Bond order angular dependence

The function/c is a smoothing function with the value 1 up to some distance Yy (typically chosen to include just the first neighbour shell) and then smoothly tapers to zero at the cutoff distance, by is the bond-order term, which incorporates an angular term dependent upon the bond angle 6yk- The Tersoff pofenfial is more broadly applicable than the Stillinger-Weber potential, but does contain more parameters. 

An expression of the type (7.101), which gives the bond order explicitly in terms of the positions of the neighbouring atoms, is called a bond order potential (BOP). Angularly dependent bond order potentials were first derived heuristically for the elemental semiconductors by TersofF (1988). We will see in the next chapter that a many-body expansion for the bond order may be derived exactly within the model. 

When angular-dependent interactions contribute significantly to the bonding, pair potentials like those described above are not sufficient and three-body or higher order terms must be included in the potential energy. That is the case of covalently bonded systems like silicon or transition metals. For the case of silicon two interatomic potentials are widely used the one developed by StiUinger and Weber [20] and the one developed by Tersoff [21]. Other empirical potentials have been developed to include the angular dependence such as the modified embedded atom method (MEAM) [22]. 

Previous NEXAFS studies on n-octadecyltrichlorosilane (OTS-C18) on silicon [29] and mica [30] indicate that monolayer films of OTS are highly oriented. The angular dependence of the resonances is indicative of a preferential molecular orientation of the polymeric C-C chains perpendicular to the surface (standing up), and the C-H bond plane is parallel to the surface. It should be noted that the orientation in the form of tilt angle or order calculated from NEXAFS are average values. 

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