Brillouin scattering in glasses

Brillouin scatteringin glassesYann VAILLS (vaills@cnrs-orleans.fr)University of OrléansCentre de Recherches sur les Matériaux à HauteTempérature (CRMHT - CNRS Orléans - FRANCE)8th ESG 2006 Sunderland 10-14 14 Sept

« La lumière est le principal personnage dans le tableau »(Light is the main subject of the picture)8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

I. Light scatteringhν 0≈ 95 to 97%≈ 3 to 5%≈ 1%Scattered lightI scatt ∝ν 048th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

hν 0Scattered light :I scatt = I(ν 0 ) + I(ν≠ν 0 )≈ 90 to 99 %Rayleigh scattering≈ 1 to 10%Raman effect8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

Vibrational spectroscopies• Brillouin scattering• Raman scattering• Infrared spectroscopy8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

II. The Raman effectI scatt = I Rayl (ν 0 ) + I Raman (ν)ν Raman = ν 0 ± ν iν 0 : incident light frequencyν i : i vibrational mode frequency8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Scattering by vibrational modes :k rinck rinck rscattq rk rscattq rantistokesAnnihilation of avibrational modeν s = ν 0 + ν phononrq≅r rk inc ≅ k scattcreation of avibrational modeν s = ν 0 - ν phononstokes8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

RAMAN scatteringExitedelectroniclevelsRAYLEIGHscatteringStokesAntistokesVirtual levelshν 0hν 0hν 1 hν 0Fondamentalelectronic level8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=VaillsVibrational levelsMaxwell – Boltzmann ⇒ Raman Stokes et anti-Stokes lines havenot the same intensities

Scattered intensityRayleighν 1 = ν 0 - 459 cm -1ν 2 = ν 0 - 314 cm -1ν 3 = ν 0 - 218 cm -1Ramanν 4 = ν 0 + 218 cm -1ν 5 = ν 0 + 314 cm -1ν 6 = ν 0 + 459 cm -1CCl 4Raman spectrum at room temperature8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

Intensity (a.u.)1,210,80,60,40,20Raman spectrum of the glass (SiO 2 ) 82 (K 2 0) 180 500 1000 1500 2000frequency shift (cm -1 )Exp.Raman8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

III. The different types of lightscattering• Rayleigh scattering• Raman scattering• Brillouin scattering8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Scattering by density inhomogeneities :Rayleigh scattering (static(static)Brillouin scattering (dynamic(dynamic)Iid=⎛ 8I ⎜π0⎝ 3 λ034⎞⎟⎠n8⎛ β⎜⎝ ρid⎞⎟⎠2Δρ2V0k TJ. Shroeder JACS 1973 ; K. Saito APL 1997B2Δρ• Static fluctuations : structural or chemical fluctuations• Dynamical fluctuations : acoustical modes of vibration8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

Rayleigh scattering : due to the staticfluctuations of refractive index8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

The case of a diatomic linear lattice, periodicity : arq≅rkinc≅rkscatt≅2 π

IV. Brillouin and Rayleigh scatteringBrillouin spectrum of the glass (SiO 2 ) 85 (K 2 O) 15Rayleigh line Brillouin longitudinal line6543Brillouin transverse line210-0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9Frequency shift (cm -1 )8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=VaillsIntensity (arbitrary unit)

• 1. Lines frequenciesFor a right scattering configurationr rk inc ⊥ k scattν0 0 2 C11ν l = n 2Vl= nνc00 44νt= n 2Vt= ncνcνcρ2 CρLongitudinal acoustic mode of vibrationTransverse acoustic mode of vibration8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• From the Brillouin frequencies we deduceThe elastic properties of materials(see for example A.K. Varshneya 2006, or Y. Vaills web page)• n and ρ : measured by classical methods• in an isotrope material : 2 independent elastic constantsBrillouin scattering ν l νt⇒ C 11 and C 44C 12 = C 11 –2C 441λ = C 12μ = C 44Lame’s constantsEχ =σp=μ(3 λ + 2 μ )λ + μ3 −=3 λ + 2 μλ=2 ( μ + λ )KYoung’s moduluscompressibilityPoisson ratio8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• 2. Lines intensitiesScattering by density fluctuations :Rayleighscattering (static)Brillouin scattering (dynamic)Freezing ofdensityfluctuationsat the glasstransitionIid=⎛ 8I ⎜π0⎝ 3 λ034⎞⎟⎠n8⎛ β⎜⎝ ρid⎞⎟⎠2Δρ2V0k TJ. Shroeder JACS 1973 ; K. Saito APL 1997BT > T gIid=⎛ 8I ⎜π0⎝ 3 λ034⎞⎟⎠n8β2idχT(T ) k TB8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

T < TgIid⎛38 ⎞I ⎜π 8 2= ⎟0n4 id T ,rel g B g3βχ⎝ λ0⎠[ χ (T ) k T + (T ) k T ]S , ∞BRayleigh scattering :- static density inhomogeneities(elastic scattering)- Incoherent atoms motions,non propagating excitations(quasielastic scattering)(R. Vacher JCP 1985)(J. Shroeder JACS 1973)Brillouin scattering : inelastic scattering(dynamic density fluctuations : mechanical waves)8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Landau-PlaczekratioR =L −PIR/2 IB LIn a viscoelastic material :(N. Laberge JACS 1973)IRayleigh∝Δρ2kν = 0=( ρ20/ V ) k T [( χBT−χS) +χrS]Isobaric-entropy fluctuationsFluctuation associated withstructural variationsin adiabatic-pressure fluctuations⇔ relaxational compressibility8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

After quenching fluctuations are frozeninto the material at the equilibriumstructural configuration corresponding tothe fictive temperature T f(N. Laberge JACS 1973)I22−1Rayleigh∝ Δρk= ( ρ0/ V ) kBTf[( χT− χS) + ( χS− Cν = 011)]IBrillouin∝Δρ2kν ≠0=(ρ20/ V )[ k TCB−111]8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

⇒ Determination of a glass fictive temperatureRL −P=TfT( C χ − 1)11TDirectly deduced from by Brillouin scattering8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

Influence of heat treatment on silica for optical fiberY. Vaills (CRMHT), P. Simon (CRMHT), G. Matzen (CRMHT),H. Cattey (post-doc CRMHT-Alcatel), G. Orcel (Alcatel)Effect of fictive de la temperatureon light scattering in silica-110 -29 I(T) sec30 SiOnucléation220100500 1000 1500 T 2000 (K)Rapport Landau-Placzek: R LP3634323028Nucleation effect1400 1500 1600 1700 1800T de recuit (K)Effect of fictivetemperatureRL −PTf=T( C χ − 1)11T8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• From the Brillouin line intensities we deduce• The photoelastic constants of the materialsCoupling between elastic waves andelectromagnetic waves• electromagnetic energy loss in materialsAttenuation of electromagnetic wave inoptical fibers• fictive temperature of glasses8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• 3. Lines shapes(See for example R. Vacher JCP 1985, J. Schreoder JNCS 1988)The experimental Brillouin line is :a convolution of the natural Brillouin line with the apparatus function• natural Brillouin line :• apparatus function :narrow lorentzian shapea convolution of several contributions- The finite frequency width of the laser- The finite acceptance angle of the light gathering- The Airy’s transmission function of the F-P interferometer8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Extraction of the natural Brillouin line :deconvolution of the spectrum : several technicsFor example :I(H.W. Leidecker J.A.S.A. 1967D. Walton S.S.C. 1982G.E. Durand IEEE J.Q.E.1968A.S. Pine PR 1969)expBrillouin( ν ) = Snat( ν ) ∗ Iapp( ν )LorentzianGaussian8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

Experimental Brillouin linewidth : convolution of- Natural Brillouin linewidth ΔΓ B (≈ 0.1 GHz)- Instrumental linewidth (≈ 1 GHz)Phonon liftime τ :τ = 1ΔΓ BPhonon attenuation coefficient α :ΔΓB =αVlπ8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• From the Brillouin line shapes we deduce• Structural informations via the lifetimeof vibrational waves• Characterization of relaxation phenomenonsbonded to rearrangements of the structure• Properties controlled by vibrational wavesThermal expansion coefficient and its anomaliesThermal expansion coefficient and its anomaliesAnharmonicity(R. Vacher ,communication at this Conference 8 th ESG 2006, and PRB 2005)8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

V. Brillouin scattering apparatussampleFabry-PérotMichelsonInterferential FilterPMPressure scannedFabry-PérotSlice λ/2PumpGas (Freon)AmplifierSynchrone detectionshopperLaserAr-514.5nmAcquisition8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

sampleTandempiezoelectricallyscannedFabry-PérotSlice eλ/2PMJ.R. SandercockAmplifierSynchrone detectionshopperLaserAr-514.5nmAcquisition8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frFigure II.9 http://crmht.cnrs-orleans.fr/: Schéma du dispositif orleans.fr/pubcrmhtexpérimental pubcrmht/ext pour l’étude ext/peoplefile.aspx?nom=Vaillsde la diffusion Brillouin

Comparison of the two devicesLow RamanfrequenciesTriple passed pressurescanned Pérot-FabryTandem Triple passedFPpiezo-elel ly controlledContrast 3 10 6 10Finesse 70 8010 11Instrumental linewidthResolutionAccessiblefrequency rangeAcquisition timefor 1 scan770 MHz 500 MHz770 MHz at e = 2.76 mmR = 760 0005 - 50 GHz0,2 - 2 cm -1500 MHz at e = 2 mmR = 1 170 0005 - 1500 THz0,2 - 500 cm -145 minutes for 4 cm -1 30 seconds for 500 cm -1Spectra accumulations Impossible Possible8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

VI. Brillouin scattering in glasses• Localisation of residual stresses insilicate binary glasses (SiO 2 ) 1-x (Na 2 O) x0,0100,0 0,1 0,2 0,3 0,4ρ(annealed)-ρ(non-annealed)(Y. Vaills JNCS 2001)Δρ(g/cm 3 )0,0080,0060,0040,0020,000-0,0020,0 0,1 0,2 0,3 0,4molar fraction of Na 2 O : x8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Localisation of residual stresses insilicate binary glasses (SiO 2 ) 1-x (Na 2 O) xHypotheses for calculation of elasticenergy• 1) We have considered that the variations of Na-O length could be the unique cause of the densitychange on annealing.• 2) Consequently, most of the elastic energy dueto residual stress before annealing is located inthe sodium atoms neighbourhood.8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Localisation of residual stresses insilicate binary glasses (SiO 2 ) 1-x (Na 2 O) x• 3) In order to simplify the evaluation of thedeformation tensor we suppose that theresidual stresses are of hydrostatic type. Thenthe relative dilatation can be calculated fromthe relative variation of the density :θ = −Δρρ• 4) We neglect the variations of interatomicbond strengths induced by the interatomic bondlength variations.8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Localisation of residual stresses insilicate binary glasses (SiO 2 ) 1-x (Na 2 O) xConsequently the elastic energy stored in thenon-annealed glass is given by :2Φ = θ6( 3C11− 4C44)Φ is the elastic energy per unit of volume, C 11 andC 44 are the elastic constants of the annealedglass (undeformed glass).8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Localisation of residual stresses insilicate binary glasses (SiO 2 ) 1-x (Na 2 O) x(J/m 3 )3,5x10 53,0x10 5elastic energy2,5x10 52,0x10 51,5x10 51,0x10 55,0x10 40,0-5,0x10 40,0 0,1 0,2 0,3 0,4mol. fract. of Na 2 O8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Localisation of residual stresses insilicate binary glasses (SiO 2 ) 1-x (Na 2 O) x• If we write the total volume of the glass as :3V = VNa−O+ VSiO= nαa +• where a = r Na-O , then we can calculate δathrough annealing by :2VSiO2δ a =−Δ ρ V2ρ n α 3a8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Localisation of residual stresses insilicate binary glasses (SiO 2 ) 1-x (Na 2 O) xUsing•the δ(r Na-O ),• the elastic energy variations through annealing• and the fact that Na is known to be fivefold coordinatedin Na 2 O-SiO 2 glasses,we calculated :1) the elastic energy for each Na-O pair2) the force constant of Na-O bond3) the frequency associated to the Na vibrational mode8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

• Localisation of residual stresses insilicate binary glasses (SiO 2 ) 1-x (Na 2 O) x(cm -1 )24020016012080IR frequencies-1ν = 1/2π[k(m +(5mONa) -1 )] 1/2-1ν = 1/2π[k(m +(6mONa) -1 )] 1/2ν = 1/2π[k/mNa] 1/2400,0 0,1 0,2 0,3 0,4 0,5molar fraction of Na 2 O : x8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills

constraint theory(J.C. Phillips JNCS 1979)Binary glasses (SiO 2 ) 1-x (Na 2 O) xSiONaNa increases ⇒connectedness decreasesQ 3n c= 2.67Q 4n c= 3.67X c = 0.28th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaillsn c =⇓3x = x c Elastic phasetransition from rigid tofloppy phase

Unsteady state and elastic free energythrow annealing : ΔΦ(Vaills J. Phys. C 2005)ΔΦ=1 ⎛ Δρ ⎞⎜ ⎟6 ⎝ ρ ⎠2( 3C − 4C )1144Random networks within meanfieldtheory : f(x)= (10x-3)/3floppy modesx = 0.238x < 0.2 ⇒ 0 floppy modesx > 0.2 ⇒ ΔΦparallels f(x)Floppy modes fraction f(x)8th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=VaillsIn amorphous silica d 2 ΔΦ/dx 2 :Cusp at x = 0.211(Thorpe JNCS 2000)

References :• THE RAMAN EFFECT, a unified treatment of the theory of Raman scattering by molecules , Derek A. Long,Wiley 2002• Infrared and Raman Spectroscopy, Mehtods and Applications, Edited d by B. SchraderVCH, 1995• Durand G.E. et al, IEEE, J. Quant. Electron. 4 (1968) 523• Laberge N. L., et al J. of the Am. Ceram. Soc. 56 (1973) 506-509509• Leidecker H.W. et al J. Acoustic Soc. Am. 43 (1967), 737• Malki M. Et al Phys. Rev. Lett. 96 (2006) 145504• Pine A.S. Phys. Rev. 185 (1969) 1187• Phillips J.C. - J. of Non-Crystalline Solids 34 (1979), 153-181181• Saito K. et al A. J. App. Phys. Lett. 70 (1997) 3504-35063506• Schroeder J., et al J. of the Am. Ceram. Soc. 56 (1973) 510-514514• Schroeder J., et al J. of Non-Crystalline Solids 40 (1980) 549-566.566.• Schroeder J. J. of Non-Crystalline Solids 102 (1988) 240-249249• Thorpe M.F. - JNCS 859 (2000), 266-269269• Vacher R. et al J. de Chimie Physique 82 (1985), 311-316316• Vacher R. et al Phys. Rev. B 72, , (2005), 214205• Vaills Y. et al JNCS 286, , (2001), 224-234234• Vaills et al J. of Physics C 17 (2005), 4889-48964896• Vaills Y. web page : http://crmht.cnrs-orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills• Varshneya A.K. Fundamentals of inorganic glasses, Soc. Of Glass Tech., 200606• Walton D. et al, Sol. St. Comm. 42 (1982), 7378th ESG 2006 Sunderland 10-14 14 SeptY. VAILLS – email : vaills@cnrs-orleans.frhttp://crmht.cnrs-orleans.fr/orleans.fr/pubcrmht/ext/peoplefile.aspx?nom=Vaills