pohl, Thermal conductivity and specific heat ofnoncrystalline solids, Phys. Rev. B . (6), 2029–2041 (1971).[9] U. Buchenau, M. Prager, N. N¨ucker, A. J. Dianoux, N. Ahmad, and W. A.Phillips, Low-frequency modes in vitreous silica, Phys. Rev. B . (8),5665–5673 (1986).[10] M. A. Ramos and U. Buchenau, Low-temperature thermal conductivity of anuary 12, 2021 2:0 ws-rv9x6 Book TitleChapter8˙Ruffle˙Foret˙Hehlen page 52 B. Ruffl´e, M. Foret & B. Hehlen glasses within the soft-potential model,
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