**6. Brillouin spectroscopy**

Aggregates of atoms, from molecules to clusters, to nanoparticles and nanocrystals, up to mesoscopic and macroscopic aggregates, can interact with electromagnetic waves either elastically or inelastically. Inelastic interactions include emission/absorption phenomena, and inelastic scattering. We consider here inelastic scattering by vibrational excitations. At the molecular scale the atomic structure of matter has a crucial role, and quantum phenomena are relevant. At this level, vibrational excitations are the vibrations of molecules, or, in a crystal, the vibrations of the internal degrees of freedom of each unit cell, which form the so called optical branches of the dispersion relation, or optical phonons. Broadly speaking, inelastic scattering by these excitations is called Raman scattering.

Aggregates above the nanometric scale also support collective vibrational excitations which begin to resemble to acoustic waves, and can be described by a continuum model. In a crystal, the vibrations of the degrees of freedom of the centre of mass of each unit cell form the so called acoustic branches of the dispersion relation, or acoustic phonons. In the long wavelength limit they are the are acoustic waves, accurately described by the continuum model (Eq. (1)). Broadly speaking, inelastic scattering by these excitations is called Brillouin scattering.

Visible light has sub micrometric wavelength. In media, either crystalline or amorphous, which are homogeneous, and therefore translationally invariant, over at least a few micrometres, vibrational excitations of sub-micrometric wavelength have a well defined

Acoustic Waves: A Probe for the Elastic Properties of Films 139

0 <sup>2</sup> <sup>2</sup> *<sup>n</sup>* π

**<sup>k</sup>** <sup>=</sup> , (12)

**<sup>k</sup>** <sup>=</sup> , (13)

ρ

*v k*( ) ; since it can also be computed as

) couple, can be determined by a standard

, (14)

χ *E*,ν

couple, and the isolevel curves of the normalized

identify the confidence region at any

is found, allowing a good identification of the

σ

(Eq. (6)) is detected, C*<sup>11</sup>*

is evaluated by Eq. (8).

identifies the most

λ

which depends on the refractive index, but depends on geometry only when the sample is

0 <sup>2</sup> 2 sin π

λ

normal). In this case the probed wavevector depends the incidence angle, but not on the refractive index, because Snell's law implies that upon refraction the optical parallel

The data analysis for Brillouin spectroscopy results is common to all the methods, like laser ultrasonics, which measure the velocity of travelling waves. In the simplest cases the velocity is a function of the elastic constants which can be given in closed form. For instance,

In other cases, and namely in the case of supported films, the mode velocity can be computed as function of the elastic constants only numerically. In that case it must be remembered that the stiffness of an elastic solid is determined by as many independent parameters as are needed to completely identify the tensor of the elastic constants. In other words the stiffness is identified by a point in a multidimensional space, the dimensionality being 2 in the simplest case of the isotropic medium, and being higher for lower symmetry

Focusing here on the isotropic case, the stiffness can be represented, among other possible

the velocity is measured for various wavevectors *k* (in laser ultrasonics it is measured for

( ) ( ) <sup>2</sup>

where, for each wavevector *k*, the sum is further extended to all the detected acoustic

predetermined confidence level (Beghi et al., 2001, 2004, 2011; Lefeuvre et al., 1999). In some

parameters (Beghi et al., 2001; Comins et al., 2000; Zhang et al., 2001a), while in other cases a broad, valley-shaped minimum is found. In such cases a good identification of the parameters is not possible (Beghi et al., 2002; Zhang et al., 1998), although sometimes some combination of the parameters can be identified with better precision than individual

,, () , *c m k v k*

ν

<sup>−</sup> <sup>=</sup>

ν

) couple (see Eqs. (3) and (4)). In Brillouin spectroscopy, for each mode

σ

( )

*vE k vk*

σ

*v k* , and its uncertainty *<sup>C</sup>*<sup>11</sup>

θ

is the incidence angle (the angle between the incident beam and the surface

anisotropic, while for surface scattering it means *<sup>i</sup>* **k q** = ±2 , i.e.

if scattering by the longitudinal bulk wave, of velocity *<sup>l</sup>* C / <sup>11</sup> *v* =

 ρω

least squares minimization procedure. The sum of squares is computed as

ρ

where

media.

( ) , , *<sup>c</sup> vE k* ν

choices, by the (*E,*

probable value ( ) *E*,

χ

θ

components **q||** remain unchanged.

is directly obtained as 2 22 C / <sup>11</sup> *<sup>l</sup>* = =

ν

ν

estimator ( ) ( ) ( ) ( ) 22 2

cases a well defined minimum of ( ) <sup>2</sup>

ν − χ

various frequencies) as *v k <sup>m</sup>* ( ) with uncertainty

, the stiffness, represented by the (*E,*

2

χ

of the ( ) *E*,

ν

*EE E* , ,/ ,

*E*

 ν

modes. Following standard estimation theory, the minimum of ( ) <sup>2</sup>

χ *E*,ν

ν

ν

χ

wavevector (see Eq. (2)). Due to translational invariance, the kinematics of scattering selects the excitations whose wavevector is close to the Brillouin zone center (**k** = 0). The acoustic and optical phonon branches have very different behaviours close to the zone center (**k** → 0). The acoustic branch frequency ω*a* goes to zero, with phase velocity ω*a*/*k* and group velocity ∂ ∂ ω*<sup>a</sup>* / *k* which tend to coincide with the sound velocity (which depends on polarization and, in an anisotropic crystal, on the wavevector direction), while the optical branch frequency ω*<sup>o</sup>*

typically goes to a maximum, with group velocity ∂ ∂ ω*<sup>o</sup>* / *k* which goes to zero.

Correspondingly, with visible light and with typical properties of solids, the two types of branches produce inelastic scattering with frequency shifts ranging from a fraction of cm-1 to a few cm-1 (i.e from a few GHz to tens of GHz) for Brillouin scattering, and from hundreds to thousands of cm-1 (i.e from THz to tens of THz) for Raman scattering. The spectral analysis of so widely different frequency ranges requires different types of spectrometer. However, for both types of scattering the experiments are performed without exciting the vibrations, but relying on the naturally occurring thermal motion.

Brillouin spectrometry thus offers a fully optical, and therefore contact-less, method to measure the dispersion relations of bulk and surface acoustic waves, whose wavelength is determined by the scattering geometry and the optical wavelength, and is typically submicrometric. The frequency results from the medium properties, and typically falls in the GHz to tens of GHz range. Measurements are performed illuminating the sample by a focused laser beam, and analyzing the spectrum of scattered light, which is dominated by the elastically scattered light, but also contains weak Stokes/anti-Stokes doublets due to inelastic scattering by thermally excited vibrations (Beghi et al., 2004; Comins, 2001; Every, 2002; Grimsditch, 2001; Sandercock, 1982).

In sufficiently transparent materials scattering can occur in the bulk, by bulk acoustic waves. The coupling mechanism is the elasto-optic (or acousto-optic) effect: the periodic modulation of the refractive index by the periodic strain of the acoustic wave. In both transparent and opaque materials scattering can also occur by surface acoustic waves, by the ripple mechanism: the periodic corrugation of the surface due to the surface wave.

In more detail: the incident beam, of angular frequency Ω*i* and wavelength λ*<sup>0</sup>*, impinges on a sufficiently transparent sample with wavevector **q***i* and is refracted into the wavevector *<sup>i</sup>* **q**′ . Scattered light, of wavevector *<sup>s</sup>* **q**′ , emerges with wavevector **q***<sup>s</sup>* . The probed wavevector, ( ) ' ' **k qq** =± −*s i* , is determined by λ*<sup>0</sup>*, the directions of **q***i* and **q***<sup>s</sup>* , and the refractive index *n*. Light inelastically scattered by a vibrational excitation of angular frequency ω*(***k***)* gives a Stokes/anti-Stokes doublet at frequencies Ω*s* = Ω*<sup>i</sup>* ± ω. Detection, in the spectrum of scattered light, of such a doublet allows to measure ω = |Ω*s* - Ω*i*| and to derive the excitation velocity *v* =ω / **k** . In both transparent and opaque samples scattering occurring by surface waves only depends on the components of wavevectors parallel to the surface: the probed wavevector is ( ) *s i* **k qq** =± −' ' , and the surface wave velocity is *v* =ω / **k** . In other words, the spontaneous thermal motion can be viewed as spatially Fourier transformed into an incoherent superposition of harmonic waves having all the possible wavevectors; the scattering geometry (the directions of wavevectors **q***i* and **q***<sup>s</sup>* ) selects a specific wavevector **k** or **k** which is probed by the inelastic light scattering event.

Although also other geometries have been exploited (Beghi et al. 2011), in Brillouin spectroscopy the most frequently adopted scattering geometry is backscattering: **q q** *s i* = − . For bulk scattering it corresponds to 2 ' **k q** = ± *<sup>i</sup>* , such that

wavevector (see Eq. (2)). Due to translational invariance, the kinematics of scattering selects the excitations whose wavevector is close to the Brillouin zone center (**k** = 0). The acoustic and optical phonon branches have very different behaviours close to the zone center (**k** → 0). The acoustic branch frequency ω*a* goes to zero, with phase velocity ω*a*/*k* and group velocity

*<sup>a</sup>* / *k* which tend to coincide with the sound velocity (which depends on polarization and, in an anisotropic crystal, on the wavevector direction), while the optical branch frequency ω*<sup>o</sup>*

Correspondingly, with visible light and with typical properties of solids, the two types of branches produce inelastic scattering with frequency shifts ranging from a fraction of cm-1 to a few cm-1 (i.e from a few GHz to tens of GHz) for Brillouin scattering, and from hundreds to thousands of cm-1 (i.e from THz to tens of THz) for Raman scattering. The spectral analysis of so widely different frequency ranges requires different types of spectrometer. However, for both types of scattering the experiments are performed without

Brillouin spectrometry thus offers a fully optical, and therefore contact-less, method to measure the dispersion relations of bulk and surface acoustic waves, whose wavelength is determined by the scattering geometry and the optical wavelength, and is typically submicrometric. The frequency results from the medium properties, and typically falls in the GHz to tens of GHz range. Measurements are performed illuminating the sample by a focused laser beam, and analyzing the spectrum of scattered light, which is dominated by the elastically scattered light, but also contains weak Stokes/anti-Stokes doublets due to inelastic scattering by thermally excited vibrations (Beghi et al., 2004; Comins, 2001; Every,

In sufficiently transparent materials scattering can occur in the bulk, by bulk acoustic waves. The coupling mechanism is the elasto-optic (or acousto-optic) effect: the periodic modulation of the refractive index by the periodic strain of the acoustic wave. In both transparent and opaque materials scattering can also occur by surface acoustic waves, by the

sufficiently transparent sample with wavevector **q***i* and is refracted into the wavevector *<sup>i</sup>* **q**′ . Scattered light, of wavevector *<sup>s</sup>* **q**′ , emerges with wavevector **q***<sup>s</sup>* . The probed wavevector,

Stokes/anti-Stokes doublet at frequencies Ω*s* = Ω*<sup>i</sup>* ± ω. Detection, in the spectrum of scattered light, of such a doublet allows to measure ω = |Ω*s* - Ω*i*| and to derive the

by surface waves only depends on the components of wavevectors parallel to the surface:

other words, the spontaneous thermal motion can be viewed as spatially Fourier transformed into an incoherent superposition of harmonic waves having all the possible wavevectors; the scattering geometry (the directions of wavevectors **q***i* and **q***<sup>s</sup>* ) selects a

Although also other geometries have been exploited (Beghi et al. 2011), in Brillouin spectroscopy the most frequently adopted scattering geometry is backscattering: **q q** *s i* = − .

ripple mechanism: the periodic corrugation of the surface due to the surface wave.

In more detail: the incident beam, of angular frequency Ω*i* and wavelength

Light inelastically scattered by a vibrational excitation of angular frequency

the probed wavevector is ( ) *s i* **k qq** =± −' ' , and the surface wave velocity is *v* =

specific wavevector **k** or **k** which is probed by the inelastic light scattering event.

λ

exciting the vibrations, but relying on the naturally occurring thermal motion.

ω

*<sup>o</sup>* / *k* which goes to zero.

λ

*<sup>0</sup>*, the directions of **q***i* and **q***<sup>s</sup>* , and the refractive index *n*.

/ **k** . In both transparent and opaque samples scattering occurring

*<sup>0</sup>*, impinges on a

*(***k***)* gives a

ω

ω/ **k** . In

typically goes to a maximum, with group velocity ∂ ∂

2002; Grimsditch, 2001; Sandercock, 1982).

( ) ' ' **k qq** =± −*s i* , is determined by

ω

For bulk scattering it corresponds to 2 ' **k q** = ± *<sup>i</sup>* , such that

excitation velocity *v* =

∂ ∂ ω

$$\left|\mathbf{k}\right| = 2\frac{2\pi}{\lambda\_0}n\quad,\tag{12}$$

which depends on the refractive index, but depends on geometry only when the sample is anisotropic, while for surface scattering it means *<sup>i</sup>* **k q** = ±2 , i.e.

$$\left|\mathbf{k}\_{\parallel}\right| = 2\frac{2\pi}{\mathcal{A}\_0}\sin\theta \,\, \mathbf{\hat{z}}\tag{13}$$

where θ is the incidence angle (the angle between the incident beam and the surface normal). In this case the probed wavevector depends the incidence angle, but not on the refractive index, because Snell's law implies that upon refraction the optical parallel components **q||** remain unchanged.

The data analysis for Brillouin spectroscopy results is common to all the methods, like laser ultrasonics, which measure the velocity of travelling waves. In the simplest cases the velocity is a function of the elastic constants which can be given in closed form. For instance, if scattering by the longitudinal bulk wave, of velocity *<sup>l</sup>* C / <sup>11</sup> *v* = ρ (Eq. (6)) is detected, C*<sup>11</sup>* is directly obtained as 2 22 C / <sup>11</sup> *<sup>l</sup>* = = ρ ρω *v k* , and its uncertainty *<sup>C</sup>*<sup>11</sup> σ is evaluated by Eq. (8). In other cases, and namely in the case of supported films, the mode velocity can be computed as function of the elastic constants only numerically. In that case it must be remembered that the stiffness of an elastic solid is determined by as many independent parameters as are needed to completely identify the tensor of the elastic constants. In other words the stiffness is identified by a point in a multidimensional space, the dimensionality being 2 in the simplest case of the isotropic medium, and being higher for lower symmetry media.

Focusing here on the isotropic case, the stiffness can be represented, among other possible choices, by the (*E,*ν) couple (see Eqs. (3) and (4)). In Brillouin spectroscopy, for each mode the velocity is measured for various wavevectors *k* (in laser ultrasonics it is measured for various frequencies) as *v k <sup>m</sup>* ( ) with uncertainty σ *v k*( ) ; since it can also be computed as ( ) , , *<sup>c</sup> vE k* ν , the stiffness, represented by the (*E,*ν) couple, can be determined by a standard least squares minimization procedure. The sum of squares is computed as

$$\mathcal{X}^2(E,\nu) = \sum\_{k} \left( \frac{\upsilon\_c(E,\nu,k) - \upsilon\_m(k)}{\sigma\_{v(k)}} \right)^2,\tag{14}$$

where, for each wavevector *k*, the sum is further extended to all the detected acoustic modes. Following standard estimation theory, the minimum of ( ) <sup>2</sup> χ *E*,ν identifies the most probable value ( ) *E*,ν of the ( ) *E*,ν couple, and the isolevel curves of the normalized estimator ( ) ( ) ( ) ( ) 22 2 χ *EE E* , ,/ , ν − χ ν χ ν identify the confidence region at any predetermined confidence level (Beghi et al., 2001, 2004, 2011; Lefeuvre et al., 1999). In some cases a well defined minimum of ( ) <sup>2</sup> χ *E*,ν is found, allowing a good identification of the parameters (Beghi et al., 2001; Comins et al., 2000; Zhang et al., 2001a), while in other cases a broad, valley-shaped minimum is found. In such cases a good identification of the parameters is not possible (Beghi et al., 2002; Zhang et al., 1998), although sometimes some combination of the parameters can be identified with better precision than individual

Acoustic Waves: A Probe for the Elastic Properties of Films 141

acoustic waves trapped within a film, which are reflected back and forth, crossing the film

Brillouin spectroscopy lends itself to the characterization of structures other than films or layers. In particular, single-walled carbon nanotubes were characterized, measuring Brillouin scattering by a free-standing film of pure, partially aligned, single-walled nanotubes, and analyzing the results in terms of continuum models (Bottani et al., 2003). The dependence of the measured spectra on the angle between the exchanged wavevector and the preferential direction of the tubes shows that the tube-tube interactions are weak: the tubes are vibrationally almost independent. The tubes are modelled as continuous membranes, at two different levels: at the first one the membrane is infinitely flexible, only able to transmit in-plane forces, while at the second level of approximation the tube wall is treated as also able to transmit shear forces and torques not belonging to the shell surface. In both cases scattering was essentially due to longitudinal waves travelling along the tubes. Taking into account that AFM images suggest that the tube segments contributing to scattering are not in the infinite tube length approximation, it was possible to derive the 2D Young modulus for the tube wall, achieving the first dynamic estimation of the stiffness of the tube wall. Scattering from carbon nanotubes was observed also in a different geometry,

Due to its intrinsic contact-less nature, Brillouin spectroscopy is the natural choice for the measurement of elastic properties in conditions, like high temperature and/or high pressure, in which physical contact with the specimen is difficult, if possible at all. Brillouin spectroscopy only requires optical access, which can be obtained by an appropriate window, and even in the extreme conditions achievable in a diamond anvil cell, optical access is

Measurements were performed at high temperatures (Pang, 1997; Stoddart, 1995; Zhang et al., 2001a), as well as at low temperatures, which were crucial to single out a particular mechanism of hypersound propagation in alkali-borate glasses (Carini et al., 2008). After pioneering experiments at high pressures (Crowhurst et al., 1999; Whitfield et al., 1976,), in recent years dedicated Brillouin spectrometers were built at synchrotron facilities, allowing simultaneous performance of high resolution x-ray diffraction and Brillouin spectroscopy (and possibly other optical investigations like Raman spectroscopy, fluorescence, absorption) on specimens subjected to extreme pressures in a diamond anvil cell, with possible heating (Murakami et al., 2009; Prakapenka et al., 2010; Sinogeikin et al., 2006). This set-up, of particular interest for geophysicists since it allows to characterize the behaviour of minerals in the conditions which are found in the Earth's interior, was exploited to perform measurements on SiO2 glass (Murakami & Bass, 2010) and other minerals, but also on

The stiffness of films, characterized by the elastic constants, depends on the film microstructure, and its precise characterization is crucial when thin layers have structural functions. The interest in the measurement of the elastic constants is witnessed by the number of new techniques, or of improvements of existing techniques, being proposed. The techniques which exploit either propagating acoustic waves or standing oscillations involve exclusively elastic strains: they therefore offer the most direct and clean access to the elastic properties, and potentially the most accurate measurements. Among the methods

with an ordered array of tubes, clamped at one end (Polomska et al., 2007).

guaranteed by the transparency of the same diamond anvils.

polymers (Stevens et al., 2007) and liquid methane (Li et al., 2010).

**7. Conclusion** 

perpendicularly to its surface (Zhang, 2001b).

parameters (Comins et al., 2000; Zhang et al., 1998, 2001a). When several acoustic modes are measured, the larger amount of available information allows a precise and complete elastic characterization, as obtained e.g. for SiC films of micrometric thickness (Djemia et al, 2004).

In Eq. (14) each value ( ) *mv k* , being obtained as ω*/ k*, has an uncertainty σ *v k*( ) which depends in turn on the uncertainty of the frequency of each spectral doublet, and on the precision of the incidence angle (see Eq. (13)) or of the refractive index (see Eq. (12)). It can be noted that the uncertainties of frequency and angle are of the random type, which affects precision but not accuracy, while the uncertainty of the refractive index affects accuracy but not precision (see Section 3.3). As with other techniques, also the uncertainties concerning the mass density and the layer thickness(es) affect accuracy but not precision. These uncertainties were the object of detailed investigations (Beghi et al., 2011; Stoddart et al., 1998). The effects of the uncertainties of the quantities which are directly measured ('primary uncertainties') on the values of the elastic constants which are finally obtained were evaluated. It was found that with appropriate sets of measurement uncertainties at the 1% level are reachable (Beghi et al., 2011).

As already noted, Brillouin spectroscopy measures the acoustic modes at frequencies of the order of GHz to tens of GHz, therefore at wavelengths much shorter than those corresponding to frequencies of tens to hundreds of MHz, typically observed with piezoelectric excitation and/or detection. This gives Brillouin spectroscopy an intrinsically higher sensitivity to the properties of films, or to the perturbation induced by the presence of very thin films. Brillouin spectroscopy was exploited to characterize tetrahedral amorphous carbon films of thicknesses of hundreds of nanometres (Chirita et al., 1999), tens of nanometres (Ferrari et al., 1999), down to a few nanometres (Beghi et al., 2002). It was also shown that inelastic light scattering can be sensitive to nanometric thickness differences (Lou et al., 2010). By Brillouin spectroscopy it was also possible to characterize buried layers in silicon-on-insulator structures (Ghislotti & Bottani, 1994).

On the other hand, the techniques which excite vibrations operate with oscillation amplitudes significantly larger than those measured by Brillouin spectroscopy; this allows more precise measurements of frequencies, which at least partially compensates for the lower intrinsic sensitivity due to the larger wavelengths. Combinations of techniques were also exploited: thicker tetrahedral amorphous carbon films films (3 micron) were characterized combining Brillouin spectroscopy and laser ultrasonics. A wide range of frequencies was thus accessible, allowing a detailed characterization of the elastic properties of the film (Berezina et al., 2004). A combination of Brillouin spectroscopy and picosecond ultrasonics was instead exploited to characterize superlattices formed by periodic multilayers of permalloy/alumina, with various periodicities at the nanometric scale (Rossignol et al., 2004). Picosecond ultrasonics characterizes the out-of-plane properties by waves travelling normal to the surface, while Brillouin spectroscopy characterizes the inplane properties by waves travelling along the surface. The combination of techniques elucidated the effects of the interfaces.

Another algorithm for data analysis, different from that outlined above, has also been recently proposed (Every et al., 2010). Both algorithms refer to the types of waves most frequently measured in Brillouin spectroscopy of films or layered structures: surface acoustic wave, or pseudo surface acoustic waves, which essentially travel parallel to the surface, or however have a significant wavevector component parallel to the surface. It can also be mentioned that it was also possible, by Brillouin spectroscopy, to detect standing

parameters (Comins et al., 2000; Zhang et al., 1998, 2001a). When several acoustic modes are measured, the larger amount of available information allows a precise and complete elastic characterization, as obtained e.g. for SiC films of micrometric thickness (Djemia et al, 2004).

in turn on the uncertainty of the frequency of each spectral doublet, and on the precision of the incidence angle (see Eq. (13)) or of the refractive index (see Eq. (12)). It can be noted that the uncertainties of frequency and angle are of the random type, which affects precision but not accuracy, while the uncertainty of the refractive index affects accuracy but not precision (see Section 3.3). As with other techniques, also the uncertainties concerning the mass density and the layer thickness(es) affect accuracy but not precision. These uncertainties were the object of detailed investigations (Beghi et al., 2011; Stoddart et al., 1998). The effects of the uncertainties of the quantities which are directly measured ('primary uncertainties') on the values of the elastic constants which are finally obtained were evaluated. It was found that with appropriate sets of measurement uncertainties at the 1% level are reachable

As already noted, Brillouin spectroscopy measures the acoustic modes at frequencies of the order of GHz to tens of GHz, therefore at wavelengths much shorter than those corresponding to frequencies of tens to hundreds of MHz, typically observed with piezoelectric excitation and/or detection. This gives Brillouin spectroscopy an intrinsically higher sensitivity to the properties of films, or to the perturbation induced by the presence of very thin films. Brillouin spectroscopy was exploited to characterize tetrahedral amorphous carbon films of thicknesses of hundreds of nanometres (Chirita et al., 1999), tens of nanometres (Ferrari et al., 1999), down to a few nanometres (Beghi et al., 2002). It was also shown that inelastic light scattering can be sensitive to nanometric thickness differences (Lou et al., 2010). By Brillouin spectroscopy it was also possible to characterize buried layers

On the other hand, the techniques which excite vibrations operate with oscillation amplitudes significantly larger than those measured by Brillouin spectroscopy; this allows more precise measurements of frequencies, which at least partially compensates for the lower intrinsic sensitivity due to the larger wavelengths. Combinations of techniques were also exploited: thicker tetrahedral amorphous carbon films films (3 micron) were characterized combining Brillouin spectroscopy and laser ultrasonics. A wide range of frequencies was thus accessible, allowing a detailed characterization of the elastic properties of the film (Berezina et al., 2004). A combination of Brillouin spectroscopy and picosecond ultrasonics was instead exploited to characterize superlattices formed by periodic multilayers of permalloy/alumina, with various periodicities at the nanometric scale (Rossignol et al., 2004). Picosecond ultrasonics characterizes the out-of-plane properties by waves travelling normal to the surface, while Brillouin spectroscopy characterizes the inplane properties by waves travelling along the surface. The combination of techniques

Another algorithm for data analysis, different from that outlined above, has also been recently proposed (Every et al., 2010). Both algorithms refer to the types of waves most frequently measured in Brillouin spectroscopy of films or layered structures: surface acoustic wave, or pseudo surface acoustic waves, which essentially travel parallel to the surface, or however have a significant wavevector component parallel to the surface. It can also be mentioned that it was also possible, by Brillouin spectroscopy, to detect standing

ω

*/ k*, has an uncertainty

σ

*v k*( ) which depends

In Eq. (14) each value ( ) *mv k* , being obtained as

in silicon-on-insulator structures (Ghislotti & Bottani, 1994).

elucidated the effects of the interfaces.

(Beghi et al., 2011).

acoustic waves trapped within a film, which are reflected back and forth, crossing the film perpendicularly to its surface (Zhang, 2001b).

Brillouin spectroscopy lends itself to the characterization of structures other than films or layers. In particular, single-walled carbon nanotubes were characterized, measuring Brillouin scattering by a free-standing film of pure, partially aligned, single-walled nanotubes, and analyzing the results in terms of continuum models (Bottani et al., 2003). The dependence of the measured spectra on the angle between the exchanged wavevector and the preferential direction of the tubes shows that the tube-tube interactions are weak: the tubes are vibrationally almost independent. The tubes are modelled as continuous membranes, at two different levels: at the first one the membrane is infinitely flexible, only able to transmit in-plane forces, while at the second level of approximation the tube wall is treated as also able to transmit shear forces and torques not belonging to the shell surface. In both cases scattering was essentially due to longitudinal waves travelling along the tubes. Taking into account that AFM images suggest that the tube segments contributing to scattering are not in the infinite tube length approximation, it was possible to derive the 2D Young modulus for the tube wall, achieving the first dynamic estimation of the stiffness of the tube wall. Scattering from carbon nanotubes was observed also in a different geometry, with an ordered array of tubes, clamped at one end (Polomska et al., 2007).

Due to its intrinsic contact-less nature, Brillouin spectroscopy is the natural choice for the measurement of elastic properties in conditions, like high temperature and/or high pressure, in which physical contact with the specimen is difficult, if possible at all. Brillouin spectroscopy only requires optical access, which can be obtained by an appropriate window, and even in the extreme conditions achievable in a diamond anvil cell, optical access is guaranteed by the transparency of the same diamond anvils.

Measurements were performed at high temperatures (Pang, 1997; Stoddart, 1995; Zhang et al., 2001a), as well as at low temperatures, which were crucial to single out a particular mechanism of hypersound propagation in alkali-borate glasses (Carini et al., 2008). After pioneering experiments at high pressures (Crowhurst et al., 1999; Whitfield et al., 1976,), in recent years dedicated Brillouin spectrometers were built at synchrotron facilities, allowing simultaneous performance of high resolution x-ray diffraction and Brillouin spectroscopy (and possibly other optical investigations like Raman spectroscopy, fluorescence, absorption) on specimens subjected to extreme pressures in a diamond anvil cell, with possible heating (Murakami et al., 2009; Prakapenka et al., 2010; Sinogeikin et al., 2006). This set-up, of particular interest for geophysicists since it allows to characterize the behaviour of minerals in the conditions which are found in the Earth's interior, was exploited to perform measurements on SiO2 glass (Murakami & Bass, 2010) and other minerals, but also on polymers (Stevens et al., 2007) and liquid methane (Li et al., 2010).
