Embedded-Atom and Related Methods for Modeling Metallic Systems
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Embedded-Atom and Related Methods for Modeling Metallic Systems Stephen M. Foiles This article discusses simple but realistic models of the bonding in metallic systems. As is discussed in the introduction, such methods are valuable both to study complex systems that are intractable with more rigorous methods and to study generic properties that do not depend on the fine details of the energetics. In addition, simple models are useful for gaining a physical understanding of a system. As discussed in the article by Vitek, a great deal of progress in the understanding of defect structures has been gained by the use of constant-volume pair potentials, that is, pair-potential models that include a volume-dependent term. In this picture, the energy is assumed to have two parts, a large density-dependent but structure-independent part and a structure-dependent part that is represented by pair interactions. This view follows from a physical picture where the metal is viewed as a uniform electron gas and the interactions between the atoms are obtained by performing perturbation theory on this reference system. This approach has a serious limitation, though, in that it is restricted to situations where the system is essentially uniform such as when the bulk or defects do not introduce significant changes in the local density. This is true for two reasons. First, there is not a clear prescription for how the structure-independent part of the energy should be treated in an inhomogeneous region. This is a serious problem since a large part of the binding energy is included in this term. Second, the pair-potential term in this picture depends on the overall density. In the vicinity of a inhomogeneity such as a surface, there is no
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MRS BULLETIN/FEBRUARY 1996
Embedded-Atom and Related Methods for Modeling Metallic Systems
that the atom forms and increases the bond length. For example, in metallic systems, the bulk nearest-neighbor spacing is generally longer than the separation of the atoms in the corresponding dimer. As will be shown in one of the examples, incorporation of this physical trend substantially improves the description of severe inhomogeneities such as surfaces. The origin of the coordination dependence of the interaction is the curvature of the function F. This can be seen by developing a pair-potential approximation to the pair-functional form. To do this, assume that the sum over the/functions will be near some value s, and then expand F in a first-order Taylor series about that value. The result is
# # m m
£ =
(2)
#
• i 0
The first sum is over energies that depend only on this typical density value s, analogous to the density-dependent energy of constant-volume pair potentials. The second term is a sum over effective pair potentials. Note, however, that the effective pair potential depends on the typical density s. For functions F with a positive curvature and a positive decreasing/, the effective pair potential becomes more repulsive as the density increases. This leads to the coordination trend de
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