Second-moment approximation

Tlie suffices i and J refer to individual atoms and S and Sj to the species of the atoms involved. The summation over j extends over those neighbors of the atom i for which ry, the separation of atoms i and J, is within the cutoff radii of these potentials. The second term in Equation (la) is the attractive many-body term and both V and are empirically fitted pair potentials. A Justification for the square root form of the many-body function is provided in the framework of a second moment approximation of the density of states to the tight-binding theory incorporating local charge conservation in this framework the potentials represent squares of the hopping integrals (Ackland, et al. 1988). 

Here V is the volume of the box and P is pressure (assumed to be zero). These trials are repeated until the system effectively reaches steady state. While several data sets have been made available to model NiAl alloys we choose the tight-binding (TB) second-moment approximation (SMA) approach adopted by Papanicolaou et al. [16], This TB-SMA approach defines the total energy of the system (U in equation 3) as... 

P(rij) is the pairwise potential, which, depending upon the model, can be considered to include electrostatic and repulsive contributions. The second term is a function of the electron density, p, and varies with the square root, in keeping with the second-moment approximation. The electron density for an atom includes contributions from the neighbouring atoms as follows ... 

Second Moment Approximation and Finnis-Sinclair Potential... 

This result is called the second-moment approximation. 

The Finnis-Sinclair analytic functional form was introduced at about the same time as two other similar forms, the embedded-atom method > and the glue model." ° However, the derivation of the Finnis-Sinclair form from the second-moment approximation is very different from the interpretation of the other empirical forms, which are based on effective medium theory as discussed later. This difference in interpretation is often ignored, and all three methods tend to be put into a single class of potential energy function. In practice, the main difference between the methods lies in the systems to which they have been traditionally applied. In developing the embedded-atom method, for example, Baskes, Daw, and Foiles emphasized close-packed lattices rather than body-centered-cubic lattices. Given that angular interactions are usually ig-... 

The functional form (11.15) was originally derived as a generalization of the effective medium theory and the second moment approximation to tight-binding theory. Later, however, it lost its close ties with original physical meaning and came to be treated as a semiempirical expression with adjustable parameters. 

DOLINSKI, K. First Order Second Moment Approximation In Reliability of Structural Systems Critical Review and Alternative Approach", Structural Safety K3) (1983) 211 -231. 

QFD quantum formation diagram SMA second-moment approximation... 

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