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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