Second nearest neighbor interactions

In their model they retained only the first- and second-nearest neighbor interactions, so that the Hamiltonian assumed the following form... 

It is possible to consider interactions between atoms separated by any distance, of course. For example, remming to the SC lattice, had it been chosen to consider second-nearest neighbor interactions as well, the result would have been ... 

The inclusion of second (and often third terms) is particularly important for certain stracmre types. For example, in the BCC and CsCl lattices the second-nearest neighbors are only 14 percent more distant than the first-nearest (Harrison, 1989). Accounting for the six second-nearest neighbor interactions in the FCC and BCC lattices, the energies of a Bloch sum of s atomic states are given by E qs. 5.42 and 5.43, respectively ... 

Since there are two atoms per primitive cell, or lattice point, and consideration is still on just s atomic orbitals, two separate Bloch sums are required. These combine to form two COs with energies given by the sum and difference in energy between the nearest neighbor and second-nearest neighbor interactions. The nearest neighbor... 

The twelve second-nearest neighbor interactions are between atoms all on the same sublattice. 

Second-nearest-neighbor interactions either follow the same laws as the nearest-neighbor interaction or are negligible. 

In order to produce large negative entropy and enthalpy of solvation, one needs to find a mechanism by means of which a solute can stabilize or enhance the formation of hydrogen bonds. This could have been achieved in the primitive model by introducing second nearest-neighbor interactions for triplets of water-solute-water in such a way that when the two water molecules are hydrogen bonded, the in-between solute molecule will stabilize the triplet, and hence enhance the formation of hydrogen bonds across the solute. This is shown schematically in Fig. 3.21. 

Table 11.1 A sample calculation of relative surface energies in a simple fee crystal based on the broken bond model when the first and second nearest-neighbor interaction are considered.

FNN and SNN represent the first and the second nearest neighbor interactions. 

Writing the energy with these effective interactions increases the number of dependent variables. Therefore several choices of these effective energies correspond to the same cluster expansion, then to the same thermodynamic and kinetic properties. If we make the assumption that second nearest neighbor interactions do not contribute to the cohesive energy of pure elements, i.e. = 0 and = 0, we obtain as many effective interactions as parameters in the truncated cluster expansion. Such an assumption does not have any physical influence and it just guarantees that homo-atomic effective... 

Figure 6.5 The five distinct alloy tetrahedra. White circles indicate A atoms, gray circles B atoms, and black circles C atoms in an Ai.xBj C type alloy. Note that each tetrahedron can be rotated to place any of the A or B atoms on a given vertex when forming the crystal. The energy of a tetrahedron is independent of its orientation. The model does not consider second-nearest neighbor interactions but can account for bond strains within each tetrahedron. A more accurate model must consider longer-range interactions.

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