Raman Spectroscopy of nanomaterials - institut de chimie et des ...

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(a)CeO 2 film onsapphire substrated g = 6 nmARTICLE IN PRESS+ MODEL32 G. Gouadec, Ph. Colomban / Progress in Crystal Growth and Characterization of Materialsxx (2007) 1e56(b)Correlation Length - L (A)(c)20018016014012010080604020L ~ d g1/3spatial correlation model00 50 100 150 200 250 300 350Grain Size - d g (nm)30 GdHW = 51.6 cm -1CeO 2single crystal300 400 500Wavenumber / cm -120 GdHW = 41.9 cm -1PureHW = 9.7 cm -1100 300 500 700 900Wavenumber / cm -1Fig. 18. (a) Grain size effect on ceria Raman signature. The fits were obtained using the PCM (Section 3.3.1). (b) Comparisonof the confinement length L obtained through the PCM with the grain size obtained by X-ray diffraction. Reprintedfrom Ref. [196, pp. 99e105] with permission from Elsevier. (c) The spectra of gadolinium-doped ceria singlecrystals depend significantly on the doping level (I. Kosacki and Ph. Colomban, private communication).on crystallized samples. Ultrasound analysis gives a dynamical view of cracking rather thanstress measurements [398e400], photoelastometry supposes perfect transparency [401,402]and SEM observations of resin grids only give an indirect measure of the surface deformations[403e405]. Other specific methods exist like instrumented (microenano)-indentation[406e410], fragmentation [411e416], ‘‘pull-out’’ [417], ‘‘push-out’’ [418] and ‘‘microdroplet’’[413] tests but the validity of the models used for data interpretation is often questionabledue to their usually large number of parameters. Besides, these methods are sometimesinapplicable. For instance, carbon fibre-reinforced composites cannot be tested by conventionalpush-in micro-indentation because the fibres have a small diameter and cleave tooeasily.5.1. Raman stress sensitivity5.1.1. The anharmonicity of atomic bondsBelow, we will assume that each stretching mode is specific to one given chemical bond. Aslong as the elongation is limited, the bond can be modelled by a spring of length l b , reducedPlease cite this article in press as: G. Gouadec, Ph. Colomban, Prog. Cryst. Growth Charact. Mater. (2007),doi:10.1016/j.pcrysgrow.2007.01.001
ARTICLE IN PRESS+ MODELG. Gouadec, Ph. Colomban / Progress in Crystal Growth and Characterization of Materials 33xx (2007) 1e56mass m and constant stiffness k b . This is the so-called harmonic model where the interatomicpotential has the form Vðl b Þ¼ðk b =2Þðl b l 0 Þ 2 :sffiffiffiffin vib ¼ 1 k bð37Þ2pc mk b ¼ v2 Vvl 2 bð38ÞThere is no l b dependence for the wavenumber in Eq. (37) and the harmonic potential doesnot predict any stress sensitivity for Raman bands. Yet such modelling rules out the phenomenonof bond dissociation (l b / N) and does not correctly account for atoms non-interpenetrability(l b / 0). In fact, the interaction potential should not be limited to a quadratic term and allreal bonds are somewhat anharmonic (Fig. 19). They include attractive and repulsive contributions,for which Mie and Grüneisen (Table 1) proposed the following expression, with A, R,a and r being positive constants [419]:Vðl b Þ¼V 0 þ V attractive ðl b ÞþV repulsive ðl b Þ¼V 0A l abþ R l rbIn spite of its very significant influence on the physical behaviour of materials (only it canexplain thermal expansion or the finite value of thermal conductivity [421]), anharmonicity isoften considered as a simple perturbation: in what is referred to as the ‘‘quasi-harmonic’’ approximation,one considers that Eqs. (37) and (38) are applicable to real potentials. Accordingto Fig. 19, atoms are bonded by strings whose stiffness ðv 2 V=vl 2 bÞ increases under compressionand decreases under tension.In the case of the ‘‘quasi-harmonic’’ approximation, Eq. (39) is equivalent to:dn vib¼ dkn 0 2k ¼ 1 aða þ 1Þða þ 2ÞA l ðaþ3Þb rðr þ 1Þðr þ 2ÞR l ðrþ3Þbdl2 rðr þ 1ÞR l ðrþ2Þb aða þ 1ÞA l ðaþ2Þbð40ÞbIntroducing the bond elongation 3 l ¼ (l b l 0 )/l 0 , a Taylor expansion of l x b ¼ lx 0 ð1 þ 3 lÞ x (smallstrain assumption) and the requirement for the potential first derivative to be nil at l b ¼ l 0 yield:ð39Þharmonic potential"realistic" potentialEnergyE dissociation0.0 1.0 2.0 3.0I b /I 0Fig. 19. Comparison of realistic and harmonic bond potentials.Please cite this article in press as: G. Gouadec, Ph. Colomban, Prog. Cryst. Growth Charact. Mater. (2007),doi:10.1016/j.pcrysgrow.2007.01.001
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