Nearest-neighbour interactions

A binary alloy of two components A and B with nearest-neighbour interactions respectively, is also isomorphic with the Ising model. This is easily seen on associating spin up with atom A and spin down with atom B. There are no vacant sites, and the occupation numbers of the site are defined by... 

Firstly, we consider 2-D square lattice with nearest-neighbour interactions. 

In a general concept of a symmetry-restricted anharmonic theory Krumhansl relates the phonon anomalies to the electron band topology. The latter is directly determined by the competition of nearest neighbour interactions which in turn can be a function of stress, composition and temperature Nagasawa, Yoshida Makita simulated the <110> ... 

In order to calculate R from equation 3, we have to evaluate the integrals iin n and mj,. n, with the MO jr and jr given in equations 5a and 5b. We assume only nearest-neighbour interactions and equal -integrals in all the integrals for the pairs (1,2) and (3,4), as justified by symmetry. The electric dipole will be described by the velocity operator V, in order to ensure origin-independent results, and equation 6 follows ... 

For ideal solutions Q. is zero and there are no extra interactions between the species that constitute the solution. In terms of nearest neighbour interactions only, the energy of an A-B interaction, mab, equals the average of the A-A, uAA, and B-B, wBB, interactions or... 

The regular model for an ionic solution is similarly analogous to the regular solution derived in Section 9.1. Recall that the energy of the regular solution model was calculated as a sum of pairwise interactions. With two sub-lattices, pair interactions between species in one sub-lattice with species in the other sub-lattice (nearest neighbour interactions) and pair interactions within each sub-lattice (next nearest neighbour interactions), must be accounted for. 

Let us first derive the regular solution model for the system AC-BC considered above. The coordination numbers for the nearest and next nearest neighbours are both assumed to be equal to z for simplicity. The number of sites in the anion and cation sub-lattice is N, and there are jzN nearest and next nearest neighbour interactions. The former are cation-anion interactions, the latter cation-cation and anion-anion interactions. A random distribution of cations and anions on each of... 

Machlin (1974, 1977) developed a semi-empirical treatment which used a constant set of nearest-neighbour interactions and was one of the earliest semi-empirical attempts to obtain the relative enthalpies of formation between different crystal structures. This successfiilly predicted the correct ground states in a substantial number of cases, but the treatment was generally restricted to transition metal combinations and a limited number of crystal structures. 

In a series of papers on cobalt and other phosphate glasses, Simpson (1970) and Simpson and Lucas (1971) showed that there is no sign of a Neel temperature down to -IK while the corresponding crystals show a Neel temperature near 20 K. In fact the 1//-T curve shows increased slope at low temperatures. Simpson pointed out that the theorem of Ziman (1952) (see Section 2)—that nearest-neighbour interaction cannot give antiferromagnetism for spherical orbitals— may be applicable here the orbitals are not spherical but are oriented at random. 

There is no adequate theory of the Neel temperature of a random distribution of centres in a dilute alloy, of indeed one exists. For higher concentrations of the magnetic matrix, with the assumption that only nearest neighbours interact, there is considerable theoretical work, giving a percolation limit , the concentration c0 at which long-range order disappears. The behaviour of TN is as (c —c0)12. For details see Brout (1965), Elliott and Heap (1962) and Klein and Brout (1963). 

If we take only nearest neighbour interactions into account (which is not too bad as a first approximation in the case of an interaction which falls off as the inverse sixth power of r) and if we assume that the mixture is perfectly random, then the internal energy (neglecting surface effects) is... 

The valence and coordination symmetry of a transition metal ion in a crystal structure govern the relative energies and energy separations of its 3d orbitals and, hence, influence the positions of absorption bands in a crystal field spectrum. The intensities of the absorption bands depend on the valences and spin states of each cation, the centrosymmetric properties of the coordination sites, the covalency of cation-anion bonds, and next-nearest-neighbour interactions with adjacent cations. These factors may produce characteristic spectra for most transition metal ions, particularly when the cation occurs alone in a simple oxide structure. Conversely, it is sometimes possible to identify the valence of a transition metal ion and the symmetry of its coordination site from the absorption spectrum of a mineral. 

Manning, P. G. (1973) Effect of second-nearest-neighbour interaction on Mn3 absorption in pink and black tourmalines. Canad. Mineral., 11,971-7. 

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